Modeling

A continuous fluid flow containing non-conservative chemical reactant A enters an ideal CSTR of volume V.

Assumptions:

• perfect or ideal mixing
• steady state ${\displaystyle {\Bigl (}{\frac {dN_{A}}{dt}}=0{\Bigr )}}$, where N_A is the number of moles of species A
• closed boundaries
• constant fluid density (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), where k is the reaction rate constant, C_A is the concentration of species A, and n is the order of the reaction
• isothermal conditions, or constant temperature (k is constant)
• single, irreversible reaction (ν_A = −1)
• All reactant A is converted to products via chemical reaction
• N_A = C_A V

Integral mass balance on number of moles N_A of species A in a reactor of volume V:${\displaystyle 1.[{\text{Net accumulation of}}~A]=[A~{\text{in}}]-[A~{\text{out}}]+[{\text{Net generation of}}~A]}$

${\displaystyle 2.{\frac {dN_{A}}{dt}}=F_{Ao}-F_{A}+V\nu _{A}r_{A}}$^[1]

where,

• F_Ao is the molar flow rate inlet of species A
• F_A is the molar flow rate outlet of species A
• v_A is the stoichiometric coefficient
• r_A is the reaction rate

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

${\displaystyle 3.\ 0=F_{Ao}-F_{A}-Vr_{A}}$

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

${\displaystyle 4.\ 0=QC_{Ao}-QC_{A}-Vr_{A}}$^[2]

Equation 4 can then be rearranged to isolate r_A and simplified:

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

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

where,

• ${\displaystyle \tau }$ is the theoretical residence time (${\displaystyle \tau ={\tfrac {V}{Q}}}$)
• 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 the order of the reaction, the reaction rate, r_A, is generally dependent on the concentration of species A 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 the Arrhenius 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 species A 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

Residence time distribution

Exit age distribution E(t) and cumulative age distribution F(t) functions for an ideal CSTR

An ideal CSTR will exhibit well-defined flow behavior that can be characterized by the reactor's residence 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.

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.^[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.

Minimizing Volume

As the number of CSTRs in series increases, the total reactor volume decreases.

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.

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

${\displaystyle {\frac {C_{j}}{C_{o}}}=1-e^{-{\frac {nt}{\bar {t}}}}[1+{\frac {nt}{\bar {t}}}+{\frac {1}{2!}}({\frac {nt}{\bar {t}}})^{2}+...+{\frac {1}{(j-1)!}}({\frac {nt}{\bar {t}}})^{j-1}]}$

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

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

${\displaystyle F(t)={\frac {C_{n}}{C_{o}}}=1-e^{-{\frac {nt}{\bar {t}}}}[1+{\frac {nt}{\bar {t}}}+{\frac {1}{2!}}({\frac {nt}{\bar {t}}})^{2}+...+{\frac {1}{(n-1)!}}({\frac {nt}{\bar {t}}})^{n-1}]}$

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 ${\displaystyle \sigma _{t}^{2}={\frac {{\bar {t}}^{2}}{n}}}$.

Cost

Cost initially decreases with the number of CSTRs as volume decreases but as operational costs increase, the total cost eventually begins to increase.

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.

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 ${\displaystyle {\ce {A -> B}}}$ and ${\displaystyle {\ce {A -> C}}}$ with constants ${\displaystyle k_{1}}$ and ${\displaystyle k_{2}}$ and rate equations ${\displaystyle {\frac {d[{\ce {B}}]}{dt}}=k_{1}[{\ce {A}}]^{n_{1}}}$ and ${\displaystyle {\frac {d[{\ce {C}}]}{dt}}=k_{2}[{\ce {A}}]^{n_{2}}}$, respectively, we can obtain a relationship between the two by dividing ${\displaystyle {\frac {d[{\ce {B}}]}{dt}}}$ by ${\displaystyle {\frac {d[{\ce {C}}]}{dt}}}$. Therefore ${\displaystyle {\frac {d[{\ce {B}}]}{d[{\ce {C}}]}}={\frac {k_{1}}{k_{2}}}[{\ce {A}}]^{n_{1}-n_{2}}}$. 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 ${\displaystyle {\ce {A}}}$ in order to maximize the concentration of ${\displaystyle {\ce {A}}}$.

For a parallel reaction with two or more reactants such as ${\displaystyle {\ce {A + D -> B}}}$ and ${\displaystyle {\ce {A + D -> C}}}$ with constants ${\displaystyle k_{1}}$ and ${\displaystyle k_{2}}$ and rate equations ${\displaystyle {\frac {d[{\ce {B}}]}{dt}}=k_{1}[{\ce {A}}]^{n_{1}}[{\ce {D}}]^{m_{1}}}$ and ${\displaystyle {\frac {d[{\ce {C}}]}{dt}}=k_{2}[{\ce {A}}]^{n_{2}}[{\ce {D}}]^{m_{1}2}}$, respectively, we can obtain a relationship between the two by dividing ${\displaystyle {\frac {d[{\ce {B}}]}{dt}}}$ by ${\displaystyle {\frac {d[{\ce {C}}]}{dt}}}$. Therefore ${\displaystyle {\frac {d[{\ce {B}}]}{d[{\ce {C}}]}}={\frac {k_{1}}{k_{2}}}[{\ce {A}}]^{n_{1}-n_{2}}[{\ce {D}}]^{m_{1}-m_{2}}}$. In the case where ${\displaystyle n_{1}>n_{2}}$ and ${\displaystyle m_{1}>m_{2}}$ and B is the desired product, a cascade of CSTRs with an inlet stream of high ${\displaystyle {\ce {[A]}}}$ and ${\displaystyle {\ce {[D]}}}$ 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 ${\displaystyle {\ce {A}}}$ in the feed and small secondary streams of ${\displaystyle {\ce {D}}}$ is favored.^[9]

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

Environmental engineering

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

Chemical engineering

• Loop reactor for production of pharmaceuticals^[12]

• Fermentation^[12]
• Biogas production