1. HW2 Example MatLab Code is posted on the course website
Due date
October 18

2. Lecture Objectives: Discuss HW 2 Finish unsteady-state heat conduction
Define building system of equations

3. Homework 2 (Similar to HW1B, but unsteady, and more realistic)
Top view
Glass
Te_i
Te_o
Tinter_surf ≠ Tair
2.5 m
Surface
radiation
Tair_in
10 m
10 m
IDIR
Ts_i
Idif
South
East
Insulation
Ts_o
Tair_out
Concrete
Surface radiation
Idif
IDIR

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

5. Physical approach (finite volume method)
For uniform grid

6. Internal node finite volume method
After some math work:
Explicit method
Implicit method

7. 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

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

9. Explicit method Accuracy when compared to explicit ?
- simple for calculation
- but unstable
Problem with stability can be fixed with
appropriate time step:
Accuracy when compared to explicit ?

10. Implicit method (internal node)
kI-1=kI+1=kI AI BI CI FI
Internal nodes B1 C1 T1 F1 A2 B2 C2 T2 F2 x = A3 B3 C3 T3 F3 1 2 3 4 5 A4 B4 C4 T4 F4 A5 B5 T5 F5

11. Implicit method (surface nodes)B0 C0 T0 F0
Surface nodes A1 B1 C1 T1 F1
T O Air
T I Air A2 B2 C2 T2 F2 x = A3 B3 C3 T3 F3 1 2 3 4 5 6
external
internal A4 B4 C4 T4 F4 A5 B5 C5 T5 F5 A6 B6 Dx T6 F6
For surface nodes: flux coming in = flux going out
Surface node: 0
Calculate B0 and C0
Surface node: 6
Calculate A6 and B6

12. 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

13. 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

14. Accuracy as a function of  and x

15. Discretization of a non-homogeneous wall structure
Section considered in the
following discussion

16. 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

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

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

19. 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

20. 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