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General homogeneous shrinking core problem

  1. Jan 1, 2014 #1
    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 [tex]a' = a[/tex] and [tex]X = 0[/tex] and differentiating with respect to [tex]c[/tex] within the following equation

    [itex]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[/itex]

    The solution given by Ishida is

    [tex]\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][/tex]

    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,

    Last edited: Jan 2, 2014
  2. jcsd
  3. Jan 2, 2014 #2
    Problem solved. Thanks for anyone who had a look.
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