EXPERIMENT #1, FOURIER’S LAW PART A – CONDUCTION ALONG A SIMPLE BAR OBJECTIVE: The objective of this experiment is to investigate and verify Fourier’s Law for linear heat conduction along a simple bar. INTRODUCTION: Conduction is defined as the transfer of energy from more energetic particles to adjacent less energetic particles as a result of interactions between the particles. In solids, conduction is the combined result of molecular vibrations and free electron mobility. Metals typically have high free electron mobility, which explains why they are good heat conductors. Conduction can be easily understood if we imagine two blocks, one very hot and the other cold. If we put these blocks in contact with one another but insulate them from the surroundings, thermal energy will be transferred from the hot to the cold block, as evidenced by the increase in temperature of the cold block. This mode of heat transfer between the two solid blocks is termed ‘conduction’. THEORY: In this experiment we will investigate conduction in an insulated long slender brass bar like the one in Figure 1. We will assume that the bar is of length L, a uniform hot temperature Th is imposed on one end, and a cold temperature Tc is imposed on the other. We will also assume, because the bar is insulated in the peripheral direction, that all the heat flows in the axial direction due to an imposed temperature differential along the bar. Figure 1: Schematic of a Long Cylindrical Insulated Bar The equation that governs the heat flow is known as Fourier's Law, and in the axial direction it is written as q x = − kAx dT dx (1) where q x is the rate of heat conduction in the x-direction, k is the thermal conductivity of the The transformer has two cords which connect it to an AC outlet and to the second item. More generally. the calibration unit. thermal conductivity is analogous to electrical conductivity and as a result. length. A x is the cross-sectional area normal to the x-direction. The negative sign indicates that heat is transferred in the direction of decreasing temperature.material. which consists of three items. and dT is the temperature gradient in dx the x-direction. Hilton H940 Heat Conduction Unit. We can then use these values in a rearranged version of Fourier's Law to find the thermal conductivity. In this experiment we will investigate Fourier's Law by finding the thermal conductivity k for brass and comparing this value to the actual value from one or more references. Figure 2: Front View of Calibration Unit and Transformer . Fourier's Law is a vector relationship which includes all directions of heat transfer: r r r r dT r dT r dT r k (2) j + Az i + Ay q = q x i + q y j + q z k = − k Ax dz dy dx The thermal conductivity k varies between different materials and can be a function of temperature. metals that are good conductors of electricity are also good conductors of heat.A. but it can be treated as a constant over small temperature ranges. Because of the enhancement of heat transfer by free electrons. To do so we will calculate the cross sectional area A x of the bar and the slope dT/ dx from a plot of measured temperatures vs. k: k=− q x dx A x dT (3) APPARATUS: The apparatus we will be using in this experiment is the P. The first item is a transformer equipped with a circuit breaker. The purpose of the cooling water running through the unit at the cold end of the bar is to remove heat that is produced at the hot end and transferred by conduction to the cold end. and an insulated disk. but we will only use the insulated bar for this experiment. It delivers power to the heater element within the test unit and it calibrates and displays the temperatures at nine locations along the test unit. as shown in Figures 2 and 3. which consists of two test geometries: an insulated brass bar which allows a sample to be placed between the two ends. Once the rate at which heat is generated is equal to the . Both geometries are equipped with a power supply. Figure 3: Rear View of Calibration Unit Insulated bar Insulated disk Figure 4: Front View of the Test Unit The third item is the test unit. keeping the cold end at a constant temperature. The test unit is also equipped with a cooling water hose. The amount of power delivered to the test unit is controlled by the power control knob on the right side of the calibration unit. to its left is the temperature selector knob which is used to select one of the nine thermocouple temperatures for display on the digital readout.The calibration unit has two basic functions. The test unit has two heater cords: one from the test bar and the other from the test disk. PROCEDURE: 1.rate at which heat is removed. 2. Connect the nine thermocouples in the appropriate order. Be sure to use only the heater cord for the bar. it is not intended to enhance heat conduction. allow the system to reach steady state (approximately 20 minutes). etc. RESULTS: 1. 6. making sure that the calibration unit is switched off before connecting the transformer to the AC outlet. Switch the calibration unit on and adjust the power control knob to deliver 10 W of power to the test unit. Insert the test brass sample (30 mm length and 25 mm diameter) into the unit and allow cooling water to flow through the test unit. Discuss the characteristics of your plots and compare them to what you would expect based on the theory. 4. Plot the temperature profile along the entire length of the bar and determine the slope. thermocouples. Apply a very small amount (a drop) of thermal conducting paste to make a thin layer on each side of the test unit surface and spread it uniformly. you will also find a box that contains the samples. Figure 4 shows the front view of the test unit. 2. Record the temperatures at each of the nine thermocouples and the power input. The thermocouples must be placed in order from 1 to 9 as shown in Figures 3 and 4. dT/ dx. which connects to the heater plug located in the lower right hand corner of the calibration unit. and conducting paste in a small syringe. 3. 5. The conducting paste is designed only to decrease contact resistance when applied to the ends of the connecting bars. Is Fourier’s Law satisfied? Also discuss the validity of the assumptions made and sources of error within the equipment or through measurements. Calculate the thermal conductivity k of the brass and determine the specific type / composition of brass by comparison with a reference value. Connect the equipment as shown in Figures 3 and 4. . The samples can be placed in the bar test unit by releasing the clamps and sliding the cold end of the bar out. steady state conditions (temperatures will be fairly constant and readings can be taken) will exist. 3. In addition to this equipment. and c denote the hot. FOURIER’S LAW PART B . In this experiment we will use the same brass test bar. In this experiment. (1). length (which is the experimental result) with the inverse ratio of the two different areas (theoretical result). THEORY: Fourier’s Law states that the rate of heat transfer is proportional to the cross-sectional area normal to the direction of heat flow. described in Part A. . INTRODUCTION: In Part A of this experiment we have seen how to calculate the thermal conductivity of bar made of the same material and uniform cross-section. In other words.A. Since the outer surface is insulated.EXPERIMENT #1. which will differ from that of the sample inserted in between the two segments. the thermal conductivity k is assumed to be constant. the heat flow rate q x is the same for each section of the bar and since it is the same material. respectively. we get (dT / dx ) s A = h/ c (dT / dx ) h / c As (2) In this experiment you will compare the gradient ratio obtained from the plot of measured temperatures vs. Hilton H940 Heat Conduction Unit. Then dT dT dT Ah (1) = As = Ac dx s dx c dx h where the subscripts h. Solving for the temperature gradient ratio in Eq.THE EFFECTS OF VARYING CROSS-SECTIONAL AREA OBJECTIVE: The objective of this experiment is to understand how variable cross-sectional area affects heat transfer by conduction. the hot and cold segments will have the same cross-sectional area. the temperature gradient is inversely proportional to the cross-sectional area. and cold segments of the bar. but instead of using the 25 mm diameter brass sample. sample. APPARATUS: The apparatus for this experiment is the P. Now we would like to see how changes in cross-sectional area affect heat transfer. we will use the brass sample with a diameter of 13 mm. s. Are the area and temperature gradient inversely proportional? Discuss possible sources of error within the equipment or the assumptions made in the theory. Connect the nine thermocouples in the appropriate order. Use these slopes to compute the experimental value of the temperature gradient ratio and compare that value to the theoretical value obtained from the crosssectional areas. 2. 3-6. Adjust the power control knob to deliver 10 W to the test unit and allow the unit to reach steady state (approximately 20 minutes). 4. 58-67. pp. To avoid burns. 7. ASSIGNED READING: Fundamentals of Heat and Mass Transfer. Apply a very small amount (a drop) of thermal conducting paste to make a thin layer on each side of the brass sample of 30 mm length and 13 mm diameter and spread it uniformly. 3. Avoid using a high cooling water flow to prevent disconnection of the hose from the test unit. RESULTS: 6. 4. Insert the sample into the test unit and allow cooling water to flow through the test unit. 5. 2. App. and do not allow the temperature to go above 100 °C at any of the thermocouple locations.PROCEDURE: 1. SAFETY GUIDELINES: 1. Do not exceed 20 W power delivery under any circumstances. pp. Avoid using too much conducting paste as this may ‘fry’ the unit. 3. Plot the temperature profile along the length of the bar and determine the slopes dT / dx of the best-fit lines for each distinct segment of the bar. . Incropera and DeWitt. A. Record the temperatures at each of the six thermocouples and the power input. do not touch any metal or plastic surfaces on the hot end of the sample or test unit. EXPERIMENT #1 RAW DATA TABLES Table 1: Raw Data for Experiment #1. Part B – Cross-Sectional Area q (W) T 1 (°C) T 2 (°C) T 3 (°C) T 4 (°C) T 5 (°C) T 6 (°C) T 7 (°C) T 8 (°C) T 9 (°C) . Part A – Simple Bar q (W) T 1 (°C) T 2 (°C) T 3 (°C) T 4 (°C) T 5 (°C) T 6 (°C) T 7 (°C) T 8 (°C) T 9 (°C) Table 2: Raw Data for Experiment #1.