Ground-Coupled Heat andMoisture Transfer fromBuildingsPart 1: Analysis and Modeling

Preprint

February 2001 � NREL/CP-550-29693

M.P. DeruNational Renewable Energy Laboratory

A.T. KirkpatrickColorado State University

To be presented at the American Solar Energy Society(ASES) National Solar Conferences Forum 2001Washington, D.C.April 21�25, 2001

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1

Ground-Coupled Heat and Moisture Transfer From BuildingsPart 1 � Analysis and Modeling

Michael P. DeruNational Renewable Energy Laboratory

1617 Cole BoulevardGolden CO 80401

303-384-7503 michael_deru@nrel.gov

Allan T. KirkpatrickMechanical Engineering Department

Colorado State UniversityFort Collins CO 80523

970-491-5500 allan@engr.colostate.edu

ABSTRACTGround-heat transfer is tightly coupled with soil-

moisture transfer. The coupling is threefold: heat istransferred by thermal conduction and by moisture transfer;the thermal properties of soil are strong functions of themoisture content; and moisture phase change includes latentheat effects and changes in thermal and hydraulic properties.A heat and moisture transfer model was developed to studythe ground-coupled heat and moisture transfer frombuildings. The model also includes detailed considerationsof the atmospheric boundary conditions, includingprecipitation. Solutions for the soil temperature distributionare obtained using a finite element procedure. The modelcompared well with the seasonal variation of measuredground temperatures.

NOMENCLATURECP = specific heat capacity [J/kg K]

�� ψC,CT = thermal and matric liquid capacitance [K-1, m-1]CTm, Cψm = thermal and matric moisture capacitance [K-1,

m-1]CTv, Cψv = thermal and matric vapor capacitance [K-1, m-1]CTT, CTψ = thermal and matric heat capacitance [J s/m4,

J/m3 K]Da = molecular diffusivity of water vapor in air

[m2/s]Dh, Dv = heat and vapor diffusivities in air

Dm = neutral stability momentum transfer coef. [m/s]DTm, Dψm = thermal and matric moisture diffusivities

[m2/s K, m/s]DTv, Dψv = thermal and matric vapor diffusivities [m/s,

m2/s K]DψT = matric potential heat diffusivity [W/m2]

e = specific internally stored energyE = evaporation rate [s-1]f = vapor diffusion correction factorg = acceleration due to gravity [m/s2]h = convective heat transfer coefficient [W/m2 K]

hfg = latent heat of water vaporization [J/kg]k = thermal conductivity [W/m K]

k* = thermal conductivity of soil with no moisturemovement [W/m K]

K = soil hydraulic conductivity [m/s]

m� = mass flow rate [kg/m2 S]P = pressure [Pa]q = heat flux [W/m2]

Rv = gas constant for water vapor [461.5 J/mole K]T = temperature [K]

(∇T)p/∇T= ratio of microscopic temperature gradient in thepores to the macroscopic temperature gradientover the soil sample

t = time [s]�

u = bulk liquid velocity [m/s]u* = frictional velocity [m/s]W = heat of wettingx = volumetric fractionz = vertical displacement, positive upward [m]

Greek Lettersβ = time-weighting factor used in the transient

solutionϕ = relative humidityΦ = total soil potential for liquid flow [m]η = soil porosityρ = density [kg/m3]�θ = volumetric liquid content [m3/m3]

θk = critical moisture content below which liquidwater is no longer continuous

ξ = temperature gradient ratio term used to calculatesoil thermal conductivity

ψ = matric liquid potential [m]

Subscriptsa = air

amb = ambient conditionsconv = convection

� = liquid waterm = moisturelw = long waveo = referencep = gas-filled poress = solid surface

sens = sensible heatsw = short wave

v = water vaporvs = saturated water vapor

2

1. IntroductionThe heat transfer between a building and the

surrounding soil is complicated by many unknowns, such asthe soil�s physical properties and complex physicalprocesses, many of which involve moisture considerations.For example, heat is transferred by thermal conduction andmoisture transfer; the thermal properties of soil are strongfunctions of the moisture content; and moisture phasechange includes latent heat effects and changes in the soil�sthermal and hydraulic properties. This paper outlines thedevelopment of a model to compute building ground-coupled heat and moisture transfer [1]. In a companionpaper, the model is applied toward determining the heattransfer from building slabs and basements.

There have been numerous works on building ground-coupled heat transfer. Some of the important numericalsolutions to the problem are by Kusuda and Achenbach [2],Wang [3], Mitalas [4], and Bahnfleth [5]. The only work toconsider heat and moisture transfer was by Shen [6].

2. Heat Transfer Paths in SoilHeat transfer in soil occurs by conduction through the

soil grains, liquid, and gases; latent heat transfer throughevaporation-condensation cycles; sensible heat transfer byvapor and liquid diffusion and convection; and radiation inthe gas-filled pores [7,8]. Conduction through the solid soilparticles is the dominant mode of heat transfer under mostcircumstances. The presence of moisture in the soil providesadditional transport mechanisms. For example, in the gas-filled pores of unsaturated soils, liquid water evaporates onthe warm side, absorbing the latent heat of vaporization.Diffusion occurs because of the vapor-pressure gradient, andthe vapor condenses on the other side of the pore, releasingthe latent heat of vaporization. This process increases theoverall thermal conductivity because the effective thermalconductivity of the vapor-distillation cycles is larger thanthat of the gas-filled pores alone. Forced convection frominfiltration of liquid at the ground surface can be significantfor a short time after a large amount of precipitation.

3. Hydraulic and Thermal Properties of SoilThere are three major soil properties that govern the

transport of heat and mass in soils. These are the soil waterpotential, the hydraulic conductivity, and the thermalconductivity. In the model presented in this paper, weassume the total soil water potential to be the sum of thegravitational and matric potentials as presented in Eq. (1),where z is taken as positive upwards. The soil matricpotential is a measure of the attractive force of the capillary

and adsorptive actions of the soil matrix. The potential isoften expressed as an equivalent head of water and thereforehas the dimension of length.

z+ψ=Φ (1)

The relationship between the matric potential and thesoil volumetric moisture content is given by the soil moistureretention curve shown in Fig. 1 for a loamy sand [9], andYolo light clay [10]. Note that the sand and clay are saturatedat moisture contents of 0.395 and 0.495. The flatness of thesandy soil curve between moisture contents of 0.1 and 0.4indicates that the moisture drains more rapidly from thesandy soil over this range relative to the clay soil, which has ahigher attraction to moisture. The temperature dependence ofthe matric potential is approximated through the temperaturedependence of the soil water surface tension. Hysteresisbetween wetting and drying is not included in this model.

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

1.E+02

1.E+03

1.E+04

1.E+05

1.E+06

0.0 0.1 0.2 0.3 0.4 0.5

Volumetric Moisture Content

Mat

ric

Pote

ntia

l (- m

)

Yolo Light Clay

Loamy Sand

Figure 1. Soil moisture retention curves for a loamysand and Yolo light clay

The hydraulic conductivity in unsaturated soil is afunction of soil and fluid properties, moisture content, andtemperature. Curves of the hydraulic conductivity of a loamysand and of Yolo light clay are shown in Fig. 2. Sands willhave a hydraulic conductivity about three orders ofmagnitude greater than clays due to their larger grain size.The temperature dependence of the hydraulic conductivity isapproximated through the temperature dependence of the soilwater kinematic viscosity. The hydraulic conductivity offrozen soil is approximated by the relationship for unfrozensoil using the moisture content that remains unfrozen.

3

Figure 2. Hydraulic conductivity of loamy sand andYolo light clay

Soil thermal conductivity is a function of the moisturecontent; temperature; and the size, shape, orientation,packing, and type of grains that make up the soil matrix.The thermal conductivity is very dependent on the moisturecontent, as changes in the moisture content can affect thethermal conductivity by almost a factor of ten. In this model,the soil thermal conductivity is approximated by a methoddeveloped by de Vries [11]. It is assumed that the soilconsists of a continuous medium with an even distributionof ellipsoidal-shaped grains. The thermal conductivity isthen given by Eq. (2). This equation represents an averageof the thermal conductivities of water w, pores p, and ntypes of soil grains (including ice), weighted by thevolumetric contents x and the ratio of the averagetemperature gradient in the constituent and the averagetemperature gradient of the medium ξ. Water is consideredto be the continuous medium, except at very low moisturecontents, where air is used.

�

�

=

=

ξ+ξ+

ξ+ξ+= n

1i iippw

n1i iiipppww

xxx

kxkxkxk (2)

The main driving force for the vapor diffusion in thesoil pores is the vapor density gradient resulting fromtemperature gradient. Because temperature is the maindriving force, this heat transfer is treated as an effective heatconduction, and the thermal conductivity of the gas-filledpores is estimated as the sum of the thermal conductivity ofthe air ka and the effective thermal conductivity of the vapordiffusion kv.

vap kkk += (3)

The effective thermal conductivity of vapor distillation cyclesfor a single pore can be expressed as

dTd

Dhk vsafgv

ρϕ= (4)

The effective thermal conductivity of Bighorn sandyloam as a function of moisture content with and without thevapor diffusion term is shown in Fig. 3 for two differenttemperatures. As the temperature increases, the contributionof the vapor diffusion term increases. The thermalconductivity of frozen soil relative to the unfrozen valuedecreases for low moisture contents and increases for highmoisture contents [12].

Figure 3. Effective thermal conductivity of sandy loam

4. Coupled Heat and Moisture Transfer ModelThe heat and mass transfer equations developed in this

work follow an approach based on physical models of theprocesses that occur in the soil [7,13]. The moisture transferequations are cast in terms of the matric potential (ψ-basedequations) because this approach can handle saturationconditions. The main simplifying assumptions made in thederivations are that the soil is assumed to be homogenous andisotropic within each defined unit of soil.

4.1. Liquid TransferDarcy�s law can be extended to unsaturated soil by using

the gradient of the total potential, Φ, and defining thehydraulic conductivity, K (m/s), as a function of the soilwater matric potential.

Φψ−= ∇∇∇∇)(K�

u (5)

Substitution of Eq. (1) into Eq. (5) yields

k�KK −ψ−= ∇∇∇∇�u (6)

1.E-23

1.E-20

1.E-17

1.E-14

1.E-11

1.E-08

1.E-05

1.E-02

0.0 0.1 0.2 0.3 0.4 0.5

Volumetric Moisture Content

Hyd

raul

ic C

ondu

ctiv

ity (m

/s)

Yolo Light Clay

Loamy Sand

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

Volumetric Moisture Content

The

rmal

Con

duct

ivity

(W/m

K)

With Vapor Diffusion (20 C)W/O Vapor Diffusion (20 C)With Vapor Diffusion (40 C)W/O Vapor Diffusion (40 C)

4

By applying the continuity equation to the liquidmoisture content of a control volume of soil, theconservation of liquid can be written as

Et

−⋅−=∂θ∂

�

� u∇∇∇∇ (7)

The liquid content is a function of the matric potential andtemperature; therefore, the time derivative of the liquidcontent can be expanded by the chain rule to give

[ ] EzKK

tTC

tC T −

∂∂+ψ⋅=

∂∂+

∂ψ∂

ψ ∇∇∇∇∇∇∇∇��(8)

4.2. Vapor TransferThe vapor diffusion in a gas-filled pore can be

approximated by modifying Fick's law of diffusion,assuming a uniform and constant total pressure, P, and thatthe vapor behaves as an ideal gas [8,14].

( ) ���

����

� ρ+ρ−= p

vvav T

TD ∇∇∇∇∇∇∇∇m� (9)

The molecular diffusivity of water vapor in air, Da, is givenby Eq. (10), and (∇∇∇∇T)P is the temperature gradient across asingle gas-filled pore [8].

n

o

oa T

TPP

cD ���

����

���

���

�= (10)

The reference pressure is Po = 1.01325 x 105 Pa, thereference temperature is To = 273.15 K, c = 2.17 x 10-5 m2/s,and n = 1.88. The vapor density in the pores can beexpressed as the product of the relative humidity, ϕ, and thesaturated vapor density, ρvs.

ϕρ=ρ vsv (11)

Assuming that the soil liquid and vapor are inthermodynamic equilibrium, and in the absence of solutes,the relative humidity in the soil pores may be written as [15]

���

����

� ψ=ϕTRgexp

v(12)

From Eq.�s (11) and (12) the vapor diffusion equationcan be written as

( )��

�

�

��

�

�

∂ψ∂+

ψ−

���

�

�+

∂ρ∂

ρ+ψϕρ−=

θp

v2

v

vs

vsvvsav

TTTR

gTRg

T1

T1

TRgD

∇∇∇∇

∇∇∇∇m�

(13)

Only the first of the four temperature gradient coefficientterms is significant for most situations. The last two terms areimportant for ψ < -104 m [1]. When applying the vapor massflux given by Eq. (13) to a porous soil system, the effects ofthe reduced cross section for diffusion, the tortuosity, and theinteractions between the vapor and liquid phases must beaddressed. A vapor diffusion correction factor, )(f

�θ , is

added to account for these effects [7].

kkaa

k

for),/()(ffor,)(f

θ>θθ−ηθθ+θ=θθ≤θη=θ

���

�� (14)

The critical moisture content, θk, is that below which thehydraulic conductivity falls to a value much lower than itsvalue at saturation. Deru [1] presents a method ofdetermining this value based on the relative humidity of thesoil moisture. The vapor mass flux of Eq. (13), divided by theliquid density for consistency with the liquid-transferequation can be written for the soil system as

TDD Tvvv ∇∇∇∇∇∇∇∇ −ψ−=

ρ ψ�

�m(15)

where the matric and thermal vapor diffusivities are

TRg

D)(fDv

vsav

ϕρρ

θ=ψ�

�(16)

( )( )T

T

TTRg

TRg

T1D)(fD

p

v2

v

vs

vs

vsaTv

∇∇∇∇

∇∇∇∇��

�

�

∂ψ∂+

ψ−

���

�

∂ρ∂

ρρρ

ϕθ=

θ

�

�

(17)

The temperature gradients across the gas-filled pores arehigher than those across the system because of the lowerthermal conductivity of the gas-filled pores, which isaccounted for by the ratio of the average temperaturegradient across the pores and the temperature gradientacross the system.

The vapor content is expressed as an equivalent liquidcontent. Assuming thermodynamic equilibrium between theliquid and vapor, we can write

�

�

ρρθ−η

=θ vv

)((18)

Expanding ( )t/v ∂θ∂ , and using the chain rule to include thedependence on moisture and temperature yields

tTC

tC

t Tvvv

∂∂+

∂ψ∂=

∂θ∂

ψ (19)

5

where the matric and thermal vapor capacitances are

��

�

�

��

�

�

���

�

�

ψ∂θ∂

−θ−η

ρϕρ

=ψT

TRg)(

Cv

vsv

��

�

(20)

��

�

�

��

�

�

���

�

�

∂θ∂

−���

�

�

∂ρ∂

ρθ−η

ρϕρ

=ψ

TT)(

C vs

vs

vsTv

��

�

(21)

The conservation of vapor content can be written as

E)/(t vv +ρ⋅−=

∂θ∂

��m∇∇∇∇ (22)

The governing equation for vapor transfer can now bederived by substitution of Eq.�s (15) and (19) into (22).

[ ] [ ] ETDDtTC

tC TvvTvv +⋅+ψ⋅=

∂∂+

∂ψ∂

ψψ ∇∇∇∇∇∇∇∇∇∇∇∇∇∇∇∇ (23)

4.3. Total Moisture TransferThe moisture transfer equation is then found by

combining the equations for liquid and vapor transport,Eq.�s (8) and (23).

[ ][ ]

zKD

DtTC

tC

Tm

mTmm

∂∂+⋅+

ψ⋅=∂∂+

∂ψ∂

ψψ

∇Τ∇Τ∇Τ∇Τ∇∇∇∇

∇∇∇∇∇∇∇∇(24)

4.4. Heat TransferBy applying the conservation of energy on the soil

system we can write

( ) qte ′′′+⋅−=

∂ρ∂ q∇∇∇∇ (25)

Neglecting the heat capacity of the air, the thermal energy ofthe soil system relative to a reference temperature, To, canbe written as [16]

�θ θθρ−

θρ+−θρ+−θρ+−η−ρ=ρ

�

�

��

���

0

0fgvov,pv

o,pos,ps

d)(W

)T(h)TT(C)TT(C)TT(C)1(e

(26)

The heat of wetting, W, is the energy released when thewater molecules adsorb on the surface of the soil grains. It isa function of the initial moisture content and surface area. Itis only significant for wetting dry clay soils and is neglectedin this work.

The heat flux in the soil can be represented by

vov,p

o,pvofg*

)TT(C)TT(C)T(hTk

mmmq

�

����

−+−++−= ∇∇∇∇ (27)

The thermal conductivity k* represents the pure heatconduction through the soil system with no moisturemovement. Note that this is different from the thermalconductivity k measured in a soil system and calculated bythe de Vries method, which includes the effect of latent heattransfer by vapor distillation.

The total volumetric heat capacity of the soil system andthe latent heat of vaporization at temperature T can bedefined as

v,pv,ps,ps CCC)1(C����

ρθ+ρθ+ρη−= (28)

)TT(C)TT(C)T(h)T(h opvopofgfg −+−−=�

(29)

With the substitution of Eq.�s (32) and (33) andsubstitutions for the liquid and vapor mass fluxes the finalversion of the governing equation for soil heat transfer is

( ) ( )qTC

DTkt

CtTC

,p

TTTT

′′′+⋅+

ψ⋅+⋅=∂ψ∂+

∂∂

ψψ

∇∇∇∇

∇∇∇∇∇∇∇∇∇∇∇∇∇∇∇∇

���m

(30)

The heat and mass transfer equations are coupled by thetemperature and moisture terms occurring in both equations.The first term on the right side of Eq. (30) represents theFourier heat conduction defined by the thermal conductivityk* and the effective heat conduction by thermally drivenvapor diffusion. The effective thermal conductivity, k, is thevalue calculated by the de Vries method. The second term isthe heat transfer by moisture diffusion due to the moisturegradient. The third term is the sensible heat transfer by bulkliquid flow. In a subsequent section of the paper the relativemagnitudes of these terms will be compared. The first termon the left hand side of Eq. (30) relates the change in thethermal energy stored to changes in temperature and thesecond term relates the change in thermal energy stored tochanges in the matric potential.

5. Surface Energy BalanceAn energy balance at the ground surface is defined by

0qqqqq vconvradsensg =++++ (31)

The first term in Eq. (31) represents heat conduction in theground due to gradients of temperature and matric potential.

nD

nTDq TTTg ∂

ψ∂−∂∂−= ψ (32)

The sensible heat, qsens, is divided into two sources: thatdue to liquid transfer across the boundary, �q , and anotherterm, qh, to account for other miscellaneous heattransfersources. The temperature of the liquid, �T , at theatmospheric boundary is assumed to be equal to the ambientdry-bulb temperature and the reference temperature is To.

6

hsens qqq +=�

(33)

)TT(Cuq op −ρ=����

(34)

The radiation term, qrad, can be divided into absorbed short-wave radiation from the sun, qsw, and the long-waveradiation exchange between the ground and the atmosphere,qlw , and is discussed further in [1]. The long-waveradiation exchange model includes the effects of cloud coverfollowing Martin and Berdahl [17].

The heat and mass transfer processes from the groundto the air behave similarly and, thus, the governingequations for qconv and qv have a similar form. The basicequations for convective heat and vapor transfer are

)TT(DC)TT(hq

sambha,Pa

sambconv

−ρ==

(35)

)(Dhq s,vamb,vvfgv ρ−ρ= (36)

The convection and evaporation depend on the wind speed,the surface conditions, and the gradients of temperature andvapor density, and are discussed in more detail in [1].

6. Surface Moisture BalanceThe moisture balance at the ground surface is

0vg,m =++ mmm ����

(37)

The ground moisture transfer is represented by Eq.(38). This term includes the liquid transfer due to thegradient of the total potential, Φ, and vapor transfer due togradients of the matric potential and temperature.

nTD

nD

nK

mTmv

g,m

∂∂−

∂ψ∂−

∂Φ∂−=

ρ ψ�

(38)

The liquid transfer at the surface is assumed to be in theform of precipitation. Moisture is assumed to accumulate upto the point of saturation, and any additional moisture isassumed to run off. The main form of vapor transfer is byevaporation from the surface.

)(Dm s,vamb,vvv ρ−ρ=� (39)

Using aerodynamic resistance alone has been shown toover-predict the evaporation from bare soil [18] and fromplant canopies [19]; therefore, a surface resistance is addedin series, as recommended by Camillo and Gurney [18].

7. Finite Element FormulationBecause of the complexity and nonlinearity of the heat

and moisture transfer equations, an analytic solution wouldbe extremely difficult to obtain. A numerical solution isobtained using a finite element method (FEM) formulation.The heat transfer equation, Eq. (30), and the moisturetransfer equation, Eq. (24), are transformed into thefollowing equations for the FEM.

0TTTTTTT =++++ ψψ fψCTCψDTD �� (40)

0mTmmTm =++++ ψψψ fψCTCψDTD �� (41)

The two sets of equations (40) and (41) can be combinedinto one system assembled by alternating equations for T andψ for each node to reduce the bandwidth of the coefficientmatrices as shown below.

0=++ fUCUD � (42)

A single-step algorithm for the transient analysis ispresented in Eq. (43), where the subscript k denotes the timestep and β is a time-weighting function, which determineswhether the method is explicit or implicit [20]. All of theanalyses completed in this work applied a time-weightingfunction of 0.5, which is analogous to a Crank-Nicolsonimplicit routine. The forcing function is averaged assuming alinear variation in time as in Eq. (44).

[ ] [ ] fUDCUDC tt)1(t k1k ∆−∆β−−=∆β+ + (43)

( )k1kk ffff −β+= + (44)

Equation (42) is nonlinear because the coefficientmatrices are functions of the dependent variables. Thesolution of the moisture transfer equation is also subject tomass balance errors due to the highly nonlinear relationshipbetween the moisture content and the matric potential[21,22]. According to Celia, et al. [22] diagonalizing orlumping the capacitance matrix is necessary to ensure a non-oscillatory and mass conservative solution. The nonlinearequations are solved using a modified Picard iterationmethod. A modified skyline storage technique is used toreduce the storage requirements. The solution of Eq. (43) iscarried out using a Gaussian elimination routine.

8. Verification and ValidationThe model was successfully verified against the analytic

solution to a two-dimensional heat conduction problem of arectangular region initialized at T = 0.0°C and boundaryconditions of T = 0.0°C for three sides and T = 100°C for thefourth side [1]. The program was also verified, with excellentagreement, against the analytical solution of an isothermalmoisture infiltration problem presented by Philip [23].

To validate the heat and moisture transfer models and theatmospheric boundary condition models, they were comparedwith field data. Weather data and ground temperatures atdepths to 1 m deep were measured from October 29, 1998, toJune 7, 1999, in an open field on the Colorado StateUniversity campus. A one-dimensional ground temperaturesimulation was conducted to a depth of 3.0 m, with SolarVillage clay [24]. The simulation period extended fromDecember 28 to June 8. This represented a time period withgood data and minor snow cover. The mesh varied from 0.74cm at the surface and increased with depth into the ground.The upper boundary was modeled using the measuredweather data. Assumptions about the surface conditions were:ground cover height = 0.1 m, ground infrared emissivity =0.9, and ground albedo = 0.23, which is representative ofaverage crop cover. The precipitation data was obtained froma nearby weather station. Prior to December 28, a largesnowfall had just melted and so the matric potential initial

7

condition was set at ψ(z) = �0.1 + z*0.05 to simulate near-saturated soil. The lower boundary condition was set with atemperature T = 10 oC and matric potential ψ = �0.25 m.

Figure 4. Atmospheric conditions and a comparison ofthe measured and computed surface temperatures

The weather conditions, along with the measured andpredicted surface temperatures for the period betweenApril 27 and May 17, are shown in Fig. 4. Predicting thesurface temperature is very sensitive to the atmosphericboundary conditions and is often too high or too low. Theover-predicted surface temperatures during the day seemto be caused by a combination of the incident solarradiation term and the convection term. During the periodfrom April 28 to May 2, a total of 12.8 cm of precipitationwas measured at the weather station. Opaque cloud coverwas assumed for one hour before, during, and for twohours after any precipitation.

Figures 5 and 6 show the measured and predictedtemperatures along with the root mean square error at depthsof z = �0.34 m and �0.95 m. The curves show the decreasein the temperature fluctuation with depth and the seasonalwarming of the soil. The predicted values at �0.34 m arevery close to the measured temperatures with the root meansquare error less than 1.0°C. The values at �0.95 m are also

good, but the error is slightly larger. The error is the leastduring the spring and summer. If the precipitation is notincluded in the model, the calculated ground temperatures arelarger than the measured temperatures, and the errorincreases in time. Incorporating the precipitation allows for amore accurate computation of the soil moisture, and thus thesoil thermal conductivity.

Figure 5. Comparison of measured and predicted soiltemperatures at z = -0.34 m

Figure 6. Comparison of measured and computed soiltemperatures at z = -0.95 m

Figures 7 and 8 show the magnitude of the four heatflux terms: heat conduction, vapor driven by temperaturegradients, vapor driven by matric potential gradients, andliquid convection at depths of z = �0.35 m and �0.95 m. Apositive value represents heat flow into the ground from theatmosphere. At z = �0.35 m, the maximum heat flux intothe ground was about 40 W/m2, while at z = �0.95 m, themaximum heat flux was about 6 W/m2 . The conductionheat transfer is usually the dominant term. Note theseasonal change in sign of the conduction term on May 10,from negative to positive heat flow. After precipitation, theheat flux by liquid convection, which is normally verysmall, dominates for a short time. The heat transfer by

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vapor distillation is small at these depths, because of therelatively cold soil temperatures, and high moisturecontents. The heat transfer of vapor driven by matricpotential gradients was two orders of magnitude smallerthan the heat transfer by vapor driven by temperaturegradients, and is not plotted here.

Figure 7. Conduction, vapor distillation, and liquidconvection heat flux terms at z = -0.35 m.

Figure 8. Conduction, vapor distillation, and liquidconvection heat flux terms at z = -0.95 m.

9. Summary and ConclusionsA heat and moisture transfer model was developed to

study the coupled heat and moisture transfer in soils. Usinga matric potential formulation, the soil properties areexpressed as functions of soil type and moisture content.The model includes a detailed treatment of the surface heatand moisture balances and includes the effects ofvaporization in the soil and surface precipitation. Theprogram compared well with measured ground temperaturesfor a one-dimensional field test case. The conduction heattransfer was usually the dominant term. After a period ofprecipitation, the heat flux by liquid convection, which wasnormally small, dominates for a short time.

ACKNOWLEDGMENTSThe funding for this work from the National Renewable

Energy Laboratory and the American Society of Heating,Refrigerating, and Air Conditioning Engineers was greatlyappreciated.

REFERENCES[1] Deru, M., 2001, �Ground-Coupled Heat and MoistureTransfer From Buildings,� Ph.D. Dissertation, ColoradoState University, Fort Collins, CO.

[2] Kusuda, T., and Achenbach, P.R., 1963, �NumericalAnalyses of the Thermal Environment of OccupiedUnderground Spaces With Finite Cover Using a DigitalComputer,� ASHRAE Transactions, 69, pp. 439�447.

[3] Wang, F.S., 1979, " Mathematical Modeling andComputer Simulation of Insulation Systems in Below GradeApplications,� Proc. ASHRAE/DOE-ORNL Conf., ThermalPerformance of the Exterior Envelopes of Buildings,Kissimmee, FL.

[4] Mitalas, G.P., 1982, �Basement Heat Loss Studies atDBR/NRC,� National Research Council of Canada, Divisionof Building Research, DBR Paper No. 1045.

[5] Bahnfleth, W.P., 1989, �Three-Dimensional Modeling ofHeat Transfer From Slab Floors,� National TechnicalInformation Service, Springfield, VA, ADA210826.

[6] Shen, L.S., 1986, �An Investigation of Transient, Two-Dimensional Coupled Heat and Moisture Flow in Soils,�Ph.D. Thesis, University of Minnesota, Minneapolis, MN.

[7] de Vries, D.A., 1958, �Simultaneous Transfer of Heatand Moisture in Porous Media,� Trans. AmericanGeophysical Union, 39(5), pp. 909�915.

[8] de Vries, D.A., 1975, �Heat Transfer in Soils,� Heat andMass Transfer in the Biosphere, Part 1 Transfer Processes inthe Plant Environment, de Vries, D.A. and Afgan, N.H. eds.,John Wiley & Sons, New York, pp. 5-28.

[9] Noborio, K., McInnes, K.J., and Heilman, J.L., 1996,�Two-Dimensional Model for Water, Heat, and SoluteTransport in Furrow-Irrigated Soil: II. Field Evaluation,� SoilScience Society of America Journal, 60, pp. 1001-1009.

[10]Moore, R.E., 1939, �Water Conduction From ShallowWater Tables,� Hilgardi, 12(6), pp. 383�426.

[11]de Vries, D.A., 1966, �Thermal Properties of Soils,�Physics of Plant Environment, Van Wijk W.R. ed., North-Holland Publishing Company, Amsterdam.

[12]Kersten, M.S., 1949, June 1, Thermal Properties ofSoils, University of Minnesota Institute of Technology,Bulletin 28, Vol. LII, n. 21, Minneapolis, MN.

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)

Liquid Convection

Conduction

Vapor (∆T)

Z = -0.35 m

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6

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[13]Philip, J.R. and de Vries, D.A., 1957, �MoistureMovement in Porous Materials Under TemperatureGradients,� Transactions American Geophysical Union,38(2), pp. 222�232.

[14]Nakano, M. and Miyazaki, T., 1979, �The Diffusionand Nonequilibrium Thermodynamic Equations of WaterVapor in Soils Under Temperature Gradients,� Soil Science,128(3), pp. 184�188.

[15]Edlefsen, N.E. and Anderson, A.B.C., 1943,�Thermodynamics of Soil Moisture,� Hilgardia, 15,pp. 31-298.

[16] Jury, W.A., 1973, Simultaneous Transport of Heat andMoisture Through a Medium Sand, Ph.D. Thesis, Universityof Wisconsin, Madison, WI.

[17]Martin, M. and Berdahl, P., 1984, �Characteristics ofInfrared Sky Radiation in the United States,� Solar Energy,Vol. 33, No. 3/4, pp. 321-336.

[18]Camillo, P.J., and Gurney, R.J., 1986, �A ResistanceParameter for Bare-Soil Evaporation Models,� Soil Science,141(2), pp. 95�105.

[19]Shuttleworth, W.J., 1993, �Evaporation,� in Handbookof Hydrology, Maidment, D.R., Ed., McGraw-Hill Inc., NewYork.

[20]Zienkiewicz, O.C. and Taylor, R.L., 1994, The FiniteElement Method, fourth edition, McGraw-Hill BookCompany, London.

[21]Milly, P.C.D., 1985, �A Mass-Conservative Procedurefor Time-Stepping in Models of Unsaturated Flow,�Advances in Water Resources, 8, pp. 32�36.

[22]Celia, M.A., Bouloutas, E.T., and Zarba, R.L., 1990,�A General Mass-Conservative Numerical Solution for theUnsaturated Flow Equation,� Water Resources Research,26(7), pp. 1483�1496.

[23]Philip, J.R., 1957, �The Theory of Infiltration: 1. TheInfiltration Equation and Its Solution,� Soil Science, 83,pp. 435�448.

[24]Hampton, D., 1989, Coupled Heat and Fluid Flow inSaturated-Unsaturated Compressible Porous Media, PhDDissertation, Colorado State University, Fort Collins, CO.

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1. AGENCY USE ONLY (Leave blank) 2. REPORT DATEFebruary 2001

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4. TITLE AND SUBTITLEGround-Coupled Heat and Moisture Transfer from Buildings; Part 1: Analysis andModeling

6. AUTHOR(S)Michael P. Deru and Allan T. Kirkpatrick

5. FUNDING NUMBERS

BET1.4001

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)National Renewable Energy Laboratory1617 Cole Blvd.

Golden, CO 80401-3393

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13. ABSTRACT (Maximum 200 words)Ground-heat transfer is tightly coupled with soil-moisture transfer. The coupling is threefold: heat is transferred by thermalconduction and by moisture transfer; the thermal properties of soil are strong functions of the moisture content; and moisturephase change includes latent heat effects and changes in thermal and hydraulic properties. A heat and moisture transfermodel was developed to study the ground-coupled heat and moisture transfer from buildings. The model also includesdetailed considerations of the atmospheric boundary conditions, including precipitation. Solutions for the soil temperaturedistribution are obtained using a finite element procedure. The model compared well with the seasonal variation of measuredground temperatures.

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buildings; heat and moisture transfer 16. PRICE CODE

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