Using the results from Example 19-3 for t and σθ² estimate values for Pe, for both open and closed boundary conditions, as far as possible, from (a) E(θ) and (b) the variance.

Question

Using the results from Example 19-3 for [latex] \bar{t} [/latex] and [latex] \sigma _ heta ^2 [/latex] estimate values for Pe, for both open and closed boundary conditions, as far as possible, from (a) E(θ) and (b) the variance.

Accepted Answer

(a) From Example 19-3, [latex] \bar{t} = 21.9 [/latex] s (trapezoidal rule) and [latex] E(\bar{t}) \simeq 0.034 s^{-1} [/latex]. Thus, [latex] E(\bar{ heta}) = \bar{t} E(\bar{t}) = 21.9(0.034) = 0.745 [/latex]. From equation 19.4-73 for open-open conditions,

[latex] Pe_, = 4\pi [E( heta = 1)]^2 [/latex] (19.4-73)

[latex] Pe_L = 4\pi (0.745)^2 = 6.97 \simeq 7 [/latex]

The same result is obtained from the Gauss distribution, equation 19.4-56, but we haven’t given the basis here for using the closed-closed and closed-open conditions.

[latex] E( heta)_{max} = (1/2)(Pe_L/\pi )^{1/2} [/latex] (19.4-56)

(b) From Example 19-3 , [latex] \sigma _t ^2 = 161.4 s^2 [/latex] .Thus, [latex] \sigma _ heta ^2 = \sigma _t ^2 /\bar{t}^2 = 161.4 /21.9^2 = 0.337[/latex] .Using the Gauss, open-open, closed-closed, and closed-open solutions, respectively, we obtain the following results (file exl9-lO.msp) :

Different estimates of Pe, are obtained if values of [latex] \bar{t}[/latex] (21.8 s) and [latex] \sigma _t ^2 ( 180 s^2 ) [/latex] from the histogram method are used.
The results again demonstrate that, since there is significant backmixing within the vessel (Pe, relatively small), the different boundary conditions and method of solution lead to markedly different values of [latex]Pe_L [/latex] .

Question 19.10: Using the results from Example 19-3 for t and σθ² estimate v...

Related Answered Questions

In an effort to determine the cause of low yields from a reactor, a tracer study was conducted. An amount mo = 3.80 kg of an inert tracer A was injected into the feed port of the 1 .9-m³ reactor. The volumetric flow rate was constant at q0 = 3.1 L s-¹. The following tracer-response data were ...

Verified Answer:

Using the data in Example 19-3, estimate the value of N for the TIS model (a) based on the mean value, and (b) based on the variance. ...

Verified Answer:

Verified Answer:

Using the data from Example 19-2, estimate values of N based on both central and backward differencing, and determine which differencing technique best describes the tracer outflow concentrations. ...

Verified Answer:

Using the data from Example 19-1, recalculate values of E(t), t; and σ²t by assuming that significant “tailing” begins at tT= 10 min. ...

Verified Answer:

Verified Answer:

For the TIS model, what are the mean and the variance of EN(θ)? ...

Verified Answer:

Calculate 5.575! given that F(1.575) = 0.8909. ...

Verified Answer:

In the TIS model, for a given value of N, what is the maximum value of EN(θ), and at what value θmax does it occur? ...

Verified Answer: