analysis of transient heat conduction in different geometries - ethesis ...

θ =
− τU
e + V
U
⎛
2B
⎞
U = ⎜ ⎟
⎜1+ B ⎟
Where 4
⎝ ⎠ ( 1+
B )
, 4
35
V =
G
(3.101)
Based on the analysis a closed form expression involving temperature, internal heat generation
parameter, Biot number and time is obtained for a tube.
3.6 TRAINSIENT HEAT CONDUCTION IN SLAB WITH DIFFERENT
PROFILES
In this previous section we have used polynomial approximation method for the analysis. We
have used both slab and cylindrical geometries. At both the geometries we have considered a
heat flux and heat generation respectively. Considering different profiles, the analysis has been
extended to both slab and cylindrical geometries. Unsteady state one dimensional temperature
distribution of a long slab can be expressed by the following partial differential equation. Heat
transfer coefficient is assumed to be constant, as illustrated in Fig 3.5. The generalized heat
conduction equation can be expressed as
∂T 1 ∂ ⎛ m ∂T
⎞
= α r m ⎜ ⎟
∂t r ∂r⎝ ∂r
⎠ (3.102)
Where, m = 0 for slab, 1 and 2 for cylinder and sphere, respectively.
Boundary conditions are
∂ T
= 0
∂ r at r = 0
(3.103)
∂ T
k =−h( T −T∞)
∂ r
at r = R
(3.104)
And initial condition: T=T0 at t=0 (3.105)
In the derivation of Equation (3.102), it is assumed that thermal conductivity is independent of