A first-order reaction, A → products, is conducted in a fluidized-bed reactor. Before fluidization, the bed is 1.8 m deep (Lpa), with a voidage (εpa) of 0.42. The feed gas (pf =0.45 kg m-³ ;μf = 4.2 X lo^-5 Pa s) flows through the bed at a velocity three times the minimum fluidization velocity. The

Question

A first-order reaction, A → products, is conducted in a fluidized-bed reactor. Before fluidization, the bed is 1.8 m deep ( $L_{pa}$ ), with a voidage ( $\epsilon _{pa}$ ) of 0.42. The feed gas ( $\rho_f = 0.45 kg m^{-3}$ ; $\mu_f = 4.2 \times 10^{-5} Pa s$ ) flows through the bed at a velocity three times the minimum fluidization velocity. The following data are also available:

$\epsilon _{mf} = 0.48 \rho_p = 3100 kg m^{-3} \gamma _b = 0$

$d_p = 0.45 mm d_b = 0.09 m k_A = 1.9 m^3 gas (m^3 cat)^{-1} s^{-1}$

Determine the fractional conversion of A ( $f_A$ ).

[latex]\epsilon _{mf} = 0.48 ho_p = 3100 kg m^{-3} \gamma _b = 0[/latex]

[latex]d_p = 0.45 mm d_b = 0.09 m k_A = 1.9 m^3 gas (m^3 cat)^{-1} s^{-1} [/latex]

Determine the fractional conversion of A ( [latex]f_A[/latex]).

Accepted Answer

First, the minimum fluidization velocity ( [latex]u_{mf}[/latex]) is calculated from 23.2-5. After substitution of the particle and fluid properties provided, the resulting quadratic equation is:

[latex]u_{mf}^2 + \frac{150(1 - \epsilon _{mf})\mu _f}{1.75\rho_fd_p^\prime}u_{mf} - \frac{g(\rho _p - \rho _f)\epsilon ^3_{mf}d_p^\prime}{1.75\rho _f} = 0 [/latex] (23.2-5)

[latex]u^2_{mf} + 9.24u_{mf} - 1.922 = 0[/latex]

which leads to a value of [latex]u_{mf}[/latex] of [latex]0.203 m s^{-1} [/latex]. Thus, [latex]u_{fl} = 3{u_{mf}} = 0.609 m s^{-1} [/latex].

Next, the bubble velocities are calculated to determine the flow regime:

[latex]u_{br} = 0.711(gd_b)^{0.5} = 0.711[(9.81)(0.09)]^{0.5} = 0.668 m s^{-1}[/latex] (23.3-2)

[latex]u_b = u_{fl} - u_{mf} + u_{br} = 0.609 - 0.203 + 0.668 = 1.074 m s^{-1}[/latex] (23.3-4)

The flow regime is determined by comparing [latex]u_{b}[/latex] with [latex]u_{mf}/\epsilon_{mf}[/latex] :

[latex]u_b/(u_{mf}/\epsilon_{mf}) = 1.704/(0.203/0.48) = 2.54[/latex]

Thus, the bubble velocity is within the intermediate-bubble regime, and equation 23.3-6b is used for [latex]f_b[/latex]. Parameters [latex]u_b^\ast[/latex] , [latex]K_{be}[/latex] , [latex]f_b[/latex] , and [latex]L_{fl}[/latex] are calculated as follows:

[latex]f_b = \frac{u_{fl} - u_{mf}}{u_b + u_{mf}} (for u_b \simeq u_{mf}/\epsilon_{mf})[/latex] (23.3-6b)

[latex]u_b^\ast = u_b + 3u_{mf} = 1.074 + 3(0.203) = 1.683 m s^{-1} [/latex] (23.4-17)

[latex]K_{be} = 4.5(u_{mf}/d_b) = 4.5(0.203/0.09) = 10.16 s^{-1} [/latex] (23.4-20)

[latex]f_b = (u_{fl} - u_{mf})/(u_b + u_{mf}) = (0.609 - 0.203)/(1.074 + 0.203) = 0.318[/latex] (23.3-6b)

[latex]L_{fl} = \frac{L_{pa}(1 - \epsilon _{pa})}{(1 - \epsilon _{mf})(1 - f_b)} = \frac{(1.8)(1 - 0.42)}{(1 - 0.48)(1 - 0.318)} = 2.94 m[/latex] (23.4-7b)

The continuity equations 23.4-18 and - 19 are applicable. Thus; Continuity equation for A in the bubble region:

[latex]f_bu_b^\ast \frac{dc_{Ab}}{dx} = - f_b\gamma_b k_Ac_{Ab} - f_bK_{be} (c_{Ab} - c_{Ae})[/latex] (23.4-18)

Continuity equation for A in the emulsion region :

[latex](1 - f_b)u_{mf}\frac{dc_{Ae}}{dx} = f_bK_{be}(c_{Ab} - c_{Ae}) - (1 - f_b)(1 - \epsilon_{mf})k_Ac_{Ae}[/latex] (23.4-19)

These equations are numerically integrated simultaneously from x = 0 to [latex] x = L_{fl} = 2.94 m[/latex] to obtain [latex]c_{Ab} [/latex] and [latex]c_{Ae}[/latex] . Using an initial (inlet) condition of [latex]c_{Ao} = c_{Ab}(0) = c_{Ae}(0) = 1.0 [/latex] , we obtain (file ex23-5.msp): [latex]c_{Ab} = 0.077 mol L^{-1} [/latex] and [latex]c_{Ae} = 0.066 mol L - 1 [/latex].

These values are substituted into equation 23.4-21 to determine [latex]f_A[/latex]:

[latex]f_A = 1 - \frac{f_bu_b^\ast c_{Ab} + (1 - f_b)u_{mf}c_{Ae}}{u_{fl}c_{Ao} }[/latex] (23.4-21)

[latex] = 1 - \frac{(0.318)(1.683)(0.077) + (1 - 0.318)(0.203)(0.066)}{(0.609)(1.0)} = 0.917 [/latex]

Thus, the fractional conversion for this system, which operates in the intermediate-bubble and -particle regime, is 0.92.

Question 23.5: A first-order reaction, A → products, is conducted in a flui...

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