Question

We are trying to solve for the temperature distribution of a sphere in boiling water. The sphere is of radius a, and it's at temperature Tc. The boiling water is at Th =Tw, and:
We understand the initial condition is:
at t=0, r: T(r,0) = Tc
And the boundary condition is:
at r=a, t: -k(dT/dr) = h[T(a,t) - Tc].

We also know that the following steps are required after this:
1. General heat equation.
2. Non-dimensionalise our heat eqn, BC, and IC.
3. Sub in our BCs to our general heat equation to solve for theta, the non-dim temperature.

We would also know if assuming the presence of a temperature gradient would be possible to solve for theta.

We are confused about what type of reaction it is. We tried working with it as an unsteady-state (i.e. theta = theta(ss) + theta(transient)) but to no avail, possibly from errors we made.

Answer 1

general HT equation for one dimentional heat transfer with no heat generation inside sphere

is

[ 1 Ꮌ , , T , 12 Ꭷr ( or = ᎧT Ꭷt

IC

T(r,0)=TC

BC1

        at r = a

k = h(T – T)

BC2:

|T(0,t)| < $\infty$

to solve this problem we will introduce dimentionless variables

T -T. 0 TC - Tu
,  
T= ata
   &
{=ra

so, partial differential equation and BC becomes

ᎧᎾ_ 1 | 2ᏊᎢ or =52 Ꭷs , as

IC:

θίξ, 0) = 1

BC1:

oᎾ + ha{k * Ꮎ

BC2: |
0(0,1)
|< $\infty$

on solving integral using separation of variable

we get solution

0(. 4) - Σκαρί-λή τοξοίη(λ)

where $\lambda _n$ are roots of

1 - Incot (An) - ha/k = 0

and

Cn =

1 (sin(& An)as 1 (sin(&An)?dE