NREL is a national laboratory of the U.S. Department of Energy, Office of Energy 

Efficiency & Renewable Energy, operated by the Alliance for Sustainable Energy, LLC.

Contract No. DE-AC36-08GO28308

Evaluation of the Effective 

Moisture Penetration Depth 

Model for Estimating Moisture 

Buffering in Buildings

J. Woods, J. Winkler, and D. Christensen

National Renewable Energy Laboratory

Technical Report

NREL/TP-5500-57441 

January 2013
NREL is a national laboratory of the U.S. Department of Energy, Office of Energy 

Efficiency & Renewable Energy, operated by the Alliance for Sustainable Energy, LLC.

National Renewable Energy Laboratory

15013 Denver West Parkway

Golden, Colorado 80401

303-275-3000 • www.nrel.gov

Contract No. DE-AC36-08GO28308

Moisture Penetration Depth 

Model for Estimating Moisture 

Buffering in Buildings 

J. Woods, J. Winkler, and D. Christensen

National Renewable Energy Laboratory

Prepared under Task No. BE12.0201

Technical Report

NREL/TP-5500-57441 

January 2013
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iii

Acknowledgments

This work was supported by the U.S. Department of Energy under Contract No. DE-AC36-08-

GO28308 with the National Renewable Energy Laboratory (NREL). The authors would like to 

thank Paulo Tabares-Velasco (NREL) for his help with EnergyPlus, and would like to thank 

Brent Griffith (NREL) and Lixing Gu (Florida Solar Energy Center) for their assistance in 

implementing the fix to the EMPD model in EnergyPlus v7.2.
iv

Nomenclature

A surface area (m2)

cp specific heat (J kg-1 K-1)

D12 moisture diffusivity, D12 = δpermRT (m2

s

-1

)

dEMPD effective moisture penetration depth of surface layer (m)

dEMPD-deep effective moisture penetration depth of deep layer (m)

du/dφ slope of moisture sorption curve ( 1

moisture dry kg kg − )

em moisture effusivity (J m-2 K-1 s

-1/2)

eth thermal effusivity (kg m-2 Pa-1

s

-1/2)

hfg enthalpy of vaporization for water (2500 kJ kg-1

)

hm boundary layer mass transfer coefficient (check)

k thermal conductivity (W m-1 K-1

)

L material thickness (m)

mair,infiltration  mass flow rate of infiltration air (kg s-1

)

v matl−sorption m moisture sorption rate into building materials (kg s-1

)

v gain m internal moisture gain (kg s-1

)

psat saturation vapor pressure (Pa)

qsorption heat transfer from moisture sorption (W)

R gas constant for water (461.5 J kg-1 K-1)

T temperature (°C)

t time (s)

u moisture capacitance ( 1

moisture dry kg kg− )

V volume (m3

)

x distance into wall (m)
v

δperm vapor permeability (kg m-1s

-1 Pa-1

)

ρair air density (kg m-3

)

ρdry dry material density (kg m-3)

φ relative humidity

φmatl effective relative humidity of material

τp humidity load cycle period (s)

Ψ EMPD1

v7.1 model correlating parameter; Eq. (A.3) (kgvapor K m2 J

-2

s

-1/2)

ω humidity ratio ( 1

vapor air kg kg − )

ωf angular frequency for sinusoidal-varying humidity (rad s-1

)

subscripts

air property of air

amb ambient condition

zone building zone
vi

Executive Summary

Building materials and furnishings play an important role in moderating relative humidity 

fluctuations. Accurately accounting for moisture buffering in building simulations is central in 

determining the need for, and the energy use from, controlling humidity. In building modeling, 

moisture buffering has typically either been ignored or has been lumped with the zone air using 

an effective moisture capacitance multiplier. Researchers have also used finite-difference models 

to simulate moisture transfer within materials, which are more physically realistic than the 

effective capacitance model, but require orders of magnitude more computation time. 

This study examines the effective moisture penetration depth (EMPD) model and its suitability 

for building simulations. The EMPD model is a compromise between the simple, inaccurate,

effective capacitance approach and the complex, yet accurate, finite-difference approach. Two 

formulations of the EMPD model were examined, including the model used in the EnergyPlus 

building simulation software. We uncovered an error in the EMPD model in EnergyPlus, which 

was fixed with the release of EnergyPlus version 7.2. The EMPD model in earlier versions of 

EnergyPlus should not be used.

Three simple building simulation cases were used to compare the two EMPD formulations, the 

effective capacitance model, and the finite-difference model. An analytical solution for the first 

case showed that the two EMPD formulations were not equal, but that both were improvements 

over the effective capacitance model. For the cases that more closely resemble real building 

loads, the improvement of the EMPD model over the effective capacitance model was small 

unless the EMPD model included two penetration depths: a surface layer for short-term humidity 

fluctuations, and a deep layer for longer term fluctuations. We are presently working to 

implement the dual-depth EMPD model in a future EnergyPlus version.
vii

Table of Contents

Acknowledgments .........................................................................................................................................iii

Nomenclature................................................................................................................................................iv

Executive Summary ......................................................................................................................................vi

List of Figures..............................................................................................................................................viii

List of Tables...............................................................................................................................................viii

1 Introduction............................................................................................................................................. 1

2 Methods .................................................................................................................................................. 3

2.1 Effective Capacitance Model ........................................................................................................... 4

2.2 Finite-Difference Heat and Moisture Transfer Model....................................................................... 4

2.3 Effective Moisture Penetration Depth Model ................................................................................... 5

2.3.1 Effective Moisture Penetration Depth Model Parameter Estimation................................... 6

2.3.2 Combining the Effective Moisture Penetration Depth Model With Conduction Transfer 

Functions: Correcting the Formulation in EnergyPlus v7.1................................................. 6

2.4 Building Simulation Test Cases ....................................................................................................... 7

3 Results and Discussion .......................................................................................................................... 9

3.1 Case 1: Internal Latent Gain (Isothermal Models)........................................................................... 9

3.2 Case 1: Internal Latent Gain (Non-Isothermal Models) ................................................................. 10

3.3 Case 2: Cyclical Ambient Humidity ................................................................................................ 13

3.4 Case 3: Ambient Humidity From Weather Data............................................................................. 15

4 Conclusions .......................................................................................................................................... 18

Appendix ..................................................................................................................................................... 22
viii

List of Figures

Figure 1. Building for test cases.................................................................................................................... 4

Figure 2. Ambient humidity used for case 3 simulations .............................................................................. 8

Figure 3. Moisture models compared to an analytical solution. EMPD1 (isothermal) and HAMT 

(isothermal) models were modified so there is no heat of sorption........................................................ 9

Figure 4. Predicted zone humidity for non-isothermal and isothermal EMPD1 models and non-isothermal 

and isothermal HAMT models .............................................................................................................. 11

Figure 5. Predicted zone humidity for non-isothermal HAMT and EMPD1 models. EMPD1 = model in 

EnergyPlus v7.2, EMPD1

v7.1 = model in EnergyPlus v7.1 and earlier. Note that the y-axis was 

expanded relative to previous figures................................................................................................... 12

Figure 6. Predicted zone humidity over a one-week period during simulations with cyclical ambient 

humidity ................................................................................................................................................ 14

Figure 7. Histogram of errors for EC, EMPD1, and EMPD2 model during simulations with cyclical ambient 

humidity. Error is calculated as deviation from HAMT model............................................................... 15

Figure 8. Predicted zone humidity over a two-week period during simulations with ambient humidity from 

weather data......................................................................................................................................... 16

Figure 9. Histogram of errors for EC, EMPD1, and EMPD2 model during simulations with ambient 

humidity from weather data. Error is calculated as deviation from HAMT model. ............................... 17

Figure A.1. Peak humidity ratio of EMPD1 (×) and EMPD1

v7.1 (○) simulations relative to HAMT simulations 

versus ψ-parameter (defined in Eq. (A.3)) ........................................................................................... 23

Figure A.2. Ψ-parameter for different materials of different thickness: (a) 15-minute time step, (b) 6-minute 

time step ............................................................................................................................................... 24

List of Tables

Table 1. Description of Four Models Analyzed in This Study ....................................................................... 3

Table 2. Properties of Test Material (aerated concrete) ............................................................................... 8

Table 3. Test Cases ...................................................................................................................................... 8
1

1 Introduction

Building simulation software can be used to predict a building’s latent and sensible loads, and 

the resulting energy use, to maintain temperature and humidity set points. As building codes 

have tightened in recent years, insulation improvements have resulted in reduced sensible load.

The latent load from internal gains and required ventilation has remained relatively constant. 

This has increased the need for dehumidification equipment and for accurately predicting the 

space relative humidity [1, 2]. 

Indoor relative humidity should be controlled to maintain occupant health and comfort, and to 

limit the likelihood of building decay such as mold growth. The humidity in a building is 

primarily affected by five factors: internal moisture gains, ventilation, infiltration, removal with 

space-conditioning equipment, and moisture sorption into (or desorption out of) the materials.

Building models often neglect this last term, but it is becoming more important because of the 

need for better humidity control and prediction. Many building simulation programs use an 

effective capacitance (EC) model, which uses a moisture capacitance multiplier to combine this 

term with the zone air [3]. This model is an improvement over neglecting the moisture buffering 

of the materials entirely. However, it does not appropriately model the physics of moisture 

transfer into materials, and the multiplier is empirical and difficult to estimate for different wall 

types and furnishings. Researchers have also used finite-difference models to simulate moisture 

transfer within materials [4-6]. Although these finite difference models are more realistic, they 

require orders of magnitude more computation time than the effective capacitance method. 

There is a need for simpler moisture prediction models with fast solution times and reasonable 

accuracy. Faster solving time enables optimization and large parametric studies to compare 

several options for the building envelope and space conditioning equipment. One possibility is 

the effective moisture penetration depth (EMPD) model, which was developed independently by 

Cunningham [7, 8] and Kerestecioglu et al. [9] in the late 1980s. The EMPD model is derived by 

assuming cyclic variations in the zone humidity, and therefore in the humidity loads. This is 

generally a reasonable assumption for hypothetical internal latent gains, which occur predictably 

on a daily basis, but is less certain for latent loads from infiltration and ventilation. Two 

formulations of the EMPD model have emerged.

One model [10] is currently a user-selectable option in the EnergyPlus building simulation 

software [11]. This non-isothermal formulation (referred to here as EMPD1) is coupled with 

conduction transfer functions for the energy equations. Although this model has been used in 

EnergyPlus for various building simulations, there has yet to be a verification study of it. This 

research revealed that the implementation of the EMPD equations in EnergyPlus version 7.1 (and 

prior versions) was incorrect. This study discusses problems with this implementation (referred 

to as EMPD1

v7.1) and assesses its accuracy. We modified the EnergyPlus source code to correct 

these problems. The fixed EMPD1 model, available starting in EnergyPlus version 7.2, is used in 

this report.

There is also a different formulation of the EMPD model (referred to here as EMPD2) that is a 

user-selectable option for the building model in the TRNSYS software. It typically includes two 

effective depths: a surface layer to account for short-term transients and a deep layer to represent 

materials’ response to longer term moisture events such as seasonal changes [12]. In contrast to 
2

EMPD1, it is an isothermal model. The building models using the EMPD2 formulation, such as 

TRNSYS, include energy balance equations, but they neglect the effects of temperature on the 

equilibrium moisture content of the materials. They also typically neglect the latent heat resulting 

from water sorption and desorption. Several researchers have verified the EMPD2 approach [13-

15], but for limited cases. Woloszyn et al. [16] performed a more in-depth study using the 

EMPD2 approach in TRNSYS. They focused on a ventilation system that was controlled based 

on relative humidity, but also reported good agreement between the humidity predictions from 

the EMPD2 model and measured humidity data from a test room.

The purpose of this study is to evaluate both formulations of the EMPD model by comparing 

them with the simple EC model and a detailed finite-difference model (referred to here, as in 

EnergyPlus, as the combined heat and moisture transfer (HAMT) model [5]). All four models 

(EC, EMPD1, EMPD2, and HAMT) are simulated for three separate cases, each with a periodic, 

internal latent gain and continuous ventilation. The ambient (outdoor) humidity is different for 

each case: (1) constant ambient humidity, (2) sinusoidal-varying ambient humidity, and (3) 

ambient humidity from weather data. These cases are used to assess the “accuracy” of the two 

EMPD model formulations compared to an analytical solution, compared to the common EC 

model, and compared to the more robust HAMT model. Thus, this study covers two of the three 

verification and validation schemes in ASHRAE Standard 140 [17]: analytical verification and 

comparative testing. These cases help the National Renewable Energy Laboratory (NREL) reach

its goals to improve the accuracy of building energy simulations [18], similar to other 

evaluations performed by NREL [19, 20]. The goal in this study is not to find the “best” model, 

but rather to compare the dynamics of each model for simple cases, and gain insight into these 

models and their use in building energy simulations.
3

2 Methods

The four moisture storage and transport models compared in this study are listed in Table 1. 

Relatively simple simulations are performed with these models using the rectangular building in 

Figure 1. These simulations are used to explore these models’ responses to different moisture 

loads. The equation governing moisture transfer in these simulations is: 

( ) gain matl sorption air,ventilation − = amb − zone + v − v

zone

air zone m m m

dt

d V  ω ω   ω

ρ (1)

where the first term is the moisture capacitance of the zone air, the second term the moisture

added with ventilation air, the third term the moisture gains, and the final term the moisture 

sorption into (or desorption out of) the building materials. In these terms, ρair is the dry-air 

density of the zone air, Vzone the zone volume, ωzone and ωamb the zone and ambient humidity 

ratios, and t time. Moisture removal with an air conditioner or dehumidifier would typically be a

term on the right-hand side of this equation, but this is not considered in this study for simplicity.

The time step (dt) for the simulations was 6 minutes for each model. Although the HAMT model 

generally needs a smaller time step than the other models, we found that all of the models were 

insensitive to the time step for time steps 6 minutes or shorter. The original EMPD1 formulation 

was strongly affected by the time step (see Figure A.2 in the Appendix), but once fixed the 

simulation results for 6-minute and 15-minute time steps were nearly the same.

The four models solve for the last term in Eq. (1), which is the focus of this study. Sections 2.1

through 2.3 describe how each of these models calculates this last term.

Table 1. Description of Four Models Analyzed in This Study

Model Description Isothermal 

Assumption?

Used in 

EnergyPlus?

Simulation 

Run Time

EC

Effective capacitance; multiplier on room air 

volume to simulate added moisture capacity 

due to walls and furnishings.

isothermal EnergyPlus 

default defined as 1

EMPD1 

Effective moisture penetration depth; based 

on cyclical humidity loads; solves for 

moisture sorption rate using Eq. (6).

nonisothermal

user 

selectable 

option a

~1.05

EMPD2

Similar to EMPD1, but solves for moisture 

sorption rate using Eq. (7). Uses two EMPD 

values (surface and deep).

isothermal not in 

EnergyPlus b ~1.05

HAMT

Combined heat and moisture transfer. Solves 

for temperature and moisture content within 

materials using finite-difference method.

nonisothermal

user 

selectable 

option

~102-104

a EMPD1 is the fixed model in EnergyPlus v7.2. The model in prior versions is referred to as EMPD1v7.1 . b Although the EMPD2 model is not currently available in EnergyPlus, we made a custom EnergyPlus 

version for this study that uses the EMPD2 model.
4

Figure 1. Building for test cases

2.1 Effective Capacitance Model

The EC model does not explicitly solve for the moisture sorption term, but instead lumps its 

effect in with the moisture capacitance of the zone air:

( ) ( ) gain EC air,ventilation amb zone v

zone

air zone m m

dt

d V =  ω −ω +  ω

ρ (2)

where EC is the effective moisture capacitance of the walls and furnishings. For example, a 

building with an EC of 15 means the building materials, plus the zone air, can absorb 15 times

more moisture than the zone air alone. A 2001 study [3] proposed EC values ranging from 10 to 

25 for different zone types (e.g., office vs. library). Values of 15 and 20 have been used in 

simulations for evaluating HVAC equipment [21, 22].

The EC values are empirical and have little physical connection to reality. The lumping of the 

room air node with the material nodes implies that the resistance to moisture transfer between the 

room air and the materials is negligible.

2.2 Finite-Difference Heat and Moisture Transfer Model

The HAMT model in EnergyPlus uses a finite-difference approach, derived from Künzel [5], for 

solving both heat and mass transfer through the building materials. The detailed moisture 

equations for the finite-difference equations are not included here, but can be found in the 

EnergyPlus Engineering Reference [11]. The following equation governs moisture transfer and 

storage inside the wall:

( )

dx

d p

dt

d

d

du sat

perm

φ δ φ

φ = (3)

where δperm (kg/m-s-Pa) is the vapor permeability, psat the saturation vapor pressure, u the 

moisture capacitance of the material (kg/kg), φ the relative humidity, and du/dφ the slope of the 

moisture capacitance curve. Eq. (3) can be discretized and solved using the finite-difference 

method.

Tamb

ωamb

gain mv  matl−adsorption mv 

m air,ventilation

8 m

6 m

2.7 m
5

The HAMT model is the most rigorous of the four models studied and is used here as a 

benchmark for comparison. It is the basis for the hygrothermal modeling program WUFI, 

developed by the Fraunhofer Institute for Building Physics, and it is expected to be more 

physically realistic in modeling moisture transport and storage through building materials than 

the other models. However, as indicated in Table 1, the simulation run time is several orders of 

magnitude longer.

In addition to vapor transfer, the HAMT model also calculates liquid transfer, which becomes 

more important at high moisture contents. In this analysis, we assume that all moisture transport 

mechanisms (vapor and/or liquid) are adequately modeled using a constant vapor permeability. 

We set the liquid transport coefficients in EnergyPlus to zero. Although this is not strictly 

accurate [5], it allows for a straightforward comparison to the EMPD models, which do not use 

liquid transport coefficients. 

2.3 Effective Moisture Penetration Depth Model

The EMPD model is based on the assumption that the moisture transfer takes place between the 

zone air and a thin fictitious layer of uniform moisture content of thickness dEMPD. This uniform 

moisture content is a function of the relative humidity, as in the HAMT model. Both EMPD 

models solve the following equation:

ρ = (ω −ω) matl EMPD hm A zone dt

du Ad (4)

where ρmatl is the bulk, dry density of the adsorbing material, dEMPD the effective moisture 

penetration depth, A the surface area, hm the airside convective mass transfer coefficient (kg/m2-

s), and ω the humidity ratio of air in equilibrium with the material. 

The first method (EMPD1), used in EnergyPlus, transforms the time derivative of the moisture 

content into [10, 11]:

dt

dT

T

u

dt

u d

dt

du

∂

∂

+

∂

∂ = ω

ω

(5)

where ∂u/∂ω and ∂u/∂T are calculated from the moisture sorption curve (u = f (φ)) and from the 

relationship between relative humidity and temperature. The non-isothermal term, which 

includes the temperature derivative (dT/dt), is calculated from the temperature change between 

the previous time step and the current time step. Substituting Eq. (5) into Eq. (4) creates an 

ordinary differential equation that can be solved for ω:

( ) dt

dT

T

u h A Ad

dt

u d

matl AdEMPD m zone dry EMPD 



 





∂

∂  = − + 

 





∂

∂

ω ω ρ

ω

ω

ρ (6)

The second method (EMPD2), used in TRNSYS, is an isothermal model; it does not include the 

effects of temperature. Although there are some variations [13, 14, 23], all calculate moisture

transfer into wall materials with an equation similar to:
6

( ) ( )

air perm

EMPD deep

air perm

EMPD

deep

air perm

EMPD

m

matl zone

matl

matl EMPD

p

d

p

d

A

p

d

h

A

dt

d

d

du Ad

δ δ

ω ω

δ

φ ω ω

φ ρ

2 2 2

1 − +

− +

+

− = . (7)

The term on the left side is the moisture accumulation in the material (φmatl is the effective 

relative humidity in the material) and the two terms on the right side are the moisture transfer 

from the zone to the surface layer and the moisture transfer from the surface layer to the deep 

layer. The deep layer takes into account the buffering of longer term humidity variations. In the 

denominators, the 1/hm term is the moisture transfer resistance in the airside boundary layer; the 

other three terms are the moisture transfer resistances for diffusion into the material. In these 

three terms, the multiplier of 2 places the nodes at the center of the buffer layers. 

2.3.1 Effective Moisture Penetration Depth Model Parameter Estimation

The EMPD model is derived based on a cyclic moisture variation, which must be estimated a 

priori to accurately estimate the effective moisture penetration depth, dEMPD. For a perfectly 

periodic boundary condition at the wall (i.e., periodic cycling of ωzone), the EMPD model will 

give an exact solution assuming that the sorption curve, u(φ), is known and the vapor 

permeability, δperm, is known and constant. In this case, the dEMPD is:

π

φ ρ

τ

d

du

δ p d

dry

perm sat p

EMPD = (8)

where τp is the humidity cycle period(s). The moisture sorption curve, u(φ), can be nonlinear, but 

dEMPD must be estimated from a linear approximation of this curve (du/dφ).

Eq. (8) is used to estimate dEMPD in the EMPD1 model and both dEMPD and dEMPD-deep in the 

EMPD2 model. The surface layer, dEMPD, is based on the short-term fluctuations of humidity, 

typically on the order of a day. The deep layer, dEMPD-deep, is based on longer term humidity 

fluctuations, which may be from short-term changes in weather (~weeks) or seasonal variations 

(~months). 

Eq. (8) is the most common method for estimating dEMPD, but the EnergyPlus Engineering 

Reference mentions a different, empirical equation [10]. This equation is not used in this report, 

but we found that it typically gives dEMPD values around 50% higher than those estimated with 

Eq. (8). Although the method used to estimate dEMPD will affect the results in this study, it does 

not affect the overall conclusions.

2.3.2 Combining the Effective Moisture Penetration Depth Model With 

Conduction Transfer Functions: Correcting the Formulation in EnergyPlus v7.1

With the simplified EMPD model equations above, EnergyPlus uses conduction transfer 

functions [24, 25] to model heat transport and storage in the building materials. The details of 

these calculations can be found in the EnergyPlus Engineering Reference [11]. The relevant 

equations for this discussion are related to the temperature calculation for the wall interior 

surface. This is where the latent heat from moisture sorption is released or absorbed, and this 

temperature also affects the equilibrium moisture content at the wall surface. 
7

The inside surface temperature is calculated iteratively with damping coefficients to ensure 

convergence. In EnergyPlus v7.1 and earlier versions, the inside surface temperature was first 

calculated without considering the latent heat released or adsorbed at the surface. The moisture 

transfer was then calculated using Eq. (5), with the surface temperature derivative term (dT/dt) 

based on the newly calculated surface temperature and the previous time step surface 

temperature. The heat of sorption was then calculated with:

qsorption = mv matl−sorption hfg  (9)

where hfg is the heat of vaporization of water. Next, the surface temperature was recalculated 

with this additional sorption heat transfer. The problem with the EMPD1

v7.1 model was the order 

in which these equations were evaluated. The dT/dt term was initially calculated incorrectly, as it 

did not consider the adsorption heat. Multiple iterations of these equations compounded the 

problem and resulted in substantial inaccuracy.

In the modified EnergyPlus implementation (EMPD1), Eq. (9) is solved before the surface 

temperature is calculated. During the first iteration, the dT/dt term is zero, but the following 

iterations use the correct dT; one that includes the effect from the sorption heat.

2.4 Building Simulation Test Cases

The roof, floor, and walls of the building used for the test cases (Figure 1) are made of concrete 

with material properties given in Table 2. The internal latent gain in all cases is 0.5 kg/h (347 W) 

from 9:00 to 17:00 and zero at all other times; the ventilation is constant at 0.5 air changes per 

hour. To avoid effects outside the scope of the moisture buffering in this study, the following 

simplifications were made: 

• The floor is not ground coupled, but is instead exposed to outside air at the exterior 

surface.

• The air in the zone is assumed to be well mixed.

• The zone has no temperature or humidity control.

• The convective mass transfer coefficient at the interior surface is a constant 2×10-8

kg/m2

-s-pa.

• The moisture transfer from the outside air through the exterior surface is zero.

• Solar radiation and radiation exchange with the sky are ignored.

• There is no furniture in the zone.

Table 3 shows that each case uses a different ambient humidity: (1) constant, (2) varying as a 

sine wave, and (3) varying based on weather data. The ambient temperatures in all three cases 

are constant. These simulations are not meant to be realistic, but their simplicity gives solutions 

that are predictable, intuitive, and can be used to evaluate the various modeling frameworks, as 

in [19, 20]. The results also provide insight into how the models may perform under more 

complex (realistic) forcing.
8

Table 2. Properties of Test Material (aerated concrete)

Material Property Value Units

Thickness, L 0.05 m

Density, ρdry 650 kg/m3

Sorption curve slope, du/dφ 0.0661 kg/kg

(HAMT) Permeability, δperm 3.0E-11 kg/m-s-Pa

(EMPD) Penetration depth, dEMPD

a 0.006706 m

(EC) Moisture capacitance multiplier 15

Thermal properties for non-isothermal models:

Specific heat, cp 800 J/kg-K

Thermal conductivity, k 0.1 W/m-K a dEMPD is calculated based on τp = 24 hours.

Table 3. Test Cases

Case Description Tamb (°C) ωamb (kg/kg) v gain m (kg/h)

1 Latent gain with constant 

ventilation (analytical solution) 20 0.00434 0.5

2 Latent gain with cyclical 

ambient humiditya 28 0.008 + sin(ωf t) 0.5

3 Latent gain with random 

ambient humidity 28 weather data 0.5

a angular frequency, ωf, used here is 0.04 rad/hr, which corresponds to a cycle period of 157 hours.

Using τp = 24 hours, the penetration depth for the EMPD1 model is 6.7 mm. For the EMPD2

model, the surface layer thickness is also 6.7 mm, and the deep layer thickness is 17 mm based 

on the 157-hour cycle period. For case 3, the deep layer thickness was estimated based on the

time between large humidity fluctuations, which occur on the order of weeks (Figure 2). Using τp

= 10 days results in a deep layer thickness of 21 mm.

Figure 2. Ambient humidity used for case 3 simulations

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0 7 14 21 28 35 42 49 56

time (days)

ωambient (kg/kg)
9

3 Results and Discussion

3.1 Case 1: Internal Latent Gain (Isothermal Models)

The simplicity of case 1 enables an analytical solution if isothermal conditions are assumed, as 

presented by Bednar and Hagentoft [26]. For an equal comparison, the EnergyPlus source code 

was modified for the EMPD1 and HAMT models to make them isothermal. These are referred to 

as EMPD1 (isothermal) and HAMT (isothermal). The results from the four models are compared 

to the analytical solution in Figure 3.

The HAMT and EMPD2 models match the analytical solution reasonably well. The EMPD1

model matches the general trend: a steep increase at the start of the latent gain, followed by a 

shallower increase shortly after. However, the EMPD1 model predicts a humidity rise1 (or 

amplitude) caused by the latent gain that is 27% below the analytical solution; both the HAMT 

and EMPD2 models predict this humidity rise within 6%. The EC model predicts the humidity 

rise within 7%, but it misses the dynamics that occur at the wall, with a nearly linear humidity 

change. In fact, this 7% “error” is somewhat arbitrary, because the choice of EC = 15 is 

somewhat arbitrary. Using EC = 10 instead results in a humidity rise 38% higher than the 

analytical solution, showing that the effective capacitance results are sensitive to the chosen 

multiplier.

Figure 3. Moisture models compared to an analytical solution. EMPD1 (isothermal) and HAMT 

(isothermal) models were modified so there is no heat of sorption

 1 The humidity rise is the difference between the minimum and maximum zone humidity ratios.

time (hr)

ωzone (kg/kg)

0.005

0.0055

0.006

0.0065

0.007

0.0075

0.008

0 4 8 12 16 20 24

analytical solution

EMPD2 

HAMT (isothermal)

EMPD1 (isothermal)

EC

EMPD2

HAMT (isothermal)

EMPD1 (isothermal) 

EC

analytical solution

internal gain = 0.5 kg/hr
10

The EMPD2 model was simulated with dEMPD-deep set to zero, but the result would be similar if 

the deep buffer layer were included. Similar results were found by researchers using the EMPD2

model without a deep-buffer layer [14, 27, 28]. The deep buffer layer is not needed because there 

is only one cycle period (24 hours) and therefore only one dEMPD. 

Therefore, the difference between EMPD1 and EMPD2 for this simple case is not due to the 

additional deep buffer of the EMPD2 formulation, but rather to the differences in the way the 

EMPD equations are solved (see Section 2.3). The EMPD1 model solves directly for the 

humidity ratio at the air-material interface (Eq. (6)) and does not include the moisture resistance 

for diffusion into the material. The EMPD2 model (Eq. (7)) includes this resistance 

(dEMPD/(2ρairδperm), which accounts for the slower moisture transfer, and therefore faster zone 

humidity change, compared to the EMPD1 model.

The deep buffer layer is used to simulate moisture sorption at longer time scales, as discussed in 

Section 3.3. Before this, we return to the HAMT and EMPD1 models as they are intended to be: 

non-isothermal models, which also illustrates the problems with the EMPD1

v7.1 model.

3.2 Case 1: Internal Latent Gain (Non-Isothermal Models)

This section repeats case 1 with the non-isothermal models: HAMT and EMPD1. These models 

are non-isothermal because they include the heat of sorption at the material surface, and they 

include the dependence of the equilibrium surface moisture content on temperature. We first 

consider the EMPD1 model in EnergyPlus v7.2, and then compare this to the model used in prior 

versions (EMPD1

v7.1).

To exaggerate the effects from the heat of sorption, we use a thermal conductivity of 0.1 W/m-K 

for concrete, which is slightly lower than estimates from Kumaran (0.2 to 0.6 W/m-K) [29]. The 

trend predicted by both models (Figure 4) is in line with expectations; the heat of sorption lowers

the zone’s minimum humidity and raises its maximum humidity. In other words, the moisture 

buffering is decreased. The root cause of this decreased moisture buffering is the inverse 

relationship between temperature and the equilibrium moisture content through the relative 

humidity. Water vapor absorbs into the material, which releases the heat of sorption and 

increases the surface temperature. The higher temperature lowers the relative humidity, which 

decreases the equilibrium moisture content. Therefore, the material absorbs less moisture from 

the room air. 

The added energy at the surface is dissipated through convection into the air and conduction into 

the material. A lower material thermal conductivity slows the rate of heat dissipation into the 

wall, which results in higher temperatures and lower moisture buffering. The process is reversed 

during vapor desorption, where sensible energy must replace the heat of sorption lost at the 

surface. This low thermal-conductivity case illustrates this effect. The humidity rise predicted 

with k = 10 W/m-K for the EMPD1 and HAMT models are within 2% and 0.5% of the 

isothermal model, respectively. For k = 0.1 W/m-K, these humidity rises are higher than the 

isothermal model by 12% and 26%, respectively. 
11

Figure 4. Predicted zone humidity for non-isothermal and isothermal EMPD1 models and nonisothermal and isothermal HAMT models

These results indicate that the difference between isothermal and non-isothermal models depends 

on the material properties, but it also depends on the zone temperature. If we change the latent 

gain to a sensible gain, as done in [14], we find another potential problem with isothermal 

models. For this case, an isothermal model predicts a higher zone temperature, but no change in 

zone humidity. In a non-isothermal model, the higher zone temperature raises the wall surface 

temperature, and therefore decreases the equilibrium moisture content of the wall. This causes 

moisture desorption from the wall into the zone, which increases the zone humidity and 

moderates the rise in surface temperature. This moderating effect essentially adds thermal 

capacitance to the building, using moisture as a phase-change material.

This means a non-isothermal model could predict a different ratio of sensible to latent cooling 

than an isothermal model at different times of the day. It could also mean an air conditioner will 

stay off (i.e., the temperature will stay below set point) for times where an isothermal model 

predicts the air conditioner will turn on. The extent of these effects is unclear, and quantifying 

them is outside of the scope of this study. It is clear, though, that there are many more factors 

that will influence the humidity response of a modeled space when considering the latentsensible coupling of moisture sorption.

Because these non-isothermal effects could be important, we need to know how to include these 

effects in a building simulation. This requires the coupling of the moisture equations with the 

energy equations. This is not always straightforward, as illustrated by the EMPD1

v7.1 model. The 

time (hr)

ωzone (kg/kg)

0.005

0.0055

0.006

0.0065

0.007

0.0075

0.008

0 4 8 12 16 20 24

HAMT (k=0.1 W/m-K)

HAMT (isothermal)

EMPD1 (k=0.1 W/m-K)

EMPD1 (isothermal)

HAMT (0.1 W/m-K)

HAMT (isothermal)

EMPD1 (0.1 W/m-K)

EMPD1 (isothermal)

0

internal gain = 0.5 kg/hr
12

differences between the EMPD1

v7.1 equations and the EMPD1 equations in version 7.2 are 

discussed in Section 2.3.2. 

Figure 5 shows the results from the EMPD1

v7.1 model (using EnergyPlus v7.1) with three 

different values for the material thermal conductivity. This clearly shows that the EMPD1

v7.1

model is incorrect, at least in some situations. The “fixed” model (EMPD1) does not show this 

problem. For k = 0.1 W/m-K, the EMPD1

v7.1 model predicts a humidity below ambient, even 

though there is no moisture sink in the model. This obviously cannot be the case. The results for 

k = 1 W/m-K are higher, but still lower than expected and the results for k = 10 W/m-K approach 

the fixed EMPD1 model. This problem is related only to the coupling of the moisture equations 

with the energy equations; the isothermal model of EMPD1 and EMPD1

v7.1 (shown as EMPD1

(isothermal) in Figure 3) are the same.

In the Appendix, we more thoroughly investigate the EMPD1

v7.1 model and assess its accuracy 

for a range of material properties. We also show that the EMPD1 model in EnergyPlus v7.2 fixes

the problem. This fix is important, because it allows for a non-isothermal EMPD model in 

EnergyPlus, whether it is the EMPD1 model, an EMPD2 model modified to be non-isothermal, 

or another similar lumped wall moisture model. This issue did not, and does not, affect the 

HAMT model because that model uses a finite-difference method to solve the energy equations, 

as opposed to conduction transfer functions.

Figure 5. Predicted zone humidity for non-isothermal HAMT and EMPD1 models. EMPD1 = model 

in EnergyPlus v7.2, EMPD1

v7.1 = model in EnergyPlus v7.1 and earlier. Note that the y-axis was 

expanded relative to previous figures.

time (hr)

ωzone (kg/kg)

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0 4 8 12 16 20 24

HAMT (k=0.1 W/m-K)

EMPD1 (0.1 W/m-K)

EMPD1* (k=10 W/m-K)

EMPD1* (k=1 W/m-K)

EMPD1* (k=0.1 W/m-K)

ωambient

HAMT (0.1 W/m-K)

EMPD1 (0.1 W/m-K)

EMPD1

v7.1

EMPD1

v7.1 (0.1 W/m-K)

(1 W/m-K)

(10 W/m-K)

internal gain = 0.5 kg/hr
13

3.3 Case 2: Cyclical Ambient Humidity

Case 1 was used to illustrate differences between the moisture models’ responses to a simple 

forcing function, compare this response to an analytical solution, and uncover problems with the 

EMPD1

v7.1 model in EnergyPlus. Case 1, though, is far from a realistic building and realistic 

loads. Case 2 moves incrementally toward reality by considering sinusoidal-varying ambient 

humidity. The frequency of this sine wave was selected to not coincide with a multiple of the 

daily fluctuations. This gives more diverse latent loads by combining the ventilation and internal 

gain in different ways throughout the two-month simulation period. 

Moving from a constant ventilation load to a sinusoidal-varying one is important for two reasons:

(1) the added complexity can amplify the small differences between the models seen in Figure 3; 

and (2) the EMPD model relies on a cyclical humidity load to estimate the penetration depth. 

The EMPD2 model uses two values: one for a short-term buffer, and one for a deep buffer. Thus, 

with this case, which has two well-defined cycle periods, both values (dEMPD and dEMPD-deep) can 

be calculated. The EMPD1 model uses only one penetration depth, and cannot account for both 

frequencies at the same time.

Figure 6 shows the zone humidity predicted by the four models over a 7-day period of the 

simulation. If, as expected, the HAMT model is assumed to be the “correct” response, the

EMPD2 model appears to be the most accurate. We quantify this accuracy by calculating, at each 

time step, the percent difference between the zone humidity predicted by the HAMT model 

and that predicted by the other models. We plot these differences, or errors, in a histogram

(Figure 7), where the bins are centered at the label on the x-axis (e.g., –5% bin is –2.5% 

to –7.5%). 

Figure 7 shows that the EMPD2 model matches the HAMT model within ±10% at all time steps, 

and is within ±2.5% for 40% of the simulation. The EC model does not match the HAMT model, 

with errors up to 25%. The EMPD1 model is not much better than the EC model. Two questions 

arise: (1) What accounts for the large difference between the EMPD1 and EMPD2 model 

formulations? and, (2) with the similar results between the EC and EMPD1 models, are there any 

advantages to the EMPD1 model over the EC model? 

For the first question, there are a few possibilities, but we found that the primary cause was that 

the EMPD2 model includes a deep buffer to account for longer term fluctuations. We verified 

that this accounted for most of the difference by simulating the EMPD2 model without a deep 

layer. The results from this simulation (not shown in figure for clarity) have the same day-to-day 

shape as the EMPD2 response in Figure 6, but are stretched to line up with the EMPD1 response. 

This indicates that a deep layer is necessary to account for long-term cycles in humidity loads in 

addition to the typical daily humidity cycles. Two other reasons for the differences are that the 

equations are solved differently, as discussed in Section 2.3, and that the EMPD2 model is 

isothermal and EMPD1 is non-isothermal. These two reasons account for some of the difference, 

but they affect the day-to-day shape of the building humidity and do not account for the 

differences in the long-term fluctuations. These day-to-day effects can be seen in Figure 3 and 

Figure 4. 

To answer the second question, note that the close overlap of the EC and EMPD1 model for this 

case is coincidental. The EC of 15 is based on empirical data for real buildings, including 
14 

furnishings; this is not the building simulated here. However, it indicates that the responses can 

be similar, depending on the parameters used. The EC model requires only an effective 

capacitance multiplier; the EMPD1 model requires dEMPD, du/dφ, and the surface area of the walls 

and furnishings. The EC multiplier is simple, but has a weak link to reality; it is hard to estimate 

without experimental data. The three parameters for the EMPD1 model can still be difficult to 

estimate, but they do have a better link with reality. Adjusting the EMPD1 model is relatively 

straightforward if, for example, the wall area doubles or if a material in the zone is replaced with 

a higher permeability material. 

The loads used in this case (internal gain and ventilation) were cyclical. In other words, dEMPD

for both the surface and deep layers can be calculated for the EMPD2 model. This is not the case 

in reality. Loads, even in a simulation, are highly variable, which is certainly the case for the 

weather. This is explored in Section 3.4. 

Figure 6. Predicted zone humidity over a one-week period during simulations with cyclical 

ambient humidity 

0.004

0.006

0.008

0.01

0.012

0.014

42 43 44 45 46 47 48 49

HAMT

EMPD2

EMPD1

EC=15

Ambient humidity

time (days)

ωzone (kg/kg)

HAMT

EMPD2

EC

EMPD1

ambient humidity
15 

Figure 7. Histogram of errors for EC, EMPD1, and EMPD2 model during simulations with cyclical 

ambient humidity. Error is calculated as deviation from HAMT model. 

3.4 Case 3: Ambient Humidity From Weather Data 

We move still further toward reality with case 3, which uses ambient humidity weather data. 

Note that the ambient temperature is still fixed at 28°C; we are imposing a latent ventilation load 

on the building, but not a sensible load. As in Section 3.3, a sample period from these 

simulations is shown in Figure 8 (two weeks in this case), and a histogram for each model 

showing their deviations from the HAMT model are shown in Figure 9. 

The results for the EMPD2 model indicate that the deep layer thickness of 21 mm seems 

appropriate, as its deviations from the HAMT model are similar to those for the sinusoidal 

humidity case. The results are relatively insensitive to the deep layer thickness, with changes of 

the deep-layer thickness by ±25% changing the zone humidity by 0.1% on average, and by a 

maximum of 3%. The EMPD1 model again shows similar results to the EC model. 

Similar to the last case, the large deviations of the EMPD1 model from the HAMT model result 

from the lack of a deep buffer layer. Thus, for the case considered here, including a deep buffer 

layer is critical. But are there instances where a deep layer is not needed? Figure 3 shows that it 

loses its importance for the case where the humidity load is a single, repeated cycle, as opposed 

to a short-term and a long-term cycle. The properties of the buffering materials also affect the 

0.00

0.15

0.30

0.45

-25% -20% -15% -10% -5% 0% 5% 10% 15% 20% 25%

0.00

0.15

0.30

0.45

-25% -20% -15% -10% -5% 0% 5% 10% 15% 20% 25%

Percent error bins

Fraction in each “percent error” bin 0.00

0.15

0.30

0.45

-25% -20% -15% -10% -5% 0% 5% 10% 15% 20% 25%

EC

EMPD1

EMPD2
16 

importance of the deep layer. For example, if the concrete used in this example were only 10 mm 

thick, a deep layer thickness of 21 mm would overestimate the buffering. Generally, this deep 

layer should always be thinner than the total material. If we calculate a surface layer thickness (τp

= 1 day) and deep layer thickness (τp = 7 days) for a few common materials, we get on the order 

of 10 mm and 25 mm for drywall, 15 mm and 30 mm for soft fabric-like materials, and 1 mm 

and 5 mm for wood. These calculations indicate that wood, like concrete, will generally need a 

deep layer thickness. Drywall will also, but it will usually be thinner than the calculated 25 mm 

because drywall is typically 12-15 mm thick. Some fabric-like materials may require a deep 

layer (e.g., carpet with pad); others may not (e.g., window curtains). 

These more nuanced calculations emphasize the difficulty of predicting the moisture properties 

to predict humidity in buildings. Future work is planned to understand the sensitivity of the 

EMPD model results to both the material properties and to the surface and deep layer 

thicknesses. 

Figure 8. Predicted zone humidity over a two-week period during simulations with ambient 

humidity from weather data 

0.004

0.006

0.008

0.01

0.012

0.014

35 36 37 38 39 40 41 42 43 44 45 46 47 48 49

HAMT

EMPD2

EMPD1

time (days)

ωzone (kg/kg)

EC

EMPD1

HAMT

EMPD2

ambient humidity
17

Figure 9. Histogram of errors for EC, EMPD1, and EMPD2 model during simulations with ambient 

humidity from weather data. Error is calculated as deviation from HAMT model.

0.00

0.15

0.30

0.45

-30% -25% -20% -15% -10% -5% 0% 5% 10% 15% 20% 25% 30%

0.00

0.15

0.30

0.45

-30% -25% -20% -15% -10% -5% 0% 5% 10% 15% 20% 25% 30%

0.00

0.15

0.30

0.45

-30% -25% -20% -15% -10% -5% 0% 5% 10% 15% 20% 25% 30%

Percent error bins

Fraction in “percent error” bin

EC

EMPD1

EMPD2
18

4 Conclusions

The goals of this research are to look at the dynamics of moisture models and discuss their

differences. Our intent is not to endorse, or condemn, a particular model. Moisture transfer 

through a building envelope is complex and not well enough understood to conclude that one 

model is more accurate than another without data from real buildings. That being said, the 

following conclusions were drawn from this research:

• The HAMT and EMPD2 models are close to the isothermal analytical solution. Assuming 

the HAMT model is “correct,” the EMPD2 model is still reasonably accurate for more 

complex cases (a varying latent ventilation load) and has the advantage of a much lower 

simulation run time. The inclusion of the deep buffer layer in the EMPD2 model was 

critical in achieving this accuracy for the case considered here. Some user insight is 

required in calculating this dEMPD-deep value. In general, the inclusion of a deep buffer 

layer is more important for thick materials with a steep slope in the moisture capacitance 

curve and a low moisture diffusivity.

• The advantage of the EC model is its simple implementation. Its response seems less 

realistic than the others based on these simple cases, but in more complicated cases other 

factors could drown out these inaccuracies, particularly for cases where a deep buffering 

layer is unimportant. 

• The EMPD1 model in EnergyPlus versions earlier than 7.2 leads to significant errors, and 

should not be used. Previous results from this model should be used with caution, 

particularly for simulations with a high Ψ-parameter (see the Appendix). The issue has 

been fixed in EnergyPlus version 7.2.

• The EMPD1 model available in EnergyPlus version 7.2 is an improvement over the EC 

model, with more realistic responses to humidity loads (Figure 3). It does not appear to 

be much improved over the EC model for more complex loads, but the parameters can be 

derived from physical entities (surface area, moisture sorption curve, and permeability), 

unlike the EC value, which is empirical (or a guess). 

These conclusions lead to the following next steps:

• There is a need to better understand the sensitivities of the EMPD model to its input 

parameters, particularly the penetration depth. This would indicate how carefully this 

penetration depth needs to be calculated or selected. Next steps in this research are to 

develop a systematic, reliable method for determining this penetration depth for different 

buildings and humidity loads.

• A more in-depth study is needed on the impacts of using an isothermal assumption (i.e., 

is an isothermal model adequate?). This will be more important for some cases than 

others. For example, a simulation that controls zone temperature to a single value will 

likely be less affected than a zone with a varying temperature, such as during the swing 

seasons or from a thermostat setback. In addition to this zone-temperature effect, the 

material properties are also important. All else being equal, low-conductivity materials 

will have a higher surface temperature during moisture sorption than high-conductivity 
19

materials. The ratio of the material’s moisture effusivity to its thermal effusivity, similar 

to the ψ-parameter in the Appendix, could likely be used to gauge the importance of the 

isothermal assumption for a given building load.

• For the cases considered, the EMPD2 model performed better than the EMPD1 model, 

and may be preferred for a model in EnergyPlus. However, the EMPD2 model, as it is 

currently formulated, uses an isothermal assumption. If an isothermal assumption is 

inappropriate, developing a non-isothermal EMPD2 model will be valuable.
20

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22

Appendix

To look more in depth at the problems with the EMPD1

v7.1 implementation, we simulated 40 

cases with different material properties. This is not an exhaustive analysis of this model, nor is it 

meant to be an evaluation of its accuracy. Rather, it is a means to illustrate the problems with the 

model and to identify simulation conditions that amplify or lessen these problems. 

The material properties were selected randomly from a uniform distribution between roughly 

40% and 250% of the nominal values in Table 2. The EMPD1

v7.1 and EMPD1 models were 

evaluated by looking at the ratio of their peak humidity prediction (ωzone,max) with the peak 

humidity prediction of the HAMT model. This differs from the comparisons based on the 

humidity rise used in previous sections.

These ratios for the 40 simulations (Figure A.1) show that the EMPD1

v7.1 model predictions are 

worse (i.e., move closer to the lowest EMPD1

v7.1 line in Figure 5) as the thermal effusivity, eth, 

decreases and the moisture effusivity, em, increases. These are defined as:

th dry p e = kρ c (A.1)

( )

sat

perm dry

m p

e

δ ρ ∂ ∂φ = u

(A.2)

where k is the material thermal conductivity, and cp the specific heat capacity.

In addition to the effusivities, the wall thickness, L, and simulation time step, ∆t, also have an 

effect. The data correlates reasonably well with the following parameter, which is used on the xaxis in Figure A.1:

e L t

e

th

m

∆

Ψ = 1 (A.3)

Figure A.1 shows that the fixed EMPD1 model is close to the HAMT model.
23

Figure A.1. Peak humidity ratio of EMPD1 (×) and EMPD1

v7.1 (○) simulations relative to HAMT 

simulations versus ψ-parameter (defined in Eq. (A.3))

Does the low humidity predicted by the EMPD1

v7.1 model, as shown in Figure A.1, mean that 

any results from the EMPD model in EnergyPlus versions earlier than 7.2 are incorrect? Not 

necessarily; it depends on the Ψ-parameter for that simulation. Low values are likely to be more 

accurate. This Ψ-parameter is linked to actual material properties in Figure A.2, which shows the 

Ψ-parameter for different materials for a range of thicknesses and two different time steps. For 

easier comparison with Figure A.1, the dependent variable (Ψ) is on the horizontal axis. Figure 

A.2 shows that the EMPD1

v7.1 implementation is inappropriate for many combinations of 

materials, especially when those materials are thin and when the simulation is solved with a 

small time step.

ωzone,max (EMPD1)

ωzone,max (HAMT)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5

EMPD1

v7.1

EMPD1

Ψ x106
24

(a)

(b)

Figure A.2. Ψ-parameter for different materials of different thickness: (a) 15-minute time step, (b) 

6-minute time step

0

0.02

0.04

0.06

0.08

0.1

0 0.1 0.2 0.3 0.4 0.5

L (m)

brick

wood 

(across grain)

gypsum

concrete

wood 

(w/grain)

particle board

insulation

∆t = 15 min.

Ψ x106

0

0.02

0.04

0.06

0.08

0.1

0 0.1 0.2 0.3 0.4 0.5

L (m)

brick wood 

(across grain)

gypsum concrete wood 

(w/grain)

particle board

insulation

Ψ x106

∆t = 6 min.