The Equation of Continuity MCQ Quiz in தமிழ் - Objective Question with Answer for The Equation of Continuity - இலவச PDF ஐப் பதிவிறக்கவும்

Concept:

Continuity Equation:

• The continuity equation states that the flow rate (Q) is constant in an incompressible fluid, i.e., A₁ × v₁ = A₂ × v₂.
• Volume flow rate Q = A × v, where A is the cross-sectional area and v is the velocity.
• Area of a pipe A = πr², where r is the radius of the pipe.

Calculation:

Given: Q = 12π L/min = 2π × 10⁻⁴ m³/s

At the point where the diameter is 2 cm, r = 1 cm = 0.01 m

A = πr² = 10⁻⁴ m²

Using the continuity equation: Q = A × v

v = Q / A = (2π × 10⁻⁴) / 10⁻⁴ = 2 m/s

∴ The velocity is 2 m/s. Option 4) is correct.

Given:

Water flows through a horizontal pipe of varying cross-section at a rate of 0.314 m³s⁻¹. The radius of the pipe at a given point is 10 cm.

Concept:

• The continuity equation for fluid flow states that the volumetric flow rate remains constant: Q = A × v.
• Here, Q is the volumetric flow rate, A is the cross-sectional area of the pipe, and v is the velocity of the fluid.
• The cross-sectional area of a pipe is given by A = πr², where r is the radius of the pipe.

Calculation:

Step-by-step calculation:

⇒ Volumetric flow rate: Q = 0.314 m³s⁻¹

⇒ Radius of the pipe: r = 10 cm = 0.1 m

⇒ Cross-sectional area: A = πr²

⇒ A = 3.14 × (0.1)² = 3.14 × 0.01 = 0.0314 m²

Using Q = A × v:

⇒ v = Q / A

⇒ v = 0.314 / 0.0314 = 10 m s⁻¹

∴ The velocity of water at the given point is 10 m s⁻¹.

The correct option is 3).

CONCEPT:

• Continuity equation: It is based on the  principle of conservation of mass, it states that mass can neither be created nor be destroyed​
  • ​The continuity equation applies to all fluids, compressible and incompressible flow, Newtonian and non-Newtonian fluids.
  • It expresses the law of conservation of mass at each point in a fluid and must therefore be satisfied at every point in a flow field.

Formula:

Q = A V

Where Q = Rate of discharge through a given tube/duct, A = Area of pipe/duct, and V = Velocity of flowing liquid 

Calculation:

Given:

d₁ = 2.5 cm, d₂ = 3.75 cm

By continuity equation,

A₁V₁ = A₂V₂

\(\frac{V_1}{V_2}=\frac{A_2}{A_1}\)

\(\frac{V_1}{V_2}=\frac{\frac{\pi}{4}d_2^2}{\frac{\pi}{4}d_1^2}\)

\(\frac{V_1}{V_2}=\frac{3.75^2}{2.5^2}\)

\(\frac{V_1}{V_2}=\frac94\)

Concept:

Incompressible fluid:

• The fluid whose density doesn't vary in any sort of flow is considered an incompressible fluid.
• Incompressible flow does not imply that the fluid itself is incompressible.
   

Example of incompressible fluid flow:

• The stream of water flowing at high speed from a garden hose pipe. Which tends to spread like a fountain when held vertically up, but tends to narrow down when held vertically down. The reason being the volume flow rate of fluid remains constant.
   

Equation of Continuity:

• According to the continuity equation, the product of the cross-sectional area of the pipe and the velocity of the fluid at any given point along the pipe is constant.​

• Equation: AV = constant, V = volume, A = area

Explanation:

From the equation of continuity,

AV = constant

Here, A = area, V = volume

So, A∝ (1/V)

• In any steady flow of an incompressible fluid, the volume flow rate of the fluid remains constant.
• When the stream of water moves up with speed, the speed decreases as it goes higher, and thus the cross-sectional area of movement of water increases to maintain the volume flow rate of water and thus spreads like a fountain.
• Similarly, when moving down, the speed of the flow of water increases, and hence the cross-sectional area of flow decreases to maintain the volume flow rate.

CONCEPT:

• Principle of Continuity: The conservation of mass is shown in a given space occupied by fluid in the principle of continuity.
  • It says that the mass discharge for steady flow in a pipe is always constant; that is, 

Q = ρVA = constant

where Q is discharge flow, A is the cross-sectional area of the pipe, ρ is the density and V is the mean velocity.

EXPLANATION:

Given that fluid is incompressible i.e. ρ = const

According to the principle of continuity,

Q = ρVA = constant

• Mass passing from one cross-section area will be the same for another cross-section area. So
• The total fluid intake in one opening will be equal to fluid output in two openings.

Q₁ = Q₂ + Q₃

\(ρV_1A_1=ρV_2A_2+ρV_3A_3\)

\(V_1A_1=V_2A_2+V_3A_3\)

\(20\times5=15 \times 5+10 \times A_3\)

A₃ = 2.5 m²

Hence the correct answer is option 1.