Unit 2 of ME 3003 (Mechanical/Automobile Engineering)

Unit 2 of ME 3003 (Mechanical/Automobile Engineering), These are short notes for revision purpose. please refer you Reference book & College study materials for complete study.

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2. FLUID FLOW

Fluid flow refers to the movement of fluid (liquids or gases) through pipes, ducts, or open spaces. Understanding how fluids flow is critical in engineering applications, especially in fluid transport systems and machinery.

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2.1 Types of Fluid Flows

Fluid flow can be classified based on several factors:

• Steady vs. Unsteady Flow:

  • Steady Flow: When the velocity of the fluid at a given point does not change with time.
  • Unsteady Flow: When the velocity at a given point changes with time.

• Laminar vs. Turbulent Flow:

  • Laminar Flow: Fluid flows in smooth, parallel layers with minimal mixing. Occurs at low velocities and is described by the Reynolds number $\text{Re} < 2000$.
  • Turbulent Flow: Fluid moves chaotically with eddies and vortices. Occurs at high velocities with $\text{Re} > 4000$.

• Compressible vs. Incompressible Flow:

  • Incompressible Flow: The fluid density remains constant. Water and most liquids are treated as incompressible in most situations.
  • Compressible Flow: The fluid density changes significantly, such as in gases.

• Uniform vs. Non-uniform Flow:

  • Uniform Flow: The velocity at any given point in the fluid does not change with time or location.
  • Non-uniform Flow: The velocity varies from one point to another.

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2.2 Continuity Equation

The Continuity Equation is based on the principle of conservation of mass. It states that for an incompressible fluid, the mass flow rate remains constant along the flow path.

• Equation: $$A_1 V_1 = A_2 V_2$$
  • Where $A$ is the cross-sectional area of the pipe and $V$ is the velocity of the fluid.
  • The equation implies that if the area of the pipe decreases, the velocity must increase, and vice versa, to keep the flow rate constant.

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2.3 Bernoulli’s Theorem

Bernoulli’s Theorem is a principle of fluid dynamics that describes the behavior of a moving fluid. It is derived from the conservation of energy for flowing fluids.

• Statement: The total mechanical energy (pressure energy, kinetic energy, and potential energy) of the fluid remains constant along a streamline, provided the flow is steady, incompressible, and frictionless.

• Equation:

  $$P + \frac{1}{2} \rho V^2 + \rho gh = \text{constant}$$
  • Where:
    • $P$ = Pressure energy
    • $\rho$ = Density of fluid
    • $V$ = Velocity of the fluid
    • $g$ = Acceleration due to gravity
    • $h$ = Height (potential energy)
  • This equation helps determine pressure, velocity, or height at different points in the flow.

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2.4 Principle of Operation of Venturimeter

A Venturimeter is a device used to measure the flow rate of a fluid through a pipe. It consists of a pipe with a gradually narrowing section (the throat) and a wider section.

• Principle: According to Bernoulli’s principle, the velocity of the fluid increases when the cross-sectional area decreases, which leads to a drop in pressure at the throat.

• Working:

  • Measure the pressure at two points: one at the large diameter section and one at the throat (smaller diameter section).
  • The difference in pressure is related to the difference in velocity, which helps determine the flow rate.

• Flow rate $Q$ is given by:

  $$Q = A_1 V_1 = A_2 V_2$$
  • Where $A_1$ and $A_2$ are the areas at the wide and narrow sections, respectively, and $V_1$ and $V_2$ are the velocities.

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2.5 Orifice Meter

An Orifice Meter is used to measure the flow rate of a fluid by creating a pressure drop as the fluid flows through an orifice (a hole in a plate).

• Principle: When the fluid flows through the orifice, the velocity increases, and the pressure decreases according to Bernoulli’s principle. The pressure difference between the pipe before the orifice and after it is measured.

• Flow Rate Equation:

  $$Q = C_d A \sqrt{2g \Delta h}$$
  • Where:
    • $C_d$ = Coefficient of discharge
    • $A$ = Area of the orifice
    • $\Delta h$ = Height difference (pressure difference converted to head)
    • $g$ = Gravitational acceleration

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2.6 Pitot Tube

A Pitot Tube is an instrument used to measure the velocity of a fluid by measuring the difference between static and dynamic pressure.

• Principle: The Pitot tube has two ports:

  • One port faces the flow and measures the total pressure (static + dynamic pressure).
  • The other port measures the static pressure (only pressure due to fluid's motion).

• Velocity Formula:

  $$V = \sqrt{\frac{2(P_t - P_s)}{\rho}}$$
  • Where:
    • $P_t$ = Total pressure
    • $P_s$ = Static pressure
    • $\rho$ = Density of the fluid
    • $V$ = Velocity of the fluid

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2.7 Numerical Problems

• Example: A pipe carries water with a velocity of 3 m/s at a cross-sectional area of 0.02 m². Calculate the velocity at another section where the area is 0.01 m².

  Solution:

  • Using the continuity equation: $$A_1 V_1 = A_2 V_2$$
    • $A_1 = 0.02 \, \text{m}^2$, $V_1 = 3 \, \text{m/s}$, $A_2 = 0.01 \, \text{m}^2$
    • $V_2 = \frac{A_1 V_1}{A_2} = \frac{0.02 \times 3}{0.01} = 6 \, \text{m/s}$

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2.8 Minor and Major Losses in Pipes

• Major Losses: These losses are due to frictional resistance in the pipes and fittings. They are primarily caused by the roughness of the pipe and the velocity of the fluid. The Darcy-Weisbach equation is used to calculate major losses:

  $$h_f = f \cdot \frac{L}{D} \cdot \frac{V^2}{2g}$$
  • Where:
    • $h_f$ = Head loss due to friction
    • $f$ = Darcy friction factor
    • $L$ = Length of the pipe
    • $D$ = Diameter of the pipe
    • $V$ = Velocity of the fluid

• Minor Losses: These occur due to fittings, valves, bends, etc., and are expressed by:

  $$h_{\text{minor}} = K \cdot \frac{V^2}{2g}$$
  • Where $K$ is the loss coefficient for the fitting.

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2.9 Numerical Problems to Estimate Major and Minor Losses

• Example 1: A pipe has a diameter of 0.1 m and length 50 m. The fluid flows at 2 m/s with a roughness factor $f = 0.02$. Calculate the major loss.

  Solution:

  • Using the Darcy-Weisbach equation: $$h_f = 0.02 \cdot \frac{50}{0.1} \cdot \frac{2^2}{2 \times 9.81} = 0.816 \, \text{m}$$

• Example 2: For a pipe bend with $K = 0.5$, and fluid velocity $V = 3 \, \text{m/s}$, calculate the minor loss.

  Solution:

  $$h_{\text{minor}} = 0.5 \cdot \frac{3^2}{2 \times 9.81} = 0.23 \, \text{m}$$

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Summary of Key Concepts

• Fluid Flow Types: Steady, Unsteady, Laminar, Turbulent, Compressible, Incompressible.
• Continuity Equation: Relates the velocity and cross-sectional area in pipe flow.
• Bernoulli’s Theorem: Describes the conservation of energy for fluid flow.
• Venturimeter, Orifice Meter, and Pitot Tube: Instruments for measuring fluid flow rate and velocity.
• Major and Minor Losses: Losses due to friction and fittings in pipe flow.