List of top Questions asked in GATE CH

Ethylene adsorbs on the vacant active sites V of a transition metal catalyst according to the following mechanism. 
 

If \(N_T\), \(N_V\) and \(N_{C_2H_4}\) denote the total number of active sites, number of vacant active sites and number of adsorbed C\(_2\)H\(_4\) molecules, respectively, the balance on the total number of active sites is given by: 
 

For a shell-and-tube heat exchanger, the clean overall heat transfer coefficient is calculated as 250 W m$^{-2}$ K$^{-1}$ for a specific process condition. It is expected that the heat exchanger may be fouled during the operation, and a fouling resistance of 0.001 m$^{2}$ K W$^{-1}$ is prescribed. The dirt overall heat transfer coefficient is \(\underline{\hspace{2cm}}\) W m$^{-2}$ K$^{-1}$.
 

A three-dimensional velocity field is given by $V = 5x^2y\,\mathbf{i} + Cy\,\mathbf{j} - 10xyz\,\mathbf{k}$, where $\mathbf{i},\mathbf{j},\mathbf{k}$ are unit vectors in $x,y,z$ directions. If $V$ describes an incompressible fluid flow, the value of $C$ is
 

A batch settling experiment is performed in a long column using a dilute dispersion containing equal number of particles of type A and type B in water (density 1000 kg m–3) at room temperature.

Type A are spherical particles of diameter 30 μm and density 1100 kg m–3.
Type B are spherical particles of diameter 10 μm and density 1900 kg m–3.
Assuming that Stokes’ law is valid throughout the duration of the experiment, the settled bed would

The function $\cos(x)$ is approximated using the Taylor series around $x = 0$ as \( \cos(x) \approx 1 + a x + b x^2 + c x^3 + d x^4. \) The values of $a, b, c$ and $d$ are: 
 

An ordinary differential equation (ODE), \( \dfrac{dy}{dx} = 2y\), with an initial condition \(y(0) = 1\), has the analytical solution \(y = e^{2x}\). Using Runge-Kutta second order method, numerically integrate the ODE to calculate y at \(x = 0.5\) using a step size of \(h = 0.5\). 
If the relative percentage error is defined as, \[ \varepsilon = \left| \frac{y_{\text{analytical}} - y_{\text{numerical}}}{y_{\text{analytical}}} \right| \times 100, \] then the value of \(\varepsilon\) at \(x = 0.5\) is \(\underline{\hspace{2cm}}\). 
 

A system has a transfer function
\[ G(s) = \frac{3e^{-4s}}{12s^2 + 1} \] When a step change of magnitude M is given to the system input, the final value of the system output is measured to be 120. The value of M is \(\underline{\hspace{1cm}}\) .
A process has a transfer function
\[ G(s) = \frac{Y(s)}{X(s)} = \frac{20}{90000s^2 + 240s + 1} \] Initially the process is at steady state with \( x(t = 0) = 0.4 \) and \( y(t = 0) = 100 \). If a step change in \( x \) is given from 0.4 to 0.5, the maximum value of \( y \) that will be observed before it reaches the new steady state is \(\underline{\hspace{1cm}}\) (round off to 1 decimal place).
Operating labor requirements L in the chemical process industry is described in terms of the plant capacity C (kg day\(^{-1}\)) over a wide range (10\(^3\) − 10\(^6\)) by a power law relationship: \[ L = \alpha C^\beta \] where \(\alpha\) and \(\beta\) are constants. It is known that \[ L = 60 \text{ when } C = 2 \times 10^4 \text{and} L = 70 \text{ when } C = 6 \times 10^4 \] The value of L when C = \(10^5\) kg day\(^{-1}\) is \(\underline{\hspace{1cm}}\) (round off to nearest integer).
A viscous liquid is pumped through a pipe network in a chemical plant. The annual pumping cost per unit length of pipe is given by \[ C_{\text{pump}} = \frac{48.13 q^2 \mu}{D^4} \] The annual cost of the installed piping system per unit length of pipe is given by \[ C_{\text{piping}} = 45.92 D \] Here, D is the inner diameter of the pipe in meter, q is the volumetric flowrate of the liquid in m\(^3\)s\(^{-1}\) and \(\mu\) is the viscosity of the liquid in Pa.s. If the viscosity of the liquid is \(20 \times 10^{-3}\) Pa.s and the volumetric flow rate of the liquid is \(10^{-4}\) m\(^3\)s\(^{-1}\), the economic inner diameter of the pipe is \(\underline{\hspace{1cm}}\) meter (round off to 3 decimal places).
A straight fin of uniform circular cross section and adiabatic tip has an aspect ratio (length/diameter) of 4. If the Biot number (based on radius of the fin as the characteristic length) is 0.04, the fin efficiency is \(\underline{\hspace{1cm}}\) % (round off to nearest integer).
A double-effect evaporator is used to concentrate a solution. Steam is sent to the first effect at 110 °C and the boiling point of the solution in the second effect is 63.3 °C. The overall heat transfer coefficient in the first effect and second effect are 2000 W m\(^{-2}\) K\(^{-1}\) and 1500 W m\(^{-2}\) K\(^{-1}\), respectively. The heat required to raise the temperature of the feed to the boiling point can be neglected. The heat flux in the two evaporators can be assumed to be equal. The temperature at which the solution boils in the first effect is \(\underline{\hspace{1cm}}\) °C (round off to nearest integer).
Consider a solid slab of thickness 2L and uniform cross section A. The volumetric rate of heat generation within the slab is \(\dot{q}\) (W m\(^{-3}\)). The slab loses heat by convection at both the ends to air with heat transfer coefficient \(h\). Assuming steady state, one-dimensional heat transfer, the temperature profile within the slab along the thickness is given by: \[ T(x) = \frac{\dot{q}L^2}{2k} \left[1 - \left(\frac{x}{L}\right)^2 \right] + T_s \text{for} -L \leq x \leq L \] where \(k\) is the thermal conductivity of the slab and \(T_s\) is the surface temperature. If \(T_s = 350\) K, ambient air temperature \(T_\infty = 300\) K, and Biot number (based on L as the characteristic length) is 0.5, the maximum temperature in the slab is \(\underline{\hspace{1cm}}\) K (round off to nearest integer).
A distillation column handling a binary mixture of A and B is operating at total reflux. It has two ideal stages including the reboiler. The mole fraction of the more volatile component in the residue (\(x_w\)) is 0.1. The average relative volatility \(\alpha_{AB}\) is 4. The mole fraction of A in the distillate (\(x_D\)) is \(\underline{\hspace{1cm}}\) (round off to 2 decimal places).
In a batch drying experiment, a solid with a critical moisture content of 0.2 kg H\(_2\)O/kg dry solid is dried from an initial moisture content of 0.35 kg H\(_2\)O/kg dry solid to a final moisture content of 0.1 kg H\(_2\)O/kg dry solid in 5 hours. In the constant rate regime, the rate of drying is 2 kg H\(_2\)O/(m\(^2\)·h). The entire falling rate regime is assumed to be uniformly linear. The equilibrium moisture content is assumed to be zero. The mass of the dry solid per unit area is \(\underline{\hspace{1cm}}\) kg/m\(^2\) (round off to nearest integer).
Seawater is passed through a column containing a bed of resin beads. Density of seawater = 1025 kg m\(^{-3}\)
Density of resin beads = 1330 kg m\(^{-3}\)
Diameter of resin beads = 50 \(\mu\)m
Void fraction of the bed at the onset of fluidization = 0.4
Acceleration due to gravity = 9.81 m s\(^{-2}\)
The pressure drop per unit length of the bed at the onset of fluidization is \(\underline{\hspace{1cm}}\) Pa m\(^{-1}\) (round off to nearest integer).
For the ordinary differential equation \[ \frac{d^3 y}{dt^3} + 6 \frac{d^2 y}{dt^2} + 11 \frac{dy}{dt} + 6y = 1 \] with initial conditions \( y(0) = y'(0) = y''(0) = y'''(0) = 0 \), the value of \( \lim_{t \to \infty} y(t) \) is \(\underline{\hspace{1cm}}\) (rounded off to 3 decimal places).
Formaldehyde is produced by the oxidation of methane in a reactor. The following two parallel reactions occur: \[ CH_4 + O_2 \rightarrow HCHO + H_2O \] \[ CH_4 + 2O_2 \rightarrow CO_2 + 2H_2O \] Methane and oxygen are fed to the reactor. The product gases leaving the reactor include methane, oxygen, formaldehyde, carbon dioxide and water vapor.
60 mol/s of methane enters the reactor. The molar flowrate (in mol/s) of CH₄, O₂ and CO₂ leaving the reactor are 26, 2, and 4, respectively. The molar flowrate of oxygen entering the reactor is \(\underline{\hspace{1cm}}\) mol/s.
The combustion of carbon monoxide is carried out in a closed, rigid and insulated vessel. 1 mol of CO, 0.5 mol of O₂ and 2 mol of N₂ are taken initially at 1 bar and 298 K, and the combustion is carried out to completion. The standard molar internal energy change of reaction (\(\Delta u^\circ_R\)) for the combustion of carbon monoxide at 298 K is -282 kJ mol\(^{-1}\). At constant pressure, the molar heat capacities of N₂ and CO₂ are 33.314 J mol\(^{-1}\) K\(^{-1}\) and 58.314 J mol\(^{-1}\) K\(^{-1}\), respectively. Assume the heat capacities are independent of temperature, and the gases are ideal. Take R = 8.314 J mol\(^{-1}\) K\(^{-1}\). The final pressure in the vessel at the completion of the reaction is \(\underline{\hspace{1cm}}\) bar (round off to 1 decimal place).
A gaseous mixture at 1 bar and 300 K consists of 20 mol% CO₂ and 80 mol% inert gas. Assume the gases to be ideal. Take R = 8.314 J mol\(^{-1}\) K\(^{-1}\). The magnitude of minimum work required to separate 100 mol of this mixture at 1 bar and 300 K into pure CO₂ and inert gas at the same temperature and pressure is \(\underline{\hspace{1cm}}\) kJ (round off to nearest integer).
A binary liquid mixture consists of two species 1 and 2. Let \( \gamma \) and \( x \) represent the activity coefficient and the mole fraction of the species, respectively. Using a molar excess Gibbs free energy model, \( \ln \gamma_1 \) vs. \( x_1 \) and \( \ln \gamma_2 \) vs. \( x_1 \) are plotted. A tangent drawn to the \( \ln \gamma_1 \) vs. \( x_1 \) curve at a mole fraction of \( x_1 = 0.2 \) has a slope = -1.728. The slope of the tangent drawn to the \( \ln \gamma_2 \) vs. \( x_1 \) curve at the same mole fraction is \(\underline{\hspace{2cm}}\) (correct to 3 decimal places).
A fluid is sheared between parallel plates in two experiments, and the required shear stresses are given. Determine the rheological behavior of the fluid.
In a solvent regeneration process, a gas is used to strip a solute from a liquid in a countercurrent packed tower operating under isothermal condition. Pure gas is used in this stripping operation. All solutions are dilute and Henry's law, $y^ = mx$, is applicable. Here, $y^$ is the mole fraction of the solute in the gas phase in equilibrium with the liquid phase of solute mole fraction $x$, and $m$ is the Henry's law constant. Let $x_1$ be the mole fraction of the solute in the leaving liquid, and $x_2$ be the mole fraction of solute in the entering liquid. When the value of the ratio of the liquid-to-gas molar flow rates is equal to $m$, the overall liquid phase Number of Transfer Units, NTU$_{OL$, is given by}
Which of the following is NOT a necessary condition for a process under closed-loop control to be stable?
The van der Waals equation in reduced form is \[ P_r = \frac{8T_r}{3v_r - 1} - \frac{3}{v_r^2}. \] If \(v_r = 3\) and \(T_r = \frac{4}{3}\), compute the compressibility factor \(Z\) (rounded to 2 decimals).