(a) Consider a gas-liquid-solid hydrogenation such as that described in (b) in which the reaction takes place within a porous catalyst particle in a trickle bed reactor. Assume that the liquid containing the compound to be hydrogenated reaches a steady state with respect to dissolved hydrogen immediately it enters the reactor and that the liquid is involatile. Show that the rate of reaction per unit volume of reactor space R [(kmol H_{2} converted)/m^{3}s] for a reaction which is pseudo first-order with respect to hydrogen is given by:
R=\frac{P_{A}}{H}\left [ \frac{1}{k_{L}a}+\frac{V_{p}}{k_{s}S_{x}(1-e)}+\frac{1}{k\eta (1-e)} \right ]^{-1}
where:
P_{A} = Pressure of hydrogen (bar)
H = Henry Law coefficient (bar m^{3}/kmol)
k_{L}a=Gas–liquid volumetric mass transfer coefficient (s^{-1})
k_{s} = Liquid–solid mass transfer coefficient (m/s)
V_{p} = Volume of single particle (m^{3})
S_{x} =External surface area of a single particle (m^{2})
e = Voidage of the bed (-)
k = First-order rate constant based on volume of catalyst [m^{3}/(m^{3} catalyst) s = s^{-1}]
\eta =Effectiveness factor (-)
(b) Crotonaldehyde is to be selectively hydrogenated to n-butyraldehyde in a process using a palladium catalyst deposited on a porous alumina support in a trickle bed reactor.
The particles will be spheres of 5 mm diameter packed into the reactor with a voidage e of 0.4. Estimated values of the parameters listed in (a) are as follows:
k_{L}a=0.02s^{-1} , k_{s}=2.1\times 10^{-4} m/s
k = 2.8 m^{3}/(m^{3}cat)s,H = 357 bar m^{3}/kmol
Also for spheres the effectiveness factor is given by:
\eta =\frac{1}{\phi }\left ( \coth 3\phi -\frac{1}{3\phi} \right ) where the Thiele modulus, \phi =\frac{V_{p}}{S_{x}}\left ( \frac{k}{D_{e}} \right )^{1/2} (equation 3.19)
For the catalyst, the effective diffusivity, D_{e}=1.9\times 10^{-9}m^{2}/s.
If the pressure of hydrogen in the reactor is 1 bar, calculate R, the rate of reaction per unit volume of reactor, and comment on the relative values of the transfer/reaction resistances involved in the process.
(c) Discuss whether the trickle bed reactor and the conditions described in (b) are the best choices for this process. What alternatives might be considered?
Question : (a) Consider a gas-liquid-solid hydrogenation such as that d...
(a) Hydrogenation takes place in a sequence of steps:
1. mass transfer takes place from the gas to the liquid,
2. mass transfer takes place from the liquid to the external surface of the particles, and 3. diffusion and reaction take place within the particles.
At steady-state, the rates of all these are the same and equal to the overall rate of reaction, R. On the basis of unit volume of reactor space that is the volume of gas, liquid and solid, each of these steps is considered in turn.
1. If k_{L}a is the volume mass transfer coefficient, that is referring to the whole reactor space, then:
R=k_{L}a(C_{i}-C_{L}).
C_{i} = P /H
and hence: R = R=k_{L}a[(P/H)-C_{L}] (i)
2. The rate of mass transfer from the liquid to the solid for one particle
=k_{s}S_{x}(C_{L}-C_{s})
Number of particles per unit volume of reactor space = (1 – e)/V_{p}
Thus: Rate of mass transfer per unit volume of reactor space,
R=(k_{s}S_{x}/V_{p})(1-e)(C_{L}-C_{s}) (ii)
3. The rate of diffusion and reaction per unit volume of particles = kC_{s}\eta . Since the particles occupy only a fraction (1 – e) of the reactor space, then:
Rate of diffusion and reaction per unit volume of reactor space,
R = kC_{s}\eta (1 – e) (iii)
From equations (i), (ii) and (iii):
R/k_{L}a = (P /R) – C_{L}
R/[(k_{s}S_{x}/V_{p})(1-e)]=C_{L}-C_{s}and: R/[k\eta (1-e)]=C_{s}
Adding: R[(1/k_{L}a)+(V_{p}/(k_{s}S_{x}(1-e)))+1/(k\eta (1-e))]=P/H
and: R=\left ( \frac{P}{H} \right )\left \{ \left ( \frac{1}{k_{L}a}\right ) +\frac{V_{p}}{k_{s}S_{x}(1-e)}+\frac{1}{k\eta (1-e)}\right \}^{-1}
(b) For the crotonaldehyde hydrogenation:
R = (1/357)[(1/0.02)+V_{p}/(2.1\times 10^{-4}S_{x} (1-0.4))+1/(2.8?(1-0.4))]^{-1}
=(1/357)[(1/0.02)+(V_{p}S_{x})/1.26\times 10^{-4}+1/(1.68\eta )]^{-1}
For spherical particles:
(V_{p}S_{x})=(\pi d_{p}^{2})=(d_{p}/6)=(0.005/6)=0.000833The effectiveness factor, \eta =(1/\phi )(\coth 3\phi -1/3\phi )
where: \phi=(V_{p}/S_{x})(k/D_{e})^{0.5}=0.000833(2.8/(1.9\times 10^{-9}))^{0.5}
= 32, which is large.
Thus: \eta =(1/32)[\coth (3\times 32)-1/(3\times 32)]= 0.031, which is low.
Substituting these values in equation (iv), then:
R = (1/357)[(1/0.02)+(1/0.151)+(1/0/052)]^{-1}
=0.0028 (50+5.5+19.2)^{-1}
(gas–liquid) (liquid–solid) (diffusion + reaction)
= 3.7 × 10^{-5} kmol/m^{3}s.