List of top Fluid Mechanics and Mechanical Operations Questions

An electrical wire of 2 mm diameter and 5 m length is insulated with a plastic layer of thickness 2 mm and thermal conductivity \( k = 0.1 \) W/(m·K). It is exposed to ambient air at 30°C. For a current of 5 A, the potential drop across the wire is 2 V. The air-side heat transfer coefficient is 20 W/(m²·K). Neglecting the thermal resistance of the wire, the steady-state temperature at the wire-insulation interface __________°C (rounded off to 1 decimal place).

GIVEN: 
Kinematic viscosity: \( \nu = 1.0 \times 10^{-6} \, {m}^2/{s} \) 
Prandtl number: \( {Pr} = 7.01 \) 
Velocity boundary layer thickness: \[ \delta_H = \frac{4.91 x}{\sqrt{x \nu}} \]

Water flowing at 70 kg/min is heated from 25°C to 65°C in a counter-flow double-pipe heat exchanger using hot oil. The oil enters at 110°C and exits at 65°C. If the overall heat transfer coefficient is 300 W/(m²·K), the heat exchanger area is ___________ m² (rounded off to 1 decimal place).

Consider the flowsheet in the figure for manufacturing C via the reaction \( A + B \rightarrow C \) in an isothermal CSTR. The split in the separator is perfect so that the recycle stream is free of C and the product stream is pure C. Let \( x_i \) denote the mole fraction of species \( i \) (where \( i = A, B, C \)) in the CSTR, which is operated in excess B with \( x_B/x_A = 4 \). The reaction is first-order in A with the reaction rate \( (-r_A) = k x_A \), where \( k_x = 5.0 \, {kmol}/({m}^3 \cdot {h}) \). The reactor volume \( V \) in \( {m}^3 \) is to be optimized to minimize the cost objective \( J = V + 0.25 R \), where \( R \) is the recycle rate in \( {kmol/h} \). For a product rate \( P = 100 \, {kmol/h} \), the optimum value of \( V \) is ____ \( {m}^3 \) (rounded off to the nearest integer). Given: \[ \frac{d}{dz} \left( \frac{z}{(1-z)^2} \right) = \frac{1}{(1 - 2z)^2} \]

It is proposed to install thermal insulation in a residence to save on the summer-monsoon season air-conditioning costs. The estimated yearly saving is 20 thousand rupees. The cost of installation of the insulation is 150 thousand rupees. The life of the insulation is 12 years. For a compound interest rate of 9% per annum, the minimum salvage value of the insulation for which the proposal is competitive _________ thousand rupees (rounded off to nearest integer).

The catalytic gas-phase reaction \( A \rightarrow {products} \) is carried out in an isothermal batch reactor of 10 L volume using 0.1 kg of a solid catalyst. The reaction is first-order with: \[ (-r_A) = k' a(t) C_A \] Where: - \( k' = \frac{1}{{kg catalyst} \cdot {h}} \) - \( C_A \) is the concentration of \( A \) in mol/L. The catalyst activity \( a(t) \) undergoes first-order decay with rate constant \( k_d = 0.01 \) per hour and \( a(0) = 1 \). The reactant conversion after 1 day of operation is __________ (rounded off to 2 decimal places).

The reaction \( A \rightarrow {products} \) with reaction rate, \( (-r_A) = k C_A^3 \), occurs in an isothermal PFR operating at steady state. The conversion (X) at two axial locations (1 and 2) of the PFR is shown in the figure. 


The value of \( l_1/l_2 \) is ________ (rounded off to 2 decimal places).

A leaf filter is operated at 1 atm (gauge). The volume of filtrate collected \(V\) (in \(m^3\)) is related with the volumetric flow rate of the filtrate \(q\) (in \(m^3/s\)) as: \[ \frac{1}{q} = \frac{1}{\frac{dV}{dt}} = 50V + 100 \] The volumetric flow rate of the filtrate at 1 hour is ___________ \( \times 10^{-3} \, m^3/s\) (rounded off to 2 decimal places).

For a steady-state, fully developed laminar flow of a Newtonian fluid through a cylindrical pipe at a constant volumetric flow rate, which of the following statements regarding the pressure drop across the pipe (\(\Delta P\)) is/are TRUE?

A catalyst particle is modeled as a symmetrical double cone solid as shown in the figure. For each conical sub-part, the radius of the base is \( r \) and the height is \( h \). The sphericity of the particle is given by:


 

The first-order irreversible liquid phase reaction \(A \to B\) occurs inside a constant volume \(V\) isothermal CSTR with the initial steady-state conditions shown in the figure. The gain, in kmol/m³·h, of the transfer function relating the reactor effluent \(A\) concentration \(c_A\) to the inlet flow rate \(F\) is:


 

The vortex shedding meter is primarily used for measuring
Schmidt number is defined as
Choose the CORRECT option for pathlines, streaklines and streamlines for a STEADY flow field.
The sum of the components of the force due to pressure and shear at the solid-fluid boundary of a solid body in the direction normal to the flow is:
In industrial heat exchanger design, the overall heat transfer coefficient \( U \) is estimated from the equation: \[ \frac{1}{U} = \frac{1}{h_i} + \frac{1}{h_o} \] where \( h_i \) and \( h_o \) are the convective heat transfer coefficients on the inner and outer side of the tube, respectively. This is valid for (i) tube of (ii) thermal conductivity.
Which one of the following is the CORRECT option to fill in the gaps (i) and (ii)?
An adiabatic pump of efficiency 40% is used to increase the water pressure from 200 kPa to 600 kPa. The flow rate of water is 600 L/min. The specific heat of water is 4.2 kJ/(kg°C). Assuming water is incompressible with a density of 1000 kg/m³, the maximum temperature rise of water across the pump is \_\_\_\_°C (rounded off to 3 decimal places).
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 _____.
large Reservoir
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 _____.
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 ____.
Venturi meter
Sulfur dioxide (SO₂) gas diffuses through a stagnant air-film of thickness 2 mm at 1 bar and 30 °C. The diffusion coefficient of SO₂ in air is 1 × 10⁻⁵ m² s⁻¹. The SO₂ partial pressures at the opposite sides of the film are 0.15 bar and 0.05 bar. The universal gas constant is 8.314 J mol⁻¹ K⁻¹. Assuming ideal gas behavior, the steady-state flux of SO₂ in mol m⁻² s⁻¹ through the air-film 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 cm. The magnitude of shear stress at the inner wall of the pipe is 0.1 N m\(^{-2}\). At a radial distance of 1.0 cm from the pipe axis, the magnitude of the shear stress, in N m\(^{-2}\), rounded off to 3 decimal places, 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 \(\Delta P/L = (P_1 - P_2)/L > 0\), 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?
fully-developed, uni-directional laminar flow of an incompressible Newtonian fluid
An infinitely long cylindrical water filament of radius \( R \) is surrounded by air. Assume water and air to be static. The pressure outside the filament is \( P_{out} \) and the pressure inside is \( P_{in} \). If \( \gamma \) is the surface tension of the water-air interface, then \( P_{in} - P_{out} \) is:
The velocity field in an incompressible flow is \( \mathbf{v} = axy\hat{i} + v_y\hat{j} + \beta\hat{k} \), where \( \hat{i} \), \( \hat{j} \), and \( \hat{k} \) are unit vectors in the \( (x, y, z) \) Cartesian coordinate system. Given that \( a \) and \( \beta \) are constants, and \( v_y = 0 \) at \( y = 0 \), the correct expression for \( v_y \) is:
A soil sample was consolidated at a cell pressure of 10 K Pa for 24 hours during a cu test. The cell pressure increases to 30. Now it results in a core water pressure increment of 1 K Pa. Soil failed when the agsist test gradually increased to 50 K Pa. Core water pressure at failure was recorded as 21 K Pa. What is the scamton core water pressure B?