Laminar and Turbulent Heat Transfer from a Uniformly Heated Plate by Dan Schwarz School of Engineering Grand Valley State University EGR 468 Heat Transfer Section 02 Instructor Dr M Sozen March 27 2008 1 Introduction Purpose A horizontal heated plate was used to experimentally verify the empirical relationships for the average surface temperature of a flat plate exposed to uniform heat flux with parallel fluid flow The average temperature difference between the plate and ambient air was measured with both laminar and turbulent air flow conditions The measured temperature differences were then compared to temperature differences calculated using both laminar and turbulent heat transfer math models The results showed that the math models consistently predicted temperature differences that were higher than the observed experimental measurements It was concluded that the heat flux which was derived experimentally using natural convection conditions was too large A smaller assumed heat flux was used in the calculations to show that the empirical relationships are valid with the correct heat flux Theory An empirical relationship which may be used to calculate the average temperature difference of a flat horizontal plate with constant heat flux is given by Equation 1 Equation 1 must only be used for laminar flow conditions See Appendix A for a sample calculation using Equation 1 Tw T qw L 0 6795k Re1L 2 Pr 1 3 1 A second empirical relationship is given by Equation 2 to calculate the average temperature difference of a flat horizontal plate with constant heat flux Equation 2 must only be used for turbulent flow conditions See Appendix B for a sample calculation using Equation 2 Tw T k1 04 Pr 1 3 qw L 0 037 Re 0L 8 871 2 Procedure An aluminum plate was placed in a wind tunnel with a ceramic heating element beneath it Four thermocouples were attached to the center line of the plate and one thermocouple was suspended above the plate Figure 1 shows the experimental system The heating element was turned on and the data acquisition was initiated without turning on the wind tunnel The data acquisition station continued taking temperature measurements until it was clear that steady state had been attained The steady state values for this free convection system were used to estimate the heat flux on the plate See Appendix C for the heat flux calculation This process was repeated several times with the wind tunnel turned on at increasing speeds Several speeds were used in both the laminar and turbulent range See Appendix D for the laminar transition calculation Finally Equation 1 and Equation 2 were used to calculate predicted temperature differences for each given set of conditions in the experiment Then the calculation results were compared to the experimental data 2 0 23m V 0 23m 0 07m Figure 1 The experimental plate and heater were contained in the test section of a wind tunnel Lab Equipment National Instruments DAQ CA 1000 National Instruments Voltage Meter NI 4350 Computer with Labview Data Acquisition Software Five K type Thermocouples Wind Tunnel Ceramic Heating Element Aluminum Plate with Wooden Ramp Results The calculation results are compared to the experimental data in Figure 2 The calculated temperature differences diverge from the experimental data by a constant factor since the curves are practically parallel See Appendix E for the raw data used to create Figure 2 The raw data in Appendix E shows that the empirical models become inaccurate when they are used for the wrong flow regime It was concluded that the heat flux value derived from the free convection portion of the experiment was too large An assumed heat flux value of 600 W m 2 was used to correct the vertical shift in the data as shown in Figure 2 With this assumption it appears that the empirical relationships given in Equation 1 and Equation 2 closely reflect our experimental results See Appendix F for the raw data used to create Figure 3 3 Figure 2 Comparison of experimental data with temperature differences calculated using a heat flux of 1910 W m2 Figure 3 Comparison of experimental data with temperature differences calculated using an assumed heat flux of 600 W m2 Conclusion Given the complexity of this experiment there are many possible sources for error The most likely source of error is caused by the unevenly distributed heat flux on the plate Table 1 shows that the temperature measurements were not symmetrical across the plate in the free convection case The back corner of the plate was much warmer than the front corner Figure 3 Steady state thermocouple readings for the free convection case T2 C T3 C T4 C T5 C 201 7 210 6 211 7 225 8 4 Another possible source of error is that the thermo couple may have been inappropriately located to measure the ambient temperature for the free convection case This error would also have an adverse effect the calculated heat flux value Ultimately it was determined that the calculated heat flux value was much too high for the system Instead an assumed heat flux value of 600 W m2 was used to show that the empirical relationships are consistent with the experimental measurements Appendix A Sample Calculation of Average Temperature Difference with Equation 5 50 Laminar Conditions Calculate the average difference between the plate temperature and the ambient temperature using equation 5 50 Tw T Tw T qw L 0 6795k Re1L 2 Pr 1 3 1910W 0 6795 0 02624W m 2 0 23m m C 196 512 1 2 0 708 1 3 62 4 C Appendix B Sample Calculation of Average Temperature Difference with Equation 5 85 5 87 Turbulent Conditions Calculate the average difference between the plate temperature and the ambient temperature using equation 5 85 and 5 87 Nu 1 04 Pr 1 3 0 037 Re 0L 8 871 kNu k1 04 Pr 1 3 0 037 Re 0L 8 871 h L L qw L Tw T 1 3 k1 04 Pr 0 037 Re 0L 8 871 1910W m 2 0 23m Tw T 35 2 C 1 3 0 8 1 04 0 02624W m C 0 708 0 037 519 921 871 Appendix C Heat Flux Calculation via Free Convection plus Radiation Heat Transfer The energy balance on the plate includes both free convection and radiation heat transfer dE q w A hA Tw T A Tw4 T 4 dt Under steady state conditions the time rate of energy change is zero q w h Tw T Tw4 T 4 5 Find the film temperature and the corresponding properties of air Tf Tw T 485K 310 K 397 5K 400 K 2 2 25 9 10 6 m 2 s k 0 03365W m C Pr 0 689 Calculate the effective length of the plate for horizontal free convection L A 0 23m 0 165m 0 048m P 2 0 23m 0 165m Calculate GrPr Gr Pr 3 g Tw T L3 Pr 9 81