Steady State Temperature Steady State Temperature Profiles in Rods - PowerPoint PPT Presentation

Steady State Temperature Steady State Temperature Profiles in Rods Profiles in Rods Amy Chan Amy Chan Anders Berliner, Catherine Chau Chau, , Erise Erise Hosoya Hosoya Anders Berliner, Catherine Case Western Reserve University Case Western Reserve University Department of Chemical Engineering Department of Chemical Engineering April 8, 2003 April 8, 2003

Outline Outline Energy Balance Energy Balance Procedure Procedure Results/Discussion Results/Discussion Conclusion Conclusion

Energy Balance Energy Balance Heat Transfer Mechanisms Heat Transfer Mechanisms � Conduction and Convection Conduction and Convection � Energy Balance: Q 1 = Q 2 + Q 3 Energy Balance: Q 1 = Q 2 + Q Q 2 3 T a T (x) Differential Equation: T s Q 1 2 d T hP Q 3 ( ) − − = T T 0 a 2 dx kA x x = 0 Δ x General Solution: 4h = + + m = T(x) Acosh(mx) Bsinh(mx) T a kd

Energy Balance Solutions Energy Balance Solutions Air Cooled Air Cooled Boundary Conditions ⎡ ⎤ h [ ] [ ] ( ) ( ) − + − cosh m L x sinh m L x ⎢ ⎥ T = T s at x = 0 mk ⎣ ⎦ = θ h [ ] [ ] + Q cond = Q conv at x = L cosh mL sinh mL mk Water Cooled Water Cooled Boundary Conditions ( ) T = T s at x = 0 − sinh[mx] sinh[m x L ] = − θ θ L sinh[mL] sinh[mL] T = T L at x = L

Experimental Setup Experimental Setup

Procedure Procedure Frequency of data acquisition: 15- -60 60 Frequency of data acquisition: 15 seconds seconds Number of Steady State Points: 5 Number of Steady State Points: 5 Order of testing: Order of testing: Air Exposed Steel Air Exposed Steel � � Air Exposed Aluminum Air Exposed Aluminum � � Water Exposed Aluminum Water Exposed Aluminum � � Water Exposed Steel Water Exposed Steel � �

Procedure for Finding k k Al Procedure for Finding Al Determining k k Al : Determining Al : θ exp θ theory Calculate θ and θ Calculate exp and theory θ exp – θ θ theory Minimize ( θ 2 : ) 2 : Minimize ( exp – theory ) � Solve for h Solve for h � � Solve for Solve for k k Al � Al − T T = θ a Analytical Methods Analytical Methods exp − T T s a End Thermocouple End Thermocouple End Rod End Rod

Summary of Aluminum Results Summary of Aluminum Results Thermocouples Ends of Rods Situation k T (W/(m K)) k R (W/(m K)) Air Exposed 186 211 Water Exposed 120 313 Percent Difference 43.1% 38.9% Uncertainties: δ h ~ 0.4 (W/m 2 K) δ k ~ 4 (W/m K)

Comparison of Heat Transfer Comparison of Heat Transfer Coefficients Coefficients Calculation Method Thermocouples Ends of Rods Situation h T (W/m 2 K) h R (W/m 2 K) % Error % Error Air Exposed 18.8 88.0% 48.3 383.0% Water Exposed 9.14 8.6% 83.3 733.0% Average - 48.3% - 558.0% Δ h i = h i – h expected Expected h ~ 10 W/m 2 K

Which Analytical Method is Better? Which Analytical Method is Better? Greater accuracy is preferred Greater accuracy is preferred Thermocouple method: Thermocouple method: � Smaller Smaller h h percent error percent error � � Fewer estimations Fewer estimations � End rod method estimations: End rod method estimations: � Length from last thermocouple to water Length from last thermocouple to water � � Length of rod into steam Length of rod into steam � Use thermocouple method data for graphs Use thermocouple method data for graphs

Unsteady State Temperature Unsteady State Temperature Profiles at Various Time Intervals Profiles at Various Time Intervals 80 Initial t = 5 min 70 t = 10 min Temperature ('C) t = 20 min 60 Steady State (t~30 min) 50 40 30 20 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 Distance down Rod (m)

Air Exposed Temperature Profiles Air Exposed Temperature Profiles 1.20 Solid Lines: Experimental 1.00 Dotted Lines: Theoretical 0.80 Aluminum 0.60 θ 0.40 0.20 Steel 0.00 0.00 0.10 0.20 0.30 0.40 0.50 -0.20 Distance from steam (m)

Steel Water Exposed Temperature Steel Water Exposed Temperature Profiles Profiles 1.20 Experimental Results 1.00 0.80 0.60 Theoretical Results 0.40 θ 0.20 0.00 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 -0.20 -0.40 Distance from Steam (m)

Aluminum Water Exposed Aluminum Water Exposed Temperature Profiles Temperature Profiles 1.20 1.00 Theoretical Results 0.80 Experimental Results 0.60 0.40 θ 0.20 0.00 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 -0.20 -0.40 Distance from Steam (m)

Conclusion Conclusion More Accurate Method: Thermocouple More Accurate Method: Thermocouple Method Method Error due to assumptions Error due to assumptions Conductivity: Conductivity: � Aluminum Aluminum � ± 4 Air: 186 ± · K m · 4 W/ W/m K Air: 186 ± 4 Water: 120 ± 2 · · K 4 W/m W/m 2 K Water: 120 · K m · � Steel: 16 W/ Steel: 16 W/m K �

Acknowledgements Acknowledgements Group Members: Group Members: � Anders Berliner Anders Berliner � � Catherine Catherine Chau Chau � � Erise Erise Hosoya Hosoya � TA: Bethany Bustard TA: Bethany Bustard Professor: Uziel Uziel Landau Landau Professor: Lab Manager: Craig Virnelson Virnelson Lab Manager: Craig