Overview: This HW set is concerned with different aspects of ground source heating. This is complicated by many factors, including (i) the presence or absence of heat exchange with the air above the soil, (ii) variation in the thermal properties of soil, both in different locations and as a function of depth, (iii) the 3-D nature ofthe heat exchange loop in the ground, and (iv) whether the heat removed from and returned to the ground is balanced over the year. The questions below focus independently on different aspects of the problem, in each case making simplifications intended to elucidate an important principle. 1. Question#1: Thermal diffusion: Consider a heat pump system connected to the ground by pipes through which a heat exchange uid M. An important aspect is how heat ows between the pipe and the surrounding soil. Analytic solutions are based on the thermal diffusion equation; we divide the thermal conductivity by the heat capacity per volume to afford the thermal m a : ic/Crp. Then we set up the time-dependent differential equation with suitable coordinates and boundary conditions and attempt to nd a closed-form solution. Numerical solutions make the calculation much simpler. The reading U2 q' q\" q" by W points out that thermal problems can be solved using a finite difference method. This is depicted in the gure as a chain of resistors and capacitors. ln this problem you will construct a one-dimensional numerical model of heat flow using an Excel spreadsheet. This will reveal how slowly heat moves! The numerical format also allows you to test different situations; for example, by changing a time scale or a boundary condition, you can see how the temperature prole changes. The procedure: (i) Consider a 1-D problem in which the heat exchange piping approximates a plane and heat flows to the right into the soil (as in the figure). Heat ow will be per unit area. (ii) Let the columns of your table represent depth as thickness increments. These should be thin enough that the temperature difference between them is modest (I use 0.1 m which seems fine). (iii) Let the first column be the boundary condition. Here, it is the temperature of the heat exchange fluid. assumed to be a constant 40C (i.e., rejected heat from air conditioning). (The real-world boundary condition also involves a heat transfer coefcient another limitation but here, to keep things simple, we assume that thermal diffusion is the only limitation.) (iv) Let the rows of the table represent time steps. The first row will be the starting temperature vs. depth, assumed to be a constant 20C. For each time step, calculate the new temperature of each column as follows. Calculate the heat flux per unit area into and out of each increment based on the temperatures of the increments on either side of it, the thermal conductivity, and the thickness of the increment. Then calculate the temperature change in each increment based on the net heat ux, the length of the time step. and the heat capacity. (v) Plot the temperature vs. depth for selected times (for example. every 5th or 10tn row). Important suggestion: define all the important variables at the top of your Excel sheet so that you can easily change them and observe the effect on your simulation. .. _ l . _ _. _ _ Assume for SOII. Cp 1840 kg_K, 1400 m3 , K 0.5 m_K. Let the time increment be I (you figure out what value to use). a. Consider the temperatures of three adjacent thickness intervals, T1, T2 and T3. Write a formula for the net heat flux per area owing into interval 2. Then write a formula for the temperature rise of interval 2 during a time step of I. b. Arbitrarily define the \"penetration depth\" of the heat as the depth at which the soil temperature has reached 21C. ln your simulation, howlong does it take for heat to penetrate 1 m deep? c. If you double the thermal diffusivity, how far does the heat penetrate during the same time as your answer to part a? d. How does the heat flow from the heat exchange fluid (the boundary condition) into the first thickness increment change overtime? You can create a new column with that information. e. _l_ngijdertg maintain a constant heat flow from the heat exchange fluid into the first thickness increment at all times, how should the temperature in the fluid change? Give a short explanation. To gain further insight (but nothing to turn in), play with this simulation. For example, as the starting condition you can insert a higher temperature in just one increment in the middle of the sample and watch it spread in time