U-Tube Manometer MCQ Quiz - Objective Question with Answer for U-Tube Manometer - Download Free PDF

Explanation:

Principle of U-tube Manometer Operation

The operation of a U-tube manometer is primarily based on the principle of hydrostatic equilibrium. This principle states that the pressure at any point in a fluid at rest is the same in all directions. In a U-tube manometer, the pressure difference between two points is determined by the height difference of the liquid column in the two arms of the tube.

Analyzing the Given Options

1. "Venturi effect." (Incorrect)

   • The Venturi effect involves the reduction in fluid pressure that results when a fluid flows through a constricted section of pipe. This is not the principle that a U-tube manometer operates on.

2. "Pascal's law." (Incorrect)

   • Pascal's law states that pressure exerted anywhere in a confined incompressible fluid is transmitted equally in all directions throughout the fluid. While relevant to fluid dynamics, it is not the primary principle behind the U-tube manometer.

3. "Hydrostatic equilibrium." (Correct)

   • Hydrostatic equilibrium is the condition in which a fluid is at rest, and the pressure at any point within the fluid is constant. This principle is the basis for how a U-tube manometer measures pressure differences.

4. "Bernoulli's principle." (Incorrect)

   • Bernoulli's principle states that an increase in the speed of a fluid occurs simultaneously with a decrease in the fluid's potential energy or pressure. This principle is not directly applied in the operation of a U-tube manometer.

Explanation:

A U-tube manometer is commonly used to measure pressure differences. In a standard U-tube manometer, the manometric liquid (such as mercury) is denser than the fluid in the pipe.

However, in an inverted U-tube manometer, the purpose is to measure small pressure differences for gases or low-density fluids. Here, the manometric liquid should have a lower specific gravity than the liquid flowing in the pipe to ensure proper differential pressure measurement.

Additional Information

• U-tube manometer (normal type): Uses a liquid with a higher specific gravity (e.g., mercury).

• Inverted U-tube manometer: Uses a liquid with a lower specific gravity than the liquid in the pipe.

• The manometric liquid must be lighter so that the pressure difference can be effectively measured.

Explanation:

A manometer is a device used to measure pressure differences using a column of liquid. The accuracy and working of manometers depend on various factors, including fluid density and temperature.

Statement I: The manometer is suitable for low-pressure applications. (True)

• Manometers are not suitable for very high-pressure applications because the liquid column required for measurement would be too large.

• They are primarily used for low to moderate-pressure measurements in laboratory and industrial applications.

Statement II: It has simple operation and construction. (True)

• A manometer is simple in construction as it consists of a U-tube filled with liquid and operates based on the principle of hydrostatic pressure.

• It requires no moving parts and operates based on the difference in liquid levels.

Statement III: The manometric fluids’ density depends on temperature. Hence, errors may result due to the change in the temperature. (True)

• Density of manometric fluid varies with temperature, leading to errors in pressure measurement.

• If the temperature increases, fluid density decreases, affecting the height of the liquid column and leading to incorrect pressure readings.

Concept:

To calculate vacuum pressure in a pipe using a U-tube manometer, we apply the principle of hydrostatic pressure balance.

At the same horizontal level (datum A-A), pressures in both limbs of the manometer are equal. The vacuum pressure is given by:

\(p = -(\rho_2 g h_2 + \rho_1 g h_1)\)

Given:

• Specific gravity of mercury, \( S_2 = 13.6 \)
• Specific gravity of fluid, \( S_1 = 0.5 \Rightarrow \rho_1 = 500 \, \text{kg/m}^3 \)
• Density of mercury, \( \rho_2 = 13.6 \times 1000 = 13600 \, \text{kg/m}^3 \)
• Height of mercury column, \( h_2 = 40 \, \text{cm} = 0.4 \, \text{m} \)
• Height of fluid column, \( h_1 = 15 \, \text{cm} = 0.15 \, \text{m} \)
• Acceleration due to gravity, \( g = 10 \, \text{m/s}^2 \) (note: question unit g in N/cm² is incorrect)

Calculation:

\(p = -(13600 \cdot 10 \cdot 0.4 + 500 \cdot 10 \cdot 0.15) \\ = -(54400 + 750) = -55150 \, \text{N/m}^2\)

Final Answer: -55150 N/m² (note: options wrongly mention N/cm²)

Differential Manometers:

• Differential manometers are instruments used to measure the difference in pressure between two points within a fluid system.
• They operate on the principle of balancing a column of fluid between two different pressures, hence providing a direct measure of the pressure difference.
• This is crucial in applications where precise pressure measurements are necessary to ensure proper system function, such as in fluid flow studies, hydraulic systems, and HVAC systems.
• Differential manometers come in various types, including U-tube, inverted U-tube, and inclined manometers, each suited for different ranges and types of pressure differences.

Concept:

In an open tube manometer

• The pressure at both the open ends is atmospheric.
• The pressure at any point inside the column can be calculated from either side.

Calculation:

Given:

ρ_water = 1000 kg/m3, l = 80 mm and d = 20 mm

So, the pressure at the bottom of the oil column can be equated from either end to find the required value of ρ_oil.

ρ_oil × g × (d + l) = ρ_water × g × l

ρ_oil × (20 + 80) = 1000 × 80

ρ_oil = 800 kg/m3

Hence the required density of oil is 800 kg/m3.

Difficulty in construction: 

• Manometers are relatively simple devices and their construction is not particularly complex. They typically consist of a U-shaped tube filled with a suitable fluid, making them much easier to construct compared to many other pressure-measuring devices.

Need for leveling:

• Manometers must be perfectly level to obtain accurate measurements.
• This is because the measurement is based on the difference in height of the liquid in the two arms of the manometer, and any tilt in the apparatus can skew this difference, leading to inaccurate measurements.

No over-range protection:

• Over-range occurs when the pressure applied to the manometer exceeds its measurement range.
• Manometers don’t have built-in protections against this, and a significant over-range can cause the liquid within the manometer to overflow, leading to faulty readings or damage to the manometer.

Large and bulky size:

• Compared to other pressure measuring devices, such as electronic pressure transducers, manometers can be bulky and take up more space because of the U-shaped tube design.
• Furthermore, the need to visually read the liquid column height can also necessitate a certain minimum size for the manometer.
• These properties can limit their use in locations where space is a constraint.

Explanation:

Given:

ρ1 = Specific gravity of mercury =  13.6, ρ2 =Specific gravity of kerosene =  0.8, h= 40 cm  

Let h₁ on the left side be the level of mercury, before pouring of kerosene

After pouring kerosene, 40 cm height is taken by kerosene, mercury is lowered by h₁ on the left side and mercury on the right side raises by h₁ from its original position, ie when kerosene was not poured 

ρ₁ = 13.6, ρ₂ = 0.8, h= 40 cm 

Writing the pressure equation for the above diagram, according to the final location of the mercury and kerosene, we get

40 × G for kerosene - 2 × h₁ × G for mercury = 0, (pressure at level B and B^' is the same, as they are at the same level)

40 × 0.8 = 2 × h₁ × 13.6

h₁ = 1.17 cm ≈ 1.2 cm

Explanation:

U-tube Manometer:

(i) It consists of a glass tube in U shape, one end of which connected to the gauge point and another end open to the atmosphere.

(ii) The tube contains a liquid of specific gravity greater than that of the fluid of which the pressure is to be measured.

(iii) The choice of the manometric liquid depends on the range of pressure to be measured For the low-pressure range, liquid of lower SG is used and for the high range, generally, mercury is used.

(iv) A U-tube manometer is used inverted if the pressure differential is small (density of the manometric fluid is less than fluid) and A U-tube manometer is used upright if the pressure differential is large (density of the manometric fluid is large)

Limitation:

• This method requires reading of fluid level at two or more points since the change in pressure causes a rise in one limb and drop in another.

Concept:

U-tube manometer:

Pressure in left limb above x-x = pressure in right limb above x-x

\(P_A+ρ_lgh_1=ρ_mgh_2\)

Where P_A = Pressure in the pipe, \(ρ_l\) = density of liquid, ρ_m = density of mercury, and g = acceleration due to gravity.

Calculation:

Given

Specific gravity of fluid = 1

Density of fluid (ρ_f) = 1000 kg/m³

Acceleration due to gravity (g) = 9.81 m/sec²

Density of mercury (ρ_m) = 13600 kg/m³

Let pressure in the pipe be P 

P + ρf × g × (0.20 - 0.12) = ρm × g × 0.20

P + 1000 × 9.81 × (0.20 - 0.12) = 13600 × 9.81 × 0.20

P + 784.8 = 26683.2

P = 25898.4 N/m²

Hence option (4) is correct.

Concept:

Differential Manometers:

• Differential manometers are devices used for measuring the difference of pressures between two points in a pipe or in two different pipes.
• A differential manometer consists of a U-tube, containing a heavy or light liquid, whose two ends are connected to the points, whose difference of pressure is to be measured.
• The most common types of differential manometers are

1. U-tube differential manometer
2. Inverted U-tube differential manometer

• The tube generally contains mercury or any other liquid whose specific gravity is greater than the specific gravity of the liquid whose pressure is to be measured.

U-tube differential manometer

• It consists of a U-tube containing a heavy liquid. 
• The two ends of the tube are connected to the points whose difference is to be measured. 
• It is used for measuring the difference of pressure between two points in the flow section to which it is connected.

Differential inverted U-tube manometer: 

• It consists of an inverted U-tube containing a light liquid.
• The two ends of the tube are connected to the points whose difference is to be measured.
• It is used for measuring the difference of pressure between two points in the flow section to which it is connected.
• It is used in two conditions
  • In underground pipeline
  • When the specific gravity of manometric fluid is less than 1

In the U-tube manometer, the height of manometric fluid is used to measure the pressure.

\(\frac{{{P_A} - {P_B}}}{\gamma } = \left( {\frac{S}{{{S_1}}} - 1} \right)h\)

S = 13.6 for mercury and S₁ = 1 for water, then (P_A – P_B)/γ = 12.6h, i.e. mercury is more advantageous as a manometric fluid when (P_A – P_B) is large.

But when pressure difference is small, S = 1.25 (for a mixture of carbon tetrachloride and kerosene) is used, because (P_A – P_B)/γ = 0.25 h, i.e. h is four times larger that actual difference in pressure heads.

A small pressure difference in a pipeline carrying water cannot be measured by water, mercury and kerosene.

Water: It will not give any difference in manometric height

Mercury: It will give a negligible difference of manometric height due to the high viscosity.

Kerosene: Its specific gravity is less than one, so it will float on water and will not give a clear value of the pressure.

Concept:

Case 1: When only mercury is filled in the U Tube manometer, then the level of mercury in both limb is the same and have reference axis x-x.

Case 2: When 13.6 cm of water is poured into the right limb of the U Tube manometer then because of water, the level of mercury down by x unit in the right limb, and that x unit of mercury rises in the left limb, and also reference axis changes from x-x to y-y. Because of this level of mercury in the left limb rises by x unit from its initial level.

Calculation:

Given:

ρn = 1000kg/m3, ρmg = 13600kg/m3, amount of water poured = 13.6 cm

Pressure Equation for the system, having y-y as the reference axis

P_atm + 2x cm of mercury = Patm + 13.6 cm of water

2x × 13600 × g = 13.6 × 1000 × g

2x = 1

x = 0.5 cm

Concept:

U tube manometers:

Gauge pressure:

Gauge pressure is the pressure relative to atmospheric pressure. Most of the pressure measuring devices are calibrated to read gauge pressure.

• Gauge pressure is positive for pressures above atmospheric pressure.
• Gauge pressure is negative for pressures below atmospheric pressure.
• Gauge pressure is zero at atmospheric pressure.

Atmospheric pressure:

Atmospheric pressure or barometric pressure is the pressure within the atmosphere of Earth. Atmospheric pressure is closely approximated by the hydrostatic pressure caused by the weight of air above the measurement point.

Patm = 1.013 bar.

Absolute pressure:

Absolute pressure is the sum of gauge pressure and atmospheric pressure.

Pabs = Pgauge + Patm .

Explanation:

Concept:

U-tube manometer:

• The U-tube manometer is a U-shape bent glass tube, whose one end is open to the atmosphere and other end is connected to a section where the pressure of the liquid is to be measured.
• The U-tube manometer is filled with a liquid whose specific gravity is more than as compared to the liquid whose pressure is to be measured.

• The gauge pressure at point B in the above-shown U-tube manometer is calculated by using hydrostatic law and is given as:

The pressure of liquid above section A-A in the right column = \(\rho_{2}gh_{2}\)

The pressure of liquid above section A-A in the left column = \(P_B+\rho_{1}gh_{1}\)

Equating the above two pressure, we get the gauge pressure at point B as:

\(P_B+\rho_{1}gh_{1}=\rho_{2}gh_{2}\)

Calculation:

Given:

In this problem, the pressure at point B is:

Pressure of liquid above A-A in right column = 0

Pressure of liquid above A-A in left column = \(P_B+\rho_{1}gh_{1}+\rho_{2}gh_{2}\)

Equating both the pressure, we get:

\(P_B+\rho_{1}gh_{1}+\rho_{2}gh_{2}=0\)

\(P_B=-(\rho_{1}gh_{1}+\rho_{2}gh_{2})\)

Thus, option (2) is correct answer.