Transient Conduction: Spatial Effects and the Role of Analytical Solutions

1. Transient Conduction:Spatial Effects and the Role ofAnalytical Solutions Chapter 5 Sections 5.4 to 5.8

2. Solution to the Heat Equation for a Plane Wall with Symmetrical Convection Conditions (5.26) (5.27) (5.29) (5.28) (5.30) Plane Wall • If the lumped capacitance approximation can not be made, consideration must • be given to spatial, as well as temporal, variations in temperature during the • transient process. • For a plane wall with symmetrical convection • conditions and constant properties, the heat • equation and initial/boundary conditions are: • Existence of seven independent variables: How may the functional dependence be simplified?

3. Dimensionless temperature difference: Dimensionless coordinate: Dimensionless time: The Biot Number: (5.39a) (5.39b,c) See Appendix B.3 for first four roots (eigenvalues ) of Eq. (5.39c) Plane Wall (cont.) • Non-dimensionalization of Heat Equation and Initial/Boundary Conditions: • Exact Solution:

4. Variation of midplane temperature (x*= 0) with time : (5.41) • Variation of temperature with location (x*) and time : (5.40b) (5.43a) (5.46) (5.44) Can the foregoing results be used if an isothermal condition is instantaneously imposed on both surfaces of a plane wall or on one surface of a wall whose other surface is well insulated? Plane Wall (cont.) • The One-Term Approximation : • Change in thermal energy storage with time: Can the foregoing results be used for a plane wall that is well insulated on one side and convectively heated or cooled on the other?

5. Heisler Charts Graphical Representation of the One-Term Approximation The Heisler Charts • Midplane Temperature:

6. Heisler Charts (cont.) • Temperature Distribution: • Change in Thermal Energy Storage:

7. One-Term Approximations: • Long Rod: Eqs. (5.49) and (5.51) • Sphere: Eqs. (5.50) and (5.52) Radial Systems Radial Systems • Long Rods or Spheres Heated or Cooled by Convection. • Graphical Representations: • Long Rod: Figs. D.4 – D.6 • Sphere: Figs. D.7 – D.9

8. (5.57) (5.58) The Semi-Infinite Solid Semi-Infinite Solid • A solid that is initially of uniform temperature Ti and is assumed to extend • to infinity from a surface at which thermal conditions are altered. • Special Cases: • Case 1: Change in Surface Temperature (Ts)

9. Case 2: Uniform Heat Flux Case 3: Convection Heat Transfer Semi-Infinite Solid (cont.) (5.59) (5.60)

10. Multidimensional Effects Multidimensional Effects • Solutions for multidimensional transient conduction can often be expressed • as a product of related one-dimensional solutions for a plane wall, P(x,t), • an infinite cylinder, C(r,t), and/or a semi-infinite solid, S(x,t). See Equations • (5.64) to (5.66) and Fig. 5.11. • Consider superposition of solutions for two-dimensional conduction in a • short cylinder:

11. Problem: Thermal Energy Storage Problem 5.66: Charging a thermal energy storagesystem consisting of a packed bed of Pyrex spheres.

12. Problem: Thermal Energy Storage

13. Problem: Thermal Energy Storage

14. Problem: Thermal Response Firewall Problem: 5.82: Use of radiation heat transfer from high intensity lamps for a prescribed duration (t=30 min) to assess ability of firewall to meet safety standards corresponding to maximum allowable temperatures at the heated (front) and unheated (back) surfaces.

15. Problem: Thermal Response of Firewall

16. Problem: Thermal Response of Firewall