An exothermic reaction, A→2B, takes place adiabati-cally in a stirred-tank reactor. This liquid reaction occurs at constant volume in a 1000-gal reactor. The reaction can be con-sidered to be first-order and irreversible with the rate constant given by k=2.4×1015e−20,000∕T(min−1) whereT is in R.(a)Using the information below, derive a transfer function relating the exit temperatureT to the inlet concentration cAi.State all assumptions that you make How sensitive is the transfer function gainKto the operating conditions? Find an expression for the gain in terms of q,T,and cAi, and evaluate the sensitivities (that is,∂K∕∂q,etc.).Available Information(i)Nominal steady-state conditions are T=150∘F,cAi=0.8 mol∕ft3,q=20 gal∕min=flow in and out of the reactor(ii)Physical property data for the mixture at the nominalsteady state: C=0.8Btu1b∘F,ρ=52 lb∕ft3,−ΔHR=500 kJ∕mol
An exothermic reaction, A→2B, takes place adiabati-cally in a stirred-tank reactor. This liquid reaction occurs at constant volume in a 1000-gal reactor. The reaction can be con-sidered to be first-order and irreversible with the rate constant given by k=2.4×1015e−20,000∕T(min−1) whereT is in R.(a)Using the information below, derive a transfer function relating the exit temperatureT to the inlet concentration cAi.State all assumptions that you make How sensitive is the transfer function gainKto the operating conditions? Find an expression for the gain in terms of q,T,and cAi, and evaluate the sensitivities (that is,∂K∕∂q,etc.).Available Information(i)Nominal steady-state conditions are T=150∘F,cAi=0.8 mol∕ft3,q=20 gal∕min=flow in and out of the reactor(ii)Physical property data for the mixture at the nominalsteady state: C=0.8Btu1b∘F,ρ=52 lb∕ft3,−ΔHR=500 kJ∕mol
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