Transient Heat Conduction in a Circular Cylinder The purely radial 2-D heat equation will be solved in cylindrical coordinates using variation of parameters. Assuming radial symmetry the solution is represented by a series of Bessel Functions. The solution is then compared with experimental results. The experimental apparatus is a circular cylinder heated from the inside with electricity and cooled on the outside with cold water. I. Background The heat equation is given by u t α 2 u = 0, (1) where u is the heat, α is the thermal diffusivity and 2 u represents u, the divergence of the gradient of u. The heat equation is used to describe the diffusion of heat in a given region over time. In one dimension the solution is represented in terms of sines and cosines by a Fourier series. On a cylinder the heat equation becomes slightly more cumbersome when written in terms of u(r, θ). However, the experimental apparatus for this project has no way to measure the θ-dependent temperature distributions, so we focus on the purely radial 2-D heat equation. The problem of purely radial heat conduction is also a one-dimensional problem but the solution is represented in terms of Bessel functions as opposed to Fourier series. The purely radial 2-D heat heat equation with forcing is given by v 1 t = α ( v ) ] r + F (r, t), R i < r < R 0, t > 0 r r r v(r i, t) = g(t) v(r 0, t) = h(t) v(r, 0) = V (r) R i < r < R 0, where v is the measured heat, α is the thermal diffusivity, R i is the radius of the inside measurement, R 0 is the radius of the outside measurement, g(t) and h(t) represent the heating and cooling as functions of time respectively (both have units c), and V (r)( c) represents the initial temperature as a function of radius. In this experiment F (r, t) = 0 and can be ignored. (2) features something that has not been accounted for in this class: time-dependent boundary conditions. The technique of variation of parameters is used to solve (2) with time-dependent boundary conditions. For the unfamiliar reader, the method of variation of parameters is explained in IV. (2) II. Materials/Equipment 1 Control Box 1 Power Regulator 1 Selector Box 1 Radial conduction apparatus 1 Water chiller For the analysis it may be important to know the dimensions of the cylinder. Cylinder Dimensions: 1
APPM 4350/5350 Projects Updated UpdatedDecember November18, 9, 2014 2013 Outer Radius: 150mm Height: 35mm. The first temperature probe is located at the center of the cylinder and the other probes are located at 10mm intervals. This means that the 6 th probe is located at r = 50mm. The probe at the center records g(t) while the probe on the outside records h(t) in equation (2). All probes are laid in the same piece of brass. The cylinder is heated from the inside by an electric heater in the middle and cooled from the outside by the cool water running along the outside of the cylinder. III. Procedure Setting Up the Apparatus Figure 1: Heat Conduction Controls A. Control Box B. Selector Box C. Power Regulator D. Heater Knob (counterclockwise is o ) off) E. Heater Power Cable 1. Make sure the Control Box (figure 1A), the Power Regulator (figure 1B), and the water chiller (figure 2) are turned OFF. Turn the Heater Power knob fully counterclockwise (this is the OFF position). 2. Plug the water chiller into one of the outlets at your lab station. 3. Make sure the inlet on the water chiller is connected to the outlet on the water chiller by a tube (figure 2). 2
APPM 4350/5350 Projects Updated December 18, 2014 Figure 2: Water Chiller (a) Front of Water Chiller (b) Back of Water Chiller. The top connection is the inlet, the bottom connection is the outlet. 4. Turn on the water chiller and set the desired temperature in the range 5 c to 10 c. Instructions on how to do this are on the chiller. Let it run for 10-15 minutes until it has reached the desired temperature. If the water chiller is making unhappy noises, seek help. 5. Check that the Control Box is plugged into the Power Regulator. 6. Connect the P-1 Military Plug to the appropriate jack on the lab station. 7. Connect the Heater Power Cable from the circular apparatus to the front of the control box. 8. Choose 6 cables from those labeled 1-7, plug these into the circular device in any orientation (figure 3). Be sure to record the orientation of the cables. Note: There are 9 cables labeled 1-9. The cords labeled 8 & 9 can only be read manually and not by the computer, do not use these. 9. Plug the Power Regulator into the other outlet next to your lab station. Note: The Power Regulator and Water Chiller must be plugged into different outlets so that they are on different circuit breakers. 10. Make sure the lab station power is turned on. The switch is located at the end of your lab station above the computer tower. Ask an ITLL assistant for help if needed. 11. Change the mode on the Selector Box to computer. Once the water chiller has reached equilibrium, the experiment can begin. 3
APPM 4350/5350 Projects Updated December 18, 2014 Figure 3: Circular Apparatus with contacts 1-6. Running the Experiment 1. Open the Heat Conduction Apparatus VI using the following path: H:\ITLL Documentation\ITLL Modules\Heat Conduction\Heat Conduction - Shortcut. When this window opens (LLB Manager), select and run Heat Conduction Apparatus.vi 2. When the VI opens, enter the following information into the User Inputs box: (a) Indicate the temperature units desired (celsius). (b) Specify the time interval between samples in minutes. This should be in the vicinity of 0.5 minutes. (c) Enter the watts applied for the experiment. This is the wattage you will dial in on the Control Box when running the experiment. It should be in the range of 15 to 20 Watts. Note: Do not set the range to be greater than 30 Watts, this will break the Control Box. 3. Turn on the Power Regulator and the Control Box. 4. Set the wattage on the control box equal to the value you entered in the VI. 5. Before connecting the Water Chiller to the apparatus, click run in the VI. The program will double check that the watts applied field was filled. Make sure it matches the display on the Control Box. 6. Each thermistor should be following the same trend after a few samples have been taken. Do not worry about the data taken from the one thermistor that is not attached. If any of the 4
thermistors seem not to be reporting the same trend in the data as the other 6, try wiggling it in the probe to change the contact. If the thermistor is still not reporting the same trend as the other 6 try switching it out for the extra thermistor (the cable not in use). 7. Once you are confident that the thermistors are providing reasonable data turn off the Water Chiller. 8. Disconnect the tube on the water chiller and attach the cooler tubes to the tubes coming from the conduction apparatus. 9. Turn the water chiller back on. Note when this occurs on your data, this is t = 0 for your experiment. 10. Continue taking measurements until the experiment reaches steady state. This should take about 10-20 minutes. When the experiment reaches steady state hit STOP and save the data with a.xls extension. 11. Repeat this procedure multiple times if desired (once should suffice, but if you want some redundancy, do it again). Before each new trial, turn off the water chiller and let the conduction apparatus reach the same temperature (or higher) than you were seeing before turning the water on in the previous trial. 12. When finished with the experiment, turn everything off and disconnect all cables. Make sure to clean up any excess water that fell on the floor. Return the module to the checkout. IV. Analysis Variation of Parameters During the semester, we learned how to solve the heat equation with fixed temperatures (or fixed heat fluxes) at the boundaries. The purpose of this section is to generalize that procedure, to handle time-dependent forcing in the equation itself, or at the boundaries, or both. A. Variation of Parameters Explanation Variation of Parameters is a method to solve linear, ordinary differential equations (ODEs) with known forcing. The heat equation is a linear partial differential equation that is first-order in time, and Variation of Parameters is especially simple for first-order (in time) ODEs. For higher order ODEs, there is one additional logical step in the method, which is discussed in many books on ODEs. Consider a first-order, linear ODE with a constant coefficient, α, and a known forcing function, f(t): dy + αy = f(t), y(0) = A (3) dt (i) Solve the homogeneous ODE (i.e., with f(t) = 0): y h (t) = ce αt (4) where c is the arbitrary constant of integration, or parameter in the solution. 5
(ii) Vary the free parameter: seek a solution of (3) of the form y(t) = c(t)e αt (5) where c(t) is an unknown function. Substitute this ansatz (5) into (3) and simplify to obtain dc dt e αt = f(t) (6) This ODE can easily be solved for c(t), in terms of an integral over known functions: c(t) = c(0) + (iii) Combine (5), (7) and (3) to obtain the full solution of (3): y(t) = Ae αt + t 0 t 0 e ατ f(τ)dτ (7) e α(t τ) f(τ)dτ (8) (iv) Check by substituting (8) into (3), making sure to differentiate the integral correctly, using Leibniz Rule. B. Solving the heat equation with known forcing (i) Going back to equation (2), to utilize variation of parameters we first change variables to move the time-dependence out of the boundary condition and into the heat equation. Define z(r, t) by ] ] ln(r) ln(r 0 ) ln(r i ) ln(r) v(r, t) = z(r, t) + g(t) + h(t) (9) ln(r i ) ln(r 0 ) ln(r i ) ln(r 0 ) where g(t) and h(t) are the functions in (2). Then show that z(r, t) satisfies z t = κ z(r i, t) = 0 z(r 0, t) = 0 1 r r ( z ) ] r + r ˆF (r, t), R i < r < R 0, t > 0 (10) where z(r, 0) = ˆV (r), R i < r < R 0 ln(r) ln(r 0 ) ˆF (r, t) = F (r, t) ln(r i ) ln(r 0 ) ] ln(r) ln(r 0 ) ˆV (r) = V (r) ln(r i ) ln(r 0 ) ] dg dt g(0) ] ln(r i ) ln(r) dh ln(r i ) ln(r 0 ) dt ] ln(r i ) ln(r) ln(r i ) ln(r 0 ) h(0) (11) (ii) Next, find the homogeneous solution of (10). Solve (10) with ˆF (r, t) = 0 by separation of variables in the usual way with one new concept: Bessel Functions. Bessel functions are a family of solutions to Bessel s Differential Equation given by z 2 d2 f dz 2 + z df dz + (λz2 n 2 )f = 0 (12) 6
The solutions to this equation are given by Bessel functions J n (z) and Y n (z). Bessel functions are used to represent the solution of differential equations in polar coordinates. The n is related to the radial dependence. Radial symmetry forces n = 0. Working with Bessel functions may require some research, use references for help. The result expresses z(r, t) in terms of an infinite series, in which each term in the series satisfies (10) with ˆF (r, t) = 0 and each term is multiplied by an arbitrary coefficient. (iii) Use variation of parameters. In the formal series solution of (10), allow each of the arbitrary coefficients in the infinite sum to depend on (t), in a way to be determined later. Substitute this generalized series into (10) with ˆF (r, t) 0, as is appropriate. The result is an equation with two (large) terms: one of the large terms is an infinite series, in which every additive term in the series contains a time-derivative of an as-yet unknown function of time, and the other term is ˆF (r, t). Label this equation ( ) for future reference. (iv) Rewrite { ˆF (r, t), ˆV (r)}. Each term in the infinite series contains a function of r, and these spatial functions comprise a mutually orthogonal set of basis functions. { ˆF (r, t), ˆV (r)} are known explicitly, so one can represent each of these functions in terms of the basis functions (Bessel Functions) in the corresponding infinite series. Do so, and determine explicit coefficients for this representation of { ˆF (r, t), ˆV (r)}. (v) Use orthogonality of the basis functions. Now combine the two terms in equation ( ) into a single infinite series, in which each term is a combination of the two coefficients from the two series in ( ). This single series vanishes, by ( ), so every coefficient must vanish. The result is an infinite set of ordinary differential equations, each like (6), with one equation for the time-derivative term of each arbitrary function in the series representation for z(r, t). Integrating these ODEs explicitly, and using the series representation of ˆV (r) provides an explicit representation of z(r, t). Then, using (9) provides an explicit representation of the solution of the original problem (2). Results 1. Solve (2) using the variation of parameters technique described above. State explicitly the boundary and initial conditions used. 2. The primary objective of the experiment is to predict the temperature value of the inner 4 temperature probes from the inside and outside probes (g(t) and h(t) respectively). Compare your theoretical predictions of the temperature at these locations with the experimental results; discuss any discrepancies. 3. An important parameter in the problem is the thermal diffusivity of the material being tested. Find a way to infer this value from your measurements. Calculate this value and compare it to the thermal diffusivity of brass. The questions above are not meant to be a comprehensive list but should serve to encourage your own ideas and analysis. V. References The following textbooks have information on Bessel Functions, 2-D Wave equation, and variation of parameters. 7
1. Asmar N. Partial Differential Equations with Fourier Series and Boundary Value Problems. 2 nd edition. Prentice Hall; 2004. 2. Haberman, R. Applied Partial Differential Equations with Fourier Series and Boundary Value Problems. 4 th edition. Pearson Prentice Hall; 2004. 3. Pinsky, M. Partial Differential Equations and Boundary-Value Problems with Applications. 3 rd edition. Waveland Press; 2003. 8