EXPERIMENT 105. REFERENCES RADIAL HEAT CONDUCTION: FOURIER LAW Incropers, Heat and mass Transfer.. Chap- ters 1 and 2, section 1.2 and 2.1 Section 80 of this lab manual OBJECTIVES In this experiment you will 1. understand the use of the Fourier Rate Equation a in determining rate of heat flow for steady state conduction of energy through the wall of a cylin- der (radial energy flow) and using the equation to determine the constant of proportionality (the thermal conductivity) of the disk material APPARATUS Heat Transfer Service Unit (HTSU: section 70) Radial Heat Conduction Unit (RHCU: section 80) Stopwatch Beaker or container can flow only in the radial direction. There is no heat flow in the angular (0) direction because the tempera- ture is the same all the way around the circumference of the cylinder. The following conditions apply: No heat flow in z-direction ยะ Uniform temperature in theta OT =0 No heat generation • Steady state (-) Constant thermal conductivity (k) direction With these conditions, the conduction equation in cylindrical coordinates becomes: 10 ()- =0. T (1) After integration and use of boundary conditions we get THEORY Consider the steady state conduction through the walls of a cylinder as shown in fig.1. When the inner and outer surfaces of a cylinder are each at a different uni- form temperature, heat flows radially through the wall of the cylinder. T=C₁ In(r) +C2. where T-T C₁ = R₁ In R₂ (3) Fig.1 Steady state heat conduction As shown in the sketch, the solid is in the form of a hollow cylinder and the outer and inner surfaces are maintained at temperatures T and Ty, respectively. The ends of the cylinder are insulated so that heat and R₁ is the inside radius with corresponding tem- perature T and R₁ is an outer radius corresponding to temperature T If a concentric layer (of thickness dr and radius r from the centre) in the wall of the cylinder is considered, the area of heat flow is 2rr for a unit length of cylin- der. The temperature gradient normal to the axis of the cylinder is dT/dr. According to Fouriers law, the heat transfer rate per unit length 9r=-2ark =2k 8T бр T-T₂ R₂ (4) In R₁ 63 APPA E Li 70 Ste Be THEORY Consider t as shown i and Cand ness L, w convect he side may b environmen infy the bes is a linear i following so Considering a cylinder of length I. and rearranging the the last equation we get: T-T Q-2rkl Ra In R In 2rkL (T-T) (6) The heat flow path in the RHCU consists of a cylinder 3.2 mm long (thickness of disk Z) with inside radius R-7 mm (where thermocouple T is located) and outer radius Re-50 mm (where thermocouple To is located). The objective of this exercise is to determine the conductivity of the disk material from measure- ments of temperature difference and heat flow through the cylindrical disk. PROCEDURE Fig. 1 Radial beat conduction 1. Connect the thermocouples to the appropriate sockets on the front of the service unit. 2. Start the computer. Start the heat transfer. service unit (HTSU) using the power switch MAINS. Set the selector switch to "Automatic", then start the ArmSoft software and select "Ex- ercise B. 3. The Mimic Diagram is the most commonly used display of the software, it gives a pictorial rep resentation of the equipment, with continuously updated display boxes for all the various sensor readings, calculated variables etc. To view the Mimic Diagram, click Diagram from the View drop down menu. 4. In order to establish communication between the software and HTSU, click on the "Power On" switch on the software diagram window 64 5. Open the water supply valve on the main sup- ply hose, then adjust the flow control valve (not the pressure regulator) to give approximately 1.5 litres/min. The actual flow can be checked using a stopwatch and a measuring cylinder. 6. Use the heater control box on the mimic diagram screen to adjust the percentage of full scale until the voltage display box reads 12 volts 7. Allow the RHCU to stabilise (at least 15 mins). Monitor the temperatures on the software mimic diagram screen to determine when the tem- peratures reach steady state (use "VIEW" and "SHOW HISTORY" buttons, as well as "VIEW" and "SHOW GAUGE" buttons). 8. When the temperatures have reached steady state, select the icon "GO" to record one set of data, which includes the following: T. Ta T, Ta. T. Te. V, I Use button "VIEW" and "PLOT RESULTS" to plot results and check if a the measured points form a straight line. You can check results by looking at the "VIEW" and "TABLE" buttons. 9. Set the heater to 17, 21, and 24 volts using the same method as before. Allow the RHCU to stabilise then repeat the above readings. Use buttons "SAMPLE" and "NEXT RESULTS" to enable the "GO" button. 10. Save the results of measurements using the com- mand "save as" from the file menu. Make sure that you specify excel format (filename.xls). 11. After completion of all measurements turn off the "Power On" button on the software screen, power button MAINS on the HTSU and turn off the water supply valve on the supply hose. Notes: For this experiment the following constants are applicable: Radius at thermocouple 7 Ry=0.007 m Radius at thermocouple Ts Thickness of disk R-0.050 m L=0.0032 m Measurement errors: Temperature is measured with accuracy of ±0.1"C. Voltage is measure with ac- curacy of ±0.1V. Current is measured with accuracy of ±0.1A. Linear dimensions are known with accuracy of 10.2mm. Position of thermocouples is known with accuracy of ±0.3mm. APPLICATION OF THEORY 1. Estimate the cumulative influence of the exper- imental errors on your calculated values for Q and k and measured values of Ry. Re and L. 2. Your results should give the approximate value of 125 W/m°C for the thermal conductivity of brass sheet. Compare the values obtained for the Thermal Conductivity k at the different settings of heat flow through the specimen. 3. Calculate the thermal conductivity of the Brass disk using two different radii and corresponding temperatures. Compare the value obtained with the original value at the same heat flow. 4. Plot a log/linear graph of radius on the loga- rithmic axis and temperature on the linear axis then draw the best straight line through the points. Calculate the average conductivity of the brass bar using the gradient of each straight line and the corresponding heat flow through the bar. Compare the value obtained with the val- ues previously obtained for each individual sec- tion of the cylinder and comment on any dif ference. Your results should give values in the range 120-140 W/m°C for the thermal conduc- tivity of Brass, assuming no heat loss from the equipment. As a small amount of heat loss is inevitable as the temperature of the cylinder in- creases, the calculated value for the conductivity will increase at higher operating temperatures. 65 EXPERIMENT 105. RADIAL HEAT CONDUCTION: FOURIER LAW-PRELAB 1. Read the Lab Manual write-up on this experiment along with the sections from your class textbook listed in the references section. NOTE: This PRELAB is due at the beginning of the lab. 2. What is the objective of this experiment? 3. Prove the result of eqs.2 and 3 67 NAME DATE GRADE