Chapter 4 Unsteady Conduction

Forsberg Heat Transfer
Chapter 4
Unsteady Conduction




Lumped Systems Analysis
For a "lumped system", all points in an object have the same
temperature at a given time. That is, T = T (t ).




The body is initially at Ti and is suddenly immersed in
a fluid at lower temperature T . There is then
convection from the body to the fluid.




1

, Rate of decrease in = Rate of heat flow from
internal energy of the body the body to the fluid
dT
−  cV = hA(T − T )
dt
V = volume of the body
A = surface area of the body
T = constant, so the above equation becomes
d (T − T ) hA
=− dt
T − T  cV
T −T
At t = 0, the body is at Ti d (T − T )
t
hA
At t = t , the body is at T 
Ti −T
T − T
=−
 cV 0
dt
Integrating, we get




 T − T  hA
ln   = − t
 Ti − T   cV
 hA 
− t
  cV 
T − T = (Ti − T ) e
 hA 
− t
  cV 
T (t ) = T + (Ti − T ) e
The instantaneous heat flow at any time t is
dT
q(t ) = −  cV = hA(T − T )
dt




2

, t
Q =  q(t ) dt is the amount of heat, Q, transferred
0

to the fluid over the period t = 0 to t = t.
We can use either
T
Q = −  cV  dT = −  cV [T (t ) − Ti ] or
Ti
t
Q = h A  [T (t ) − T ] dt
0

  hA 
− t 
The result is: Q =  cV (Ti − T ) 1 − e   cV  
 




Application Criterion for Lumped Analysis
The Biot number for lumped systems is
hL
Bilumped =
k
where:
h = convective coefficient at the surface
L = characteristic length for the body
= volume V / surface area A
k = thermal conductivity of the body
If Bilumped < 0.1 Lumped-System Analysis is applicable




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