Effect of moisture transfer on internal surface temperature

Energy and Buildings 60 (2013) 83–91

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Effect of moisture transfer on internal surface temperature Yanfeng Liu ∗ , Yingying Wang, Dengjia Wang, Jiaping Liu School of Environmental and Municipal Engineering, Xi’an University of Architecture and Technology, No. 13 Yanta Road, Xi’an 710055, China

a r t i c l e

i n f o

Article history: Received 8 June 2012 Received in revised form 15 January 2013 Accepted 23 January 2013 Keywords: Heat transfer Moisture transfer Internal surface temperature Energy conservation

a b s t r a c t In order to study the influence of moisture transfer on the wall internal surface temperature, a coupled heat and moisture transfer transient model was proposed in this paper. Relative humidity and temperature were chosen as the driving potentials. The accuracy of the proposed model was verified by comparing the simulated results with the experimental data. By employing the developed model, the internal surface temperature of walls made of different materials was evaluated under the constant and variable boundary conditions. The results show that the effect of moisture transfer on the wall internal surface temperature is significant. When the moisture transfer is taken into account, the internal surface temperature has cooling effect on the indoor air and is beneficial to improve the indoor thermal environment during the working hours in summer, which lays a scientific basis for low energy building design. © 2013 Elsevier B.V. All rights reserved.

1. Introduction At present, air-conditioning is commonly used to satisfy thermal comfort in buildings in summer all over the world. However, the energy consumption of air-conditioning is very high. Therefore, how to reduce air-conditioning running time is worth researching. When the indoor air temperature keeps at a certain value, the wall internal surface temperature will have a great impact on the thermal comfort. When the wall internal surface temperature is lower than the indoor air temperature, it will have cooling effect. The reduction of the wall internal surface temperature is beneficial to improve the body’s radiation heat transfer and enhance the indoor thermal comfort effectively in summer. Therefore, the cooling effect of the internal surface temperature may reduce the air-conditioning running time, to achieve the goal of energy saving. The wall surface temperature is calculated only by heat transfer theory, and the effect of moisture transfer is usually neglected. In fact, the moisture migration process has a great impact on the heat transfer, especially in humid climates. The periodic changes of the indoor and outdoor air relative humidity can induce moisture adsorption–desorption of the wall surface day and night. When the moisture content of the wall surface is higher than that of the ambient environment, the wall surface will absorb moisture from the ambient environment. Otherwise, the wall surface will release moisture to the ambient environment. The moisture adsorption–desorption process will accompany with the latent heat caused by the moisture composition phase change at the same time. The latent heat has heating or cooling effect on the wall

∗ Corresponding author. Tel.: +86 29 82201514; mobile: +86 13909261178. E-mail addresses: [email protected], [email protected] (Y. Liu). 0378-7788/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.enbuild.2013.01.019

surface under the different conditions and leads the surface temperature to rise or fall. Therefore, the wall surface temperature can not be calculated accurately without taking the moisture transfer into account. According to the analysis above, the complex mechanism of heat and moisture transfer through the porous medium should be understood firstly. The most often used and accepted macroscopic models for studying heat and moisture transfer through a porous medium are the Luikov model [1] or Phillip and de Vries model [2], which use the temperature and moisture content as driving potentials. However, the heat and moisture diffusion coefficients of the coupled equations are difficult to determine and the moisture content is not continuous at the interface between different materials. Pedersen [3] and Künzel [4] used the capillary pressure and the relative humidity as the moisture migration potential, respectively. But they made them impossible to be solved by means of analytical methods. Mikhailov and Ozisik [5] have got the analytical solutions for linear problems, based on the classical integral transform approach. Qin [6] proposed an analytical solution by using Transfer Function Method. However, the properties of the porous materials are not constant and vary greatly with environmental parameters, so the coupled equations are highly non-linear and only can be solved numerically. At present, the numerical solution methods mainly include finite difference method, finite element method, transfer function method, control volume method, boundary element method and effective heat conduction coefficient method and so on. The applicability of numerical solutions is powerful, especially for the problems which are difficult to be solved by analytical methods. As long as the discrete analysis and the solving method are proper, the numerical solutions are accurate. Meng Qinglin has researched the attenuation of outer surface temperature wave and the cooling process caused by passive water

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Y. Liu et al. / Energy and Buildings 60 (2013) 83–91

Nomenclature T Tk ϕ u w Jv Jl  l v C ıv Ml hr hc,p  Dv Dl o eff Pv Pl Pv,sat L(T ) R Rv hc hm  Dϕ 

DT aT aϕ to MRT

temperature (◦ C) thermodynamic temperature (K) relative humidity moisture content in mass (kg/kg) moisture content in volume (kg/m3 ) vapor diffusion flux (kg/m2 s) liquid conduction flux (kg/m2 s) density of material (kg/m3 ) density of water (kg/m3 ) density of water vapor (kg/m3 ) specific heat capacity of material (J/kg K) vapor permeability (kg/Pa m s) molar mass of water (g/mol) radiant heat-transfer coefficient for body (W/m2 K) convective heat transfer coefficient for body (W/m2 K) sorption capacity of the porous material water vapor diffusion coefficient (m2 /s) liquid water transmission coefficient (kg/Pa m s) thermal conductivity (W/m K) effective thermal conductivity (W/m K) vapor pressure (Pa) suction pressure (Pa) saturated vapor pressure (Pa) heat of vaporization (J/kg) universal gas constant (J/kg K) gas constant of water vapor (J/kg K) convective heat transfer coefficient (W/m2 K) convective mass transfer coefficient (m/s) mass transport coefficient associated with a relative humidity gradient (m2 /s) mass transport coefficient associated with a temperature gradient (m2 /s◦ C) heat transport coefficient associated with a temperature gradient (m2 /s) heat transport coefficient associated with a relative humidity gradient (W/m) operative temperature (◦ C) Mean Radiant Temperature(◦ C)

Subscripts 0 indoor e outdoor x=0 internal boundary x=l external boundary

evaporation of building outer surface. The experimental results have proved that the passive evaporative cooling method can reduce the building external surface temperature effectively [7,8]. However, to ensure the better cooling effect, this technology has to provide water source for humid porous medium constantly. Hans Janssen and Staf Roels introduced the production-adaptive characterization of the moisture buffer potential of interior elements. It formed an alternative basis for quantitative evaluation of interior moisture buffering by the effective moisture penetration depth and effective capacitance models [9]. Olalekan F. Osanyintola studied the effect of initial conditions, boundary conditions and thickness on the moisture buffering capacity of plywood. The results showed that the buffering capacity of plywood depended on the initial conditions and thickness of the plywood as well as the surface film coefficient and humidity cycle [10]. They only studied the

moisture buffering capacity of internal materials. The internal surface temperature needs to be researched for further study. In this paper, a coupled heat and moisture transfer transient model is proposed. Relative humidity and temperature are chosen as driving potentials. The heat and moisture diffusion coefficients are simplified as the functions of the relative humidity and temperature. The accuracy of the proposed model is verified by comparing the simulated results with the experimental data from the literature. The effect of the relative humidity and temperature on the heat and moisture transfer coefficients is also analyzed. The influence of the moisture transfer on the internal surface temperature is calculated under the constant and varying boundary conditions. The guiding suggestions are obtained for energy-saving building design at the same time. 2. Theoretical model 2.1. Governing equations 2.1.1. Moisture transfer The total moisture flux consists of vapor flux and liquid flux. More complicated descriptions of the total moisture flow have also been suggested [11,12]. However, the required material parameters are difficult to determine. Furthermore, the vapor flow and liquid flow usually occur in the same direction, and can not be easily separated in an experiment. For the isothermal case, neglecting some thermodiffusion, a simple description of the total moisture flow Jm proposed by Nilsson [13] is adopted in this research. Jm = Jv + Jl = −ıv

∂Pv ∂P ∂Pv  ∂Pv − Dl l = −(ıv + ıl ) = −ıv ∂x ∂x ∂x ∂x

(1)

where Jm is total moisture flux, kg/m2 s; Jv is vapor diffusion flux, kg/m2 s; Jl is liquid conduction flux, kg/m2 s; ıv is the vapor permeability, kg/Pa m s; Dl is the liquid water transmission coefficient, kg/Pa m s; Pv and Pl are the vapor pressure and the suction pres sure, Pa; ıv is the equivalent total moisture diffusion coefficient. The suction pressure can be expressed as a function of both relative humidity and temperature by using the Kelvin’s equation,  RT Pl = lM k ln(Pv /Pv,sat )where Tk is the thermodynamic temperature, l

K; l is the density of water, kg/m3 ; Ml is the molar mass of water, g/mol; R is the universal gas constant, J/kg K; Pv,sat is the saturated vapor pressure, Pa. Since the main gradients for both temperature and relative humidity are on the x-axis direction, the 1-D modeling hypothesis is considered in this paper. Based on the Philip and De Vries theory [2], the coupled heat and moisture transfer equations are given. The moisture composition equilibrium equation in the porous material can be expressed by: ∂w + ∇ (Jv + Jl ) = 0 ∂t Or ∂w ∂ = ∂t ∂x





ıv

∂Pv ∂x

(2)

 (3)

where w is the volumetric moisture content, kg/m3 . Theoretically, all the moisture component parameters can be used as the moisture driving potential. For example, moisture content, vapor pressure and absolute humidity, and so on. As shown in Eq. (3), w is used as storage quantity and Pv is used as the only driving potential according to Eq. (1). For convenience of calculations, w and Pv can be expressed in terms of relative humidity. The relative humidity is continuous at the interface of different materials having different moisture storage properties. According to Eq.

Y. Liu et al. / Energy and Buildings 60 (2013) 83–91

(4), ∂Pv /∂x can be transformed a function of relative humidity and temperature. According to the definition of the slope of the moisture retention curve, ∂w/∂t in Eq. (3) can be transformed (∂ϕ/∂t) in Eq. (6). Therefore, relative humidity is chosen as the moisture driving potential. The vapor pressure gradient ∂Pv /∂x is expressed as a function of both relative humidity and saturated vapor pressure as follows: ∂Pv,sat ∂Pv,sat ∂T ∂(ϕPv,sat ) ∂ϕ ∂ϕ ∂Pv = =ϕ + Pv,sat =ϕ + Pv,sat ∂x ∂x ∂x ∂x ∂T ∂x ∂x (4) where T is the temperature, ◦ C; Pv,sat is expressed as a function of temperature [14]: Pv,sat (T ) = 610.5 exp

 17.269T 

(5)

237.3 + T

Therefore, substituting the above equations into Eq. (3), the moisture transfer process can be expressed by: ∂ϕ ∂  = ∂t ∂x 

∂ϕ ∂ = ∂t ∂x

  

ıv

 Dϕ

∂Pv,sat ∂T ∂ϕ ϕ + Pv,sat ∂T ∂x ∂x

∂ϕ ∂x

 +

∂ ∂x

 DT

∂T ∂x



(6)

 (7)

where  = ∂w/∂ϕ = (∂u/∂ϕ) is the slope of the moisture retention curve; u is the qualitative moisture content, kg/kg. 



Dϕ = ıv Pv,sat and DT = ıv ϕ

∂Pv,sat ∂T

(8)

C





(9)

 is the mass transport coefficient associated with a relawhere Dϕ tive humidity gradient, m2 /s; DT is the mass transport coefficient associated with a temperature gradient, m2 /s◦ C. They are functions of both relative humidity and temperature. Thus, the mass conservation equation can be written as:

∂ϕ ∂ = ∂t ∂x



 Dϕ

∂ϕ ∂x



+

∂ ∂x



DT

∂T ∂x



∂T ∂ C = ∂t ∂x



∂T o ∂x



where C is the specific heat capacity of material, J/kg K; L(T) is the heat of vaporization, J/kg; o is the thermal conductivity, W/m K. The last term represents the heat sources or heat sinks due to liquidto-vapor phase change and to the adsorption or desorption process. According to the equations above, the total moisture flux Jm can be expressed by

 

Jm = −

ıv

∂Pv,sat ∂T ∂ϕ ϕ + Pv,sat ∂T ∂x ∂x



(12)

Substituting Eq. (12) into Eq. (11), C

∂T ∂ = ∂t ∂x



o

∂T ∂x



+ L(T )

∂ ∂x

  ıv

ϕ

∂Pv,sat ∂T ∂ϕ + Pv,sat ∂T ∂x ∂x

 (13)

+ L(T )

∂ ∂x

 ıv Pv,sat

∂ϕ ∂x

 (14)

aϕ =

L(T )ıv Pv,sat C

(16)

where aT is the heat transport coefficient associated with a temperature gradient, m2 /s; aϕ is the heat transport coefficient associated with a relative humidity gradient, W/m. Thus, the energy conservation equation can be written as: ∂ ∂T = ∂t ∂x



aT

∂T ∂x



+

∂ ∂x



aϕ

∂ϕ ∂x



(17)

2.2. Boundary conditions At the two sides of the wall (x = 0 and x = l), the heat and moisture diffusion which are caused by both relative humidity gradient and temperature gradient affect the heat and mass balance. At the interfaces between two materials inside the wall, the distribution of the temperature and relative humidity are continuous. According to the analysis above, the boundary conditions are given as follows: For the internal surface (x = 0):



∂ϕ ∂T −Dϕ − DT ∂x ∂x

 −

eff

   

= hm0 (v,x=0 − v,0 )



−

(18)

x=0

∂T ∂ϕ + L(T )ıv Pv,sat ∂x ∂x

   

x=0

= hc0 (Tx=0 − T0 ) + L(T )hm0 (v,x=0 − v,0 )

(19)

For the external surface (x = l): ∂ϕ ∂T − DT ∂x ∂x



(11)



(15)

(10)

∂Jm − L(T ) ∂x

∂T ∂x

o + L(T )ıv ϕ(∂Pv,sat /∂T ) C

−Dϕ

2.1.2. Heat transfer The phase change occurring within porous materials acts as a heat source or sink, which results in the coupled relationship between moisture transfer and heat transfer. The energy conservation equation in porous materials can be given by:

eff

aT =



Dϕ ı Pv,sat ı ϕ(∂Pv,sat /∂T ) DT  = v = v and DT = (∂u/∂ϕ) (∂u/∂ϕ)  



where eff is the effective thermal conductivity, W/m K. eff = o + L(T )ıv ϕ(∂Pv,sat /∂T ). We define that:

In order to make the equations form looks more concise, we define that: Dϕ =

∂ ∂T = ∂t ∂x

85

eff

   

= hme (v,e − v,x=l )

(20)

x=l

∂T ∂ϕ + L(T )ıv Pv,sat ∂x ∂x

   

x=l

= hce (Te − Tx=l ) + L(T )hme (v,e − v,x=l )

(21)

The external and internal surface convective heat transfer coefficients hc are 23 and 8.7 W/m2 K, respectively. The convective mass transfer coefficients hm are related to hc by the Lewis’ relation [15]. hm =

hc a Cp

(22)

Eqs. (18) and (20) express the moisture balance at the surfaces (x = 0 and x = l). The two terms on the left-hand side of the equal sign describe the supply of the moisture flux under the influence of a temperature gradient and a relative humidity gradient. The terms on the right side of the equal sign describe the moisture amount drawn off from (or into) the surfaces. Eqs. (19) and (21) express the heat flux in terms of convection heat transfer and latent heat caused by phase change at the surfaces (x = 0 and x = l). The two terms on the left-hand side of the equal sign describe the supply of the heat flux under the influence of a temperature gradient and a relative humidity gradient.

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Y. Liu et al. / Energy and Buildings 60 (2013) 83–91

66

(b) Experiment value x=9mm Experiment value x=18mm Simulation value x=9mm Simulation value x=18mm

62 60

23.30 23.25

Temperature (

Relative humidity (%)

64

)

(a)

58 56 54

23.20 23.15 23.10

Experiment value x=9mm Experiment value x=18mm Simulation value x=9mm Simulation value x=18mm

23.05

52 23.00

50 48

22.95 0

4

8

12

16

20

24

28

32

36

40

44

48

0

4

8

12

16

Time (h)

20

24

28

32

36

40

44

48

Time (h)

Fig. 1. Measured and simulated relative humidity (a) and temperature (b) within the spruce plywood at x = 9 mm and x = 18 mm.

v =

ϕPv,sat Pv = (kg/m3 ) Rv Tk Rv (T + 273)

(23)

where the heat of vaporization L(T) can be expressed by [17]: L(T ) = (2500 − 2.4T ) × 103 (J/kg)

(24)

3. Validation of the numerical model In order to verify whether the relative humidity used as moisture driving potential is correct or not, the simulated results calculated by proposed mathematical model in this paper and the experimental values are presented in Fig. 1. All experimental data and material parameters are from literature [18] and [19]. The properties for dry spruce plywood are: the density 0 = 445 (kg/m3 ), the thermal conductivity  = 0.082 (W/m K), the specific heat c = 1880 (J/kg K), the vapor permeability ıv = 0.5

(−2.3573 × 10−25 + (−8.1601 × 10−24 ϕ/(ln ϕ))) (kg/m s Pa). The physical properties values are used in the numerical calculation. The measured spruce plywood consists of three pieces of plywood which are held together with nylon screws to reduce the air gap between the pieces. The spruce plywood is 600 mm long, 280 mm wide and 27 mm deep. There are five sensors and one sensor at x = 9 mm and x = 18 mm, respectively. The spruce plywood is initially conditioned to equilibrium with the air relative humidity at 50% and the temperature at 23 ◦ C, the external air relative humidity is 85% and the temperature is 23.1 ◦ C. The simulated results and measured values are shown in Fig. 1. The agreement between them is very good. 4. Influence of temperature and relative humidity on the heat and moisture transport coefficient Fig. 2 shows the moisture retention curves of brick, pine board and concrete. For brick and pine board, the slope of the moisture retention curve does not change significantly when the relative humidity is at between 0 and 70%. As seen in Fig. 2, the moisture retention curves become steep at the end. For concrete, the slope of the moisture retention curve changes obviously under the different relative humidity. The moisture retention curves are obtained by testing the equilibrium moisture content of materials under the different relative humidity environment. The testing points may be several cases, for example, at the relative humidity of 10%, 30%, 70%, 95% and so on. The testing results are fitted by using the nonlinear least squares method, the function relationship between moisture

content and relative humidity is obtained at the same time. When the relative humidity is at 100%, the moisture content of material can be calculated by function. In this section, in order to evaluate the influence of temperature and relative humidity on the heat and moisture transport coefficients, the pine board is chosen as the study object. Figs. 3–6 show that the heat and moisture diffusion coefficients change with the changes of both relative humidity and temperature.  increases with the increasing of temperaAs shown in Fig. 3, Dϕ  will increase obviously. The value ture. Especially at above 20 ◦ C, Dϕ  of Dϕ is nearly the same when the temperature keeps at between  0 ◦ C and 20 ◦ C. In a given temperature condition, the change of Dϕ is only related with the slope of the moisture retention curve, so  changes uncertainly. At 50% and 100% relative humidity, D will Dϕ ϕ reach the largest value and the smallest value, respectively. At 100%  is the slowest with the change relative humidity, the change of Dϕ of temperature. Fig. 4 shows that DT increases with the increasing of temperature. In the case of different temperature, DT is largest at about 70% relative humidity. When not considering the influence of temperature and relative humidity on aT , aT is 2.31 × 10−7 m2 /s. Fig. 5 shows that aT increases with the increasing of both relative humidity and temperature. The influence of relative humidity on aT will be small when the temperature is lower than 20 ◦ C. The higher the temperature is, the more obvious the change of aT is with the change of relative 0.30

Equilibrium moisture content (kg/kg)

The water-vapor density can also be expressed as a function of both relative humidity and temperature [16]:

0.25

Brick Pine board Concrete

0.20

0.15

0.10

0.05

0.00 0.0

0.2

0.4

0.6

0.8

1.0

Relative humidity Fig. 2. Sorption isotherms curves for brick, pine board and concrete.

Y. Liu et al. / Energy and Buildings 60 (2013) 83–91

87

3.0

7

1.0

RH=10% RH=30% RH=50% RH=80% RH=100%

2

2.0

-8

-7

4

1.5

3

' D

10 ( m /s)

5

D'

2.5

T=10 T=20 T=40 T=60

2

10 ( m /s)

6

2 0.5

1 0 0.0

0.2

0.4

0.6

0.8

0.0

1.0

0

20

40

60

80

100

80

100

Temperature ( )

Relative humidity 

Fig. 3. Dϕ changes with relative humidity and temperature.

11

2.8

10

T=10 T=20 T=40 T=60

)

-10

1.2

5

D D

4 3 2

0.4

1

0.0 -0.4 0.0

7 6

0.8

RH=10% RH=30% RH=50% RH=80% RH=100%

8

2

2

10 ( m /s

1.6

10 ( m /s

)

2.0

9

-10

2.4

0 -1 0.2

0.4

0.6

0.8

1.0

0

20

40

60

Temperature ( )

Relative humidity 

Fig. 4. DT changes with relative humidity and temperature.

humidity. Therefore, in order to calculate accurately especially at high temperature circumstance, the influence of temperature and relative humidity on the equation coefficients must be taken into account. Fig. 6 shows that the change of aϕ is only related with temperature. The higher the temperature is, the larger the value of aϕ is.

5. Influence of moisture transfer on internal surface temperature 5.1. Using constant conditions for air temperature and relative humidity In this section we evaluate the influence of moisture transfer on the internal surface temperature under the constant boundary conditions, with the goal of simplifying the heat transfer equation by setting aϕ = 0. Three monolithic walls made of pine board, concrete and brick with the thickness of 10 cm were studied, respectively. The basic material properties are given in Table 1. The indoor and outdoor air temperature was kept at 25 ◦ C and 35 ◦ C. The indoor air relative humidity was maintained at 50%. Case 0 was the simulation case without taking moisture transfer into account. To obtain the different relative humidity gradients, the outdoor air relative humidity of case 1, 2, 3, 4, 5 were assumed at 90%, 80%, 70%, 60% and 50%, respectively.

Table 1 Basic material properties for pine board, concrete and brick.

Pine board Concrete Brick

/kg m−3

/W (m K) −1

c/J (kg K) −1

ıv /kg (Pa m2 s) −1

500 1800 1800

0.29 0.814 0.93

2510 879 1050

4.7 × 10−11 4.8 × 10−12 2.9 × 10−11

Table 2 shows that the internal surface temperature of case 0 is highest for three walls. The larger the difference value between outdoor and indoor air moisture content is, the more the moisture flux is. Therefore, the internal surface temperature will be reduced more obviously due to the moisture evaporation at the internal surface. For the concrete, the internal surface temperature difference is very small for all cases due to its smaller vapor permeability. Compared to the case 0, the reducing maximum values of the internal Table 2 The internal surface temperature for different cases. Pine board

Case 0 Case 1 Case 2 Case 3 Case 4 Case 5

Concrete

Brick

aϕ = / 0

aϕ = 0

aϕ = / 0

aϕ = 0

aϕ = / 0

aϕ = 0

27.3 25.9 26.2 26.5 26.8 27.2

25.8 26.1 26.5 26.8 27.1

29.1 28.9 29 29 29 29.1

28.9 29 29 29 29.1

29.3 28.6 28.7 28.9 29 29.2

28.5 28.7 28.8 29 29.1

88

Y. Liu et al. / Energy and Buildings 60 (2013) 83–91

3.2

5.6

3.1

5.2

T=10 T=20 T=40 T=60

4.4

3.2

4.0

-7

2.6 2.5

3.6

2.4

2.8

2.3

2.4

2.2 0.0

RH=10% RH=30% RH=50% RH=80% RH=100%

2

10 ( m /s)

2.7

4.8

aT

2.8

aT

2

2.9

-7

10 ( m /s)

3.0

0.2

0.4

0.6

0.8

2.0 0

1.0

20

Relative humidity

40

60

80

100

Temperature ( )

9

1.8

8

1.6

7

1.2 1.0

RH=10% RH=30% RH=50% RH=80% RH=100%

6 5

-6

T=10 T=20 T=40 T=60

10

1.4

(W/m)

2.0

4

a

10

-6

(W/m)

Fig. 5. aT changes with relative humidity and temperature.

3

a

0.8 0.6 0.4

2

0.2

1

0.0 0.0

0 0.2

0.4

0.6

0.8

1.0

0

20

40

60

80

100

Temperature ( )

Relative humidity

Fig. 6. aϕ changes with relative humidity and temperature.

surface temperature for pine board, concrete and brick are 1.4 ◦ C, 0.2 ◦ C and 0.7 ◦ C, respectively. The internal surface temperatures of case 5 for three walls are slightly lower than those of case 0 due to the less moisture evaporation amount. As shown in Table 2, whether the heat transport coefficient associated with a relative humidity gradient equals to 0 or not, the maximum difference value of temperature is 0.1 ◦ C. Therefore, the Eq. (17) can be simplified as ∂T/∂t = ∂/∂x(aT (∂T/∂x)) under the constant boundary condition. As shown in Table 3, QL is the latent heat flux induced by moisture composition phase change, QT is the total heat flux (latent heat flux and sensible heat flux induced by temperature difference). The more the moisture flux is, the larger the latent heat factor QL /QT is. Table 3 shows that the maximum latent heat factors for pine board, concrete and brick are 72.9%, 8.1% and 35.6% for case 1, respectively. Though the internal surface temperature difference between case 0 and 5 is small for three walls, the latent heat factor of pine board and brick wall can still reach 11.8% and 9.1%. Therefore, to ensure the Table 3 The latent heat factor (QL /QT ) of three walls for different cases.

Case 1 Case 2 Case 3 Case 4 Case 5

Pine board

Concrete

Brick

72.9% 61.6% 48.6% 33.6% 11.8%

8.1% 4.9% 3.9% 3.1% 0

35.6% 30.8% 23.6% 17.7% 9.1%

accuracy of calculating conduction load, the latent heat flux can not be ignored, especially for materials with the stronger performance of moisture adsorption–desorption. 5.2. Using varying boundary conditions for outdoor and indoor air 5.2.1. Using cosine functions for air temperature and relative humidity In order to calculate the proposed models under the periodic boundary conditions, cosine functions are introduced for temperature and for relative humidity. 5.2.2. Influence of outdoor relative humidity on internal surface temperature In this section, we evaluate the influence of the outdoor relative humidity on the internal surface temperature. Two walls made of pine board and brick with the thickness of 10 cm were studied. The outdoor and indoor temperature was kept at 35 ◦ C and 28 ◦ C. The indoor relative humidity was assumed at 70% for humid climate and 40% for dry climate. The outdoor relative humidity was assumed for three cases. Case 1 and case 2 stood for hot humid climate with different relative humidity amplitude variation. Case 3 stood for hot dry climate with smaller relative humidity amplitude variation. Case 1: ϕe (t) = 0.7 − 0.2 cos

 t 12



+ 2.6

(a)

2.0

36

1.5

0.0

89

Case 1 Internal surface Case 2 Internal surface Case 3 Internal surface Case 1 External surface Case 2 External surface Case 3 External surface

2

37

10 ( kg/( m ·s) )

(a)

Moisture flux

Y. Liu et al. / Energy and Buildings 60 (2013) 83–91

34

-5

Temperature ( )

35

33 32

Case 1 Internal surface Case 2 Internal surface Case 3 Internal surface

31

Case 1 External surface Case 2 External surface Case 3 External surface

1.0 0.5

-0.5 -1.0

30 -1.5 0

29 0

2

4

6

8

10

12

14

16

18

20

4

6

4

3

2

-5

33

Case 1 Internal surface Case 2 Internal surface Case 3 Internal surface

Case 1 External surface Case 2 External surface Case 3 External surface

31

Moisture flux

o

Temperature ( C)

34

10

12

14

16

18

20

22

18

20

22

Case 1 Internal surface Case 2 Internal surface Case 3 Internal surface Case 1 External surface Case 2 External surface Case 3 External surface

2

(b) 10 ( kg/( m ·s) )

35

32

8

Time (h)

Time (h)

(b)

2

22

1

0

-1

-2 0

2

4

6

8

10

12

14

16

Time (h)

30 0

2

4

6

8

10

12

14

16

18

20

22

Time (h) Fig. 7. The external and internal surface temperature of pine wood (a) and brick (b) for three cases.

 t  Case 2: ϕe (t) = 0.7 − 0.1 cos 12 + 2.6  t  Case 3: ϕe (t) = 0.4 − 0.05 cos

12

+ 2.6

As shown in Fig. 7, the indoor and outdoor temperature and indoor relative humidity are constant. Therefore, the changes of the external and internal surface temperature are caused by the outdoor air relative humidity. When the outdoor air relative humidity changes periodically, the moisture exchange will exist at the interface between the outdoor air and external surface. The external surface temperature will change periodically at the same time due to the heat absorption–desorption caused by moisture absorption–desorption. For the internal surface, when the moisture-desorption amount increases, the surface temperature should be reduced, but the fact is opposite. The reason is that: the internal surface moisture-desorption amount caused by the change of the outdoor relative humidity is small, and the influence on the internal surface temperature can be ignored. When the moisture transfer is not taken into account, the internal surface temperatures of pine board and brick walls are 29.6 ◦ C and 31 ◦ C, the external surface temperatures are 34.4 ◦ C and 33.9 ◦ C. Fig. 7 (a) shows that the time of the external surface temperature below 34.4 ◦ C accounts for about 47% of the whole cycle for three cases. Fig. 7 (b) shows that the time of the external surface temperature below 33.9 ◦ C accounts for about 54% of the whole cycle for case 1, case 3 and 42% for case 2. As shown in Fig. 8, the negative values represent that the moisture migrates from surface to surrounding environment and the

Fig. 8. The external and internal surface moisture flux of pine wood (a) and brick (b) for three cases.

positive value is opposite, the moisture flux of the internal surface is far less than that of the external surface. The internal surface always releases moisture, while the external surface absorbs or releases moisture alternately during the whole cycle. When the moistureabsorption amount of the external surface reduces gradually, the internal surface moisture-desorption amount through the wall will reduce also. In conclusion, the change of outdoor air relative humidity has a great effect on the external surface temperature and has smaller effect on the internal surface moisture flux. The change of internal surface temperature caused by moisture composition phase change can be ignored and is mainly affected by the external surface temperature. 5.2.3. Influence of indoor relative humidity change on internal surface temperature When the humidity level of the internal surface is higher or lower than that of the indoor air, the internal surface will absorb or release moisture from or to the indoor air. Meanwhile, the moisture adsorption–desorption process always accompanies phase change latent heat and affects the internal surface temperature. In this section we evaluate the effect of heat and moisture transfer on the internal surface temperature under periodic boundary conditions in different climate areas. We assumed five cases for outdoor and indoor air relative humidity as shown in Table 4. The outdoor and indoor air temperature can be expressed by cosine functions as follows: Outdoor temperature: Te (t) = 30 + 3 cos

 t 12



− 161

90

Y. Liu et al. / Energy and Buildings 60 (2013) 83–91

Table 4 Outdoor and indoor air relative humidity for different cases. Outdoor relative humidity

Indoor relative humidity

ϕe (t) = 0.7 − 0.1 cos

ϕ0 (t) = 0.7 − 0.05 cos

 t  + 2.6  12  t ϕe (t) = 0.7 − 0.1 cos 12  t + 2.6 ϕe (t) = 0.7 − 0.1 cos 12 + 2.6  t  ϕe (t) = 0.4 − 0.1 cos 12  t + 2.6

Case 1 Case 2 Case 3 Case 4

ϕe (t) = 0.4 − 0.1 cos

Case 5

Indoor temperature: T0 (t) = 26 + 2.5 cos

12

 t 12

 t  + 2.6  12  t ϕ0 (t) = 0.7 − 0.15 cos 12  t + 2.6 ϕ0 (t) = 0.7 − 0.25 cos 12 + 2.6  t  ϕ0 (t) = 0.4 − 0.05 cos 12  t + 2.6

+ 2.6

ϕ0 (t) = 0.4 − 0.1 cos



− 161

Case 0 is the simulation case without taking moisture transfer into account. Cases 1, 2 and 3 represent humid climates with same outdoor air relative humidity and different indoor air relative humidity variation amplitudes. Case 4 and 5 represent dry climates. The internal surface moisture flux caused by the change of outdoor air relative humidity is small as seen in Section 5.2.2, so internal surface moisture amount is mainly affected by the change of indoor air relative humidity. As shown in Fig. 9 (a), when the moisture transfer is not taken into account, the internal surface temperature is always higher than indoor temperature and has heating effect on indoor air. However, the internal surface temperature is lower than the indoor temperature in a period of time with taking moisture transfer into account. The larger the variation amplitude of indoor relative humidity is,

(a)

29

Temperature ( )

28

27

26

Indoor Case 0 Case 1 Case 2 Case 3 Case 4 Case 5

25

24

12

+ 2.6

the more the moisture absorption–desorption amount of internal surface is. Though the outdoor and indoor air relative humidity of case 1 and case 4 are different, the internal surface temperature changes are similar. The reason is that the indoor air relative humidity variation amplitudes of them are the same and the moisture adsorption–desorption amount of the internal surface is similar. The internal surface temperatures have cooling effect on the indoor air from 10:00 to 17:00 for case 2 and case 5, and from 8:00 to 18:00 for case 3. Moreover, this period of time coincides with the working hours, so the internal surface temperature is beneficial to improve the indoor thermal environment and enhance the thermal comfort. Therefore, people may still feel comfortable without air-conditioning and the running time of air-conditioning may be reduced at the same time. The internal surface temperatures of case 2, case 3 and case 5 are higher than that of case 0 at night. However, people sleep or rest just in this time and have lower requirements on the thermal comfort. Fig. 9 (b) shows that the internal surface temperature is always higher than indoor temperature without taking moisture transfer into account and has heating effect on indoor air. Only the internal surface temperature of case 4 has cooling effect on the indoor air from 9:00 to 17:00 with taking moisture transfer into account. The internal surface temperatures of other cases are lower than that of case 0 in a period of time, but it still has heating effect on indoor air. In order to claim how much the energy consumption of an air conditioning can be reduced with taking moisture transfer into account. A practical example of case 0 and case 3 for pine board house is taken. Under the condition of the natural ventilation, the people will feel comfortable when the operative temperature to at about 26 ◦ C.

23 0

2

4

6

8

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12

14

16

18

20

22

Time (h)

(b)

30 29

Temperature ( )

28 27

Indoor Case 0 Case 1 Case 2 Case 3 Case 4 Case 5

26 25 24

indoor temperature MRT for case 3 operative temperature for case 3 MRT for case 0 operative temperature for case 0

23 0

2

4

6

8

10

12

14

16

18

20

22

Time (h) Fig. 9. The internal surface temperature of pine wood (a) and brick (b) for different cases.

Fig. 10. Profile of operative temperature of case0 and case3.

Y. Liu et al. / Energy and Buildings 60 (2013) 83–91

The formula of operative temperature to is [20] to =

hr MRT + hc,p T0 hr + hc,p

91

Acknowledgement (25)

Where MRT is mean radiant temperature, ◦ C. hr is radiant heattransfer coefficient for body, W/m2 K, with values of 5.43 in winter and 5.8 in summer [21]. hc,p is convective heat transfer coefficient for body, W/m2 K, with values of 3.73 in winter and 2.64 in summer [22]. As seen in Fig. 10, the time range that the operative temperature of case 3 lower than 26 ◦ C is from 8:00 to 12:00, so the indoor temperature can meet the thermal comfort of the human body without air conditioning. However, the operative temperature of case 0 is always higher than 26 ◦ C and an air conditioning is needed. Assuming that the working hours is from 8:00 to 18:00. Therefore, the energy consumption of an air conditioning can be reduced about 40% for case 3. 6. Conclusions With an aim at the effect of the moisture transfer on the internal surface temperature, the coupled heat and moisture transfer model through the walls was developed. Relative humidity and temperature were chosen as the driving potentials. The effects of both relative humidity and temperature on the heat and moisture transport coefficients are also discussed. At a given relative humidity, the heat and moisture transport coefficients can be considered as constant value when the temperature keeps at between 0 ◦ C and 20 ◦ C. However, in order to calculate accurately especially at high temperature circumstance, the influence of temperature and relative humidity on the equation coefficients must be taken into account. The coupled heat and moisture transfer through walls under the constant and the periodic boundary conditions were simulated, and the influence of moisture transfer on the internal surface temperature was also analyzed. The following conclusions are drawn. 1) The effect of relative humidity gradient on the heat transfer process can be ignored under the constant boundary condition. To ensure the accuracy of calculating conduction load, the latent heat flux can not be ignored. 2) The influence of the change of outdoor air relative humidity on the internal surface temperature is small and can be ignored. 3) For the periodic boundary condition of indoor air relative humidity, the internal surface temperature of the wall has cooling effect on indoor air during working hours and is beneficial to improve the indoor thermal environment and enhance the thermal comfort. 4) The running time of air-conditioning can be reduced with taking moisture transfer into account. The energy consumption of an air conditioning can be reduced about 40% for case 3 of pine board house.

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