1. HW2 Due date Next Tuesday (October 14)

2. Lecture Objectives: Unsteady-state heat transfer - conduction Solve unsteady state heat transfer equation for a wall

3. Implicit methods - example  =0 To Tw Ti  =36 system of equation Tw Ti  =72 system of equation Tw Ti After rearranging: 2 Equations with 2 unknowns!

4. Explicit methods - example  =0 To Tw Ti  =360 To Tw Ti  =720 To Tw Ti   =36 sec UNSTABILITY There is NO system of equations! Time

5. Explicit method Problems with stability !!! Often requires very small time steps

6. Explicit methods - example  =0 To Tw Ti  =36 To Tw Ti  =72 To Tw Ti   =36 sec Stable solution obtained by time step reduction 10 times smaller time step Time

7. Unsteady-state conduction - Wall q Nodes for numerical calculation xx

8. Discretization of a non-homogeneous wall structure Section considered in the following discussion Discretization in space Discretization in time

9. Internal node Finite volume method For node “I” - integration through the control volume Boundaries of control volume Surface node

10. Left side of equation for node “I” Right side of equation for node “I” Mathematical approach (finite volume method) - Discretization in Time - Discretization in Space

11. Mathematical approach (finite volume method) Explicit method and for uniform grid Implicit method By Substituting left and right side terms of equation we get the following equation for Using

12. Physical approach (finite volume method) Boundaries of control volume Change of energy in  x Change of heat flux along x qxqx = Change of energy in  x Sum of energy that goes in and out of control volume  x = or For finite volume  x:

13. For uniform grid Physical approach (finite volume method)

14. Internal node finite volume method Explicit method Implicit method By defining time step on the right side we get

15. Internal node finite volume method Explicit method Implicit method Rearranging:

16. Implicit method (internal node) AIAI BIBI CICI FIFI Internal nodes 12 345 B1C1 A2B2C2 A3B3C3 A4B4C4 A5B5 x = F1 F2 F3 F4 F5 T1 T2 T3 T4 T5 k I-1 =k I+1 =k I

17. Implicit method (surface nodes) B1C1 A2B2C2 A3B3C3 A4B4C4 A5B5 x = F1 F2 F3 F4 F5 Surface nodes 12 345 external internal 06 T O Air T I Air F0 F6 B0C0 A1 C5 B6 A6 T1 T2 T3 T4 T5 T0 T6 Surface node: 0 xx For surface nodes: flux coming in = flux going out Surface node: 6 Calculate B0 and C0 Calculate A6 and B6

18. Discretization of a non-homogeneous wall structure Section considered in the following discussion