Cascades of CSTRs, also known as a series of CSTRs, are used to decrease the volume of a system.
Minimizing volume As seen in the graph with one CSTR, where the inverse rate is plotted as a function of
fractional conversion, the area in the box is equal to \frac{V}{F_{Ao}} where V is the total reactor volume and F_{Ao} is the molar flow rate of the feed. When the same process is applied to a cascade of CSTRs as seen in the graph with three CSTRs, the volume of each reactor is calculated from each inlet and outlet fractional conversion, therefore resulting in a decrease in total reactor volume. Optimum size is achieved when the area above the rectangles from the CSTRs in series that was previously covered by a single CSTR is maximized. For a
first order reaction with two CSTRs, equal volumes should be used. As the number of ideal CSTRs (n) approaches infinity, the total reactor volume approaches that of an ideal
PFR for the same reaction and fractional conversion.
Ideal cascade of CSTRs From the design equation of a single CSTR where \tau = \frac{C_{Ao} - C_A}{-r_A} , we can determine that for a single CSTR in series that \tau_i = \frac{C_{A(i-1)}-C_{Ai}}{-r_{Ai}} , where \tau is the
space time of the reactor, C_{Ao} is the feed concentration of A, C_{A} is the outlet concentration of A, and -r_{A} is the
rate of reaction of A.
First order For an
isothermal first order, constant density reaction in a cascade of identical CSTRs operating at
steady state For one CSTR: C_{A1} = \frac{C_{Ao}}{1 + k\tau} , where k is the
rate constant and C_{A1} is the outlet concentration of A from the first CSTR Two CSTRs: C_{A1} = \frac{C_{Ao}}{1 + k\tau} and C_{A2} = \frac{C_{A1}}{1 + k\tau} Plugging in the first CSTR equation to the second: C_{A2} = \frac{C_{Ao}}{(1 + k\tau)^2} Therefore, for m identical CSTRs in series: C_{Am} = \frac{C_{Ao}}{(1 + k\tau)^m} When the volumes of the individual CSTRs in series vary, the order of the CSTRs does not change the overall conversion for a first order reaction as long as the CSTRs are run at the same temperature.
Zeroth order At steady state, the general equation for an isothermal
zeroth order reaction at in a cascade of CSTRs is given by C_{Am} = C_{Ao} - \sum_{i=1}^{m} k_i\tau_i When the cascade of CSTRs is isothermal with identical reactors, the concentration is given by C_{Am} = C_{Ao} - mk_i\tau_i
Second order For an isothermal
second order reaction at steady state in a cascade of CSTRs, the general design equation is C_{Ai} = \frac{-1 + \sqrt{1 + 4k_i\tau_iC_{A(i-1)}}}{2k_i\tau_i}
Non-ideal cascade of CSTRs With non-ideal reactors,
residence time distributions can be calculated. At the concentration at the jth reactor in series is given by \frac{C_j}{C_o} = 1 -e^{-\frac{nt}{\bar{t}}}[1 + \frac{nt}{\bar{t}} + \frac{1}{2!}(\frac{nt}{\bar{t}})^2 + ... + \frac{1}{(j-1)!}(\frac{nt}{\bar{t}})^{j-1}] where n is the total number of CSTRs in series, and \bar{t} is the average residence time of the cascade given by \bar{t} = \frac{V}{Q} where Q is the
volumetric flow rate. From this, the cumulative residence time distribution (F(t)) can be calculated as F(t) = \frac{C_n}{C_o} = 1 -e^{-\frac{nt}{\bar{t}}}[1 + \frac{nt}{\bar{t}} + \frac{1}{2!}(\frac{nt}{\bar{t}})^2 + ... + \frac{1}{(n-1)!}(\frac{nt}{\bar{t}})^{n-1}] As n → ∞, F(t) approaches the ideal PFR response. The
variance associated with F(t) for a
pulse stimulus into a cascade of CSTRs is \sigma_{t}^2 = \frac{\bar{t}^2}{n} .
Cost When determining the cost of a series of CSTRs,
capital and
operating costs must be taken into account. As seen above, an increase in the number of CSTRs in series will decrease the total reactor volume. Since cost scales with volume, capital costs are lowered by increasing the number of CSTRs. The largest decrease in cost, and therefore volume, occurs between a single CSTR and having two CSTRs in series. When considering operating cost, operating cost scales with the number of
pumps and controls, construction, installation, and maintenance that accompany larger cascades. Therefore, as the number of CSTRs increases, the operating cost increases. Therefore, there is a minimum cost associated with a cascade of CSTRs.
Zeroth order reactions From a rearrangement of the equation given for identical isothermal CSTRs running a
zeroth order reaction: \tau = \frac{C_{Ao} - C_{Am}}{mk} , the volume of each individual CSTR will scale by \frac{1}{m}. Therefore, the total reactor volume is independent of the number of CSTRs for a zeroth order reaction. Therefore, cost is not a function of the number of reactors for a zeroth order reaction and does not decrease as the number of CSTRs increases.
Selectivity of parallel reactions When considering
parallel reactions, utilizing a cascade of CSTRs can achieve greater
selectivity for a desired product. For a given parallel reaction A -> B and A -> C with constants k_1 and k_2 and rate equations \frac{d[\ce B]}{dt}=k_1[\ce A]^{n_1} and \frac{d[\ce C]}{dt}=k_2[\ce A]^{n_2}, respectively, we can obtain a relationship between the two by dividing \frac{d[\ce B]}{dt} by \frac{d[\ce C]}{dt}. Therefore \frac{d[\ce B]}{d[\ce C]} = \frac{k_1}{k_2}[\ce A]^{n_1 - n_2} . In the case where n_1 > n_2 and B is the desired product, the cascade of CSTRs is favored with a fresh secondary feed of A in order to maximize the concentration of A . For a parallel reaction with two or more reactants such as A + D -> B and A + D -> C with constants k_1 and k_2 and rate equations \frac{d[\ce B]}{dt}=k_1[\ce A]^{n_1}[\ce D]^{m_1} and \frac{d[\ce C]}{dt}=k_2[\ce A]^{n_2}[\ce D]^{m_12}, respectively, we can obtain a relationship between the two by dividing \frac{d[\ce B]}{dt} by \frac{d[\ce C]}{dt}. Therefore \frac{d[\ce B]}{d[\ce C]} = \frac{k_1}{k_2}[\ce A]^{n_1 - n_2}[\ce D]^{m_1 - m_2} . In the case where n_1 > n_2 and m_1 > m_2 and B is the desired product, a cascade of CSTRs with an inlet stream of high [A] and [D] is favored. In the case where n_1 > n_2 and m_1 and B is the desired product, a cascade of CSTRs with a high concentration of A in the feed and small secondary streams of D is favored.
Series reactions such as A -> B -> C also have selectivity between B and C but CSTRs in general are typically not chosen when the desired product is B as the back mixing from the CSTR favors C . Typically a
batch reactor or
PFR is chosen for these reactions. == Applications ==