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conversion within a reactor depends on the active surface area of the catalyst, which can be time-variant. A classic opt imization problem oft)en encountered involves deciding on the most efficient schedule for replacing or reactivating catalyst.
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Reaction Rate
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Equilibrium occurs when the rates of forward and reverse reactions are equal. These rates are proportional to concent rations of reactants and products respectively. Let k, and k, be designated as forward- and reverse-rate coefficients. Then at equilibrium, k,[S0,1[0,15 L = k,[SO,i (10.3)
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Icrom Eqs. (10.2) and (10.3), the equilibrium constant is the ratio of t he forward- to reverse-rate coefficients: +! I (10.4)
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The rate of reaction can be identified as the rate of change of comeutrat ion of one of the reactants or products in approaching equilibrium:
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k,[so,][o,p - k,[SOa]
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This forward reaction is l..j-order, indicated by the sum of the exponeuts, while t hc reverse reaction is first-order. If one of the reactants, for example, 02, is in considerable excess, the rate of reaction will depend principally on the concentration of the other, and therefore will approach first-order. This is, in fact, a very common occurrence, so the majority of reactions can be treated as first-order. Furthermore, if any of the four steps previously given to promote conversion are employed, the rate of t he reverse reaction is usually negligible. So a general equation may be applied to describe the rate of most reactions, relative to the concentrnt ion . : of the controlling reactant:
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~1 batch of reactant will change its concentration exponentially with t ime from an initia1 value I ,) to a current value X, according to the integration of E:q. (10.5):
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Controlling Chemical Reactions
Converting the natural logarithms to exponents yields: x = xoeckt Kate t hat the t ime constant for a first-order reaction is l/k; thus the units of the reaction-rate coefficient are in inverse time. Fractional conversion of reactant into product will be identified as TV, varying with Gnie: x0 - x f/= ~ = 1 - e-kt x0 (10.6)
In a continuous plug-jlow reactor, the reaction mixture flows through a pipe without back-mixing. This type of reactor is dominated by dead time. The residence time of the reactants traveling through a volume 1 at a flow F is V/F. Thus the concentration at the exit of the vessel is x = xOe-kVIF Conversion also varies exponentially with flow: y = 1 - e-kvlF (10.7)
A continuous back-mixed reactor is one throughout which the reactant is uniformly distributed by means of agitation. It approaches a singlecapacity system. Reaction rate is constant throughout, given by IQ. (10.5). The rate of consumpt,ion of the reactant is the volume of the vessel times the reaction rate, which equals the flow times the loss in csoncentration between inlet and outlet: V dx - __ = dt ICT X = F(x0 - x)
Solving for exit concentration,
x=1+:l;iOV,F
Conversion in a back-mixed reactor varies inversely with residence time: 1 kV/F Y = - 1 + kJ,,T,F = 1 + IcV/F (10.8) A plug-flow reactor is dominated by dead time equal to the residence time. A back-mixed reactor, hoxvever, has a time constant which is a function of both k and T /F. To illustrate this, a differential equat ion will be written to describe the dynamic material balance: F(x,, - x ) - vkx = v @ dt
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Solving for x, V dx +F+Vk 0 if-i Fxo -___ =F+Vk
The concentration time constant is the coefficient of the second term: (10.9) The reaction-rate coefficient k increases sharply with temperatureperhaps its most outstanding characteristic:
1~ = ae-EIRT
(10.10)
where a and E = constants peculiar to the reaction R = universal gas constant T = absolute temperature To ihustrate this strong dependency, k is plotted vs. T in Fig. 10.1 for a typical reaction whose parameters are: a = ezg min-I, E/R = 20,000 Rankine. The conversion in a plug-flow reactor varies with temperature in a double exponent ial, combining Eqs. (10.7) and (10.10). Conversion versus temperature for a back-mixed reactor is found by combining Eqs. (10.8) and (10.10). In Fig. 10.2, conversion is plotted against t emperat,ure for t,hree values of V/F in both types of continuous reactors using values taken from Fig. 10.1. The plug-flow reactor delivers higher conversion than a back-mixed reactor operating under the same conditions. Differentiation of the conversion vs. temperature relationships for each reactor yields expressions for their slopes. For the plug-flow reactor, dT = RT2
Jc (1 - y)
(1 - y)
(10.11)
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