1. Announcement MATHCAD for solving system of equation for HW1b
Today at 5:00 pm this studio (computer lab)

2. Lecture Objectives: Discuss HW1b
Answer your questions
Analyze the unsteady-state heat transfer numerical calculation methods

3. 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

4. Explicit – Implicit methods example
Conservation of energy equations:
Wall:
Air:
After substitution:
For which time
step to solve:
+  or  ?
Wall:
Air:
+  Implicit method
 Explicit method

5. 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

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

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

8. 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

9. Explicit methods information progressing during the calculationTw Ti To

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

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

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

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

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

15. 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