CHEMICAL ENGINEERING KINETCS J$.$ M$.$ SMITH SECOND EDITION

$13-8.$ Design a reactor system to produce styrene by the vapor$-$phase catalytic dehydrogenation of ethyl benzene. The reaction is endothermic, so that elevated temperatures are necessary to obtain reasonable conversions. The plant capacity is to be $20$ tons of crude styrene $($styrene$,$ benzene, and toluene$)$ per day. Determine the bulk volume of catalyst and number of tubes in the reactor by the one$-$dimensional method. Assume that two reactors will be needed for continuous production of $20$ tons$/$day$,$ with one reactor in operation

while the catalyst is being regenerated in the other. Also determine the composition of the crude styrene product.

With the catalyst proposed for the plant, three reactions may be significant:

$C_6{H}_5{C}_2{H}_5{}^{k_1}{C}_6{H}_{5}CH=C{H}_2+H_2$

$C_6{H}_5{C}_2{H}_5{}^{k_2}{C}_6{H}_6+C_2{H}_4$

$H_2+C_6{H}_5{C}_2{H}_5{}^{k_3}{C}_6{H}_{5}C{H}_3+C{H}_4$

The mechanism of each reaction follows the stoichiometry indicated by these reactions. The forward rate constants determined for this catalyst by Wenner and Dybdal are:

$log{k}_1=\frac{-11,370}{4\mathrm{.}575T}+0\mathrm{.}883$

$log{k}_2=\frac{-50,800}{4\mathrm{.}575T}+9\mathrm{.}13$

$log{k}_3=\frac{-21,800}{4\mathrm{.}575T}+2\mathrm{.}78$

where $T$ is in degrees Kelvin, and

$k_1=lb$ moles styreme$/($hr$)($atm$)($lb catalyst$)$

$k_2=$ Ib moles benzene$/($hr$)($atm$)($lb catalyst$)$

$k_3=16$ moles toluene$/($hr$)at{m}^2($lb catalyst$)$

The overall equilibrium constants for the three reversible reactions are as follows: $(TABLE)$

The reactor will be heated by flue gas passed at a rate of hr$)($tube$)$ countercurrent $($outside the tubes$)$ to the reaction mixture in the tubes. The flue gas leaves the reactor at a temperature of $1600F.$The reactant streamentering the reactor will be entirely ethyl benzene. The reactor tubes are $4\mathrm{.}03in.ID,4\mathrm{.}50in.OD,$ and $15ft$ long. The feed, $425lb$ of ethyl benzene$/($hr$)($tube$),$ enters the tubes at a temperature of $550C$ and a pressure of $44$ psia; it leaves the reactor at a pressure of $29$ psia. The heat$-$transfer coefficient between the reaction mixture and the flue gas is $)(ft2($based on ouside area$)$ and$p_B=61l\frac{b}{f{t}^3}.$

The thermodynamic data are as follows:

Average specific heat of reaction mixture $=0\mathrm{.}63$ Btu$/(lb)(F)$

Average specific heat of flue gas$=0\mathrm{.}28Bt\frac{u}{lb}(F)$

Average heat of reaction $1,H_1=53,600Bt\frac{u}{l}b$ mole

Average heat of reaction $2,H_2=43,900Bt\frac{u}{l}b$ mole

Average heat of reaction $3,H_2=-27,700Bt\frac{u}{l}b$ mole

To simplify the calculations assume that the pressure drop is directly proportional to the length of catalyst tube. Point out the possibility for error in this assumption, and describe a more aocurate approach.