Ethyl acetate (A) undergoes a reaction with sodium hydroxide (B) to form
sodium acetate and ethyl alcohol:
The reaction is carried out at steady state in a series of stirred-tank
reactors. The output from the ith reactor is the input to the \((i+1)\) st
reactor. The volumetric flow rate between the reactors is constant at
\(\dot{V}(\mathrm{L} / \mathrm{min}),\) and the volume of each tank is
\(V(\mathrm{L})\)
The concentrations of \(\mathrm{A}\) and \(\mathrm{B}\) in the feed to the first
tank are \(C_{\mathrm{A} 0}\) and \(C_{\mathrm{B} 0}(\mathrm{mol} / \mathrm{L})\).
The tanks are stirred sufficiently for their contents to be uniform
throughout, so that \(C_{\mathrm{A}}\) and \(C_{\mathrm{B}}\) in a tank equal
\(C_{\mathrm{A}}\) and \(C_{\mathrm{B}}\) in the stream leaving that tank. The
rate of reaction is given by the expression where \(k[\mathrm{L} /(\mathrm{mol}
\cdot \mathrm{min})]\) is the reaction rate constant.
(a) Write a material balance on \(A\) in the \(i\) th tank, and show that it
yields
where \(\tau=V / \dot{V}\) is the mean residence time in each tank. Then write a
balance on \(\mathrm{B}\) in the ith tank and subtract the two balances, using
the result to prove that
(b) Use the equations derived in Part (a) to prove that
and from this relation derive an equation of the form
where \(\alpha, \beta,\) and \(\gamma\) are functions of \(k, C_{\mathrm{A} 0},
C_{\mathrm{B} 0}, C_{\mathrm{A}, i-1},\) and \(\tau .\) Then write the solution
of this equation for \(C_{\mathrm{A} i}\)
(c) Write a spreadsheet or computer program to calculate \(N\), the number of
tanks needed to achieve a fractional conversion \(x_{\mathrm{A} N} \geq
x_{\mathrm{Af}}\) at the outlet of the final reactor. Your program should
implement the following procedure:
(i) Take as input values of \(k, \dot{V}, V, C_{\mathrm{A} 0}(\mathrm{mol} /
\mathrm{L}), C_{\mathrm{B} 0}(\mathrm{mol} / \mathrm{L}),\) and
\(x_{\mathrm{Af}}\)
(ii) Use the equation for \(C_{A i}\) derived in Part ( \(b\) ) to calculate
\(C_{\mathrm{Al}}\); then calculate the corresponding fractional conversion
\(x_{\mathrm{A} 1}\)
(iii) Repeat the procedure to calculate \(C_{\mathrm{A} 2}\) and \(x_{\mathrm{A}
2},\) then \(C_{\mathrm{A} 3}\) and \(x_{\mathrm{A} 3},\) continuing until
\(x_{\mathrm{A} i} \geq x_{\mathrm{Af}}\) Test the program supposing that the
reaction is to be carried out at a temperature at which \(k=36.2 \mathrm{L}
/(\mathrm{mol} \cdot \mathrm{min}),\) and that the other process variables have
the following values: Use the program to calculate the required number of
tanks and the final fractional conversion for the following values of the
desired minimum final fractional conversion, \(x_{\mathrm{Af}}:
0.50,0.80,0.90,0.95\) 0.99, 0.999. Briefly describe the nature of the
relationship between \(N\) and \(x_{\mathrm{Af}}\) and what probably happens to
the process cost as the required final fractional conversion approaches \(1.0
.\) Hint: If you write a spreadsheet, it might appear in part as follows:
(d) Suppose a 95\% conversion is desired. Use your program to determine how
the required number of tanks varies as you increase (i) the rate constant, \(k
;\) (ii) the throughput, \(\dot{V} ;\) and (iii) the individual reactor volume,
\(V\). Then briefly explain why the results you obtain make sense physically.
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