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

Question

For the TIS model, what are the mean and the variance of [latex]E_N( heta)[/latex]?

Accepted Answer

This involves obtaining the mean-residence time, [latex]\bar{ heta } [/latex] , and the variance, [latex]\sigma _ heta ^2[/latex], of the distribution represented by equation 19.4-14. Since, in general, these are related to the first and second moments, respectively, of the distribution, it is convenient to connect the determination of moments in the time domain to that in the Laplace domain. By definition of a Laplace transform ,

[latex]E_N( heta) = \frac { N^N heta ^{N - 1} e^{ - N heta }} {(N - 1)!} (N = 1 , 2 , ... , N) [/latex] (19.4-14)

[latex] L[E_N( heta )] = \int_{0}^{\infty}{e^{-s heta}E_N( heta)d heta} [/latex] (19.4-17)

and

[latex]\frac {dL[E_N( heta)]}{ds} = - \int_{0}^{\infty } heta e^{- s heta}E_N( heta)d heta [/latex] (19.4-18)

[latex]\underset{s ightarrow 0 }{\lim }(-1) \frac {dL[E_N( heta)]}{ds} = \int^{\infty }_{0} heta E_N( heta)d heta = \mu_1 = \bar{ heta} [/latex] (19.4-19)

from equation 13.3-14, where [latex]\mu _1[/latex] is the first moment about the origin, which, in turn, is the location expressed as the mean residence time [latex]\bar{ heta} [/latex] .

[latex]\mu _1 = \int_{0}^{\infty } tf(t)dt = \bar{t}[/latex] (13.3-14)

From equation 19.4-12, with [latex]C_{AN}[/latex] replaced by its equivalent [latex]E_N( heta) [/latex]

[latex]L (C_{AN}) = 1/(1 + s/N)^N [/latex] (19.4-12)

[latex]L [E_N( heta)] = \frac {1}{(1 + s/N)^N} [/latex] (19.4-20)

[latex]\frac {dL [E_N( heta)] }{ds} = \frac {- 1}{(1 + s/N)^{N + 1} } [/latex] (19.4-21)

Therefore ,

[latex]\bar{ heta} = \underset{s ightarrow 0 }{\lim } \frac {( -1 )(-1)}{(1 + s/N)^{N + 1}} = 1[/latex] (19.4-22)

From equation 19.4-17,

[latex]\frac {d^2L[E_N( heta)]}{ds^2} = -\int_{0}^{\infty } heta^2 e^{-s heta}E_N( heta)d heta[/latex] (19.4-23)

[latex]\underset{s ightarrow 0 }{\lim } \frac {d^2L[E_N( heta)]}{ds^2} = \int _{0}^{\infty}{ heta^2E_N( heta)d heta} = \mu_2 [/latex] (19.4-24)

rom equation 13.3-12, where [latex]\mu _2[/latex] is the second moment about the origin. From equation 13.3-16a, the variance, which is the second moment about the mean, is related to [latex]\mu _1[/latex] and [latex]\mu _2[/latex] by

[latex]\mu _k = \int _{0}^{\infty} {t^kf(t)dt } ; t \geq 0 [/latex] (13.3-12)

[latex]\sigma _{t}^{2} = \left[\int_{0}^{\infty}{t^2f(t)dt} ight] - \bar{t}^2[/latex] (13.3-16a)

[latex]\sigma _ heta^2 = \mu_2 - \mu_1^2[/latex] (19.4-25)

From equations 19.4-20, -22, -24, and -2.5,

[latex]\sigma _ heta^2 = \underset{s ightarrow 0 }{\lim } \frac {d^2L[E_N( heta)]}{ds^2} - 1 [/latex]

[latex] = lii \frac {N + 1}{N}\frac {1}{(1 + s/N)^{N + 2}} - 1 = \frac {N + 1} {N} - 1 = \frac {1}{N} [/latex]

Thus ,

[latex]\sigma _ heta^2 = 1/N (N = 1 , 2 , ... , N) [/latex] (19.4-26)

Note that the determination of moments in the Laplace domain in this case is much easier than in the time domain, since tedious integration of a complicated function is replaced by the easier operation of differentiation. The result for [latex]\bar{ heta} [/latex] we could have anticipated, but not the very simple result for [latex]\sigma _ heta^2 [/latex] in equation 19.4-2 .

Question 19.6: For the TIS model, what are the mean and the variance of EN(...

Related Answered Questions

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. ...

Verified Answer:

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:

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: