Announcement MATHCAD for solving system of equation for HW1b Today at 5:00 pm this studio (computer lab)

Lecture Objectives: Discuss HW1b Answer your questions Analyze the unsteady-state heat transfer numerical calculation methods

Unsteady-state heat transfer (Explicit – Implicit methods) Example: To - known and changes in time Tw - unknown Ti - unknown Ai=Ao=6 m2 (mcp)i=648 J/K (mcp)w=9720 J/K Initial conditions: To = Tw = Ti = 20oC Boundary conditions: hi=ho=1.5 W/m2 Tw Ti To Ao=Ai Conservation of energy: Time [h] 0.1 0.2 0.3 0.4 0.5 0.6 0.7 To 20 30 35 32 10 15 Time step Dt=0.1 hour = 360 s

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

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

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

Problems with stability !!! Often requires very small time steps 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 Stable solution obtained by time step reduction 10 times smaller time step Time  =36 sec

Explicit methods information progressing during the calculation Tw Ti To

Unsteady-state conduction - Wall q Dx Nodes for numerical calculation

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

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

Internal node finite volume method After some math work: Explicit method Implicit method

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

Unsteady-state conduction Implicit method b1T1 + +c1T2+=f(Tair,T1,T2) a2T1 + b2T2 + +c2T3+=f(T1 ,T2, T3) Air 1 2 3 4 5 6 Air a3T2 + b3T3+ +c3T4+=f(T2 ,T3 , T4) ……………………………….. a6T5 + b6T6+ =f(T5 ,T6 , Tair) Matrix equation M × T = F for each time step M × T = F