Chapter 11.
Modeling Heat Transfer
This chapter provides details about the heat transfer models available in
FLUENT.
Information is presented in the following sections:
• Section 11.1: Overview of Heat Transfer Models in FLUENT
• Section 11.2: Convective and Conductive Heat Transfer
• Section 11.3: Radiative Heat Transfer
• Section 11.4: Periodic Heat Transfer
• Section 11.5: Buoyancy-Driven Flows
11.1
Overview of Heat Transfer Models in FLUENT
The flow of thermal energy from matter occupying one region in space
to matter occupying a different region in space is known as heat transfer.
Heat transfer can occur by three main methods: conduction, convection,
and radiation. Physical models involving only conduction and/or convection are the simplest, while buoyancy-driven flow, or natural convection,
and radiation models are more complex. Depending on your problem,
FLUENT will solve a variation of the energy equation that takes into
account the heat transfer methods you have specified. FLUENT is also
able to predict heat transfer in periodically repeating geometries, thus
greatly reducing the required computational effort in certain cases.
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Modeling Heat Transfer
11.2
Convective and Conductive Heat Transfer
FLUENT allows you to include heat transfer within the fluid and/or solid
regions in your model. Problems ranging from thermal mixing within a
fluid to conduction in composite solids can thus be handled by FLUENT
using the physical models and inputs described in this section. Radiation
modeling is described in Section 11.3 and natural convection is described
in Section 11.5.
Information about heat transfer is presented in the following subsections:
• Section 11.2.1: Theory
• Section 11.2.2: User Inputs for Heat Transfer
• Section 11.2.3: Solution Process for Heat Transfer
• Section 11.2.4: Reporting and Displaying Heat Transfer Quantities
• Section 11.2.5: Exporting Heat Flux Data
11.2.1
Theory
The Energy Equation
FLUENT solves the energy equation in the following form:

∂
(ρE) + ∇ · (~v (ρE + p)) = ∇ · keff ∇T −
∂t
X

hj J~j + (τ eff · ~v ) + Sh
j
(11.2-1)
where keff is the effective conductivity (k + kt , where kt is the turbulent
thermal conductivity, defined according to the turbulence model being
used), and J~j is the diffusion flux of species j. The first three terms on
the right-hand side of Equation 11.2-1 represent energy transfer due to
conduction, species diffusion, and viscous dissipation, respectively. Sh
includes the heat of chemical reaction, and any other volumetric heat
sources you have defined.
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11.2 Convective and Conductive Heat Transfer
In Equation 11.2-1,
E =h−
p v2
+
ρ
2
(11.2-2)
where sensible enthalpy h is defined for ideal gases as
X
h=
Yj hj
(11.2-3)
j
and for incompressible flows as
h=
X
Yj hj +
j
p
ρ
(11.2-4)
In Equations 11.2-3 and 11.2-4, Yj is the mass fraction of species j and
Z
T
hj =
cp,j dT
(11.2-5)
Tref
where Tref is 298.15 K.
The Energy Equation for the Non-Premixed Combustion Model
When the non-adiabatic non-premixed combustion model is enabled,
FLUENT solves the total enthalpy form of the energy equation:
∂
(ρH) + ∇ · (ρ~v H) = ∇ ·
∂t
kt
∇H
cp
!
+ Sh
(11.2-6)
Under the assumption that the Lewis number (Le) = 1, the conduction
and species diffusion terms combine to give the first term on the righthand side of the above equation while the contribution from viscous
dissipation appears in the non-conservative form as the second term.
The total enthalpy H is defined as
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Modeling Heat Transfer
H=
X
Yj H j
(11.2-7)
j
where Yj is the mass fraction of species j and
Z
T
Hj =
Tref,j
cp,j dT + h0j (Tref,j )
(11.2-8)
h0j (Tref,j ) is the formation enthalpy of species j at the reference temperature Tref,j .
Inclusion of Pressure Work and Kinetic Energy Terms
Equation 11.2-1 includes pressure work and kinetic energy terms which
are often negligible in incompressible flows. For this reason, the segregated solver by default does not include the pressure work or kinetic
energy when you are solving incompressible flow. If you wish to include
these terms, use the define/models/energy? text command to turn
them on.
Pressure work and kinetic energy are always accounted for when you are
modeling compressible flow or using one of the coupled solvers.
Inclusion of the Viscous Dissipation Terms
Equations 11.2-1 and 11.2-6 include viscous dissipation terms, which
describe the thermal energy created by viscous shear in the flow.
When the segregated solver is used, FLUENT’s default form of the energy
equation does not include them (because viscous heating is often negligible). Viscous heating will be important when the Brinkman number,
Br, approaches or exceeds unity, where
Br =
µUe2
k∆T
(11.2-9)
and ∆T represents the temperature difference in the system.
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When your problem requires inclusion of the viscous dissipation terms
and you are using the segregated solver, you should activate the terms using the Viscous Heating option in the Viscous Model panel. Compressible
flows typically have Br ≥ 1. Note, however, that when the segregated
solver is used, FLUENT does not automatically activate the viscous dissipation if you have defined a compressible flow model.
When one of the coupled solvers is used, the viscous dissipation terms
are always included when the energy equation is solved.
Inclusion of the Species Diffusion Term
Equations 11.2-1 and 11.2-6 both include the effect of enthalpy transport
due to species diffusion.
When the segregated solver is used, the term

∇·
X

hj J~j 
j
is included in Equation 11.2-1 by default. If you do not want to include
it, you can turn off the Diffusion Energy Source option in the Species
Model panel.
When the non-adiabatic non-premixed combustion model is being used,
this term does not explicitly appear in the energy equation, because it
is included in the first term on the right-hand side of Equation 11.2-6.
When one of the coupled solvers is used, this term is always included in
the energy equation.
Energy Sources Due to Reaction
Sources of energy, Sh , in Equation 11.2-1 include the source of energy
due to chemical reaction:
Sh,rxn = −
X
j
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Z T
h0j
+
cp,j dT
Mj
Tref,j
!
Rj
(11.2-10)
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Modeling Heat Transfer
where h0j is the enthalpy of formation of species j and Rj is the volumetric
rate of creation of species j.
In the energy equation used for non-adiabatic non-premixed combustion
(Equation 11.2-6), the heat of formation is included in the definition
of enthalpy (see Equation 11.2.1), so reaction sources of energy are not
included in Sh .
Energy Sources Due To Radiation
When one of the radiation models is being used, Sh in Equation 11.2-1 or
11.2-6 also includes radiation source terms. See Section 11.3 for details.
Interphase Energy Sources
It should be noted that the energy sources, Sh , also include heat transfer
between the continuous and the discrete phase. This is discussed further
in Section 19.5.
Boundary Conditions for Heat Transfer at Walls
Heat transfer boundary conditions at walls are discussed in Section 10.8.2.
Energy Equation in Solid Regions
In solid regions, the energy transport equation used by FLUENT has the
following form:
∂
(ρh) + ∇ · (~v ρh) = ∇ · (k∇T ) + Sh
∂t
where
ρ
h
k
T
Sh
=
=
=
=
=
(11.2-11)
density
R
sensible enthalpy, TTref cp dT
conductivity
temperature
volumetric heat source
The second term on the left-hand side of Equation 11.2-11 represents
convective energy transfer due to rotational or translational motion of
the solids. The velocity field ~v is computed from the motion specified
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11.2 Convective and Conductive Heat Transfer
for the solid zone (see Section 6.18). The terms on the right-hand side
of Equation 11.2-11 are the heat flux due to conduction and volumetric
heat sources within the solid, respectively.
Anisotropic Conductivity in Solids
When you use the segregated solver, FLUENT allows you to specify
anisotropic conductivity for solid materials. The conduction term for
an anisotropic solid has the form
∇ · (kij ∇T )
(11.2-12)
where kij is the conductivity matrix. See Section 7.4.5 for details on
specifying anisotropic conductivity for solid materials.
Diffusion at Inlets
The net transport of energy at inlets consists of both the convection
and diffusion components. The convection component is fixed by the
inlet temperature specified by you. The diffusion component, however,
depends on the gradient of the computed temperature field. Thus the
diffusion component (and therefore the net inlet transport) is not specified a priori.
In some cases, you may wish to specify the net inlet transport of energy
rather than the inlet temperature. If you are using the segregated solver,
you can do this by disabling inlet energy diffusion. By default, FLUENT
includes the diffusion flux of energy at inlets. To turn off inlet diffusion,
use the define/models/energy? text command.
Inlet diffusion cannot be turned off if you are using one of the coupled
solvers.
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Modeling Heat Transfer
11.2.2
User Inputs for Heat Transfer
When your FLUENT model includes heat transfer you need to activate
the relevant models, supply thermal boundary conditions, and input material properties that govern heat transfer and/or may vary with temperature. These inputs are described in this section.
The procedure for setting up a heat transfer problem is described below.
(Note that this procedure includes only those steps necessary for the heat
transfer model itself; you will need to set up other models, boundary
conditions, etc. as usual.)
1. To activate the calculation of heat transfer, turn on the Energy
Equation option in the Energy panel (Figure 11.2.1).
Define −→ Models −→Energy...
Figure 11.2.1: The Energy Panel
2. (optional, segregated solver only) If you are modeling viscous flow
and you want to include the viscous heating terms in the energy
equation, turn on the Viscous Heating option in the Viscous Model
panel. As noted in Section 11.2.1, the viscous heating terms in
the energy equation are (by default) ignored by FLUENT when
the segregated solver is used. (They are always included when
one of the coupled solvers is used.) Viscous dissipation should be
enabled when the shear stress in the fluid is large (e.g., in lubrication problems) and/or in high-velocity, compressible flows (see
Equation 11.2-9).
Define −→ Models −→Viscous...
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11.2 Convective and Conductive Heat Transfer
3. Define thermal boundary conditions at flow inlets, flow outlets, and
walls.
Define −→Boundary Conditions...
At flow inlets and exits you will set the temperature; at walls you
may use any of the following thermal conditions:
• Specified heat flux
• Specified temperature
• Convective heat transfer
• External radiation
• Combined external radiation and external convective heat
transfer
Section 6.13.1 provides details on the model inputs that govern
these thermal boundary conditions. The default thermal boundary
condition at inlets is a specified temperature of 300 K; at walls the
default condition is zero heat flux (adiabatic). See Chapter 6 for
details about boundary condition inputs.
!
If your heat transfer application involves two separated fluid regions, see the information provided below.
4. Define material properties for heat transfer.
Define −→Materials...
Heat capacity and thermal conductivity must be defined, and you
can specify many properties as functions of temperature, as described in Chapter 7.
!
If your heat transfer application involves two separated fluid regions, see the information provided below.
The Temperature Floor and Ceiling
For stability reasons, FLUENT includes a limit on the predicted temperature range. The purpose of the temperature ceiling and floor is to
improve the stability of calculations in which the temperature should
physically lie within known limits. Sometimes intermediate solutions of
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the equations give rise to temperatures beyond these limits for which
property definitions, etc. are not well defined. The temperature limits
keep the temperatures within the expected range for your problem. If the
FLUENT calculation predicts a temperature above the maximum limit,
the stored temperature values are “pegged” at this maximum value. The
default for the temperature ceiling is 5000 K. If the FLUENT calculation
predicts a temperature below the minimum limit, the stored temperature values are “pegged” at this minimum value. The default for the
temperature minimum is 1 K.
If you expect the temperature in your domain to exceed 5000 K, you
should use the Solution Limits panel to increase the Maximum Temperature.
Solve −→ Controls −→Limits...
Modeling Heat Transfer in Two Separated Fluid Regions
If your heat transfer application involves two fluid regions separated by
a solid zone or a wall, as illustrated in Figure 11.2.2, you will need to
define the problem with some care. Specifically:
• You should not use outflow boundary conditions in either fluid.
• You can establish separate fluid properties by selecting a different
fluid material for each zone. (For species calculations, however, you
can only select a single mixture material for the entire domain.)
➞
fluid 2
➞
fluid 1
Figure 11.2.2: Typical Counterflow Heat Exchanger Involving Heat
Transfer Between Two Separated Fluid Streams
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11.2 Convective and Conductive Heat Transfer
11.2.3
Solution Process for Heat Transfer
Although many simple heat transfer problems can be successfully solved
using the default solution parameters assumed by FLUENT, you may
accelerate the convergence of your problem and/or improve the stability
of the solution process using some of the guidelines provided in this
section.
Under-Relaxation of the Energy Equation
When you use the segregated solver, FLUENT under-relaxes the energy
equation using the under-relaxation parameter defined by you in the
Solution Controls panel, as described in Section 22.9.
Solve −→ Controls −→Solution...
If you are using the non-adiabatic non-premixed combustion model, you
will set the energy under-relaxation factor as usual, but you will also
set an under-relaxation factor for temperature, which will be used as
described below.
FLUENT uses a default under-relaxation factor of 1.0 for the energy
equation, regardless of the form in which it is solved (temperature or
enthalpy). In problems where the energy field impacts the fluid flow
(via temperature-dependent properties or buoyancy) you should use a
lower value for the under-relaxation factor, in the range of 0.8–1.0. In
problems where the flow field is decoupled from the temperature field (no
temperature-dependent properties or buoyancy forces), you can usually
retain the default value of 1.0.
Under-Relaxation of Temperature When the Enthalpy Equation
is Solved
When the enthalpy form of the energy equation is solved (i.e., when you
are using the non-adiabatic non-premixed combustion model), FLUENT
also under-relaxes the temperature, updating the temperature by only a
fraction of the change that would result from the change in the (underrelaxed) enthalpy values. This second level of under-relaxation can be
used to good advantage when you would like to let the enthalpy field
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change rapidly, but the temperature response (and its effect on fluid
properties) to lag. FLUENT uses a default setting of 1.0 for the underrelaxation on temperature and you can modify this setting using the
Solution Controls panel.
Disabling the Species Diffusion Term
If you are solving for species transport using the segregated solver and
you encounter convergence difficulties, you may want to consider turning
off the Diffusion Energy Source option in the Species Model panel.
Define −→ Models −→Species...
When this option is disabled, FLUENT will neglect the effects of species
diffusion on the energy equation.
Note that species diffusion effects are always included when one of the
coupled solvers is used.
Step-by-Step Solutions
Often the most efficient strategy for predicting heat transfer is to compute an isothermal flow first and then to add the calculation of the energy
equation. The procedure differs slightly, depending on whether or not
the flow and heat transfer are coupled.
Decoupled Flow and Heat Transfer Calculations
If your flow and heat transfer are decoupled (no temperature-dependent
properties or buoyancy forces), you can first solve the isothermal flow
(energy equation turned off) to yield a converged flow-field solution and
then solve the energy transport equation alone.
! Since the coupled solvers always solve the flow and energy equations
together, the procedure for solving for energy alone applies only to the
segregated solver.
You can temporarily turn off the flow equations or the energy equation
by deselecting Energy in the Equations list in the Solution Controls panel.
(See also Section 22.19.2.)
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11.2 Convective and Conductive Heat Transfer
Solve −→ Controls −→Solution...
Coupled Flow and Heat Transfer Calculations
If your flow and heat transfer are coupled (i.e., your model includes
temperature-dependent properties or buoyancy forces), you can first
solve the flow equations before turning on energy. Once you have a
converged flow-field solution, you can turn on energy and solve the flow
and energy equations simultaneously to complete the heat transfer simulation.
11.2.4
Reporting and Displaying Heat Transfer Quantities
FLUENT provides several additional reporting options for simulations
involving heat transfer. You can generate graphical plots or reports of
the following variables/functions:
• Static Temperature
• Total Temperature
• Enthalpy
• Relative Total Temperature
• Rothalpy
• Wall Temperature (Outer Surface)
• Wall Temperature (Inner Surface)
• Total Enthalpy
• Total Enthalpy Deviation
• Entropy
• Total Energy
• Internal Energy
• Total Surface Heat Flux
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Modeling Heat Transfer
• Surface Heat Transfer Coef.
• Surface Nusselt Number
• Surface Stanton Number
The first 12 variables listed above are contained in the Temperature...
category of the variable selection drop-down list that appears in postprocessing panels, and the remaining variables are in the Wall Fluxes...
category. See Chapter 27 for their definitions.
Definition of Enthalpy and Energy in Reports and Displays
The definitions of the reported values of enthalpy and energy will be different depending on whether the flow is compressible or incompressible.
See Section 27.4 for a complete list of definitions.
Reporting Heat Transfer Through Boundaries
You can use the Flux Reports panel to compute the heat transfer through
each boundary of the domain, or to sum the heat transfer through all
boundaries to check the heat balance.
Report −→Fluxes...
It is recommended that you perform a heat balance check to ensure
that your solution is truly converged. See Section 26.2 for details about
generating flux reports.
Reporting Heat Transfer Through a Surface
You can use the Surface Integrals panel (described in Section 26.5) to
compute the heat transfer through any boundary or any surface created
using the methods described in Chapter 24.
Report −→Surface Integrals...
To report the flow rate of enthalpy
Z
Q=
11-14
~
Hρ~v · dA
(11.2-13)
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choose the Mass Flow Rate option in the Surface Integrals panel, select
Enthalpy (in the Temperature... category) as the Field Variable, and pick
the surface(s) on which to integrate.
Reporting Averaged Heat Transfer Coefficients
The Surface Integrals panel can also be used to generate
a report of
R
averaged heat transfer coefficient h on a surface ( A1 h dA).
Report −→Surface Integrals...
In the Surface Integrals panel, choose the Area-Weighted Average option,
select Surface Heat Transfer Coef. (in the Wall Fluxes... category) as the
Field Variable, and pick the surface.
11.2.5
Exporting Heat Flux Data
It is possible to export heat flux data on wall zones (including radiation)
to a generic file that you can examine or use in an external program. To
save a heat flux file, you will use the custom-heat-flux text command.
file −→ export −→custom-heat-flux
Heat transfer data will be exported in the following free format for each
face zone that you select for export:
zone-name nfaces
x_f y_f z_f A
.
.
.
Q
T_w
T_c
Each block of data starts with the name of the face zone (zone-name)
and the number of faces in the zone (nfaces). Next there is a line for
each face (i.e., nfaces lines), each containing the components of the
face centroid (x f, y f, and, in 3D, z f), the face area (A), the total
heat transfer including radiation heat transfer (Q), the face temperature
(T w), and the adjacent cell temperature (T c).
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11.3
Radiative Heat Transfer
Information about radiation modeling is presented in the following sections:
• Section 11.3.1: Introduction to Radiative Heat Transfer
• Section 11.3.2: Choosing a Radiation Model
• Section 11.3.3: The Discrete Transfer Radiation Model (DTRM)
• Section 11.3.4: The P-1 Radiation Model
• Section 11.3.5: The Rosseland Radiation Model
• Section 11.3.6: The Discrete Ordinates (DO) Radiation Model
• Section 11.3.7: The Surface-to-Surface (S2S) Radiation Model
• Section 11.3.8: Radiation in Combusting Flows
• Section 11.3.9: Overview of Using the Radiation Models
• Section 11.3.10: Selecting the Radiation Model
• Section 11.3.11: Defining the Ray Tracing for the DTRM
• Section 11.3.12: Computing or Reading the View Factors for the
S2S Model
• Section 11.3.13: Defining the Angular Discretization for the DO
Model
• Section 11.3.14: Defining Non-Gray Radiation for the DO Model
• Section 11.3.15: Defining Material Properties for Radiation
• Section 11.3.16: Setting Radiation Boundary Conditions
• Section 11.3.17: Setting Solution Parameters for Radiation
• Section 11.3.18: Solving the Problem
• Section 11.3.19: Reporting and Displaying Radiation Quantities
• Section 11.3.20: Displaying Rays and Clusters for the DTRM
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11.3.1
Introduction to Radiative Heat Transfer
FLUENT provides five radiation models which allow you to include radiation, with or without a participating medium, in your heat transfer
simulations:
• Discrete transfer radiation model (DTRM) [30, 208]
• P-1 radiation model [35, 210]
• Rosseland radiation model [210]
• Surface-to-surface (S2S) radiation model [210]
• Discrete ordinates (DO) radiation model [37, 183]
Heating or cooling of surfaces due to radiation and/or heat sources or
sinks due to radiation within the fluid phase can be included in your
model using one of these radiation models.
Radiative Transfer Equation
The radiative transfer equation (RTE) for an absorbing, emitting, and
scattering medium at position ~r in the direction ~s is
dI(~r, ~s)
σT 4
σs
+ (a + σs )I(~r, ~s) = an2
+
ds
π
4π
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Z
4π
0
I(~r, ~s 0 ) Φ(~s · ~s 0 ) dΩ0
(11.3-1)
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Modeling Heat Transfer
where
~r
~s
~s 0
s
a
n
σs
σ
I
=
=
=
=
=
=
=
=
=
T
Φ
Ω0
=
=
=
position vector
direction vector
scattering direction vector
path length
absorption coefficient
refractive index
scattering coefficient
Stefan-Boltzmann constant (5.672 × 10−8 W/m2 -K4 )
radiation intensity, which depends on position (~r)
and direction (~s)
local temperature
phase function
solid angle
(a + σs )s is the optical thickness or opacity of the medium. The refractive index n is important when considering radiation in semi-transparent
media. Figure 11.3.1 illustrates the process of radiative heat transfer.
Absorption and
scattering loss:
I (a+σs) ds
Outgoing radiation
I + (dI/ds)ds
Incoming
radiation (I)
Gas emission:
(aσT 4/π) ds
Scattering
addition
ds
Figure 11.3.1: Radiative Heat Transfer
The DTRM and the P-1, Rosseland, and DO radiation models require
the absorption coefficient a as input. a and the scattering coefficient σs
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11.3 Radiative Heat Transfer
can be constants, and a can also be a function of local concentrations
of H2 O and CO2 , path length, and total pressure. FLUENT provides
the weighted-sum-of-gray-gases model (WSGGM) for computation of a
variable absorption coefficient. See Section 11.3.8 for details. The discrete ordinates implementation can model radiation in semi-transparent
media. The refractive index n of the medium must be provided as a part
of the calculation for this type of problem.
Applications of Radiative Heat Transfer
Typical applications well suited for simulation using radiative heat transfer include the following:
• Radiative heat transfer from flames
• Surface-to-surface radiant heating or cooling
• Coupled radiation, convection, and/or conduction heat transfer
• Radiation through windows in HVAC applications, and cabin heat
transfer analysis in automotive applications
• Radiation in glass processing, glass fiber drawing, and ceramic processing
You should include radiative heat transfer in your simulation when the
4
4 ), is large compared to the heat
radiant heat flux, Qrad = σ(Tmax
− Tmin
transfer rate due to convection or conduction. Typically this will occur
at high temperatures where the fourth-order dependence of the radiative
heat flux on temperature implies that radiation will dominate.
11.3.2
Choosing a Radiation Model
For certain problems, one radiation model may be more appropriate than
the others. When deciding which radiation model to use, consider the
following:
• Optical thickness: The optical thickness aL is a good indicator of
which model to use in your problem. Here, L is an appropriate
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length scale for your domain. For flow in a combustor, for example, L is the diameter of the combustion chamber. If aL 1,
your best alternatives are the P-1 and Rosseland models. The P-1
model should typically be used for optical thicknesses > 1. For
optical thickness > 3, the Rosseland model is cheaper and more
efficient. The DTRM and the DO model work across the range
of optical thicknesses, but are substantially more expensive to use.
So you should use the “thick-limit” models, P-1 and Rosseland, if
the problem allows it. For optically thin problems (aL < 1), only
the DTRM and the DO model are appropriate.
• Scattering and emissivity: The P-1, Rosseland, and DO models
account for scattering, while the DTRM neglects it. Since the
Rosseland model uses a temperature slip condition at walls, it is
insensitive to wall emissivity.
• Particulate effects: Only the P-1 and DO models account for exchange of radiation between gas and particulates (see Equation
11.3-15).
• Semi-transparent media and specular boundaries: Only the DO
model allows specular reflection (e.g., for mirrors) and calculation
of radiation in semi-transparent media such as glass.
• Non-gray radiation: Only the DO model allows you to compute
non-gray radiation using a gray band model.
• Localized heat sources: In problems with localized sources of heat,
the P-1 model may overpredict the radiative fluxes. The DO model
is probably the best suited for computing radiation for this case,
although the DTRM, with a sufficiently large number of rays, is
also acceptable.
• Enclosure radiative transfer with non-participating media: The
surface-to-surface (S2S) model is suitable for this type of problem. The radiation models used with participating media may, in
principle, be used to compute the surface-to-surface radiation, but
they are not always efficient.
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11.3 Radiative Heat Transfer
External Radiation
If you need to include radiative heat transfer from the exterior of your
physical model, you can include an external radiation boundary condition in your model (see Section 6.13.1). If you are not concerned with
radiation within the domain, this boundary condition can be used without activating one of the radiation models.
Advantages and Limitations of the DTRM
The primary advantages of the DTRM are threefold: it is a relatively
simple model, you can increase the accuracy by increasing the number
of rays, and it applies to a wide range of optical thicknesses.
You should be aware of the following limitations when using the DTRM
in FLUENT:
• The DTRM assumes that all surfaces are diffuse. This means that
the reflection of incident radiation at the surface is isotropic with
respect to solid angle.
• The effect of scattering is not included.
• The implementation assumes gray radiation.
• Solving a problem with a large number of rays is CPU-intensive.
Advantages and Limitations of the P-1 Model
The P-1 model has several advantages over the DTRM. For the P-1
model, the RTE (Equation 11.3-1) is a diffusion equation, which is easy
to solve with little CPU demand. The model includes the effect of scattering. For combustion applications where the optical thickness is large,
the P-1 model works reasonably well. In addition, the P-1 model can
easily be applied to complicated geometries with curvilinear coordinates.
You should be aware of the following limitations when using the P-1
radiation model:
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• The P-1 model assumes that all surfaces are diffuse. This means
that the reflection of incident radiation at the surface is isotropic
with respect to the solid angle.
• The implementation assumes gray radiation.
• There may be a loss of accuracy, depending on the complexity of
the geometry, if the optical thickness is small.
• The P-1 model tends to overpredict radiative fluxes from localized
heat sources or sinks.
Advantages and Limitations of the Rosseland Model
The Rosseland model has two advantages over the P-1 model. Since it
does not solve an extra transport equation for the incident radiation (as
the P-1 model does), the Rosseland model is faster than the P-1 model
and requires less memory.
The Rosseland model can only be used for optically thick media. It is
recommended for use when the optical thickness exceeds 3. Note also
that the Rosseland model is not available when one of the coupled solvers
is being used; it is available only with the segregated solver.
Advantages and Limitations of the DO Model
The DO model spans the entire range of optical thicknesses, and allows
you to solve problems ranging from surface-to-surface radiation to participating radiation in combustion problems. It also allows the solution
of radiation in semi-transparent media. Computational cost is moderate
for typical angular discretizations, and memory requirements are modest.
The current implementation is restricted to either gray radiation or nongray radiation using a gray-band model. Solving a problem with a fine
angular discretization may be CPU-intensive.
The non-gray implementation in FLUENT is intended for use with participating media with a spectral absorption coefficient aλ that varies in
a stepwise fashion across spectral bands, but varies smoothly within the
band. Glass, for example, displays banded behavior of this type. The
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current implementation does not model the behavior of gases such as
carbon dioxide or water vapor, which absorb and emit energy at distinct
wave numbers [161]. The modeling of non-gray gas radiation is still an
evolving field. However, some researchers [66] have used gray-band models to model gas behavior by approximating the absorption coefficients
within each band as a constant. The implementation in FLUENT can be
used in this fashion if desired.
The non-gray implementation in FLUENT is compatible with all the models with which the gray implementation of the DO model can be used.
Thus, it is possible to include scattering, anisotropy, semi-transparent
media, and particulate effects. However, the non-gray implementation
assumes a constant absorption coefficient within each wavelength band.
The weighted sum of gray gases model (WSGGM) cannot be used to
specify the absorption coefficient in each band. The implementation allows the specification of spectral emissivity at walls. The emissivity is
assumed to be constant within each band.
Advantages and Limitations of the S2S Model
The surface-to-surface radiation model is good for modeling the enclosure radiative transfer without participating media (e.g., spacecraft heat
rejection system, solar collector systems, radiative space heaters, and
automotive underhood cooling). In such cases, the methods for participating radiation may not always be efficient. As compared to the DTRM
and the DO radiation model, the S2S model has a much faster time per
iteration, although the view factor calculation itself is CPU-intensive.
You should be aware of the following limitations when using the S2S
radiation model:
• The S2S model assumes that all surfaces are diffuse.
• The implementation assumes gray radiation.
• The storage and memory requirements increase very rapidly as the
number of surface faces increases. This can be minimized by using
a cluster of surface faces, although the CPU time is independent
of the number of clusters that are used.
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• The S2S model cannot be used to model participating radiation
problems.
• The surface clustering method does not work with sliding meshes
or hanging nodes.
• The S2S model cannot be used if your model contains periodic or
symmetry boundary conditions.
11.3.3
The Discrete Transfer Radiation Model (DTRM)
The main assumption of the DTRM is that the radiation leaving the
surface element in a certain range of solid angles can be approximated
by a single ray. This section provides details about the equations used
in the DTRM.
The DTRM Equations
The equation for the change of radiant intensity, dI, along a path, ds,
can be written as
dI
aσT 4
+ aI =
ds
π
where
a
I
T
σ
=
=
=
=
(11.3-2)
gas absorption coefficient
intensity
gas local temperature
Stefan-Boltzmann constant (5.672 × 10−8 W/m2 -K4 )
Here, the refractive index is assumed to be unity. The DTRM integrates
Equation 11.3-2 along a series of rays emanating from boundary faces.
If a is constant along the ray, then I(s) can be estimated as
I(s) =
σT 4
(1 − e−as ) + I0 e−as
π
(11.3-3)
where I0 is the radiant intensity at the start of the incremental path,
which is determined by the appropriate boundary condition (see the description of boundary conditions, below). The energy source in the fluid
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11.3 Radiative Heat Transfer
due to radiation is then computed by summing the change in intensity
along the path of each ray that is traced through the fluid control volume.
The “ray tracing” technique used in the DTRM can provide a prediction
of radiative heat transfer between surfaces without explicit view-factor
calculations. The accuracy of the model is limited mainly by the number
of rays traced and the computational grid.
Ray Tracing
The ray paths are calculated and stored prior to the fluid flow calculation.
At each radiating face, rays are fired at discrete values of the polar and
azimuthal angles (see Figure 11.3.2). To cover the radiating hemisphere,
θ is varied from 0 to π2 and φ from 0 to 2π. Each ray is then traced to
determine the control volumes it intercepts as well as its length within
each control volume. This information is then stored in the radiation
file, which must be read in before the fluid flow calculations begin.
n
θ
φ
P
t
Figure 11.3.2: Angles θ and φ Defining the Hemispherical Solid Angle
About a Point P
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Clustering
DTRM is computationally very expensive when there are too many surfaces to trace rays from and too many volumes crossed by the rays. To
reduce the computational time, the number of radiating surfaces and
absorbing cells is reduced by clustering surfaces and cells into surface
and volume “clusters”. The volume clusters are formed by starting from
a cell and simply adding its neighbors and their neighbors until a specified number of cells per volume cluster is collected. Similarly, surface
clusters are made by starting from a face and adding its neighbors and
their neighbors until a specified number of faces per surface cluster is
collected.
The incident radiation flux, qin , and the volume sources are calculated
for the surface and volume clusters respectively. These values are then
distributed to the faces and cells in the clusters to calculate the wall and
cell temperatures. Since the radiation source terms are highly non-linear
(proportional to the fourth power of temperature), care must be taken
to calculate the average temperatures of surface and volume clusters and
distribute the flux and source terms appropriately among the faces and
cells forming the clusters.
The surface and volume cluster temperatures are obtained by area and
volume averaging as shown in the following equations:
P
Tsc =
Tvc =
Af Tf4
P
Af
!1/4
f
!
P
4 1/4
c Vc Tc
P
Vc
(11.3-4)
(11.3-5)
where Tsc and Tvc are the temperatures of the surface and volume clusters
respectively, Af and Tf are the area and temperature of face f , and Vc
and Tc are the volume and temperature of cell c. The summations are
carried over all faces of a surface cluster and all cells of a volume cluster.
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Boundary Condition Treatment for the DTRM at Walls
The radiation intensity approaching a point on a wall surface is integrated to yield the incident radiative heat flux, qin , as
Z
qin =
~
s·~
n>0
Iin~s · ~ndΩ
(11.3-6)
where Ω is the hemispherical solid angle, Iin is the intensity of the incoming ray, ~s is the ray direction vector, and ~n is the normal pointing
out of the domain. The net radiative heat flux from the surface, qout , is
then computed as a sum of the reflected portion of qin and the emissive
power of the surface:
qout = (1 − w )qin + w σTw4
(11.3-7)
where Tw is the surface temperature of the point P on the surface and w
is the wall emissivity which you input as a boundary condition. FLUENT
incorporates the radiative heat flux (Equation 11.3-7) in the prediction of
the wall surface temperature. Equation 11.3-7 also provides the surface
boundary condition for the radiation intensity I0 of a ray emanating from
the point P , as
I0 =
qout
π
(11.3-8)
Boundary Condition Treatment for the DTRM at Flow Inlets
and Exits
The net radiative heat flux at flow inlets and outlets is computed in the
same manner as at walls, as described above. FLUENT assumes that the
emissivity of all flow inlets and outlets is 1.0 (black body absorption)
unless you choose to redefine this boundary treatment.
FLUENT includes an option that allows you to use different temperatures
for radiation and convection at inlets and outlets. This can be useful
when the temperature outside the inlet or outlet differs considerably
from the temperature in the enclosure. See Section 11.3.16 for details.
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11.3.4
The P-1 Radiation Model
The P-1 radiation model is the simplest case of the more general P-N
model, which is based on the expansion of the radiation intensity I into
an orthogonal series of spherical harmonics [35, 210]. This section provides details about the equations used in the P-1 model.
The P-1 Model Equations
As mentioned above, the P-1 radiation model is the simplest case of
the P-N model. If only four terms in the series are used, the following
equation is obtained for the radiation flux qr :
qr = −
1
∇G
3(a + σs ) − Cσs
(11.3-9)
where a is the absorption coefficient, σs is the scattering coefficient, G
is the incident radiation, and C is the linear-anisotropic phase function
coefficient, described below. After introducing the parameter
Γ=
1
(3(a + σs ) − Cσs )
(11.3-10)
qr = −Γ∇G
(11.3-11)
Equation 11.3-9 simplifies to
The transport equation for G is
∇ (Γ∇G) − aG + 4aσT 4 = SG
(11.3-12)
where σ is the Stefan-Boltzmann constant and SG is a user-defined radiation source. FLUENT solves this equation to determine the local radiation intensity when the P-1 model is active.
Combining Equations 11.3-11 and 11.3-12, the following equation is obtained:
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11.3 Radiative Heat Transfer
−∇qr = aG − 4aσT 4
(11.3-13)
The expression for −∇qr can be directly substituted into the energy
equation to account for heat sources (or sinks) due to radiation.
Anisotropic Scattering
Included in the P-1 radiation model is the capability for modeling anisotropic scattering. FLUENT models anisotropic scattering by means of a
linear-anisotropic scattering phase function:
Φ(~s 0 · ~s) = 1 + C~s 0 · ~s
(11.3-14)
Here, ~s is the unit vector in the direction of scattering, and ~s 0 is the
unit vector in the direction of the incident radiation. C is the linearanisotropic phase function coefficient, which is a property of the fluid.
C ranges from −1 to 1. A positive value indicates that more radiant
energy is scattered forward than backward, and a negative value means
that more radiant energy is scattered backward than forward. A zero
value defines isotropic scattering (i.e., scattering that is equally likely in
all directions), which is the default in FLUENT. You should modify the
default value only if you are certain of the anisotropic scattering behavior
of the material in your problem.
Particulate Effects in the P-1 Model
When your FLUENT model includes a dispersed second phase of particles,
you can include the effect of particles in the P-1 radiation model. Note
that when particles are present, FLUENT ignores scattering in the gas
phase. (That is, Equation 11.3-15 assumes that all scattering is due to
particles.)
For a gray, absorbing, emitting, and scattering medium containing absorbing, emitting, and scattering particles, the transport equation for
the incident radiation can be written as
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!
σT 4
∇ · (Γ∇G) + 4π a
+ Ep − (a + ap )G = 0
π
(11.3-15)
where Ep is the equivalent emission of the particles and ap is the equivalent absorption coefficient. These are defined as follows:
Ep = lim
V →0
N
X
pn Apn
n=1
4
σTpn
πV
(11.3-16)
and
ap = lim
V →0
N
X
pn
n=1
Apn
V
(11.3-17)
In Equations 11.3-16 and 11.3-17, pn , Apn , and Tpn are the emissivity,
projected area, and temperature of particle n. The summation is over
N particles in volume V . These quantities are computed during particle
tracking in FLUENT.
The projected area Apn of particle n is defined as
Apn =
πd2pn
4
(11.3-18)
where dpn is the diameter of the nth particle.
The quantity Γ in Equation 11.3-15 is defined as
Γ=
1
3(a + ap + σp )
(11.3-19)
where the equivalent particle scattering factor is defined as
σp = lim
V →0
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N
X
(1 − fpn )(1 − pn )
n=1
Apn
V
(11.3-20)
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11.3 Radiative Heat Transfer
and is computed during particle tracking. In Equation 11.3-20 fpn is the
scattering factor associated with the nth particle.
Heat sources (sinks) due to particle radiation are included in the energy
equation as follows:
!
σT 4
−∇qr = −4π a
+ Ep + (a + ap )G
π
(11.3-21)
Boundary Condition Treatment for the P-1 Model at Walls
To get the boundary condition for the incident radiation equation, the
dot product of the outward normal vector ~n and equation 11.3-11 is
computed:
qr · ~n = −Γ∇G · ~n
∂G
qr,w = −Γ
∂n
(11.3-22)
(11.3-23)
Thus the flux of the incident radiation, G, at a wall is −qr,w . The wall
radiative heat flux is computed using the following boundary condition:
Iw (~r, ~s) = fw (~r, ~s)
σT 4
fw (~r, ~s) = w w + ρw I(~r, −~s)
π
(11.3-24)
(11.3-25)
where ρw is the wall reflectivity. The Marshak boundary condition is
then used to eliminate the angular dependence [171]:
Z
0
2π
Iw (~r, ~s) ~n · ~s dΩ =
Z
0
2π
fw (~r, ~s) ~n · ~s dΩ
(11.3-26)
Substituting Equations 11.3-24 and 11.3-25 into Equation 11.3-26 and
performing the integrations yields
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4
qr,w
4πw σTπw − (1 − ρw )Gw
=−
2(1 + ρw )
(11.3-27)
If it is assumed that the walls are diffuse gray surfaces, then ρw = 1 − w ,
and Equation 11.3-27 becomes
qr,w = −
w
4σTw4 − Gw
2 (2 − w )
(11.3-28)
Equation 11.3-28 is used to compute qr,w for the energy equation and for
the incident radiation equation boundary conditions.
Boundary Condition Treatment for the P-1 Model at Flow Inlets
and Exits
The net radiative heat flux at flow inlets and outlets is computed in the
same manner as at walls, as described above. FLUENT assumes that the
emissivity of all flow inlets and outlets is 1.0 (black body absorption)
unless you choose to redefine this boundary treatment.
FLUENT includes an option that allows you to use different temperatures
for radiation and convection at inlets and outlets. This can be useful
when the temperature outside the inlet or outlet differs considerably
from the temperature in the enclosure. See Section 11.3.16 for details.
11.3.5
The Rosseland Radiation Model
The Rosseland or diffusion approximation for radiation is valid when the
medium is optically thick ((a+σs )L 1), and is recommended for use in
problems where the optical thickness is greater than 3. It can be derived
from the P-1 model equations, with some approximations. This section
provides details about the equations used in the Rosseland model.
The Rosseland Model Equations
As with the P-1 model, the radiative heat flux vector in a gray medium
can be approximated by Equation 11.3-11:
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11.3 Radiative Heat Transfer
qr = −Γ∇G
(11.3-29)
where Γ is given by Equation 11.3-10.
The Rosseland radiation model differs from the P-1 model in that the
Rosseland model assumes that the intensity is the black-body intensity
at the gas temperature. (The P-1 model actually calculates a transport
equation for G.) Thus G = 4σT 4 . Substituting this value for G into
Equation 11.3-29 yields
qr = −16σΓT 3 ∇T
(11.3-30)
Since the radiative heat flux has the same form as the Fourier conduction
law, it is possible to write
q = qc + qr
(11.3-31)
= −(k + kr )∇T
kr = 16σΓT
3
(11.3-32)
(11.3-33)
where k is the thermal conductivity and kr is the radiative conductivity. Equation 11.3-31 is used in the energy equation to compute the
temperature field.
Anisotropic Scattering
The Rosseland model allows for anisotropic scattering, using the same
phase function (Equation 11.3-14) described for the P-1 model in Section 11.3.4.
Boundary Condition Treatment for the Rosseland Model at Walls
Since the diffusion approximation is not valid near walls, it is necessary
to use a temperature slip boundary condition. The radiative heat flux
at the wall boundary, qr,w , is defined using the slip coefficient ψ:
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qr,w = −
σ Tw4 − Tg4
(11.3-34)
ψ
where Tw is the wall temperature, Tg is the temperature of the gas at
the wall, and the slip coefficient ψ is approximated by a curve fit to the
plot given in [210]:


 1/2
ψ=


2x3 +3x2 −12x+7
54
0
Nw < 0.01
0.01 ≤ Nw ≤ 10
Nw > 10
(11.3-35)
where Nw is the conduction to radiation parameter at the wall:
Nw =
k(a + σs )
4σTw3
(11.3-36)
and x = log10 Nw .
Boundary Condition Treatment for the Rosseland Model at Flow
Inlets and Exits
No special treatment is required at flow inlets and outlets for the Rosseland model. The radiative heat flux at these boundaries can be determined using Equation 11.3-31.
FLUENT includes an option that allows you to use different temperatures
for radiation and convection at inlets and outlets. This can be useful
when the temperature outside the inlet or outlet differs considerably
from the temperature in the enclosure. See Section 11.3.16 for details.
11.3.6
The Discrete Ordinates (DO) Radiation Model
The discrete ordinates (DO) radiation model solves the radiative transfer equation (RTE) for a finite number of discrete solid angles, each
associated with a vector direction ~s fixed in the global Cartesian system
(x, y, z). The fineness of the angular discretization is controlled by you,
analogous to choosing the number of rays for the DTRM. Unlike the
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11.3 Radiative Heat Transfer
DTRM, however, the DO model does not perform ray tracing. Instead,
the DO model transforms Equation 11.3-1 into a transport equation for
radiation intensity in the spatial coordinates (x, y, z). The DO model
solves for as many transport equations as there are directions ~s. The
solution method is identical to that used for the fluid flow and energy
equations.
The implementation in FLUENT uses a conservative variant of the discrete ordinates model called the finite-volume scheme [37, 183], and its
extension to unstructured meshes [165].
The DO Model Equations
The DO model considers the radiative transfer equation (RTE) in the
direction ~s as a field equation. Thus, Equation 11.3-1 is written as
∇ · (I(~r, ~s)~s) + (a + σs )I(~r, ~s) = an2
σT 4
σs
+
π
4π
Z
4π
0
I(~r, ~s 0 ) Φ(~s · ~s 0 ) dΩ0
(11.3-37)
FLUENT also allows the modeling of non-gray radiation using a grayband model. The RTE for the spectral intensity Iλ (~r, ~s) can be written
as
σs
∇·(Iλ (~r, ~s)~s)+(aλ +σs )Iλ (~r, ~s) = aλ n Ibλ +
4π
Z
2
0
4π
Iλ (~r, ~s 0 ) Φ(~s ·~s 0 ) dΩ0
(11.3-38)
Here λ is the wavelength, aλ is the spectral absorption coefficient, and Ibλ
is the black body intensity given by the Planck function. The scattering
coefficient, the scattering phase function, and the refractive index n are
assumed independent of wavelength.
The non-gray DO implementation divides the radiation spectrum into N
wavelength bands, which need not be contiguous or equal in extent. The
wavelength intervals are supplied by you, and correspond to values in
vacuum (n = 1). The RTE is integrated over each wavelength interval,
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resulting in transport equations for the quantity Iλ ∆λ, the radiant energy contained in the wavelength band ∆λ. The behavior in each band
is assumed gray. The black body emission in the wavelength band per
unit solid angle is written as
[F (0 → nλ2 T ) − F (0 → nλ1 T )]n2
σT 4
π
(11.3-39)
where F (0 → nλT ) is the fraction of radiant energy emitted by a black
body [161] in the wavelength interval from 0 to λ at temperature T in a
medium of refractive index n. λ2 and λ1 are the wavelength boundaries
of the band.
The total intensity I(~r, ~s) in each direction ~s at position ~r is computed
using
I(~r, ~s) =
X
Iλk (~r, ~s)∆λk
(11.3-40)
k
where the summation is over the wavelength bands.
Boundary conditions for the non-gray DO model are applied on a band
basis. The treatment within a band is the same as that for the gray DO
model.
Angular Discretization and Pixelation
Each octant of the angular space 4π at any spatial location is discretized
into Nθ ×Nφ solid angles of extent ωi , called control angles. The angles θ
and φ are the polar and azimuthal angles respectively, and are measured
with respect to the global Cartesian system (x, y, z) as shown in Figure 11.3.3. The θ and φ extents of the control angle, ∆θ and ∆φ, are
constant. In two-dimensional calculations, only four octants are solved
due to symmetry, making a total of 4Nθ Nφ directions in all. In threedimensional calculations, a total of 8Nθ Nφ directions are solved. In the
case of the non-gray model, 4Nθ Nφ or 8Nθ Nφ equations are solved for
each band.
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z
s
θ
φ
y
x
Figure 11.3.3: Angular Coordinate System
When Cartesian meshes are used, it is possible to align the global angular
discretization with the control volume face, as shown in Figure 11.3.4.
For generalized unstructured meshes, however, control volume faces do
not in general align with the global angular discretization, as shown in
Figure 11.3.5, leading to the problem of control angle overhang [165].
Essentially, control angles can straddle the control volume faces, so that
they are partially incoming and partially outgoing to the face. Figure 11.3.6 shows a 3D example of a face with control angle overhang.
The control volume face cuts the sphere representing the angular space
at an arbitrary angle. The line of intersection is a great circle. Control
angle overhang may also occur as a result of reflection and refraction.
It is important in these cases to correctly account for the overhanging
fraction. This is done through the use of pixelation [165].
Each overhanging control angle is divided into Nθp ×Nφp pixels, as shown
in Figure 11.3.7. The energy contained in each pixel is then treated as
incoming or outgoing to the face. The influence of overhang can thus be
accounted for within the pixel resolution. FLUENT allows you to choose
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incoming
directions
C0
●
●
n
C1
outgoing
directions
face f
Figure 11.3.4: Face with No Control Angle Overhang
overhanging
control angle
incoming
directions
C0
n
●
● C1
outgoing
directions
face f
Figure 11.3.5: Face with Control Angle Overhang
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11.3 Radiative Heat Transfer
outgoing
directions
z
overhanging
control
angle
y
x
control
volume
face
incoming
directions
Figure 11.3.6: Face with Control Angle Overhang (3D)
control angle ω i
si
control
volume
face
pixel
Figure 11.3.7: Pixelation of Control Angle
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the pixel resolution. For problems involving gray-diffuse radiation, the
default pixelation of 1 × 1 is usually sufficient. For problems involving
symmetry, periodic, specular, or semi-transparent boundaries, a pixelation of 3 × 3 is recommended. You should be aware, however, that
increasing the pixelation adds to the cost of computation.
Anisotropic Scattering
The DO implementation in FLUENT admits a variety of scattering phase
functions. You can choose an isotropic phase function, a linear anisotropic
phase function, a Delta-Eddington phase function, or a user-defined
phase function. The linear anisotropic phase function is described in
Equation 11.3-14. The Delta-Eddington function takes the following
form:
Φ(~s · ~s 0 ) = 2f δ(~s · ~s 0 ) + (1 − f )(1 + C~s · ~s 0 )
(11.3-41)
Here, f is the forward-scattering factor and δ(~s · ~s 0 ) is the Dirac delta
function. The f term essentially cancels a fraction f of the out-scattering;
thus, for f = 1, the Delta-Eddington phase function will cause the intensity to behave as if there is no scattering at all. C is the asymmetry
factor. When the Delta-Eddington phase function is used, you will specify values for f and C.
When a user-defined function is used to specify the scattering phase
function, FLUENT assumes the phase function to be of the form
Φ(~s · ~s 0 ) = 2f δ(~s · ~s 0 ) + (1 − f )Φ∗ (~s · ~s 0 )
(11.3-42)
The user-defined function will specify Φ∗ and the forward-scattering factor f .
The scattering phase functions available for gray radiation can also be
used for non-gray radiation. However, the scattered energy is restricted
to stay within the band.
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Particulate Effects in the DO Model
The DO model allows you to include the effect of a discrete second phase
of particulates on radiation. In this case, FLUENT will neglect all other
sources of scattering in the gas phase.
The contribution of the particulate phase appears in the RTE as:
∇·(I~s)+(a+ap +σp )I(~r, ~s) = an2
σT 4
σp
+Ep +
π
4π
Z
4π
0
I(~r, ~s 0 ) Φ(~s ·~s 0 ) dΩ0
(11.3-43)
where ap is the equivalent absorption coefficient due to the presence of
particulates, and is given by Equation 11.3-17. The equivalent emission
Ep is given by Equation 11.3-16. The equivalent particle scattering factor
σp , defined in Equation 11.3-20, is used in the scattering terms.
For non-gray radiation, absorption, emission, and scattering due to the
particulate phase are included in each wavelength band for the radiation
calculation. Particulate emission and absorption terms are also included
in the energy equation.
Boundary Condition Treatment at Gray-Diffuse Walls
For gray radiation, the incident radiative heat flux, qin , at the wall is
Z
qin =
~
s·~
n>0
Iin~s · ~ndΩ
(11.3-44)
The net radiative flux leaving the surface is given by
qout = (1 − w )qin + n2 w σTw4
(11.3-45)
where n is the refractive index of the medium next to the wall. The
boundary intensity for all outgoing directions ~s at the wall is given by
I0 =
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qout
π
(11.3-46)
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Modeling Heat Transfer
For non-gray radiation, the incident radiative heat flux qin,λ in the band
∆λ at the wall is
Z
qin,λ = ∆λ
~
s·~
n>0
Iin,λ~s · ~ndΩ
(11.3-47)
The net radiative flux leaving the surface in the band ∆λ is given by
qout,λ = (1 − wλ )qin,λ + wλ [F (0 → nλ2 Tw ) − F (0 → nλ1 Tw )]n2 σTw4
(11.3-48)
where wλ is the wall emissivity in the band. The boundary intensity for
all outgoing directions ~s in the band ∆λ at the wall is given by
I0λ =
qout,λ
π∆λ
(11.3-49)
Boundary Condition Treatment at Semi-Transparent Walls
FLUENT allows the specification of both diffusely and specularly reflecting semi-transparent walls. You can prescribe the fraction of the incoming radiation at the semi-transparent wall which is to be reflected and
transmitted diffusely; the rest is treated specularly.
For non-gray radiation, this treatment is applied on a band basis. The
radiant energy within a band ∆λ is transmitted, reflected, and refracted
as in the gray case; there is no transmission, reflection, or refraction of
radiant energy from one band to another.
Specular Semi-Transparent Walls
Consider a ray traveling from a semi-transparent medium a with refractive index na to a semi-transparent medium b with a refractive index nb
in the direction ~s, as shown in Figure 11.3.8. Side a of the interface is the
side that faces medium a; similarly, side b faces medium b. The interface
normal ~n is assumed to point into side a. We distinguish between the
intensity Ia (~s), the intensity in the direction ~s on side a of the interface,
and the corresponding quantity on the side b, Ib (~s).
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11.3 Radiative Heat Transfer
s’
st
θb
medium b
medium a
θa
n
s
sr
nb > na
Figure 11.3.8: Reflection and Refraction of Radiation at the Interface
Between Two Semi-Transparent Media
A part of the energy incident on the interface is reflected, and the rest is
transmitted. The reflection is specular, so that the direction of reflected
radiation is given by
~sr = ~s − 2 (~s · ~n) ~n
(11.3-50)
The radiation transmitted from medium a to medium b undergoes refraction. The direction of the transmitted energy, ~st , is given by Snell’s
law:
sin θb =
na
sin θa
nb
(11.3-51)
where θa is the angle of incidence and θb is the angle of transmission, as
shown in Figure 11.3.8. We also define the direction
~s 0 = ~st − 2 (~st · ~n) ~n
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(11.3-52)
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Modeling Heat Transfer
shown in Figure 11.3.8.
The interface reflectivity on side a [161]
ra (~s) =
1
2
na cos θb − nb cos θa
na cos θb + nb cos θa
2
+
1
2
na cos θa − nb cos θb 2
na cos θa + nb cos θb
(11.3-53)
represents the fraction of incident energy transferred from ~s to ~sr .
The boundary intensity Iw,a (~sr ) in the outgoing direction ~sr on side a of
the interface is determined from the reflected component of the incoming
radiation and the transmission from side b. Thus
Iw,a (~sr ) = ra (~s)Iw,a (~s) + τb (~s 0 )Iw,b (~s 0 )
(11.3-54)
where τb (~s 0 ) is the transmissivity of side b in direction ~s 0 . Similarly, the
outgoing intensity in the direction ~st on side b of the interface, Iw,b (~st ),
is given by
Iw,b (~st ) = rb (~s 0 )Iw,b (~s 0 ) + τa (~s)Iw,a (~s)
(11.3-55)
For the case na < nb , the energy transmitted from medium a to medium
b in the incoming solid angle 2π must be refracted into a cone of apex
angle θc (see Figure 11.3.9) where
θc = sin−1
na
nb
(11.3-56)
Similarly, the transmitted component of the radiant energy going from
medium b to medium a in the cone of apex angle θc is refracted into the
outgoing solid angle 2π. For incident angles greater than θc , total internal
reflection occurs and all the incoming energy is reflected specularly back
into medium b.
When medium b is external to the domain, Iw,b (~s 0 ) is given in Equation 11.3-54 as a part of the problem specification. This boundary specification is usually made by providing the incoming radiative flux and the
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11.3 Radiative Heat Transfer
medium b
medium a
θc
θb
θa
n
nb > na
Figure 11.3.9: Critical Angle θc
solid angle over which the radiative flux is to be applied. The refractive
index of the external medium is assumed to be unity.
Diffuse Semi-Transparent Walls
In many
a diffuse
assumed
value rd .
engineering problems, the semi-transparent interface may be
reflector. For such a case, the interfacial reflectivity r(~s) is
independent of ~s, and equal to the hemispherically averaged
For n = na /nb > 1, rd,a and rd,b are given by [211]
(1 − rd,b )
(11.3-57)
n2
n−1
1 (3n + 1)(n − 1) n2 (n2 − 1)2
+
ln
−
+
2
6(n + 1)2
(n2 + 1)3
n+1
2n3 (n2 + 2n − 1)
8n4 (n4 + 1)
ln(n)
(11.3-58)
+
(n2 + 1)(n4 − 1)
(n2 + 1)(n4 − 1)2
rd,a = 1 −
rd,b =
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The boundary intensity for all outgoing directions on side a of the interface is given by
Iw,a =
rd,a qin,a + τd,b qin,b
π
(11.3-59)
Iw,b =
rd,b qin,b + τd,a qin,a
π
(11.3-60)
Similarly for side b,
where
qin,a = −
Z
Z
qin,b =
4π
4π
Iw,a~s · ~ndΩ, ~s · ~n < 0
Iw,b~s · ~ndΩ, ~s · ~n ≥ 0
(11.3-61)
(11.3-62)
As before, if medium b is external to the domain, qin,b is given as a part
of the boundary specification.
Beam Irradiation
As mentioned above, FLUENT allows the specification of the irradiation
at semi-transparent boundaries. The irradiation is specified in terms
of an incident radiant heat flux (W/m2 ). You can specify the solid
angle over which the irradiation is distributed, as well as the vector of
the centroid of the solid angle. To indicate whether the irradiation is
reflected specularly or diffusely, you can specify the diffuse fraction.
For non-gray radiation, FLUENT allows you to specify the irradiation at
semi-transparent boundaries on a band basis. The irradiation is specified
as an incident heat flux (W/m2 ) for each wavelength band. As in the
gray case, you can specify the solid angle over which the irradiation is
distributed, as well as the vector of the centroid of the solid angle.
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11.3 Radiative Heat Transfer
The Diffuse Fraction
At semi-transparent boundaries, FLUENT allows you to specify the fraction of the incoming radiation that is treated as diffuse. The diffuse
fraction is reflected diffusely, using the treatment described above; the
transmitted portion is also treated diffusely. The remainder of the incoming energy is treated in a specular fashion.
For non-gray radiation, you can specify the diffuse fraction separately
for each band.
Boundary Condition Treatment at Specular Walls and Symmetry
Boundaries
At specular walls and symmetry boundaries, the direction of the reflected ray ~sr corresponding to the incoming direction ~s is given by
Equation 11.3-50. Furthermore
Iw (~sr ) = Iw (~s)
(11.3-63)
Boundary Condition Treatment at Periodic Boundaries
When rotationally periodic boundaries are used, it is important to use
pixelation in order to ensure that radiant energy is correctly transferred
between the periodic and shadow faces. A pixelation between 3 × 3 and
10 × 10 is recommended.
Boundary Condition Treatment at Flow Inlets and Exits
The treatment at flow inlets and exits is described in Section 11.3.3.
11.3.7
The Surface-to-Surface (S2S) Radiation Model
The surface-to-surface radiation model can be used to account for the
radiation exchange in an enclosure of gray-diffuse surfaces. The energy
exchange between two surfaces depends in part on their size, separation distance, and orientation. These parameters are accounted for by a
geometric function called a “view factor”.
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The main assumption of the S2S model is that any absorption, emission,
or scattering of radiation can be ignored; therefore, only “surface-tosurface” radiation need be considered for analysis.
Gray-Diffuse Radiation
FLUENT’s S2S radiation model assumes the surfaces to be gray and
diffuse. Emissivity and absorptivity of a gray surface are independent of
the wavelength. Also, by Kirchoff’s law [161], the emissivity equals the
absorptivity ( = α). For a diffuse surface, the reflectivity is independent
of the outgoing (or incoming) directions.
The gray-diffuse model is what is used in FLUENT. Also, as stated earlier,
for applications of interest, the exchange of radiative energy between surfaces is virtually unaffected by the medium that separates them. Thus,
according to the gray-body model, if a certain amount of radiant energy
(E) is incident on a surface, a fraction (ρE) is reflected, a fraction (αE)
is absorbed, and a fraction (τ E) is transmitted. Since for most applications the surfaces in question are opaque to thermal radiation (in the
infrared spectrum), the surfaces can be considered opaque. The transmissivity, therefore, can be neglected. It follows, from conservation of
energy, that α + ρ = 1, since α = (emissivity), and ρ = 1 − .
The S2S Model Equations
The energy flux leaving a given surface is composed of directly emitted
and reflected energy. The reflected energy flux is dependent on the incident energy flux from the surroundings, which then can be expressed in
terms of the energy flux leaving all other surfaces. The energy reflected
from surface k is
qout,k = k σTk4 + ρk qin,k
(11.3-64)
where qout,k is the energy flux leaving the surface, k is the emissivity,
σ is Boltzmann’s constant, and qin,k is the energy flux incident on the
surface from the surroundings.
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11.3 Radiative Heat Transfer
The amount of incident energy upon a surface from another surface is
a direct function of the surface-to-surface “view factor,” Fjk . The view
factor Fjk is the fraction of energy leaving surface k that is incident on
surface j. The incident energy flux qin,k can be expressed in terms of the
energy flux leaving all other surfaces as
Ak qin,k =
N
X
Aj qout,j Fjk
(11.3-65)
j=1
where Ak is the area of surface k and Fjk is the view factor between
surface k and surface j. For N surfaces, using the view factor reciprocity
relationship gives
Aj Fjk = Ak Fkj for j = 1, 2, 3, . . . N
(11.3-66)
so that
qin,k =
N
X
Fkj qout,j
(11.3-67)
j=1
Therefore,
qout,k =
k σTk4
+ ρk
N
X
Fkj qout,j
(11.3-68)
j=1
which can be written as
Jk = Ek + ρk
N
X
Fkj Jj
(11.3-69)
j=1
where Jk represents the energy that is given off (or radiosity) of surface
k, and Ek represents the emissive power of surface k. This represents N
equations, which can be recast into matrix form as
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Modeling Heat Transfer
KJ = E
(11.3-70)
where K is an N × N matrix, J is the radiosity vector, and E is the
emissive power vector.
Equation 11.3-70 is referred to as the radiosity matrix equation. The
view factor between two finite surfaces i and j is given by
1
Fij =
Ai
Z
Ai
Z
Aj
cos θi cos θj
δij dAi dAj
πr 2
(11.3-71)
where δij is determined by the visibility of dAj to dAi . δij = 1 if dAj is
visible to dAi and 0 otherwise.
Clustering
The S2S radiation model is computationally very expensive when there
are a large number of radiating surfaces. To reduce the computational
time as well as the storage requirement, the number of radiating surfaces
is reduced by creating surface “clusters”. The surface clusters are made
by starting from a face and adding its neighbors and their neighbors until
a specified number of faces per surface cluster is collected.
The radiosity, J, is calculated for the surface clusters. These values
are then distributed to the faces in the clusters to calculate the wall
temperatures. Since the radiation source terms are highly non-linear
(proportional to the fourth power of temperature), care must be taken to
calculate the average temperature of the surface clusters and distribute
the flux and source terms appropriately among the faces forming the
clusters.
The surface cluster temperature is obtained by area averaging as shown
in the following equation:
P
Tsc =
11-50
Af Tf4
P
Af
f
!1/4
(11.3-72)
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11.3 Radiative Heat Transfer
where Tsc is the temperature of the surface cluster, and Af and Tf are
the area and temperature of face f . The summation is carried over all
faces of a surface cluster.
Smoothing
Smoothing can be performed on the view factor matrix to enforce the
reciprocity relationship and conservation.
The reciprocity relationship is represented by
Ai Fij = Ai Fji
(11.3-73)
where Ai is the area of surface i, Fij is the view factor between surfaces
i and j, and Fji is the view factor between surfaces j and i.
Once the reciprocity relationship has been enforced, a least-squares smoothing method [123] can be used to ensure that conservation is satisfied, i.e.,
X
11.3.8
Fij = 1.0
(11.3-74)
Radiation in Combusting Flows
The Weighted-Sum-of-Gray-Gases Model
The weighted-sum-of-gray-gases model (WSGGM) is a reasonable compromise between the oversimplified gray gas model and a complete model
which takes into account particular absorption bands. The basic assumption of the WSGGM is that the total emissivity over the distance s can
be presented as
=
I
X
a,i (T )(1 − e−κi ps )
(11.3-75)
i=0
where a,i are the emissivity weighting factors for the ith fictitious gray
gas, the bracketed quantity is the ith fictitious gray gas emissivity, κi is
the absorption coefficient of the ith gray gas, p is the sum of the partial
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pressures of all absorbing gases, and s is the path length. For a,i and κi
FLUENT uses values obtained from [41] and [219]. These values depend
on gas composition, and a,i also depend on temperature. When the
total pressure is not equal to 1 atm, scaling rules for κi are used (see
Equation 11.3-81).
The absorption coefficient for i = 0 is assigned a value of zero to account
for windows in the spectrum between spectral regions of high absorption
P
( Ii=1 a,i < 1) and the weighting factor for i = 0 is evaluated from [219]:
a,0 = 1 −
I
X
a,i
(11.3-76)
i=1
The temperature dependence of a,i can be approximated by any function, but the most common approximation is
a,i =
J
X
b,i,j T j−1
(11.3-77)
j=1
where b,i,j are the emissivity gas temperature polynomial coefficients.
The coefficients b,i,j and κi are estimated by fitting Equation 11.3-75 to
the table of total emissivities, obtained experimentally [41, 49, 219].
The absorptivity α of the radiation from the wall can be approximated in
a similar way [219], but, to simplify the problem, it is assumed that =
α [160]. This assumption is justified unless the medium is optically thin
and the wall temperature differs considerably from the gas temperature.
Since the coefficients b,i,j and κi are slowly varying functions of ps and
T , they can be assumed constant for a wide range of these parameters.
In [219] these constant coefficients are presented for different relative
pressures of the CO2 and H2 O vapor, assuming that the total pressure pT
is 1 atm. The values of the coefficients shown in [219] are valid for 0.001 ≤
ps ≤ 10.0 atm-m and 600 ≤ T ≤ 2400 K. For T > 2400 K, coefficient
values suggested by [41] are used. If κi ps 1 for all i, Equation 11.3-75
simplifies to
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11.3 Radiative Heat Transfer
=
I
X
a,i κi ps
(11.3-78)
i=0
Comparing Equation 11.3-78 with the gray gas model with absorption
coefficient a, it can be seen that the change of the radiation intensity
over the distance s in the WSGGM is exactly the same as in the gray
gas model with the absorption coefficient
a=
I
X
a,i κi p
(11.3-79)
i=0
which does not depend on s. In the general case, a is estimated as
a=−
ln(1 − )
s
(11.3-80)
where the emissivity for the WSGGM is computed using Equation 11.3-75.
a as defined by Equation 11.3-80 depends on s, reflecting the non-gray
nature of the absorption of thermal radiation in molecular gases. In FLUENT, Equation 11.3-79 is used when s ≤ 10−4 m and Equation 11.3-80
is used for s > 10−4 m. Note that for s ≈ 10−4 m, the values of a predicted by Equations 11.3-79 and 11.3-80 are practically identical (since
Equation 11.3-80 reduces to Equation 11.3-79 in the limit of small s).
FLUENT allows you to specify s as the characteristic cell size or the mean
beam length. The model based on the mean beam length is appropriate
if you have a nearly homogeneous medium and you are interested mainly
in the radiation exchange between the walls of the enclosure. You can
specify the mean beam length or have FLUENT compute it. If you are
primarily interested in the radiation heat exchange between neighboring
cells (e.g., the distribution of radiation in the vicinity of a heater), which
is very common for the optically thick media for which the P-1 model is
primarily designed, using the characteristic cell size as s is more appropriate. Note that the values of a predicted by the WSGGM based on
the characteristic cell size can be somewhat grid-dependent, if s is small.
This grid dependence, however, will not necessarily affect the predicted
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Modeling Heat Transfer
temperature distribution, since the radiation energy is proportional to
T 4 . The characteristic-cell-size approach may give a better temperature
distribution, while the mean-beam-length approach can give more accurate fluxes at the boundaries. See Section 7.6.1 for details about setting
properties for the WSGGM.
! The WSGGM cannot be used to specify the absorption coefficient in
each band when using the non-gray DO model. If the WSGGM is used
with the non-gray DO model, the absorption coefficient will be the same
in all bands.
When pT 6= 1 atm
The WSGGM, as described above, assumes that pT —the total (static)
gas pressure—is equal to 1 atm. In cases where pT is not unity (e.g.,
combustion at high temperatures), scaling rules suggested in [59] are
used to introduce corrections. When pT < 0.9 atm or pT > 1.1 atm, the
values for κi in Equations 11.3-75 and 11.3-79 are rescaled:
κi → κi pm
T
(11.3-81)
where m is a nondimensional value obtained from [59], which depends
on the partial pressures and temperature T of the absorbing gases, as
well as on pT .
The Effect of Soot on the Absorption Coefficient
When soot formation is computed, FLUENT can include the effect of the
soot concentration on the radiation absorption coefficient. The generalized soot model estimates the effect of the soot on radiative heat transfer
by determining an effective absorption coefficient for soot. The absorption coefficient of a mixture of soot and an absorbing (radiating) gas is
then calculated as the sum of the absorption coefficients of pure gas and
pure soot:
as+g = ag + as
11-54
(11.3-82)
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11.3 Radiative Heat Transfer
where ag is the absorption coefficient of gas without soot (obtained from
the WSGGM) and
as = b1 cm [1 + bT (T − 2000)]
(11.3-83)
with
b1 = 1232.4 m2 /kg and bT ≈ 4.8 × 10−4 K−1
cm is the soot concentration in kg/m3 .
The coefficients b1 and bT were obtained [199] by fitting Equation 11.3-83
to data based on the Taylor-Foster approximation [240] and data based
on the Smith et al. approximation [219].
See Sections 7.6 and 17.2.3 for information about including the sootradiation interaction effects.
The Effect of Particles on the Absorption Coefficient
FLUENT can also include the effect of discrete phase particles on the
radiation absorption coefficient, provided that you are using either the
P-1 or the DO model. When the P-1 or DO model is active, radiation
absorption by particles can be enabled. The particle emissivity, reflectivity, and scattering effects are then included in the calculation of the
radiative heat transfer. See Section 19.11 for more details on the input
of radiation properties for the discrete phase.
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11.3.9
Overview of Using the Radiation Models
The procedure for setting up and solving a radiation problem is outlined
below, and described in detail in Sections 11.3.10–11.3.18. Steps that
are relevant only for a particular radiation model are noted as such.
Remember that only the steps that are pertinent to radiation modeling
are shown here. For information about inputs related to other models
that you are using in conjunction with radiation, see the appropriate
sections for those models.
1. Select the radiation model, as described in Section 11.3.10.
2. If you are using the DTRM, define the ray tracing as described
in Section 11.3.11. If you are using the S2S model, compute or
read the view factors as described in Section 11.3.12. If you are
using the DO model, define the angular discretization as described
in Section 11.3.13 and, if relevant, define the non-gray radiation
parameters as described in Section 11.3.14.
3. Define the material properties, as described in Section 11.3.15.
4. Define the boundary conditions, as described in Section 11.3.16. If
your model contains a semi-transparent medium, see the information below on setting up semi-transparent media.
5. Set the solution parameters (DTRM, DO, S2S, and P-1 only). See
Section 11.3.17 for details.
6. Solve the problem, as described in Section 11.3.18.
Setup of Semi-Transparent Media
As part of step 4 above, you will take the following steps in order to set
up a semi-transparent medium such as glass in your domain.
1. If your semi-transparent material is a solid, enable the calculation
of radiation in the solid cell zone, as described in Section 11.3.16.
(If your semi-transparent material is a fluid, this step is not necessary.)
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11.3 Radiative Heat Transfer
2. At internal boundaries between the semi-transparent medium and
adjacent fluids (or between adjacent semi-transparent media), set
the two-sided wall to be semi-transparent, as described in Section 11.3.16. This will enable radiation to pass through the internal boundary and will account for effects such as reflection and
refraction.
3. At external semi-transparent boundaries, set the external wall radiation boundary condition to be semi-transparent, as described in
Section 11.3.16. This will allow the externally specified radiative
flux to enter the domain, and also allow the transmission of radiation from the interior to the outside. Both the radiation from the
exterior of the domain and the radiation being transmitted from
the interior to the outside will be reflected and refracted at the
boundary appropriately.
4. Specify the degree to which interior and exterior walls reflect diffusely or specularly by setting the diffuse fraction, as described in
Section 11.3.16.
5. Specify the appropriate refractive index for the material associated
with the solid cell zone (in the Materials panel).
If you are not interested in the detailed temperature distribution inside
your semi-transparent medium, you can use a thin semi-transparent wall
instead of a semi-transparent solid zone, as described in Section 11.3.16.
11.3.10
Selecting the Radiation Model
You can enable radiative heat transfer by selecting a radiation model in
the Radiation Model panel (Figure 11.3.10).
Define −→ Models −→Radiation...
Select Rosseland, P1, Discrete Transfer (DTRM), Surface to Surface (S2S),
or Discrete Ordinates as the Model. To disable radiation, select Off. Note
that when the DTRM or the DO or S2S model is activated, the Radiation
Model panel will expand to show additional parameters (described in
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Figure 11.3.10: The Radiation Model Panel (DO Model)
Sections 11.3.12, 11.3.13, 11.3.14, and 11.3.17). These parameters will
not appear if you select one of the other radiation models.
! The Rosseland model can be used only with the segregated solver.
When the radiation model is active, the radiation fluxes will be included
in the solution of the energy equation at each iteration. If you set up
a problem with the radiation model turned on, and you then decide to
turn it off completely, you must select the Off button in the Radiation
Model panel.
Note that, when you enable a radiation model, FLUENT will automatically turn on solution of the energy equation; you need not turn on the
energy equation first.
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11.3.11
Defining the Ray Tracing for the DTRM
When you select the Discrete Transfer model and click OK in the Radiation Model panel, the Ray Tracing panel (Figure 11.3.11) will open
automatically. (Should you need to modify the current settings later
in the problem setup or solution procedure, you can open this panel
manually using the Define/Ray Tracing... menu item.)
Figure 11.3.11: The Ray Tracing Panel
In this panel you will set parameters for and create the rays and clusters
discussed in Section 11.3.3.
The procedure is as follows:
1. To control the number of radiating surfaces and absorbing cells,
set the Cells Per Volume Cluster and Faces Per Surface Cluster. (See
the explanation below.)
2. To control the number of rays being traced, set the number of
Theta Divisions and Phi Divisions. (Guidelines are provided below.)
3. When you click OK in the Ray Tracing panel, a Select File dialog
box will open, prompting you for the name of the “ray file”. After
you have specified the file name and written the ray file, FLUENT
will read the file back in again for use in the calculation. See below
for details.
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! If you cancel the Ray Tracing panel without writing and reading the ray
file, the DTRM will be disabled.
Controlling the Clusters
Your inputs for Cells Per Volume Cluster and Faces Per Surface Cluster will
control the number of radiating surfaces and absorbing cells. By default,
each is set to 1, so the number of surface clusters (radiating surfaces)
will be the number of boundary faces, and the number of volume clusters
(absorbing cells) will be the number of cells in the domain. For small 2D
problems, these are acceptable numbers, but for larger problems you will
want to reduce the number of surface and/or volume clusters in order
to reduce the ray-tracing expense. (See Section 11.3.3 for details about
clustering.)
Controlling the Rays
Your inputs for Theta Divisions and Phi Divisions will control the number
of rays being traced from each surface cluster (radiating surface).
Theta Divisions defines the number of discrete divisions in the angle θ
used to define the solid angle about a point P on a surface. The solid
angle is defined as θ varies from 0 to 90 degrees (Figure 11.3.2), and the
default setting of 2 for the number of discrete settings implies that each
ray traced from the surface will be located at a 45◦ angle from the other
rays.
Phi Divisions defines the number of discrete divisions in the angle φ used
to define the solid angle about a point P on a surface. The solid angle
is defined as φ varies from 0 to 180 degrees in 2D and from 0 to 360
degrees in 3D (Figure 11.3.2). The default setting of 2 implies that each
ray traced from the surface will be located at a 90 ◦ angle from the other
rays in 2D calculations, and in combination with the default setting
for Theta Divisions, above, implies that 4 rays will be traced from each
surface control volume in your 2D model. Note that the Phi Divisions
should be increased to 4 for equivalent accuracy in 3D models. In many
cases, it is recommended that you at least double the number of divisions
in θ and φ.
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Writing and Reading the DTRM Ray File
After you have activated the DTRM and defined all of the parameters
controlling the ray tracing, you must create a ray file which will be read
back in and used during the radiation calculation. The ray file contains
a description of the ray traces (path lengths, cells traversed by each
ray, etc.). This information is stored in the ray file, instead of being
recomputed, in order to speed up the calculation process.
By default, a binary ray file will be written. You can also create text
(formatted) ray files by turning off the Write Binary Files option in the
Select File dialog box.
! Do not write or read a compressed ray file, because FLUENT will not be
able to access the ray tracing information properly from a compressed
ray file.
The ray filename must be specified to FLUENT only once. Thereafter, the
filename is stored in your case file and the ray file will be automatically
read into FLUENT whenever the case file is read. FLUENT will remind
you that it is reading the ray file after it finishes reading the rest of the
case file by reporting its progress in the text (console) window.
Note that the ray filename stored in your case file may not contain the full
name of the directory in which the ray file exists. The full directory name
will be stored in the case file only if you initially read the ray file through
the GUI (or if you typed in the directory name along with the filename
when using the text interface). In the event that the full directory name
is absent, the automatic reading of the ray file may fail (since FLUENT
does not know in which directory to look for the file), and you will need
to manually specify the ray file, using the File/Read/Rays... menu item.
The safest approaches are to use the GUI when you first read the ray
file or to supply the full directory name when using the text interface.
! You should recreate the ray file whenever you do anything that changes
the grid, such as:
• change the type of a boundary zone
• adapt or reorder the grid
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• scale the grid
• change from 2D space to axisymmetric space or vice versa
You can open the Ray Tracing panel directly with the Define/Ray Tracing... menu item.
Displaying the Clusters
Once a ray file has been created or read in manually, you can click on the
Display Clusters button in the Ray Tracing panel to graphically display the
clusters in the domain. See Section 11.3.20 for additional information
about displaying rays and clusters.
11.3.12
Computing or Reading the View Factors for the S2S
Model
When you select the Surface to Surface (S2S) model, the Radiation Model
panel will expand (see Figure 11.3.12). In this section of the panel,
you will compute the view factors for your problem or read previously
computed view factors into FLUENT.
The S2S radiation model is computationally very expensive when there
are a large number of radiating surfaces. To reduce the computational
time as well as the storage requirement, the number of radiating surfaces
is reduced by creating surface clusters. The surface cluster information
(coordinates and connectivity of the nodes, surface cluster IDs) is used
by FLUENT to compute the view factors for the surface clusters.
! You should recreate the surface cluster information whenever you do
anything that changes the grid, such as:
• change the type of a boundary zone
• reorder the grid
• scale the grid
• change from 2D space to axisymmetric space or vice versa
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Figure 11.3.12: The Radiation Model Panel (S2S Model)
Note that you do not need to recalculate view factors after shell conduction at any wall has been enabled or disabled. See Section 6.13.1 for
more information about shell conduction.
Computing View Factors
FLUENT can compute the view factors for your problem in the current
session and save them to a file for use in the current session and future
sessions. Alternatively, you can save the surface cluster information and
view factor parameters to a file, calculate the view factors outside FLUENT, and then read the view factors into FLUENT. These methods for
computing view factors are described below.
! For large meshes or complex models, it is recommended that you calculate the view factors outside FLUENT and then read them into FLUENT
before starting your simulation.
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Computing View Factors Inside FLUENT
To compute view factors in your current FLUENT session, you must first
set the parameters for the view factor calculation in the View Factor
and Cluster Parameters panel (see below for details). When you have set
the view factor and surface cluster parameters, click Compute/Write...
under Methods in the Radiation Model panel. A Select File dialog box
will open, prompting you for the name of the file in which FLUENT
should save the surface cluster information and the view factors. After
you have specified the file name, FLUENT will write the surface cluster
information to the file. FLUENT will use the surface cluster information
to compute the view factors, save the view factors to the same file, and
then automatically read the view factors.
Computing View Factors Outside FLUENT
To compute view factors outside FLUENT, you must save the surface
cluster information and view factor parameters to a file.
File −→ Write −→Surface Clusters...
FLUENT will open the View Factor and Cluster Parameters panel, where
you will set the view factor and surface cluster parameters (see below for
details). When you click OK in the View Factor and Cluster Parameters
panel, a Select File dialog box will open, prompting you for the name of
the file in which FLUENT should save the surface cluster information and
view factor parameters. After you have specified the file name, FLUENT
will write the surface cluster information and view factor parameters
to the file. If the specified Filename ends in .gz or .Z, appropriate file
compression will be performed. (See Section 3.1.5 for details about file
compression.)
To calculate the view factors outside FLUENT, enter one of the following
commands:
• For the serial solver:
utility viewfac inputfile
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where inputfile is the filename, or the correct path to the filename,
for the surface cluster information and view factor parameters file
that you saved from FLUENT. You can then read the view factors
into FLUENT, as described below.
• For the network parallel solver:
utility viewfac -p -tn -cnf=host1,host2,. . .,hostn inputfile
where n is the number of compute nodes, and host1, host2,. . . are
the names of the machines being used.
!
Note that host1 must be a host machine.
• For a dedicated parallel machine with multiple processors:
utility viewfac -tn inputfile
! Note that for parallel runs (dedicated or network) using n processors,
the problem is duplicated for each processor. For example, if the view
factor calculation requires 100 MB of RAM using a single CPU, it will
require 100n MB of RAM to run the calculation on n processors.
Reading View Factors into FLUENT
If the view factors for your problem have already been computed (either
inside or outside FLUENT) and saved to a file, you can read them into
FLUENT. To read in the view factors, click Read... under Methods in the
Radiation Model panel. A Select File dialog box will open where you can
specify the name of the file containing the view factors. You can also
manually specify the view factors file, using the File/Read/View Factors...
menu item.
Setting View Factor and Surface Cluster Parameters
You will use the View Factor and Cluster Parameters panel (Figure 11.3.13)
to set view factor and cluster parameters for the S2S model. To open
this panel, click Set... under Parameters in the Radiation Model panel or
use the File/Write/Surface Clusters... menu item.
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Figure 11.3.13: The View Factor and Cluster Parameters Panel
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Controlling the Clusters
Your input for Faces Per Surface Cluster will control the number of radiating surfaces. By default, it is set to 1, so the number of surface clusters
(radiating surfaces) will be the number of boundary faces. For small 2D
problems, this is an acceptable number. For larger problems, you may
want to reduce the number of surface clusters to reduce both the size
of the view-factor file and the memory requirement. Such a reduction
in the number of clusters, however, comes at the cost of some accuracy.
(See Section 11.3.7 for details about clustering.)
In some cases, you may wish to modify the cutoff or “split” angle between
adjacent face normals for the purpose of controlling surface clustering.
The split angle sets the limit for which adjacent surfaces are clustered. A
smaller split angle allows for a better representation of the view factor.
By default, no surface cluster will contain any face that has a face normal
greater than 20◦ . To modify the value of this parameter, you can use
the split-angle text command:
define −→ models −→ radiation −→ s2s-parameters −→split-angle
or
file −→ write-surface-clusters −→split-angle
Specifying the Orientation of Surface Pairs
View factor calculations depend on the geometric orientations of surface
pairs with respect to each other. Two situations may be encountered
when examining surface pairs:
• If there is no obstruction between the surface pairs under consideration, then they are referred to as “non-blocking” surfaces.
• If there is another surface blocking the views between the surfaces
under consideration, then they are referred to as “blocking” surfaces. Blocking will change the view factors between the surface
pairs and require additional checks to compute the correct value of
the view factors.
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For cases with blocking surfaces, select Blocking under Surfaces in the
View Factor and Cluster Parameters panel. For cases with non-blocking
surfaces, you can choose either Blocking or Nonblocking without affecting
the accuracy. However, it is better to choose Nonblocking for such cases,
as it takes less time to compute.
Selecting the Method for Smoothing
In order to enforce reciprocity and conservation (see Section 11.3.7),
smoothing can be performed on the view factor matrix. To use the
least-squares method for smoothing of the view factor matrix, select
Least Square under Smoothing in the View Factor and Cluster Parameters
panel. If you do not wish to smooth the view factor matrix, select None
under Smoothing.
Selecting the Method for Computing View Factors
FLUENT provides two methods for computing view factors: the hemicube
method and the adaptive method. The hemicube method is available
only for 3D cases.
The adaptive method calculates the view factors on a pair-by-pair basis
using a variety of algorithms (analytic or Gauss quadrature) that are
chosen adaptively depending on the proximity of the surfaces. To maintain accuracy, the order of the quadrature increases the closer the faces
are together. For surfaces that are very close to each other, the analytic
method is used. FLUENT determines the method to use by performing
a visibility calculation. The Gaussian quadrature method is used if none
of the rays from a surface are blocked by the other surface. If some of
the rays are blocked by the other surface, then either a Monte Carlo
integration method or a quasi-Monte Carlo integration method is used.
To use the adaptive method to compute the view factors, select Adaptive
in the View Factor and Cluster Parameters panel. It is recommended that
you use the adaptive method for simple models, because it is faster than
the hemicube method for these types of models.
The hemicube method uses a differential area-to-area method and calculates the view factors on a row-by-row basis. The view factors calculated
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from the differential areas are summed to provide the view factor for the
whole surface. This method originated from the use of the radiosity
approach in the field of computer graphics [40].
To use the hemicube method to compute the view factors, select Hemicube
in the View Factor and Cluster Parameters panel. It is recommended that
you use the hemicube method for large complex models, because it is
faster than the adaptive method for these types of models.
The hemicube method is based upon three assumptions about the geometry of the surfaces: aliasing, visibility, and proximity. To validate
these assumptions, you can specify three different hemicube parameters,
which can help you obtain better accuracy in calculating view factors.
In most cases, however, the default settings will be sufficient.
• Aliasing—The true projection of each visible face onto the hemicube
can be accurately accounted for by using a finite-resolution hemicube.
As described above, the faces are projected onto a hemicube. Because of the finite resolution of the hemicube, the projected areas
and resulting view factors may be over- or under-estimated. Aliasing effects can be reduced by increasing the value of the Resolution
of the hemicube under Hemicube Parameters.
• Visibility—The visibility between any two faces does not change.
In some cases, face i has a complete view of face k from its centroid, but some other face j occludes much of face k from face i.
In such a case, the hemicube method will overestimate the view
factor between face i and face k calculated from the centroid of
face i. This error can be reduced by subdividing face i into smaller
subfaces. You can specify the number of subfaces by entering a
value for Subdivision under Hemicube Parameters.
• Proximity—The distance between faces is great compared to the effective diameter of the faces. The proximity assumption is violated
whenever faces are close together in comparison to their effective
diameter or are adjacent to one another. In such cases, the distances between the centroid of one face and all points on the other
face vary greatly. Since the view factor dependence on distance is
non-linear, the result is a poor estimate of the view factor.
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Under Hemicube Parameters, you can set a limit for the Normalized Separation Distance, which is the ratio of the minimum face
separation to the effective diameter of the face. If the computed
normalized separation distance is less than the specified value, the
face will then be divided into a number of subfaces until the normalized distances of the subfaces are greater than the specified
value. Alternatively, you can specify the number of subfaces to
create for such faces by entering a value for Subdivision.
11.3.13
Defining the Angular Discretization for the DO Model
When you select the Discrete Ordinates model, the Radiation Model panel
will expand to show inputs for Angular Discretization (see Figure 11.3.10).
In this section, you will set parameters for the angular discretization and
pixelation described in Section 11.3.6.
Theta Divisions (Nθ ) and Phi Divisions (Nφ ) will define the number of
control angles used to discretize each octant of the angular space (see
Figure 11.3.3). For a 2D model, FLUENT will solve only 4 octants (due to
symmetry); thus, a total of 4Nθ Nφ directions ~s will be solved. For a 3D
model, 8 octants are solved, resulting in 8Nθ Nφ directions ~s. By default,
the number of Theta Divisions and the number of Phi Divisions are both
set to 2. For most practical problems, these settings are adequate. A
finer angular discretization can be specified to better resolve the influence
of small geometric features or strong spatial variations in temperature,
but larger numbers of Theta Divisions and Phi Divisions will add to the
cost of the computation.
Theta Pixels and Phi Pixels are used to control the pixelation that accounts for any control volume overhang (see Figure 11.3.7 and the figures
and discussion preceding it). For problems involving gray-diffuse radiation, the default pixelation of 1 × 1 is usually sufficient. For problems
involving symmetry, periodic, specular, or semi-transparent boundaries,
a pixelation of 3 × 3 is recommended. You should be aware, however,
that increasing the pixelation adds to the cost of computation.
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11.3.14
Defining Non-Gray Radiation for the DO Model
If you want to model non-gray radiation using the DO model, you can
specify the Number Of Bands (N ) under Non-Gray Model in the expanded
Radiation Model panel (Figure 11.3.14). For a 2D model, FLUENT will
solve 4Nθ Nφ N directions. For a 3D model, 8Nθ Nφ N directions will be
solved. By default, the Number of Bands is set to zero, indicating that
only gray radiation will be modeled. Because the cost of computation
increases directly with the number of bands, you should try to minimize
the number of bands used. In many cases, the absorption coefficient or
the wall emissivity is effectively constant for the wavelengths of importance in the temperature range of the problem. For such cases, the gray
DO model can be used with little loss of accuracy. For other cases, nongray behavior is important, but relatively few bands are necessary. For
typical glasses, for example, two or three bands will frequently suffice.
When a non-zero Number Of Bands is specified, the Radiation Model panel
will expand once again to show the Wavelength Intervals (Figure 11.3.14).
You can specify a Name for each wavelength band, as well as the Start
and End wavelength of the band in µm. Note that the wavelength bands
are specified for vacuum (n = 1). FLUENT will automatically account
for the refractive index in setting band limits for media with n different
from unity.
The frequency of radiation remains constant as radiation travels across a
semi-transparent interface. The wavelength, however, changes such that
nλ is constant. Thus, when radiation passes from a medium with refractive index n1 to one with refractive index n2 , the following relationship
holds:
n 1 λ1 = n 2 λ2
(11.3-84)
Here λ1 and λ2 are the wavelengths associated with the two media. It is
conventional to specify the wavelength rather than frequency. FLUENT
requires you to specify wavelength bands for an equivalent medium with
n = 1.
For example, consider a typical glass with a step jump in the absorption
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Figure 11.3.14: The Radiation Model Panel (Non-Gray DO Model)
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coefficient at a cut-off wavelength of λc . The absorption coefficient is
a1 for λ ≤ λc µm and a2 for λ > λc µm. The refractive index of the
glass is ng . Since nλ is constant across a semi-transparent interface, the
equivalent cut-off wavelength for a medium with n = 1 is ng λc using
Equation 11.3-84. You should choose two bands in this case, with the
limits 0 to ng λc and ng λc to 100. Here, the upper wavelength limit
has been chosen to be a large number, 100, in order to ensure that
the entire spectrum is covered by the bands. When multiple materials
exist, you should convert all the cut-off wavelengths to equivalent cutoff wavelengths for an n = 1 medium, and choose the band boundaries
accordingly.
The bands can have different widths and need not be contiguous. You
can ensure that the entire spectrum is covered by your bands by choosing
λmin = 0 and nλmax Tmin ≥ 50, 000. Here λmin and λmax are the minimum
and maximum wavelength bounds of your wavelength bands, and Tmin
is the minimum expected temperature in the domain.
11.3.15
Defining Material Properties for Radiation
When you are using the P-1, DO, or Rosseland radiation model in FLUENT, you should be sure to define both the absorption and scattering
coefficients of the fluid in the Materials panel. If you are modeling semitransparent media using the DO model, you should also define the refractive index for the semi-transparent fluid or solid material. For the
DTRM, you need to define only the absorption coefficient.
Define −→Materials...
If your model includes gas phase species such as combustion products,
absorption and/or scattering in the gas may be significant. The scattering coefficient should be increased from the default of zero if the fluid
contains dispersed particles or droplets which contribute to scattering.
FLUENT provides the facility for input of a composition-dependent absorption coefficient for CO2 and H2 O mixtures, using the WSGGM. The
method for computing a variable absorption coefficient is described in
Section 11.3.8. Section 7.6 provides a detailed description of the procedures used for input of radiation properties.
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Absorption Coefficient for a Non-Gray DO Model
If you are using the non-gray DO model, you can specify a different
constant absorption coefficient for each of the bands used by the grayband model, as described in Section 7.6. You cannot, however, compute
a composition-dependent absorption coefficient in each band. If you use
the WSGGM to compute a variable absorption coefficient, the value will
be the same for all bands.
11.3.16
Setting Radiation Boundary Conditions
When you set up a problem that includes radiation, you will set additional boundary conditions at walls, inlets, and exits.
Define −→Boundary Conditions...
Inlet and Exit Boundary Conditions
Emissivity
When radiation is active, you can define the emissivity at each inlet
and exit boundary when you are defining boundary conditions in the
associated inlet or exit boundary panel (Pressure Inlet panel, Velocity
Inlet panel, Pressure Outlet panel, etc.). Enter the appropriate value for
Internal Emissivity. The default value for all boundary types is 1.
For non-gray DO models, the specified constant emissivity will be used
for all wavelength bands.
! The Internal Emissivity boundary condition is not available with the
Rosseland model.
Black Body Temperature
FLUENT includes an option that allows you to take into account the influence of the temperature of the gas and the walls beyond the inlet/exit
boundaries, and specify different temperatures for radiation and convection at inlets and exits. This is useful when the temperature outside
the inlet or exit differs considerably from the temperature in the enclosure. For example, if the temperature of the walls beyond the inlet is
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2000 K and the temperature at the inlet is 1000 K, you can specify the
outside-wall temperature to be used for computing radiative heat flux,
while the actual temperature at the inlet is used for calculating convective heat transfer. To do this, you would specify a radiation temperature
of 2000 K as the black body temperature.
Although this option allows you to account for both cooler and hotter
outside walls, you must use caution in the case of cooler walls, since
the radiation from the immediate vicinity of the hotter inlet or outlet
almost always dominates over the radiation from cooler outside walls.
If, for example, the temperature of the outside walls is 250 K and the
inlet temperature is 1500 K, it might be misleading to use 250 K for the
radiation boundary temperature. This temperature might be expected
to be somewhere between 250 K and 1500 K; in most cases it will be
close to 1500 K. (Its value depends on the geometry of the outside walls
and the optical thickness of the gas in the vicinity of the inlet.)
In the flow inlet or exit panel (Pressure Inlet panel, Velocity Inlet panel,
etc.), select Specified External Temperature in the External Black Body
Temperature Method drop-down list, and then enter the value of the
radiation boundary temperature as the Black Body Temperature.
! If you want to use the same temperature for radiation and convection,
retain the default selection of Boundary Temperature as the External Black
Body Temperature Method.
! The Black Body Temperature boundary condition is not available with
the Rosseland model.
Wall Boundary Conditions for the DTRM, and the P-1, S2S, and
Rosseland Models
The DTRM and the P-1, S2S, and Rosseland models assume all walls to
be gray and diffuse. The only radiation boundary condition required in
the Wall panel is the emissivity. For the Rosseland model, the internal
emissivity is 1. For the DTRM and the P-1 and S2S models, you can
enter the appropriate value for Internal Emissivity in the Radiation section
of the Wall panel. The default value is 1.
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Wall Boundary Conditions for the DO Model
When the DO model is used, you can model diffuse, specular, and semitransparent walls, as discussed in Section 11.3.6.
You can use a diffuse wall to model wall boundaries in many industrial
applications since, for the most part, surface roughness makes the reflection of incident radiation diffuse. For highly polished surfaces, such
as reflectors or mirrors, the specular boundary condition is appropriate.
The semi-transparent boundary condition is appropriate for modeling
glass panes in air, for example.
Diffuse Wall Boundary Conditions for the DO Model
In the Radiation section of the Wall panel, select diffuse in the BC Type
drop-down list to specify a diffuse wall. Diffuse walls are treated as gray
if gray radiation is being computed, or non-gray if the non-gray DO
model is being used. Once you have selected diffuse as the BC Type,
the only radiation boundary condition required in the Wall panel is the
emissivity.
For gray-radiation DO models, enter the appropriate value for Internal
Emissivity. (The default value is 1.) For non-gray DO models, specify a
constant Internal Emissivity for each wavelength band. (The default value
in each band is 1.)
Specular Wall Boundary Conditions for the DO Model
In the Radiation section of the Wall panel, select specular in the BC Type
drop-down list to specify a specular wall. No additional inputs are required.
Semi-Transparent Wall Boundary Conditions for the DO Model
In the Radiation section of the Wall panel, select semi-transparent in the
BC Type drop-down list to specify a semi-transparent wall.
For an external semi-transparent wall, you can define an external irradiation flux in the Wall panel (see Figure 11.3.15). For an internal
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semi-transparent wall, see the discussion on multiple-zone domains, below.
Figure 11.3.15: The Wall Panel for a Semi-Transparent Wall
The inputs for an external semi-transparent wall are as follows:
1. Specify the value of the irradiation flux under Irradiation. If the
non-gray DO model is being used, a constant Irradiation can be
specified for each band.
2. Define the Beam Width by specifying the beam Theta and Phi extents.
3. Specify the (X,Y,Z) vector that defines the Beam Direction.
4. Specify the fraction of the irradiation that is to be treated as diffuse. By default, the Diffuse Fraction is set to 1, indicating that
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all of the irradiation is diffuse. If you specify a value less than 1,
the diffuse fraction will be reflected diffusely (as described in Section 11.3.6), the transmitted portion will also be reflected diffusely,
and the remainder will be reflected specularly.
If the non-gray DO model is being used, the Diffuse Fraction can
be specified for each band.
! Note that the refractive index of the external medium is assumed to be 1.
! If Heat Flux conditions are specified in the Thermal section of the Wall
panel, the specified heat flux is considered to be only the conduction and
convection portion of the boundary flux. The given irradiation specifies
the incoming exterior radiative flux; the radiative flux transmitted from
the domain interior to the outside is computed as a part of the calculation
by FLUENT.
Enabling Radiation in Specific Cell Zones (DO Model Only)
With the DO model, you can specify whether or not you want to solve for
radiation in each cell zone in the domain. By default, the DO equations
are solved in all fluid zones, but not in any solid zones. If you want to
model semi-transparent media, for example, you can enable radiation in
the solid zone(s). To do so, turn on the Participates In Radiation option
in the Solid panel (Figure 11.3.16).
! In general, you should not turn off the Participates In Radiation option
for any fluid zones.
Two-Sided-Wall Boundary Conditions for the DO Model in
Multiple-Zone Domains
For the DO model, you can specify the boundary condition on each side
of a two-sided wall independently to be either diffuse or specular. Note
that the two fluid zones bordering the wall will not be radiatively coupled
(although you can choose them to be thermally coupled.)
You can also choose to couple the contiguous fluid or solid zones radiatively by making the two-sided wall between them semi-transparent. In
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Figure 11.3.16: The Solid Panel
this case, radiation will pass through the wall. You can specify the twosided wall to be semi-transparent only if both the neighboring cell zones
participate in radiation; both sides of the wall will be semi-transparent
if you define one side to be semi-transparent. You can, however, specify
a different diffuse fraction for each side.
It is also possible to associate a thickness with the two-sided wall. In this
case, the refraction due to the wall thickness is accounted for as radiation
travels through the boundary. You can specify a Wall Thickness and a
wall Material Name in the Wall panel (as described in Section 6.13.1).
The refractive index and the absorption coefficient are those of the specified wall material. Only a constant absorption coefficient is allowed
for a solid material. The effective reflectivity and transmissivity of the
wall are computed assuming a planar layer of the given thickness with
absorption but no emission. The refractive indices of the surrounding
media correspond to those of the surrounding fluid materials. (When an
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external wall is specified to be semi-transparent, the refractive index of
the external medium is assumed to be 1.)
Thermal Boundary Conditions
In general, any well-posed combination of thermal boundary conditions
can be used when any of the radiation models is active. The radiation
model will be well-posed in combination with fixed temperature walls,
conducting walls, and/or walls with set external heat transfer boundary
conditions (Section 6.13.1). You can also use any of the radiation models
with heat flux boundary conditions defined at walls, in which case the
heat flux you define will be treated as the sum of the convective and radiative heat fluxes. The exception to this is the case of semi-transparent
walls for the DO model. Here, FLUENT allows you to specify the convective and radiative portions of the heat flux separately, as explained
above. Also, the fixed-temperature boundary condition is not allowed at
a semi-transparent wall.
11.3.17
Setting Solution Parameters for Radiation
For the DTRM and the DO, S2S, and P-1 radiation models, there are
several parameters that control the radiation calculation. You can use
the default solution parameters for most problems, or you can modify
these parameters to control the convergence and accuracy of the solution.
There are no solution parameters to be set for the Rosseland model, since
it impacts the solution only through the energy equation.
DTRM Solution Parameters
When the DTRM is active, FLUENT updates the radiation field during the calculation and computes the resulting energy sources and heat
fluxes via the ray-tracing technique described in Section 11.3.3. FLUENT
provides several solution parameters that control the solver and the solution accuracy. These parameters appear in the expanded portion of
the Radiation Model panel (Figure 11.3.17).
You can control the maximum number of sweeps of the radiation calculation during each global iteration by changing the Number of DTRM
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Figure 11.3.17: The Radiation Model Panel (DTRM)
Sweeps. The default setting of 1 sweep implies that the radiant intensity
will be updated just once. If you increase this number, the radiant intensity at the surfaces will be updated multiple times, until the tolerance
criterion is met or the number of radiation sweeps is exceeded.
The Tolerance parameter (0.001 by default) determines when the radiation intensity update is converged. It is defined as the maximum normalized change in the surface intensity from one DTRM sweep to the
next (see Equation 11.3-85).
You can also control the frequency with which the radiation field is updated as the continuous phase solution proceeds. The Flow Iterations Per
Radiation Iteration parameter is set to 10 by default. This implies that
the radiation calculation is performed once every 10 iterations of the solution process. Increasing the number can speed the calculation process,
but may slow overall convergence.
S2S Solution Parameters
For the S2S model, as for the DTRM, you can control the frequency
with which the radiosity is updated as the continuous-phase solution
proceeds. See the description of Flow Iterations Per Radiation Iteration
for the DTRM, above.
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If you are using the segregated solver and you first solve the flow equations with the energy equation turned off, you should reduce the Flow
Iterations Per Radiation Iteration from 10 to 1 or 2. This will ensure
the convergence of the radiosity. If the default value of 10 is kept in
this case, it is possible that the flow and energy residuals may converge
and the solution will terminate before the radiosity is converged. See
Section 11.3.18 for more information about residuals for the S2S model.
You can control the maximum number of sweeps of the radiation calculation during each global iteration by changing the Number of S2S Sweeps.
The default setting of 1 sweep implies that the radiosity will be updated
just once. If you increase this number, the radiosity at the surfaces will
be updated multiple times, until the tolerance criterion is met or the
number of radiation sweeps is exceeded.
The Tolerance parameter (0.001 by default) determines when the radiosity update is converged. It is defined as the maximum normalized change
in the radiosity from one S2S sweep to the next (see Equation 11.3-86).
DO Solution Parameters
For the discrete ordinates model, as for the DTRM, you can control the
frequency with which the surface intensity is updated as the continuous phase solution proceeds. See the description of Flow Iterations Per
Radiation Iteration for the DTRM, above.
For most problems, the default under-relaxation of 1.0 for the DO equations is adequate. For problems with large optical thicknesses (aL > 10),
you may experience slow convergence or solution oscillation. For such
cases, under-relaxing the energy and DO equations is useful. Underrelaxation factors between 0.9 and 1.0 are recommended for both equations.
P-1 Solution Parameters
For the P-1 radiation model, you can control the convergence criterion
and under-relaxation factor. You should also pay attention to the optical
thickness, as described below.
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The default convergence criterion for the P-1 model is 10−6 , the same
as that for the energy equation, since the two are closely linked. See
Section 22.16.1 for details about convergence criteria. You can set the
Convergence Criterion for p1 in the Residual Monitors panel.
Solve −→ Monitors −→Residual...
The under-relaxation factor for the P-1 model is set with those for other
variables, as described in Section 22.9. Note that since the equation
for the radiation temperature (Equation 11.3-12) is a relatively stable
scalar transport equation, in most cases you can safely use large values
of under-relaxation (0.9–1.0).
For optimal convergence with the P-1 model, the optical thickness
(a + σs )L must be between 0.01 and 10 (preferably not larger than 5).
Smaller optical thicknesses are typical for very small enclosures (characteristic size of the order of 1 cm), but for such problems you can safely
increase the absorption coefficient to a value for which (a + σs )L = 0.01.
Increasing the absorption coefficient will not change the physics of the
problem because the difference in the level of transparency of a medium
with optical thickness = 0.01 and one with optical thickness < 0.01 is
indistinguishable within the accuracy level of the computation.
11.3.18
Solving the Problem
Once the radiation problem has been set up, you can proceed as usual
with the calculation. Note that while the P-1 and DO models will solve
additional transport equations and report residuals, the DTRM and the
Rosseland and S2S models will not (since they impact the solution only
through the energy equation). Residuals for the DTRM and S2S model
sweeps are reported by FLUENT every time a DTRM or S2S model iteration is performed, as described below.
Residual Reporting for the P-1 Model
The residual for radiation as calculated by the P-1 model is updated
after each iteration and reported with the residuals for all other variables. FLUENT reports the normalized P-1 radiation residual as defined
in Section 22.16.1 for the other transport equations.
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Residual Reporting for the DO Model
After each DO iteration, the DO model reports a composite normalized residual for all the DO transport equations. The definition of the
residuals is similar to that for the other transport equations (see Section 22.16.1).
Residual Reporting for the DTRM
FLUENT does not include a DTRM residual in its usual residual report
that is issued after each iteration. The effect of radiation on the solution
can be gathered, instead, via its impact on the energy field and the
energy residual. However, each time a DTRM iteration is performed,
FLUENT will print out the normalized radiation error for each DTRM
sweep. The normalized radiation error is defined as
X
E=
(Inew
all radiating surfaces
N (σT 4 /π)
− Iold )
(11.3-85)
where the error E is the maximum change in the intensity (I) at the current sweep, normalized by the maximum surface emissive power, and N
is the total number of radiating surfaces. Note that the default radiation
convergence criterion, as noted in Section 11.3.17, defines the radiation
calculation to be converged when E decreases to 10−3 or less.
Residual Reporting for the S2S Model
FLUENT does not include an S2S residual in its usual residual report that
is issued after each iteration. The effect of radiation on the solution can
be gathered, instead, via its impact on the energy field and the energy
residual. However, each time an S2S iteration is performed, FLUENT
will print out the normalized radiation error for each S2S sweep. The
normalized radiation error is defined as
X
E=
11-84
all radiating surface clusters
N σT 4
(Jnew − Jold )
(11.3-86)
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where the error E is the maximum change in the radiosity (J) at the
current sweep, normalized by the maximum surface emissive power, and
N is the total number of radiating surface clusters. Note that the default
radiation convergence criterion, as noted in Section 11.3.17, defines the
radiation calculation to be converged when E decreases to 10−3 or less.
Disabling the Update of the Radiation Fluxes
Sometimes, you may wish to set up your FLUENT model with the radiation model active and then disable the radiation calculation during the
initial calculation phase. For the P-1 and DO models, you can turn off
the radiation calculation temporarily by deselecting P1 or Discrete Ordinates in the Equations list in the Solution Controls panel. For the DTRM
and the S2S model, there is no item in the Equations list. You can instead set a very large number for Flow Iterations Per Radiation Iteration
in the expanded portion of the Radiation Model panel.
If you turn off the radiation calculation, FLUENT will skip the update of
the radiation field during subsequent iterations, but will leave in place
the influence of the current radiation field on energy sources due to absorption, wall heat fluxes, etc. Turning the radiation calculation off in
this way can thus be used to initiate your modeling work with the radiation model inactive and/or to focus the computational effort on the
other equations if the radiation model is relatively well converged.
11.3.19
Reporting and Displaying Radiation Quantities
FLUENT provides several additional reporting options when your model
includes solution of radiative heat transfer. You can generate graphical
plots or alphanumeric reports of the following variables/functions:
• Absorption Coefficient (DTRM, P-1, DO, and Rosseland models
only)
• Scattering Coefficient (P-1, DO, and Rosseland models only)
• Refractive Index (DO model only)
• Radiation Temperature (P-1 and DO models only)
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• Incident Radiation (P-1 and DO models only)
• Incident Radiation (Band n) (non-gray DO model only)
• Surface Cluster ID (S2S model only)
• Radiation Heat Flux
The first seven variables are contained in the Radiation... category of
the variable selection drop-down list that appears in postprocessing panels, and the last one is contained in the Wall Fluxes... category. See
Chapter 27 for their definitions.
! Note the sign convention on the radiative heat flux: heat flux from the
wall surface is a positive quantity.
Note that it is possible to export heat flux data on wall zones (including
radiation) to a generic file that you can examine or use in an external
program. See Section 11.2.5 for details.
Reporting Radiative Heat Transfer Through Boundaries
You can use the Flux Reports panel to compute the radiative heat transfer
through each boundary of the domain, or to sum the radiative heat
transfer through all boundaries.
Report −→Fluxes...
See Section 26.2 for details about generating flux reports.
Overall Heat Balances When Using the DTRM
The DTRM yields a global heat balance and a balance of radiant heat
fluxes only in the limit of a sufficient number of rays. In any given
calculation, therefore, if the number of rays is insufficient you may find
that the radiant fluxes do not obey a strict balance. Such imbalances
are the inevitable consequence of the discrete ray tracing procedure and
can be minimized by selecting a larger number of rays from each wall
boundary.
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11.3.20
Displaying Rays and Clusters for the DTRM
When you use the DTRM, FLUENT allows you to display surface or volume clusters, as well as the rays that emanate from a particular surface
cluster. You will use the DTRM Graphics panel (Figure 11.3.18) for all
of these displays.
Display −→DTRM Graphics...
Figure 11.3.18: The DTRM Graphics Panel
Displaying Clusters
To view clusters, select Cluster under Display Type and then select either
Surface or Volume under Cluster Type.
To display all of the surface or volume clusters, select the Display All
Clusters option under Cluster Selection and click the Display button.
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To display only the cluster (surface or volume) nearest to a specified
point, deselect the Display All Clusters option and specify the coordinates
under Nearest Point. You may also use the mouse to choose the nearest
point. Click on the Select Point With Mouse button and then right-click
on a point in the graphics window.
Displaying Rays
To display the rays emanating from the surface cluster nearest to the
specified point, select Ray under Display Type. Set the appropriate values
for Theta and Phi Divisions under Ray Parameters (see Section 11.3.11 for
details), and then click on the Display button. Figure 11.3.19 shows a
ray plot for a simple 2D geometry.
DTRM Rays
Figure 11.3.19: Ray Display
Including the Grid in the Display
For some problems, especially complex 3D geometries, you may want
to include portions of the grid in your ray or cluster display as spatial
reference points. For example, you may want to show the location of an
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inlet and an outlet along with displaying the rays. This is accomplished
by turning on the Draw Grid option in the DTRM Graphics panel. The
Grid Display panel will appear automatically when you turn on the Draw
Grid option, and you can set the grid display parameters there. When
you click on Display in the DTRM Graphics panel, the grid display, as
defined in the Grid Display panel, will be included in the ray or cluster
display.
11.4
Periodic Heat Transfer
FLUENT is able to predict heat transfer in periodically repeating geometries, such as compact heat exchangers, by including only a single
periodic module for analysis.
This section discusses streamwise-periodic heat transfer. The treatment
of streamwise-periodic flows is discussed in Section 8.3, and a description
of no-pressure-drop periodic flow is provided in Section 6.15.
Information about streamwise-periodic heat transfer is presented in the
following sections:
• Section 11.4.1: Overview and Limitations
• Section 11.4.2: Theory
• Section 11.4.3: Modeling Periodic Heat Transfer
• Section 11.4.4: Solution Strategies for Periodic Heat Transfer
• Section 11.4.5: Monitoring Convergence
• Section 11.4.6: Postprocessing for Periodic Heat Transfer
11.4.1
Overview and Limitations
Overview
As discussed in Section 8.3.1, streamwise-periodic flow conditions exist
when the flow pattern repeats over some length L, with a constant pressure drop across each repeating module along the streamwise direction.
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Periodic thermal conditions may be established when the thermal boundary conditions are of the constant wall temperature or wall heat flux type.
In such problems, the temperature field (when scaled in an appropriate
manner) is periodically fully-developed [173]. As for periodic flows, such
problems can be analyzed by restricting the numerical model to a single
module or periodic length.
Constraints for Periodic Heat Transfer Predictions
In addition to the constraints for streamwise-periodic flow discussed in
Section 8.3.1, the following constraints must be met when periodic heat
transfer is to be considered:
• The segregated solver must be used.
• The thermal boundary conditions must be of the specified heat flux
or constant wall temperature type. Furthermore, in a given problem, these thermal boundary types cannot be combined: all boundaries must be either constant temperature or specified heat flux.
(You can, however, include constant-temperature walls and zeroheat-flux walls in the same problem.) For the constant-temperature
case, all walls must be at the same temperature (profiles are not
allowed) or zero heat flux. For the heat flux case, profiles and/or
different values of heat flux may be specified at different walls.
• When constant-temperature wall boundaries are used, you cannot
include viscous heating effects or any volumetric heat sources.
• In cases that involve solid regions, the regions cannot straddle the
periodic plane.
• The thermodynamic and transport properties of the fluid (heat
capacity, thermal conductivity, viscosity, and density) cannot be
functions of temperature. (You cannot, therefore, model reacting
flows.) Transport properties may, however, vary spatially in a periodic manner, and this allows you to model periodic turbulent flows
in which the effective turbulent transport properties (effective conductivity, effective viscosity) vary with the (periodic) turbulence
field.
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Sections 11.4.2 and 11.4.3 provide more detailed descriptions of the input
requirements for periodic heat transfer.
11.4.2
Theory
Streamwise-periodic flow with heat transfer from constant-temperature
walls is one of two classes of periodic heat transfer that can be modeled
by FLUENT. A periodic fully-developed temperature field can also be
obtained when heat flux conditions are specified. In such cases, the
temperature change between periodic boundaries becomes constant and
can be related to the net heat addition from the boundaries as described
in this section.
! Periodic heat transfer can be modeled only if you are using the segregated
solver.
Definition of the Periodic Temperature for ConstantTemperature Wall Conditions
For the case of constant wall temperature, as the fluid flows through the
periodic domain, its temperature approaches that of the wall boundaries.
However, the temperature can be scaled in such a way that it behaves
in a periodic manner. A suitable scaling of the temperature for periodic
flows with constant-temperature walls is [173]
θ=
T (~r) − Twall
Tbulk,inlet − Twall
(11.4-1)
The bulk temperature, Tbulk,inlet , is defined by
R
Tbulk,inlet = RA
~
T |ρ~v · dA|
~
|ρ~v · dA|
(11.4-2)
A
where the integral is taken over the inlet periodic boundary (A). It is
the scaled temperature, θ, which obeys a periodic condition across the
domain of length L.
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Definition of the Periodic Temperature Change σ for Specified
Heat Flux Conditions
When periodic heat transfer with heat flux conditions is considered, the
form of the unscaled temperature field becomes analogous to that of the
pressure field in a periodic flow:
~ − T (~r)
~ − T (~r + L)
~
T (~r + L)
T (~r + 2L)
=
= σ.
L
L
(11.4-3)
~ is the periodic length vector of the domain. This temperature
where L
gradient, σ, can be written in terms of the total heat addition within the
domain, Q, as
σ=
Tbulk,exit − Tbulk,inlet
Q
=
ṁcp L
L
(11.4-4)
where ṁ is the specified or calculated mass flow rate.
11.4.3
Modeling Periodic Heat Transfer
Overview of Streamwise-Periodic Flow and Heat Transfer
Modeling Procedures
A typical calculation involving both streamwise-periodic flow and periodic heat transfer is performed in two parts. First, the periodic velocity
field is calculated (to convergence) without consideration of the temperature field. Next, the velocity field is frozen and the resulting temperature
field is calculated. These periodic flow calculations are accomplished using the following procedure:
1. Set up a grid with translationally periodic boundary conditions.
2. Input constant thermodynamic and molecular transport properties.
3. Specify either the periodic pressure gradient or the net mass flow
rate through the periodic boundaries.
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4. Compute the periodic flow field, solving momentum, continuity,
and (optionally) turbulence equations.
5. Specify the thermal boundary conditions at walls as either heat
flux or constant temperature.
6. Define an inlet bulk temperature.
7. Solve the energy equation (only) to predict the periodic temperature field.
These steps are detailed below.
User Inputs for Periodic Heat Transfer
In order to model the periodic heat transfer, you will need to set up your
periodic model in the manner described in Section 8.3.3 for periodic
flow models with the segregated solver, noting the restrictions discussed
in Sections 8.3.1 and 11.4.1. In addition, you will need to provide the
following inputs related to the heat transfer model:
1. Activate solution of the energy equation in the Energy panel.
Define −→ Models −→Energy...
2. Define the thermal boundary conditions according to one of the
following procedures:
Define −→Boundary Conditions...
• If you are modeling periodic heat transfer with specified-temperature boundary conditions, set the wall temperature Twall
for all wall boundaries in their respective Wall panels. Note
that all wall boundaries must be assigned the same temperature and that the entire domain (except the periodic boundaries) must be “enclosed” by this fixed-temperature condition,
or by symmetry or adiabatic (q=0) boundaries.
• If you are modeling periodic heat transfer with specified-heatflux boundary conditions, set the wall heat flux in the Wall
panel for each wall boundary. You can define different values
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of heat flux on different wall boundaries, but you should have
no other types of thermal boundary conditions active in the
domain.
3. Define solid regions, if appropriate, according to one of the following procedures:
Define −→Boundary Conditions...
• If you are modeling periodic heat transfer with specified-temperature conditions, conducting solid regions can be used within
the domain, provided that on the perimeter of the domain
they are enclosed by the fixed-temperature condition. Heat
generation within the solid regions is not allowed when you
are solving periodic heat transfer with fixed-temperature conditions.
• If you are modeling periodic heat transfer with specified-heatflux conditions, you can define conducting solid regions at
any location within the domain, including volumetric heat
addition within the solid, if desired.
4. Set constant material properties (density, heat capacity, viscosity,
thermal conductivity), not temperature-dependent properties, using the Materials panel.
Define −→Materials...
5. Specify the Upstream Bulk Temperature in the Periodicity Conditions
panel.
Define −→Periodic Conditions...
!
If you are modeling periodic heat transfer with specified-temperature
conditions, the bulk temperature should not be equal to the wall
temperature, since this will give you the trivial solution of constant
temperature everywhere.
11.4.4
Solution Strategies for Periodic Heat Transfer
After completing the inputs described in Section 11.4.3, you can solve
the flow and heat transfer problem to convergence. The most efficient
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approach to the solution, however, is a sequential one in which the periodic flow is first solved without heat transfer and then the heat transfer
is solved leaving the flow field unaltered. This sequential approach is
accomplished as follows:
1. Disable solution of the energy equation under Equations in the Solution Controls panel.
Solve −→ Controls −→Solution...
2. Solve the remaining equations (continuity, momentum, and, optionally, turbulence parameters) to convergence to obtain the periodic flow field.
!
When you initialize the flow field before beginning the calculation,
use the mean value between the inlet bulk temperature and the
wall temperature for the initialization of the temperature field.
3. Return to the Solution Controls panel and turn off solution of the
flow equations and turn on the energy solution.
4. Solve the energy equation to convergence to obtain the periodic
temperature field of interest.
While you can solve your periodic flow and heat transfer problems by
considering both the flow and heat transfer simultaneously, you will find
that the procedure outlined above is more efficient.
11.4.5
Monitoring Convergence
If you are modeling periodic heat transfer with specified-temperature
conditions, you can monitor the value of the bulk temperature ratio
θ=
Twall − Tbulk,inlet
Twall − Tbulk,exit
(11.4-5)
during the calculation using the Statistic Monitors panel to ensure that
you reach a converged solution. Select per/bulk-temp-ratio as the variable
to be monitored. See Section 22.16.2 for details about using this feature.
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11.4.6
Postprocessing for Periodic Heat Transfer
The actual temperature field predicted by FLUENT in periodic models will not be periodic, and viewing the temperature results during
postprocessing will display this actual temperature field (T (~r) of Equation 11.4-1). The displayed temperature may exhibit values outside the
range defined by the inlet bulk temperature and the wall temperature.
This is permissible since the actual temperature profile at the inlet periodic face will have temperatures that are higher or lower than the inlet
bulk temperature.
Static Temperature is found in the Temperature... category of the variable
selection drop-down list that appears in postprocessing panels.
Figure 11.4.1 shows the temperature field in a periodic heat exchanger
geometry.
4.00e+02
3.87e+02
3.74e+02
3.61e+02
3.48e+02
3.35e+02
3.22e+02
3.09e+02
2.96e+02
2.83e+02
2.70e+02
Contours of Static Temperature (k)
Figure 11.4.1: Temperature Field in a 2D Heat Exchanger Geometry
With Fixed Temperature Boundary Conditions
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11.5
Buoyancy-Driven Flows
When heat is added to a fluid and the fluid density varies with temperature, a flow can be induced due to the force of gravity acting on
the density variations. Such buoyancy-driven flows are termed naturalconvection (or mixed-convection) flows and can be modeled by FLUENT.
11.5.1
Theory
The importance of buoyancy forces in a mixed convection flow can be
measured by the ratio of the Grashof and Reynolds numbers:
Gr
∆ρgh
=
ρv 2
Re2
(11.5-1)
When this number approaches or exceeds unity, you should expect strong
buoyancy contributions to the flow. Conversely, if it is very small, buoyancy forces may be ignored in your simulation. In pure natural convection, the strength of the buoyancy-induced flow is measured by the
Rayleigh number:
Ra =
gβ∆T L3 ρ
µα
(11.5-2)
where β is the thermal expansion coefficient:
β=−
1
ρ
∂ρ
∂T
(11.5-3)
p
and α is the thermal diffusivity:
α=
k
ρcp
(11.5-4)
Rayleigh numbers less than 108 indicate a buoyancy-induced laminar
flow, with transition to turbulence occurring over the range of
108 < Ra < 1010 .
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11.5.2
Modeling Natural Convection in a Closed Domain
When you model natural convection inside a closed domain, the solution
will depend on the mass inside the domain. Since this mass will not be
known unless the density is known, you must model the flow in one of
the following ways:
• Perform a transient calculation. In this approach, the initial density will be computed from the initial pressure and temperature, so
the initial mass is known. As the solution progresses over time, this
mass will be properly conserved. If the temperature differences in
your domain are large, you must follow this approach.
• Perform a steady-state calculation using the Boussinesq model (described in Section 11.5.3). In this approach, you will specify a constant density, so the mass is properly specified. This approach is
valid only if the temperature differences in the domain are small;
if not, you must use the transient approach.
For a closed domain, you cannot use the incompressible ideal gas law
with a fixed operating pressure. You can use the compressible ideal gas
law with a fixed operating pressure, but the incompressible ideal gas law
can be used only with a floating operating pressure. See Section 8.5.4
for information about the floating operating pressure option.
11.5.3
The Boussinesq Model
For many natural-convection flows, you can get faster convergence with
the Boussinesq model than you can get by setting up the problem with
fluid density as a function of temperature. This model treats density as
a constant value in all solved equations, except for the buoyancy term in
the momentum equation:
(ρ − ρ0 )g ≈ −ρ0 β(T − T0 )g
(11.5-5)
where ρ0 is the (constant) density of the flow, T0 is the operating temperature, and β is the thermal expansion coefficient. Equation 11.5-5 is
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obtained by using the Boussinesq approximation ρ = ρ0 (1 − β∆T ) to
eliminate ρ from the buoyancy term. This approximation is accurate as
long as changes in actual density are small; specifically, the Boussinesq
approximation is valid when β(T − T0 ) 1.
Limitations of the Boussinesq Model
The Boussinesq model should not be used if the temperature differences
in the domain are large. In addition, it cannot be used with species
calculations, combustion, or reacting flows.
11.5.4
User Inputs for Buoyancy-Driven Flows
You must provide the following inputs to include buoyancy forces in the
simulation of mixed or natural convection flows:
1. Turn on solution of the energy equation in the Energy panel.
Define −→ Models −→Energy...
2. Turn on Gravity in the Operating Conditions panel (Figure 11.5.1)
and set the Gravitational Acceleration in each Cartesian coordinate
direction by entering the appropriate values in the X, Y, and (for
3D) Z fields.
Define −→Operating Conditions
Note that the default gravitational acceleration in FLUENT is zero.
3. If you are using the incompressible ideal gas law, check that the
Operating Pressure is set to an appropriate (non-zero) value in the
Operating Conditions panel.
4. Depending on whether or not you use the Boussinesq approximation, specify the appropriate parameters described below:
• If you are not using the Boussinesq model, the inputs are as
follows:
(a) If necessary, enable the Specified Operating Density option in the Operating Conditions panel, and specify the
Operating Density. See below for details.
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Figure 11.5.1: The Operating Conditions Panel
(b) Define the fluid density as a function of temperature, as
described in Sections 7.1.3 and 7.2.
Define −→Materials...
• If you are using the Boussinesq model (described in Section 11.5.3),
the inputs are as follows:
(a) Specify the Operating Temperature (T0 in Equation 11.5-5)
in the Operating Conditions panel.
(b) Select boussinesq as the method for Density in the Materials panel, as described in Sections 7.1.3 and 7.2.
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(c) Also in the Materials panel, set the Thermal Expansion
Coefficient (β in Equation 11.5-5) for the fluid material
and specify a constant density.
Note that, if your model involves multiple fluid materials, you can
choose whether or not to use the Boussinesq model for each material. As a result, you may have some materials using the Boussinesq
model and others not. In such cases, you will need to set all the
parameters described above in this step.
5. The boundary pressures that you input at pressure inlet and outlet
boundaries are the redefined pressures as given by Equation 11.5-6.
In general you should input equal pressures, p0 , at the inlet and
exit boundaries of your FLUENT model if there are no externallyimposed pressure gradients.
Define −→Boundary Conditions...
6. Select Body Force Weighted or Second Order as the Discretization
method for Pressure in the Solution Controls panel.
Solve −→ Controls −→Solution...
You may also want to add cells near the walls to resolve boundary
layers.
If you are using the segregated solver, selecting PRESTO! as the
Discretization method for Pressure is another recommended approach.
See also Section 11.2.2 for information on setting up heat transfer calculations.
Definition of the Operating Density
When the Boussinesq approximation is not used, the operating density, ρ0 , appears in the body-force term in the momentum equations as
(ρ − ρ0 )g.
This form of the body-force term follows from the redefinition of pressure
in FLUENT as
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p0s = ps − ρ0 gx
(11.5-6)
The hydrostatic pressure in a fluid at rest is then
p0s = 0
(11.5-7)
The definition of the operating density is thus important in all buoyancydriven flows.
Setting the Operating Density
By default, FLUENT will compute the operating density by averaging
over all cells. In some cases, you may obtain better results if you explicitly specify the operating density instead of having the solver compute
it for you. For example, if you are solving a natural-convection problem
with a pressure boundary, it is important to understand that the pressure you are specifying is p0s in Equation 11.5-6. Although you will know
the actual pressure ps , you will need to know the operating density ρ0
in order to determine p0s from ps . Therefore, you should explicitly specify the operating density rather than use the computed average. The
specified value should, however, be representative of the average value.
In some cases, the specification of an operating density will improve
convergence behavior, rather than the actual results. For such cases,
use the approximate bulk density value as the operating density, and
be sure that the value you choose is appropriate for the characteristic
temperature in the domain.
Note that, if you are using the Boussinesq approximation for all fluid
materials, the operating density is not used, so you need not specify it.
11.5.5
Solution Strategies for Buoyancy-Driven Flows
For high-Rayleigh-number flows, you may want to consider the solution guidelines below. In addition, the guidelines presented in Section 11.2.3 for solving other heat transfer problems can also be applied
to buoyancy-driven flows. Note, however, that for some laminar, highRayleigh-number flows, no steady-state solution exists.
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11.5 Buoyancy-Driven Flows
Guidelines for Solving High-Rayleigh-Number Flows
When you are solving a high-Rayleigh-number flow (Ra > 108 ), you
should follow one of the procedures outlined below for best results.
The first procedure uses a steady-state approach:
1. Start the solution with a lower value of Rayleigh number (e.g., 107 )
and run to convergence using the first-order scheme.
2. To change the effective Rayleigh number, change the value of gravitational acceleration (e.g., from 9.8 to 0.098 to reduce the Rayleigh
number by two orders of magnitude).
3. Use the resulting data file as an initial guess for the higher Rayleigh
number, and start the higher-Rayleigh-number solution using the
first-order scheme.
4. After you obtain a solution with the first-order scheme, you may
continue the calculation with a higher-order scheme.
The second procedure uses a time-dependent approach to obtain a steadystate solution [89]:
1. Start the solution from a steady-state solution obtained for the
same or a lower Rayleigh number.
2. Estimate the time constant as [16]
τ=
L
L
L2
∼
(PrRa)−1/2 = √
U
α
gβ∆T L
(11.5-8)
where L and U are the length and velocity scales, respectively. Use
a time step ∆t such that
∆t ≈
τ
4
(11.5-9)
Using a larger time step ∆t may lead to divergence.
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3. After oscillations with a typical frequency of f τ = 0.05–0.09 have
decayed, the solution reaches steady state. Note that τ is the
time constant estimated in Equation 11.5-8 and f is the oscillation
frequency in Hz. In general, this solution process may take as many
as 5000 time steps to reach steady state.
11.5.6
Postprocessing for Buoyancy-Driven Flows
The postprocessing reports of interest for buoyancy-driven flows are the
same as for other heat transfer calculations. See Section 11.2.4 for details.
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