BOUNDARY CONDITIONS FOR CONVECTIVE
HEAT TRANSFER IN OpenFOAM®
Document created by Alex Roqué Velasco
with the knowledge aquired during the development of the Final Thesis entitled “Thermal analysis
with OpenFOAM® of a neutron flux furnace in order to study the thermal gradients in the
sample”.
A summarized version of this thesis can be seen at http://foamingtime.wordpress.com/
BASIC CONVECTION
Knowing that
q conv =q cond
q conv =h⋅A⋅(T ∞−T B )
(2)
q cond = λ
δ ⋅A⋅(T B−T X )
(3)
A
(1)
λ
T∞
TB
qconv
We get
TX
qcond
h
q
= h⋅(T ∞ −T B )= λ⋅(T B−T X )
δ
A
(4)
δ
In OpenFOAM values at boundaries are calculated as follows:
[
X B= f⋅X ref + (1− f )⋅ X X +
Grad ref ( X )
deltaCoeffs
]
(5)
Being Xref a reference value for the variable at boundary, Xx the value of the variable in the cell
center, Gradref(X) the reference gradient of the variable, deltaCoeffs the inverse of the face­center to
1
cell­center distance (deltaCoeffs= ) and f a weighted factor that defines the BC type, beeing a
δ
Dirichlet BC when f=1, a Neumann BC when f=0 and a Robin BC when 1>f>0.
Eq. (4) has to be converted into something similar to Eq. (5), so let's do it! First we have to isolate
TB from Eq. (4)
(
λ + h ⋅T = h⋅T + λ⋅T
∞
B
δ
δ X
)
→
T B=
h
(
λ +h
δ
⋅T ∞ +
)
(
λ
δ ⋅T
X
λ +h
δ
)
If we rename some terms of the equation
f=
h
( λδ + h )
λ
; 1− f = δ
λ +h
δ
(
)
Thus, we obtain
T B = f⋅T ∞ + (1− f )⋅T X being T∞ the reference value.
We can even rewrite f in a better looking entity
f=
1
1+ λ
h⋅δ
Document written by Alex Roqué Velasco
CONVECTION WITH ADDITION OF A CONSTANT HEAT FLUX
Now the balance satisfies that
A
q conv + qext =q cond
Thus,
λ
T∞
q
= h⋅(T ∞ −T B ) + q ext = λ⋅(T B −T X )
δ
A
TX
qcond
h
As it was said before, TB must be isolated,
(
TB
qconv
qext
[W/m²]
λ +h ⋅T = h⋅T + λ⋅T + q
∞
B
ext
δ
δ X
)
δ
↓
T B=
h
⋅T ∞ +
( λδ + h )
(
λ
δ ⋅T +
X
λ +h
δ
)
(
qext
λ +h
δ
)
↓
T B=
Again
h
(
λ +h
δ
f=
⋅T ∞ +
)
1
1+ λ
h⋅δ
(
λ
δ ⋅T +
X
λ +h
δ
)
q
⋅ ext
λ +h
h
δ
h
(
)
→
T B=
but this time the reference value is T ∞ +
q
⋅ T ∞ + ext
λ +h
h
δ
(
)
h
(
q ext
h
) (
+
λ
δ ⋅T
X
λ +h
δ
)
.
Note: By doing another trick it could have been obtained a reference value of T∞ and a reference
q ext
gradient of
instead.
λ
Document written by Alex Roqué Velasco
CONVECTION WITH ADDITION OF A CONSTANT HEAT FLUX
AND AN EXTRA EXTERNAL SOLID LAYER
Remember that the external solid layer does not belong to the calculation domain!
In this case
A
q conv + qext =q cond1 =q cond2
Therefore,
λ1
λ1
q
= h⋅(T ∞ −T α ) + q ext =
⋅(T −T B)=
δ1 α
A
= λ⋅(T B−T X )
δ
T∞
qconv
Tα
λ
TB
h
TX
qcond
qext
[W/m²]
Tα is a variable that we cannot manage, so we need to get rid of it. Firstδ 1of all, weδ will make pass q ext
within the brackets.
q
q
λ1
q
= h⋅ T ∞ −T α + ext = h⋅ T ∞ + ext −T α =
⋅(T α−T B )= λ⋅( T B−T X )
δ
δ
A
h
h
1
) ((
(
) )
As it is known,
q=
T A−T B
T −T C
= B
R AB
R BC
→ q=
T A−T C
R AC
with
R AC = R AB + R BC
Therefore,
q
=
A
(
q ext
−T B
h
= λ⋅(T B−T X )
δ
δ
1 1
+λ
h 1
T ∞+
)
Again, TB has to be isolated, let's do it!
T ∞+
q ext
λ +h⋅δ1
−T B= λ⋅ 1
⋅(T B−T X )
δ
h
λ1⋅h
To make it cleaner,
β= λ⋅
δ
λ 1+ h⋅δ1
λ 1⋅h
Document written by Alex Roqué Velasco
Thus,
T ∞+
q ext
−T B= β⋅( T B−T X )
h
→
T B=
→
T B⋅(1+β)= T ∞ +
q ext
+β⋅T X
h
→
q ext
1
β
⋅ T ∞+
+
⋅T
1+β
h
1+β X
(
)
Hence it is deduced that this is the same case than the previous one, except for the value of f, which
λ 1+ h⋅δ1
1
now it has a value of f =
being β= λ⋅
.
δ
1+β
λ 1⋅h
Summary
This is the general expression for a boundary condition:
[
X B= f⋅refValue + (1− f )⋅ X X +
refGrad
deltaCoeffs
]
The following table shows the values that each term has to take for every one of the 3 modes of
convection studied.
Convection
type
f
refValue
refGrad
OPTION 1 OPT. 2 OPT. 1
Basic
1
1+β
T∞
Constant Heat
Flux
1
1+β
T ∞+
Heat Flux +
Extra Layer
1
1+β
T ∞+
OPTION 2
β
­
0
­
λ
h⋅δ
q ext
h
T∞
0
q ext
λ
λ
h⋅δ
q ext
h
T∞
0
q ext
λ1
⋅
λ λ 1+ h⋅δ1
λ ⋅ λ1 +h⋅δ1
λ1
h⋅δ
Document written by Alex Roqué Velasco