convection diffusion equation

7/30/2019 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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International Journal on Architectural Science

69

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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70

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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71

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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72

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+++

+++

+++

=

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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73

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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74

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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75

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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76

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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77

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:


exterior side:

temperature -15 C air pressure 10 Pa

interior side:


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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78

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

REFERENCES

1. K.H. Huebner and E.A. Thornton, The finite

element method for engineers, John Wiley & Sons,

New York (1982).

2. J. Claesson, Forced convective-ciffusive heat

flow in insulations, A new analytical technique,

Proc. of the 3rd Symposium Building Physics in

the Nordic Countries, Vol. 1, Thermal insulation

Laboratory, TU of Denmark, Copenhagen, pp.137-

144 (1993).

3. P.J. Dimbylow, The solution of the pressure

driven flow equation for radon ingress through

cracks in concrete foundantions, Radiation

Protection Dosimetry, Vol. 18, pp. 163-167 (1987).

4. C.E. Hagentoft, Thermal effects due to air flows

in cracks, Proc. of the 3rd Symposium BuildingPhysics in the Nordic Countries, Vol. 1, Thermal


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Bauverlag GmBH, Wiesbaden, pp. 88-96 (1991).

6. International Energy Agency ANNEX 24, Heat,

air and moisture transfer through new and

retrofitted insulated envelope parts, Final report,

Task 1 Modelling (1993).

7. W.W. Nazaroff and R.G. Sextro, Technique for

measuring the indoor 222Rn source potential in

soil, Environmental Science and Technology, Vol.

23, pp. 451-458 (1989).

8. W.W. Nazaroff and A.V. Nero, Radon and its

decay products in indoor air, McGraw-Hill, New

York (1988).

9. O.C. Zienkiewicz and R.L. Taylor, The finiteelement method, 4th edition, Vol. 1 & 2, McGraw-

Hill Book Company, London (1991).

10. C.E. Andersen, D. Albarracn, I. Csige, E.R. van

der Graaf, M. Jirnek, B. Rehs, Z. Svoboda and L.

Toro, ERRICCA radon model intercomparison

exercis, Riso National Laboratory, Roskilde

Denmark (1999).

11. M. Jirnek and Z. Svoboda, The verification of

radon protective measures by means of a computer

model, Proceedings of the 5th Building

Simulation Conference, Prague, Vol. II, pp. 165-

172 (1997).

12. M. Jirnek and Z. Svoboda, The computer model

for simulation of soil ventilation systems

performance, Proceedings of European

Conference on Protection against Radon at Home

and at Work, Prague, Part II, pp. 110-113 (1997).

13. Z. Svoboda, The analysis of the convective-

conductive heat transfer in the building

constructions, Proceedings of the 6th Building

Simulation Conference, Kyoto, Vol. I, pp. 329-335

(1999).

convection diffusion equation

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