Research Papers

Constructal Underground Designs for Ground-Coupled Heat Pumps

[+] Author and Article Information
A. Bejan

Department of Mechanical
Engineering and Materials Science,
Duke University,
Durham, NC 27708-0300

S. Lorente

LMDC (Laboratoire Matériaux et
Durabilité des Constructions),
Université de Toulouse,
135, Avenue de Rangueil,
Toulouse Cedex 04 F-31 077, France

R. Anderson

239 Powderhorn Trail,
Broomfield, CO 80020

Contributed by the Solar Energy Division of ASME for publication in the JOURNAL OF SOLAR ENERGY ENGINEERING. Manuscript received May 14, 2013; final manuscript received August 28, 2013; published online November 19, 2013. Assoc. Editor: Aldo Steinfeld.

J. Sol. Energy Eng 136(1), 010903 (Nov 19, 2013) (8 pages) Paper No: SOL-13-1135; doi: 10.1115/1.4025699 History: Received May 14, 2013; Revised August 28, 2013

In this paper, we review the main advances made by our research group on the heat transfer performance of complex flow architectures embedded in a conducting solid. The immediate applications of this work include the design of ground-coupled heat pumps, seasonal thermal energy storage systems, and district heating and cooling systems. Various configurations are considered: U-shaped ducts with varying spacing between the parallel portions of the U, serpentines with three elbows, and trees with T-shaped and Y-shaped bifurcations. In each case, the volume ratio of fluid to soil is fixed. We found the critical geometric features that allow the heat transfer density of the stream-solid configuration to be the highest. In the case of U-tubes and serpentines, the best spacing between parallel portions is discovered, whereas the vascular designs morph into bifurcations and angles of connection that provide progressively greater heat transfer rate per unit volume. We show that the flow of heat into or out of a solid volume must have an S-shaped history curve that is entirely deterministic. This constructal-design principle unites a wide variety of previously disconnected S-curve phenomena (ground heat storage and retrieval, population growth, cancer, chemical reactions, contaminants, languages, news, information, innovations, technologies, economic activity).

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Bejan, A., and Lorente, S., 2008, Design With Constructal Theory, Wiley, Hoboken, NJ.
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Bejan, A., and Lorente, S., 2013, “Constructal Law of Design and Evolution: Physics, Biology, Technology, and Society,” J. Appl. Phys., 113, p. 151301. [CrossRef]
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Jung, J., Lorente, S., Anderson, R., and Bejan, A., 2011, “Configuration of Heat Sources or Sinks in a Finite Volume,” J. Appl. Phys., 110, p. 023502. [CrossRef]
Combelles, L., Lorente, S., Anderson, R., and Bejan, A., 2012, “Tree-Shaped Fluid Flow and Heat Storage in a Conducting Solid,” J. Appl. Phys., 111, p. 014902. [CrossRef]
Kobayashi, H., Lorente, S., Anderson, R., and Bejan, A., 2012, “Freely Morphing Tree Structure in a Conducting Body,” Int. J. Heat Mass Transfer, 55, pp. 4744–4753. [CrossRef]
Kobayashi, H., Lorente, S., Anderson, R., and Bejan, A., 2012, “Trees and Serpentines in a Conducting Body,” Int. J. Heat Mass Transfer, 56, pp. 488–494. [CrossRef]
Rocha, L. A. O., Lorente, S., Bejan, A., and Anderson, R., 2012, “Constructal Design of Underground Heat Sources or Sinks for the Annual Cycle,” Int. J. Heat Mass Transfer, 55, pp. 7832–7837. [CrossRef]
Kobayashi, H., Lorente, S., Anderson, R., and Bejan, A., 2013, “Underground Heat Flow Patterns for Dense Neighborhoods With Heat Pumps,” Int. J. Heat Mass Transfer, 62, pp. 632–637. [CrossRef]
Bejan, A., and Lorente, S., 2011, “The Constructal Law Origin of the Logistics S Curve,” J. Appl. Phys., 110, p. 024901. [CrossRef]
Bejan, A., and Lorente, S., 2012, “The S-Curves are Everywhere,” Mech. Eng., 134, pp. 44–47.
Cetkin, E., Lorente, S., and Bejan, A., 2012, “The Steepest S Curve of Spreading and Collecting: Discovering the Invading Tree, Not Assuming it,” J. Appl. Phys., 111, p. 114903. [CrossRef]
Bejan, A., and Lorente, S., 2012, “The Physics of Spreading Ideas,” Int. J. Heat Mass Transfer, 55, pp. 802–807. [CrossRef]


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Fig. 1

Constructal design of densely populated areas: heat pump with tree-shaped ground heat exchanger

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Fig. 2

U-shaped pipe inside a cube with S/D = 5 [11]

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Fig. 3

Four pipe shapes in a solid cube [11]

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Fig. 4

(a) The outlet temperature in the four configurations of Fig. 3 and (b) the relation between the time of approach to equilibrium and the Peclet number of the fluid [11]

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Fig. 5

Tree-shaped configurations [14]

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Fig. 6

Three-dimensional tree-shaped flow architecture. The patterns of isotherms in the solid when α˜=0.36,ρ=0.5,k˜=0.6, Pr = 7.1, Re1 = 390; 450, Pes = 1.5 × 104; 2.01 × 104, φ=0.002, and t˜=1.5×10-6 [14].

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Fig. 7

The effect of complexity (n) on the evolution of the system average temperature in a three-dimensional configuration [14]

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Fig. 8

The optimization of the bifurcation angle β4 when β1 = 101°, β2 = 81°, and β3 = 85° in a larger cube [15]

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Fig. 9

The temperature field around the tree in a large solid: (a) trunk; (b) tree when β1 = 95 deg; (c) tree when β1 = 95 deg and β2 = 80 deg (β2A = β2B = 40 deg); (d) tree when the optimization is repeated and β1 = 100 deg, β2 = 80 deg (β2A = β2B = 40 deg); (e) tree when β1 = 100 deg, β2 = 80 deg (β2A = β2B = 40 deg), and β3 = 85 deg (β3A = β3B = 42.5 deg); (f) tree when the optimization is repeated and β1 = 101 deg, β2 = 81 deg (β2A = β2B = 40.5 deg), and β3 = 85 deg (β3A = β3B = 42.5 deg); (g) tree when β1 = 101 deg, β2 = 81 deg (β2A = β2B = 40.5 deg), β3 = 85 deg (β3A = β3B = 42.5 deg), and β4 = 70 deg (β4A = 40 deg, β4B = 30 deg); and (h) tree when the optimization is repeated and β1 = 105 deg, β2 = 82 deg (β2A = β2B = 41 deg), β3 = 85 deg (β3A = β3B = 42.5 deg), and β4 = 70 deg (β4A = 40 deg, β4B = 30 deg) [15]

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Fig. 10

Thermal performance of Y-shaped, T-shaped, and classical designs [16]

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Fig. 11

The effect of porosity on thermal performance when t* = 1 [16]

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Fig. 12

Comparison between the Y and T designs when t* = 0.2 and 1 [16]

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Fig. 13

Examples of S-curve phenomena: the growth of brewer's yeast, the spreading of radios and TVs, and the growth of the readership of scientific publications [19]

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Fig. 14

Line-shaped invasion, followed by consolidation by transversal diffusion. The predicted history of the area covered by diffusion reveals the S-shaped curve [19].

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Fig. 15

Tree-shaped invasion, showing the narrow regions covered by diffusion in the immediate vicinity of the invasion lines [19]



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