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HVAC and Building Physics Numerical Methods Lecture

Learn about HVAC connection with building physics, solve homework problems using MatCad Equation Solver, study heat transfer methods, and analyze numerical calculation techniques. Explore air balance, convection, ventilation, infiltration, and energy balance.

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HVAC and Building Physics Numerical Methods Lecture

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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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