Forms of Energy Generation : Degradation of electrical energy to heat

1. Overall Shell Energy Balance Forms of Energy Generation: Degradation of electrical energy to heat Heat from nuclear source (by fission) Heat from viscous dissipation (Se) (Sn) (Sv) Energy Generation Let S = rate of heat production per unit volume (W/m3)

2. Electrical Heat Source Consider an electrical wire (solid cylinder): Shell Heat Balance: Rate of Heat IN: Rate of Heat OUT: Generation:

3. Electrical Heat Source Rate of Heat IN Area perpendicular to qr at r = r The Shell: Rate of Heat IN: Rate of Heat OUT: Generation:

4. Electrical Heat Source Rate of Heat OUT Area perpendicular to qr at r = r + dr The Shell: Rate of Heat IN: Rate of Heat OUT: Generation:

5. Electrical Heat Source Generation = Volume X Se Too small The Shell: Rate of Heat IN: Rate of Heat OUT: Generation:

6. Electrical Heat Source Consider an electrical wire (solid cylinder): Shell Heat Balance: Dividing by Q: Why did we divide by and not by ?

7. Electrical Heat Source Consider an electrical wire (solid cylinder): We now have: Taking the limit as : Q: Is this correct? NO!

8. Electrical Heat Source Consider an electrical wire (solid cylinder): We now have: We must adhere to the definition of the derivative:

9. Electrical Heat Source Consider an electrical wire (solid cylinder): We now have: Integrating: Boundary conditions: Note: The problem statement will tell you hints about what boundary conditions to use.

10. Electrical Heat Source Consider an electrical wire (solid cylinder): We now have: Applying B.C. 1: Because q has to be finite at r = 0, all the terms with radius, r, below the denominator must vanish. Therefore:

11. Electrical Heat Source Consider an electrical wire (solid cylinder): We now have: Substituting Fourier’s Law:

12. Electrical Heat Source Consider an electrical wire (solid cylinder): We now have: Applying B.C. 2: This is it! But, we rewrite it into a nicer form…

13. Electrical Heat Source Temperature Profile: Consider an electrical wire (solid cylinder): Important assumptions: Temperature rise is not large so that k and Se are constant & uniform. The surface of the wire is maintained at T0. Heat flux is finite at the center.

14. Electrical Heat Source Other important notes… Let: electrical conductivity current density voltage drop over a length These imply the following :

15. Electrical Heat Source Heat flux profile: Temperature Profile: The stress profile versus the temperature profile:

16. Electrical Heat Source Quantities that might be asked for: Maximum Temperature Average Temperature Rise Heat Outflow Rate at the Surface Substituting r = 0 to the profile T(r):

17. Electrical Heat Source Examples for Review: Example 10.2-1 and Example 10.2-2 Bird, Stewart, and Lightfoot, Transport Phenomena, 2nd Ed., p. 295

18. Nuclear Heat Source Consider a spherical nuclear fuel assembly (solid sphere): Before doing a balance, let: volumetric heat rate of production within the fissionable material only volumetric heat rate of production at r = 0 Sn depends on radius parabolically: a dimensionless positive constant

19. Nuclear Heat Source Consider a spherical nuclear fuel assembly (solid sphere): Before doing a balance, let: temperature profile in the fissionable sphere temperature profile in the Alcladding heat flux in the fissionable sphere heat flux in the Al cladding

20. Nuclear Heat Source Consider a spherical nuclear fuel assembly (solid sphere): For the fissionable material: Rate of Heat IN: Rate of Heat OUT: Generation:

21. Electrical Heat Source Generation = Volume X Sn Too small Rate of Heat IN: Rate of Heat OUT: Generation:

22. Nuclear Heat Source For the fissionable material: No generation here! For the Al cladding: Dividing by : Dividing by :

23. Nuclear Heat Source For the fissionable material: No generation here! For the Al cladding: Taking : Taking :

24. Nuclear Heat Source For the fissionable material: No generation here! For the Al cladding: Taking : Taking :

25. Nuclear Heat Source For the fissionable material: No generation here! For the Al cladding: Integrating: Integrating:

26. Nuclear Heat Source Boundary Conditions: Boundary Conditions: Integrating: Integrating: For the fissionable material For the Al cladding

27. Nuclear Heat Source For the fissionable material For the Al cladding Inserting Fourier’s Law: Inserting Fourier’s Law:

28. Nuclear Heat Source For the fissionable material For the Al cladding Boundary Conditions: Boundary Conditions: R(C) At r = R(F), T(F) = T(C) At r = R(C), T(C) = T0 R(F)

29. Nuclear Heat Source For the fissionable material For the Al cladding

30. Overall Shell Energy Balance Recall the Overall Shell Energy Balance: Q by Convective Transport Q by Molecular Transport W by Molecular Transport W by External Forces Energy Generation Steady-State!

31. Overall Shell Energy Balance We need all these terms for viscous dissipation: Q by Convective Transport Q by Molecular Transport W by Molecular Transport How can we account for all these terms at once?

32. Combined Energy Flux Vector We introduce something new to replace q: Combined Energy Flux Vector: Heat Rate from Molecular Motion Convective Energy Flux Work Rate from Molecular Motion

33. Combined Energy Flux Vector We introduce something new to replace q: Combined Energy Flux Vector: Recall the molecular stress tensor: When dotted with v: Substituting into e:

34. Combined Energy Flux Vector We introduce something new to replace q: Combined Energy Flux Vector: Simplifying the boxed expression: Finally:

35. Viscous Dissipation Source Consider the flow of an incompressible Newtonian fluid between 2 coaxial cylinders:

36. Viscous Dissipation Source Consider the flow of an incompressible Newtonian fluid between 2 coaxial cylinders: We now make a shell balance shown in red on the left. Rate of Energy IN: Rate of Energy OUT: When the combined energy flux vector is used, the generation term will automatically appear from e.

37. Viscous Dissipation Source Consider the flow of an incompressible Newtonian fluid between 2 coaxial cylinders: We now make a shell balance shown in red on the left. Rate of Energy IN: Rate of Energy OUT: When the combined energy flux vector is used, the generation term will automatically appear from e.

38. Viscous Dissipation Source Consider the flow of an incompressible Newtonian fluid between 2 coaxial cylinders: Fourier’s Law: Newton’s Law: When the combined energy flux vector is used, the generation term will automatically appear from e.

39. Viscous Dissipation Source Consider the flow of an incompressible Newtonian fluid between 2 coaxial cylinders: Substituting the velocity profile: Integrating: When the combined energy flux vector is used, the generation term will automatically appear from e.

40. Viscous Dissipation Source Consider the flow of an incompressible Newtonian fluid between 2 coaxial cylinders: Boundary Conditions: After applying the B.C.: When the combined energy flux vector is used, the generation term will automatically appear from e.

41. Viscous Dissipation Source Consider the flow of an incompressible Newtonian fluid between 2 coaxial cylinders: Q: So where is Sv? After applying the B.C.: When the combined energy flux vector is used, the generation term will automatically appear from e.

42. Viscous Dissipation Source Consider the flow of an incompressible Newtonian fluid between 2 coaxial cylinders: Temperature Profile: New Dimensionless Number:

43. Viscous Dissipation Source Scenarios when viscous heating is significant: Flow of lubricant between rapidly moving parts. Flow of molten polymers through dies in high-speed extrusion. Flow of highly viscous fluids in high-speed viscometers. Flow of air in the boundary layer near an earth satellite or rocket during reentry into the earth’s atmosphere.