The performance of a reactor for a gas-solid reaction (A(g) + bB(s)→ products) is to be analyzed based on the following model: solids in BMF, uniform gas composition, and no overhead loss of solid as a result of entrainment. Calculate the fractional conversion of B (fB) based on the following

Question

The performance of a reactor for a gas-solid reaction (A(g) + bB(s)→ products) is to be analyzed based on the following model: solids in BMF, uniform gas composition, and no overhead loss of solid as a result of entrainment. Calculate the fractional conversion of B ([latex]\bar{f_B}[/latex] ) based on the following information and assumptions: T = 800 K; [latex]P_A = 2[/latex] bar; the particles are cylindrical with a radius of 0.5 mm; from a batch-reactor study, the time for 100% conversion of 2-mm particles is 40 min at 600 K and [latex]P_A = 1[/latex] bar. Compare results for [latex]\bar{f_B}[/latex] assuming (a) gas-film (mass-transfer) control; (b) surface-reaction Control; and (c) ash-layer diffusion control. The solid flow rate is 1000 kg min-¹, and the solid holdup ([latex]{W_B}[/latex] )
in the reactor is 20,000 kg. Assume also that the SCM is valid, and the surface reaction is first-order with respect to A .

Accepted Answer

A flow diagram for the reactor is shown schematically in Figure 22.3, together with the given data and assumptions.
Since the particles are all of one size and are in BMF, the reactor model is given by equation 22.2-13. The expression for [latex][1 - f_B(t)] [/latex] in the integral depends on the relevant kinetics (for a single particle), and is obtained from Table 9.1.

[latex]1 - \bar{f_B} = \frac{1}{\bar{t}} \overset{\underset{}{ t_1} }{\underset{I_{0}}{}}[1 - f_B(t)]e^{- t/\bar{t}}dt[/latex] (22.2-13)

(a) From Table 9.1, for a cylindrical particle with gas-film control

[latex]t(f_B) = t_1f_B[/latex] (22.2-14)

or

[latex]1 - f_B(t) = 1 - t/t_1[/latex] (22.2-14a)

where [latex]t_1 = \rho _{Bm}R/2bk_{Ag}c_{Ag}[/latex] (22.2-15)

With equation 22.2-14a substituted in equation 22.2-13, the reactor model is

[latex]1 - \bar{f_B} = \frac{1}{\bar{t}}\int_{0}^{t_1}{\left(1 - \frac{t}{t_1}\right)e^{-t/\bar{t}}dt } = 1 - (\bar{t}/t_1)(1 - e^{-t_1/\bar{t}}) [/latex] (22.2-16)

(cylindrical particles BMF, SCM, gas-film control)

on integration and insertion of the limits. (The two terms in the integrand, after multiplication, can be integrated separately, but the second requires integration by parts.) Equation 22.2- 16 may be written as

[latex]\bar{f_B} = (\bar{t}/t_1)(1 - e^{-t_1/\bar{t}}) [/latex] (22.2-16a)

Information is given directly for [latex]\bar{t}[/latex] ; but indirectly for [latex]t_1[/latex]

[latex]\bar{t} = W_B /F_{Bo} = 20,000/1000 = 20 min [/latex]

To obtain [latex]t_1[/latex] at flow-reactor conditions from [latex]t_{1B}[/latex] obtained in the batch reactor, we use equation 22.2-15, together with ideal-gas behavior for reactant A :

[latex]c_{Ag} = p_A/RT[/latex] (4.2-3a)

Denoting BR conditions by subscript 1 for radius R, T and [latex]p_A[/latex], and flow-reactor conditions by subscript 2, from 22.2-15, for fixed values of the other parameters, we have

[latex]\frac{t_1}{t_{1B}} - \frac{R_2/c_{Ag2}}{R_1/c_{Ag1}} - \frac{R_2p_{A1}T_2}{R_1p_{A2}T_1} = \frac{0.5(1)800}{2(2)600} = \frac{1}{6}[/latex]

Thus , [latex] t_1 = 40/6 = 6.67 min [/latex]

and from equation 22.2-16a, [latex] \bar{f_B} (gas - film control ) = 0.850 [/latex]

(b) From Table 9.1, for a cylindrical particle with surface-reaction control,

[latex]t(f_B) = t_1[1 - (1 - f_B)^{1/2}][/latex] (22.2-17)

or

[latex] 1 - f_B(t) = (1 - t/t_1)^2 [/latex] (22.2-17a)

where [latex] t_1 = \rho _{Bm}R/bc_{Ag}k_{As} [/latex] (9.1-31a)

With equation 22.2-17a substituted into 22.2-13, the reactor model becomes, after integration,

[latex]\bar{f_B} = (2\bar{t}/t_1)[1 - (\bar{t}/t_1)(1 - e^{-t_1/\bar{t}})] [/latex] (22.2-18)

(cylindrical particles, BMF, SCM, n = 1, reaction control)

Equation 22.2- 18 corresponds to equation 22.2-16a for gas-film control. The values of both [latex]\bar{t}[/latex] and [latex]{t_1}[/latex] are the same as in part (a), 20 and 6.67 min, respectively. The latter is because, for reaction and gas-film control, the forms of the expressions for [latex]{t_1}[/latex] are the same with respect to the parameters that change. Thus, with these values used in 22.2-18,

[latex]\bar{f_B} (reaction control) = 0.898[/latex]

This value of [latex]\bar{f_B}[/latex] may also be obtained, by means of the E-Z Solve software, by simultaneous solution of equation 22.2-17 and numerical integration of 22.2-13 (with user-defined function fbcr(t, [latex]{t_1}[/latex] ), for cylindrical particles with reaction control; see file ex22-3.msp). This avoids the need for analytical integration leading to equation 22.2-18.

(c) From Table 9.1, for a cylindrical particle with ash-layer diffusion control

[latex] t(f_B) = t_1[f_B + (1 - f_B) \ln (1 - f_B)] [/latex] (22.2-19)

where [latex] t_1 = \rho_{Bm} R^2/4bc_{Ag}D_e [/latex] (22.2-20)

Unlike the situation in (a) and (b), equation 22.2-19 cannot be inverted functionally to obtain [latex]1 - {f_B}[/latex] or [latex]{f_B}[/latex] explicitly in terms of t, for ultimate use in equation 22.2- 13. Rather than an analytical expression corresponding to 22.2-16a or -18, we must use the alternative numerical procedure described in part (b). Furthermore, for ash-layer diffusion control, since [latex]t_1 \propto R^2 , t_1 = 40/24 = 1.667 mm[/latex]. Using this value for [latex]{t_1}[/latex] and [latex]\bar{t} = 20 [/latex] as before, we obtain (see file ex22-3.msp):

[latex]\bar{f_B} ( ash-layer control) = 0.980[/latex]

Note that in the numerical solution of equation 22.2-19, [latex]{f_B}[/latex] must be less than 1 to avoid ln(0); in the use of E-Z Solve, an initial guess of 0.5 is entered, and the upper and lower limits may be set to 0 and 0.9999, respectively. Since upper and lower limits must be specified, a user-defined function is not applicable to this case.

Table 9.1 SCM: Summary of [latex]t(f_B) [/latex] ffor various shapes of particle '

Particle shape	Size Parameters	[latex]f_B[/latex](size parameters	[latex]t(f_B)[/latex]
flat plate (one face permeable)	length l (of Zone unreacted ) L(of particale)	[latex]1 - \frac{1}{L}[/latex]	[latex]\frac{\rho _{Bm}Lf_B}{bc_{Ag}}\left(\frac{1}{k_{Ag}} + \frac{Lf_B}{2D_e} + \frac{1}{k_{As}}\right) [/latex]
cylinder (ends sealed)	radius [latex]r_c[/latex] (of core unreacted R(of particle)	[latex]1 - \frac{r_c}{_0R}^2[/latex]	[latex]\frac{\rho _{Bm}R}{bc_{Ag}}\left\{\frac{f_B}{2k_{Ag}} + \frac{R}{4D_e}\left[f_B + (1 - f_B)\ln(1 f_B)\right] + \frac{1}{k_{As}}\left[1 - (1 - f_B)^{1/2}\right] \right\} [/latex]
sphere	same as for cylinder	[latex]1-\left(\frac{r_c}{R}\right) ^3[/latex]	[latex]\frac{\rho _{Bm}R}{bc_{Ag}}\left\{\frac{f_B}{3k_{Ag}} + \frac{R}{6D_e}[1 - 3(1 f_B)^{2/3} + 2(1 -f_B) + 1\frac{1}{k_{As}}1\left[ - (1 - f_B)^{1/3}\right] \right\}[/latex] (9.1-28)
Reaction : A(g k bB(s) -) product (s),(g)]; first order with respect to A at core surface. Particle(B) : constant-size (L, R constant); isothermal. For [latex]t_1[/latex] (time for complete reaction of particle), set [latex]f_B = 1 ; [(1 f_B)\ln(1 f_B) \rightarrow 0 ][/latex] . Symbols : see text and Nomenclature.

Question 22.3: The performance of a reactor for a gas-solid reaction (A(g) ...

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