Convection Diffusion Equation
Transcript of Convection Diffusion Equation
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International Journal on Architectural Science, Volume 1, Number 2, p.68-79, 2000
68
THE CONVECTIVE-DIFFUSION EQUATION AND ITS USE IN BUILDING
PHYSICS
Z. SvobodaFaculty of Civil Engineering, Czech Technical University, Thakurova 7, 166 29 Prague 6, Czech Republic
ABSTRACT
The convective-diffusion equation is the governing equation of many important transport phenomena in
building physics. The paper deals in its first part with the general formulation of the convective-diffusion
equation and with the numerical solution of this equation by means of the finite element method. The second
part of the paper contains the analysis of one typical combined transport problem the combined heat transfer
through the building constructions caused by conduction and convection. The finite element solution of this
problem is presented in the paper together with the numerical stability analysis and one practical example of the
numerical analysis of a model lightweight building construction. The results of the model analysis demonstrate
that the lightweight constructions insulated with permeable mineral wool are very sensitive to the convective
heat transfer.
1. INTRODUCTION
Combined transport of a substance through a
porous medium caused by diffusion and convection
is a relatively frequent problem in building physics.
Characteristic examples are the heat transfer
through a permeable medium, the transport of a
pollutant through the atmosphere or the transport
of a fluid through the porous medium.
The governing equations of such building physics
problems are generally called the convective-
diffusion equations. Such equations are the centre
of many recent investigations [e.g. 1-6] not only
due to their importance to building physics
analyses, but also due to some problems with
numerical stability in their solution.
2. THE CONVECTIVE - DIFFUSIONEQUATION
The complex steady-state transport of a substance
through the porous medium caused by diffusion
and convection is described by the partial
differential equation:
02
2
2
2
2
2
=+
+
+
+
+
Q
z
Uw
y
Uv
x
Uu
z
U
y
U
x
U
(1)
Boundary conditions for equation (1), which are
the most useful for building physics problems, are
usually of Dirichlet type defined as:
UU = (2)
or of Newton type defined as:
( ) ( )UUUUvn
Un =+
(3)
The Newton type boundary condition (3) is the
most frequently used boundary condition for
common problems such as heat transfer. This type
of condition could also be easily converted within
the computer programs to Dirichlet type byassigning a large number to the boundary transfer
coefficient. The Newton type condition will
therefore be chosen as the basic boundary
condition in the following derivations.
In this paper, the following assumptions were taken
to obtain the numerical solution of equation (1)
with the boundary condition (3):
convection of fluid through the porousmedium is caused only by a pressure
difference
the moving fluid is incompressible the fluid flow is linear according to Darcys
Law
Pk
v =r
r
(4)
Equation (4) could be used only if the fluid
flow is laminar. That could be reached
according to W. Nazaroff [2,3] if the
Reynolds number defined for fluid flow in
the porous medium as:
vdRe = (5)
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does not exceed the value of 4. This condition
could be satisfied for most building materials
with permeability lower than 10-8
m2
when
the operating pressure gradient is not higher
then 5 Pa [7,8].
pressure distribution is governed by Laplaceequation:02 = Pk (6)
the pressure losses of cracks are considered inthe model in a simplified way by means of an
equivalent permeability of the air in the
crack, which is defined as:
3
2b
ka = (7)
Equation (7) was derived from the equality of
the air flow velocity defined by Darcys Law
and the mean velocity of the laminar air flow
in the crack defined as:
3
2b
L
Pvm
= (8)
3. NUMERICAL SOLUTION OF THECONVECTIVE - DIFFUSION
EQUATION
The search for the numerical solution of the
convective-diffusion equation is always more
complicated than the search for the solution of the
related diffusion equation. The main cause is the
convective transport term - second term in equation
(1) - which can introduce under certain conditions
instabilities in the numerical solution.
The finite element method was used in this paper to
find the solution of equation (1). The general finite
element formulation was derived by means of the
Petrov-Galerkin process. As the Petrov-Galerkinprocess is one of the weighted residuals methods,
the derivation of the finite element formulation
starts with the following equation:
02
2
2
2
2
2
=
+
+
+
+
+
dW
Qz
Uw
y
Uv
x
Uu
z
U
y
U
x
U
)e(
i
The Petrov-Galerkin approach, which is also
known as streamline balancing diffusion orstreamline Petrov-Galerkin process, is based on the
special selection of the weighting functions
different from the interpolation functions. The
identity of the weighting and interpolation
functions is characteristic for the standard Galerkin
method which is the most commonly used method
in the finite element solution of the field problems.
Unfortunately, this method cannot be applied toequation (1), because it leads to the numerical
oscillations mentioned above, as was already
shown by several researchers (Zienkiewicz [9],
Huebner [1]).
Equation (9) is a mathematical expression of the
requirement that the residual of the numerical
solution of equation (1) must be orthogonal to the
weighting functions Wi.
The unknown function Uin equation (9) is taken as
an approximation:
iT
i UNU= (10)
The interpolation functionsNi are known functions
closely connected to the type of the chosen finite
elements.
The definition of the weighting functions Wi is very
important in this case. The approach recommended
by Zienkiewicz [9] takes the weighting functions as:
u
z
N
wy
N
vx
N
uhNW
iii
ii
+
+
+=2
(11)
Now, if the value of is chosen as:
PePecoth
1= (12)
and Peclet numberPe as:
2
huPe
= (13)
then according to Zienkiewicz [9] numerical
oscillations will not arise for any possible rate
between convective and conductive transport.
The weighting functions defined in equation (11)
are constructed to be different from zero only in the
direction of the velocity vector. This fact is the
reason why the terms streamline Petrov-Galerkin
process or streamline balancing diffusion are
used as synonyms for Petrov-Galerkin method.
Equations (10) and (11) can be substituted intoequation (9). Integration by parts can be then
applied to the first term in equation (9) and
(9)
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subsequently the boundary conditions (3) can be
introduced into the equation. The general finite
element formulation which was derived by means
of this approach can be finally written as:
( ) Qiv qqUKKK +=++ (14)
The conductance matrixK is defined as:
dz
N
z
W
y
N
y
W
x
N
x
WK
)e(
Tii
Tii
Tii
+
+
=
(15)
the convective transport matrixKv as:
dz
NwW
y
NvW
x
NuWK
)e(
Ti
i
Ti
i
Ti
iv
+
+
=
(16)
the boundary conditions matrixKas:
( )
dNWvK)e(
Tiin = (17)
the boundary conditions vectorq as:
( )
dUWvq)e(
in = (18)
and the substance generation rate vectorqQ as:
dQWq)e(
iQ = (19)
Note that the convective transport matrix Kv is
asymmetrical, which is caused by the fact that the
differential operator in equation (1) is not self-
adjoint. This leads to the asymmetrical matrix of
the system of linear equations for unknown nodal
values Ui.
4. COMBINED HEAT TRANSFERCAUSED BY CONDUCTION AND
CONVECTION
Several building physics problems governed by the
convective-diffusion equation have been
mentioned already in the introduction of this paper.
The architects and building engineers usually pay
the greatest attention to two important combined
transport phenomena taking place in the building
constructions, which are the radon transport caused
by diffusion and convection and the heat transferdue to conduction and convection. The first
transport problem mentioned has been discussed in
detail in several papers [10-12]. The second
problem - the combined heat transfer caused by
conduction and convection - is currently actively
discussed in the building physics field [2,4-6,13].
This type of heat transfer appears in building
constructions loaded by the temperature and the
pressure gradient between the interior and theexterior.
The importance of the convective-conductive heat
transfer depends mainly on the type of the
construction and on its tightness against the air
flow. Typical construction for which the
convective part of the heat transfer is of high
importance is a modern lightweight construction
with permeable thermal insulation, such as mineral
wool, covered by thin layers of plasterboards.
Traditional constructions, such as brick walls, are
not so sensitive to the convective heat transfer and
the convective component of the total heat loss isusually negligible in comparison with the
conductive component.
5. GOVERNING EQUATION OF THECONVECTIVE - CONDUCTIVE
HEAT TRANSFER AND ITS
NUMERICAL SOLUTION
The analysis of many combined convective-
conductive heat transfer problems could be based
on the partial differential equation for the two-dimensional steady-state heat transport in a porous
medium, which is a member of convective-
diffusion equations family. This equation could be
expressed as:
02
2
2
2
=
+
+
y
Tv
x
Tuc
y
T
x
Taa (20)
The Newton type boundary condition connected to
equation (20) is defined as:
( ) ( )TThTTcvnT aan =+
(21)
Note that equation (20) is a subtype of the general
convective-diffusion equation (1) and the boundary
condition (21) is of the same type as condition (3).
The numerical solution of equation (20) could be
derived using the same assumptions and the same
method as was already shown for the general
convective-diffusion equation. The finite element
solution of equation (20) could be reached by
means of Petrov-Galerkin process with weighting
functions Widefined for this two dimensional caseas:
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u
y
Nv
x
Nu
hNW
ii
ii
+
+=2
(22)
and the Peclet number defined as:
2
hucPe
aa = (23)
The derived general finite element solution of the
equation (20) could be finally written as:
( ) qTKKK iv =++ (24)
The conductance matrixK is defined as:
dy
N
y
W
x
N
x
WK
)e(
T
ii
T
ii
+
= (25)
the convective transport matrixKv as:
dy
NvW
x
NuWcK
)e(
Ti
i
Ti
iaav
+
= (26)
the boundary conditions matrixKas:
( )
dNWcvhK
)e(
Tiiaan = (27)
and the boundary conditions vectorq as:
( )
dTWcvhq)e(
iaan = (28)
Equation (24) is the basis for the computer
program called WIND developed by Z. Svoboda.
This program calculates the pressure field within
the porous building construction, the air flow
velocity field, the temperature field and the heat
flow rate due to conduction and due to convection.This calculation tool uses the simple triangular
finite elements with three nodes (Fig. 1) and with
the linear interpolation functions defined as:
( )ycxbaA
N iiii ++=2
1i = 1, 2, 3 (29)
where the values a, b and c are expressed as:
jkkji yxyxa =
kji yyb =
jki xxc =
with the indices i,j, ktaken as 1, 2, 3 in a cycle.
The weighting functions derived from equation (22)
could be written for this simple finite element with
the linear interpolation as:
( )
+
+++= iiiiii vcub
u
hycxba
AW
22
1
i = 1, 2, 3 (30)
1
2
3
L3
L1
L2
Fig. 1: Triangular finite element with 3 nodes
and linear interpolation
Now, if the interpolation and weighting functions
are defined in the way as shown in equations (29)
and (30), it is possible to derive analytical
expressions for all matrixes and vectors in the
finite element formulation (24).
The analytical expression for the conductance
matrixKis:
+
++
+++
=2
32
3
32322
22
2
313121212
12
1
4cbsymm.
ccbbcb
ccbbccbbcb
AK
(31)
For the convective transport matrixKvis:
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+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+++
+++
+++
=
23
2
33
2
3
232
2
32
32
322
22
2
22
22
2
312
31
31
312
212
21
21
212
21
2
11
21
2
3
3
2
2
1
1
3
3
2
2
1
1
3
3
2
2
1
1
2
2
2
8
6
cv
cuvbbu
symm.
ccv
cuvb
buvc
bbu
cv
cuvb
bu
ccv
cuvb
buvc
bbu
ccv
cuvb
buvc
bbu
cv
cuvb
bu
uA
ch
vc
ub
vc
ub
vc
ub
vc
ub
vc
ub
vc
ub
vc
ub
vc
ub
vc
ub
cK
aa
aav
(32)
and for the boundary conditions matrixK is:
+
+
+
+
+
+
+
=
4
44
444
3
63
663
21
131
2332
21
131
2332
BBsymm.
BBB
BBBB
BBsymm.
BBB
BBBB
K
(33)
with Bi defined as ( iaai,nii LcvhB = and i=1,2, 3, where i is the number of the triangular finite
element boundary (side of the triangle). The
definition of the sides of the triangular finite
element is shown in Fig. 1. If no boundary
condition is defined at the finite element boundary,
all terms corresponding to that boundary in
equation (33) are taken as 0.
The boundary conditions matrix K could also be
defined in a way different from equation (33). If
the interpolation functions on the finite element
boundaryLi are taken as partly continuous (Fig. 2),
which is acceptable according to Zienkiewicz [9]
as far as there are no derivatives of interpolation
functions Ni in equation (27), the resulting
boundary conditions matrixKcan be defined as:
+
+
+
+
+
+
+
=
2
02
002
2
02
002
21
31
32
21
31
32
BBsymm.
BB
BB
BBsymm.
BB
BB
K
(34)
with ( iaai,nii LcvhB = and index i rangingfrom 1 to 3.
1
23
LL
L1
3
2
N N23
x
1
1
23
LL
L1
3
2
N N23
x
a) b)
The use of the boundary conditions matrix defined
in equation (34) leads to higher numerical stability
of the calculation results, especially in the parts of
the construction where two boundary conditions
are in connection (e.g. at the external surface of the
wallecorners .
(a) (b)
Fig. 2: (a) Linear and (b) partly continuous interpolation functions on the element boundary L 1
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The last term in equation (24) is the boundary
conditions vector q which could be analytically
expressed as:
+
+
+
+
+
+
+
=
2
2
2
2
2
2
2211
3311
3322
2211
3311
3322
TBTB
TBTB
TBTB
TBTB
TBTB
TBTB
q (35)
with ( iaai,nii LcvhB = and index i rangingfrom 1 to 3.
6. NUMERICAL STABILITYANALYSIS
The analysis of the numerical stability of the
solution obtained by the computer program
WIND could be realised in two ways. The first
method is to calculate the exact analytical solution
of a simple one dimensional problem and to
compare it with the numerical solution obtained by
the program. The second method of the numerical
stability analysis could be based on the evaluation
of the functional corresponding to the governing
equation (20).
The first method of the numerical stability analysis
was studied based on a simple one dimensionalproblem:
02
2
=dx
dTuc
dx
Tdaa (36)
with boundary conditions:
10 =)(T , 01 =)(T (37)
The analytical solution of the equation (36) is:
B
BBx
e
ee)x(T
=
1(38)
with the value ofB defined as:
ucB aa= (39)
The numerical solution of equation (36) forB = 4
and for various number of the finite elements in
comparison with the exact solution could be seen
in Table 1. The analysis shows clearly that the
results of the numerical solution converge to theexact solution with increasing number of the finite
elements covering the solved area.
Table 1: Results of the exact and the numerical
solution
Distance Temperature
exact
solution
numerical
solution
x (m) T (K) T1 (K) T2 (K) T3 (K)0,0 1,000 1,000 1,000 1,000
0,2 0,977 0,978 0,977 0,977
0,4 0,926 0,928 0,927 0,926
0,6 0,813 0,815 0,814 0,813
0,8 0,561 0,564 0,562 0,561
1,0 0,000 0,000 0,000 0,000
No. of
elements
--- 20 40 80
The second method of the numerical stability
analysis - the evaluation of the functional
corresponding to equation (20) - is more
complicated but it is more appropriate for theanalysis of the two or three dimensional problems.
The basic idea of this approach is based on the fact
that the finite element method is one of the
variation methods which means that the exact
solution of the equation (20) obtained by the finite
element method must minimise the functional
corresponding to the equation (20). The functional
connected with the equation (20) could be derived
by means of the Guymon process cited by
Zienkiewicz [9]. The first step in this process is to
adjust the linear differential operatorA in equation
(20):
yvc
xuc
yxA aaaa
+
=
2
2
2
2
(40)
which is not self-adjoint, in a special way so that
the self-adjointness is achieved without altering the
equation (20). Let us expect that q(x,y) is a general
function and let us multiply both sides of equation
(20) with this function. Equation (20) could be
rewritten after this operation as:
0=TqA (41)
The test for symmetry of the operatorqA could be
expressed for any two functions and as:
( ) ( )
dqAdqA = (42)
Now, if we take the operatorqA from equation (41),
substitute it into equation (42) and integrate the
result by parts, we could obtain following equation
(b.t. denoting boundary terms):
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b.t.d
y
qvqc
y
xquqc
xyq
yxq
x
b.t.d
y
qvqc
y
x
quqc
xyq
yxq
x
aa
aa
aa
aa
+
+
+
=+
+
+
The operator qA will be symmetric and therefore
self-adjoint if the equations:
0
0
=
+
=
+
y
qvqc
,x
quqc
aa
aa
(43)
are fulfilled. The solution of equations (43), which
is the function q we search for, could be found as:
( )vyuxcaa
eq+
=
(44)
If we take the function q defined according to
equation (44) and multiply equation (20) with it,
we finally obtain the symmetric operator qA.
Subsequently, we could use a well-known
expression:
( )
dTqATF21 = (45)
for the derivation of the functionalFcorresponding
to equation (20). The resulting functional
connected to equation (20) could be obtained after
some derivations in the following form:
( )( )
dTTTcvhq
dy
Tq
x
TqF
aan
+
=
2
21
22
21
(46)
where the function q is defined in equation (44).
The building construction shown in Fig. 3 was
analysed from the point of view of the numerical
stability using the functional evaluation. The
characteristics of the materials involved are shown
in Table 2. The boundary conditions were taken as
follows:
interior: temperature 20 C, pressure 0 Pa exterior: temperature -15 C, pressure 10 Pa
The initial mesh system with 580 finite elements is
shown in Fig. 4. The mesh system was refined
twice and each time the functional (46) was
calculated by means of the Gauss numerical
integration. The results of the analysis are
presented in Fig. 5. It could be clearly seen that the
values of the functional decrease with theincreasing number of finite elements. This shows
that the calculated results of equation (20)
converge to the exact solution and the numerical
stability is reached.
plasterboard 13 mm
mineral wool 120 mm
plasterboard 13 mm 1 mm wide crack
1 m
Fig. 3: The model building construction for
the functional analysis
Table 2: Used material characteristics
Material Permeability(m
2)
Thermalconductivity
(Wm-1
K-1
)
Plasterboard 1.10-12 0.220
Mineral wool 1.10-9
0.040
7. ANALYSIS OF A MODEL
CONSTRUCTION
The use of the program WIND for the purposes
of the heat transfer calculations could be
demonstrated on the analysis of a typical building
slope roof construction. A cross-sectional view of
the analysed construction and the boundary
conditions are shown in Fig. 6. The material
characteristics are recapitulated in Table 2.
The calculation has been performed several times
for various widths of the crack in the internal
plasterboard cladding, including perfectly tight
construction with no cracks. For each width of the
crack and for each loading pressure gradient, the
air pressure field, the air flow velocity field and the
temperature field have been calculated.
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Fig. 4: The initial mesh system with 580 elements
0,00040
0,00045
0,00050
0,00055
0,00060
0,00065
580 1160 2320
Number of elements
Functionalvalues
Fig. 5: The calculated functional values
1000 mm
mineral wool 160 mm
closed air layer 40 mm
plasterboard 12 mm
exterior side:
temperature -15 C air pressure from 0 to 10 Pa
interior side:
temperature 20 C air pressure 0 Pa
crack of
various
width
roof tiles
ventilated air layer
(300 mm to 800 mm)
inlet
outlet
Fig. 6: Cross-section of the analysed construction
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The influence of the convective heat transport
through the crack has been expressed for each
analysed case by means of the convective linear
thermal transmittance according to the following
equation:
ccei
p,v lUtt
= (47)
The linear thermal transmittance is used in ISO and
European standards as a value showing the
influence of a thermal bridge on the heat loss. In
this paper, the crack is taken as a convective
bridge and the convective linear thermal
transmittance is used to show the influence of the
air flow through the convective bridge on the
heat loss, which can be finally expressed as:
+= tltUAQ p,vvT (48)
The convective linear thermal transmittance is
always related to the length of the crack and to the
operating pressure difference.
The results of the convective linear thermal
transmittance calculation for various pressure
differences and various widths of the crack in theplasterboard are shown in Table 3 and Fig. 7.
Another interesting calculation result is the
temperature distribution in the analysed
construction. Fig. 8 shows the temperature fields in
the construction with 1 mm wide crack in the
plasterboard for pressure differences 0 and 10 Pa.
Note the deformation of the temperature field
caused by the air flow through the permeable
thermal insulation not covered by any air-tight
layer and through the crack in the plasterboard in
the case of pressure gradient of 10 Pa.
Table 3: The results of the convective linear thermal transmittance analysis
Operating Width of the crack
pressure difference no crack 1 mm 2 mm 3 mm
0 Pa 0.000 Wm-1
K-1
0.000 Wm-1
K-1
0.000 Wm-1
K-1
0.000 Wm-1
K-1
1 Pa 0.003 Wm-1K-1 0.188 Wm-1K-1 0.231 Wm-1K-1 0.247 Wm-1K-1
5 Pa 0.015 Wm-1
K-1
1.296 Wm-1
K-1
1.565 Wm-1
K-1
1.628 Wm-1
K-1
10 Pa 0.031 Wm-1
K-1
2.748 Wm-1
K-1
3.203 Wm-1
K-1
3.385 Wm-1
K-1
0,00
0,50
1,00
1,50
2,00
2,50
3,00
3,50
0 2 4 6 8 10
The operating pressure difference [Pa]
Theconvectivelinearthermaltransmittance[W/mK]
no crack
1 mm wide crack
2 mm wide crack
3 mm wide crack
Fig. 7: The convective linear thermal transmittance for various widths of the crack
Convectivelinear
thermaltransmittance(Wm-1K-1)
Operating pressure difference (Pa)
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Fig. 9 demonstrates three temperature distributions
in the cross-section of the construction. The first
distribution, which was calculated for no operating
pressure difference, is a typical temperature profile
in the construction exposed to the heat transfer
caused exclusively by conduction. The second
distribution calculated for the pressure differenceof 10 Pa shows clearly how the air flow from the
exterior side changes the temperature profile in the
construction, even if the analysed construction has
no crack in the plasterboard covering. The last
temperature distribution shows the extraordinary
influence of the crack in the plasterboard. This
temperature profile is calculated for the pressure
gradient of 5 Pa and is taken from the cross-sectionleading through the centre of the 1 mm wide crack.
exterior side:
temperature -15 C air pressure 0 Pa
interior side:
temperature 20 C air pressure 0 Pa
exterior side:
temperature -15 C air pressure 10 Pa
interior side:
temperature 20 C air pressure 0 Pa
Fig. 8: The temperature distribution in the construction with 1 mm wide crack in the plasterboard
-15
-10
-5
0
5
10
15
20
0 53 106 159 212
Cross-section of the construction [mm]
Tem
perature[C]
no crack, 0 Pa
no crack, 10 Pa
1 mm wide crack, 5 Pa
mineral woolair layerplasterboard
Fig. 9: The temperature distribution in the cross-section of the analysed construction
Cross-section of the construction (mm)
Temperature(oC)
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The results of the model analysis show these major
conclusions:
The modern constructions containing permeable
thermal insulation, such as mineral wool, are very
sensitive to the convective component of the heat
transfer. Any crack in the covering of a lightweightconstruction filled with permeable thermal
insulation could cause air flow into the thermal
insulation and consequently modifications to the
temperature distribution. The result of such
temperature field deformation is a considerable
increase in the heat loss through the construction.
ACKNOWLEDGEMENT
This paper was supported by the research program
VZ CEZ J04/98:210000001.
NOMENCLATURE
A area of a triangular finite element, m2
b one half of the width of the crack, m
ca thermal capacity of the air, 1010 Jkg-1
K-1
d characteristic diameter of the pore in the
porous medium, m
h size of a finite element in the velocity
direction, m
h heat transfer coefficient, Wm-2K-1
k permeability of the porous medium, m2K conductance matrix
Kv convective transport matrix
K boundary conditions matrix
lc width associated with the U value, m
lv length of the crack associated with the
convective linear thermal transmittance, m
L length of the crack in the direction of the air
flow, m
Li length of a triangular finite element side, m
Ni vector of the interpolation functions
P pressure of the fluid in the porous medium,
Pa
P pressure difference between the inlet and the
outlet of the crack, Pa
q boundary conditions vector
T unknown temperature, K
Ti vector of unknown nodal temperature values,
K
T known temperature in the environment in the
connection with the element boundary, K
ti interior temperature, K
te exterior temperature, K
t temperature difference, K
u, v, w velocity vector components in the x-axis,
y-axis and z-axis direction respectively, ms-1u velocity vector magnitude, ms-1
U unknown concentration of a substance in the
medium
Ui vector of unknown nodal concentrations of
the substance
Uc U value, thermal transmittance coefficient,
Wm-2
K-1
U known concentration of the substance at partof the boundary
U known concentration of the substance in theenvironment neighbouring with the boundary
vn velocity component normal to the boundary,
ms-1
v fluid flow velocity in the porous medium,ms-1Q amount of the internal source of the substance
QT transmission heat loss, W
Wi vector of the weighting functions
x, y co-ordinates of the mesh node
boundary transfer coefficient of the substanceat the discussed boundary
heat flow rate, Wm-1
parameter describing the convective
properties of the medium
e boundary of a finite element
parameter describing the diffusive properties
of the medium
thermal conductivity, Wm-1K-1
viscosity of the fluid (for air = 1.7 x 10-5Pas),
kinematic viscosity of the fluid (for air =
1.4 x 10-5
m2s
-1),
e area of a finite element, m2
a density of the air, 1.2 kgm
-3
v,p convective linear thermal transmittance at a
given pressure difference, Wm-1
K-1
n derivative in the direction of the external
normal to the boundary
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