Ideal reactors formulas

Ideal reactors: batch, mixed flow and plug flow. 18 formulas for GATE BT Fundamentals of Biological Engineering, each with when to use it. Each is explained in the topic notes.

All 18 Ideal reactors formulas

CONVERSION (CONSTANT DENSITY)

X_{A} = \frac{C _{A 0} - C _{A}}{C _{A 0}}, C_{A} = C_{A 0} (1 - X_{A})

The fraction of fed A that has reacted. The concentration form holds only at constant density (liquid-phase systems).

SPACE TIME AND SPACE VELOCITY

τ = \frac{V}{v _{0}} = \frac{C _{A 0} V}{F _{A 0}}, s = \frac{1}{τ}

The time needed to process one reactor volume of feed, measured at feed conditions (h). The space velocity $s$ is its reciprocal (h $^{- 1}$ ).

DILUTION RATE

D = \frac{F}{V} = \frac{1}{τ}

The bioprocess name for the reciprocal space time of a continuous stirred vessel. $F$ is the volumetric feed rate (L h $^{- 1}$ , the same quantity as $v_{0}$ ) and $D$ is in h $^{- 1}$ .

CHEMOSTAT STEADY STATE (MONOD GROWTH)

S = \frac{K _{s} D}{μ _{m a x} - D}, x = Y_{X / S} (S_{0} - S), D_{crit} = \frac{μ _{m a x} S _{0}}{K _{s} + S _{0}}

Steady-state chemostat with Monod growth $μ = μ_{m a x} S / (K_{s} + S)$ , a sterile feed, no cell death or maintenance, and a constant yield $Y_{X / S} = - d x / d S$ (g cells per g substrate). $x$ is the cell concentration (g L $^{- 1}$ ), written in lower case here so that it is not confused with conversion. $S$ , $S_{0}$ and $K_{s}$ are in g L $^{- 1}$ ; $μ_{m a x}$ and $D$ are in h $^{- 1}$ . The residual substrate $S$ does not depend on $S_{0}$ , and the cell productivity is $D x$ (g L $^{- 1}$ h $^{- 1}$ ). These results hold for $D < D_{crit}$ ; at $D \geq D_{crit}$ the only steady state is washout, $x = 0$ and $S = S_{0}$ .

BATCH REACTOR

t t = N_{A 0} \int_{0}^{X_{A}} \frac{d X _{A}}{( - r _{A} ) V} = C_{A 0} \int_{0}^{X_{A}} \frac{d X _{A}}{- r _{A}} = \int_{C_{A}}^{C_{A 0}} \frac{d C _{A}}{- r _{A}} (constant volume)

$N_{A 0}$ is the moles of A charged. This is the reaction time needed to reach $X_{A}$ . It does not include the time to charge, empty or clean the vessel.

MIXED FLOW REACTOR (CSTR)

\frac{V}{F _{A 0}} = \frac{X _{A}}{( - r _{A} ) _{f}}, τ = \frac{C _{A 0} X _{A}}{( - r _{A} ) _{f}} = \frac{C _{A 0} - C _{A}}{( - r _{A} ) _{f}}

Steady state, with a feed that contains no converted A. The rate is evaluated at exit (f) conditions, which equal the reactor contents. The last form holds only at constant density.

PLUG FLOW REACTOR (PFR)

\frac{V}{F _{A 0}} = \int_{0}^{X_{A}} \frac{d X _{A}}{- r _{A}}, τ = C_{A 0} \int_{0}^{X_{A}} \frac{d X _{A}}{- r _{A}} = \int_{C_{A}}^{C_{A 0}} \frac{d C _{A}}{- r _{A}}

Steady state, with no mixing along the length and a feed that contains no converted A. The rate varies along the reactor, so it stays inside the integral. The concentration form holds only at constant density.

FIRST ORDER: BATCH AND PFR

k τ = ln \frac{C _{A 0}}{C _{A}} = - ln (1 - X_{A}), X_{A} = 1 - e^{- k τ}

For $- r_{A} = k C_{A}$ with $k$ in h $^{- 1}$ . For a batch reactor, replace $τ$ with $t$ . The dimensionless group $k τ$ is the first-order Damköhler number. The result does not depend on $C_{A 0}$ .

FIRST ORDER: CSTR

k τ = \frac{X _{A}}{1 - X _{A}}, X_{A} = \frac{k τ}{1 + k τ}, \frac{C _{A}}{C _{A 0}} = \frac{1}{1 + k τ}

For $- r_{A} = k C_{A}$ at constant density and steady state. The result does not depend on $C_{A 0}$ .

SECOND ORDER ($-R_A = KC_A^2$)

PFR or batch: k C_{A 0} τ CSTR: k C_{A 0} τ = \frac{X _{A}}{1 - X _{A}} = \frac{X _{A}}{( 1 - X _{A} ) ^{2}}

Also covers $A + B \to$ products with $- r_{A} = k C_{A} C_{B}$ and equimolar feed ( $C_{A 0} = C_{B 0}$ ), because the 1 : 1 stoichiometry then keeps $C_{B} = C_{A}$ . $k$ is in L mol $^{- 1}$ h $^{- 1}$ .

MICHAELIS–MENTEN RATE

- r_{S} = \frac{V _{m a x} S}{K _{m} + S}

$V_{m a x} = k_{c a t} E_{0}$ stays constant only if the enzyme concentration is constant and the enzyme does not deactivate.

MICHAELIS–MENTEN: BATCH AND PFR

V_{m a x} τ = (S_{0} - S) + K_{m} ln \frac{S _{0}}{S} = S_{0} X + K_{m} ln \frac{1}{1 - X}

The integrated Michaelis–Menten equation. For a batch reactor, replace $τ$ with $t$ . The first term is the zero-order part and the second is the first-order part.

MICHAELIS–MENTEN: CSTR

τ = \frac{( S _{0} - S ) ( K _{m} + S )}{V _{m a x} S}

Steady state. The rate is evaluated at the exit substrate concentration $S$ .

CSTR-TO-PFR VOLUME RATIO

First order: \frac{V _{CSTR}}{V _{PFR}} Second order: \frac{V _{CSTR}}{V _{PFR}} = \frac{X _{A} / ( 1 - X _{A} )}{- ln ( 1 - X _{A} )} = \frac{1}{1 - X _{A}}

Same feed and same conversion, constant density. Neither ratio depends on $k$ , and the second-order ratio does not depend on $C_{A 0}$ either.

N EQUAL CSTRS IN SERIES, FIRST ORDER

\frac{C _{A N}}{C _{A 0}} = \frac{1}{( 1 + k τ _{i} ) ^{N}}, τ_{i} = \frac{τ}{N}

First order, constant density, with $N$ equal tanks each of space time $τ_{i}$ . The total space time is $τ = N τ_{i}$ . For unequal tanks, multiply the stage factors: $C_{A N} / C_{A 0} = 1/ [(1 + k τ_{1}) (1 + k τ_{2}) \dots (1 + k τ_{N})]$ .

SIZE OF EACH TANK FOR A TARGET CONVERSION

k τ_{i} = (1 - X_{A})^{- 1/ N} - 1

First order, with $N$ equal CSTRs in series reaching an overall conversion $X_{A}$ .

PFR WITH RECYCLE (GENERAL)

τ = \frac{C _{A 0} V}{F _{A 0}} X_{A 1} = (R + 1) C_{A 0} \int_{X_{A 1}}^{X_{A f}} \frac{d X _{A}}{- r _{A}} = \frac{R}{R + 1} X_{A f}

$τ$ is based on the fresh feed $v_{0}$ , which contains no converted A. $X_{A 1}$ is the conversion at the reactor inlet after mixing with the recycle stream.

PFR WITH RECYCLE, FIRST ORDER

\frac{k τ}{R + 1} = ln [\frac{C _{A 0} + R C _{A f}}{( R + 1 ) C _{A f}}]

First order, constant density, with $τ = V / v_{0}$ based on the fresh feed. $R = 0$ gives the PFR result, and $R \to \infty$ gives $k τ = X_{A} / (1 - X_{A})$ .

Where this was asked: 2022 (1), 2026 (1). See the 2 previous-year questions on Ideal reactors. This shows what was asked, not what will be asked.

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Ideal reactors: the notes and 2 solved previous-year questions