List of top Questions asked in GATE CH

For the block diagram shown in the figure, the correct expression for the transfer function \( G_d = \frac{y_2(s){d(s)} \) is:} \includegraphics[width=0.7\linewidth]{q48 CE.PNG}
Consider the line integral \( \int_C \mathbf{F}(\mathbf{r}) \cdot d\mathbf{r \), with \( \mathbf{F}(\mathbf{r}) = x \mathbf{i} + y \mathbf{j} + z \mathbf{k} \), where \( \mathbf{i}, \mathbf{j}, \mathbf{k} \) are unit vectors in the \( (x, y, z) \) Cartesian coordinate system. The path \( C \) is given by \( \mathbf{r}(t) = \cos(t) \mathbf{i} + \sin(t) \mathbf{j} + t \mathbf{k} \), 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 \( y(x=2) = 17/2 \). The solution \( y(x) \) at \( x = 3/2 \), rounded off to 2 decimal places, is:
Methane combusts with air in a furnace as \( \text{CH}_4 + 2\text{O}_2 \rightarrow \text{CO}_2 + 2\text{H}_2\text{O} \). The heat of reaction \( \Delta H_\text{r} = -880 \, \text{kJ/mol CH}_4 \), and is assumed to be constant. The furnace is well-insulated, and no other side reactions occur. All components behave as ideal gases with a constant molar heat capacity \( c_p = 40 \, \text{J mol}^{-1} \, \text{°C}^{-1} \). Air may be considered as 20 mol\% \( \text{O}_2 \) and 80 mol\% \( \text{N}_2 \). The air-fuel mixture enters the furnace at \( 50 \, \text{°C} \). The methane conversion \( X \) varies with the air-to-methane mole ratio, \( r \), as: \[ X = 1 - 0.1e^{-2(r-r_s)}, \, \text{with} \, 0.9r_s \leq r \leq 1.3r_s, \] where \( r_s \) is the stoichiometric air-to-methane mole ratio. For \( r = 1.05r_s \), the exit flue gas temperature in \( \, \text{°C} \), rounded off to 1 decimal place, is:
For purchasing a batch reactor, three alternatives \( P \), \( Q \), and \( R \) have emerged, as summarized in the table below. For a compound interest rate of \( 10\% \) per annum, choose the correct option that arranges the alternatives, in order, from the least expensive to the most expensive. \begin{table}[h!] \centering \begin{tabular}{|c|c|c|c|} \hline & P & Q & R
\hline Installed Cost (lakh rupees) & 15 & 25 & 35
\hline Equipment Life (years) & 3 & 5 & 7
\hline Maintenance Cost (lakh rupees per year) & 4 & 3 & 2
\hline \end{tabular} \end{table}
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 \( (\mathbf{i} + \mathbf{j}) \), where \( \mathbf{i} \) and \( \mathbf{j} \) are unit vectors in the \( x \)- and \( y \)-directions, respectively, rounded off to 2 decimal places, is:
The Newton-Raphson method is used to solve \( f(x) = 0 \), where \( f(x) = e^x - 5x \). If the 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 \( 100 \, \text{kmol/h} \), - 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 \( \text{kmol/h} \), rounded off to 1 decimal place, is: \includegraphics[width=0.5\linewidth]{q54 CE.PNG}
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 vapor leaving the 1\textsuperscript{st, 2\textsuperscript{nd}, and 3\textsuperscript{rd} 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:}
A first-order heterogeneous reaction \( \text{A} \rightarrow \text{B} \) is carried out using a porous spherical catalyst. Assume isothermal conditions, and that intraphase diffusion controls the reaction rate. At a bulk \( \text{A} \) concentration of \( 0.3 \, \text{mol/L} \), the observed reaction rate in a \( 3 \, \text{mm} \) diameter catalyst particle is \( 0.2 \, \text{mol/s} \cdot \text{L}^{-1} \, \text{catalyst volume} \). At a bulk \( \text{A} \) concentration of \( 0.1 \, \text{mol/L} \), the observed reaction rate, in \( \text{mol/s} \cdot \text{L}^{-1} \, \text{catalyst volume} \), in a \( 6 \, \text{mm} \) diameter catalyst particle, is:
Let \( r \) and \( \theta \) be the polar coordinates defined by \( x = r \cos\theta \) and \( y = r \sin\theta \). The area of the cardioid \( r = a(1 - \cos\theta) \), \( 0 \leq \theta \leq 2\pi \), is: \includegraphics[width=0.3\linewidth]{q47 CE.PNG}
The residence time distribution, \( E \), for a non-ideal flow reactor is given in the figure. A first-order liquid phase reaction with a rate constant \( 0.2 \, \text{min}^{-1} \) is carried out in the reactor. For an inlet reactant concentration of \( 2 \, \text{mol/L} \), the reactant concentration (in \( \text{mol/L} \)) in the exit stream is: \includegraphics[width=0.5\linewidth]{q46 CE.PNG}
A first-order liquid phase reaction \( \text{A} \rightarrow \text{B} \) is carried out in two isothermal plug flow reactors (PFRs) of volume \( 1 \, \text{m}^3 \) each, connected in series. The feed flow rate and concentration of \( \text{A} \) to the first reactor are \( 10 \, \text{m}^3/\text{h} \) and \( 1 \, \text{kmol}/\text{m}^3 \), respectively. At steady-state, the concentration of \( \text{A} \) at the exit of the second reactor is \( 0.2 \, \text{kmol}/\text{m}^3 \). If the two PFRs are replaced by two equal-volume continuously stirred tank reactors (CSTRs) to achieve the same overall steady-state conversion, the volume of each CSTR, in \( \text{m}^3 \), is:
Alumina particles with an initial moisture content of \( 5 \, \text{kg} \, \text{moisture}/\text{kg dry solid} \) are dried in a batch dryer. For the first two hours, the measured drying rate is constant at \( 2 \, \text{kg} \, \text{m}^{-2} \, \text{h}^{-1} \). Thereafter, in the falling-rate period, the rate decreases linearly with the moisture content. The equilibrium moisture content is \( 0.05 \, \text{kg}/\text{kg dry solid} \), and the drying area of the particles is \( 0.5 \, \text{m}^2/\text{kg dry solid} \). The total drying time, in hours, to reduce the moisture content to half its initial value is:
A solid slab of thickness \( H_1 \) is initially at a uniform temperature \( T_0 \). At time \( t = 0 \), the temperature of the top surface at \( y = H_1 \) is increased to \( T_1 \), while the bottom surface at \( y = 0 \) is maintained at \( T_0 \) 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 \( y = H_1/2 \) to reach 99\% of its final steady value is \( \tau_1 \). If the thickness of the slab is doubled to \( H_2 = 2H_1 \), and the time required for the temperature at \( y = H_2/2 \) to reach 99\% of its final steady value is \( \tau_2 \), then \( \tau_2 / \tau_1 \) is:
The temperatures of two large parallel plates of equal emissivity are \( 900 \, \text{K} \) and \( 300 \, \text{K} \). A reflective radiation shield of low emissivity and negligible conductive resistance is placed parallelly between them. The steady-state temperature of the shield, in \( \text{K} \), is:
Consider a steady, fully-developed, uni-directional laminar flow of an incompressible Newtonian fluid (viscosity \( \mu \)) between two infinitely long horizontal plates separated by a distance \( 2H \) as shown in the figure. The flow is driven by the combined action of a pressure gradient and the motion of the bottom plate at \( y = -H \) in the negative \( x \)-direction. Given that \( \frac{\Delta P}{L} = \frac{P_1 - P_2}{L}>0 \), where \( P_1 \) and \( P_2 \) are the pressures at two \( x \)-locations separated by a distance \( L \). The bottom plate has a velocity of magnitude \( V \) with respect to the stationary top plate at \( y = H \). Which one of the following represents the \( x \)-component of the fluid velocity vector?
The opposite faces of a metal slab of thickness \( 5 \, \text{cm} \) and thermal conductivity \( 400 \, \text{W} \, \text{m}^{-1} \, ^\circ\text{C}^{-1} \) are maintained at \( 500 \, ^\circ\text{C} \) and \( 200 \, ^\circ\text{C} \). The area of each face is \( 0.02 \, \text{m}^2 \). 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 \( \text{kW} \), rounded off to the nearest integer, is \_\_\_\_\_\_\_.
The capital cost of a distillation column is Rs. 90 lakhs. The cost is to be fully depreciated (salvage value is zero) using the double-declining balance method over 10 years. At the end of two years of continuous operation, the book value of the column, in lakhs of rupees, rounded off to 1 decimal place, is:
Sulfur dioxide (SO\(_2\)) gas diffuses through a stagnant air-film of thickness \( 2 \, \text{mm} \) at \( 1 \, \text{bar} \) and \( 30^\circ \text{C} \). The diffusion coefficient of SO\(_2\) in air is \( 1 \times 10^{-5} \, \text{m}^2/\text{s} \). The SO\(_2\) partial pressures at the opposite sides of the film are \( 0.15 \, \text{bar} \) and \( 0.05 \, \text{bar} \). The universal gas constant is \( 8.314 \, \text{J}/\text{mol} \cdot \text{K} \). Assuming ideal gas behavior, the steady-state flux of SO\(_2\) in \( \text{mol}/\text{m}^2 \cdot \text{s} \) is:
A gas stream containing 95 mol\% CO\(_2\) and 5 mol\% ethanol is to be scrubbed with pure water in a counter-current, isothermal absorption column to remove ethanol. The desired composition of ethanol in the exit gas stream is 0.5 mol\%. The equilibrium mole fraction of ethanol in the gas phase, \( y^* \), is related to that in the liquid phase, \( x \), as \( y^* = 2x \). Assume CO\(_2\) is insoluble in water and neglect evaporation of water. If the water flow rate is twice the minimum, the mole fraction of ethanol in the spent water is:
Hot oil at \( 110^\circ\text{C} \) heats water from \( 30^\circ\text{C} \) to \( 70^\circ\text{C} \) in a counter-current double-pipe heat exchanger. The flow rates of water and oil are \( 50 \, \text{kg/min} \) and \( 100 \, \text{kg/min} \), respectively, and their specific heat capacities are \( 4.2 \, \text{kJ/kg}^\circ\text{C} \) and \( 2.0 \, \text{kJ/kg}^\circ\text{C} \), respectively. Assume the heat exchanger is at steady state. If the overall heat transfer coefficient is \( 200 \, \text{W/m}^2{}^\circ\text{C} \), the heat transfer area in \( \text{m}^2 \) is:
Consider a matrix \( A = \begin{bmatrix} -5 & a
-2 & -2 \end{bmatrix} \), where \( a \) is a constant. If the eigenvalues of \( A \) are \( -1 \) and \( -6 \), then the value of \( a \), rounded off to the nearest integer, is \_\_\_\_\_\_\_.
Consider the reaction \( \text{N}_2(g) + 3\text{H_2(g) \rightarrow 2\text{NH}_3(g) \) in a continuous flow reactor under steady-state conditions. The component flow rates at the reactor inlet are: \[ F_{\text{N}_2}^0 = 100 \, \text{mol/s}, \quad F_{\text{H}_2}^0 = 300 \, \text{mol/s}, \quad F_{\text{inert}}^0 = 1 \, \text{mol/s}. \] If the fractional conversion of \( \text{H}_2 \) is \( 0.60 \), the outlet flow rate of \( \text{N}_2 \), in mol/s, rounded off to the nearest integer, is \_\_\_\_\_\_\_.}
Consider the steady, uni-directional, fully-developed, pressure-driven laminar flow of an incompressible Newtonian fluid through a circular pipe of inner radius \( 5.0 \, \text{cm} \). The magnitude of shear stress at the inner wall of the pipe is \( 0.1 \, \text{N} \, \text{m}^{-2} \). At a radial distance of \( 1.0 \, \text{cm} \) from the pipe axis, the magnitude of the shear stress, in \( \text{N} \, \text{m}^{-2} \), rounded off to 3 decimal places, is \_\_\_\_\_\_\_.