APPSC AEE PAPER-2: FLUID MECHANICS & HYDRAULIC MACHINERY (PART - B)
This document contains quick revision notes tailored for the APPSC Assistant Executive Engineers (AEE) examination, covering the Fluid Mechanics and Hydraulic Machinery syllabus.
1. Fluid Statics
Hydrostatic Forces on Surfaces
- Total Pressure ($F$): Force exerted by a static fluid on a surface.
- $F = \rho \cdot g \cdot A \cdot \bar{h}$ (For both horizontal and vertical/inclined planes)
- Where: $\rho$ = density, $A$ = Area, $\bar{h}$ = depth of centroid from free surface.
- Centre of Pressure ($h_p$): The point of application of the total hydrostatic force.
- $h_p = \bar{h} + \frac{I_G \cdot \sin^2\theta}{A \cdot \bar{h}}$
- Note: Centre of pressure always lies below the centroid ($h_p > \bar{h}$), except for a purely horizontal surface where $h_p = \bar{h}$.
Buoyancy and Floatation
- Metacentre ($M$): Point about which a body starts oscillating when given a small angular displacement.
- Metacentric Height ($GM$): Distance between the center of gravity ($G$) and the metacentre ($M$).
- $GM = BM - BG = \frac{I}{V} - BG$
- Stability of Floating Bodies:
- Stable Equilibrium: $M$ is above $G$ ($GM > 0$).
- Unstable Equilibrium: $M$ is below $G$ ($GM < 0$).
- Neutral Equilibrium: $M$ coincides with $G$ ($GM = 0$).
2. Fluid Dynamics
Acceleration
- Local Acceleration: Change of velocity with respect to time at a given point ($\frac{\partial V}{\partial t}$). Zero for steady flow.
- Convective Acceleration: Change of velocity with respect to space/position ($\frac{\partial V}{\partial s}$). Zero for uniform flow.
Governing Equations
- Euler's Equation of Motion: Based on the conservation of momentum along a streamline (inviscid fluid). $$ \frac{dp}{\rho} + g \cdot dz + V \cdot dV = 0 $$
- Bernoulli's Equation: Obtained by integrating Euler's equation (assumes steady, ideal, incompressible, irrotational flow).
$$ \frac{P}{\rho g} + \frac{V^2}{2g} + z = \text{Constant} $$
- $\frac{P}{\rho g}$ = Pressure head
- $\frac{V^2}{2g}$ = Kinetic / Velocity head
- $z$ = Datum / Potential head
Vortex Flow
- Forced Vortex: Fluid rotates under an external torque (e.g., stirring liquid). $V \propto r$ or $\frac{V}{r} = \omega = \text{constant}$.
- Free Vortex: Fluid rotates without external torque (e.g., whirlpool, washbasin drain). $V \propto \frac{1}{r}$ or $V \cdot r = \text{constant}$.
3. Flow Measurements
- Venturimeter: Measures rate of flow in a pipe. Converging cone, throat, diverging cone.
- Discharge $Q = C_d \frac{a_1 a_2}{\sqrt{a_1^2 - a_2^2}} \sqrt{2gh}$
- $C_d$ is high (0.96 - 0.98).
- Orifice Meter / Nozzle Meter: Cheaper alternatives to venturimeter. $C_d$ is lower (around 0.62 for orifice).
- Pitot Tube: Measures velocity of flow at a specific point by converting kinetic energy into pressure energy (stagnation point).
- $V = C_v \sqrt{2gh}$
- Notches and Weirs: Used to measure discharge in open channels.
- Rectangular Notch: $Q = \frac{2}{3} C_d \cdot L \cdot \sqrt{2g} \cdot H^{3/2}$
- Triangular (V) Notch: $Q = \frac{8}{15} C_d \cdot \tan\left(\frac{\theta}{2}\right) \cdot \sqrt{2g} \cdot H^{5/2}$
- Current Meter: Mechanical device used to measure velocity of flow in rivers and open channels.
4. Compressible Flow
Flow where density ($\rho$) changes with pressure. * Velocity of Pressure Wave (Sound Wave): $C = \sqrt{\frac{K}{\rho}}$ (where $K$ is Bulk Modulus). * Isothermal Process: $C = \sqrt{\frac{RT}{M}}$ or $\sqrt{\frac{p}{\rho}}$ * Adiabatic Process: $C = \sqrt{\frac{\gamma RT}{M}}$ or $\sqrt{\frac{\gamma p}{\rho}}$ * Mach Number ($M$): $M = \frac{V}{C}$ (Velocity of fluid / Velocity of sound) * $M < 1$: Subsonic flow * $M = 1$: Sonic flow * $M > 1$: Supersonic flow * Continuity Equation (1D Compressible): $\rho \cdot A \cdot V = \text{constant} \implies \frac{d\rho}{\rho} + \frac{dA}{A} + \frac{dV}{V} = 0$
5. Laminar and Turbulent Flow in Pipes
Reynolds Experiment
- Reynolds Number ($Re$): Ratio of inertia force to viscous force.
- $Re = \frac{\rho V D}{\mu} = \frac{V D}{\nu}$
- $Re < 2000$: Laminar Flow
- $2000 < Re < 4000$: Transition
- $Re > 4000$: Turbulent Flow
Laminar Flow (Circular Pipes)
- Velocity distribution is parabolic.
- Maximum velocity is twice the average velocity ($u_{max} = 2\bar{V}$).
- Head loss: $h_f = \frac{32 \mu \bar{V} L}{\rho g D^2}$ (Hagen-Poiseuille equation)
Turbulent Flow
- Darcy-Weisbach Equation: Universal equation for friction head loss.
$$ h_f = \frac{f \cdot L \cdot V^2}{2gD} $$
- Where: $f$ is the Darcy friction factor.
- Friction Factor ($f$):
- For laminar flow: $f = \frac{64}{Re}$
- For turbulent flow (smooth pipes up to $Re = 10^5$): $f = \frac{0.316}{Re^{1/4}}$
- Moody's Diagram: Graphical representation of the friction factor $f$ as a function of Reynolds Number ($Re$) and relative roughness ($\epsilon / D$).
6. Hydraulic Machinery: Turbines
Turbines convert hydraulic energy into mechanical energy.
Classification
- Based on Energy at Inlet:
- Impulse Turbine: Only kinetic energy at inlet (e.g., Pelton Wheel).
- Reaction Turbine: Both kinetic and pressure energy at inlet (e.g., Francis, Kaplan).
- Based on Flow Direction:
- Tangential: Pelton Wheel
- Radial: Old Francis
- Axial: Kaplan, Propeller
- Mixed: Modern Francis
Specific Speed ($N_s$)
Speed of a geometrically similar turbine producing 1 kW power under 1 m head. $$ N_s = \frac{N \sqrt{P}}{H^{5/4}} $$ (Pelton: 8.5 to 30, Francis: 50 to 250, Kaplan: 250 to 850)
Velocity Triangles & Work Done
- For Pelton Wheel: $Work = \rho Q (V_{w1} \pm V_{w2}) u$
- Condition for max efficiency (Pelton): Bucket velocity ($u$) is half of jet velocity ($V$). $\eta_{max} = \frac{1 + \cos\phi}{2}$
7. Hydraulic Machinery: Pumps
Pumps convert mechanical energy into hydraulic energy (pressure energy).
Centrifugal Pumps
- Principle: Forced vortex flow. Rotation imparts centrifugal head to the liquid.
- Work Done by Impeller: $W = \frac{W}{g} V_{w2} u_2$ (Assuming radial entry where $V_{w1} = 0$).
- Specific Speed ($N_s$): Speed of a geometrically similar pump delivering 1 m³/s discharge under 1 m head. $$ N_s = \frac{N \sqrt{Q}}{H^{3/4}} $$
Important Terms
- Minimum Starting Speed: The speed at which centrifugal head matches the static head, enabling the pump to start delivering water.
- Efficiencies:
- Manometric Efficiency ($\eta_{man}$) = $\frac{g H_m}{V_{w2} u_2}$
- Volumetric Efficiency ($\eta_v$) = $\frac{Q}{Q + \Delta Q}$
- Overall Efficiency ($\eta_o$) = $\frac{\rho g Q H_m}{1000 \cdot P_{shaft}}$
Characteristic Curves
- Main Characteristics: Head vs Discharge ($H \sim Q$), Power vs Discharge ($P \sim Q$), Efficiency vs Discharge ($\eta \sim Q$) plotted at a constant speed ($N$).
- Centrifugal pump $H-Q$ curve drops parabolically as discharge increases.