Thecontinuous stirred-tank reactor (CSTR), also known asvat- orbackmix reactor,mixed flow reactor (MFR), or acontinuous-flow stirred-tank reactor (CFSTR), is a common model for achemical reactor inchemical engineering andenvironmental engineering. A CSTR often refers to a model used to estimate the key unit operation variables when using a continuous agitated-tank reactor to reach a specified output. The mathematical model works for all fluids: liquids, gases, andslurries.

The behavior of a CSTR is often approximated or modeled by that of an ideal CSTR, which assumesperfect mixing. In a perfectly mixed reactor, reagent is instantaneously and uniformly mixed throughout the reactor upon entry. Consequently, the output composition is identical to composition of the material inside the reactor, which is a function of residence time and reaction rate. The CSTR is the ideal limit of complete mixing in reactor design, which is the complete opposite of aplug flow reactor (PFR). In practice, no reactors behave ideally but instead fall somewhere in between the mixing limits of an ideal CSTR and PFR.

A continuous fluid flow containing non-conservative chemical reactantA enters an ideal CSTR of volumeV.

• perfect or ideal mixing
• steady state${\displaystyle (\frac{d{N}_A}{dt}=0)}$, whereN_A is the number of moles of speciesA
• closed boundaries
• constant fluiddensity (valid for most liquids; valid for gases only if there is no net change in the number of moles or drastic temperature change)
• n^th-order reaction (r =kC_Aⁿ), wherek is the reaction rate constant,C_A is the concentration of speciesA, andn is the order of the reaction
• isothermal conditions, or constant temperature (k is constant)
• single, irreversiblereaction (ν_A = −1)
• All reactantA is converted to products via chemical reaction
• N_A =C_AV

Integral mass balance on number of molesN_A of speciesA in a reactor of volumeV:

1.${\displaystyle [\text{Net accumulation of}A]=[A\text{in}]-[A\text{out}]+[\text{Net generation of}A]}$

2.${\displaystyle \frac{d{N}_A}{dt}=F_{Ao}-F_A+V{\nu }_A{r}_A}$^[1]

where

• F_Ao is the molar flow rate inlet of speciesA
• F_A is the molar flow rate outlet of speciesA
• v_A is thestoichiometric coefficient
• r_A is the reaction rate

Applying the assumptions of steady state andν_A = −1, Equation 2 simplifies to:

3.${\displaystyle 0=F_{Ao}-F_A-V{r}_A}$

The molar flow rates of speciesA can then be rewritten in terms of the concentration ofA and the fluid flow rate (Q):

4.${\displaystyle 0=Q{C}_{Ao}-Q{C}_A-V{r}_A}$^[2]

Equation 4 can then be rearranged to isolater_A and simplified:

5.${\displaystyle r_A=\frac{Q}{V}(C_{Ao}-C_A)}$^[2]

6.${\displaystyle r_A=\frac{1}{\tau }(C_{Ao}-C_A)}$

where

• ${\displaystyle \tau }$ is the theoretical residence time (${\displaystyle \tau =\frac{V}{Q}}$)
• C_Ao is the inlet concentration of species A
• C_A is the reactor/outlet concentration of species A

Residence time is the total amount of time a discrete quantity of reagent spends inside the reactor. For an ideal reactor, the theoretical residence time,${\displaystyle \tau }$, is always equal to the reactor volume divided by the fluid flow rate.^[2] See the next section for a more in-depth discussion on the residence time distribution of a CSTR.

Depending on theorder of the reaction, the reaction rate,r_A, is generally dependent on the concentration of speciesA in the reactor and the rate constant. A key assumption when modeling a CSTR is that any reactant in the fluid is perfectly (i.e. uniformly) mixed in the reactor, implying that the concentration within the reactor is the same in the outlet stream.^[3] The rate constant can be determined using a known empirical reaction rate that is adjusted for temperature using theArrhenius temperature dependence.^[2] Generally, as the temperature increases so does the rate at which the reaction occurs.

Equation 6 can be solved by integration after substituting the proper rate expression. The table below summarizes the outlet concentration of speciesA for an ideal CSTR. The values of the outlet concentration and residence time are major design criteria in the design of CSTRs for industrial applications.

Outlet Concentration for an Ideal CSTR

Reaction Order	CA
n=0	${\displaystyle C_A=C_{Ao}-k\tau }$
n=1	${\displaystyle C_A=\frac{C_{Ao}}{1+k\tau }}$[1]
n=2	${\displaystyle C_A=\frac{-1+\sqrt{1+4k\tau C_{Ao}}}{2k\tau }}$
Other n	Numerical solution required

An ideal CSTR will exhibit well-defined flow behavior that can be characterized by the reactor'sresidence time distribution, or exit age distribution.^[4] Not all fluid particles will spend the same amount of time within the reactor. The exit age distribution (E(t)) defines the probability that a given fluid particle will spend time t in the reactor. Similarly, the cumulative age distribution (F(t)) gives the probability that a given fluid particle has an exit age less than time t.^[3] One of the key takeaways from the exit age distribution is that a very small number of fluid particles will never exit the CSTR.^[5] Depending on the application of the reactor, this may either be an asset or a drawback.

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.^[2] 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.^[6] 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.^[6] 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.^[2] Any deviation from ideal flow will result in a residence time distribution different from the ideal distribution, as seen at right.

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 andplug flow reactors (PFRs) either in series or in parallel.^[6] For examples, an infinite series of ideal CSTRs is hydraulically equivalent to an ideal PFR.^[2] Reactor models combining a number of CSTRs in series are often termed tanks-in-series (TIS) models.^[7]

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, also known as a series of CSTRs, are used to decrease the volume of a system.^[8]

As seen in the graph with one CSTR, where the inverse rate is plotted as a function offractional 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 afirst 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 idealPFR for the same reaction and fractional conversion.

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 thespace 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 therate of reaction of A.

For anisothermal first order, constant density reaction in a cascade of identical CSTRs operating atsteady state

For one CSTR:${\displaystyle C_{A1}=\frac{C_{Ao}}{1+k\tau }}$, where k is therate 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.

At steady state, the general equation for an isothermalzeroth 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${\displaystyle C_{Am}=C_{Ao}-m{k}_i{\tau }_i}$

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

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 thevolumetric 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. Thevariance associated with F(t) for apulse stimulus into a cascade of CSTRs is${\displaystyle {\sigma }_t^2=\frac{{\overline{t}}^2}{n}}$.

When determining the cost of a series of CSTRs,capital andoperating 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 ofpumps 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.

From a rearrangement of the equation given for identical isothermal CSTRs running azeroth 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.

When consideringparallel reactions, utilizing a cascade of CSTRs can achieve greaterselectivity 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 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.^[9]

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 abatch reactor orPFR is chosen for these reactions.

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.^[3] 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:

• Activated sludge process forwastewater treatment^[2]
• Lagoon treatment systems for natural wastewater treatment^[2]
• Anaerobic digesters for the stabilization of wastewater biosolids^[10]
• Treatment wetlands for wastewater andstorm water runoff^[11]

• Loop reactor for production of pharmaceuticals^[12]

• Fermentation^[12]
• Biogas production

• Laminar flow reactor
• Microreactor
• Oscillatory baffled reactor
• Plug flow reactor model

• Chemical reactors

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