Lecture Objectives: Continue with linearization of radiation and convection Example problem Modeling steps

Linearization of radiation equations Surface to surface radiation Exact equations for internal surfaces - closed envelope Ti Tj Linearized equations: Calculate h based on temperatures from previous time step Or for your HW2

Discretization 1-D Boundaries of control volume 2-D 3-D

Conservation of energy or Finite volume method Boundaries of control volume For node “I” - integration through the control volume

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

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

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

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

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

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

Energy balance for air unsteady-state heat transfer QHVAC

Example System of equation for 2D room with 13 nodes: L –left wall , C ceiling, F – floor, R – right wall, A – air node

Example System of equation for case when air temperature is “fluctuating” Tair is unknown

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

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 x Air equation

System of equations for a building Matrix 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

Linear equation solver M × t = f M matrix for element, room, or building (n x n) t - temperature vector (1 x n) f - free vector (1 x n) M × t = f / multiply left and right side by M-1 from left side M-1 × M = I I × t = t t = M-1 × f 1 1 1 1 U= 1 1 1 1 How to find M-1 ?

Linear equation solver To solve the temperature field in a given room: calculate coefficients of the matrix and free vector enter the coefficients to your favorite software call a function that solves the system

Initial condition and “Warming-up” Solution Internal air Day 1 Day 2 Day 3

Your HW2 Assumed zero ºC initial temperature

Modeling

Modeling

Modeling

Modeling 1) External wall (north) node 2) Internal wall (north) node Qsolar+C1·A(Tsky4 - Tnorth_o4)+ C2·A(Tground4 - Tnorth_o4)+hextA(Tair_out-Tnorth_o)=Ak/(Tnorth_o-Tnorth_in) A- wall area [m2] - wall thickness [m] k – conductivity [W/mK]  - emissivity [0-1] - absorbance [0-1] =  - for radiative-gray surface, esky=1, eground=0.95 Fij – view (shape) factor [0-1] h – external convection [W/m2K] s – Stefan-Boltzmann constant [5.67 10-8 W/m2K4] Qsolar=asolar·(Idif+IDIR) A C1=esky·esurface_long_wave·s·Fsurf_sky C2=eground·esurface_long_wave·s·Fsurf_ground 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_ considered _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 for homework assume yij = Fijei

Modeling Matrix equation M × t = f for each time step b1T1 + +c1T2+=f(Tair,T1,T2) a2T1 + b2T2 + +c2T3+=f(T1 ,T2, T3) 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

Modeling

Modeling steps Define the domain Analyze the most important phenomena and define the most important elements Discretize the elements and define the connection Write energy and mass balance equations Solve the equations (use numeric methods or solver) Present the result