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

2. 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

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

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

5. 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

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

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

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

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

10. 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

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

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

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

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

15. 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

16. 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

17. 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 ?

18. 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

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

20. Your HW2
Assumed zero ºC initial temperature

21. Modeling

22. Modeling

23. Modeling

24. 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

25. 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

26. Modeling

27. 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