Interphase Transport in
Nonisothermal Systems
§14.1 Definitions of heat transfer coefficients
§14.2 Analytical calculations of heat transfer coefficients
for forced convection through tubes and slits
§14.3 Heat transfer coefficients for forced convection in
tubes
§14.4 Heat transfer coefficients for forced convection
around submerged objects
§14.5 Heat transfer coefficients for forced convection
through packed beds
1
14.1 DEFINITIONS OF HEAT TRANSFER
COEFFICIENTS
Let’s consider a flow system with the fluid flowing either in a
conduit or around a solid object. Suppose that the solid surface
is warmer than the fluid, so that heat is being transferred from
the solid to the fluid. Then the rate of heat flow across the
solid-fluid interface would be expected to depend on the area
of the interface and on the temperature drop between the fluid
and the solid. It is customary to define a proportionality factor
h (the heat transfer coefficient) by:
which Q is the heat flow into the fluid (J/hr or Btu/hr), A is a
characteristic area, and ∆T is a characteristic temperature
difference. This equation can also be used when the fluid is cooled.
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14.1 DEFINITIONS OF HEAT TRANSFER
COEFFICIENTS
As an example of flow in conduits, we consider a fluid flowing
through a circular tube of diameter D, in which there is a
heated wall section of length L and varying inside surface
temperature T0 (z), going from T01 to T02. Suppose that the
bulk temperature Tb of the fluid increases from Tb1 to Tb2 in
the heated section. Then there are three conventional
definitions of heat transfer coefficients for the fluid in the
heated section:
Figure 14.1-1 Heat transfer in a
circular tube.
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14.1 DEFINITIONS OF HEAT TRANSFER
COEFFICIENTS
If the wall temperature distribution is initially unknown, or if the fluid
properties change appreciably along the pipe, it is difficult to predict
the heat transfer coeffcients.
Under some conditions, we can rewrite this equation:
In the differential form:
Here dQ is the heat added to the fluid over a distance dz along the pipe,
∆Tloc is the local temperature difference (at position z), and hloc is the local
heat transfer coefficient.
Actually, the definition of hloc and ∆Tloc is not complete without specifying
the shape of the element of area.
In equation above we have set dA = πDdz, which means that hloc
and ∆Tloc are the mean values for the shaded area dA in figure 1.
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14.1 DEFINITIONS OF HEAT TRANSFER
COEFFICIENTS
As an example of flow around submerged objects, consider a fluid
flowing around a sphere of radius R, whose surface
temperature is maintained at a uniform value T0 . Suppose
that the fluid approaches the sphere with a uniform
temperature Tx . Then we may define a mean heat transfer
coefficient, hm , for the entire surface of the sphere by the
relation:
A local coefficient can also be defined for submerged objects by
This coefficient is more informative than hm because it predicts how
the heat flux is distributed over the surface. However, most
experimentalists report only hm , which is easier to measure.
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14.1 DEFINITIONS OF HEAT TRANSFER
COEFFICIENTS
In the calculation of heat transfer rates between two fluid streams
separated by one or more solid layers, it is convenient to use
an overall heat transfer coefficient, U o , which expresses the
combined effect of the series of resistances through which the
heat flows.
To calculate U0 in the special case of heat exchange between two
coaxial streams with bulk temperatures Th („hot”) and Tc
(„cold”), separated by a cylindrical tube of inside diameter D0
and outside diamenter D1:
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14.1 DEFINITIONS OF HEAT TRANSFER
COEFFICIENTS
Last two equations are, of course, restricted to thermal resistances
connected in series.
Fig. 2. Series of experiments
for measuring heat transfer coefficients.
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14.2 ANALYTICAL CALCULATIONS OF HEAT TRANSFER
COEFFICIENTS FOR FORCED CONVECTION
THROUGH TUBES AND SLITS
The difference between the wall temperature and the bulk
temperature:
in which R and D are the radius and diameter of the tube. Solving
for the wall flux we get:
Then making use of the definition of the local heat transfer
coefficient hloc —namely, that q0 = hloc(T0-Tb) we find that:
or
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14.2 ANALYTICAL CALCULATIONS OF HEAT TRANSFER
COEFFICIENTS FOR FORCED CONVECTION
THROUGH TUBES AND SLITS
Some Nusselt numbers for Newtonian fluids with constant physical
properties are shown in figure below.
Fig. 3 The Nusselt number for fully
developed, laminar flow of Newtonian
fluids with constant physical
properties;
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14.2 ANALYTICAL CALCULATIONS OF HEAT TRANSFER
COEFFICIENTS FOR FORCED CONVECTION
THROUGH TUBES AND SLITS
For turbulent flow in a circular tube with constant heat flux, the
Nusselt number can be obtained from equation:
This is valid only for αz/<vz> D²>>1, for fluids with constant
physical properties, and for tubes with no roughness. It has been
applied successfully over the Prandtl-number
range 0.7 < Pr < 590. Note that, for very large Prandtl numbers,
equation above gives:
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14.2 ANALYTICAL CALCULATIONS OF HEAT TRANSFER
COEFFICIENTS FOR FORCED CONVECTION
THROUGH TUBES AND SLITS
For the turbulent flow of liquid metals, for which the Prandtl
numbers are generally much less than unity, there are two
results of importance. Notter and Sleicher solved the energy
equation numerically, using a realistic turbulent velocity
profile, and obtained the rates of heat transfer through the
wall. The final results were curve-fitted to simple analytical
expressions for two cases:
Constant wall temperature:
COnstant wall heat flux:
These equations are limited to L/D > 60 and constant physical properties.
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14.2 ANALYTICAL CALCULATIONS OF HEAT TRANSFER
COEFFICIENTS FOR FORCED CONVECTION
THROUGH TUBES AND SLITS
When there are large temperature differences in the system, it is
necessary to take into account the temperature dependence of
the viscosity, density, heat capacity, and thermal conductivity.
Usually this is done by means of an empiricism—namely, by
evaluating the physical properties at some appropriate average
temperature.
All physical properties are to be calculated at the film
temperature T f defined as follows:
a. For tubes, slits, and other ducts,
b. For submerged objects with uniform surface temperature
T0 in a stream of liquid approaching with uniform
temperature T∞
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14.2 ANALYTICAL CALCULATIONS OF HEAT TRANSFER
COEFFICIENTS FOR FORCED CONVECTION
THROUGH TUBES AND SLITS
For flow systems involving more complicated geometries, it is
preferable to use experimental correlations of the heat transfer
coefficients.
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14.3. HEAT TRANSFER COEFFICIENTS FOR
14.
FORCED CONVECTION IN TUBES
First we extend the dimensional analysis given to obtain a
general form for correlations of heat transfer
coefficients in forced convection.
Consider the steadily driven laminar or turbulent flow of
a Newtonian fluid through a straight tube of inner
radius R, as shown in Fig 4. The fluid enters the tube
at z = 0 with velocity uniform out to very near the
wall, and with a uniform inlet temperature T1 (= Tb1).
The tube wall is insulated except in the region 0 ≤z
≤L, where a uniform inner-surface temperature T0 is
maintained by heat from vapor condensing on the
outer surface.
For the moment, we assume constant physical properties
ρ, µ, k and Ĉp.
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14.3. HEAT TRANSFER COEFFICIENTS FOR
14.
FORCED CONVECTION IN TUBES
We start by writing the expression for the instantaneous heat flow
from the tube wall into the fluid.
which is valid for laminar or turbulent flow (in laminar flow, Q would,
of course, be independent of time). The + sign appears here
because the heat is added to the system in the negative r direction.
Equating the expressions for Q and solving for h1, we get:
Next we introduce the dimensionless quantities ř= r/D, ž= z/D, and Ť
=(T- T0)/(Tb1– T0), and multiply by D/k to get an expression for the
Nusselt number NU1 =h1D/k:
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14.3. HEAT TRANSFER COEFFICIENTS FOR
14.
FORCED CONVECTION IN TUBES
The Nusselt number is basically a dimensionless temperature
gradient averaged over the heat transfer surface.
Fig. 4. Heat transfer in the entrance region of a tube.
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14.3. HEAT TRANSFER COEFFICIENTS FOR
14.
FORCED CONVECTION IN TUBES
From equations:
and these boundary conditions,
we conclude that the dimensionless instantaneous temperature
distribution must be of the following form:
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14.3. HEAT TRANSFER COEFFICIENTS FOR
14.
FORCED CONVECTION IN TUBES
Substitution of this relation into
leads to the conclusion that Nu1 (Ť) = Nu1 (Re,Pr, Br, L/D, Ť).
When time-averaged over an interval long enough to include
all the turbulent disturbances, this becomes
Nu1 = Nu1 (Re, Pr, Br, L/D)
Therefore, dimensional analysis tells us that, for forced-convection
heat transfer in circular tubes with constant wall temperature,
experimental values of the heat transfer coefficient h1 can be
correlated by giving Nu1 as a function of the Reynolds number, the
Prandtl number, and the geometric ratio L/D.
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14.3. HEAT TRANSFER COEFFICIENTS FOR
14.
FORCED CONVECTION IN TUBES
Thus far we have assumed that the physical properties are
constants over the temperature range encountered in the flow
system.
However, for very large temperature differences, the viscosity
variations may result in such a large distortion of the velocity
profiles that it is necessary to account for this by introducing an
additional dimensionless group,µb/ µ0 where µb is the viscosity at
the arithmetic average bulk temperature and µ0 is the viscosity at
the arithmetic average wall temperature.
Then we may write:
Nu = Nu(Re, Pr, L/D, µb/ µ0 )
This type of correlation seems to have first been presented by Sieder and Tate. If, in
addition, the density varies significantly, then some free convection may occur. This
effect can be accounted for in correlations by including the Grashof number along
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with the other dimensionless groups.
14.3. HEAT TRANSFER COEFFICIENTS FOR
14.
FORCED CONVECTION IN TUBES
The heat transfer coefficient h depends on eight physical quantities
(D, (v),ρ, µb/ µ0, Ĉp, k, L). However, this dependence can be
expressed more concisely by giving Nu as a function of only four
dimensionless groups (Re, Pr, L/D, µb/ µ0).
A good global view of heat transfer in circular tubes with nearly
constant wall temperature can be obtained from the Sieder and Tate
correlation shown in figure below.
Fig. 5. Heat transfer coefficients for
fully developed flow in smooth tubes.
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14.3. HEAT TRANSFER COEFFICIENTS FOR
14.
FORCED CONVECTION IN TUBES
It has been found empirically that transition to turbulence usually
begins at about Re = 2100, even when the viscosity varies
appreciably in the radial direction.
For highly turbulent flow, the curves for L/D > 10 converge to a
single curve. For Re b> 20,000 this curve is described by the
equation.
This equation reproduces available experimental data within about
±20% in the ranges 10⁴< Reb< 10⁵and 0.6 < Pr < 100.
For laminar flow, the descending lines at the left are given by the
equation
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14.3. HEAT TRANSFER COEFFICIENTS FOR
14.
FORCED CONVECTION IN TUBES
The transition region, roughly 2100 < Re < 8000 in Fig. 5, is not well
understood and is usually avoided in design if possible.
For a heated section of given L and D and a fluid of given physical
properties, the ordinate is proportional to the dimensionless
temperature rise of the fluid passing
through. Under these conditions, as the flow rate is increased, the
exit fluid temperature will first decrease until Re reaches about 2100,
then increase until Re reaches about 8000, and then finally decrease
again. The influence of L/D on hln is marked in laminar flow but
becomes insignificant for Re > 8000 with L/D > 60.000
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14.4 HEAT TRANSFER COEFFICIENTS FOR FORCED
14.
CONVECTION AROUND SUBMERGED OBJECTS
Flow Along a Flat Plate
We first examine the flow along a flat plate, oriented parallel to the
flow, with its surface maintained at T0 and the approaching stream
having a uniform temperature T∞ and a uniform velocity v∞ .
For the laminar region, which normally exists near the leading edge of
the plate, the theoretical expressions are obtained:
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14.4 HEAT TRANSFER COEFFICIENTS FOR FORCED
14.
CONVECTION AROUND SUBMERGED OBJECTS
Flow Around a Sphere
The Nusselt number for a sphere in a stationary fluid is
2. For the sphere with constant surface temperature T0 in a flowing
fluid approaching with a uniform velocity v∞, the mean Nusselt
number is given by the following empiricism:
Flow Around a Cylinder
A cylinder in a stationary fluid of infinite extent does not admit a steadystate solution. Therefore the Nusselt number for a cylinder does not
have the same form as that for a sphere. Whitaker recommends for the
mean Nusselt number:
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in the range 1.0 < Re < 1.0 X 10⁵, 0.67 < Pr < 300, and 0.25 < µ∞/µ
0
< 5.2.
14.4 HEAT TRANSFER COEFFICIENTS FOR FORCED
14.
CONVECTION AROUND SUBMERGED OBJECTS
Flow Around Other Objects
The heat transfer coefficients can be obtained by using the relation:
in which Num,0 is the mean Nusselt number at zero Reynolds
number. This generalization, which is shown in figure below, is often
useful in estimating the heat transfer from irregularly shaped
objects.
Fig. 6. Graph comparing the
Nusselt numbers for flow around flat
plates, spheres, and cylinders with
equation above.
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14.5 HEAT TRANSFER COEFFICIENTS FOR FORCED
CONVECTION THROUGH PACKED BEDS
The velocity profiles in packed beds exhibit a strong maximum near
the wall, attributable partly to the higher void fraction there and
partly to the more ordered interstitial passages along this smooth
boundary. The resulting segregation of the flow into a fast outer
stream and a slower interior one, which mix at the exit of the bed,
leads to complicated behavior of mean Nusselt numbers in deep
packed beds, unless the tube-to-particle diameter ratio Dt /Dp is very
large or close to unity. Experiments with wide, shallow beds show
simpler behavior and are used in the following discussion.
We define hloc for a representative volume Sdz of particles and fluid
by the following equation:
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14.5 HEAT TRANSFER COEFFICIENTS FOR FORCED
CONVECTION THROUGH PACKED BEDS
Extensive data on forced convection for the flow of gases and
liquids through shallow packed beds have been critically analyzed to
obtain the following local heat transfer correlation,
Here the Chilton-Colburn jH factor and the Reynolds number are
defined by:
In this equation the physical properties are all evaluated at the film
temperature Tf=½(T0- Tb) and G0=w/S is the superficial mass flux. The
quantity ψ is a particle-shape factor, with a defined value of 1 for
spheres and a fitted value of 0.92 for cylindrical pellets. The present
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factor ψ is used in Re only.
14.6 HEAT TRANSFER COEFFICIENTS FOR
FREE AND MIXED CONVECTION
We know that for the free convection near a vertical flat plate, the
principal dimensionless group is GrPr, which is often called the
Rayleigh number, Ra. If we define the area mean Nusselt number as
Num= hH/k= qavgH/k(T0-T1), then we can write an equation:
where С was found to be a weak function of Pr. The heat transfer
behavior at moderate values of Ra = GrPr is governed, and the
results of those discussions are normally used directly.
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14.6 HEAT TRANSFER COEFFICIENTS FOR
FREE AND MIXED CONVECTION
No Buoyant Forces
The limiting Nusselt number for vanishingly small free and forced
convection is obtained by solving the heat conduction equation for
constant, uniform temperature over the solid surface and a
different constant temperature at infinity. The mean Nusselt
number then has the general form:
With К equal to zero for all objects with at least one infinite dimension.
For finite bodies К is nonzero:
Thin Laminar Boundary Layers
For thin laminar boundary layers, the isothermal vertical flat plate is
a representative system, conforming to generalized equation:
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14.6 HEAT TRANSFER COEFFICIENTS FOR
FREE AND MIXED CONVECTION
For heated horizontal flat surfaces facing downward and cooled
horizontal flat surfaces facing upward, the following correlation is
recommended:
Turbulent Boundary Layers
The effects of turbulence increase gradually, and it is common
practice to combine the laminar and turbulent contributions as
follows:
Thus for the vertical isothermal flat plate, one writes:
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14.6 HEAT TRANSFER COEFFICIENTS FOR
FREE AND MIXED CONVECTION
Mixed Free and Forced Convection
Finally, one must deal with the problem of simultaneous free and
forced convection, and this is again done through the use of an
empirical combining rule:
This rule appears to hold reasonably well for all geometries and
situations, provided only that the forced and free convection have
the same primary flow direction.
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