Experiment 4
1 Objective
The objectives of this experiment are:
1. Use Fourier’s Law of thermal conduction to determine the rate of heat transfer through a metal rod.
2. To estimate the thermal contact resistance between two metallic surfaces.
3. Measure and plot the temperature distribution in a 2D surface with prescribed temperature boundary conditions and compare it with analytical and numerical solutions.
2 Background
2.1 Fourier’s law of thermal conduction Fourier’s law is an empirical relation based on experimental observation. It relates the rate of heat transfer through a surface to its temperature gradient and the surface area. In functional form, it is shown in Eq. 1
dT Q˙ = −kA (1) dx
Here, k [W/m.K] is the proportionality constant known as the coefficient of thermal conductivity. Here, the unit of Q˙ is Joules per s [J/s] or Watts [W ]
2.2 1-D and 2-D heat conduction equations For one-dimensional (1D) heat conduction at steady state without heat generation, the simplified form of Eqn. 1 can be written as follows, T − T Q˙ = kA 1 2 (2) x1 − x2
As observed in Eqn. 2, by measuring the temperature T1 and T2 at two different locations x1 and x2 of a metal rod (1D approximation) and knowing the cross sectional area (A), thermal conductivity k of the material, the heat transfer rate, Q˙ , can be calculated.
For a more complex two-dimensional (2D) body, assuming steady state and no heat generation, Eq.1 combined with the conservation of energy can be written as follows:
d2T d2T + = 0 (3) dx2 dy2
Equation 3 is also called Laplace’s equation. This can be solved for temperature distribution in a 2- dimensional body with four known boundary conditions. A schematic of the heat transfer domain and boundary conditions are shown in Fig. 1.
1 Thermal Conduction 2
Figure 1: The domain and boundary conditions for 2D heat conduction
2.3 Analytical Solution In order to solve Laplace’s equation using the method of separation of variables, three of the four boundary conditions are required to be homogeneous. Laplace’s equation (Eq. 3) can be solved analytically using the principles of superposition as illustrated below. Principle of superposition is shown in Eq. 4. A Matlab code can be used (will be provided) for solving the Laplace’s equation for given constant temperature boundary conditions for a 2D problem.
Figure 2: The principle of linear superposition of solutions
T (x, y) = T1(x, y) + T2(x, y) + T3(x, y) + T4(x, y) (4)
2.4 Thermal Contact Resistance When heat flows between two surfaces in contact, the imperfections in the contact may lead to a temperature drop. Thermal contact resistance, R, is defined as the ratio of this temperature drop to the mean rate of heat flow through the surface. i.e., ∆T R = (5) Q˙ where ∆T the temperature drop across the surface, and Q˙ is the mean heat transfer rate through the surface. Experiment 4 3
Figure 3: Measurement of thermal contact resistance between two materials
3 Apparatus Used in Experiments
The hardware of this experiment consists of the following,
1. Constant temperature hot and cold baths
2. A Submersible pump
3. An insulated metal bar (1-D) and plate (2-D)
4. Thermocouples, thermocouple selector, and display
Note: A description of some of the hardware used in this experiment is provided below.
3.1 Constant Temperature Baths Two water baths, one heated to a prescribed temperature (60 - 80oC) and the other maintained at 0oC are used in this experiment. A pump will circulate this water through prescribed paths (boundaries) and provide temperature boundary conditions to the domains under consideration. The cold bath is maintained at 0oC by keeping enough ice in the cold water bath.
3.2 Insulated Metal Bar & Plate The bar for 1-D conduction consists of two different metals. The metal on the hot side is brass and the metal on the cold side is copper. Each bar has a length of 5.5 inches, providing a total test length of 11 inches. The two metal bars are welded and joined together as shown in Fig. 4a. This bar is insulated by 0.25 inches thick corkboard on all sides. On each end, there is a 4in. × 3/8in. diameter copper pipe welded onto the bar to circulate water at known temperatures. The top surface of the bar has 10 holes, drilled 1 inch apart from each other. These holes are provided to measure the temperature distribution of the bar using a thermocouple. The thermocouple allows a perfect surface contact with the metals in the bar. A schematic of the 1D bar is shown in Fig. 4a. The 2-D conduction model, as shown in Fig. 4b, is a solid copper plate with a thickness of 1/8 inch. There are 3/8 inch diameter copper pipes welded onto each side of the plate for circulating water at known temperatures to provide temperature boundary conditions. The top and bottom surfaces of the plate are Thermal Conduction 4
(a)
(b)
Figure 4: Schematic of (a) 1-D metal bar used for contact resistance measurement (b) 2-D conduction plate with hot and cold boundary conditions
Figure 5: Picture of insulated 1D metal bar and 2D plate used in this study insulated with corkboard that has a thickness of 0.25 inch. There are 100 holes drilled approximately 1 inch apart on the top of the plate to measure the surface temperature at various locations. Figure 5 shows a picture of the 1-D metal bar and the 2-D plate used in this study.
3.3 Temperature measurement Temperatures at various locations of the metal bar and plate is measured using a K type thermocouple. In order to obtain more accurate readings, the thermocouple must be in contact with the surface for at least 10-15 seconds.
4 Experimental Procedure
1. Maintain the hot and cold baths at constant temperatures.
2. Circulate hot and cold water through copper piping attached to both specimens shown in Fig. 5. Make sure that the piping is routed correctly for hot and cold boundaries of the specimens.
3. Run the system for half an hour and wait for the system to reach steady state Experiment 4 5
4. Measure the temperatures at different points. The steady state temperatures will represent the tem- perature distribution of the plate. Make sure that there is good contact between the thermocouple and the plate and watch the temperature output until it reaches an equilibrium value. This may take 15 seconds or more. 5. Record these temperatures in the data table and move on to the next point in the cork. It does not matter if the data is collected in rows or in columns, as long as the procedure is done systematically. Make sure that equal pressure is applied to the thermocouple against the plate for each collected data point. The temperature output is easily altered by varying the pressure applied, making consistency very important. Allow 45 minutes to 1 hour for data collection. 6. Once all temperatures for both bar and plate are collected, unplug the submersible pump and turn off the heater.