1. Thermal diffusion: Consider a heat pump system connected to the ground by pipes through which a heat exchange fluid flows. An important aspect is how heat flows 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 diffusivity a = k/Cpp. Then we set up the time-dependent differential equation with suitable coordinates and boundary conditions and attempt to find a closed-form solution. Numerical solutions make the calculation much simpler. The reading 91 92 9n-1 by Formhals points out that thermal problems can be solved using a finite difference method. This is depicted in the figure as a chain of resistors and capacitors. In 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 profile 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 flow 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 40*C (i.e., rejected heat from air conditioning). (The real-world boundary condition also involves a heat transfer coefficient - 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 20*C. For each time step, calculate the new temperature of eachcolumn 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 ofthe increment. Then calculate the temperature change in each increment based on the net heat flux. the length of the time step, and the heat capacity. (v) Plot the temperature vs. depth for selected times (for example, every 5th or 10\"\" row). Important suggestion: define all the important variables at the top ofyour Excel sheet so that you can easily change them and observe the effect on your simulation. As an example. i copy a portion of my Excel sheet with one row plotted out. This took about 30 minutes to create and debug (not too bad)... time stepts) 86400 Cp*rho 2.60E+06 K 0.5 d 0.1 01:0) 40 20 20 20 20 20 T[t.i] 40 23.3 20.6 20.1 20.0 20.0 40 25.6 21.3 , 40 27.3 22.1 Chart T' 40 23.6 22.9 m 40 29.5 23.7 353 ' 40 30.3 24.4 m ' , 40 30.9 25.0 25: - . 40 31.5 25.6 ' . . . _ . 40 31.9 25.1 m ' ' ' ' 40 32.3 26.6 153 40 32.6 27.0 193 40 32.9 27.4 5.-:- 40 33.2 27.3 9,: 40 33.4 28.1 :1 5 1c 15 40 33.6 28.4 24.; 22.4 21.2 40 33.3 28.7 25.0 22.7 21.3 - . l . kg . W AssumetorsoH. ifp 1840k 7 . . p 1400 1.x 0.5 7 g R m m A Let the time increment be r (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 flowing into interval 2. Then write a formula for the temperature rise of interval 2 during a time step of 1