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 C initial temperature Modeling Modeling Modeling Modeling 1) External wall (north) node 2) Internal wall (north) node
Qsolar+C1A(Tsky4 - Tnorth_o4)+ C2A(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 [ W/m2K4] Qsolar=asolar(Idif+IDIR) A C1=eskyesurface_long_wavesFsurf_sky C2=egroundesurface_long_wavesFsurf_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_insynorth_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