GATE 2022 BT Q.59: solution and answer

GATE 2022 · BTQ.59 · 2 marks · NAT

An enzyme catalyzes the conversion of substrate A into product B. The rate equation for this reaction is

- r_{A} = \frac{C _{A}}{5 + C _{A}} mol L^{- 1} min^{- 1}

Substrate A at an initial concentration of 10 $mol L^{- 1}$ enters an ideal mixed flow reactor (MFR) at a flow rate of 10 $L min^{- 1}$ . The volume of the MFR required for 50% conversion of substrate to product is ________ L.

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Answer and solution

ANSWER

The MFR volume required for 50% conversion is 100 L.

Official answer: 100Source: Official answer key, GATE 2022 BT (IIT Kharagpur)

Symbols: $C_{A 0} = 10 mol L^{- 1}$ is the feed concentration of A, $v_{0} = 10 L min^{- 1}$ the volumetric flow rate, $X_{A} = 0.5$ the fractional conversion, $C_{A}$ the concentration of A in the tank (and at its exit), and $- r_{A}$ the rate of disappearance of A. The rate law is of Michaelis-Menten form $- r_{A} = \frac{V _{ma x} C _{A}}{K _{m} + C _{A}}$ with maximum rate $V_{ma x} = 1 mol L^{- 1} min^{- 1}$ and Michaelis constant $K_{m} = 5 mol L^{- 1}$ .
Exit concentration (no density change in a liquid enzyme reaction): $C_{A} = C_{A 0} (1 - X_{A}) = 10 (1 - 0.5) = 5 mol L^{- 1}$ .
An ideal MFR is perfectly mixed, so the concentration everywhere in the tank equals the exit concentration, and the whole tank reacts at the exit rate: $- r_{A} = \frac{5}{5 + 5} = 0.5 mol L^{- 1} min^{- 1}$ .
MFR design equation (steady-state mole balance on A, in = out + consumed): $v_{0} C_{A 0} = v_{0} C_{A} + (- r_{A}) V$ , so $V = \frac{v _{0} ( C _{A 0} - C _{A} )}{- r _{A}} = \frac{F _{A 0} X _{A}}{( - r _{A} ) _{exit}}$ , where $F_{A 0} = v_{0} C_{A 0} = 100 mol min^{- 1}$ is the molar feed rate of A.
$V = \frac{10 \times ( 10 - 5 )}{0.5} = \frac{50}{0.5} = 100 L$
Check: space time $τ = V / v_{0} = 100/10 = 10 min$ , and $τ (- r_{A}) = 10 \times 0.5 = 5 mol L^{- 1} = C_{A 0} - C_{A}$ , so the mole balance closes. The required volume is 100 L.

KEY IDEA

In an ideal mixed flow reactor the contents are uniform and identical to the outlet stream, so the whole volume reacts at the rate evaluated at the EXIT concentration: $V = v_{0} (C_{A 0} - C_{A}) / (- r_{A})_{exit}$ . No integration is needed, even for a Michaelis-Menten rate law.

WATCH OUT FOR

Evaluating the rate at the feed concentration ( $- r_{A} = 10/15 = 0.667$ , giving 75 L) instead of at the exit concentration, or using the integrated plug-flow/batch expression $τ = [K_{m} ln (C_{A 0} / C_{A}) + (C_{A 0} - C_{A})] / V_{ma x} = 5 ln 2 + 5 \approx 8.47 min$ (about 84.7 L), which applies to a PFR, not an MFR.

From the GATE 2022 BT paper. Every question of this paper · Download the full paper · All 2 Ideal reactors questions, with notes