2.3 Varying Volume Batch Reactor (VVBR)

  • \(\varepsilon_A\): fractional change in volume of the system (or) volume expansion / contraction factor. It is defined as follows. \[\varepsilon_A=\frac{V_{X_A=1}-V_{X_A=0}}{V_{X_A=0}}\] It is also defined as below: \[\varepsilon_A = y_{A0}\delta\] where

    \(y_{A0}\) = initial mole fraction of the limiting reactant (\(A\))
    \(\delta\) = \(\dfrac{d+c-b-a}{a}\)  for \(aA+bB\rightarrow cC+dD\)

    e.g.: 1 \[A \rightarrow 3R \qquad \text{(isothermal gas-phase reaction)}\] Pure \(A\) was fed. \[\varepsilon_A =\frac{V_{X_A=1}-V_{X_A=0}}{V_{X_A=0}} = \frac{3-1}{1}= 2\] Also, for pure \(A\), \(y_{A0}=1\); and \(\delta=\dfrac{3-1}{1}=2\). So, \(\varepsilon_A = y_{A0}\delta = 1\times2=2\).
     
    e.g.: 2 \[A\rightarrow 3R \qquad \text{(feed consists of 50% $A$ & 50% inerts)}\] \[\varepsilon_A = \frac{(3\times0.5+0.5)-(0.5+0.5)}{0.5+0.5} = 1\] Also, \(y_{A0}=0.5\), and \(\delta=\frac{3-1}{1}=2\). So, \(\varepsilon_A = y_{A0}\delta = 0.5\times2=1\).

  • Note:

    • The term \(\varepsilon_A\) accounts for both the stoichiometry and the presence of inerts.

    • For all liquid-phase reactions, the volume change is not significant. Hence \(\varepsilon_A=0\).

    • For gas-phase reactions, one has to calculate the value of \(\varepsilon_A\) and same will be used with design calculations.

  • The variation of volume of a batch system is given by a linear relationship: \[V = V_0(1+\varepsilon_AX_A)\left(\frac{P_0}{P}\right)\left(\frac{T}{T_0}\right)\] In case of isothermal constant pressure systems, for which \(T = T_0\) and \(P=P_0\), the above equation reduces to \[V = V_0(1+\varepsilon_A X_A)\] where

    \(V_0\) = initial volume of the reactor
    \(V\) = volume of the reactor at time ‘\(t\)’
    \(X_A\) = fractional conversion at time ‘\(t\)’

    Note: \[X_A = \frac{V-V_0}{\varepsilon_AV_0} \qquad \text{(for isothermal and constant-pressure operation)}\]

  • Definition and expression for concentration in a batch system of varying volume: \[\left.\begin{array}{c}\text{The concentration of the} \\ \text{limiting reactant, $A$, at any time `$t$'} \end{array}\right\} \equiv C_A\] \[\begin{align*} C_A &= \frac{N_A}{V} \\ \text{Also,} \\ N_A &= N_{A0}(1-X_A) \\ \text{and} \\ V &= V_0(1+\varepsilon_AX_A) \\ \text{Hence,} \\ C_A &= \frac{N_{A0}(1-X_A)}{V_0(1+\varepsilon_AX_A)} = \frac{C_{A0}(1-X_A)}{(1+\varepsilon_AX_A)} \\ \Longrightarrow \quad \frac{C_A}{C_{A0}} &= \frac{1-X_A}{1+\varepsilon_AX_A} \\ \text{By further reductions, we can get} \\ X_A &= \frac{1-(C_A/C_{A0})}{1+\varepsilon_A(C_A/C_{A0})}\end{align*}\]

  • Expression for the rate of reaction (VVBR):

    The rate of disappearance of the limiting reactant, \(A\), is defined for homogeneous systems as follows: \[-r_A = \frac{-1}{V}\frac{dN_A}{dt}\] Now, \(N_A=N_{A0}(1-X_A) \qquad \Longrightarrow \quad dN_A = -N_{A0}dX_A\).

    Also, \(V=V_0(1+X_A)\).

    So, \[\begin{align*} -r_A &= \frac{-1}{V_0(1+\varepsilon_A)}(-N_{A0})\frac{dX_A}{dt} \\ \text{We know, $N_{A0}/V_0=C_{A0}$. Therefore,} \\ \Longrightarrow \quad -r_A &= \frac{C_{A0}}{(1+\varepsilon_AX_A)}\frac{dX_A}{dt} \qquad \text{(rate of reaction in terms of $X_A$)}\end{align*}\] We have, \(V=V_0(1+\varepsilon_AX_A) \qquad \Longrightarrow \quad dV = V_0\varepsilon_AdX_A \qquad \Longrightarrow \quad dX_A = \dfrac{dV}{V_0\varepsilon_A}\). Thus, the rate of reaction takes the form, in terms of volume as follows: \[\begin{align*} -r_A &= \frac{C_{A0}}{(1+\varepsilon_AX_A)}\frac{1}{V_0\varepsilon_A}\frac{dV}{dt} \\ \Longrightarrow \quad -r_A &= \frac{C_{A0}}{V\varepsilon_A}\frac{dV}{dt} = \frac{C_{A0}}{\varepsilon_A}\frac{d(\ln V)}{dt}\end{align*}\]