One-Dimensional, Steady-State Conduction with Thermal Energy Generation

1. One-Dimensional, Steady-State Conduction with Thermal Energy Generation Chapter Three Section 3.5, Appendix C

2. Alternative Conduction Analysis  Standard approach is useful for constant k and A. Alternative method may be needed for changing k(T) or A(x) as long as qx is constant. Refer to Figure 3.6, A(x) is a function of x, k(T) changes with T qx = -k(T)*A(x)*(dT/dx)=constant

3. Alternative Conduction Analysis 

4. Alternative Conduction Analysis  qx = -k(T)*A(x)*(dT/dx)=constant The last eqn. applies to uniform A and constant k. 1-D, Steady-state, no heat generation

5. Examples 3.5 (pages 133-135) The diagram shows a conical section fabricated from pyroceram. It is of a circular cross section with the diameter D = ax, where a = 0.25. The smaller end at x1=50 mm and large end at x2=250 mm. The end Ts are T1=400K and T2=600K, while the lateral surface is well insulated. Derive an expression for the T(x) in symbolic form, assume 1-D condition. Sketch the T distribution. Calculate the heat rate qx through the cone.

6. Examples 3.5

7. Examples 3.5 Known: Conduction in circular conical section having a diameter D=ax, where a=0.25. Find: 1. T(x), 2. Heat transfer rate qx Schematic:

8. Examples 3.5 Assumptions: Steady-state; 1-D conduction in x direction; No internal heat generation; Constant properties. Properties: Table A.2 (page 988), pyroceram (500K): k=3.46 W/mK

9. Examples 3.5 Analysis: 1-D, steady-state without heat generation Where A = ? D2/4=? a2x2/4, separating variables,

10. Examples 3.5 Analysis: Hence: Although qx is a constant, yet, an unknown. We need the second b.c. to evaluate qx.

11. Examples 3.5 Analysis: At x=x2, T=T2 (2nd b.c.)

12. Examples 3.5 Analysis: Substituting numerical values into the foregoing eqn. Comments: The heat transfers in the direction of decreasing temp.

13. Examples 3.6 (pages 138-141) 

14. Examples 3.6 Known: Liquid N2 is stored in a spherical container that is insulated and exposed to ambient air Find: 1 The rate of heat transfer to N2 2 The mass rate of N2 boil-off Schematic:

15. Examples 3.6 

16. Examples 3.6 Assumptions: Steady-state; 1-D conduction through radial direction; Negligible resistance to heat transfer through the container wall and from the container to N2; Constant properties; Negligible radiation exchange between outer surface of insulation and the surroundings.

17. Examples 3.6 Properties: From Table A.3 (page 936), evacuated silica powder (300K): k=0.0017W/mK Analysis: The thermal circuit involves a conduction and convection resistance in series and is of the form:

18. Examples 3.6 Analysis: From Eqn. 3.36 From Eqn. 3.9 Heat transfer rate to the liquid N2:

19. Examples 3.6 Analysis: Heat transfer rate to the liquid N2: Hence

20. Examples 3.6 Analysis: q = 223/(17.02+0.05)=13.06 W (2). Energy balance for a control surface about N2

21. Examples 3.6 Analysis: The loss per day (liters/day):

22. Examples 3.6 Comments: Rt,cond >>Rt,conv, to reduce the boil-off, need to look for better or thicker insulator. Daily loss is about 10.8% of the total volume in the container. Doubling thickness of insulator can reduce 45% loss. If a specific boil-off rate is required, insulator thickness can be determined to meet the needs.

23. One-Dimensional, Steady-State Conduction with Thermal Energy Generation Chapter Three Section 3.5, Appendix C

24. Implications

25. Conduction with Heat Generation Plane Wall with Thermal Energy Generation Steady-state, 1-D conduction through x direction

26. The Plane Wall

27. Plane wall (cont.)

28. Radial Systems

29. Radial systems (cont.)

30. Problem: Nuclear Fuel Rod

31. Problem: Nuclear fuel rod (cont.)

32. Problem: Nuclear fuel rod (cont.)

33. Problem: Nuclear fuel rod (cont.)

34. Problem: Nuclear fuel rod (cont.)

35. Problem: Nuclear fuel rod (cont.)