GATE Chemical Engineering Chemical Reaction Engineering Syllabus

What is the order of the Michaelis-Menten rate law in the limits of very low and very high substrate concentration?

For $C_S \ll K_M$: $-r_S \approx \dfrac{V_{max}}{K_M}C_S$ (first order). For $C_S \gg K_M$: $-r_S \approx V_{max}$ (zero order, enzyme saturated).

Write the Lineweaver-Burk (double-reciprocal) linearization of the Michaelis-Menten equation.

$\dfrac{1}{-r_S} = \dfrac{1}{V_{max}} + \dfrac{K_M}{V_{max}}\cdot\dfrac{1}{C_S}$; a plot of $1/(-r_S)$ vs $1/C_S$ gives intercept $1/V_{max}$ and slope $K_M/V_{max}$.

What is the mechanism underlying the Michaelis-Menten model?

$\ce{E + S <=> ES -> E + P}$: enzyme and substrate reversibly form a complex $ES$, which then breaks down irreversibly to product and free enzyme. The steady-state (or rapid-equilibrium) assumption on $ES$ yields the rate law.

Write the Monod equation for microbial growth.

$\mu = \dfrac{\mu_{max}\,C_S}{K_S + C_S}$, where $\mu$ is the specific growth rate, $\mu_{max}$ the maximum specific growth rate, $C_S$ the limiting-substrate concentration, and $K_S$ the saturation constant.

How are the Monod and Michaelis-Menten models related in form, and how do they differ in application?

They are mathematically identical hyperbolic saturation expressions. Michaelis-Menten describes a single enzyme-catalyzed reaction rate, while Monod describes whole-cell (microbial) population growth rate as a function of the limiting substrate.

Define the saturation constant $K_S$ in the Monod model.

$K_S$ is the substrate concentration at which the specific growth rate equals half of $\mu_{max}$, i.e. $\mu = \mu_{max}/2$.

How is the cell-doubling time related to the specific growth rate in exponential (balanced) growth?

$t_d = \dfrac{\ln 2}{\mu}$, where $\mu$ is the specific growth rate; during exponential growth $\dfrac{dC_X}{dt} = \mu C_X$.

Define the Residence Time Distribution (RTD) function $E(t)$ and its normalization.

$E(t)$ is the exit-age distribution: the fraction of fluid leaving with age between $t$ and $t+dt$ is $E(t)\,dt$. It is normalized so that $\displaystyle\int_0^{\infty} E(t)\,dt = 1$.

How is $E(t)$ obtained from a pulse (delta) tracer experiment?

$E(t) = \dfrac{C(t)}{\displaystyle\int_0^{\infty} C(t)\,dt}$, where $C(t)$ is the measured exit tracer concentration following an instantaneous pulse input.

Relate the step-response cumulative function $F(t)$ to $E(t)$.

$F(t) = \displaystyle\int_0^{t} E(t)\,dt$ and $E(t) = \dfrac{dF(t)}{dt}$; $F$ is the fraction of exit stream younger than age $t$.

Give the mean residence time and variance from the RTD.

Mean: $\bar{t} = \displaystyle\int_0^{\infty} t\,E(t)\,dt$. Variance: $\sigma^2 = \displaystyle\int_0^{\infty} (t-\bar{t})^2 E(t)\,dt = \int_0^{\infty} t^2 E(t)\,dt - \bar{t}^{\,2}$.

Write the RTD function $E(t)$ for an ideal CSTR (mixed flow reactor).

$E(t) = \dfrac{1}{\tau}e^{-t/\tau}$, where $\tau = V/v_0$ is the mean residence time; equivalently $E_\theta(\theta) = e^{-\theta}$ with $\theta = t/\tau$.

What is the RTD for an ideal plug flow reactor (PFR)?

A Dirac delta: $E(t) = \delta(t - \tau)$; all fluid elements have exactly the same residence time $\tau$, and the variance is zero.

What is the dimensionless variance $\sigma_\theta^2$ and its values for ideal PFR and CSTR?

$\sigma_\theta^2 = \dfrac{\sigma^2}{\bar{t}^{\,2}}$. For an ideal PFR $\sigma_\theta^2 = 0$; for an ideal CSTR $\sigma_\theta^2 = 1$.

Name the two common single-parameter models used to characterize non-ideal flow.

(1) The dispersion (axial dispersion) model, characterized by the vessel dispersion number $D/uL$; and (2) the tanks-in-series model, characterized by the number of equal-size CSTRs $N$.

In the tanks-in-series model, how is $N$ related to the dimensionless variance?

$N = \dfrac{1}{\sigma_\theta^2} = \dfrac{\bar{t}^{\,2}}{\sigma^2}$; large $N$ approaches PFR behavior, $N=1$ is a single CSTR.

Write the RTD for $N$ equal CSTRs in series (tanks-in-series model).

$E(t) = \dfrac{t^{N-1}}{(N-1)!\,\tau_i^{N}}e^{-t/\tau_i}$, where $\tau_i$ is the residence time of one tank and total $\tau = N\tau_i$.

Define the vessel dispersion number and what its limits represent.

$\dfrac{D}{uL}$ where $D$ is the axial dispersion coefficient, $u$ velocity, $L$ length. $\dfrac{D}{uL}\to 0$ means negligible dispersion (plug flow); $\dfrac{D}{uL}\to\infty$ means large dispersion (mixed flow).

For small extents of dispersion, relate dimensionless variance to the dispersion number.

$\sigma_\theta^2 = 2\left(\dfrac{D}{uL}\right)$ for the small-dispersion (closed-closed, nearly plug flow) case.

What is the Bodenstein / Peclet number in the dispersion model?

The axial Peclet number $Pe = \dfrac{uL}{D}$, the reciprocal of the vessel dispersion number; large $Pe$ corresponds to plug flow, small $Pe$ to well-mixed flow.

Write the steady-state energy balance for an adiabatic reactor relating temperature and conversion.

$T = T_0 + \dfrac{(-\Delta H_R)\,C_{A0}X_A}{\sum \theta_i C_{p_i}\,\rho}$ (per mole fed); temperature rises linearly with conversion for an adiabatic exothermic reaction.

Define the adiabatic temperature rise for complete conversion.

$\Delta T_{ad} = \dfrac{(-\Delta H_R)\,C_{A0}}{\sum\theta_i C_{p_i}}$; it is the maximum temperature change attainable when all reactant converts with no heat exchange.

What characterizes multiple steady states in a non-isothermal CSTR?

The S-shaped heat-generation curve $Q_g(T)$ can intersect the linear heat-removal line $Q_r(T)$ at up to three points; the upper and lower are stable steady states and the middle one is unstable (ignition-extinction behavior).

State the stability criterion for a steady state in a non-isothermal CSTR.

A steady state is stable when the slope of the heat-removal line exceeds the slope of the heat-generation curve at the operating point: $\dfrac{dQ_r}{dT} > \dfrac{dQ_g}{dT}$.

Distinguish exothermic and endothermic reactions in terms of optimum temperature progression.

For an irreversible reaction, use the highest allowable temperature. For a reversible exothermic reaction, decrease temperature as conversion proceeds (locus of maximum rates). For a reversible endothermic reaction, use the highest allowable temperature throughout.

Define the Thiele modulus and what it measures in catalysis.

For a first-order reaction in a sphere, $\phi = R\sqrt{\dfrac{k}{D_e}}$ (general $M_T = L\sqrt{k C_A^{n-1}/D_e}$). It is the ratio of surface (intrinsic) reaction rate to intraparticle diffusion rate.

Define the internal effectiveness factor $\eta$ and its limiting forms.

$\eta = \dfrac{\text{actual rate with diffusion}}{\text{rate if entire interior were at surface concentration}}$. When $\phi \ll 1$ (kinetics control) $\eta \to 1$; when $\phi \gg 1$ (strong diffusion limitation) $\eta \approx 1/\phi$.

For strong pore-diffusion resistance, how do the observed reaction order and activation energy compare with the true values?

The observed order becomes $\dfrac{n+1}{2}$ (so first order appears first order, second order appears 1.5 order) and the observed activation energy is about half the true value, $E_{obs}\approx \tfrac{1}{2}E_{true}$.

What is the Weisz-Prater criterion and its use?

$C_{WP} = \eta\phi^2 = \dfrac{(-r_A')_{obs}\,\rho_c R^2}{D_e C_{As}}$, computed from observable quantities. $C_{WP} \ll 1$ means no internal diffusion limitation; $C_{WP} \gg 1$ means severe internal diffusion control.

Classify the mechanisms of catalyst deactivation.

(1) Sintering/aging - loss of active surface area by thermal agglomeration; (2) Fouling/coking - deposition of carbon or material blocking sites; (3) Poisoning - chemisorption of impurities on active sites. They may be parallel, series, side-by-side, or independent with respect to the main reaction.

Write a general rate expression for catalyst deactivation in terms of an activity term.

$-r_A' = a(t)\,k\,f(C_A)$ with deactivation rate $-\dfrac{da}{dt} = k_d\,a^{\,d}\,h(C_{poison})$, where activity $a(t)=\dfrac{-r_A'(t)}{-r_A'(t=0)}$ ranges from 1 (fresh) to 0 (fully deactivated) and $d$ is the deactivation order.