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

Based on the analysis a closed form expression involving temperature, heat source parameter,
Biot number and time is obtained for a slab.
3.3 TRANSIENT ANALYSIS ON A TUBE WITH SPECIFIED HEAT FLUX
We consider the heat conduction in a tube of diameter 2R, initially at a uniform temperature T0,
having heat flux at one side and exchanging heat by convection at another side. A constant heat
transfer coefficient (h) is assumed on the other side and the ambient temperature (T∞) is assumed
to be constant. Assuming constant physical properties, k and α, the generalized transient heat
conduction valid for slab, cylinder and sphere can be expressed as:
∂T 1 ∂ ⎛ m ∂T
⎞
= α r m ⎜ ⎟
∂t r ∂r⎝ ∂r
⎠
Where, m = 0 for slab, 1 and 2 for cylinder and sphere, respectively. Here we have considered
tube geometry. Putting m=1, equation (3.1) reduces to
∂T 1 ∂ ⎛ ∂T
⎞
= α ⎜r⎟ ∂t r ∂r⎝ ∂r
⎠ (3.23)
Fig 3.2: Schematic of a tube with heat flux
Subjected to boundary conditions
∂ T
k = −q"
∂ r at
r = R1
23
(3.24)