My Air Conditioner? You’re Standing on it!

Installing a ground source heat pump as a means of heating and cooling a house is becoming an increasingly popular alternative to the traditional methods. However, the installation costs being very high, an in-depth study of the configuration of such systems is needed in order to minimize these costs. The relationship between power output and controlable parameters, such as pipe radius and length was investigated to this end. A derivation of the temperature profile of the soil was done in order to make the most of the temperature difference. The flow of water was modeled first as plug flow, and then as Poisseuille flow, with both models yielding sensibly the same results for the optimal length of the pipe at small radii. The diffusion of heat through the soil was modeled numerically in order to derive a confortable distance at which pipes can be packed while using the heat capacity of the soil efficiently. A numerical simulation of the whole system was performed and the results for optimal length of the pipe were compared to those obtained from the theoretical model. Further, a value of the heat transfer coefficient of the soil was computed using the numerical simulation. 1 Problem Description The annual residential energy consumption of the United States reaches up to 9.9 quadrillion BTU, of which 6.3 are used for spacial and water heating and another 0.6 for air-conditioning. The total bill adds up to 160 billion dollars. On everage, this means that every household invests $ 1100 in heating and cooling air and water. These figures reflect the fact that more than 62% of households use natural gas as a source of energy, whereas only 14% use wood, and even fewer use alternate sources. One very promising such alternative is the exploitation of geothermal energy. Indeed, installing a ground source heat pump as a means of heating and cooling a house costs only $400 per year, instead of the typical $1100. However its large installation costs ($10,000 to $30,000) are a limiting factor to its widespread use. In this case, it takes about 13 years of operation before the installation of such a system can become profitable for its owner. If the break-even time could be reduced, the use of geothermal pumps could be more appealing to more houseowners, who would start saving money faster. The geothermal heating system takes advantage of the fact that the temperature of the soil fluctuates slower than that of air, and in fact is almost stable at a certain depth. A series of pipes is buried in the ground following different configurations, and water is circulated through them. Thus the water either heats up or cools down depending on the season. A heat exchanger installed in the house then uses this water to either heat or cool the house and the water re-enters the cycle. The configuration of the pipes through the ground can be either vertical or horizontal. As shown in figures 1, the pipes can be streched out or coiled together. A variant of the system is to put run the pipes through a pond of Air Conditioning GSMMC Page 2 of 11 Figure 1: Diagram showing a house regulating its temperature using geothermal energy. water, for better conductivity. Right now, the choice of the configuration is more or less arbitrary, depending only on such factors as the shape of the yard or the experience of the entrepreneur. A detailed analysis of each of them however could reveal the main differences and thus allow for better choices of the most appropriate configuration. The efficiency of such a system relies on how much heat can be exchanged with the soil. Generally speaking, the longer the pipe carrying water through the ground, the better the heat exchange. However, other factors, such as flow rate, pipe radius and geometry of the pipe are also to be considered in calculating the heat transfer occurring between water and soil. Modelling the heat transport in a ground source heat pump will allow for the optimization of its installation costs. As these depend largely on the amount of soil that has to be dug up, the model will be used to minimize the length of the coil and determine the optimal configuration in which it should be laid into the ground for optimal heat transfer to cost ratio. 2 Soil Temperature Profile The premise of the geothermal heating system is that the soil remains at almost constant temperature at a certain depth. In order to take the best advantage Air Conditioning GSMMC Page 3 of 11 Table 1: Dimensional Quantities for Soil Volumetric density ρs 1600 Kg/m 3 Heat transfer coefficient h From 150W/Km to 1200 W/Km Specific heat cs 800 J/KgK Thermal conductivity ks = QL tA∆T 0.3 W/Km Themal diffusivity αs = kw ρcw 2.34 · 10m/s Table 2: Dimensional Quantities for Water Volumetric density ρw 1,000 Kg/m 3 Dynamic viscosity μw 8.90 · 10Kg/ms Kinematic viscosity νw 8.90 · 10m/s Specific heat cw 4,180 J/KgK Thermal conductivity kw = QL tA∆T 0.6 W/Km Themal diffusivity αw = kw ρcw 1.44 · 10m/s Table 3: Dimensionless Quantities Pe = Qρwcw Rkw Bi = hR k of that, one needs to know exactly how the soil responds to the seasonal temperature changes in the air and calculate that depth. The variation of the temperature in function of the depth in the soil can be set up as a partial differential equation [1]. Indeed, it can be assumed that Θ(x, t), the temperature in function of the depth and of the time, respects the heat diffusion equation: Θt = kΘxx. (1) The seasonal variation of the soil surface temperature yields a periodic boundary condition: Θ(0, t) = TA +∆Te , (2) where σ is proportional to a month. Now, a trial function to transform the PDE in an ODE can be used: Θ(x, t) = TA +Ae W (x). It then follows that W ′′(x) = iσ k W (x) (3) with W (0) = 1 and limx→∞ W (x) = 0. Trying then W (x) = e mx gives that m = ± √ σ 2k (1 + i). Since W (x) decays with x increasing, the negative value of m must be used. Finally, the result lies in the real part of Θ(x, t): R{Θ(x, t)} = TA + e− √ σ 2k x cos (