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 xx

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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