I tried solving unsteady heat transfer in radial direction for a cylindrical insulation case .Solving this equation requires two BC in radius r and one in time .But i have two in time and one in r.
BC at t=0 the inside surface temperature=-40 C
t=20min the outer surface temperature=-20 C
at radius r=outer radius : h(Tsurf(t)-Tamb)=k*DT/Dr
Hi,
The initial condition should describe the state of the complete domain at a given instant (usually t = 0), and not just at one radial location.
It will be of the form : T(t = 0, r) = f(r)
Usually it is taken as a uniform distribution (no dependence on r).
In your case, it is probably good to take the entire storage vessel to be at an equilibrium temperature of -40 deg C at time t = 0.
Likewise the boundary condition at a certain radial location should hold good at all later instants of time.
In this case, convective b.c. is most appropriate for the outer surface, as rightly mentioned in your post.
For the inner surface, you may assume the gas to be lumped system and find out the variation in its temperature with time (an exponential increase). This can be specified as the time-variant fluid temperature which will appear in another convective equation.
Alternatively, if you are solving this numerically, the temperature variation of the gas can be obtained using energy balance, i.e. by relating the quantity of heat transferred inward during each time step with the temperature rise:
DT_gas = Q_in / [m_gas x cp_gas]
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