Peube J-L. Fundamentals of fluid mechanics and transport phenomena
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432 Fundamentals of Fluid Mechanics and Transport Phenomena
properties are different, depending on the elliptic, parabolic or hyperbolic nature of
the system.
We will here consider problems of thermal conduction in homogenous media.
The method can also be applied analogously in heterogenous media with physical
properties which depend on the spatial variables.
8.3.2.2.
Wall problem
8.3.2.2.1.
Equations and solution of homogenous systems
Consider a wall of thickness 2
A
comprising a homogenous material whose faces
are supposed, for example, to be at a given, equal temperature T
1
(t). The thermal
diffusivity a of the material is supposed constant. The distribution of temperature on
the interval [–
A
,
A
] satisfies the heat equation:
AA dd
w
w
w
w
x
x
T
a
t
T
2
2
[8.32]
The given data are:
– the initial conditions:
xTxT
0
,0 ;
– the boundary conditions (fixed temperature):
tTtTtT
1
,, AA .
As equation [8.32] is linear, the methodology is the same as before: the complete
solution can be obtained by superposition of the particular solution
txT
e
, of
equation [8.32] satisfying the non-zero boundary conditions and a general solution
of the heat equation with zero boundary conditions for the temperature. As in section
8.2.2.2, the coefficients of this solution are calculated such that the complete
solution satisfies the initial conditions.
In order to simplify the notation of the general solution, let us take the non-
dimensional variables:
2
~
;
~
AA tatxx [8.33]
Equation [8.32] becomes:
1
~
1
~
~
2
2
dd
w
w
w
w
x
x
T
t
T
[8.34]
Thermal Systems and Models 433
We seek a group of solutions with separated variables of the form:
)
~
()
~
()
~
,
~
( xgtftxT
. Substituting this expression into [8.34], we obtain
2
:
2
cte
)
~
(
)
~
("
)
~
(
)
~
('
/
xg
xg
tf
tf
[8.35]
We derive from this the function f:
)
~
exp()
~
(
2
ttf / [8.36]
and the eigenvalue problem for the function g:
01;0)
~
()
~
("
2
r / gxgxg [8.37]
The integration of equation [8.37] shows that the eigenfunctions are of the form
BxAxg /
~
cos)
~
( . The expression of boundary conditions [8.37] at 1
~
r x
gives the values of the constants
/
and B. We thus obtain:
integers):',('
2
;
2
pppBpB
S
S
S
S
/ /
or:
2
'
2
;
2
'
SSS
ppBpp /
In the preceding expressions p – p' and p + p' have the same parity:
– if p – p'=2k is even, we have
S
k
k
/
2
and the corresponding eigenfunctions
are odd:
>@
xkg
k
~
sin
2
S
;
– if p – p' = 2k + 1 is odd, we have
2
12
2
S
/ k
k
and the eigenvalues are
even:
»
¼
º
«
¬
ª
xkg
k
~
2
12cos
12
S
.
The first eigenfunctions of the wall problem at zero wall temperature are
represented in Figure 8.12.
2
The system being damped, the constant of equation [8.36] is necessarily negative.
434 Fundamentals of Fluid Mechanics and Transport Phenomena
x
~
x
~
-1 1
g
i
-1 1
g
i
: g
3
: g
4
: g
5
: g
6
: g
7
: g
8
: g
1
: g
2
Figure 8.12.
First eigenfunctions of the homogenous problem
inside a wall with constant temperature faces
The preceding eigenfunctions form a representation basis of functions which are
zero at the extremities of the interval [-1,1] and which possess sufficient regularity
properties. Taking account of the condition
01
r
g , the eigenfunctions of
problem [8.37] are orthogonal. In effect, an integration by parts immediately gives:
³³³
1
1
''
1
1
'
1
1
''
~~
xdggdggxdgg
qpqpqp
hence:
1
'' ''
1
0.
pq qp
gg gg dx
¯
¡°
¢±
¨
The operator of problem [8.37] is self-adjoint (Appendix 4), showing the
orthogonality property. T
e
being the established solution (section 8.2.2.2), we thus
express the solution in the form:
¦
f
/
0
2
)
~
()
~
exp(
~
,
~
)
~
,
~
( xgtctxTetxT
iii
[8.38]
The coefficients c
i
are calculated at the instant 0
~
t :
¦
f
0
0
)
~
(0,
~~
)0,
~
( xgcxTxTxT
iie
hence (Appendix 4):
Thermal Systems and Models 435
>@
³
³
1
1
2
1
1
0
~
)
~
(
~
)
~
(0,
~
)
~
(
xdxg
xdxgxTxT
c
i
ie
i
[8.39]
8.3.2.2.2.
Thermal shocks on the walls
Let us now consider the simple example of the formation of thermal shocks on
the two faces of a wall whose initial temperature
xT
0
is uniform and whose
temperatures
tT
~
1
on the wall faces are constant and equal to T
1
for positive t
~
.
The established solution
txT
e
~
,
~
is thus constant and equal to T
1
. The coefficients
c
2k
corresponding to the odd eigenfunctions are zero. Formula [8.39] gives, for the
coefficients of the even eigenfunctions:
S
12
41
1012
k
TTc
k
k
From this we can derive the solution of the thermal shock problem T
1
– T
0
applied on the two faces of a wall of thickness 2
A
:
»
¼
º
«
¬
ª
¦
f
xke
kTT
TtxT
k
tk
k
~
2
12cos.
12
41
1
)
~
,
~
(
0
~
4
12
01
0
2
2
S
S
S
[8.40]
The mean
3
temperature
³
1
1
~
)
~
,
~
(
2
1
xdtxTT
m
can be expressed:
¦
f
0
~
4
12
2
2
01
0
2
2
12
8
1
)
~
(
k
tk
m
e
k
TT
TtT
S
S
[8.41]
This expression allows us to know the contribution of each mode (Table 8.1) to
the thermal energy supplied by conduction, this being proportional to the relative
amplitude of the mode (we have
2
2
0
82 1 1):
k
k
d
3
Defined as the space mean temperature for homogenous media.
436 Fundamentals of Fluid Mechanics and Transport Phenomena
Mode number 0 1 2 3 4 5 6
Mode amplitude 0.8106 0.0901 0.0324 0.0165 0.0100 0.0067 0.0048
Table 8.1.
Distribution of thermal energy in the seven first modes
The wall behaves in a manner similar to a first order system, since only 20% of
the heat is exchanged more rapidly in modes higher than the first mode. The density
of the thermal flux
A
A
ww
x
Tp
xTtq
O
,
on the face of the wall at -
A can be
derived from [8.40]. Infinite at time t = 0, it decays very rapidly in the first instants
and it can immediately be written in non-dimensional form:
.2
,
0
4
12
01
2
2
2
¦
f
A
AA
at
k
Tp
e
TT
tq
S
O
[8.42]
x
A
T
1
T
0
–
A
1
2
3
4
5
T
0
f
Figure 8.13.
Evolution of the temperature distribution during
a thermal shock on the two faces of a wall
Figure 8.13 shows the evolution of the temperature profiles with time (curves
from 0 to 5). The curve 0 is the distribution at the initial instant where we impose the
temperature T
1
on the walls ( Ar x ). The diffusion of this condition occurs
progressively from the walls: the curves 1 and 2 represent the thermal shocks in a
quasi-infinite medium from the walls. The thermal diffusion zones are then rejoined
on the other curves (3 to 5). On curves 4 and 5, observed after a non-dimensional
time in the order of 1//
1
, only the first mode g
1
remains, and it damps until its
amplitude falls to zero.
The temperature distribution of the thermal shock in a semi-infinite domain from
a wall has already been obtained (section 5.4.5.4 and formula [5.52]). It can be
written here for the thermal boundary layer on the wall
A x
, and by taking non-
dimensional variables [8.33]:
Thermal Systems and Models 437
due
t
x
erf
at
x
erf
TT
TT
tx
u
³
¸
¸
¹
·
¨
¨
©
§
¸
¸
¹
·
¨
¨
©
§
~
2/1
~
0
01
0
2
2
1
~
2
1
~
1
2
1
S
A
The complete solution which represents the two boundary layers can be written:
¸
¸
¹
·
¨
¨
©
§
¸
¸
¹
·
¨
¨
©
§
t
x
erf
t
x
erf
TT
TT
~
2
1
~
~
2
1
~
2
01
0
[8.43]
The thermal flux density at the wall
1
~
x
can be derived from [8.43]:
01
TT
ta
x
T
q
x
Tp
¸
¸
¹
·
¨
¨
©
§
w
w
S
O
O
A
It can be written in non-dimensional form (section 4.6.1.3.4), with the reduced
time
t
~
[8.33]:
t
TT
tq
Tp
~
1
~
,
01
S
O
AA
[8.44]
The mean temperature of the wall in the thermal boundary layer regime can be
obtained by performing the energy balance in the wall between instants 0 and t:
a
t
TT
au
du
TTduuqCTT
tt
Tpm
S
O
S
O
U
2
01
0
01
0
0
³³
A
or:
t
TT
TtT
m
~
2
01
0
S
[8.45]
8.3.2.2.3.
Composite representation by matched asymptotic expansions
The series expansion of eigenfunctions of the thermal shock problem on the
faces of a wall is a poorly adapted representation in the first instants of the thermal
shock, whereas the representation by thermal boundary layers captures the physics
of the problem more satisfactorily. For small time values, the modal representation
of the mean temperature is nearly acceptable with very few modes (taking account
of a discontinuity at t = 0); on the other hand, modal expression [8.42] for the
438 Fundamentals of Fluid Mechanics and Transport Phenomena
thermal flux density is unusable and we thus have recourse to expression [8.44]
which is equivalent to the sum of the series [8.42] in these conditions.
We are thus led to search for a formula which contains the two different
asymptotic expressions f
1
(t) and f
2
(t) of the function f(t), which are valid for small
and large values of t respectively following the time value. This can be obtained by
means of a matching formula or a weight between the two temporal domains which
gives exact values for the function f and its temporal derivative at the origin, and
which respects the asymptotic behavior at infinity. A simple means consists of
weighting the two formulae by a suitable auxiliary function
M
(t) close to 1 for small t
and tending quite quickly to zero for t equal to infinity. The expression:
tfttfttf
21
1
M
M
satisfies these conditions if the function
M
(t) at least satisfies the relations
00',10 f
M
M
M
and if
M
'(t) tends at infinity faster to zero than
tf
2
; we
have:
f ff f
'
2
'
2
'
11
00'00 ffffffff
The simplest weighting function is the Gaussian
2
~
~
t
et
D
M
.
It remains to write a matching condition which can be defined at a point where
the two approximations differ very little and where we require that the value of the
function
M
(t) is equal to 0.5 (here, the function f(t) is the mean of the values f
1
(t) and
f
2
(t)).
Let us apply this procedure to obtain a quite simple expression of the solution
valid all over the interval [-
A,+A]. We take as our asymptotic expression at infinity
the modal solution limited to the first mode
, and for small t, the boundary layer
solution. We will choose the mean temperature in order to determine the matching
condition. A simple numerical calculation shows that the difference between the
values of expression [8.41] limited to the first mode and formula [8.45] is minimal
in the vicinity of
20.0
~
t . Taking as a weighting function the Gaussian
2
~
~
t
et
D
M
and taking
M
(0.2) equal to 0.5, we find
D
= 7.5.
The mixed representation thus obtained for the first mode of the temperature
T
m
can be written:
Thermal Systems and Models 439
2
22
0
7.5. 7.5.
4
2
10
28
11.
t
m
tt
Tt T
ete e
TT
¯
¡°
¡°
¡°
¢±
[8.46]
The corresponding expression for the wall’s thermal flux density is thus:
2
22
7.5. 7.5.
4
10
,
1
12.
t
Tp
tt
qt
eee
TT
t
AA
[8.47]
The preceding formulae represent the exact solution to 1% accuracy.
The spatio-temporal temperature distribution can also be written in the same
manner from expressions [8.40] and [8.43]:
2
2
2
7.5.
0
10
7.5.
4
11
2
22
4
11cos
2
t
t
t
TT
xx
eerferf
TT
tt
x
ee
¯
¬ ¬
¡°
¡°
® ®
¡°
¢±
¯
¡°
¡°
¡°
¢±
The interest in simple analytic expressions is clear; furthermore, the precision of
the approximations effected can be improved as much as we desire by conserving
additional terms of the modal solution in the preceding composite solution.
8.3.2.3.
Thermal systems with continuous components
8.3.2.3.1. Conduction in two walls separated by a thermal resistance
The separation of variables method can be applied to continuous media by
individual segments which comprise discontinuities. We will reconsider the linear
invariant system with two identical components discussed in section 8.3.1.2 with a
continuous medium model of constant thermal diffusivity
a; the sub-systems are
separated by the thermal resistance
R per unit cross-section, located at the origin
(Figure 8.14). We will assume that the
thermal flux densities are given on the
external faces,
the external thermal resistance R’ being taken as zero.
With equation [8.32] on the intervals [–
A,0
–
] and [0
+
,A], we must associate:
– the matching condition involving the thermal resistance between the two sub-
systems:
R
TT
x
T
x
T
)0()0(
00
¸
¸
¹
·
¨
¨
©
§
w
w
¸
¸
¹
·
¨
¨
©
§
w
w
OO
[8.48]
440 Fundamentals of Fluid Mechanics and Transport Phenomena
– the boundary conditions of the system: thermal flux densities given (or
eventually the temperature or thermal resistance) at the points
Ar x ;
– the initial temperature distribution
0,xT .
T
1ex
T(x)
T
2
ex
M
2w
M
1
w
M
1,2
E
1
E
2
A
–
A
x
O
T
T(0
+
)
T(0
-
)
Figure 8.14.
Thermal conduction in a system of two continuous media
separated by a thermal resistance
As before, the complete solution can be obtained by superposition of a solution
which satisfies the non-zero boundary conditions and a general solution of the heat
equation with homogenous boundary conditions. The coefficients of this general
solution are calculated such that the complete solution satisfies the initial conditions.
8.3.2.3.2.
General solution of the homogenous problem
As an example, we will treat the
problem where a thermal flux is imposed at the
interval extremities
[–A,+A]. The condition at x = ± A of the homogenous problem is
thus a zero thermal flux. As above, we write the homogenous problem with non-
dimensional variables [8.33]. We add to equation [8.34] the boundary conditions and
thermal resistance matching condition [8.48], which can be written in non-
dimensional form as:
>@
)0()0(
~~
;0
~
001
r
¸
¸
¹
·
¨
¨
©
§
w
w
¸
¸
¹
·
¨
¨
©
§
w
w
¸
¸
¹
·
¨
¨
©
§
w
w
TT
Rx
T
x
T
x
T
O
A
[8.49]
The problem depends on the parameter
RP
O
A , which represents the
importance of the thermal resistance
O
A of a component with respect to the thermal
resistance
R.
The solutions to equation [8.34] where the variables are separated as
xgtftxT
~
~
~
,
~
again satisfy relations [8.35] and [8.36] for the function
)
~
exp()
~
(
2
ttf / .
Thermal Systems and Models 441
The eigenvalue problem for the eigenfunction g can now be written:
>@
)0()0()0(')0(';01'
;0)
~
()
~
("
2
r
/
gg
R
ggg
xgxg
O
A
[8.50]
The integration of equation [8.50] leads to eigenfunctions of the form
() cos
g
xA xB /
, the constants (A, B) taking on, respectively, the values (A
,
B
) and (A
+
, B
+
) on the intervals (-1,0) and (0,1):
/ d
/ d
BxAxgx
BxAxgx
~
cos)
~
(:1
~
0
~
cos)
~
(:0
~
1
The expression of the boundary conditions at
1
~
r x gives:
0sin.sin. // //
BABA
or:
integers:',' pppBpB
S
S
/
[8.51]
Matching condition [8.49] for the thermal flux densities at x = 0 can be written:
>@
/ / BABAPBABA coscossinsin [8.52]
Equations [8.51] and [8.52] possess two groups of solution
4
:
1) Even eigenfunctions. We find immediately the solutions to equations [8.51]
and [8.52]:
integer) (;;0
2
kkAABB
k
S
/
[8.53]
The evenness of the eigenfunctions
xkxg
k
~
cos
~
2
S
of the two-wall
ensemble leads to the absence of thermal transfer between the two sub-systems
which behave as a single symmetric block. The first four eigenfunctions (k = 0, 1, 2,
3) are represented in Figure 8.15a (we have taken
1
AA ).
4
We will number the functions with even numbers (respectively odd) for even eigenvalues
(respectively odd) with increasing order of the eigenvalues.