Chapter 5, Problem 5.70P

Methanol is synthesized from carbon monoxide and hydrogen in the reaction

   ${\text{CO+2H}}_{\text{2}}\rightleftharpoons {\text{CH}}_{\text{3}}\text{OH}$

A process ?owchart is shown below.

The fresh feed to the system, which contains only CO and H₂, is blended with a recycle stream containing the same species. The combined stream is heated and compressed to a temperature T(K) and a pressure P(kPa) and fed to the reactor. The percentage excess hydrogen in this stream is H_xs. The reactor ef?uent—also at T and P—goes to a separation unit where essentially all of the methanol produced in the reactor is condensed and removed as product. The unreacted CO and H₂constitute the recycle stream blended with the fresh feed.

Provided that the reaction temperature (and hence the rate of reaction) is high enough and the ideal- gas equation of state is a reasonable approximation at the reactor outlet conditions (a questionable assumption), the ratio

   $K_{pc}=\frac{p_{{\text{CH}}_{\text{3}}\text{OH}}}{p_{\text{CO}}{p}_{{\text{H}}_2}^2}$

   $K_p\left(\text{T}\right)=1.390\times {10}^{-4}\exp \left(21.225+\frac{9143.6}{T}-7.492\text{In}T+4.076\times {10}^{-3}T-7.161\times {10}^{-8}{T}^2\right)$

In these equations, p_iis the partial pressure of species i in kilopascals $\left(i={\text{CH}}_{\text{3}}{\text{OH,CO,H}}_{\text{2}}\right)$and T is in Kelvin.

Suppose P = 5000 kPa, T = 500 K, and the percentage excess of hydrogen in the feed to the reactor (H_xs) = 5.0%. Calculate ${\stackrel{\dot{}}{n}}_4,{\stackrel{\dot{}}{n}}_5$, and ${\stackrel{\dot{}}{n}}_6$, the component ?ow rates (kmol/h) in the reactor effluent. [Suggestion: Use the known value of H_xs, atomic balances around the reactor, and the equilibrium relationship, K_pc= K_p(T), to write four equations in the four variables ${\stackrel{\dot{}}{n}}_3$to ${\stackrel{\dot{}}{n}}_6$; use algebra to eliminate all but ${\stackrel{\dot{}}{n}}_6$; and use Goal Seek or Solver in Excel to solve the remaining nonlinear equation for ${\stackrel{\dot{}}{n}}_6$.] Then calculate component fresh feed rates ( ${\stackrel{\dot{}}{n}}_1$and ${\stackrel{\dot{}}{n}}_2$

1. ) and the ?ow rate (SCMH) of the recycle stream.

Prepare a spreadsheet to perform the calculations of Part (a) for the same basis of calculation (100 kmol CO/h fed to the reactor) and different speci?ed values of P(kPa), T(K), and H_xs(%). The spreadsheet should have the following columns:

P(kPa)

T(K)

H_xs(%)

   $K_p\left(T\right)\times {10}^8$. (The given function of T multiplied by 10⁸. When T = 500 K, the value in this column should be 91.113.)

K_pP²

   ${\stackrel{\dot{}}{n}}_3$. The rate (kmol/h) at which H₂enters the reactor.

   ${\stackrel{\dot{}}{n}}_4$. The rate (kmol/h) at which CO leaves the reactor.

   ${\stackrel{\dot{}}{n}}_5$. The rate (kmol/h) at which H₂leaves the reactor.

   ${\stackrel{\dot{}}{n}}_6$. The rate (kmol/h) at which methanol leaves the reactor.

   ${\stackrel{\dot{}}{n}}_{\text{tot}}$. The total molar ?ow rate (kmol/h) of the reactor ef?uent.

$K_{pc}\times {10}^8$. The ratio $y_{\text{M}}/\left(y_{\text{CO}}{y}_{{\text{H}}_2}^2\right)$multiplied by 10⁸. When the correct solution has been attained, this value should equal the one in Column E.

   $K_p{P}^2-K_{pc}{P}^2$. Column E-Column K. which equals zero for the correct solution.

   ${\stackrel{\dot{}}{n}}_1$. The molar ?ow rate (kmol/h) of CO in the fresh feed.

   ${\stackrel{\dot{}}{n}}_2$. The molar ?ow rate (kmol/h) of H₂in the fresh feed.

   ${\stackrel{\dot{}}{V}}_{\text{rec}}\left(\text{SCHMH}\right)$. The ?ow rate of the recycle stream in m³(STP)/h.When the correct formulas have been entered, the value in Column l should be varied until the value in Column L equals 0.

Run the program for the following nine conditions (three of which are the same):

$T=500\text{K},H_{\text{xs}}=5\%$, and $P=1000\text{kPa},5000\text{kPa}$

, and 10,000 kPa.

$P=5000\text{kPa},H_{\text{xs}}=5\%$, and $T=400\text{K},500\text{K}$

, and 600 K.

$T=500\text{K},P=5000\text{kPa}$, and $H_{\text{xs}}=10\%,5\%$

, and 10%

Summarize the effects of reactor pressure, reactor temperature, and excess hydrogen on the yield of methanol (kmol M produced per 100 kmol CO fed to the reactor).

(c) You should ?nd that the methanol yield increases with increasing pressure and decreasing temperature. What cost is associated with increasing the pressure?

(d) Why might the yield be much lower than the calculated value if the temperature is too low?

(e) If you actually ran the reaction at the given conditions and analyzed the reactor ef?uent, why might the spreadsheet values in Columns F—M be signi?cantly different from the measured values of these quantities? (Give several reasons, including assumptions made in obtaining the spreadsheet values.)