Diffusion
MEL 807
Computational Heat Transfer (2-0-4)
Dr. Prabal Talukdar
Assistant Professor
Department of Mechanical Engineering
IIT Delhi
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The diffusion Equation
• Finite volume method will be applied for steady
state diffusion equation on Cartesian structured
meshes
• As a diffusion problem, we have the best
example of heat conduction
• There are other physical process which closely
follow this formulation of diffusion
v
v
v
v
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Mass diffusion
Potential flow
Flow through porous media
Some fully developed flow
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Introduction (cont’d)
•
•
•
•
We will discuss 1-D, 2-D, 3-D diffusion problem
We will consider both steady and unsteady
We will discuss implicit and explicit scheme
We will discuss about stability issues related to
diffusion problem
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1-D Steady Diffusion
d  dφ 
Γ + S = 0
dx  dx 
Consider the diffusion
equation
Integrate over the control volume
d d  e
∫ Γ + ∫Sdx = 0
w dx  dx  w
e
Cell
äxw
BC
W
Face
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äxe
Äx
1D solution domain
E
P
Control volume central point
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Insulated
BC
1-D Diffusion (cont’d)
 dφ   dφ  e
Γ  − Γ  + ∫Sdx = 0
 dx e  dx w w
Average value of
S over a CV
Make linear profile assumption
Γe (φE − φP ) Γw (φP − φW )
between cell centroids for ö.
−
+ S ∆x = 0
(δx e )
(δx w )
Assume S varies linearly over CV
Rearranging the terms results
the algebraic equation
a P φP = a E φE + a W φW + b
a E = Γe /( δx e )
a W = Γw /( δx w )
aP = aE +aW
b = S ∆x
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2-D Steady Diffusion
N
An
W
Ae
Aw
P
äxw
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As
S äx
e
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E
äyn
äys
2-D Steady Diffusion
• Consider steady diffusion with a source term
• Here
∇ ⋅J = S
J = −Γ∇ φ
∂
∂
∇=
i+
j
∂x
∂y
• Integrating over a CV yields
∫∇ ⋅JdV
∆V
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= ∫SdV
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∆V
2-D Steady Diffusion
• Applying Divergence
theorem,
∫J ⋅dA = ∫SdV
A
∆V
or,
(J ⋅A) e + (J ⋅A) w + (J ⋅A) n + (J ⋅A) s
= S ∆V
The face areas Ae and Aw are given by
Ae = ∆y i
A w = −∆y i
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Discrete Flux Balance
∂φ 
( J ⋅A) e = J e ⋅A e = −Γe ∆y
∂y 

 e
∂φ 
( J ⋅A) w = J w ⋅A w = −Γw ∆y
∂y 

 w
Assuming a piece wise linear profile
φE − φP
J e ⋅A e = −Γe ∆y
(δx ) e
φP − φW
J w ⋅A w = Γw ∆y
(δx ) w
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Discrete Flux Balance
• Source function could be a function of φ
• Linearization of source term
constant
S = SC + S P φP
With SP
0
Will discuss about it later
Substituting all these fluxes and source term
in the flux balance equation, (next page)
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Final Discretized Equation
a P φP = a E φE + a W φW + a N φN + a S φS + b
Γe ∆y
aE =
,
(δx ) e
Γw ∆y
=
(δx ) w
aW
Γn ∆x
aN =
,
(δy) n
Γs ∆x
aS =
(δy) s
a P = a E + a W + a N + a S − S P ∆ x∆ y
b = SC ∆x∆y
a P φP = ∑ a nb φnb + b
nb
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nb denotes the cell neighbor
E, W, N, and S
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Discussion about discretization
• In deriving φ/ x, we assumed the simplest
linear profile assumption-but we are free to
assume any profile
• It is not necessary to assume same profile
for all quantities like S, ke and φ
• Even for a given variable it is not
necessary to assume same profile for all
the terms in a equation.
• For example, if there is a term with φ alone in the
diffusion equation, it is possible to assume a step
wise profile for φ
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Guiding Principles
• Freedom of choice gives rise to a variety of discretization
formulations
• As the number of grid points increased, all formulations
are expected to give the same solution
• However, an additional requirements is imposed that will
enable us to narrow down the number of acceptable
formulations
• We shall require that even the coarse grid solution
should always have
– Physically realistic behavior
– Overall balance
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Guiding Principles (cont’d)
• Physically realistic behavior
– A realistic behavior should have the same qualitative trend as
the exact variation
– Example: In heat conduction without source no temperature can
lie outside the range of temperature established by the boundary
temperature
• The requirement of overall balance implies integral
conservation over the whole calculation domain - heat
•
flux, mass flow rates, momentum fluxes must correctly be balanced
in overall for any grid size- not just for a finer grid
Constraint of physical realism and overall balance will be used to
guide our choices of profile assumptions and related practices-on
the basis of these practice we will develop some basic rule that will
enable us to discriminate between available formulations and to
invent new ones
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Four Basic Rules-(1/4)
• Consistency at control volume faces-same flux
expression at the faces common to neighboring
CVs.
P
kP
Je = Jw
(TP − TE )
(δx ) e
E
kE
(TP − TE )
(δx ) e
Not consistent
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Four Basic Rules-(2/4)
• All coefficients must have same sign say positive.
Implies that if neighbor φ goes up, φP also goes up
φW
φP
W
φe
P
E
a P φP = a E φE + a W φW + b
• If an increase in φE must lead to an increase in φP, it
follows that aE and aP must have same sign
a E = Γe /(δx e )
a W = Γw /(δx w )
aP = aE + aW
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Our formulation gives
+ve coefficients
Four Basic Rules-(2/4)
• But there are numerous formulations that
frequently violates this rule. Usually, the
consequence is a physically unrealistic
solution.
• The presence of a negative neighbor
coefficient can lead to the situation in
which an increase in a boundary
temperature causes the temperature at the
adjacent grid point to decrease
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Four Basic Rules (3/4)
• Negative-slope linearization of the source term
negative
• Source term linearization: S = Sc + SPφP
– See the expression a P = a E + a W + a N + a S − S P ∆x∆y
– aP can be negative via the source term SP if kept positive, hence
could violate Rule 2
– Increase of S with increase of φP could bring unstable resultsalways good to make SP negative to have a successful
computation
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Four Basic Rules (4/4)
• Sum of the neighboring coefficients
• We require a P = ∑ a nb for situations (SP = 0) where
the DE continues to be remain satisfied after a constant
is added to the dependent variable
If
Show
a P φP = a E φE + a W φW + b
a P (φP + c) = a E (φE + c) + a W (φW + c) + b
• Even satisfied by both φ and φ+c, the desired φ field is
not multi valued or indeterminate - BC gives appropriate
values
• If the source term SP is non-zero, this rule is not
applicable. Even the differential equation is not satisfied
by both φ and φ+c when source term depends on φ
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Four Basic Rules (4/4)
• Implications: if boundary temperature is increased
by a constant, all temperature would increase by
the same constant
• If all φnb are equal, φP should be same to all of
them
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• Formulation is conservative: Discrete
equation was derived by enforcing
conservation. Fluxes balance source term
regardless of mesh density
• For a structured mesh, each point P is
coupled to its four nearest neighbors.
Corner points do not enter formulation
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