A step increase in the concentration of helium (tracer A), from 1.0 to 2.0 mm01 L^-1, was used to determine the mixing pattern in a fluidized-bed reactor. The response data were as follows: thin: 0 5 1 0 1 5 20 30 45 60 90 120 150 cA,out/mmol L^-1: 1.00 1.005 1.02 1.06 1.20 1.41 1.61 1.77 1.92 1.96

Question

A step increase in the concentration of helium (tracer A), from 1.0 to 2.0 mm01 L[latex]^{-1}[/latex], was used to determine the mixing pattern in a fluidized-bed reactor. The response data were as follows:

thin: 0 5 1 0 1 5 20 30 45 60 90 120 150

[latex]c_{A,out} / mmol L^{-1}[/latex] : 1.00 1.005 1.02 1.06 1.20 1.41 1.61 1.77 1.92 1.96 2.00
Determine F(t), E(t), [latex] \bar{t} [/latex] ; and [latex] \sigma^2_t[/latex] for flow through the vessel, using both backward and forward differencing for E(t), and calculating [latex] \bar{t} [/latex] and [latex] \sigma^2_t[/latex] from E(t)

Accepted Answer

The results for backward differencing are given in Table 19.2 in the form of a spread-sheet analysis, analogous to that in Example 19-1. Column (4) gives F(t) calculated from

Table 19.2 Spreadsheet analysis for Example 19-2 using backward differencing

(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)
i	t/min	c/ mmol L[latex]^{-1}[/latex]	F(t) 19.3-18	E(t) / min[latex]^{-1}[/latex] 19.3-19	ΔF(t)	Δt/min	[latex] [tE(t)]_{a u} \Delta t/\min [/latex]	[latex] [t^2E(t)]_{a u} \Delta t/\min ^2[/latex]
1	0	1.000	0.000
					0.005	5	0.01	0.1
2	5	1.005	0.005	0.0010
					0.015	5	0.09	0.8
3	10	1.020	0.020	0.0030
					0.040	5	0.38	5.3
4	15	1.060	0.060	0.0080
					0.140	5	1.70	32.5
5	20	1.200	0.200	0.0280
					0.210	10	5.95	150.5
6	30	1.410	0.410	0.0210
					0.200	15	9.23	344.2
7	45	1.610	0.610	0.0133
					0.160	15	9.3	490.5
8	60	1.770	0.770	0.0107
					0.150	30	16.35	1183.5
9	90	1.920	0.920	0.0050
					0.040	30	9.15	895.5
10	120	1.960	0.960	0.0013
					0.040	30	5.4	738.0
11	150	2.000	1.000	0.0013
Σ					1.0000		57.600	3840.9

Table 19.3 Spreadsheet analysis for Example 19-2 using central differencing (columns (2), (3), (4), (6), and (7) are the same as in Table 19.2)

(1)	(5)	(8)	(9)
	E(t) / min[latex]^{-1}[/latex] 19.3-20	[latex] [tE(t)]_{a u} \Delta t/\min [/latex]	[latex] [t^2E(t)]_{a u} \Delta t/\min ^2[/latex]
1
		0.03	0.1
2	0.0020
		0.16	1.5
3	0.0055
		0.81	11.5
4	0.0180
		1.84	33.5
5	0.0233
		4.79	120.5
6	0.0164
		7.74	293.0
7	0.1200
		7.15	368.3
8	0.0069
		10.48	756.7
9	0.0032
		6.67	672.7
10	0.0013
		2.50	303.3
11
Σ		42.18	2561.1

equation 19.3-18, and column (5) gives E(t) from equation 19.3-19. The sum of Column (6) merely confirms that the material balance on the tracer is correct. Columns (8) and (9) are used to calculate [latex]\bar{t} [/latex] and [latex] \sigma ^2_t [/latex] , respectively, by means of the trapezoid rule, as in Example 19-1; only the final quantity required is tabulated. From Table 19.2,

[latex] F(t) = \frac{c(t) - c_1}{c_2 - c_1} [/latex] (19.3-18)

[latex] E(t) = \frac{dF(t)}{dt} = \frac{F_i - F_{i - 1}}{t_i - t_{i - 1}} ; i = 2 , 3 , ...[/latex] (19.3-19)

[latex] \bar{t} = 57.6 \min [/latex]

[latex] \sigma ^2_t = 3841 - (57.6)^2 = 529 \min^2 [/latex]

he results for central differencing are given similarly in Table 19.3. Only the columns which are different from those in Table 19.2 are shown. Thus, F(t) is the same as in Table 19.2, but E(t), [latex]\bar{t} [/latex] ; and [latex] \sigma ^2_t [/latex] are different. From Table 19.3,

[latex] \bar{t} = 42.2 \min [/latex]

[latex] \sigma ^2_t = 2561 - (42.2)^2 = 780 \min^2 [/latex]

Estimation of [latex]\bar{t} [/latex] and [latex] \sigma ^2_t [/latex] by nonlinear regression (file exl9-2.msp) leads to the following values: [latex]\bar{t} = 43.1 \min [/latex] , and [latex] \sigma ^2_t = 715 \min^2 [/latex] . The significant variation between the estimates of [latex]\bar{t} [/latex] and [latex] \sigma ^2_t [/latex] from the central differencing and backward differencing methods is mainly due to infrequent sampling. The sampling frequency is less likely to affect the results obtained from nonlinear regression, provided that enough samples have been collected to characterize properly the tracer profile. To determine which method provides the “correct” estimate of mixing behavior (and [latex]\bar{t} [/latex] ), it is necessary to compare the experimental data with tracer curves predicted from the values of [latex]\bar{t} [/latex] and [latex] \sigma ^2_t [/latex] , using one of the RTD models discussed in Section 19.4 (see Example 19-9 below).

Question 19.2: A step increase in the concentration of helium (tracer A), f...

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