List of top Questions asked in GATE CH

If a matrix \(M\) is defined as \(M=\begin{bmatrix}10 & 6 \\ 6 & 10\end{bmatrix}\), the sum of all the eigenvalues of \(M^3\) is equal to ________________ (in integer).

A measuring instrument has transfer function \[ G_m(s)=\frac{1.05}{2s+1}\,e^{-s}. \] At \(t=0\), a step of \(+1\) unit is applied. Find the time taken for the output to increase by \(1\) unit.

A design engineer needs to purchase a membrane module (M) for a plant. Two options are available: M1 and M2. The overall plant life is 7 years. Interest rate is \(8%\) per annum, compounded annually. The table gives purchase cost (lakhs of rupees) and expected life (years): \[ \begin{array}{c|cc} & \text{M1} & \text{M2}\\ \hline \text{Purchase cost (lakhs)} & 10 & 5\\ \text{Expected life (years)} & 5 & 3 \end{array} \] Find the difference in the net present value (NPV) of these two options over the 7-year horizon (lakhs of rupees).

A distillation column has a purchase cost of ₹ 10 lakhs with an installation factor \(5.8\). The installed capital is annualized over \(6\) years at \(5%\) per annum, compounded annually. Find the annual cost (in lakhs of rupees) of the installed capital.

Pumps A and B are being considered for purchase in a chemical plant. Cost details are: Pump A — installed cost \(16000\), uniform end-of-year maintenance \(2400\), salvage value \(1000\), service life \(1\) year; Pump B — installed cost \(32000\), uniform end-of-year maintenance \(1600\), salvage value \(S\) (unknown), service life \(2\) years. The interest rate is \(10%\) per annum, compounded annually. For both pumps to have the same capitalized cost, find the salvage value \(S\) of pump B (in Rs., rounded to the nearest integer).

A fluid is flowing steadily under laminar conditions over a thin rectangular plate at temperature \(T_s\) as shown. The free-stream velocity and temperature are \(u_\infty\) and \(T_\infty\), respectively. When the fluid flow is only in the \(x\)-direction, \(h_x\) is the local 64ac0493b52af67589bd410c coefficient. Similarly, when the fluid flow is only in the \(y\)-direction, \(h_y\) is the corresponding local 64ac0493b52af67589bd410c coefficient. Use the correlation \(\mathrm{Nu}=0.332\,\mathrm{Re}^{1/2}\,\mathrm{Pr}^{1/3}\) for the local 64ac0493b52af67589bd410c coefficient, where Nu, Re and Pr are the appropriate Nusselt, Reynolds and Prandtl numbers. The average 64ac0493b52af67589bd410c coefficients are defined as \(\displaystyle \overline{h}_l=\frac{1}{l}\int_0^l h_x\,dx\) and \(\displaystyle \overline{h}_w=\frac{1}{w}\int_0^w h_y\,dy\). If \(w=1\ \text{m}\) and \(l=4\ \text{m}\), the value of the ratio \(\overline{h}_w/\overline{h}_l\) is ____________ (in integer).

A perfectly insulated, concentric tube countercurrent heat exchanger is used to cool lubricating oil using water as a coolant. Oil enters the outer annulus at a mass flow rate of \(2\ \mathrm{kg\,s^{-1}}\) with a temperature of \(100^{\circ}\mathrm{C}\) and leaves at \(40^{\circ}\mathrm{C}\). Water enters the inner tube at \(1\ \mathrm{kg\,s^{-1}}\) with a temperature of \(20^{\circ}\mathrm{C}\) and leaves at \(80^{\circ}\mathrm{C}\). Specific heats: \(c_{p,\text{oil}}=2089\ \mathrm{J\,kg^{-1}K^{-1}}\), \(c_{p,\text{water}}=4178\ \mathrm{J\,kg^{-1}K^{-1}}\). No phase change. Under steady state, the number of transfer units (NTU) is ________ (in integer).

Partially saturated air at \(1\ \text{bar}\) and \(50^\circ\text{C}\) is contacted with water in an adiabatic saturator. The air is cooled and humidified to saturation, and exits at \(25^\circ\text{C}\) with an absolute humidity of \(0.02\ \text{kg water per kg dry air}\). Use latent heat of vaporization of water as \(2450\ \text{kJ}\,\text{kg}^{-1}\), and average specific heats \(c_{p,a}=1.01\ \text{kJ}\,\text{kg}^{-1}\text{K}^{-1}\) (dry air) and \(c_{p,w}=4.18\ \text{kJ}\,\text{kg}^{-1}\text{K}^{-1}\) (liquid water). If the absolute humidity of air entering the adiabatic saturator is \(\mathcal{H}\times 10^{-3}\ \text{kg water per kg dry air}\), the value of \(\mathcal{H}\) is ____________ (rounded off to two decimal places).

Distillation of a non-reactive binary mixture with components \(A\) and \(B\) is carried out in a batch still as shown. The initial charge in the still is \(1\ \text{kmol}\). The initial and final amounts of \(A\) in the still are \(0.1\ \text{kmol}\) and \(0.01\ \text{kmol}\), respectively. The relative volatility is constant at \(\alpha = 4.5\). The mole fraction of \(B\) remaining in the vessel is __________ (rounded off to three decimal places).

Fresh catalyst is loaded into a reactor for the reaction \(A \rightarrow \text{products}\). The catalyst deactivates with time. The instantaneous activity \(a(t)\) is defined as the ratio of the rate \(-r_A'(t)\) (with deactivated catalyst) to the rate with fresh catalyst. Experimental correlation: \[ -r_A'(t) = -0.5\,t + 10 \quad \text{(mol\,(g cat)\(^{-1}\)\,hr\(^{-1}\))}, \] with \(t\) in hours. The activity of the catalyst at \(t=10\ \text{hr}\) is __________ (rounded off to one decimal place).

A unimolecular, irreversible liquid-phase reaction \(A \rightarrow P\) is carried out in an ideal batch reactor at temperature \(T\). The rate of reaction \((-r_A)\) (in \(\text{mol}\,\text{m}^{-3}\,\text{s}^{-1}\)) measured at different conversions \(X_A\) is given: \[ \begin{array}{c|cccccc} X_A & 0 & 0.1 & 0.2 & 0.4 & 0.6 & 0.8\\\hline -r_A & 0.45 & 0.35 & 0.31 & 0.18 & 0.11 & 0.05 \end{array} \] The reaction is also carried out in an ideal CSTR at the same \(T\) with a feed concentration \(C_{A0}=1\ \text{mol}\,\text{m}^{-3}\) under steady state. For a conversion of \(0.8\), the space time (in s) of the CSTR is ____________ (in integer).

An irreversible liquid-phase second-order reaction \(A \xrightarrow{k} B\) with rate constant \(k=0.2\ \mathrm{L\,mol^{-1}\,min^{-1}}\) is carried out isothermally in a non-ideal reactor. A tracer test gives the residence time distribution \(E(t)\) shown: three rectangles (i), (ii), (iii) of equal areas; rectangles (i) and (ii) span \(t=5\)–\(10\ \mathrm{min}\) (stacked), and (iii) spans \(t=10\)–\(15\ \mathrm{min}\). Pure \(A\) of concentration \(C_{A0}=1.5\ \mathrm{mol\,L^{-1}}\) is fed. Using the segregated model, determine the percentage conversion of \(A\) at the exit (rounded to the nearest integer).

The outlet concentration \(C_A\) of a plug flow reactor (PFR) is controlled by manipulating the inlet concentration \(C_{A0}\). The transfer function is \[ \frac{C_A(s)}{C_{A0}(s)}=\exp\!\left[-\left(\frac{V}{F}\right)(k+s)\right], \] with \(V=1\ \text{m}^3\), \(F=0.1\ \text{m}^3\text{min}^{-1}\), \(k=0.5\ \text{min}^{-1}\). Measurement and valve transfer functions are 1. Find the ultimate gain (proportional controller gain for sustained oscillations).

The first derivative of the function \[ U(r)=4\left[\left(\frac{1}{r}\right)^{12}-\left(\frac{1}{r}\right)^{6}\right] \] evaluated at \(r=1\) is ______________ (in integer).

Wet air containing 10 mole percent water vapor is dried by continuously passing it through a column of CaCl\(_2\) pellets. The pellets remove 50 percent of water from wet air entering the column. The mole percent of water vapor in the product stream exiting the column is ____________________ (rounded off to two decimal places).

Orsat analysis of a stack gas on a dry basis is: \(\mathrm{N_2}=65%\), \(\mathrm{CO_2}=15%\), \(\mathrm{CO}=10%\), \(\mathrm{O_2}=10%\). The measured mole fraction of \(\mathrm{H_2O}\) in the stack gas (wet basis) is \(0.07\). The mole fraction of \(\mathrm{N_2}\) on a wet basis is ____________ (rounded off to two decimal places).

A pump draws water (density \(=1000\ \mathrm{kg\,m^{-3}}\)) at a steady rate of \(10\ \mathrm{kg\,s^{-1}}\). The pressures at the suction and discharge sides of the pump are \(-20\ \mathrm{kPa}\) (gauge) and \(350\ \mathrm{kPa}\) (gauge), respectively. The pipe diameters at the suction and discharge side are \(70\ \mathrm{mm}\) and \(50\ \mathrm{mm}\), respectively. The suction and discharge lines are at the same elevation, and the pump operates at an efficiency of \(80%\). Neglecting frictional losses in the system, the power (in kW) required to drive the pump is __________ (rounded off to two decimal places).

A cylindrical tank with a diameter of \(500\ \text{mm}\) contains water (density \(=1\ \text{g}\cdot\text{cm}^{-3}\)) up to a height \(h\). A \(5\ \text{mm}\) diameter round nozzle, whose center is \(1\ \text{cm}\) above the base of the tank, has its exit open to the atmosphere. The pressure above the water level in the tank is maintained at \(2\ \text{bar (abs)\). Neglect all frictional and entry/exit losses. Take \(g=10\ \text{m}\,\text{s}^{-2}\) and atmospheric pressure \(=1\ \text{bar}\). The absolute value of initial \(\dfrac{dh}{dt}\) (in \(\text{mm}\,\text{s}^{-1}\)) when \(h=51\ \text{cm}\) is ____________ (rounded off to two decimal places).}

A large tank is filled with water (density \(=1\ \mathrm{g\,cm^{-3}}\)) up to a height of \(5\ \mathrm{m}\). A \(100\ \mu\mathrm{m}\) diameter solid spherical particle (density \(=0.8\ \mathrm{g\,cm^{-3}}\)) is released at the bottom of the tank. The particle attains its terminal velocity \((v_t)\) after traveling to a certain height in the tank. Use acceleration due to gravity as \(10\ \mathrm{m\,s^{-2}}\) and water viscosity as \(10^{-3}\ \mathrm{Pa\,s}\). Neglect wall effects on the particle. If Stokes’ law is applicable, the absolute value of \(v_t\) (in \(\mathrm{mm\,s^{-1}}\)) is ________________ (rounded off to two decimal places).

Which one of the following represents the CORRECT effects of concentration polarization in a reverse osmosis process?

CO and H\(_2\) participate in a catalytic reaction. The partial pressures (in atm) of the reacting species CO and H\(_2\) in the feed stream are \(p_{CO}\) and \(p_{H_2}\), respectively. While CO undergoes molecular adsorption, H\(_2\) adsorbs via dissociative adsorption (as H atoms). The equilibrium constants (in atm\(^{-1}\)) corresponding to adsorption of CO and H\(_2\) to the catalyst sites are \(K_{CO}\) and \(K_{H_2}\), respectively. Total molar concentration of active sites per unit mass of the catalyst is \(C_t\) (in mol\(\cdot\)(g cat)\(^{-1}\)). Both the adsorption steps are at equilibrium. Which one of the following expressions is the CORRECT ratio of the concentration of catalyst sites occupied by CO to that by hydrogen atoms?

Level (\(h\)) in a steam boiler is controlled by manipulating the flow rate (\(F\)) of the make-up water using a proportional (P) controller. The transfer function is \[ \frac{h(s)}{F(s)} = \frac{0.25(1-s)}{s(2s+1)}. \] The measurement and valve transfer functions are both 1. A process engineer wants the closed-loop response to give decaying oscillations under servo mode. Which one of the following is the CORRECT value of the controller gain?

Which of the following statements is/are CORRECT?

An exhibition was held in a hall on 15 August 2022 between 3 PM and 4 PM during which any person was allowed to enter only once. Visitors who entered before 3:40 PM exited the hall exactly after 20 minutes from their time of entry. Visitors who entered at or after 3:40 PM, exited exactly at 4 PM. The probability distribution of the arrival time of any visitor is uniform between 3 PM and 4 PM. Two persons \(X\) and \(Y\) entered the exhibition hall independent of each other. Which one of the following values is the probability that their visits to the exhibition overlapped with each other?

Simpson’s one-third rule is used to estimate the definite integral \(I=\displaystyle\int_{-1}^{1}\sqrt{1-x^{2}}\,dx\) with an interval length of \(0.5\). Which one of the following is the CORRECT estimate of \(I\) obtained using this rule?