1. Lecture Objectives:
Analysis of unsteady state heat transfer
HW3

2. Unsteady-state heat transfer (Explicit – Implicit methods)
Example:
To - known and changes in time
Tw - unknown
Ti - unknown
Ai=Ao=6 m2
(mcp)i=648 J/K
(mcp)w=9720 J/K
Initial conditions:
To = Tw = Ti = 20oC
Boundary conditions:
hi=ho=1.5 W/m2 Tw Ti To
Ao=Ai
Conservation of energy:
Time [h]
0.1
0.2
0.3
0.4
0.5
0.6
0.7 To 20 30 35 32 10 15
Time step Dt=0.1 hour = 360 s

3. Implicit methods - example
After rearranging:
2 Equations with 2 unknowns!
 = To Tw Ti
 =36 system of equation Tw Ti
 =72 system of equation Tw Ti

4. Explicit methods - example
 =360 sec
 = To Tw Ti
 =360 To Tw Ti
 =720 To Tw Ti
Time
There is NO system of equations!
UNSTABILE

5. Problems with stability !!! Often requires very small time steps
Explicit method
Problems with stability !!!
Often requires very small time steps

6. Explicit methods - example
 = To Tw Ti
 =36 To Tw Ti
 =72 To Tw Ti
Stable solution obtained
by time step reduction
10 times smaller time step
Time
 =36 sec

7. Explicit methods information progressing during the calculationTw Ti To

8. Unsteady-state conduction - Wallq Dx
Nodes for numerical
calculation

9. Discretization of a non-homogeneous wall structure
Section considered in the
following discussion
Discretization in space
Discretization in time

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

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

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

13. 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

14. Announcement Help with solving system of equation for HW 3
Next Tuesday at 6:00 pm Computer lab in the 2nd floor in ECJ

15. Homework 3 (Similar to HW2, but unsteady, and more realistic)
Top view
Surface
radiation
Tinter_surf ≠ Tair
Te_i
Te_o
2.5 m
Tair_in
Idif
8 m
8 m
IDIR
Glass
South
East
Insulation Ts
Tair_out
Concrete
Surface radiation
IDIR
IDIR

16. HW3

17. 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 ?

18. 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

19. 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

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

21. 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

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

23. 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

24. 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