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

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:

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

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

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

Using MathCad

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

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] ToTo Time step  =0.1 hour = 360 s Conservation of energy:

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

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!

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

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

Explicit methods information progressing during the calculation TiTi ToTo TwTw

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

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

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

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

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

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

Internal node finite volume method Explicit method Implicit method Rearranging:

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

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:

Unsteady-state conduction Implicit method 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

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

Unsteady-state conduction Homogeneous Wall

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