GATE Chemical Engineering Fluid Mechanics and Mechanical Operations Syllabus

What is the equation of mechanical energy and which term represents irreversible loss?

Obtained by dotting the equation of motion with velocity, it tracks kinetic + potential energy. The term $-(\tau:\nabla v)$ representing viscous dissipation $\Phi_{v}$ is always positive and converts mechanical energy irreversibly into internal energy (heat).

Define the Fanning friction factor and relate pressure drop to it.

$f = \frac{\tau_{w}}{\frac{1}{2}\rho u^{2}}$ (wall shear basis). Pressure drop: $$\Delta p = \frac{2 f \rho u^{2} L}{D}$$ The Darcy friction factor is $f_{D} = 4 f_{Fanning}$.

Give the Fanning friction factor for laminar pipe flow and the laminar-turbulent transition Reynolds number.

Laminar: $f = \frac{16}{Re}$ (so $f_{Darcy} = \frac{64}{Re}$). Transition in a pipe occurs around $Re \approx 2100$; flow is generally fully turbulent above $Re \approx 4000$.

State the Hagen-Poiseuille equation for laminar flow in a circular pipe.

$$Q = \frac{\pi \Delta p\, R^{4}}{8 \mu L} = \frac{\pi \Delta p\, D^{4}}{128 \mu L}$$ Volumetric flow rate is proportional to the fourth power of the radius.

State the Blasius correlation for the turbulent friction factor in smooth pipes and its range.

$$f = \frac{0.079}{Re^{0.25}}$$ (Fanning), valid for smooth pipes over roughly $4000 < Re < 10^{5}$.

What is dimensional analysis and what does the Buckingham $\pi$ theorem state?

Dimensional analysis reduces variables into dimensionless groups. The Buckingham $\pi$ theorem: a physical relation among $n$ variables involving $m$ fundamental dimensions can be expressed in terms of $(n-m)$ independent dimensionless $\pi$ groups.

Define geometric, kinematic, and dynamic similarity in similitude.

Geometric similarity: model and prototype have the same shape (constant length-scale ratio). Kinematic similarity: similar velocity fields (same velocity-ratio at corresponding points). Dynamic similarity: corresponding forces are in the same ratio, requiring equality of the relevant dimensionless numbers (e.g. $Re$, $Fr$).

Define the Reynolds number and state its physical meaning.

$$Re = \frac{\rho u D}{\mu} = \frac{u D}{\nu}$$ It is the ratio of inertial forces to viscous forces and governs the laminar/turbulent flow regime.

Define the Froude, Euler, and Weber dimensionless numbers.

Froude: $Fr = \frac{u}{\sqrt{gL}}$ (inertia/gravity). Euler: $Eu = \frac{\Delta p}{\rho u^{2}}$ (pressure/inertia). Weber: $We = \frac{\rho u^{2} L}{\sigma}$ (inertia/surface tension).

For pipeline systems, how are major and minor losses expressed?

Major (friction) loss: $h_{f} = \frac{2 f L u^{2}}{D g}$ (Fanning) or $\frac{f_{D} L}{D}\frac{u^{2}}{2g}$. Minor losses (fittings, bends, valves): $h_{m} = K\frac{u^{2}}{2g}$, where $K$ is the loss coefficient, or via equivalent length $L_{eq}$.

How do head losses combine for pipes in series versus pipes in parallel?

Series: the same flow rate passes through each pipe and total head loss adds: $h_{f}=\sum h_{f,i}$, $Q$ constant. Parallel: head loss across each branch is equal, and total flow is the sum: $Q=\sum Q_{i}$, $h_{f}$ same across branches.

Give the velocity profile and mean-to-maximum velocity ratio for laminar flow in a pipe.

Parabolic profile: $u(r) = u_{max}\left[1-\left(\frac{r}{R}\right)^{2}\right]$. The average velocity is half the maximum: $\frac{u_{avg}}{u_{max}} = \frac{1}{2}$.

What is the velocity profile and mean-to-maximum ratio for turbulent pipe flow?

Approximated by the $\frac{1}{7}$-power law $\frac{u}{u_{max}}=\left(\frac{y}{R}\right)^{1/7}$. The profile is much flatter than laminar, with $\frac{u_{avg}}{u_{max}} \approx 0.8$.

What is the kinetic energy correction factor $\alpha$ for laminar and turbulent flow?

$\alpha$ corrects the average kinetic energy term $\alpha\frac{u_{avg}^{2}}{2}$. For laminar (parabolic) flow $\alpha = 2$; for turbulent flow $\alpha \approx 1.0$ (commonly taken as 1).

State the working principle and equation for a Venturi meter.

A converging-diverging tube creates a pressure drop measured between inlet and throat: $$Q = C_{d} A_{2}\sqrt{\frac{2(p_{1}-p_{2})/\rho}{1-(A_{2}/A_{1})^{2}}}$$ $C_{d}$ is high (0.95-0.98) due to low losses; it measures flow rate.

Compare an orifice meter with a Venturi meter.

Orifice meter: a thin plate with a hole; cheap, simple, but high permanent pressure loss and low discharge coefficient ($C_{d}\approx 0.6$). Venturi meter: smoothly contoured; expensive but low permanent loss and high $C_{d}\approx 0.98$. Both use the same head-flow relation.

What does a Pitot tube measure and what is its governing equation?

A Pitot tube measures local (point) velocity from the difference between stagnation and static pressure: $$u = C_{v}\sqrt{\frac{2(p_{0}-p_{s})}{\rho}} = C_{v}\sqrt{2 g \Delta h}$$

What is a rotameter and what type of flow meter is it?

A rotameter is a variable-area (constant pressure-drop) flow meter: fluid flows up a tapered vertical tube, raising a float to a height where its weight is balanced by drag and buoyancy. The float position indicates the flow rate.

Classify pumps into the two major categories with examples.

Positive-displacement pumps: deliver a fixed volume per cycle (reciprocating e.g. piston/plunger; rotary e.g. gear, screw, vane). Dynamic/kinetic pumps: add energy via velocity, mainly centrifugal pumps. PD pumps suit high-head/low-flow; centrifugal suit high-flow/moderate-head.

Define NPSH and the condition to avoid cavitation in a pump.

Net Positive Suction Head is the suction-side head above the liquid's vapor pressure. To avoid cavitation: $$NPSH_{available} \geq NPSH_{required}$$ where $NPSH_{a} = \frac{p_{s}-p_{v}}{\rho g} + \frac{u_{s}^{2}}{2g}$.

State the centrifugal pump affinity laws (for speed change at constant impeller diameter).

$$\frac{Q_{2}}{Q_{1}}=\frac{N_{2}}{N_{1}},\quad \frac{H_{2}}{H_{1}}=\left(\frac{N_{2}}{N_{1}}\right)^{2},\quad \frac{P_{2}}{P_{1}}=\left(\frac{N_{2}}{N_{1}}\right)^{3}$$ Flow $\propto N$, head $\propto N^{2}$, power $\propto N^{3}$.

Give the formula for pump power and define overall efficiency.

Hydraulic (fluid) power: $P = \rho g Q H$. Brake (shaft) power $P_{shaft} = \frac{\rho g Q H}{\eta}$, where the overall efficiency $\eta = \frac{\text{fluid power}}{\text{shaft power}}$.

Compare reciprocating compressors with centrifugal/rotary compressors.

Reciprocating (positive-displacement): high pressure ratios, lower flow, pulsating discharge, used for high-pressure duties. Centrifugal/rotary (dynamic): large continuous flow, lower pressure ratio per stage, smooth delivery, used for high-volume gas handling.

Give the ideal work for isothermal versus adiabatic (isentropic) gas compression.

Isothermal: $W = RT \ln\frac{p_{2}}{p_{1}}$ (minimum work). Adiabatic/isentropic: $$W = \frac{\gamma}{\gamma-1}RT_{1}\left[\left(\frac{p_{2}}{p_{1}}\right)^{\frac{\gamma-1}{\gamma}}-1\right]$$ Isothermal requires the least work; adiabatic the most.

What is the boundary layer and the concept of boundary layer thickness $\delta$?

The boundary layer is the thin region near a surface where viscous effects are significant and velocity rises from zero at the wall to ~free-stream value. The thickness $\delta$ is conventionally defined as the distance where $u = 0.99\,U_{\infty}$.

Give the Blasius results for the laminar boundary layer on a flat plate (thickness and local skin friction).

Thickness: $\frac{\delta}{x} = \frac{5}{\sqrt{Re_{x}}}$. Local skin-friction coefficient: $C_{f,x} = \frac{0.664}{\sqrt{Re_{x}}}$, where $Re_{x} = \frac{\rho U_{\infty} x}{\mu}$.

What is the critical Reynolds number for transition of a flat-plate boundary layer?

For flow over a flat plate, transition from laminar to turbulent boundary layer occurs around $Re_{x} = \frac{\rho U_{\infty} x}{\mu} \approx 5\times 10^{5}$.

Define displacement thickness and momentum thickness of a boundary layer.

Displacement thickness: $\delta^{*} = \int_{0}^{\infty}\left(1-\frac{u}{U_{\infty}}\right)dy$ (deficit of mass flow). Momentum thickness: $\theta = \int_{0}^{\infty}\frac{u}{U_{\infty}}\left(1-\frac{u}{U_{\infty}}\right)dy$ (deficit of momentum flux).

What is boundary layer separation and what causes it?

Separation is the detachment of the boundary layer from the surface, occurring when an adverse pressure gradient ($\frac{dp}{dx}>0$) decelerates near-wall fluid until $\left.\frac{\partial u}{\partial y}\right|_{wall}=0$ and flow reverses, creating a wake and increased form drag.

Give the Darcy-Weisbach equation and relate the Darcy friction factor to the Fanning factor.

$$h_{f} = f_{D}\frac{L}{D}\frac{u^{2}}{2g}$$ where $f_{D}$ is the Darcy friction factor. The relation to Fanning is $f_{D} = 4 f_{Fanning}$, so for laminar flow $f_{D}=\frac{64}{Re}$.

State the hydraulic diameter and its use for non-circular ducts.

$$D_{h} = \frac{4 A_{c}}{P_{w}}$$ where $A_{c}$ is the cross-sectional flow area and $P_{w}$ the wetted perimeter. It replaces $D$ in $Re$ and friction-factor correlations for non-circular conduits.