Chapter 13: Unsteady State Nonisothermal Reactor Design
Professional Reference Shelf
R13.7 Unsteady Operation of Plug-Flow Reactors
We plan to reduce the energy balance into a more usable form. To achieve this form for plug-flow reactors, we begin by applying the balance to a small differential volume,, in which there are no spatial variations (Figure R13.7-1). The number of moles of species i inis |
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Figure CD13.7 |
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Substituting for Ni and dividingby gives |
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Taking the limit asand noting that gives |
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Rearranging, we have |
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(R13.7-1) |
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Comparing Equations (R13.7-1) and (R13.7-2) |
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(R13.7-2) |
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we note that the term in parentheses is just ri. The rate of reaction of species i is related to the rate of disappearance of species A through the stoichiometric coefficient, |
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Then |
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Finally, |
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where a is the heat exchange area per unit volume. Differentiating yields |
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Recalling that |
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we substitute these equations to obtain |
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Neglecting shaft work and changes in pressure with respect to time, we obtain |
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Transient energy |
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(R13.7-3) |
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This equation must be coupled with the mole balances, |
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(R13.7-4) |
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Numerical solution |
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and the rate law, |
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(R13.7-5) |
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and solved numerically. A variety of numerical techniques
for solving equations of this type can be found in the book Applied Numerical
Methods.1 For steady-state operation in which no work is done by the system, Equation (R13.7-3) reduces to |
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(R13.7-6) |
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Substitution for the molar flow rates Fi in terms of conversion gives Equation (R13.7-7). |
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Use this |
(R13.7-7) |
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As stated previously, this equation is solved simultaneously with the mole balance. |