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Lecture Objectives:

Explore the analysis of unsteady-state heat transfer using explicit and implicit methods, with a focus on conductive heat transfer, boundary conditions, and stability. Learn to solve system of equations and simulate energy transfers in complex systems. Join us for a hands-on session to enhance your understanding.

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Lecture Objectives:

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  1. Lecture Objectives: • Analysis of unsteady state heat transfer • HW3

  2. Conductive heat transfer k - conductivity of material • Steady-state • Unsteady-state • Boundary conditions • Dirichlet Tsurface = Tknown • Neumann TS1 TS2 L h Tair

  3. Example: Unsteady-state heat transfer(Explicit – Implicit methods) 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 step Dt=0.1 hour = 360 s

  4. 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

  5. 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

  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 Stable solution obtained by time step reduction 10 times smaller time step Time  =36 sec

  8. Explicit methods information progressing during the calculation Tw Ti To

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

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

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

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

  14. Unsteady-state conductionImplicit method b1T1 + +c1T2+=f(Tair,T1,T2) a2T1+b2T2 + +c2T3+=f(T1 ,T2, T3) Air 1 4 3 2 5 Air 6 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

  15. Announcement • Help with solving system of equation for HW 3 • Next Tuesday at 6:00 pm Computer lab in the 2nd floor in ECJ

  16. Top view Homework 3 (Similar to HW2, but unsteady, and more realistic) Surface radiation Tinter_surf ≠ Tair 2.5 m Tair_in Idif 10 m 10 m IDIR South East Insulation Tair_out Concrete IDIR IDIR Surface radiation

  17. HW3

  18. Explicit method - simple for calculation - but unstable Problem with stability can be fixed with appropriate time step: Accuracy when compared to explicit ?

  19. Tj Ti Linearization of radiation equationsSurface to surface radiation Equations for internal surfaces - closed envelope Linearized equations: Calculate h based on temperatures from previous time step Or for your HW3

  20. Linearized radiation means linear system of equations Calculated based on temperature values from previous time step T0 F0 B0 C0 A1 B1 C1 T1 F1 T2 F2 A2 B2 C2 x These coefficient will have Some radiation convection coefficients = T3 F3 A3 B3 C3 A4 B4 C4 T4 F4 T5 F5 C5 A5 B5 A6 B6 T6 F6

  21. Energy balance for air unsteady-state heat transfer QHVAC

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

  23. Example Tair is unknown and it is solved by system of equation :

  24. System of equations (matrix) for single zone (room) 8 elements Three diagonal matrix for each element x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x Air equation x

  25. System of equations for a building Matrix for the whole building 4 rooms Rom matrixes Connected by common wall elements and airflow in-between room – Airflow simulation program (for example CONTAM) Energy Simulation program “meet” Airflow simulation program

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