List of top Questions asked in GATE Chemical Engineering- 2024

Methane combusts with air in a furnace according to the reaction \( \text{CH}_4 + 2\text{O}_2 \rightarrow \text{CO}_2 + 2\text{H}_2\text{O} \). The heat of reaction \( \Delta H_{\text{rxn}} \) is -880 kJ per mol of CH4 and is assumed constant. The furnace is well-insulated, and no side reactions occur. All components behave as ideal gases with a constant molar heat capacity of 44 J mol\(^{-1}\) °C\(^{-1}\). Air is composed of 20 mol% O2 and 80 mol% N2. The air-fuel mixture enters the furnace at 50°C. The methane conversion \( X \) varies with the air-to-methane mole ratio \( r \) as \( X = 1 - 0.1 e^{-2(r-r_s)} \) with \( 0.9 r_s \leq r \leq 1.1 r_s \), where \( r_s \) is the stoichiometric air-to-methane mole ratio. For \( r = 1.05 r_s \), the exit flue gas temperature in °C, rounded off to 1 decimal place, is _____.

A solid slab of thickness \( H1 \) is initially at a uniform temperature \( T0 \). At time \( t = 0 \), the temperature of the top surface at \( y = H1 \) is increased to \( T1 \), while the bottom surface at \( y = 0 \) is maintained at \( T0 \) for \( t \geq 0 \). Assume heat transfer occurs only in the \( y \)-direction, and all thermal properties of the slab are constant. The time required for the temperature at \( y = \frac{H1}{2} \) to reach 99% of its final steady value is \( T1 \). If the thickness of the slab is doubled to \( H2 = 2H1 \), and the time required for the temperature at \( y = \frac{H2}{2} \) to reach 99% of its final steady value is \( T2 \), then \( \frac{T2}{T1} \) is .

The temperatures of two large parallel plates of equal emissivity are 900 K and 300 K. A reflection radiation shield of low emissivity and negligible conductive resistance is placed parallelly between them. The steady-state temperature of the shield, in K, is

Consider a vapour-liquid mixture of components A and B that obeys Raoult's law. The vapour pressure of A is half that of B. The vapour phase concentrations of A and B are 3 mol m\(^{-3}\) and 6 mol m\(^{-3}\), respectively. At equilibrium, the ratio of the liquid phase concentration of A to that of B is:

A simple distillation column separates a binary mixture of A and B. The relative volatility of A with respect to B is 2. The steady-state composition of A in the vapour leaving the 1st, 2nd and 3rd trays in the rectifying section are 94, 90 and 85 mol%, respectively. For ideal trays and constant molal overflow, the reflux-to-distillate ratio is

Two parallel first-order liquid phase reactions \( A \xrightarrow{k_1} B \) and \( A \xrightarrow{k_2} C \) are carried out in a well-mixed isothermal batch reactor. The initial concentration of A in the reactor is \( 1 \) kmol m\(^{-3}\), while that of B and C is zero. After 2 hours, the concentration of A reduces to half its initial value, and the concentration of B is twice that of C. The rate constants \( k_1 \) and \( k_2 \), in h\(^{-1}\), are, respectively,

A homogeneous azeotropic distillation process separates an azeotropic AB binary feed using a heavy entrainer, E, as shown in the figure. The loss of E in the two product streams is negligible so that E circulates around the process in a closed- circuit. For a distillation column with fully specified feed(s), given operating pressure, a single distillate stream and a single bottoms stream, the steady-state degrees of freedom equals 2. For the process in the figure with a fully specified AB feed stream and given column operating pressures, the steady-state degrees of freedom equals

Alumina particles with an initial moisture content of 5 kg per kg dry solid are dried in a batch dryer. For the first two hours, the measured drying rate is constant at 2 kg m\(^{-2}\) h\(^{-1}\). Thereafter, in the falling-rate period, the rate decreases linearly with the moisture content. The equilibrium moisture content is 0.05 kg per kg dry solid, and the drying area of the particles is 0.5 m\(^2\) per kg dry solid. The total drying time, in hours, to reduce the moisture content to half its initial value is:

An isolated system consists of two perfectly sealed cuboidal compartments A and B separated by a movable rigid wall of cross-sectional area 0.1 m\(^2\) as shown in the figure. Initially, the movable wall is held in place by latches L1 and L2 such that the volume of compartment A is 0.1 m\(^3\). Compartment A contains a monoatomic ideal gas at 5 bar and 400 K. Compartment B is perfectly evacuated and contains a massless Hookean spring of force constant 0.3 N m\(^{-1}\) at its equilibrium length (stored elastic energy is zero). The latches L1 and L2 are released, the wall moves to the right by 0.2 m, where it is held at the new position by latches L3 and L4. Assume all the walls and latches are massless. The final equilibrium temperature of the gas in compartment A, in K, rounded off to 1 decimal place, is ______.

A metallic spherical particle of density 7001 kg/m³ and diameter 1 mm is settling steadily due to gravity in a stagnant gas of density 1 kg/m³ and viscosity \( 10^{-5} \) kg/(m·s). Take \( g = 9.8 \) m/s². Assume that the settling occurs in the regime where the drag coefficient \( C_D \) is independent of the Reynolds number and equals 0.44. The terminal settling velocity of the particle, in m/s, rounded off to 2 decimal places, is _____.

The opposite faces of a metal slab of thickness 5 cm and thermal conductivity 400 W m\(^{-1}\) °C\(^{-1}\) are maintained at 500 °C and 200 °C. The area of each face is 0.02 m\(^2\). Assume that the heat transfer is steady and occurs only in the direction perpendicular to the faces. The magnitude of the heat transfer rate, in kW, rounded off to the nearest integer, is ____.

The rate of a reaction \( A \rightarrow B \) is \( 0.2 \) mol m\(^{-3}\) s\(^{-1}\) at a particular concentration \( C_{A1} \). The rate constant of the reaction at a given temperature is \( 0.1 \) m\(^3\) mol\(^{-1}\) s\(^{-1}\). If the reactant concentration is increased to \( 10 C_{A1} \) at the same temperature, the reaction rate, in mol m\(^{-3}\) s\(^{-1}\), is

Consider a tray-column with a diameter of 120 cm. Each downcomer has a cross-sectional area of 575 cm². For a tray, the percentage of the column cross-sectional area not available for vapor flow due to the downcomers, rounded off to 1 decimal place, is _____.

Ethylene obeys the truncated virial equation-of-state \( \frac{PV}{RT} = 1 + \frac{BP}{RT} \) where \( P \) is the pressure, \( V \) is the molar volume, \( T \) is the absolute temperature, and \( B \) is the second virial coefficient. The universal gas constant \( R = 83.14 \) bar cm\(^3\) mol\(^{-1}\) K\(^{-1}\). At 340 K, the slope of the compressibility factor (\( Z \)) vs. pressure curve is \( -3.538 \times 10^{-3} \) bar\(^{-1}\). Let \( G_R \) denote the molar residual Gibbs free energy. At these conditions, the value of \( \left( \frac{\partial G_R}{\partial P} \right)_T \), in cm\(^3\) mol\(^{-1}\), rounded off to 1 decimal place, is _____.

A Venturi meter with a throat diameter \( d = 2 \) cm measures the flow rate in a pipe of diameter \( D = 6 \) cm, as shown in the figure. A U-tube manometer is connected to measure the pressure drop. Assume the discharge coefficient is independent of the Reynolds number and geometric ratios. If the volumetric flow rate through the pipe is doubled (\( Q_2 = 2Q_1 \)), the corresponding ratio of the manometer readings \( \frac{\Delta_{h2}}{\Delta_{h1}} \), rounded off to the nearest integer, is ____.

Heat is available at a rate of 2 kW from a thermal reservoir at 400 K. A two-stage process harnesses this heat to produce power. Stages 1 and 2 reject heat at 360 K and 300 K, respectively. Stage 2 is driven by the heat rejected by Stage 1. If the overall process efficiency is 50% of the corresponding Carnot efficiency, the power delivered by the process, in kW, rounded off to 2 decimal places, is ____.

Water of density 1000 kg/m³ is pumped at a volumetric flow rate of \( 3.14 \times 10^{-2} \) m³/s through a pipe of inner diameter 10 cm and length 100 m, from a large Reservoir 1 to another large Reservoir 2 at a height of 50 m above Reservoir 1, as shown in the figure. The flow in the pipe is in the turbulent regime with a Darcy friction factor \( f = 0.06 \) and a kinetic energy correction factor \( \alpha = 1 \). Take \( g = 9.8 \) m/s². If all minor losses are negligible and the pump efficiency is 100%, the pump power, in kW, rounded off to 2 decimal places, is _____.

A chemostat with cell recycle is shown in the figure. The feed flow rate and culture volume are \( F = 75 \) L/h and \( V = 200 \) L, respectively. The glucose concentration in the feed \( C_{S0} = 15 \) g/L. Assume Monod kinetics with specific cell growth rate \(\mu_g=\)\( \frac{1}{C_c} \frac{dC_c}{dt} = \frac{\mu_m C_S}{K_s + C_S} \),, where \( \mu_m = 0.25 \) h\(^{-1}\) and \( K_s = 1 \) g/L. Assume  maintenance, and death rates to be zero, input feed to be sterile (\( C_{S0} = 0 \)), and steady-state operation. The glucose concentration in the recycle stream, \( C_{S1} \), in g/L, rounded off to 1 decimal place, is ____.

Consider the surge drum in the figure. Initially, the system is at steady state with a hold-up \( V = 5 \) m³, which is 50% of the full tank capacity, \( V_{\text{full}} \), and volumetric flow rates \( \vec{F}_{\text{in}} = \vec{F}_{\text{out}} = 1 \) m³/h. The high hold-up alarm limit \( V_{\text{high}} = 0.8 V_{\text{full}} \) while the low hold-up alarm limit \( V_{\text{low}} = 0.2 V_{\text{full}} \). A proportional (P-only) controller manipulates the outflow to regulate the hold-up \( V \) as \( \vec{F}_{\text{out}} = K_c (V - \vec{V}) + \vec{F}_{\text{out}} \). At \( t = 0 \), \( \vec{F}_{\text{in}} \) increases as a step from 1 m³/h to 2 m³/h. Assume linear control valves and instantaneous valve dynamics. Let \( K_c^{\text{min}} \) be the minimum controller gain that ensures \( V \) never exceeds \( V_{\text{high}} \). The value of \( K_c^{\text{min}} \), in h\(^{-1}\), rounded off to 2 decimal places, is _____.

A PD controller with transfer function \( G_c \) is used to stabilize an open-loop unstable process with transfer function \( G_p \), where \( G_c = K_c \frac{(T_D s + 1)}{(\frac{T_D}{20})s + 1} \) and \( G_p = \frac{1}{(s - 1)(10s + 1)} \), and time is in minutes. From the necessary conditions for closed-loop stability, the maximum feasible value of \( T_D \), in minutes, rounded off to 1 decimal place, is _____.

The Newton-Raphson method is used to solve \( f(x) = 0 \), where \( f(x) = e^x - 5x \). Starting with an initial guess \( x^{(0)} = 1.0 \), the value of the next iterate \( x^{(1)} \), rounded off to 2 decimal places, is _____

Consider the process in the figure for manufacturing \( B \). The feed to the process is 90 mol% \( A \) and a close-boiling inert component \( I \). At a particular steady-state:
- \( B \) product rate is 100 kmol \(h^{-1}\)
- Single-pass conversion of \( A \) in the reactor is 50%
- Recycle-to-purge stream flow ratio is 10
The flow rate of \( A \) in the purge stream in kmol h^-1, rounded off to 1 decimal place, is ___________ 

Consider the function \( f(x,y,z) = x^4 + 2y^3 + z^2 \). The directional derivative of the function at the point \( P(-1,1,-1) \) along \( (\hat{i} + \hat{j}) \), where \( \hat{i} \) and \( \hat{j} \) are unit vectors in the \( x \) and \( y \) directions, respectively, rounded off to 2 decimal places, is ___________

Consider the line integral \( \int_C \mathbf{F}(\mathbf{r}) \cdot d\mathbf{r} \), where \( \mathbf{F}(\mathbf{r}) = x \hat{\mathbf{i}} + y \hat{\mathbf{j}} + z \hat{\mathbf{k}} \), with \( \hat{\mathbf{i}}, \hat{\mathbf{j}}, \) and \( \hat{\mathbf{k}} \) as unit vectors in the (x, y, z) Cartesian coordinate system. The path \( C \) is given by \( \mathbf{r}(t) = \cos(t) \hat{\mathbf{i}} + \sin(t) \hat{\mathbf{j}} + t \hat{\mathbf{k}} \), where \( 0 \leq t \leq \pi \). The value of the integral, rounded off to 2 decimal places, is _____

Consider the ordinary differential equation \[x^2 \frac{d^2y}{dx^2} - x \frac{dy}{dx} - 3y = 0,\]
with the boundary conditions \( y(x = 1) = 2 \) and \( y(x = 2) = \frac{17}{2} \). The solution \( y(x) \) at \( x = \frac{3}{2} \), rounded off to 2 decimal places, is __________.