the convective-diffusion equation and its use in ... - Semantic Scholar

International Journal on Architectural Science, Volume 1, Number 2, p.68-79, 2000

THE CONVECTIVE-DIFFUSION EQUATION AND ITS USE IN BUILDING PHYSICS Z. Svoboda Faculty 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.

or of Newton type defined as:

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 convectivediffusion 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.

−η

The complex steady-state transport of a substance through the porous medium caused by diffusion and convection is described by the partial differential equation:

• •

  ∂U ∂U ∂U  − γ u +w +v   ∂x ∂z ∂y  

  + Q = 0  (1)

Boundary conditions for equation (1), which are the most useful for building physics problems, are usually of Dirichlet type defined as: U =U′

68

(2)

)

(3)

In this paper, the following assumptions were taken to obtain the numerical solution of equation (1) with the boundary condition (3): •

 ∂ 2U ∂ 2 U ∂ 2U η  2 + 2 + 2 ∂y ∂z  ∂x

) (

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 by assigning 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.

THE CONVECTIVE - DIFFUSION EQUATION

2.

(

∂U + vnγ U − U = α U − U ∂n

convection of fluid through the porous medium is caused only by a pressure difference the moving fluid is incompressible the fluid flow is linear according to Darcy’s Law r k r v = − ∇P

µ

(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:

Re =

dv

ν

(5)

International Journal on Architectural Science

•

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].

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.

pressure distribution is governed by Laplace equation:

Unfortunately, this method cannot be applied to equation (1), because it leads to the numerical oscillations mentioned above, as was already shown by several researchers (Zienkiewicz [9], Huebner [1]).

k ⋅∇2 P = 0

•

(6)

the pressure losses of cracks are considered in the model in a simplified way by means of an “equivalent” permeability of the air in the crack, which is defined as: ka =

b2 3

(7)

Equation (7) was derived from the equality of the air flow velocity defined by Darcy´s Law and the mean velocity of the laminar air flow in the crack defined as: vm =

3.

∆P b 2 µL 3

(8)

NUMERICAL SOLUTION OF THE CONVECTIVE - 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.

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 U in equation (9) is taken as an approximation: U = NiTU i

(10)

The interpolation functions Ni 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:

Wi = N i + ε

∆h

u

2

∂N i ∂N i ∂N +v +w i ∂x ∂y ∂z u

(11)

Now, if the value of ε is chosen as:

ε = coth Pe −

1 Pe

(12)

and Peclet number Pe as: 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-Galerkin process is one of the weighted residuals methods, the derivation of the finite element formulation starts with the following equation:   ∂ 2U ∂ 2 U ∂ 2U  η  2 + 2 + 2  ∂y ∂z    ∂x   ∂U ∂U ∂U Ω(e)   − γ  u ∂x + v ∂y + w ∂z  

∫

   Wi d Ω = 0   + Q     (9)

The Petrov-Galerkin approach, which is also known as streamline balancing diffusion or streamline Petrov-Galerkin process, is based on the special selection of the weighting functions

Pe =

γ ⋅ u ⋅ ∆h 2η

(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 into equation (9). Integration by parts can be then applied to the first term in equation (9) and

69


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:

(K λ + K v + K α ) ⋅ U i

= qα + qQ

(14)

The conductance matrix Kλ is defined as:

 ∂W ∂N i T ∂Wi ∂N i T ∂Wi ∂N i T  d Ω + + Kλ = η i  ∂x ∂y ∂y ∂z ∂z  ( e )  ∂x Ω (15)

∫

the convective transport matrix Kv as:

∫

Kv =

Ω(e)

  

γ  uWi

 d Ω   (16)

∂N i T ∂N i T ∂N i T + vWi + wWi ∂x ∂y ∂z

the boundary conditions matrix Kα as: Kα = Γ

∫ (α − v nγ )Wi N i

T

dΓ

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 the exterior. 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 is usually negligible in comparison with the conductive component.

(17)

5.

(18)

The analysis of many combined convectiveconductive heat transfer problems could be based on the partial differential equation for the twodimensional steady-state heat transport in a porous medium, which is a member of convectivediffusion equations family. This equation could be expressed as:

(e)

the boundary conditions vector qα as: qα = Γ

∫ (α − vnγ )Wi U d Γ (e)

and the substance generation rate vector qQ as: qQ =

∫ QWi dΩ

(19)

Ω(e)

Note that the convective transport matrix Kv is asymmetrical, which is caused by the fact that the differential operator in equation (1) is not selfadjoint. This leads to the asymmetrical matrix of the system of linear equations for unknown nodal values Ui.

 ∂ 2T

λ 

 ∂x

COMBINED HEAT TRANSFER CAUSED 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 transfer due to conduction and convection. The first transport problem mentioned has been discussed in

70

2

+

 ∂T ∂ 2T  ∂T − ρ a c a  u +v 2  ∂y ∂y   ∂x

  = 0 

(20)

The Newton type boundary condition connected to equation (20) is defined as: −λ

4.

GOVERNING EQUATION OF THE CONVECTIVE - CONDUCTIVE HEAT TRANSFER AND ITS NUMERICAL SOLUTION

∂T + vn ρ a ca T − T = h T − T ∂n

(

) (

)

(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 Wi defined for this two dimensional case as:


Wi = N i + ε

∆h

u

2

bi = y j − y k

∂N i ∂N +v i ∂x ∂y u

(22)

and the Peclet number defined as:

with the indices i, j, k taken as 1, 2, 3 in a cycle.

ρ a c a ⋅ u ⋅ ∆h 2λ

Pe =

(23)

The derived general finite element solution of the equation (20) could be finally written as:

(K λ + K v + Kα )Ti

ci = x k − x j

= qα

(24)

The weighting functions derived from equation (22) could be written for this simple finite element with the linear interpolation as: Wi =

 ε ⋅ ∆h 1  (ubi + vci ) ai + bi x + ci y + 2 A  2u 

i = 1, 2, 3

(30)

The conductance matrix Kλ is defined as:  ∂W ∂N i T ∂Wi ∂N i T + Kλ = λ i  ∂x ∂x ∂y ∂y Ω(e) 

∫

 d Ω  

y

(25)

3

the convective transport matrix Kv as:

∫

Kv =

Ω(e)

  

ρ a c a  uWi

∂N i T ∂N i T + vWi ∂x ∂y

L1   d Ω (26)  

L2 2

the boundary conditions matrix Kα as: Kα =

T ∫ (h − vn ρ a c a )Wi N i d Γ

L3 (27)

Γ (e)

1

and the boundary conditions vector qα as: qα =

∫ (h − v n ρ a c a )Wi T d Γ

x

(28)

Γ(e)

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: Ni =

1 (ai + bi x + ci y ) 2A

i = 1, 2, 3

where the values a, b and c are expressed as: ai = x j y k − x k y j

(29)

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 matrix Kλ is: b1 2 + c1 2 λ  Kλ =  4A  symm. 

b1b2 + c1c 2 b2 2 + c 2 2

b1b3 + c1c3   b2 b3 + c 2 c3  b3 2 + c3 2   (31)

For the convective transport matrix Kv is:

71


ub1  + vc1 ρ a c a ub1 Kv = 6 + vc1  ub1 + vc1 

ub2 + vc 2 ub2 + vc 2 ub2 + vc 2

 2 2 u b1  + 2uvb1c1 + v 2 c 2 1    ε ⋅ ∆h ρ a c a   + 8A u       symm.    

with Bi defined as Bi = (hi − v n ,i ρ a c a )Li and i=1,

ub3   + vc 3  ub3  + vc 3   ub3  + vc 3 

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.

  + uvc1b3  + uvb1 c 3   + v 2 c1 c 3   u 2 b2 b3  + uvc 2 b3   + uvb2 c 3   + v 2 c 2 c3   u 2 b3 2  + 2uvb3 c 3   + v 2 c3 2    (32) u 2 b1b3

u 2 b1b2 + uvc1b2 + uvb1c 2 + v 2 c1c 2 u 2 b2 2 + 2uvb2 c 2 + v 2c2 2

The boundary conditions matrix Kα could also be defined in a way different from equation (33). If the interpolation functions on the finite element boundary Li 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 matrix Kα can be defined as:   B 2 + B3 0 0   2   B1 + B3 Kα =  0    2  B1 + B 2    symm. 2    B 2 + B3  0 0   2   B1 + B3 +ε 0    2  B1 + B 2   symm.  2  

and for the boundary conditions matrix Kα is: B3 B2   B 2 + B3  3 6 6   B1 + B3 B1  Kα =   3 6   B1 + B 2    symm. 3   B3 B2   B 2 + B3  4 4 4   B1 + B3 B1  +ε  4 4   B1 + B 2    symm. 4  

3

with Bi = (hi − v n ,i ρ a c a )Li and index i ranging

(33)

from 1 to 3. 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).

x

x

N3

N3

N2 3

2

L1

L2

L3

L2

N2

1

2

L1

L3 1

1

(a) a)

(b) b)

Fig. 2: (a) Linear and (b) partly continuous interpolation functions on the element boundary L1

72

(34)


The last term in equation (24) is the boundary conditions vector qα which could be analytically expressed as:  B2T2 + B3T3   B2T2 + B3T3      2 2      B1T1 + B3T3   B1T1 + B3T3  qα =   +ε ⋅ 2 2      B1T1 + B2T2   B1T1 + B2T2      2 2    

(35)

with Bi = (hi − v n ,i ρ a c a )Li and index i ranging from 1 to 3.

6.

NUMERICAL ANALYSIS

STABILITY

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 dimensional problem: d 2T

dT λ 2 − ρ a ca u =0 dx dx

T (1 ) = 0

(37)

The analytical solution of the equation (36) is: T( x ) =

e Bx − e B 1− eB

ρ a ca u λ

x (m) 0,0 0,2 0,4 0,6 0,8 1,0 No. of elements

exact solution T (K) 1,000 0,977 0,926 0,813 0,561 0,000 ---

Temperature numerical solution T1 (K) T2 (K) 1,000 1,000 0,978 0,977 0,928 0,927 0,815 0,814 0,564 0,562 0,000 0,000 20

40

T3 (K) 1,000 0,977 0,926 0,813 0,561 0,000 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 the analysis 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 operator A in equation (20): A=λ

∂2 ∂x

2

+λ

∂2 ∂y

2

− ρ a ca u

∂ ∂ − ρ a ca v ∂x ∂y

(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: qA ⋅ T = 0

(38)

with the value of B defined as: B=

Distance

(36)

with boundary conditions: T( 0 ) = 1 ,

Table 1: Results of the exact and the numerical solution

(39)

The numerical solution of equation (36) for B = 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 the exact solution with increasing number of the finite elements covering the solved area.

(41)

The test for symmetry of the operator qA could be expressed for any two functions γ and ψ as:

∫ψ (qAγ ) dΩ = ∫ γ (qAψ ) dΩ

Ω

(42)

Ω

Now, if we take the operator qA from equation (41), substitute it into equation (42) and integrate the result by parts, we could obtain following equation (b.t. denoting boundary terms):

73


 − λ   Ω − ψ 

∫

∂ψ ∂γ ∂ψ ∂γ ∂γ  ∂q  q q −λ −ψ  ρ a c a uq + λ  ∂x ∂x ∂y ∂y ∂x  ∂x  dΩ + b.t. =  ∂γ  ∂q    ρ a c a vq + λ  ∂y  ∂y  

 ∂γ ∂ψ ∂γ ∂ψ ∂ψ  ∂q  q q −λ −γ  ρ a c a uq + λ  − λ ∂x ∂x ∂y ∂y ∂x  ∂x   dΩ + b.t.   ∂ψ  ∂q  Ω − γ   ρ a c a vq + λ  ∂y  ∂y   

∫

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 the increasing number of finite elements. This shows that the calculated results of equation (20) converge to the exact solution and the numerical stability is reached.

The operator qA will be symmetric and therefore self-adjoint if the equations: ∂q = 0, ∂x ∂q ρ a c a vq + λ =0 ∂y

plasterboard 13 mm mineral wool 120 mm plasterboard 13 mm

ρ a c a uq + λ

1 mm wide crack

(43)

are fulfilled. The solution of equations (43), which is the function q we search for, could be found as: 1m

q=e

−

ρ a ca (ux + vy ) λ

(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:

F=

∫ ( 21 T ⋅ qA ⋅ T )dΩ

(45)

Ω

for the derivation of the functional F corresponding to equation (20). The resulting functional connected to equation (20) could be obtained after some derivations in the following form: F = − 12

  ∂T  2  ∂T qλ   + qλ    ∂x   ∂y Ω

∫

(

)

  

2

dΩ  

(46)

− q(h − v n ρ a c a ) T − TT dΓ

∫

Γ

1 2

2

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:

• •

74

interior: temperature 20 °C, pressure 0 Pa exterior: temperature -15 °C, pressure 10 Pa

Fig. 3: The model building construction for the functional analysis Table 2: Used material characteristics Material

Permeability (m2)

Plasterboard Mineral wool

1.10-12 1.10-9

7.

ANALYSIS OF CONSTRUCTION

Thermal conductivity (Wm-1K-1) 0.220 0.040

A

MODEL

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.


Fig. 4: The initial mesh system with 580 elements

Functional values

0,00065 0,00060 0,00055 0,00050 0,00045 0,00040 580

1160

2320

Number of elements

Fig. 5: The calculated functional values

roof tiles outlet

ventilated air layer (300 mm to 800 mm)

inlet exterior side: • temperature -15 °C • air pressure from 0 to 10 Pa

interior side: • temperature 20 °C • air pressure 0 Pa

mineral wool 160 mm closed air layer 40 mm plasterboard 12 mm

crack of various width 1000 mm

Fig. 6: Cross-section of the analysed construction

75


The convective linear thermal transmittance is always related to the length of the crack and to the operating pressure difference.

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:

ψ v ,∆p =

Φ ti − te

− U c ⋅ lc

The results of the convective linear thermal transmittance calculation for various pressure differences and various widths of the crack in the plasterboard are shown in Table 3 and Fig. 7.

(47)

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.

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: QT =

∑ A ⋅ U ⋅ ∆t + ∑ lv ⋅ψ v ,∆p ⋅ ∆t

(48)

Table 3: The results of the convective linear thermal transmittance analysis no crack 0.000 Wm-1K-1 0.003 Wm-1K-1 0.015 Wm-1K-1 0.031 Wm-1K-1

Width of the crack 1 mm 2 mm 0.000 Wm-1K-1 0.000 Wm-1K-1 0.188 Wm-1K-1 0.231 Wm-1K-1 -1 -1 1.296 Wm K 1.565 Wm-1K-1 -1 -1 2.748 Wm K 3.203 Wm-1K-1

3 mm 0.000 Wm-1K-1 0.247 Wm-1K-1 1.628 Wm-1K-1 3.385 Wm-1K-1

3,50

-1

-1

Convective linear thermal transmittance (Wm K ) The convective linear thermal transmittance [W/mK]

Operating pressure difference 0 Pa 1 Pa 5 Pa 10 Pa

3,00

2,50 no crack

2,00

1 mm wide crack 2 mm wide crack

1,50

3 mm wide crack

1,00

0,50

0,00 0

2

4

6

8

10

Operating pressure difference (Pa) The operating pressure difference [Pa]

Fig. 7: The convective linear thermal transmittance for various widths of the crack

76


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 difference of 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-section leading through the centre of the 1 mm wide crack.

exterior side: temperature -15 °C air pressure 0 Pa

exterior side: temperature -15 °C air pressure 10 Pa

interior side: temperature 20 °C air pressure 0 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

20

15

5

no crack, 0 Pa no crack, 10 Pa 1 mm wide crack, 5 Pa

0

plasterboard

Temperature(o[C] Temperature C)

10

-5

mineral wool

air layer

-10

-15 0

53

106

159

212

Cross-sectionof ofthe theconstruction construction(mm) [mm] Cross-section

Fig. 9: The temperature distribution in the cross-section of the analysed construction

77


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 lightweight construction 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 area of a triangular finite element, m2 one half of the width of the crack, m thermal capacity of the air, 1010 Jkg-1K-1 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, m2 Kλ conductance matrix Kv convective transport matrix Kα 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 qα boundary conditions vector T unknown temperature, K Ti vector of unknown nodal temperature values, K known temperature in the environment in the T 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-1 u velocity vector magnitude, ms-1

A b ca d

78

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-2K-1 U ′ known concentration of the substance at part of the boundary

U

known concentration of the substance in the environment neighbouring with the boundary vn velocity component normal to the boundary, ms-1 v fluid flow velocity in the porous medium,ms-1 Q 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 α boundary transfer coefficient of the substance at the discussed boundary Φ heat flow rate, Wm-1 γ parameter describing the convective properties of the medium Γe boundary of a finite element η parameter describing the diffusive properties of the medium λ thermal conductivity, Wm-1K-1 µ viscosity of the fluid (for air µ = 1.7 x 10-5 Pas), ν kinematic viscosity of the fluid (for air ν = 1.4 x 10-5 m2s-1), Ωe area of a finite element, m2 ρa density of the air, 1.2 kgm-3 Ψv,∆p convective linear thermal transmittance at a given pressure difference, Wm-1K-1 ∂ derivative in the direction of the external ∂n normal to the boundary

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