Steady-State Component of Three-Dimensional Pirawas ... for ThaiScience/Article/3/10003225.pdf ·...
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Pirawas Chuangchid Moncef Krarti Joint Center for Energy Management, CEAE Dept., CB 428 University of Colorado at Boulder, Boulder, CO 80309 Steady-State Component of Three-Dimensional Slab-on-Grade Foundation Heat Transfer General semi-analytical solutions for the steady-state heat conduction problems for cir- cular and three-dimensional rectangular slab-on-grade floors with uniform insulation are presented. The soil temperature field, and the total slab heat loss are presented and analyzed using the Interzone Temperature Profile Estimation (ITPE) technique. A para- metric analysis is conducted to determine the effect of thermal insulation U-value, slab size, and water table depth on the total slab heat loss. In particular, it was found that the total slab heat loss is independent of its shape but is strongly affected by the slab size and thermal characteristics. [DOI: 10.1115/1.1346677] 1 Introduction Most of the common methods available for calculating ground heat losses are based on two-dimensional solutions. For instance, the decrement factors of Akridge and Poulos [1] as well as the correlations of Yard et al. [2] and Ship [3] were generated from two-dimensional finite difference simulations. Similarly, Mitalas's method [4,5] was generated from a two-dimensional finite- element computer program. Only a few computer codes were de- veloped for three-dimensional earth-coupling heat flow. Kusuda and Achenbach [6] developed a three-dimensional explicit finite difference computer program using several simplifying assump- tions. Their program analyzes the simultaneous heat and water vapor transport from an underground shelter to earth. With a com- bination of finite element and finite difference methods, Metz [7] developed a computer program called GROCS (GROund Coupled Systems) to analyze three-dimensional underground heat flow. Walton [8] developed a procedure to approximate the three- dimensional comer effects of foundation heat transfer from rect- angular basement and slabs by using a two-dimensional finite dif- ference simulation. Finally, Bahnfleth and Petersen [9] developed a three-dimensional model of heat transfer from slab floors using a finite difference approach. Lachenbrunch [10] and Delsante et al. [11] have attempted to find an analytical solution of the three-dimensional slab-on-grade problem. Although their work is a major contribution to the earth- coupling heat transfer theory, both Lachenbruch and Delsante et al. [10,11] do not allow inclusion of any thermal insulation on the floor surface. In fact, for his Green's function solution, Lachenbruch has assumed a prescribed slab temperature (i.e., a first-kind boundary condition at the slab surface). The same as- sumption was made for the Fourier-type solution of Delsante. However, to avoid large temperature gradients at the edges, Del- sante considered a perimeter zone around the slab across which the temperature changes linearly from the soil surface temperature to the prescribed floor temperature. A semi-analytical solution for three-dimensional rectangular slab-on-grade floors was developed by Krarti et al. [12] using the Interzone Temperature Profile Estimation (ITPE) technique. In his work, the boundary condition of the soil surface was assumed to be constant and equal to the soil surface temperature. Moreover, Contributed by the Solar Energy Division of THE AMERICAN SOCIETY OF ME- CHANICAL ENGINEERS for publication in the ASME JOURNAL OF SOLAR ENERGY ENGINEERING. Manuscript received by the ASME Solar Energy Division, Sept. 1998; final revision, Dec. 2000. Associate Editor: D. E. Claridge. Krarti et al. [12] limited the soil medium to the zone just beneath the slab. The temperature profiles at the boundary surfaces of that zone were estimated based on the two-dimensional solution devel- oped by Krarti et al. [13]. Using the same technique as Krarti et al. [12], this paper devel- ops analytical solutions for the three-dimensional heat transfer beneath slab foundation. Two types of slab shape are considered in this paper, including circular and three-dimensional rectangular slab-on-grade floors. The second section of this paper develops the circular slab solution. The soil temperature profile and the total slab heat loss are presented for various insulation configura- tions applied to a circular slab. A parametric analysis is conducted to determine the effect of thermal insulation U-value, slab size, and water table depth on the circular slab heat loss. In the third section, the solution of the three-dimensional rectangular slab-on- grade floor is provided. Analysis of the temperature gradient as well as of the total slab heat loss for several insulation configura- tions is presented. Finally, the fourth section presents the variation of the foundation heat loss as a function of the ratio of slab area and slab perimeter for all the slab shapes considered in this paper. 2 Circular Slab-on-Grade Floor 2.1 Formulation of the Problem. Figure 1 shows a model of a circular slab-on-grade floor with uniformly insulated slab. The footings of the foundation are not shown because they are usually made up of concrete, which has a thermal conductivity that, in most cases, is similar to that of the ground. Therefore, the foundation elements are considered as integral parts of the soil medium. The analysis presented in this paper considers the annual aver- age thermal performance of the foundation and thus, is limited to steady-state conditions. Time-varying analysis such as monthly and/or seasonal calculation of foundation heat loss/gain are the subject of another paper [14]. Under these conditions, the tem- perature distribution within the ground medium, T(r,z) is subject to the Laplace equation: with the following boundary conditions:
Transcript of Steady-State Component of Three-Dimensional Pirawas ... for ThaiScience/Article/3/10003225.pdf ·...
Pirawas Chuangchid
Moncef Krarti
Joint Center for Energy Management,
CEAE Dept., CB 428
University of Colorado at Boulder,
Boulder, CO 80309
Steady-State Component ofThree-DimensionalSlab-on-Grade Foundation HeatTransferGeneral semi-analytical solutions for the steady-state heat conduction problems for cir-cular and three-dimensional rectangular slab-on-grade floors with uniform insulation arepresented. The soil temperature field, and the total slab heat loss are presented andanalyzed using the Interzone Temperature Profile Estimation (ITPE) technique. A para-metric analysis is conducted to determine the effect of thermal insulation U-value, slabsize, and water table depth on the total slab heat loss. In particular, it was found that thetotal slab heat loss is independent of its shape but is strongly affected by the slab size andthermal characteristics. [DOI: 10.1115/1.1346677]
1 Introduction
Most of the common methods available for calculating groundheat losses are based on two-dimensional solutions. For instance,the decrement factors of Akridge and Poulos [1] as well as thecorrelations of Yard et al. [2] and Ship [3] were generated fromtwo-dimensional finite difference simulations. Similarly, Mitalas'smethod [4,5] was generated from a two-dimensional finite-element computer program. Only a few computer codes were de-veloped for three-dimensional earth-coupling heat flow. Kusudaand Achenbach [6] developed a three-dimensional explicit finitedifference computer program using several simplifying assump-tions. Their program analyzes the simultaneous heat and watervapor transport from an underground shelter to earth. With a com-bination of finite element and finite difference methods, Metz [7]developed a computer program called GROCS (GROund CoupledSystems) to analyze three-dimensional underground heat flow.Walton [8] developed a procedure to approximate the three-dimensional comer effects of foundation heat transfer from rect-angular basement and slabs by using a two-dimensional finite dif-ference simulation. Finally, Bahnfleth and Petersen [9] developeda three-dimensional model of heat transfer from slab floors usinga finite difference approach.
Lachenbrunch [10] and Delsante et al. [11] have attempted tofind an analytical solution of the three-dimensional slab-on-gradeproblem. Although their work is a major contribution to the earth-coupling heat transfer theory, both Lachenbruch and Delsanteet al. [10,11] do not allow inclusion of any thermal insulation onthe floor surface. In fact, for his Green's function solution,Lachenbruch has assumed a prescribed slab temperature (i.e., afirst-kind boundary condition at the slab surface). The same as-sumption was made for the Fourier-type solution of Delsante.However, to avoid large temperature gradients at the edges, Del-sante considered a perimeter zone around the slab across whichthe temperature changes linearly from the soil surface temperatureto the prescribed floor temperature.
A semi-analytical solution for three-dimensional rectangularslab-on-grade floors was developed by Krarti et al. [12] using theInterzone Temperature Profile Estimation (ITPE) technique. In hiswork, the boundary condition of the soil surface was assumed tobe constant and equal to the soil surface temperature. Moreover,
Contributed by the Solar Energy Division of THE AMERICAN SOCIETY OF ME-CHANICAL ENGINEERS for publication in the ASME JOURNAL OF SOLAR ENERGYENGINEERING. Manuscript received by the ASME Solar Energy Division, Sept.1998; final revision, Dec. 2000. Associate Editor: D. E. Claridge.
Krarti et al. [12] limited the soil medium to the zone just beneaththe slab. The temperature profiles at the boundary surfaces of thatzone were estimated based on the two-dimensional solution devel-oped by Krarti et al. [13].
Using the same technique as Krarti et al. [12], this paper devel-ops analytical solutions for the three-dimensional heat transferbeneath slab foundation. Two types of slab shape are consideredin this paper, including circular and three-dimensional rectangularslab-on-grade floors. The second section of this paper developsthe circular slab solution. The soil temperature profile and thetotal slab heat loss are presented for various insulation configura-tions applied to a circular slab. A parametric analysis is conductedto determine the effect of thermal insulation U-value, slab size,and water table depth on the circular slab heat loss. In the thirdsection, the solution of the three-dimensional rectangular slab-on-grade floor is provided. Analysis of the temperature gradient aswell as of the total slab heat loss for several insulation configura-tions is presented. Finally, the fourth section presents the variationof the foundation heat loss as a function of the ratio of slab areaand slab perimeter for all the slab shapes considered in this paper.
2 Circular Slab-on-Grade Floor
2.1 Formulation of the Problem. Figure 1 shows a modelof a circular slab-on-grade floor with uniformly insulated slab.The footings of the foundation are not shown because they areusually made up of concrete, which has a thermal conductivitythat, in most cases, is similar to that of the ground. Therefore, thefoundation elements are considered as integral parts of the soilmedium.
The analysis presented in this paper considers the annual aver-age thermal performance of the foundation and thus, is limited tosteady-state conditions. Time-varying analysis such as monthlyand/or seasonal calculation of foundation heat loss/gain are thesubject of another paper [14]. Under these conditions, the tem-perature distribution within the ground medium, T(r,z) is subjectto the Laplace equation:
with the following boundary conditions:
3 Rectangular Slab-on-Grade Floor
4 General Three-Dimensional Total Slab Heat LossAnalysis
5 Summary and Conclusions
