# General homogeneous shrinking core problem

1. ### jpmann

2
Hi Guys,

First post here. I'm just wondering if anyone could lend a helping hand in the following derivation. It is taken from Ishida AIChE J 14 (1968) 311 (also very similar to that derived by Ausman Chem Eng Sci 17 (1962) 323) and concerns the derivation of the general non-catalytic shrinking core model.

The step which is confusing me concerns the derivation of the transient behavior of the retreating interface. This is achieved through setting $$a' = a$$ and $$X = 0$$ and differentiating with respect to $$c$$ within the following equation

$X = 1 - \frac{{\sinh \left( {ab} \right)}}{{a\sinh \left( b \right)}} - \frac{{\sinh \left( {ab} \right)}}{a}\int_{c1}^{c} {\frac{{{{a'} \mathord{\left/ {\vphantom {{a'} {\sinh \left( {a'b} \right)}}} \right. \kern-\nulldelimiterspace} {\sinh \left( {a'b} \right)}}}}{{1 + d\left[ {1 - a + \frac{a}{{Sh}}} \right]\left[ {a'b\coth \left( {a'b} \right) - 1} \right]}}} dc$

The solution given by Ishida is

$$\frac{{dc}}{{da'}} = - \frac{1}{{a'}}\left[ {a'b\coth\left( {a'b} \right) - 1} \right]\left[ {1 + d\left[ {1 - a + \frac{a}{{Sh}}} \right]\left[ {a'b\coth\left( {a'b} \right) - 1} \right]} \right]$$

however, no matter how hard I try, I can't seem to arrive at their answer. I know I'm missing something simple, but I just can't see it. Any help on a way forward with this problem would be greatly appreciated.

Thanks and kind regards,

Jason

Last edited: Jan 2, 2014
2. ### jpmann

2
Problem solved. Thanks for anyone who had a look.