Effect of moisture transfer across building components on room temperature

Building and Environment 37 (2002) 11–17

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Eect of moisture transfer across building components on room temperature R.C. Gaur, N.K. Bansal ∗ Centre for Energy Studies, Indian Institute of Technology, Hauz Khas, New Delhi-110016, India Received 20 October 1998; received in revised form 2 August 2000; accepted 24 October 2000

Abstract Analytical closed-form solutions for the heat and moisture distribution given by the Luikov system of partial dierential equations have been obtained by using a periodic solution approach. The energy balance of a model room has been considered with dierent thermophysical properties produced by a change in the moisture content, of the spruce walls, along with the eect of vapor diusion. The results show that the humidity present in ambient air and room air aects room temperature and in the hygroscopic region it can alter ◦ the room temperature by 2–3 C depending upon the amount and direction of temperature and moisture gradients, which is in conformity c 2001 Elsevier Science Ltd. All rights reserved. with the earlier results of Hall.  Keywords: Heat and Moisture; Temperature; Building

1. Introduction Substantial work has been done for prediction of thermal environment inside a building considering only the heat transfer [1,2] and for simulation, many programs [3] and codes [4] are available. Literature on moisture transfer in buildings is also available [5 –7] but the study of combined heat and moisture transfer aecting the temperature and humidity of room air is still lacking. Any porous building material consists of a solid matrix containing pores. The pores may contain moisture (water vapor, absorbed or capillary-bound water, and ice) and dry air. In the rainy season the rainwater pouring on the walls and roof penetrates inside the fabric and causes uncomfortable, unhygienic conditions and changes heating=cooling loads. In other days also, the porous building material absorbs the water vapor present in the surrounding air according to its equilibrium sorption moisture content. In turn, the equilibrium sorption relation depends on the characteristics of the porous system, such as the porosity, the speci
work of the body and the material bound within the pores (vapor, gas, and liquid) [9]. The moisture accumulated either by rain or absorbed from the atmosphere is normally transferred from the pores of the building material to the surrounding air by, unsaturated Bow of liquid within the porous solid, the liquid–vapor phase change, and convective diusive transfer of vapor from the surface of the solid to the surroundings [10]. The Bow of moisture and heat is dependent on each other such that the solution to the thermal part of a problem should be accompanied by a simultaneous solution to the moisture part [11]. Luikov [9] has given the governing equations for simultaneous heat and mass (moisture) transfer but the solutions are either numerical or complicated involving complex eigenvalues [12]. In an earlier paper [13] we have used the periodic solutions of the heat and mass (moisture) transfer equations to obtain closed-form solutions for the temperature and moisture distribution subject to the boundary condition that there was no moisture at the boundary in contact with the room air. This condition is not usual; normally in most of the developing countries, there is no vapor barrier, and the moisture diuses through at a rate depending upon the building material and the surrounding conditions. In an earlier paper [14] we have considered those usual boundary conditions and obtained the closed-form solutions for the temperature and moisture distribution across the building fabric considering the periodic variation of ambient temperature on one

c 2001 Elsevier Science Ltd. All rights reserved. 0360-1323/02/$ - see front matter
PII: S 0 3 6 0 - 1 3 2 3 ( 0 0 ) 0 0 0 9 4 - 9

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R.C. Gaur, N.K. Bansal / Building and Environment 37 (2002) 11–17

Nomenclature

ui

aq

ug

coeFcient of thermal diusivity, m2 =s ATl total area of all building elements, m2 Ap area of the element, m2 Ag area of the window, m2 am mass transfer coeFcient for vapor and liquid inside the body, m2 =s C moisture content, kg=kg Cm moisture capacity, kg (moisture)=kg (dry body) M Cq heat capacity of constituent, J=kg K Cr heat capacity of room air, J=kg K g solar gain factor of single glazed window GOC gain due to occupancy, W hins inside surface heat transfer coeFcient, W=m2 K outside surface heat transfer coeFhout cient, W=m2 K I solar radiation, W=m2 K constant coeFcient ( Km =Cq ) moisture conductivity coeFcient, Km Kg=m s M Kq thermal conductivity coeFcient, W=m k L thickness of wall, m ◦ U moisture potential, M Ua moisture potential of ambient air, ◦ M ◦ Ur moisture potential of room air, M N number of air changes per hour ◦ Ta temperature of ambient air, C ◦ temperature of room air, C Tr ◦ solar temperature, C Tsa T time, s

side of the wall and assuming a
V Wr Greek symbols  m out m ins 

  ! Subscripts and Superscripts a avg e ins m n nr o out p q r R S w

heat loss coeFcient of building element, W=m2 K heat loss coeFcient of single glazed window, W=m2 K volume of room, m3 mass of room air, kg absorption coeFcient of outside surface outside convective mass transfer coeFcient, kg=m2 s M inside convective mass transfer coeFcient, kg=m2 s M thermogradient coeFcient, ◦ M=K ratio of vapor diusion coeFcient to total moisture diusion coeFcient heat of phase change, J=kg dry body density, kg=m3 relative humidity, % angular frequency, h−1 ambient air average east inside mass (moisture) transfer harmonic part north average part outside building element heat transfer room roof south west

room air and considering moisture transport properties of the building material.

2. Mathematical modelling Considering homogeneous walls and roof in plane geometry and assuming one-dimensional movement of heat and moisture the Luikov equations as given by Liu and Cheng [12] can be written as follows: Cq

@T @2 T @U = Kq 2 + Cm ; @t @x @t

(1)

R.C. Gaur, N.K. Bansal / Building and Environment 37 (2002) 11–17

Cq

@U @2 U @2 T = Km  2 + Km 2 : @t @x @X

(2)

Consider a room with four walls and roof elements each of thickness L, exposed on one side to ambient conditions, with hourly variable temperature Ta and constant (assumed) relative humidity out . The solar radiation on each wall and roof has been taken into account in the form of soliar temperature of the medium [9], Tsa = Ta + I=hout , where I is the radiant energy Bow absorbed by the surface and hout is the total coeFcient of heat transfer between outside air and the surface of wall=roof. The moisture content C = Cm U , where U is the mass transfer potential (analogues to the heat transfer potential temperature). The mass transfer potential is some function of the moisture content of thermodynamic equilibrium, must be the same in all parts of the body or system of bodies [9]. In a state of equilibrium the moisture content corresponding to relative humidity equal to 100%, is known as the maximum sorptional moisture content or maximum hygroscopic humidity. The maximum hygroscopic moisture content of any body is signi
(3) (4) (5)

where To(x) ; Uo(x) , and Tro are the steady parts and Tn(x) ; Un(x) , and Trn are the harmonic components of the temperature and the moisture. ! is the frequency of variation normally over a 24 h cycle, i.e. ! = 2
=24 h−1 . Eqs. (1) and (2) can be written as @2 T @2 U @T ; = aq 2 + K @t @X @X 2

(6)

@U @2 U @2 T = a m 2 + am  2 ; @t @X @X

(7)

where aq = Kq + Km =Cq ; K = Km =Cq ; am = Km =Cm : The transfer coeFcients are assumed constant initially. Their dependence on moisture and temperature is considered

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during the calculation at each hour. Substituting for T (X; t) and U (X; t) in Eqs. (6) and (7) and separating the steady state and harmonic parts one gets (a) For steady part aq

@2 To @ 2 Uo + K = 0; @X 2 @x2

(8)

am

@2 Uo @2 To + a  = 0: m @X 2 @X 2

(9)

(b) For Buctuating part aq

@2 Tn @ 2 Un + K = in!Tn ; @X 2 @X 2

(10)

am

@2 Un @2 Tn + am  2 = in!Un : 2 @X @X

(11)

The solution of the above equations should satisfy the following boundary conditions for free diusion as given by Luikov: 
@T −Kq out + hout (Tx=0 − Tsa ) @X X =0 +(1 − out ) out m out (Ux=0 − Ua ) = 0; (12)  
@U @T + km out out @X X =0 @X x=0 +m out (Ux=0 − Ua ) = 0;

(13)

 @T + hins (Tr − Tx=L ) @X X =L −(1 − ins ) ins m ins (Ux=L − Ur ) = 0;

(14)

 
@U @T + Km ins ins t @X X =L @X x=L +m ins (Ux=L − Ur ) = 0:

(15)




Km out




−Kq ins




Km ins

The energy balance equation for the room temperature can be written as 5

W r Cr

@Tr  = hins Ap (Tx=L − Tr ) @t p=1

+

5 

(1 − ) Ap m ins (Ux=L − Ur )

p=1

−

5 

up Ap (Tr − Ta ) − ug Ag (Tr − Ta )

p=1

− 0:33NV (Tr − Ta ) + gAg Is + GOC:

(16)

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R.C. Gaur, N.K. Bansal / Building and Environment 37 (2002) 11–17

The solutions of Eqs. (8) and (9) are obtained as

3. Numerical example and discussion

Uo(x) = Fp X + Fp ;

(17)

To(x) = Fp X + Fp ;

(18)

where p = 1; : : : ; 4 for East wall, p = 5; : : : ; 8 for West wall, p = 9; : : : ; 12 for North wall, p = 13; : : : ; 16 for South wall, p = 17; : : : ; 20 for roof. From the separated steady part of boundary conditions (12) – (15) four equations will be available for each building element, giving rise to 20 equations for given surfaces. Eq. (16) combines the eect of heat and moisture transfer from all surfaces to room air temperature. The separated steady part of this equation gives one combined equation including Tro . Solving these 21 equations simultaneously by the matrix solution method, one obtains the values of F1 –F21 . The matrix equation formed is as follows: [F]21×1 = [D]−1 21×21 [E]21×1 ;

(19)

where the elements of the matrices are as de
(21)

Solution of Eq. (21) is found as 

Un(x) =

4  cj exp (j X ) j=1

−in!

The thermo-physical and mass transport properties play a major role in the analysis. These properties are dependent on temperature and moisture contents. For building materials the eect of temperature variation is less signi
(22)

j ’s are the roots given by the following equation: E1 4 + E2 3 + E3 2 + E4  + E5 = 0;

(23)

where E1 =(am K −am aq )=in!; E2 =0:0; E3 =am +aq ; E4 = 0:0, and E5 = −in!. The coeFcient cj s (1,4) are obtained by using the separated Buctuating parts of the boundary conditions (12) – (15) giving rise to 20 equations for
(24)

where the elements of the matrices are de
Fig. 1. Room temperature variation with time, for average outside humidity and without humidity, in June.

R.C. Gaur, N.K. Bansal / Building and Environment 37 (2002) 11–17

Fig. 2. Room temperature variation with time, for other outside humidities, in June.

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Fig. 4. Room temperature variation with time, for other outside humidities, in January.

Fig. 5. Heat Bux through dierent building components with moisture, in June.

Fig. 3. Room temperature variation with time, for average outside humidity and without humidity, in January.

◦

outside (r:h: = 50%; 24:7 M) and compares it with the corresponding case of no humidity. It is clearly shown that consideration of moisture transfer results in higher temperature Buctuations inside the room. Increasing the air changes per hour increases Buctuations of inside temperature further as expected. Fig. 2 plotted for

three air changes shows that the higher dierences in the inside and outside relative humidity increase the temperature Buctuation even more, though a higher humidity inside decreases the peak of temperature slightly. Figs. 3 and 4 plotted for the month of January show that the changes in ventilation rate have a greater eect on temperature variation in winter than in summer. Fig. 4 shows that the moisture dierences between outside and inside increase the temperature Buctuations further. The change in temperature due to moisture eect is due to the combined eect of changed conduction and diusion of vapor. Hall et al. [10] have found that the moisture aects ◦ the heat transfer from walls by 2–3 C, so the above results

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R.C. Gaur, N.K. Bansal / Building and Environment 37 (2002) 11–17

Fig. 6. Heat Bux through dierent building components without moisture, in June.

Fig. 8. Heat Bux through dierent building components without moisture, in January.

groscopic media (true for most building materials) in order to have a true assessment of the room air temperature variations in a building. The periodic solutions provide a fairly good estimate about these variations. Future work would concentrate on the temperature as well as humidity variations of room air inside a building. Appendix The elements of the matrices in Eqs. (19) and (24) are zero except those de
are in good agreement with those of Hall et al. [10]. This analysis has been done for the hygroscopic region only. Similar analysis may be done for moist conditions with modi
D(1; 3) = −Kq out p ; D(1; 4) = hout ; D(2; 3) = Km out p ; D(2; 2) = m out ; D(2; 3) = Km out out p ; D(3; 1) = Rm ins Lp ; D(3; 2) = Rm ins ; D(3; 3) = hins Lp + Kq ins p ; D(3; 4) = hins ; D(3; 21) = −hins ; D(4; 1) = m ins Lp+ Km ins p ; D(4; 2) = m ins ; D(4; 3) = Km ins p + ins p :

4. Conclusion

The above elements are written for p = East wall. On similar lines the elements D(5; 21)–D(8; 21) for West wall, D(9; 21)–D(12; 21) for North wall, D(13; 21)–D(16; 21) for South wall and D(17; 21)–D(20; 21) for roof can be written:

The results presented in this paper show that moisture transfer should necessarily be considered especially in hy-

D(21; 1) = Rm ins Ap Lp ; D(21; 2) = Rm ins Ap ;

R.C. Gaur, N.K. Bansal / Building and Environment 37 (2002) 11–17

D(21; 3) = Ap hins Lp ; D(20; 4) = Ap hins : The above elements are written for p = East wall. On similar lines the elements D(21; 5)–D(21; 8) for West wall, D(21; 9)–D(21; 12) for North wall, D(21; 13)–D(21; 16) for South wall, D(21; 17)–D(21; 20) for roof can be written:

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written: A(21; 21) = (−Wr Cr ln ! − hins ATL − ui ATL −0:33NV − ug Ag )E5 : The elements of matrix B are zero except those de
D(21; 21) = −ATL hins − 0:33NV − ug Ag − uI ATL

B(5) = hout Tsnw E5 ;

The elements of matrix E are as follows:

B(9) = hout Tsnnr E5 ;

E(1) = Rm out Ua + hout Ts op ; E(2) = Ua m out ; E(3) = Rm ins Ur ; E(4) = m ins Ur :

B(21) = (−ui ATL Tan − ug Ag Tan − 0:33NVTan − gAg Isn )E5 ;

The above elements are written for p=East wall. On similar lines E(5)–E(8) for West wall, E(9)–E(12) for North wall, E(13)–E(16) for South wall, and E(17)–E(20) for roof can be written: E(21) = (−0:33NV − ug Ag − uI ATL )Tao −gAg Iso + Rm ins ATL Ur − GOC; where Tso; Tao are the average parts of the solar and ambient temperatures, respectively. Iso is the average part of the solar radiation on South wall. The elements of matrix A are zero except those de
B(13) = hout Tsns E5 ; B(17) = hout TsnR E5 ; where Tan and Ian are the harmonic parts of the ambient temperature and solar radiation (on South wall), respectively. References [1] Markus TA, Moris EN. Buildings, climate and energy. Great Britain: Pitman Publishing Limited, 1980. [2] Sodha MS, Bansal NK, Kumar A, Bansal PK, Malik MA. Solar passive buildings: science and design. Oxford: Pergamon Press, 1986. [3] Milbank NO, Harrington-Lynn. Thermal response and the admittance procedure. Building Service Engineer (J.IHVE) 1974;42:38–51. [4] DIN 4108. Thermal insulation in buildings. Berlin: Beuth Verlag GMbh, 1986 [in German]. [5] Kohonen R. Transient analysis of the thermal and moisture physical behavior of building construction. Building Environment 1984;19: 1–11. [6] Cunnigngham MJ. Modeling of moisture transfer in structures — II. A comparison of a numerical model, an analytical model and some experimental results. Building Environment 1990;25:85–94. [7] Cunningham MJ. Moisture diusion due to periodic moisture and temperature boundary conditions — an approximate steady analytical solution with non-constant diusion coeFcients. Building Environment 1992;27:367–77. [8] Huang CLD, Siang HH, Best CH. Heat and moisture transfer in concrete slabs. International Journal of Heat and Mass Transfer 1979;22:257–66. [9] Luikov AV. Heat mass transfer in capillary-porous bodies. oxford: Pergamon Press, 1966. [10] Hall C, Ho WD, Nixon MR. Water movement in porous building materials — VI. Evaporation and drying in brick and block materials. Building Environment 1984;19:3–20. [11] Cunningham MJ. Modelling of moisture transfer in structures — I. A description of a