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Lecture Objectives:

Lecture Objectives:. Discuss the HW1b solution Learn about the connection of building physics with HVAC Solve part of the homework problem Introduce Mat Cad Equation Solver Analyze the unsteady-state heat transfer numerical calculation methods Explicit – Implicit methods.

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Lecture Objectives:

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  1. Lecture Objectives: • Discuss the HW1b solution • Learn about the connection of building physics with HVAC • Solve part of the homework problem • Introduce Mat Cad Equation Solver • Analyze the unsteady-state heat transfer numerical calculation methods • Explicit – Implicit methods

  2. Air balance - Convection on internal surfaces + Ventilation + Infiltration Uniform Air Temperature Assumption! What affects the air temperature? - h and corresponding Q - as many as surfaces Energy balance: Tsupply -maircp.airΔTair= Qconvective+ Qventilation Qconvective= ΣAihi(TSi-Tair) Ts1 mi Qventilation= Σmicp,i(Tsupply-Tair) Q2 Q1 Tair h1 h2

  3. Air balance – steady stateConvection on internal surfaces + Infiltration = Load Uniform temperature Assumption • h, and Qsurfaces as many as surfaces • infiltration – mass transfer (mi – infiltration) • Qair= Qconvective+ Qinfiltration T outdoor air Qconvective= ΣAihi(TSi-Tair) Ts1 mi Qinfiltration= Σmicp(Toutdoor_air-Tair) Q2 Q1 In order to keep constant air Temperate, HVAC system needs to remove cooling load Tair h1 h2 QHVAC= Qair= m·cp(Tsupply_air-Tair) HVAC

  4. Top view Glass Twest_oi Twest_i Tinter_surf Tair_in Surface radiation IDIR Idif Tnorth_i conduction Tnorth_o Tair_out Styrofoam Surface radiation Idif IDIR Homework assignment 1 2.5 m 10 m 10 m North West

  5. Homework assignment 1 Surface energy balance 1) External wall (north) node Qsolar+C1·A(Tsky4 - Tnorth_o4)+ C2·A(Tground4 - Tnorth_o4)+hextA(Tair_out-Tnorth_o)=Ak/(Tnorth_o-Tnorth_in) Qsolar=asolar·(Idif+IDIR)A C1=e·asurface_long_wave·s·Fsurf_sky 2) Internal wall (north) node C3A(Tnorth_in4- Tinternal_surf4)+C4A(Tnorth_in4- Twest_in4)+hintA(Tnorth_in-Tair_in)= =kA(Tnorth_out--Tnorth_in)+Qsolar_to_int_surf Qsolar_to int surf =portion of transmitted solar radiation that is absorbed by internal surface C3=eniort_in·s· ynorth_in_to_ internal surface

  6. Using MathCad

  7. Air balance steady state vs. unsteady state For steady state we have to bring or remove energy to keep the temperature constant QHVAC= Qconvection+ Qinfiltration If QHVAC= 0 temperature is changing – unsteady state maircp(DTair/Dt)= Qconvection+ Qinfiltration mi Q2 Q1 Tair HVAC

  8. Example: Unsteady-state problemExplicit – 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

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

  10. 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

  11. Explicit methods - example  =36 sec  =0 To Tw Ti  =360 To Tw Ti  =720 To Tw Ti Time There is NO system of equations! UNSTABILITY

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

  13. 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

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

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

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

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

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

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

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

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

  22. Dx Dx/2 Energy balance for element’s surface node Implicit equation: Or if TSi and TA are known:

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

  24. 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

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

  26. Unsteady-state conductionHomogeneous Wall

  27. 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

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