Lecture Objectives: Analyze the unsteady-state heat transfer Conduction Introduce numerical calculation methods Explicit – Implicit methods
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] ToTo Time step =0.1 hour = 360 s Conservation of energy:
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
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!
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
Explicit method Problems with stability !!! Often requires very small time steps
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
Explicit methods information progressing during the calculation TiTi ToTo TwTw
Unsteady-state conduction - Wall q Nodes for numerical calculation xx
Discretization of a non-homogeneous wall structure Section considered in the following discussion Discretization in space Discretization in time
Internal node Finite volume method For node “I” - integration through the control volume Boundaries of control volume
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
Internal node finite volume method Explicit method For uniform grid Implicit method
Internal node finite volume method Explicit method Implicit method Substituting left and right sides:
Internal node finite volume method Explicit method Implicit method Rearranging:
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:
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:
Unsteady-state conduction Implicit method 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
Stability of numerical scheme Explicit method - simple for calculation - unstable Implicit method - complex –system of equations (matrix) - Unconditionally stabile What about accuracy ?
Unsteady-state conduction Homogeneous Wall
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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