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 h1 Q1 h2 Q2 What affects the air temperature? - h and corresponding Q - as many as surfaces mimi Ts1 Tair Uniform Air Temperature Assumption! Q convective = ΣA i h i (T Si -T air ) Q ventilation = Σm i c p,i (T supply -T air ) Tsupply -m air c p.air Δ T air = Qc onvective + Q ventilation Energy balance:

3. Air balance – steady state Convection on internal surfaces + Infiltration = Load h1 Q1 h2 Q2 - h, and Q surfaces as many as surfaces - infiltration – mass transfer (m i – infiltration) Q air = Q convective + Q infiltration mimi Ts1 Tair Uniform temperature Assumption Q convective = ΣA i h i (T Si -T air ) Q infiltration = Σm i c p (T outdoor_air -T air ) Q HVAC = Q air = m·c p (T supply_air -T air ) T outdoor air HVAC In order to keep constant air Temperate, HVAC system needs to remove cooling load

4. Homework assignment 1 North 10 m 2.5 m West conduction T air_in I DIR I dif Glass T inter_surf T north_i T north_o T west_i T west_oi T air_out Styrofoam I DIR I dif Surface radiation Surface radiation Top view

5. Homework assignment 1 Surface energy balance 1) External wall (north) node 2) Internal wall (north) node Q solar =  solar ·(I dif +I DIR ) A Q solar +C 1 ·A(T sky 4 - T north_o 4 )+ C 2 ·A(T ground 4 - T north_o 4 )+h ext A(T air_out -T north_o )=Ak/  (T north_o -T north_in ) C 1 =  ·  surface  long_wave ·  ·F surf_sky Q solar_to int surf =  portion of transmitted solar radiation that is absorbed by internal surface C 3 A(T north_in 4 - T internal_surf 4 )+C 4 A(T north_in 4 - T west_in 4 )+ h int A(T north_in -T air_in )= =kA(T north_out-- T north_in )+Q solar_to_int_surf C 3 =  niort_in ·  ·  north_in_to_ internal surface

6. Using MathCad

7. Air balance steady state vs. unsteady state Q1 Q2 Q HVAC = Q convection + Q infiltration mimi Tair HVAC For steady state we have to bring or remove energy to keep the temperature constant If Q HVAC = 0 temperature is changing – unsteady state m air c p  air  = Q convection + Q infiltration

8. Unsteady-state problem 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]00.10.20.30.40.50.60.7 ToTo 2030353220101510 Time step  =0.1 hour = 360 s Conservation of energy:

9. 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

10. 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!

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

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

14. Explicit methods information progressing during the calculation TiTi ToTo TwTw

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

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

18. 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

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

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

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

22. Energy balance for element’s surface node  x/2 xx Implicit equation: Or if T Si and T A are known:

23. 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:

24. Unsteady-state conduction Implicit method 1 2 3 4 5 6 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

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 conduction Homogeneous Wall

27. 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