1. 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

2. System of equations for wall assemblyC0 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)

3. 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

4. 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

5. 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

6. Energy balance for air unsteady-state heat transfer
QHVAC

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

8. 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

9. 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

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

11. 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.

12. Laplace transformation table

13. 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 or

14. 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?

15. Modeling

16. Modeling

17. Modeling

18. 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 [ 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

19. 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

20. Modeling

21. 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