- #1
jpmann
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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/<br /> {\vphantom {{a'} {\sinh \left( {a'b} \right)}}} \right.<br /> \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
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/<br /> {\vphantom {{a'} {\sinh \left( {a'b} \right)}}} \right.<br /> \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
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