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 Anders Berliner, Catherine Chau Chau, , Erise Erise Hosoya Hosoya 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 Energy Balance: Q1

1 = Q

= Q2

2 + Q

+ Q3

3

Q1 Q2 Q3 Ta Ts x Δx x = 0 T (x)

( )

T T kA hP dx T d

a 2 2

= − −

a

T Bsinh(mx) Acosh(mx) T(x) + + =

Differential Equation: General Solution:

kd 4h m =

Energy Balance Solutions Energy Balance Solutions

Air Cooled Air Cooled Water Cooled Water Cooled

( )

sinh[mL] ] L x sinh[m sinh[mL] sinh[mx] θ θ

L

− − =

( )

[ ]

( )

[ ] [ ] [ ]

mL sinh mk h mL cosh x L m sinh mk h x L m cosh θ + ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − + − =

Boundary Conditions T = Ts at x = 0 Qcond = Qconv at x = L T = Ts at x = 0 T = TL at x = L Boundary Conditions

Experimental Setup Experimental Setup

Procedure Procedure

Frequency of data acquisition: 15 Frequency of data acquisition: 15-

• 60

60 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 Procedure for Finding k kAl

Al

Determining Determining k kAl

Al:

: Calculate Calculate θ θexp

exp and

and θ θtheory

theory

Minimize ( Minimize (θ θexp

exp –

– θ θtheory

theory)

)2

2:

:

• Solve for h

Solve for h

• Solve for

Solve for k kAl

Al

Analytical Methods Analytical Methods End Thermocouple End Thermocouple End Rod End Rod

a s a exp

T T T T θ − − =

Summary of Aluminum Results Summary of Aluminum Results

Uncertainties: δh ~ 0.4 (W/m2K) δk ~ 4 (W/m K)

Thermocouples Ends of Rods kT (W/(m K)) kR (W/(m K)) Air Exposed 186 211 Water Exposed 120 313 Percent Difference 43.1% 38.9% Situation

Comparison of Heat Transfer Comparison of Heat Transfer Coefficients Coefficients

Calculation Method Thermocouples Ends of Rods hT (W/m2K) % Error hR (W/m2K) % 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%

Situation

Δhi = hi – hexpected Expected h ~ 10 W/m2K

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

20 30 40 50 60 70 80 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) Temperature ('C) Initial t = 5 min t = 10 min t = 20 min Steady State (t~30 min)

Air Exposed Temperature Profiles Air Exposed Temperature Profiles

• 0.20

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

Aluminum

Steel Water Exposed Temperature Steel Water Exposed Temperature Profiles Profiles

• 0.40
• 0.20

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

Aluminum Water Exposed Aluminum Water Exposed Temperature Profiles Temperature Profiles

• 0.40
• 0.20

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

Conclusion Conclusion

More Accurate Method: Thermocouple More Accurate Method: Thermocouple Method Method Error due to assumptions Error due to assumptions Conductivity: Conductivity:

• Aluminum

Aluminum

Air: 186 Air: 186 ± ± 4 4 W/ W/m m·

·K

K Water: 120 Water: 120 ± ± 4 4 W/m W/m2

2· ·K

K

• Steel: 16 W/

Steel: 16 W/m m·

·K

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: Professor: Uziel Uziel Landau Landau Lab Manager: Craig Lab Manager: Craig Virnelson Virnelson