3.6 Size Comparison of Single Ideal Flow Reactors

(a) Size comparison of MFR and PFR for a Zero Order Reaction

For the isothermal zero order reaction: \(-r_A=k; \quad \varepsilon_A=0\) (constant density system)
the volume of PFR, \[\begin{align*} \frac{V_p}{F_{A0}} =\int_0^{X_A}\frac{dX_A}{-r_A }=\int_0^{X_A}\frac{dX_A}{k}&=\frac{1}{k} \int_0^{X_A}dX_A \\ \Longrightarrow \quad V_p\left(\frac{k}{F_{A0}}\right) &= X_A\end{align*}\] the volume of MFR, \[\begin{align*} \frac{V_m}{F_{A0}} =\frac{X_A}{-r_A}&=\frac{X_A}{k} \\ \Longrightarrow \quad V_m\left(\frac{k}{F_{A0}}\right) &= X_A\end{align*}\] For the same feed conditions and temperature, the ratio of the volumes of MFR and PFR is given by: \[\begin{align*} \frac{V_m}{V_p} = 1\end{align*}\]

(b) Size Comparison of MFR and PFR for a Half Order Reaction

For the isothermal half order reaction: \(-r_A=kC_A^{1/2}; \ \varepsilon_A=0\) (constant density system)
the volume of PFR, \[\begin{align*} \frac{V_p}{F_{A0}} =\int_0^{X_A}\frac{dX_A}{-r_A}=\int_0^{X_A}\frac{dX_A}{kC_A^{1/2}}=\int_0^{X_A}\frac{dX_A}{kC_{A0}^{1/2} (1-X_A)^{1/2}}&=\frac{1}{kC_{A0}^{1/2}} \int_0^{X_A}\frac{dX_A}{(1-X_A)^{1/2}} \\ \Longrightarrow \quad V_p\left(\frac{kC_{A0}^{1/2}}{F_{A0}}\right)&=2\left[1-\sqrt{1-X_A}\right]\end{align*}\] the volume of MFR, \[\begin{align*} \frac{V_m}{F_{A0}} =\frac{X_A}{-r_A}=\frac{X_A}{kC_A^{1/2}}&=\frac{X_A}{kC_{A0}^{1/2}(1-X_A)^{1/2}} \\ \Longrightarrow \quad V_m \left(\frac{kC_{A0}^{1/2}}{F_{A0}}\right)&=\frac{X_A}{\sqrt{1-X_A}}\end{align*}\] For the same feed conditions and temperature, the ratio of the volumes of MFR and PFR are given by: \[\begin{align*} \frac{V_m}{V_p}=\frac{\dfrac{X_A}{\sqrt{1-X_A}}}{2\left[1-\sqrt{1-X_A}\right]} =\frac{X_A}{2\left[\sqrt{1-X_A}-1+X_A\right] }\end{align*}\]

(c) Size Comparison of MFR and PFR for a First Order Reaction

For the isothermal first order reaction: \(-r_A=kC_A; \ \varepsilon_A=0\) (constant density system)
the volume of PFR, \[\begin{align*} \frac{V_p}{F_{A0}} =\int_0^{X_A}\frac{dX_A}{-r_A}=\int_0^{X_A}\frac{dX_A}{kC_A}&=\int_0^{X_A}\frac{dX_A}{kC_{A0}(1-X_A)} \\ \Longrightarrow \quad V_p \left(\frac{kC_{A0}}{F_{A0}} \right)&=-\ln(1-X_A) \end{align*}\] the volume of MFR, \[\begin{align*} \frac{V_m}{F_{A0}} =\frac{X_A}{-r_A}=\frac{X_A}{kC_A }&=\frac{X_A}{kC_{A0}(1-X_A)} \\ \Longrightarrow \quad V_m \left(\frac{kC_{A0}}{F_{A0}}\right)&=\frac{X_A}{1-X_A}\end{align*}\] For the same feed conditions and temperature, the ratio of the volumes of MFR and PFR is given by: \[\begin{align*} \frac{V_m}{V_p} &=\frac{\left(\dfrac{X_A}{1-X_A}\right)}{-\ln(1-X_A)} \end{align*}\]

(d) Size Comparison of MFR and PFR for a Second Order Reaction

For the isothermal second order reaction: \(-r_A=kC_A^2; \ \varepsilon_A=0\) (constant density system)
the volume of PFR, \[\begin{align*} \frac{V_p}{F_{A0}} =\int_0^{X_A}\frac{dX_A}{-r_A }=\int_0^{X_A}\frac{dX_A}{kC_A^2} =\int_0^{X_A}\frac{dX_A}{kC_{A0}^2 (1-X_A)^2} &=\frac{1}{kC_{A0}^2}\int_0^{X_A}\frac{dX_A}{(1-X_A)^2} \\ \Longrightarrow \quad V_p \left(\frac{kC_{A0}^2}{F_{A0}} \right)&=\frac{X_A}{1-X_A} \end{align*}\] the volume of MFR, \[\begin{align*} \frac{V_m}{F_{A0}} =\frac{X_A}{-r_A}=\frac{X_A}{kC_A^2}&=\frac{X_A}{kC_{A0}^2(1-X_A)^2} \\ \Longrightarrow \quad V_m \left(\frac{kC_{A0}^2}{F_{A0}} \right)&=\frac{X_A}{(1-X_A)^2} \end{align*}\] For the same feed conditions and temperature, the ratio of the volumes of MFR and PFR is given by: \[\frac{V_m}{V_p} =\frac{1}{1-X_A}\]

Therefore, the ratio of volume of MFR to that of PFR for the isothermal half order, first order and second order reactions (constant density system) can be calculated easily, as shown in the table below:

\(X_A\)
  \(n=0.5\) \(n=1\) \(n=2\)
0.01 1.003 1.01 1.01
0.05 1.01 1.03 1.05
0.20 1.06 1.12 1.25
0.50 1.21 1.44 2.00
0.80 1.62 2.49 5.00
0.90 2.08 3.91 10.00
0.95 2.74 6.34 20.00
0.99 2.50 21.50 100.00
0.999 16.30 145.00 1,000.00

Points to note based on the above discussion (for the same feed conditions and temperature):

  • For a given conversion of a zero order reaction, both MFR and PFR would be of same size.

  • For all positive orders of reaction, the volume of MFR is always larger than the volume of PFR in order to achieve the same conversion (note that every number in this table is greater than 1).

  • The ratio of volume of MFR to that of PFR increases with increase in the order of reaction.

  • The size of MFR is only slightly bigger than the size of PFR when the conversion is small (\(X_A\le5\%\)).

  • The size ratio increases rapidly for high conversions (\(X_A>80\%\)), especially for the second order reaction.