All questions of Fluid Dynamics for Civil Engineering (CE) Exam

Applying Bernoulli’s equation between point ① and ②

True: Venturimeter is based on Bernoulli’s equation.

Consider the cross-section of wing

At section 1-1,

Consider relative motion of water with respect to submarines.

Consider the horizontal streamline along the centerline of flow passage with two points shown. Point 1 is a far away point from where the fluid particle starts moving and point 2 is the gauge point in the passage where the manometer is installed.

The manometer will read the pressure of the air in the wind tunnel because of its taping position. So equating pressure at the same level of water (in manometer)

Considering steady state conditions, any fluid particle shall move along a streamline from 1 to 2, the streamline passing through the turbine. applying Bernoulli’s equation between 1 and 2

The pipe is of uniform cross sectional area. Therefore, the flow velocities at section 1-1 and section 2-2 are the same.

Conservation Laws in Fluid Mechanics

Fluid mechanics is the study of how fluids (liquids and gases) behave under different conditions. It is governed by several fundamental principles, known as conservation laws, which describe the behavior of fluids in various physical processes. These conservation laws include the conservation of mass, momentum, and energy.

The Equation of Conservation of Mass

The conservation of mass, also known as the continuity equation, states that the mass of a fluid remains constant within a control volume, unless there are sources or sinks of mass. Mathematically, it can be expressed as:

ρ * A * v = constant

where ρ is the density of the fluid, A is the cross-sectional area of flow, and v is the velocity of the fluid. The continuity equation ensures that mass is conserved in any fluid flow, and it is applicable to both incompressible and compressible flows.

The Equation of Conservation of Momentum

The conservation of momentum states that the total momentum of a fluid remains constant within a control volume, unless there are external forces acting on it. Mathematically, it can be expressed as:

ρ * A * v * V + ∫∫∫ ρ * g * dV = constant

where V is the velocity vector of the fluid, ρ is the density of the fluid, A is the cross-sectional area of flow, g is the acceleration due to gravity, and the integral term represents the external forces acting on the fluid. The conservation of momentum ensures that the motion of fluid particles is governed by the forces acting on them.

The Equation of Conservation of Energy

The conservation of energy states that the total energy of a fluid remains constant within a control volume, unless there are energy transfers or conversions. Mathematically, it can be expressed as:

ρ * A * v * (E + 0.5 * V^2) + ∫∫∫ ρ * g * V * dV = constant

where E is the internal energy per unit mass of the fluid and the other terms have the same meanings as in the conservation of momentum equation. The conservation of energy ensures that the total energy of the fluid, including its internal and kinetic energy, is conserved in any physical process.

Conclusion

Among the three conservation equations, the conservation of energy (option A) is the primary equation that must be satisfied for a flow to be physically possible. This is because energy is a fundamental property of a fluid and any flow must obey the principle of energy conservation. The conservation of mass and momentum are also important in fluid mechanics, but they are derived from the conservation of energy equation and are secondary to it. Therefore, the correct answer is option A, the equation of conservation of energy.

The coefficient of discharge (Cd) is a dimensionless factor that represents the efficiency of an orifice meter in measuring the flow rate of a fluid. It is defined as the ratio of the actual flow rate to the theoretical flow rate. The Cd value depends on various factors such as the shape and size of the orifice, the fluid properties, and the flow conditions.

The correct answer to the given question is option 'C', which states that the coefficient of discharge (Cd) in the orifice meter usually ranges between 0.61 and 0.65. This range is commonly accepted for orifice meters and is based on experimental data and industry standards. Let's discuss this in more detail:

1. Definition and Significance of Cd:
- The coefficient of discharge (Cd) is a dimensionless factor that relates the actual flow rate through an orifice to the theoretical flow rate.
- It takes into account the effect of various factors on the flow measurement accuracy, such as friction, turbulence, and contraction.

2. Factors Affecting Cd:
- Size and shape of the orifice: The Cd value depends on the shape and size of the orifice. Different shapes and sizes result in different flow patterns and affect the discharge coefficient.
- Fluid properties: The properties of the fluid, such as density and viscosity, affect the Cd value. Different fluids have different Cd values for the same orifice.
- Flow conditions: The flow conditions, such as Reynolds number and flow regime (turbulent or laminar), also influence the Cd value.

3. Experimental Data and Industry Standards:
- The range of Cd values mentioned in option 'C' (0.61-0.65) is commonly accepted in practice.
- These values are based on extensive experimental data and have been adopted in industry standards such as ISO 5167.
- The Cd values within this range provide a reasonable level of accuracy for flow measurements using orifice meters.

4. Variation in Cd Values:
- It is important to note that the Cd values can vary for different orifice designs, fluid properties, and flow conditions.
- The given range (0.61-0.65) is an average range and may not be applicable in all cases.
- Manufacturers and researchers may provide specific Cd values for their orifice designs based on calibration and testing.

In conclusion, the coefficient of discharge (Cd) in the orifice meter usually ranges between 0.61 and 0.65, as stated in option 'C'. This range is commonly accepted based on experimental data and industry standards. However, it is important to consider specific factors such as orifice design, fluid properties, and flow conditions, which may cause variations in Cd values.

Solution:

Given, diameters of two pipes as d1 and d2 and the diameter of the third pipe as 2d.

Let the flow velocity in the third pipe be v.

According to the principle of continuity, the volume of liquid flowing per unit time through the three pipes must be the same.

Mathematically,

v1 * A1 = v2 * A2 = v * A3

where A1, A2, and A3 are the cross-sectional areas of pipes 1, 2, and 3 respectively.

The cross-sectional area of a pipe is proportional to the square of its diameter.

Mathematically,

A1 = π(d1/2)²

A2 = π(d2/2)²

A3 = π(2d/2)² = 4π(d²)

Substituting the values of A1, A2, and A3 in the continuity equation, we get

v1(d1/2)² = v2(d2/2)² = v(d²)

Simplifying the above equation, we get

v = (v1(d1/2)² + v2(d2/2)²)/d²

v = (v1d1² + v2d2²)/(4d²)

v = (v1/v2)(d1/d2)²(v2/4)

v = (v1v2/4)(d1/d2)²

v = v1v2/4(d2/d1)²

v = v1v2/4(1/(d1/d2)²)

v = v1v2/4(1/(d2/d1)²)

v = v1v2/4(1/(d2²/d1²))

v = v1v2/4(d1²/d2²)

v = v1v2/4(1/d2²)(d1²)

v = v1v2/4(d1²/d2²)

v = v1v2/4(4/1)

v = v1v2

Therefore, the flow velocity in the third pipe is v1v2/4. Hence, option C is the correct answer.

Factors affecting the coefficient of bend in a pipe are the angle of bend, radius of curvature of the bend, and the radius of the pipe.

1. Angle of bend:
The angle of bend refers to the amount of change in direction that the pipe makes at the bend. It is measured in degrees. The greater the angle of bend, the more significant the impact on the coefficient of bend. A sharper bend will result in a higher coefficient of bend.

2. Radius of curvature of the bend:
The radius of curvature of the bend is the radius of the imaginary circle formed by the bend. It is measured in meters. The smaller the radius of curvature, the more severe the bend, and therefore, the higher the coefficient of bend. A smaller radius of curvature indicates a tighter bend, which results in a higher resistance to flow and a higher coefficient of bend.

3. Radius of the pipe:
The radius of the pipe refers to the distance from the center of the pipe to its inner or outer surface. It is also measured in meters. The larger the radius of the pipe, the less impact it has on the coefficient of bend. A larger radius of the pipe indicates a more gradual bend, resulting in a lower resistance to flow and a lower coefficient of bend.

Interaction between factors:
The coefficient of bend depends on the interaction between these factors. For example, a larger angle of bend will increase the resistance to flow, leading to a higher coefficient of bend. Similarly, a smaller radius of curvature will also increase the resistance to flow, resulting in a higher coefficient of bend. The radius of the pipe, on the other hand, has an inverse relationship with the coefficient of bend. A larger radius of the pipe will decrease the resistance to flow and lower the coefficient of bend.

In conclusion, the coefficient of bend in a pipe depends on the angle of bend, radius of curvature of the bend, and the radius of the pipe. These factors interact with each other to determine the resistance to flow and ultimately affect the coefficient of bend.

Introduction:
The orifice meter is a type of flow meter used to measure the flow rate of fluid through a pipe. It consists of a flat rectangular plate with a small hole, known as an orifice, through which the fluid passes. This question is asking whether the given statement about the orifice meter is true or false.

Explanation:
The correct answer is option 'B', which states that the given statement is false. Let's understand why this is the correct answer.

Orifice Plate:
The orifice meter typically consists of a circular orifice plate rather than a flat rectangular plate. The orifice plate is installed perpendicular to the flow direction and creates a pressure difference across itself. This pressure difference is then used to determine the flow rate of the fluid.

Working Principle:
When the fluid passes through the orifice plate, it experiences a reduction in cross-sectional area, leading to an increase in fluid velocity. According to Bernoulli's principle, this increase in velocity corresponds to a decrease in pressure. The pressure difference across the orifice plate is measured using pressure taps located upstream and downstream of the plate.

Calculation of Flow Rate:
The flow rate through the orifice meter can be calculated using various equations, such as the Bernoulli equation, the continuity equation, and the discharge coefficient. These equations take into account the pressure difference, the properties of the fluid, and the dimensions of the orifice plate.

Advantages and Disadvantages:
The orifice meter has several advantages, including simplicity, low cost, and wide applicability. However, it also has limitations, such as pressure loss, sensitivity to changes in fluid properties, and the need for accurate calibration.

Conclusion:
In conclusion, the orifice meter used for measuring fluid flow does not consist of a flat rectangular plate. Instead, it employs a circular orifice plate. Therefore, the given statement is false.

False

Explanation:
In case of any orifice, the velocity does not always remain constant. The velocity of fluid flowing through an orifice depends on various factors such as the size of the orifice, the pressure difference across the orifice, and the properties of the fluid. Therefore, the discharge cannot be calculated solely based on the assumption that the velocity remains constant.

Factors affecting the velocity and discharge through an orifice:

1. Size of the orifice: The size of the orifice affects the velocity and discharge of the fluid. A smaller orifice will result in a higher velocity and discharge, while a larger orifice will result in a lower velocity and discharge.

2. Pressure difference across the orifice: The pressure difference across the orifice also affects the velocity and discharge. A higher pressure difference will result in a higher velocity and discharge, while a lower pressure difference will result in a lower velocity and discharge.

3. Properties of the fluid: The properties of the fluid, such as its density and viscosity, also play a role in determining the velocity and discharge. A more viscous fluid will have a lower velocity and discharge compared to a less viscous fluid.

Calculation of the discharge through an orifice:

The discharge through an orifice can be calculated using the following equation:

Q = Cd * A * √(2gh)

Where:
Q = Discharge
Cd = Coefficient of discharge
A = Area of the orifice
g = Acceleration due to gravity
h = Head difference across the orifice

The coefficient of discharge takes into account the various factors that affect the velocity and discharge, such as the shape of the orifice, the roughness of the edges, and the contraction and expansion of the flow. Therefore, it is necessary to consider these factors and use the appropriate value of the coefficient of discharge in the calculation.

Conclusion:

In conclusion, the velocity does not always remain constant in case of any orifice. The velocity and discharge through an orifice depend on various factors such as the size of the orifice, the pressure difference across the orifice, and the properties of the fluid. Therefore, it is important to consider these factors and use the appropriate equations, such as the discharge equation mentioned above, to calculate the discharge accurately.

Corresponding to nozzle exit, the co-ordinates of the point P where the jet strikes the window are: x = 5 m and y = 7.5 − 1.5 = 6 m (i) The path of trajectory is prescribed by the equation,

(i)

Formula is used to calculate the pressure drop in pipes.b)Darcy-Weisbach formula is used to calculate the flow rate in pipes.

a) Darcy-Weisbach formula is used to calculate the pressure drop in pipes.

Impulse momentum equation gives

A venturi meter is preferable to an orifice meter because the energy loss is less in a venturi meter compared to an orifice meter. The energy loss in a flow measurement device is an important consideration as it affects the accuracy and efficiency of the measurement.

Energy Loss in Venturi Meter:
1. The venturi meter operates on the principle of Bernoulli's equation, which states that the total energy of a fluid is constant along a streamline. This means that the sum of the potential energy, kinetic energy, and static pressure remains constant.

2. In a venturi meter, the fluid flowing through a converging section experiences an increase in velocity and a decrease in pressure. This change in velocity and pressure allows for the measurement of the flow rate.

3. The gradual change in velocity and pressure in a venturi meter minimizes energy losses due to turbulence and friction. The smooth transition from the converging section to the throat and then to the diverging section helps maintain the total energy of the fluid.

Energy Loss in Orifice Meter:
1. An orifice meter consists of a plate with a hole (orifice) inserted in a pipeline. The fluid passing through the orifice experiences a reduction in pressure and an increase in velocity.

2. However, due to the sudden contraction and expansion of the fluid flow, energy losses occur in the form of turbulence and eddies. These losses result in a decrease in the accuracy and efficiency of the flow measurement.

Comparison:
1. The venturi meter has a smoother flow path compared to the orifice meter, reducing the energy losses. This leads to higher accuracy and efficiency in flow measurement.

2. The orifice meter is cheaper and easier to assemble compared to a venturi meter. However, the energy loss in an orifice meter can be significant, impacting the reliability of the measurement.

Conclusion:
In conclusion, a venturi meter is preferable to an orifice meter because it has lower energy losses. While the orifice meter may be cheaper and easier to assemble, the accuracy and efficiency of the flow measurement are compromised due to higher energy losses. Therefore, in applications where accuracy is crucial, such as in industrial processes or water supply systems, a venturi meter is the preferred choice.