Predicting heat and moisture transfer in buildings: a comparison ...

DRAFT A comparison exercise for calculating heat and moisture transfers using TRNSYS and PowerDomus M. Abadie, J.P. Deblois and N. Mendes Pontifical Catholic University of Paraná – PUC/CCET - Curitiba – PR – 80215-901 – Brazil Many of the now well-known building energy simulation programs use the response factor method developed in the early 70’s by Stephenson and Mitalas. These are TRNSYS, EnergyPlus, Blast and DOE-2, to name a few of them. Others, such as PowerDomus, ESP-r and BSim, perform finite-volume or finite-difference calculations to solve the heat and mass transfer through the building envelope. These two different approaches are known to have strength and weakness. The main objective of the present exercise is to compare the prediction of both methods using TRNSYS 15.3 (2003) and PowerDomus 1.5 (2005). A two-step procedure is employed here. The first deals with pure thermal problem i.e. without moisture calculation. Three different cases of increasing complexity are studied and compared to analytical solutions. The second step focus on the moisture problem alone by comparing the Buffer Storage Humidity Model used in TRNSYS to Umidus model implemented in PowerDomus. Results show that time step values are determinant even for pure thermal cases where the classical value of 1 hour can lead to notable errors. For problems with moisture sorption in the wall, its value has to be set to unusual small values to achieve a good response. Keywords: heat, moisture, response factor, finite-difference, TRNSYS, PowerDomus.

1. Introduction The use of building energy analysis computer programs is becoming more and more prevalent in the field of research. This increased adoption of these simulation tools places an even greater demand on building energy simulation software developers to ensure that the simulation models that they develop are accurate, and that the implementation of these models within simulation engines is correct. As stated by Judkoff (1988), there are three ways to evaluate the accuracy of these simulation programs: empirical validation in which calculated results are compared to monitored data from a real building, test cell such as PASSYS (1993) and ETNA (1999), analytical verification in which outputs from a program are compared to results from a known analytical solution (ASHRAE Test Suite, 2001), and comparative testing in which a program is compared to itself or to other programs such as BESTEST (1995), HVAC BESTEST (2002) and HAMSTAD (2002). The present study lies on the second way of evaluation. We present a comparison exercise between TRNSYS 15.3 (2003) and PowerDomus 1.5 (2005). This exercise aims to compare these two softwares with analytical solution for very simple and highly constrained boundary conditions of heat and moisture transfers through walls. In a first part, the main theory and methodology of both softwares are summed-up. Then three cases of heat transfer and two cases of moisture transfer are described. Results are presented and discussed in a third part.

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DRAFT

2. Theory and methodology involved in TRNSYS and PowerDomus The classical partial equation describing conduction heat transfer based on Fourier’s law and its general boundaries conditions are given by equations (1) and (2).

∂ 2T

1 ∂T ∂x 2 α ∂t ∂T at the wall boundaries q& = −k ∂x =

(1) (2)

In building simulation software, this equation needs to be solved in conjunction with many building process, such as heating, ventilation and air conditioning among others. In sake of clarity, only the solution of heat transfers though the building envelope is presented here. Specific procedures are briefly introduced in each numerical method.

2.1.

TRNSYS

TRNSYS heat transfer calculations are based on the transfer function methodology that has been introduced by Stephenson and Mitalas (1971). The transfer function links the present surface heat flux to the past and present surface temperature and heat flux. The relationships of Mitalas and Arseneault (1971) are used in TRNSYS:

q& s,i =

n bS

∑ bsk Tsk,o −

k =0

q& s, o =

n aS

∑

a sk Tsk, o k =0

n cS

n dS

∑ csk Tsk,i − ∑ d sk q& sk,i

(3)

n dS k k − b s Ts, i − d sk q& sk, o k =1 k =0

(4)

k =0 n bS

∑

k =1

∑

Where k refers to the term in the time series (current time k=0, previous time k=1…). The time base on which these calculations are based is specified by the user. The time base value has to be lower than the thermal time-constant of the wall and higher than the transfer function algorithm stability criteria. Typical values for building walls are within 0.25 to 2 hours. The coefficients of the time series (a, b, c and d) are determined from the material layers properties of the wall (i.e. thermal conductivity, capacity, density and thickness) using the z-transfer function algorithm presented in Mitalas and Arseneault (1971). A comparison of other methods to find these coefficients can be found in Iu and Fisher (2004). It should be pointed out that in TRNSYS, the calculation of these coefficients is performed only once at the beginning of the simulation. As a consequence, variable material properties induced for example by variable moisture content in the material, cannot be taken into account in the heat transfer calculation. As already stated, heat and moisture transfers within the building envelope are decoupled in TRNSYS but moisture influence on the energy balance of the zone air can be performed via two models. The first one considers sorption effects at walls with an enlarged moisture capacity of the zone. The main problem with this model is to correctly evaluate the new air capacity of the zone air based on the materials that composed the envelope. The second, more sophisticated, model is the so-called Buffer Storage Humidity Model. This model describes a separate humidity buffer divided into a surface and a deep storage portion. Each buffer is defined by three parameters: the gradient of the sorptive isothermal line of the material (κ), the mass of the material (M) and the moisture exchange coefficient

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DRAFT between the two regions (β) and the zone air. The difficulty of this simple model relies on the determination of the mass of the material of each wall regions which depends on the value of the effective moisture penetration depth dEMPD. This point will be discussed in the section concerning the moisture transfers. It should be noticed that this model allows taking into account the moisture transfer between an external wall (with dSURF = dEMPD), an internal wall or a furniture (dSURF = 2×dEMPD) and the zone air only. No moisture transfers with the exterior are modeled here; as a result the outdoor climate has no effect on the moisture transfer within the building envelope. A third moisture model based on the Moisture Transfer Function (which is equivalent to the heat transfer function presented before but applied for moisture) has been developed by Transsolar but is not part of the TRNSYS 15 standard package and thus has not been tested in the present study.

2.2.

PowerDomus

PowerDomus calculates the coupled heat and moisture transfers through the building walls. The governing partial differential equations derived from conservation of energy and mass flow in an elemental volume of porous material are presented below.

∂T ∂ ∂T ∂ = (λ(T, θ) ) − L(T ) ( jV ) ∂t ∂x ∂x ∂x ∂θ ∂ ⎛ j ⎞ = − ⎜⎜ ⎟⎟ ∂t ∂x ⎝ ρ l ⎠

ρ 0 c m (T, θ)

(5) (6)

Note that the first equation differs from Fourier’s equation for transient heat flow by an added convective transport term (due to moisture diffusion associated with evaporation and condensation of water in the pores of the medium) and by a dependence on the moisture content (so that it is coupled to the second equation). The driving forces for convective transport are temperature and moisture gradients. The vapor flow is expressed in terms of transport coefficients, D, associated with the thermal and moisture gradients. According to Philip and DeVries (1957), the equation for vapor flow is:

jv ∂T ∂θ = −D Tv (T, θ) − D θv (T, θ) ρl ∂x ∂x

(7)

Nevertheless, vapor diffusion is normally written in the literature in terms of a vapor pressure gradient and data are provided for the water vapor permeability. The vapor transport coefficients are evaluated from the following relations:

D θV = δ p

D Tv =

p s, v

1

(8)

ξ u ρ mat

δp ρH 2O

φ

∂p s, v

(9)

∂T

The liquid transport coefficients are based on capillary pressure data, which are ignored in the present

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DRAFT study. The governing partial differential equations are discretized using a fully-implicit scheme and the equations system is solved by means of the Multi-TriDiagonal Matrix Algorithm – MTDMA by Mendes et al. (2002) that was shown to be more stable, accurate and time-efficient than the classical Tri-Diagonal Matrix Algorithm - TDMA.

2.3.

Simulation parameters

In both methods, the precision of the calculations can be adjusted via two parameters. In reality there are also the convergence parameters of the different iterative calculations but, in this study, these parameters have been set constant and small enough to not represent a potential source of errors. In TRNSYS, the first parameter is the time base that is used to determine the transfer function coefficients. The second parameter is the time step of the simulation. The time step has to be lower than or equal to the time base. If it is lower, a linear interpolation is performed which can cause small errors. As a consequence, it is always better to keep the same value for these two parameters. This fact limits the range of possible time step values for a particular wall. In all the presented simulations, the time step is chosen equal to the time base value. In PowerDomus, the parameters are the number of nodes used to spatially discretize the wall thickness and the time step of the simulation. Even if the method is stable, the values of both parameters have to be set sufficiently small to reach correct solution. No specific rule exists to determine the right values of these parameters; so they have to be adjusted for each case.

3. Description of the studied configurations The BESTEST building geometry without windows is chosen as a reference for all cases. Its internal dimensions are 8m × 6m × 2.7m. Heat and moisture transfers are only possible at the external surfaces, the floor internal surface is considered adiabatic and impermeable. Convective heat transfer coefficients are 3.2 W.m-2.K-1 at the internal surface and 24.7 W.m-2.K-1 at the external one.

3.1.

Thermal transfer through the building envelope

The three basic cases of thermal perturbation are based on the ASHRAE Test Suite (2001) that provides the analytical solution for an external step perturbation with an adiabatic internal wall (TC1), an external step perturbation with a constant zone air temperature (TC2) and an external sinusoidal perturbation with a constant zone air temperature (TC3). Table 1: Thermal properties of the materials. Material Concrete Insulation

k (W.m-1.K-1) 1.13 0.04

ρ (kg.m-3) 1400 10

c (J.kg-1.K-1) 1000 1400

Two materials (Table 1) and two thicknesses (L=0.04m and L=0.25m) were chosen in order to test the range of thermal inertia encountered in real buildings.

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DRAFT

3.2.

Moisture transfer through the building envelope

In this first studied case, all the material’s properties are constant. This case corresponds to the Exercise 0 (Case 0A and 0B) of the IEA Annex 41. The wall is made up of aerated concrete whose thermal properties are presented in Table 2. Table 2: Thermal properties of the dry aerated concrete. Material Aerated Concrete

k (W.m-1.K-1) 0.18

ρ (kg.m-3) 650

c (J.kg-1.K-1) 840

The thickness of the wall is 0.15m. The material’s properties for moisture transfers are: − Water vapour permeability: δp = 3.0×10-11 kg.m-1.s-1.Pa-1 − Mean sorption curve: u = 0.0661ϕ − Convective surface resistance to vapour transfer: R = 5.0×107 Pa.m2.s.kg-1 Internal moisture gain is 0.5 kg.h-1 from 9:00 to 17:00 every day. There are no moisture gains outside these hours and no heat gains at any time. External air enters the room at a constant rate of 0.5 ach. Outside and initial indoor temperature and relative humidity are 20°C and 30% respectively. These are also the initial conditions of materials in the constructions. In case 0A, the interior surfaces are clad with a vapour tight material whereas these surfaces are open in case 0B. All constructions are supposed to be covered on the outside by a moisture tight layer, which prevents loss of vapour through the constructions from the inside of the building to the outside. Analytical solutions have been provided by Bednar and Hagentoft (2005). In the second studied case, the material’s properties depend on the moisture content. Two wall thicknesses (L=0.04m and L=0.25m) of lime mortar are studied here. Material’s properties derived from Perrin (1985) are presented in Table 3. Other properties are: − − −

Mean water vapour permeability: δp = 3.2×10-12 kg.m-1.s-1.Pa-1 Mean sorption curve: u = 0.0445ϕ Convective surface resistance to vapour transfer: R = 5.0×107 Pa.m2.s.kg-1

Internal moisture gain, outside and initial temperature and humidity are the same. Moisture transfer can occur at the external wall surfaces. No analytical solution exists for this complete case. Table 3: Thermal properties of the dry lime mortar. Material

Lime mortar

k (W.m-1.K-1) 1.9

5

ρ (kg.m-3) 2500

c (J.kg-1.K-1) 932

DRAFT

4. Heat transfer through the building envelope For the three thermal transfer cases, the time step for both programs was set to 60 min. PowerDomus nodes number was 40 that corresponds to a 5mm space for the thicker wall.

4.1.

Test TC1: Transient Conduction – Step response

The external surface is exposed to an environment at 35°C while the interior surface is adiabatic. Concrete: thickness = 0.25m 36

34

34

32

32 Temperature (°C)

Temperature (°C)

Concrete: thickness = 0.04m 36

30 28 TSI - Analytical Sol.

26

TSI - PowerDomus TSI - TRNSYS

24

30 28 TSI - Analytical Sol.

26


24

TSO - Analytical Sol.

22


22

TSO - PowerDomus

TSO - PowerDomus

TSO - TRNSYS

20 2160

2161

2162

2163

2164

2165

TSO - TRNSYS

20 2160

2166

2168

2176

Time (h)

34

34

32

32

30 28 TSI - Analytical Sol. TSI - PowerDomus TSI - TRNSYS

24

TSI - Analytical Sol.

26


24


22

TSO - PowerDomus

2161

2162

2163

2208

28

TSO - PowerDomus

TSO - TRNSYS

20 2160

2200

30


22

2192

Insulation: thickness = 0.25m 36

Temperature (°C)

Temperature (°C)


26

2184 Time (h)

2164

2165

TSO - TRNSYS

20 2160

2166

Time (h)

2162

2164

2166

2168

2170

2172

Time (h)

Figure 1: Temperature evolution for the TC1 configuration. Figure 1 presents the temperature evolution for the four walls. On the whole, the two programs give the same response. They are both very close to the analytical solution with greater differences obtained with TRNSYS right after the temperature step for the thickest walls. For the smallest Biot number case (0.04m concrete wall), both programs completely miss the temperature increase. Additional simulations have been performed to test the temperature response versus the simulation

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DRAFT time step (Figure 2). TRNSYS seems to react faster than PowerDomus to a time step decrease but it seems that a limit exists for TRNSYS under which it tends to overestimate the temperature and to increase the error. At the opposite, the smaller the time step is, the higher the accuracy is for PowerDomus. Concrete: thickness = 0.04m 36

34

34

32


Temperature (°C)


30 28 TSI - TS=03 min - Analytical Sol.

26

TSI - TS=60 min - PowerDomus

30 28 TSI - TS=03 min - Analytical Sol.

26

TSI - TS=60 min - TRNSYS

24


24


22


22



20 2160

2161

2162

2163

2164

2165

TSI - TS=3.75 min - TRNSYS

20 2160

2166

2161

2162

Time (h)

2163

2164

2165

2166

Time (h)

Figure 2: Temperature evolution for the TC1 configuration (0.04m concrete wall).

4.2.

Test TC2: Transient Conduction – Step response

The external surface is exposed to an environment at 35°C while the interior zone stays at 20°C. Again, the two programs give similar response. They are both very close to the analytical solution except for the 0.04m concrete wall where smaller time step is needed for both softwares to match the analytical result.

4.3.

Test TC3: Transient Conduction – Sinusoidal driving temperature

The external surface is exposed to an environment at a 24h periodic sinusoid temperature while the interior zone stays at 20°C. Concrete: thickness = 0.25m 28

26

26

24


Temperature (°C)


22 20 18 16

22 20 18 16

TSI - Analytic Sol. TSI - PowerDomus

14

TSI - PowerDomus

14

TSI - TRNSYS TEXT

12 168

172

TSI - Analytic Sol. TSI - TRNSYS TEXT

176

180

184

188

12 168

192

Time (h)

172

176

180 Time (h)

7

184

188

192

DRAFT


26

26

24


Temperature (°C)


22 20 18 16

22 20 18 16

TSI - Analytic Sol.

TSI - Analytic Sol.

TSI - PowerDomus

14

TSI - PowerDomus

14

TSI - TRNSYS

TSI - TRNSYS

TEXT

12 168

172

TEXT

176

180

184

188

12 168

192

172

176

Time (h)

180

184

188

192

Time (h)

Figure 3: Temperature evolution for the TC3 configuration. Results presented in Figure 3 clearly show that both programs give correct temperature evolutions. The greatest errors are found to the 0.25m concrete wall for PowerDomus simulation. Figure 4 presents the effect of decreasing the time step value on the response. Once again, the finite difference method needs smaller time step than the function transfer method to achieve a good convergence. Concrete: thickness = 0.25m 22.0 21.5

Temperature (°C)

21.0 20.5 20.0 19.5 TSI - Analytic Sol. TSI - TS=60min - PowerDomus TSI - TS=30min - PowerDomus TSI - TS=15min - PowerDomus TSI - TRNSYS

19.0 18.5 18.0 168

172

176

180

184

188

192

Time (h)

Figure 4: Temperature evolution for the TC3 configuration (0.25m concrete wall).

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DRAFT

5. Moisture transfer through the building envelope For all results presented in this section, calculations have been performed until quasi-periodic state condition for relative humidity profiles was reached. The quasi-periodic state condition is defined as a relative difference between two consecutive day relative humidity lower than 0.001% at each hour of the 24 hour period.

5.1.

IEA Revised Cases 0A and 0B

A TRNSYS time step of 15 min was used for these two cases. PowerDomus nodes number and time step are 40 and 1 min respectively. Both softwares match the analytical solution when surfaces are clad with a vapour tight material (Case 0A). When the internal surfaces are open (Case 0B), there is a small but notable difference between the simulation and the analytical results. PowerDomus follows better the real evolution whereas TRNSYS tends to overestimate the relative humidity increase at 10:00 and its decrease at 18:00. Case 0A

Case 0B 50

80

49

70 Relative Humidity (%)

Relative Humidity (%)

48 60 50 40 30 Analytical Sol.

20

47 46 45 44 43 Analytical Sol.

42

PowerDomus

PowerDomus

41

TRNSYS

10

TRNSYS

40 0

4

8

12

16

20

0

24

4

8

12

16

20

24

Time (h)

Time (h)

Figure 5: Case 0A and 0B relative humidity evolution. It should be added that the Buffer Storage Humidity Model used in TRNSYS only relies on the evaluation of the surface exchange coefficient which depends on the Effective Moisture Penetration Depth. This penetration depth can be evaluated from the vapor diffusion coefficient in the porous media and the period of the moisture perturbation. Because different relations can be found in the literature, a parametric study has been carried to evaluate the penetration depth. Figure 6 presents the evolution of the relative humidity for different values of depth in the case 0B. The parameter d in this figure is defined by:

d = D θv × τ

(11)

Where Dθv is vapor diffusion coefficient due to vapor gradient in the porous media (m.s-2) and τ is the period of the moisture perturbation (= 1day = 86400s). Results clearly show that none of these values is in total accordance with the analytical solution.

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DRAFT Some agree well in the first decrease (from 0 to 9) but miss the increase (from 10 to 17), and inversely. It should be noticed that the relation provided by Cunningham (1992), which corresponds to d π in the figure, gives a reasonable result. Cunningham’s relation will be used for the following simulations so the penetration depth is now calculated by:

D θv × T π

d EMPD =

(12) Case 0B

51 0.3d 0.4d 0.5d (1/PI)^0.5d 0.6d 0.7d 1.0d Analytical


49

47

45

43

41 0

4

8

12

16

20

24

Time (h)

Figure 6: TRNSYS relative humidity for different value of the EMPD (Case 0B).

5.2.

Additional cases with walls made up of lime mortar

In order to avoid false results linked to the time step value, TRNSYS time step was decreased to 3 min and PowerDomus time step was kept to 1 min. The node number is chosen to obtain a distance of 1 mm between the nodes. Thickness: 0.25m 54

51

51

48

48



Thickness: 0.04m 54

45 42 39 36

PowerDomus

33

45 42 39 36

PowerDomus

33

TRNSYS

30

TRNSYS

30 0

4

8

12

16

20

24

0

Time (h)

4

8

12

16

20

24

Time (h)

Figure 7: Zone relative humidity for a lime mortar thickness of 0.04m (left) and 0.25m (right).

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DRAFT Figure 7 presents the calculated relative humidity for a lime mortar thickness of 0.04m and 0.25m. There is a huge difference between the relative humidity profiles. TRNSYS predicts higher level of relative humidity and higher amplitude for both thicknesses. Its model is not very sensitive to the total thickness of the wall the transfer coefficient between the surface layer and the deep one is three times lower than that between the surface layer and the zone air. Thus, the time constant is not of the same order. In the opposite, PowerDomus predicts an increase of the relative humidity level with an increase of the wall depth. To investigate the reason of such a difference between the two predictions, the parameters used in the PowerDomus model have been modified in order to bring this model to the simplified model of TRNSYS. Three simplifications have been done (Figure 8): − − −

PowerDomus A: no moisture transfer at the external surface. PowerDomus B: simplification A + constant diffusion coefficient Dθv = 6.7×10-11 m2.s-1 calculated as the mean value for relative humidity between 40% and 80%. PowerDomus C: simplification B + linear sorption curve calculated for the same relative humidity range.

With no moisture transfers at the external surface, PowerDomus model gives a mean relative humidity (45%) similar to that predicted by TRNSYS model but the amplitude stays lower. The amplitude difference is due to the variable vapor diffusion coefficient. There is almost no effect of the variable sorption curve gradient because it’s already constant for the humidity range involved in the present case (between 40% and 80%). Thickness: 0.25m 54

51

51

48

48



Thickness: 0.04m 54

45 42 39 36

PowerDomus PowerDomus - A PowerDomus - B PowerDomus - C TRNSYS

33 30 0

4

8

12

16

20

45 42 39 PowerDomus PowerDomus - A PowerDomus - B PowerDomus - C TRNSYS

36 33 30

24

0

Time (h)

4

8

12

16

20

Time (h)

Figure 8: Zone relative humidity obtained after simplifications in PowerDomus.

11

24

DRAFT

6. Conclusion By simulating simple cases involving heat or moisture transfers through walls with TRNSYS and PowerDomus, several points came out. First, the time step values are determinant even for pure thermal cases where the classical value of 1 hour can lead to notable errors and a value of 15 minutes has to be preferred. Second, the simplified model used in TRNSYS for moisture sorption at wall gives reasonable results as soon as effects of the outside conditions are negligible i.e. in the case in which the zone moisture is decoupled from the outside moisture by an impermeable layer within the wall or by a painted external surface. In the opposite case or if the variation with the moisture content of the vapor diffusion coefficient in the material is important, a more detailed model such as that implemented in PowerDomus has to be employed in order to avoid huge errors on the prediction of the relative humidity of a zone. It should be noticed that the time step for moisture transfer problems has to be set lower than 3 minutes to achieve a good convergence. No case of heat and moisture perturbation has been investigate in the present study but, from our results regarding the decoupled phenomena, it seems that a time step lower than 3 minutes has to be applied with PowerDomus. With TRNSYS, it will probably depend on the kind of thermal perturbation (step, sinusoid…) because, even if it improves the moisture transfers evaluation, decreasing the time step value does not necessarily lead to a better response for the heat transfer.

References: Bednar T. and Hagentoft C.E. “Analytical Solution for Moisture Buffering Effect - Validation Exercises for Simulation Tools”, presented in Annex 41, 2005. Cunningham M.J. “Effective Penetration Depth and Effective Resistance in Moisture Transfer”, Building and Environment, volume 27(3), 1992. HAMSTAD Project. “Determination of liquid water transfer properties of porous building materials and development of numerical assessment methods”, Chalmers University of Technology, December, 2002. Iu I. and Fisher D. “Application of Conduction Transfer Functions and Periodic Response Factors in Cooling Load Calculation Procedures”, ASHRAE Transactions, 110(2): 829-841, 2004. Jensen S. “Empirical Whole Model Validation Case Study: the PASSYS Reference Wall”, Proceedings of Building Simulation ’93. International Building Performance Simulation Association, August 16-18, Adelaide, Australia, 1993. Judkoff R. “Validation of Building Energy Analysis Simulation Programs at the Solar Energy Research Institute”. Energy and Buildings, Vol. 10, No. 3, p. 235. Lausanne, Switzerland, 1988. Judkoff R.D. and Neymark J.S. “Building Energy Simulation Test (BESTEST) and Diagnostic Method”, NREL/TP-472-6231, Golden, Colorado National renewable Energy Laboratory, 1995. Mendes N., Philippi P.C. and Lamberts R. “A new Mathematical Method to Solve Highly Coupled Equations of Heat and Mass Transfer in Porous Media”, International Journal of Heat and Mass Transfer, V. 45, p. 509-518, 2002. Mitalas G.P. and Arseneault J.G. “FORTRAN IV Program to Calculate z-Transfer Functions for the Calculation of Transient Heat Transfer Through Walls and Roofs”, Proceedings of the Conference on Use of Computers for Environmental Engineering Related to Buildings, Gaithersburg, MD, NBS Building Science Series 39, October, 1971. Moinard S. and Guyon G.“Empirical Validation of EDF ETNA and GENEC Test-Cell Models - Final Report”. IEA SHC Task 22 Building Energy Analysis Tools Project A.3. Moret sur Loing, France: Electricité de France. http://www.ieashc.org/task22/deliverables.htm, 1999. Neymark J. and Judkoff R. “International Energy Agency Building Energy Simulation Test and Diagnostic Method for Heating, Ventilating, and Air-Conditioning Equipment Models (HVAC BESTEST) Volume 1: Cases E100-E200”. NREL/TP-550-30152. Golden, CO: National Renewable Energy Laboratory. http://www.nrel.gov/docs/fy02osti/30152.pdf, 2002. Perrin B. “Etude des tranferts couplés de chaleur et de masse dans des matériaux poreux consolidés non saturés en génie civil”, Thèse Docteur D’Etat, Université Paul Sabatier, Toulouse, p. 267, 1985. Philip J.R. and DeVries D.A. “Moisture Movement in Porous Materials under Temperature Gradients”, Trans. Amer. Geophys. Union 38(2), p. 22-232, 1957. PowerDomus Version 1.5: www.pucpr.br/lst, September, 2005. Spitler J.D., Rees S.J. and Dongyi X. “Development of an Analytical Verification Test Suite for Whole Building Energy Simulation Programs – Building Fabric”, ASHRAE 1052-RP, April, 2001. Stephenson D.G. and Mitalas G.P. “Calculation of Heat Conduction Transfer Functions for multi-Layer Slabs”, ASHRAE Annual Meeting, Washington, D.C., August 22-25, 1971. Transsolar: http://www.trnsys.de/ts/english/indexSen.htm. TRNSYS Version 15.3.00: www.sel.me.wisc.edu/trnsys, March, 2003.

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