November 7, 2019

**Zhengen Ren, Omid Motlagh, Dong Chen**

Heat loss of a building through the ground becomes relatively important as the above-grade components of the building fabric are built more energy efficient. While analytical methods can provide fast and accurate results for some simple foundation configurations, detailed 3D numerical models can be used for complex configurations with long computation time. For annual space heating and cooling load calculations that require short computation time with reasonable accuracy for modelling heat losses through foundations of interest, this study presents a new correlation-based model. The correlation algorithms were derived from the results of a commercial 3D steady-state and transient heat transfer program (HEAT3, http://buildingphysics.com/heat3-3) using Artificial Neural Network, and the model is known as 3DANN. 3D numerical simulations of ground heat transfer were carried out for a broad range of foundation configurations to generate the data base for the 3DANN correlations.

In general, the annual ground-coupled heat loss, *Q(t)*, varies sinusoidally with time *t* and can be expressed as Equation 1:

*Q(t)* = *Q _{m }+ Q_{a }*cos (ωt – Φ) (1)

Where *Q _{m }* and

*Q*are annual mean heat loss (W) and annual ground-coupled heat loss amplitude (W) respectively,

_{a}*ω*is angular velocity (rad/s),

*t*is time (s) and

*Ф*is phase lag between foundation heat loss and soil surface temperature (rad).

With the knowledge of the three constants (*Q _{m}*,

*Q*and

_{a}*Ф*), the foundation heat loss can be calculated at any time of the year by Equation 1.

*Q*,

_{m}*Q*and

_{a}*Ф*are constants that are functions of various parameters, including building dimensions, insulation configurations, insulation R-values, indoor temperatures, ground surface temperatures, soil thermal properties, etc. As mentioned in [1], the annual mean heat loss (

*Q*), annual amplitude of heat loss (

_{m}*Q*) and the phase lag angle (

_{a}*Φ*) can be estimated as [1]

*Q _{m}* =

*K*× Δ

_{s}*T*×

*F*(

_{Qm}*X*) (2)

*Q _{a}* =

*K*× Δ

_{s}*T*×

_{a}*F*(

_{Qa}*X*) (3)

*Φ* = *F*_{Φ} (*X*) (4)

where *K _{s}*
is slab (or basement wall) thermal conductivity,

*ΔT*annual mean temperature difference between indoor air and ground surface, and

*ΔT*annual amplitude value of ground surface temperature. The three

_{a}*F*functions (

*F*,

_{Qm}(X)*F*and

_{Qa}(X)*F*) need to be determined using regression approach based on the results of HEAT3. In this study, Supervised ANN (.i.e, the Backpropagation network) was used for the regression analysis.

_{Φ}(X)To evaluate the new model, for heat loss through slab-on-ground floor, the results from 3DANN model are compared with the CSIRO analytical solution [2] for steady-state conditions, and the difference between the 3DANN model results and the CSIRO analytical solution is within 6.8%.

For
transient condition, for both of slab-on-ground and basement the calculated
results of *Q _{m}*,

*Q*and

_{a}*Ф*from the 3DANN (Equations 2-4) are compared with values of the HEAT3 results. In all the insulation configurations the linear relationship between the modelled data of 3DANN and HEAT3 is presented, and the R-squared value is close to 1, which shows the 3DANN model agrees with HEAT3 well. We can imagine heat losses calculated by these two models will be close to each other. Figure 1 presents an example of the hourly heat losses for a period of one year calculated by HEAT3 and 3DANN for basement with floor of 400 m

^{2}and wall 1.5m high and slab-on-ground with floor of 300 m

^{2}with two types of soil (average soil properties:

*Ksoil=0.837 W/m.K, Csoil=1.36×10*.K, and fine quartz flour:

^{6}J/m^{3}*Ksoil=2.218 W/m.K, Csoil=2.66×10*) and three levels of horizontal insulation (

^{6}J/m^{3}.K*R0, R1.0*and

*R3.0*) under an indoor air temperature of 21ᵒC and the outdoor ground surface temperature

*T*(ᵒC) of

_{o}*T*. For heat loss through the basement floor the largest difference between the two model is 5.4 W occurred at the ground with

_{o}=15-17cos(ωt)*Ksoil=2.218 W/m.K*and no insulation (i.e.,

*R0*) to the basement floor, which is around 0.6% of the annual mean heat loss at 894.5 W. For slab-on-ground the largest difference of the hourly heat losses between these two models is 11.5 W occurred at the ground with

*Ksoil=0.837 W/m.K*and

*R1.0*, which is less than 1.8% related to the annual mean heat loss of 646.8W. Results show that the thermal conductivity of soil has significant impacts on the heat losses. For the basement with soil conductivity from 0.837 to 2.218 W/m.K, the annual mean heat loss through the floor increased by 159.6% (from 344.6 to 894.5 W), 114.4% (from 291.2 to 624.4 W) and 83.3% (from 233.2 to 427.5 W) for the basement floor horizontally insulated with

*R0*,

*R1*and

*R3*respectively. For slab-on-ground the annual mean heat losses increased by136.2% (from 646.8 to 1527.8 W), 94.3% (from 560.9 to 1089.7 W) and 78.2% (from 471 to 839.4 W) for the floor horizontally insulated with

*R0*,

*R1*and

*R3*respectively.

In
general, soil thermal conductivity depends on soil type, composition,
structure, bulk density, porosity, moisture content, and temperature. In
particular, the moisture content has a strong impact on *Ksoil*, and it varies most substantially under field conditions.
With the development of new ground heat loss model for Chenath engine, the
impacts of *Ksoil* will also be
addressed.

**References**

[1] M. Krarti, S. Choi, Simplified method for foundation heat loss calculation, ASHRAE Transactions 102 (1996) 140-151.

[2] A. Delsante, A. Stokes, P. Walsh, Application of Fourier transforms to periodic heat flow into the ground under a building, International Journal of Heat and Mass Transfer 26 (1983) 121–132.