Mixing-Cup Averages

Example 8.1 Find the mixing-cup average outlet concentration for an isothermal, first-order reaction with rate constant k that is occurring in a laminar flow reactor with a parabolic velocity profile as given by Equation (8.1). 

The double integral in Equation (8.4) is a fairly general definition of the mixing-cup average. It is applicable to arbitrary velocity profiles and noncircular cross sections but does assume straight streamlines of equal length. Treatment of curved streamlines requires a precise and possibly artificial definition of the system boundaries. See Nauman and Buffham. ... 

The hnal step in the design calculations for a laminar flow reactor is determination of mixing-cup averages based on Equation (8.4). The trapezoidal rule is recommended for this numerical integration because it is easy to implement and because it converges O(Ar ) in keeping with the rest of the calculations. 

Both F 0) and F R) vanish for a velocity profile with zero slip at the wall. The mixing-cup average is determined when the integral of F(r) is normalized by Q = 7tR u. There is merit in using the trapezoidal rule to calculate Q by integrating dQ = InrVzdr. Errors tend to cancel when the ratio is taken. 

Solution Example 8.1 laid the groundwork for this case of laminar flow without diffusion. The mixing-cup average is... 

Example 8.5 Use the method of lines combined with Euler s method to determine the mixing-cup average outlet for the reactor of Example 8.4. ... 

Turn now to the flat-plate geometry. The coefficients A, B, and C, and the mixing-cup averaging technique must be revised. This programming exercise is left to the reader. Run the modified program with ki = I but without... 

When an axial position corresponding to four mixing elements is reached, calculate the mixing-cup average composition... 

Restart the solution of Equation (8.12) using a uniform concentration profile equal to the mixing-cup average, a(z) =... 

Turn off the reaction by setting RateConst =0. The resulting final value for the mixing-cup average, avgC, is 1, confirming the material balance. 

In practice, it may be more simple to evaluate E(t) either through the washout function or the cumulative RTD F(t) by making a step change in inlet tracer concentration and measuring outlet concentration by the mixing cup average. Such a procedure is advocated by Nauman [4]. 

Solution For a first-order reaction, we can arbitrarily set am = 1 so that the results can be interpreted as the fraction unreacted. The choices for 7 and J determined in Example 8.4 will be used. The marching-ahead procedure uses Equations (8.25), (8.26), and (8.27) to calculate concentrations. The trapezoidal rule is used to calculate the mixing-cup average at the end of the reactor. The results are... 

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