Continuous stirred-tank reactor

Non-ideal CSTR

While the ideal CSTR model is useful for predicting the fate of constituents during a chemical or biological process, CSTRs rarely exhibit ideal behavior in reality. More commonly, the reactor hydraulics do not behave ideally or the system conditions do not obey the initial assumptions. Perfect mixing is a theoretical concept that is not achievable in practice. For engineering purposes, however, if the residence time is 5–10 times the mixing time, the perfect mixing assumption generally holds true.

Non-ideal hydraulic behavior is commonly classified by either dead space or short-circuiting. These phenomena occur when some fluid spends less time in the reactor than the theoretical residence time, ${\displaystyle \tau }$. The presence of corners or baffles in a reactor often results in some dead space where the fluid is poorly mixed. Similarly, a jet of fluid in the reactor can cause short-circuiting, in which a portion of the flow exits the reactor much quicker than the bulk fluid. If dead space or short-circuiting occur in a CSTR, the relevant chemical or biological reactions may not finish before the fluid exits the reactor. Any deviation from ideal flow will result in a residence time distribution different from the ideal distribution, as seen at right.

Modeling non-ideal flow

Although ideal flow reactors are seldom found in practice, they are useful tools for modeling non-ideal flow reactors. Any flow regime can be achieved by modeling a reactor as a combination of ideal CSTRs and plug flow reactors (PFRs) either in series or in parallel. For examples, an infinite series of ideal CSTRs is hydraulically equivalent to an ideal PFR. Reactor models combining a number of CSTRs in series are often termed tanks-in-series (TIS) models.

To model systems that do not obey the assumptions of constant temperature and a single reaction, additional dependent variables must be considered. If the system is considered to be in unsteady-state, a differential equation or a system of coupled differential equations must be solved. Deviations of the CSTR behavior can be considered by the dispersion model. CSTRs are known to be one of the systems which exhibit complex behavior such as steady-state multiplicity, limit cycles, and chaos.

Cascades of CSTRs

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 ${\displaystyle \frac{V}{F_{Ao}}}$ where V is the total reactor volume and ${\displaystyle F_{Ao}}$ 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 ${\displaystyle \tau =\frac{C_{Ao}-C_A}{-r_A}}$, we can determine that for a single CSTR in series that ${\displaystyle {\tau }_i=\frac{C_{A(i-1)}-C_{Ai}}{-r_{Ai}}}$, where ${\displaystyle \tau }$ is the space time of the reactor, ${\displaystyle C_{Ao}}$ is the feed concentration of A, ${\displaystyle C_A}$ is the outlet concentration of A, and ${\displaystyle -r_A}$ 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: ${\displaystyle C_{A1}=\frac{C_{Ao}}{1+k\tau }}$, where k is the rate constant and ${\displaystyle C_{A1}}$ is the outlet concentration of A from the first CSTR

Two CSTRs: ${\displaystyle C_{A1}=\frac{C_{Ao}}{1+k\tau }}$ and ${\displaystyle C_{A2}=\frac{C_{A1}}{1+k\tau }}$

Plugging in the first CSTR equation to the second: ${\displaystyle C_{A2}=\frac{C_{Ao}}{(1+k\tau {)}^2}}$

Therefore, for m identical CSTRs in series: ${\displaystyle C_{Am}=\frac{C_{Ao}}{(1+k\tau {)}^m}}$

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 ${\displaystyle C_{Am}=C_{Ao}-\sum _{i=1}^m{k}_i{\tau }_i}$

When the cascade of CSTRs is isothermal with identical reactors, the concentration is given by ${\displaystyle C_{Am}=C_{Ao}-m{k}_i{\tau }_i}$

Second order

For an isothermal second order reaction at steady state in a cascade of CSTRs, the general design equation is ${\displaystyle C_{Ai}=\frac{-1+\sqrt{1+4k_i{\tau }_i{C}_{A(i-1)}}}{2k_i{\tau }_i}}$

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

${\displaystyle \frac{C_j}{C_o}=1-e^{-\frac{nt}{\overline{t}}}[1+\frac{nt}{\overline{t}}+\frac{1}{2!}(\frac{nt}{\overline{t}}{)}^2+...+\frac{1}{(j-1)!}(\frac{nt}{\overline{t}}{)}^{j-1}]}$

where n is the total number of CSTRs in series, and ${\displaystyle \overline{t}}$ is the average residence time of the cascade given by ${\displaystyle \overline{t}=\frac{V}{Q}}$ where Q is the volumetric flow rate.

From this, the cumulative residence time distribution (F(t)) can be calculated as

${\displaystyle F(t)=\frac{C_n}{C_o}=1-e^{-\frac{nt}{\overline{t}}}[1+\frac{nt}{\overline{t}}+\frac{1}{2!}(\frac{nt}{\overline{t}}{)}^2+...+\frac{1}{(n-1)!}(\frac{nt}{\overline{t}}{)}^{n-1}]}$

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 ${\displaystyle {\sigma }_t^2=\frac{{\overline{t}}^2}{n}}$.

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: ${\displaystyle \tau =\frac{C_{Ao}-C_{Am}}{mk}}$, the volume of each individual CSTR will scale by ${\displaystyle \frac{1}{m}}$. 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 ${\displaystyle \text{A}⟶\text{B}}$ and ${\displaystyle \text{A}⟶\text{C}}$ with constants ${\displaystyle k_1}$ and ${\displaystyle k_2}$ and rate equations ${\displaystyle \frac{d[\text{B}]}{dt}=k_1[\text{A}{]}^{n_1}}$ and ${\displaystyle \frac{d[\text{C}]}{dt}=k_2[\text{A}{]}^{n_2}}$, respectively, we can obtain a relationship between the two by dividing ${\displaystyle \frac{d[\text{B}]}{dt}}$ by ${\displaystyle \frac{d[\text{C}]}{dt}}$. Therefore ${\displaystyle \frac{d[\text{B}]}{d[\text{C}]}=\frac{k_1}{k_2}[\text{A}{]}^{n_1-n_2}}$. In the case where ${\displaystyle n_1>n_2}$ and B is the desired product, the cascade of CSTRs is favored with a fresh secondary feed of ${\displaystyle \text{A}}$ in order to maximize the concentration of ${\displaystyle \text{A}}$.

For a parallel reaction with two or more reactants such as ${\displaystyle \text{A}+\text{D}⟶\text{B}}$ and ${\displaystyle \text{A}+\text{D}⟶\text{C}}$ with constants ${\displaystyle k_1}$ and ${\displaystyle k_2}$ and rate equations ${\displaystyle \frac{d[\text{B}]}{dt}=k_1[\text{A}{]}^{n_1}[\text{D}{]}^{m_1}}$ and ${\displaystyle \frac{d[\text{C}]}{dt}=k_2[\text{A}{]}^{n_2}[\text{D}{]}^{m_{1}2}}$, respectively, we can obtain a relationship between the two by dividing ${\displaystyle \frac{d[\text{B}]}{dt}}$ by ${\displaystyle \frac{d[\text{C}]}{dt}}$. Therefore ${\displaystyle \frac{d[\text{B}]}{d[\text{C}]}=\frac{k_1}{k_2}[\text{A}{]}^{n_1-n_2}[\text{D}{]}^{m_1-m_2}}$. In the case where ${\displaystyle n_1>n_2}$ and ${\displaystyle m_1>m_2}$ and B is the desired product, a cascade of CSTRs with an inlet stream of high ${\displaystyle [\text{A}]}$ and ${\displaystyle [\text{D}]}$ is favored. In the case where ${\displaystyle n_1>n_2}$ and ${\displaystyle m_1<m_2}$ and B is the desired product, a cascade of CSTRs with a high concentration of ${\displaystyle \text{A}}$ in the feed and small secondary streams of ${\displaystyle \text{D}}$ is favored.

Series reactions such as ${\displaystyle \text{A}⟶\text{B}⟶\text{C}}$ also have selectivity between ${\displaystyle \text{B}}$ and ${\displaystyle \text{C}}$ but CSTRs in general are typically not chosen when the desired product is ${\displaystyle \text{B}}$ as the back mixing from the CSTR favors ${\displaystyle \text{C}}$. Typically a batch reactor or PFR is chosen for these reactions.

Applications

CSTRs facilitate rapid dilution of reagents through mixing. Therefore, for non-zero-order reactions, the low concentration of reagent in the reactor means a CSTR will be less efficient at removing the reagent compared to a PFR with the same residence time. Therefore, CSTRs are typically larger than PFRs, which may be a challenge in applications where space is limited. However, one of the added benefits of dilution in CSTRs is the ability to neutralize shocks to the system. As opposed to PFRs, the performance of CSTRs is less susceptible to changes in the influent composition, which makes it ideal for a variety of industrial applications:

Environmental engineering

• Activated sludge process for wastewater treatment
• Lagoon treatment systems for natural wastewater treatment
• Anaerobic digesters for the stabilization of wastewater biosolids
• Treatment wetlands for wastewater and storm water runoff

Chemical engineering

• Loop reactor for production of pharmaceuticals
• Fermentation
• Biogas production