Lecture Objectives: Finish with system of equation for Wall, Room, and whole Building Discuss HW2 Discus alternative conduction equation solution method Define basic modeling steps

System of equations for wall assembly C0 T0 F0 brick A1 B1 C1 T1 F1 insulation dry wall A2 B2 C2 T2 F2 x = A3 B3 C3 T3 F3 A4 B4 C4 T4 F4 Air 1 2 3 4 5 6 Air A5 B5 C5 T5 F5 A6 B6 T6 F6 Matrix equation M × T = F for each time step b0T0 + +c0T1+=f(Tair,T0,T1) a1T0 + b1T1 + +c1T2+=f(T0 ,T1, T2) a2T1 + b2T2+ +c2T3+=f(T1 ,T2 , T3) ……………………………….. a6T5 + b6T6+ =f(T5 ,T6 , Tair)

Linearization of radiation equations Surface to surface radiation Equations for internal surfaces - closed envelope Ti Tj Linearized equations: Calculate h based on temperatures from previous time step Or for your HW3

Linearized radiation means linear system of equations Calculated based on temperature values from previous time step B0 C0 T0 F0 A1 B1 C1 T1 F1 A2 B2 C2 T2 F2 These coefficient will have Some radiation convection coefficients x = A3 B3 C3 T3 F3 A4 B4 C4 T4 F4 A5 B5 C5 T5 F5 A6 B6 T6 F6

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

Energy balance for air unsteady-state heat transfer QHVAC

System of equation for more than one element Tair is unknown and it is solved by system of equation :

General System of equations (matrix) for a 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 for 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

Alternative: Response function methods for conduction calculation NOTATION: θ(x,t)=T(x,)

Laplace transformation Laplace transform is given by Where p is a complex number whose real part is positive and large enough to cause the integral to converge.

Laplace transformation table

Principles of Response function methods The basic strategy is to predetermine the response of a system to some unit excitation relating to the boundary conditions anticipated in reality. Reference: JA Clarke http://www.esru.strath.ac.uk/Courseware/Class-16458/ or http://www.hvac.okstate.edu/research/documents/iu_fisher_04.pdf

Response functions Computationally inexpensive Accuracy ? Flexibility ???? What if we want to calculate the moisture transport and we need to know temperature distribution in the wall elements?

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 the energy and mass balance equations Solve the equations (use numeric methods or solver) Present the result