Demonstration of Fourier’s Law of Conduction

University of Michigan-Dearborn Demonstration of Fourier’s Law of Conduction ME 379 Thermal Fluid Laboratory Experiment conducted May XX, 2008 Report due May XX, 2008 “I have neither given nor received aid on this graded assignment.” Bill Campus _ Jane University _ Brian Michigan _ Bob Dearborn _ Abstract: In this experiment, two conductors were heated with a constant heat flux from the bottom, and cooled from water at the top Temperatures were measured at time equal to zero and every ten minutes after that until steady state was reached The temperatures were measured from thermocouples located along the length of the two conductors Theoretical temperature relations are developed from Fourier’s law, and compared to the experimental temperatures at steady state The theoretical temperatures that are calculated are very close to those that are measured during the experiment Objective: The purpose of this experiment was to compare measured and analytical temperature distributions for two copper rods The analytical temperature distribution is calculated using a one–dimensional heat conduction relation for the case of a constant heat flux One rod has a constant cross sectional area, and the other has a varying cross sectional area The temperature distributions that have reached steady state are to be compared to the theoretical model to show whether or not the theoretical model is a good approximation for predicting the temperature distribution in the rods Theory: The temperature distribution in the copper rods is developed from Fourier’s law in the form: q  k T  x, t  x (1) where k is the proportionality constant known as the thermal conductivity It was assumed that the rods had a constant thermal conductivity, were subjected to a constant heat flux, and that they reached steady state For unit 3, the rod has a varying cross sectional area, and equation (1) reduces to: d  dT  A 0 dx  dx  (2) Further manipulation of equation (2) yields the temperature distribution as a function of x:   0.005614 x  0.000783     ln 0.002027   T  T  x  T2  T1  0.9512       (3) This derivation can be seen in the appendix For unit 4, the rod has a constant cross sectional area, and equation (1) reduces to: d 2T 0 dx (4) Equation (4) can also be manipulated to yield the temperature distribution as a function of x:  T  T1  T  x    x  T1  0.2216  (5) This derivation can be found in the appendix as well Experimental Apparatus: A thermal conduction system (Model 9051) was used for the experiment The major components of the system are illustrated in figure below Figure 1: Experimental apparatus The system consisted of a tube furnace with two integrated heat paths One is the tapered cross sectional area conductor and the other is a constant cross sectional conductor Each conductor was mounted on an individual heat source A PC data acquisition system was set up to record the data and print it out The dimensions of the conductors and locations of the thermocouples used to measure the temperature can be seen in figure Figure 2: Conductors shown with the thermocouple locations Experimental Procedure: The computer system was turned on and the temperature readings were printed out The water was turned on as a cooling source for the conductor and entered at the tops of the units The water was set at a flow that was 50% of the full scale The heat sources that were located at the bottom of the units were set to 300  C Then, every ten minutes the temperature readings were printed out again until steady state was reached Once steady state was reached, the system was shut down, and the data analyzed Discussion of Results: The main part of this experiment was to allow the conductors to heat up from a constant heat source, and reach steady state The units sufficiently reached steady state after one hour and twenty minutes The experimental data in the appendix shows that nine readings were taken, one at t = 0, and the rest at ten minute increments Each of the readings included ten temperatures that were measured from thermocouples at different locations along the length of the conductors It can be seen that the units reached steady state by comparing the temperatures of the last three or four readings The temperature readings were used to generate graphs (Figures and 4), to show how the temperature varied with position and time These graphs each have nine curves corresponding to each of the nine readings taken, as discussed above Using Fourier’s law, the temperature distribution in each unit was developed (Appendix) Two more graphs were generated of temperature vs position for each unit (Figures and 6) Each of these graphs has two curves One curve is the steady state temperature distribution from the experimental readings that were taken The other curve is the theoretical temperature distribution that was developed as mentioned above The experimental curve shows error bars that correspond to the uncertainty in the experiment The uncertainty is  1° C This is due to the thermocouples not being accurate to more than this uncertainty There are a few points that the theoretical curve fell slightly outside of the range of uncertainty for unit (Figure 5) This is most likely due to the rounding of numbers from the large amount of calculations necessary to develop the theoretical curve for unit In looking at a comparison between theoretical and experimental, the maximum difference in temperature for unit is 2.1 °C, and for unit is 1°C Aside from these few differences, the experimental curve correlated very well with the theoretical curve, as can be seen by the graphs The temperature distribution in unit was a straight-line relationship This unit had a constant cross sectional area The temperature distribution in unit was not a straight-line relationship, due to the fact that it had a varying cross sectional area Conclusion: The theoretical temperature distribution based on steady state heat conduction with no internal energy sources and constant thermal conductivity is a valid model for a copper bar with varying cross section, and with constant cross sectional area The theoretical temperature distribution developed from Fourier’s law was very close to the temperature distribution from the experimental data This lab proves that Fourier’s law is a valid model for determining the temperature distribution for the two cases mentioned above  ... corresponding to each of the nine readings taken, as discussed above Using Fourier’s law, the temperature distribution in each unit was developed (Appendix) Two more graphs were generated of temperature... theoretical curve fell slightly outside of the range of uncertainty for unit (Figure 5) This is most likely due to the rounding of numbers from the large amount of calculations necessary to develop... temperature distribution developed from Fourier’s law was very close to the temperature distribution from the experimental data This lab proves that Fourier’s law is a valid model for determining