1. Lecture Objectives: Analyze the unsteady-state heat transfer Conduction Introduce numerical calculation methods Explicit – Implicit methods

2. Example: TiTi ToTo TwTw A o =A i T o - known and changes in time T w - unknown T i - unknown A i =A o =6 m 2 (mc p ) i =648 J/K (mc p ) w =9720 J/K Initial conditions: T o = T w = T i = 20 o C Boundary conditions: hi=ho=1.5 W/m 2 Time [h]00.10.20.30.40.50.60.7 ToTo 2030353220101510 Time step  =0.1 hour = 360 s Conservation of energy:

3. Explicit – Implicit methods example Conservation of energy equations: Wall: Air: Wall: Air: After substitution: For which time step to solve:  +   or  ?  +   Implicit method  Explicit method

4. 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!

5. Explicit methods - example  =0 To Tw Ti  =360 To Tw Ti  =720 To Tw Ti   =360 sec NON-STABILE There is NO system of equations! Time

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

7. 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

8. Explicit methods information progressing during the calculation TiTi ToTo TwTw

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

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

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

12. Left side of equation for node “I” Right side of equation for node “I” Internal node finite volume method - Discretization in Time - Discretization in Space

13. Internal node finite volume method Explicit method For uniform grid Implicit method

14. Internal node finite volume method Explicit method Implicit method Substituting left and right sides:

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

16. Energy balance for element’s surface node Implicit equation for node I (node with thermal mass): Implicit equation for node S (node without thermal mass): After formatting:

17. Energy balance for element’s surface node General form for each internal surface node: After rearranging the elements for implicit equation for surface equations: General form for each external surface node:

18. Unsteady-state conduction Implicit method 1 2 3 4 5 6 Matrix equation M × T = F for each time step Air b 1 T 1  +  +c 1 T 2  +  =f(T air,T 1 ,T 2  ) a 2 T 1  +  b 2 T 2  +  +c 2 T 3  +  =f(T 1 ,T 2 , T 3  ) a 3 T 2  +  b 3 T 3  +  +c 3 T 4  +  =f(T 2 ,T 3 , T 4  ) a 6 T 5  +  b 6 T 6  +  =f(T 5 ,T 6 , T air ) ……………………………….. M × T = F

19. Stability of numerical scheme Explicit method - simple for calculation - unstable Implicit method - complex –system of equations (matrix) - Unconditionally stabile What about accuracy ?

20. Unsteady-state conduction Homogeneous Wall

21. System of equation for more than one element air Left wall Roof Right wall Floor Elements are connected by: 1)Convection – air node 2)Radiation – surface nodes

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