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Set8Solutions - ChE 101 2012 Set 8 Solutions 1(a The PFR...

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ChE 101 2012 Set 8 Solutions 1. (a) The PFR mass balance is given by Equation 5.22 as u dc A dz = - k 0 c A exp - E RT , where we have explicitly used the Arrhenius temperature dependence of k . The energy balance is given in Equation 5.31, u dT dz = - Δ H R ρC p k 0 c A exp - E RT - Up w ρC p A t ( T - T c ) Note that in our convention, Δ H R is negative for an exothermic reaction. Now, since we would like the independent variable z to range from 0 to 1, the obvious choice of nondimensional distance is to take ˜ z = z L The most appropriate dimensionless concentration variable is one we have been using all term, namely the fractional conversion X , since this precisely ranges from 0 to 1 over the course of reaction completion. Thus, we define X = 1 - c A c A, 0 The choice of nondimensional temperature is less obvious, since there are two temper- ature scales in the system: T 0 and T c . However, notice that as long as heat transfer is sufficiently fast, fluid will enter the reactor at temperature T 0 and leave at temperature T c , so these two quantities provide the natural range of temperature variation within the system. Thus, in order to have the temperature variable range from 0 to 1, the natural combination to use is Θ = T - T 0 T c - T 0 , which should be very reminiscent of variables we often use in heat transport calculations. Notice that this temperature scale only works if T c and T 0 are unequal, but this condition is often met in any cooled reactor. Substituting these expressions into the mass balance, we find that - c A, 0 u L dX d ˜ z = - k 0 c A, 0 (1 - X ) exp - E/R ( T c - T 0 ) Θ + T 0 1
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To clean this up, we introduce the following dimensionless groups: α = k 0 L u = k 0 V V 0 = k 0 τ β = T c - T 0 T 0 = T c T 0 - 1 γ = E R T 0 Thus, the mass balance nondimensionalizes as follows: dX d ˜ z = α (1 - X ) exp - γ 1 + β Θ Now we work on the energy balance: ( T c - T 0 ) u L d Θ d ˜ z = - k 0 c A, 0 Δ H R ρC p (1 - X ) exp - γ 1 + β Θ - Up w ρC p A t ( T c - T 0 ) ( θ - 1) To massage this formula further, we define the following two dimensionless variables, ˜ J = - c A, 0 Δ H R ρC p ( T c - T 0 ) ˜ K = Up w L ρC p A t u Notice that we have incorporated a negative sign into our definition of ˜ J in anticipation of the fact that many reactions of interest are exothermic. With these additional definitions, we obtain the dimensionless energy balance for the system, d Θ d ˜ z = ˜ (1 - X ) exp - γ 1 + β Θ - ˜ K ( θ - 1) (b) The parameter α can be viewed as a dimensionless residence time in the reactor, weighted by the reaction rate constant. Hence, α provides a measure of the extent to which the reaction is allowed to proceed relative to its intrinsic rate. If α is large, then we will approach complete conversion at the end of the reactor. The interpretation of β is fairly simple: it is simply the relative magnitude of the characteristic temperature variation in the reactor induced by the coolant. The parameter γ is one of the most important determinants of the reactor dynamics. Intuitively, γ is a comparison of the activation energy for the reaction relative to the thermal energy in the reactor. Thus, γ sets the
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