Nozzle and Diffuser MCQ Quiz - Objective Question with Answer for Nozzle and Diffuser - Download Free PDF

Explanation:

Critical Pressure Ratio in a Convergent Nozzle

• The critical pressure ratio of a convergent nozzle is defined as the ratio of the outlet pressure to the inlet pressure of the nozzle when the flow through the nozzle reaches its maximum mass flow rate per unit area. This ratio is a critical parameter in the design and operation of convergent nozzles, particularly in applications involving compressible fluid flow, such as in turbines, jet engines, and various industrial processes.

Convergent nozzle:

• A convergent nozzle is a device used to accelerate a fluid by decreasing its cross-sectional area. When a compressible fluid, such as a gas, flows through a convergent nozzle, the velocity of the fluid increases as the cross-sectional area decreases, in accordance with the principle of conservation of mass and energy. The relationship between the pressure, velocity, and density of the fluid is governed by the isentropic flow equations for compressible fluids.
• The critical pressure ratio occurs when the flow at the throat of the nozzle (the point of smallest cross-sectional area) reaches the speed of sound, also known as the Mach number equal to 1. At this point, the flow is said to be "choked," and the mass flow rate through the nozzle becomes maximum for the given inlet conditions. Any further decrease in the outlet pressure below this critical value will not increase the mass flow rate.

The critical pressure ratio of a convergent nozzle is the ratio at which the flow becomes choked and mass flow rate per unit area is maximum.

At this point, further decrease in outlet pressure does not increase the mass flow rate.

\( \left(\frac{P_2}{P_1}\right)_{\text{critical}} = \left(\frac{2}{\gamma + 1} \right)^{\frac{\gamma}{\gamma - 1}} \)

Where \( \gamma \) is the ratio of specific heats, \( P_1 \) is inlet pressure, and \( P_2 \) is outlet pressure.

Explanation:

Mach Number Across a Normal Shock Wave

• A normal shock wave is a sudden and nearly discontinuous change in the flow properties of a compressible fluid (usually a gas) that occurs when the flow transitions from supersonic to subsonic speeds. The Mach number (M) is a crucial parameter in this context, defined as the ratio of the fluid velocity to the speed of sound in the medium.
• In an adiabatic and steady flow situation, the Mach number cannot increase across a normal shock wave. This behavior is governed by the fundamental principles of gas dynamics and is mathematically expressed using the Rankine-Hugoniot relation.

Rankine-Hugoniot Relation

• The Rankine-Hugoniot relation is a set of equations derived from the conservation laws of mass, momentum, and energy across a shock wave. These equations describe the relationship between the pre-shock and post-shock states of the fluid. Specifically, they ensure the continuity of mass, momentum, and energy across the shock front. Here’s how the Rankine-Hugoniot relation explains why the Mach number cannot increase across a normal shock wave:

1. Conservation of Mass:

The mass flow rate is conserved across the shock wave. Mathematically, this is expressed as:

ρ₁U₁ = ρ₂U₂

where:

• ρ₁, ρ₂: Densities of the fluid before and after the shock
• U₁, U₂: Velocities of the fluid before and after the shock

2. Conservation of Momentum:

The momentum equation across the shock wave is given by:

P₁ + ρ₁U₁² = P₂ + ρ₂U₂²

where:

• P₁, P₂: Static pressures before and after the shock

3. Conservation of Energy:

The total energy (including internal energy and kinetic energy) is also conserved across the shock wave, expressed as:

h₁ + (U₁² / 2) = h₂ + (U₂² / 2)

where:

• h₁, h₂: Specific enthalpies before and after the shock

4. Implications for the Mach Number:

Combining these conservation equations reveals that the post-shock Mach number (M₂) is always less than 1 (subsonic), while the pre-shock Mach number (M₁) is greater than 1 (supersonic). This means the flow transitions from supersonic to subsonic across the normal shock wave.

In essence, the Rankine-Hugoniot relation establishes the fundamental physics that prevents the Mach number from increasing across a normal shock wave. Instead, the Mach number decreases as a result of the increase in pressure, temperature, and density, and the decrease in velocity

Explanation:

To calculate the velocity of steam at the outlet of the nozzle, we will use the principle of energy conservation, specifically the Steady Flow Energy Equation (SFEE). The SFEE states:

h₁ + (v₁²/2) = h₂ + (v₂²/2)

Here:

• h₁ = Specific enthalpy at the inlet (kJ/kg)
• h₂ = Specific enthalpy at the outlet (kJ/kg)
• v₁ = Velocity at the inlet (m/s)
• v₂ = Velocity at the outlet (m/s)

\( \frac{V_2^2}{2} - \frac{V_1^2}{2} = \Delta h \Rightarrow V_2 = \sqrt{V_1^2 + 2\Delta h} \)

Calculation:

Given:

Enthalpy drop, \( \Delta h = 50~\text{kJ/kg} = 50000~\text{J/kg} \)

Inlet velocity, \( V_1 = 5~\text{m/s} \)

Substitute into equation:

\( V_2 = \sqrt{5^2 + 2 \times 50000} = \sqrt{25 + 100000} = \sqrt{100025} \approx 316.2~\text{m/s}\)

Concept:

Pressure distribution in a converging-diverging nozzle

• Let us assume that flow is taking place in a converging-diverging nozzle.
• Initially, the outlet valve is closed and there is no flow.
• Thus, the pressure at inlet and outlet is the same, and we get a straight line pressure throughout the nozzle.
• As the valve is opened gradually the pressure drops in the converging section and increases in the diverging section.
• Once the flow becomes supersonic at the throat (minimum area section), pressure variation becomes the same in the converging section.
• Further decrease in pressure at the outlet, in the diverging part of the flow, tries to maintain the supersonic level.
• But due to insufficient energy, it becomes subsonic and generates a shock wave during the transition.
• With more drop in pressure, the location of the shock wave moves towards the exit.
• The nozzle is designed to have shock-free flow.

We know that,

\(\frac{{dA}}{A} = \frac{{dP}}{{\rho {v^2}}}\left( {1 - {M^2}} \right)\)

The value of M = 1 gives dA = 0 which means that area is constant, i.e. at the point where the change in area is zero (at the throat), M becomes 1 (sonic) and the flow-through nozzle reaches its maximum value and the nozzle gets chocked.

When the flow becomes maximum, the pressure at the throat is called critical pressure P*.

Explanation:

After normal shock:

• Stagnation temperature remains the same.

• Density after shock will increase, i.e., ρE < ρF

• Mach number will decrease, i.e., ME > MF.

• Entropy across a shockwave always increases as it is a highly irreversible process.

Explanation:

Throat:

1. The section of the minimum cross-sectional area is known as the throat. These are designed to be obtained the desired pressure ratio as well as the maximum discharge transfer. 
2. Flow velocity at the throat of nozzle operating at designed for maximum pressure ratio is the velocity of sound.
3. Flow velocity if it designed for the maximum discharge then,

• Up to the throat, the flow velocity is subsonic.
• Flow velocity after the throat is supersonic.​

​So, Flow velocity at the throat of a converging-diverging nozzle is Sonic when it designed for the maximum discharge.

Additional Information

Mach number (M):

We can say the speed of sound can be equated to Mach 1 speed. The Mach number due to the local speed of sound is dependent on the surrounding mediums in specific temperature and pressure. Flow can be determined as an incompressible flow with the help of the Mach number. The medium can either be a liquid or a gas. The medium can be flowing, whereas the boundary may be stable or the boundary may be traveling in a medium that is at rest.

M = \(\frac{u}{c}\)

Where, u = Local flow velocity, c = Speed of sound

Its classification:

Explanation:

Nozzle: A nozzle is a flow passage of varying cross-sectional area in which the velocity of fluid increases and pressure drops in the direction of flow.

Choked Flow: 

• Back pressure (p_b) is the pressure applied at nozzle discharge region. Expansion occurs in the nozzle from pressure p₁ to p_b. Initially, when back pressure p_b is equal to p₁ there shall be no flow through the nozzle but as backpressure p_b is reduced the mass flow through nozzle increases.
• With the reduction in backpressure, a situation comes when pressure ratio equals to a critical pressure ratio (back pressure attains critical pressure value) then mass flow through the nozzle is found maximum.
• Further reduction in back pressure beyond critical pressure value does not affect the mass flow i.e. mass flow rate does not increase beyond its’ limiting value at critical pressure ratio.
• A nozzle operating with maximum mass flow rate condition is called choked flow nozzle.
• At the critical pressure ratio, the velocity at exit is equal to the velocity of sound.
• If the back pressure is reduced below critical pressure then too the mass flow remains at maximum value and exit pressure remains as critical pressure and the fluid leaving nozzle at critical pressure expands violently down to the reduced back pressure value.

Concept:

For the flow of compressible fluid through orifices and nozzles,

For the maximum mass flow rate,

\(\frac{{{P_2}}}{{{P_1}}} = {\left( {\frac{2}{{n + 1}}} \right)^{\frac{n}{{n - 1}}}}\)and \(\frac{{{T_2}}}{{{T_1}}} = \frac{2}{{n + 1}}\)

where, n is the index of polytropic expansion.

Calculation:

Given:

Entry pressure P₁ = 8 bar, n = 1.135

\(\frac{{{P_2}}}{{{P_1}}} = {\left( {\frac{2}{{n + 1}}} \right)^{\frac{n}{{n - 1}}}}\)

\(\frac{{{P_2}}}{{{P_1}}} = {\left( {\frac{2}{{1.135 + 1}}} \right)^{\frac{{1.135}}{{1.135 - 1}}}} = 0.577\)

Explanation:

Nozzle:

• A nozzle is a device, a duct of smoothly varying cross-section area, that increases the velocity of a fluid at the expense of pressure.
• The chief use of nozzle is to produce a jet of steam (or gas) of high velocity to produce thrust for the propulsion of rocket motors and jet engines and to drive steam or gas turbines.

Friction losses in a nozzle depend upon various aspects, the effects of nozzle friction are as follows:

Reduction in enthalpy drop:

• Friction in nozzle affects its efficiency. As the efficiency of the nozzle is the ratio of actual enthalpy drop to ideal enthalpy drop in the nozzle, the friction in nozzle decreases the enthalpy drop.

Reduction in exit velocity:

• The kinetic energy of the steam increases at the expense of its pressure energy in a steam nozzle. Some kinetic energy gets lost to overcome the friction in the nozzle. Therefore, the exit velocity of steam decreases due to nozzle friction.

Increase in specific volume:

• The specific volume of steam can be defined as the volume of steam per unit weight of the steam. Specific volume increases due to nozzle friction.The specific volume of steam is increased due to frictional reheating

Decrease in mass flow rate:

• As the friction in the nozzle slows down the flow of steam in it, the mass flow rate also decreases due to nozzle friction.

Reheating of steam i.e., improving the quality of vapour at the exit:

Explanation:

Convergent-divergent nozzle:

• Convergent-Divergent nozzles are used to increase the flow of gas to supersonic speeds (as in the case of rockets).
• Their cross-sectional area first decreases and then increases. The area where the diameter is minimum is called the throat.
• As the gas enters the converging section, its velocity increases, considering the mass flow rate to be constant.
• As the gas passes through the throat, it attains sonic velocity (Mach number = 1).
• As the gas passes through the divergent nozzle, the velocity increases to supersonic (Mach number >1).

​

• The flow rate is maximum for a given nozzle if the flow is sonic at the throat. This condition is achieved by managing the back-pressure. If the mass flow rate is reached the maximum value at the throat then the flow is called choked flow.
• For the compressible fluid flow, the Mach number is an important dimensionless parameter. On the basis of the Mach number, the flow is defined.
• Choked flow is a limiting condition where the mass flow will not increase with a further decrease in the downstream pressure environment while upstream pressure is fixed.
• For chocked flow in the convergent-divergent nozzle, the Mach number at the throat is equal to 1 and the pressure at the throat is equal to the critical pressure.
• Critical pressure ratio for a choked nozzle:
• \(\frac{{{p^*}}}{{{p_o}}} = {\left( {\frac{2}{{\gamma + 1}}} \right)^{\frac{\gamma }{{\gamma - 1}}}}\)

where p* is the critical pressure and p0 is the inlet pressure.

Concept:

For high speed flows, such as those on encountered in jet engines, the potential energy of the fluid is negligible, therefore for convenience, the enthalpy and kinetic energy of the fluid is combined into a single term called stagnation enthalpy h_o defined per unit mass as

Stagnation enthalpy is given by, 

\({h_0} = {h_1} + \frac{{V_1^2}}{2}\)  

h₀ → stagnation enthalpy

\(\Rightarrow {C_p}{T_0} = {C_p}{T_1} + \frac{{V_1^2}}{2}\)

\({T_0} = {T_1} + \frac{{V_1^2}}{{2\;{C_p}}}\)

Calculation:

To = Temperature at the exit of diffuser = 36°C

T₁ = Ambient temperature at the sea level = 18°C, C_p = 1 kJ/kg = 1000 J/kg

\(36 = 18 + \frac{{V_1^2}}{{2 \times 1000}}\)

Explanation:

Critical pressure ratio:

• The critical pressure ratio is the pressure ratio required for sonic or choked flow at the throat.
• The critical pressure ratio is the pure function of the specific heat ratio. The range of specific heat ratio encountered in propellant gases is narrow. The lowest is 1.2 and the maximum is 1.67.
• The corresponding pressure ratios are 1.77 and 2.12 for an engine operating at sea level a chamber pressure of about 35 psi is adequate to assure supersonic flow in the nozzle.
• The higher the altitude lower the chamber pressure for sonic flow.Because the chamber pressure of real engines is usually greater than 100 psi so it is safe to assume sonic flow at the throat of a rocket nozzle.

Additional Information

A nozzle is a passage of smoothly varying cross-section by means of which the pressure energy of working fluid is converted into kinetic energy.

• A nozzle whose flow area decreases in the flow direction is called a Convergent nozzle.
• It is designed such that a drop in pressure from inlet to outlet accelerates the flow. 
• The location of the smallest flow area of a nozzle is called the throat.
• The ratio of the pressure at the section where sonic velocity is attained to the inlet pressure of a nozzle is called the critical pressure ratio.
• It happens at Mach number equals 1.

Mach number < 1 → subsonic velocity

Mach number = 1 → sonic velocity

Mach number > 1 → supersonic velocity

• If a convergent nozzle is operating under choked condition, the exit Mach number is unity.
• To determine whether a nozzle is choked or not, we calculate the actual pressure ratio and then compare this with the critical pressure ratio.
• If the actual pressure ratio > critical pressure ratio, the nozzle is said to be choked.

Explanation:

Sonic speed:

• When the local flow velocity is equal to the speed of the sound (the speed at which the disturbance propagates) is known as the sonic speed. 
• The mach number for the sonic speed is 1.
• It is also known as the disturbance propagates speed through the medium.

Additional Information

Mach number (M):

We can say the speed of sound can be equated to Mach 1 speed. The Mach number due to the local speed of sound is dependent on the surrounding mediums in specific temperature and pressure. Flow can be determined as an incompressible flow with the help of the Mach number. The medium can either be a liquid or a gas. The medium can be flowing, whereas the boundary may be stable or the boundary may be traveling in a medium that is at rest.

M = \(\frac{u}{c}\)

Where, u = Local flow velocity, c = Speed of sound

Its classification:

Concept:

A nozzle is a passage of smoothly varying cross-section by means of which the pressure energy of working fluid is converted into kinetic energy.

Convergent-Divergent nozzle: In a convergent-divergent nozzle the cross-section of the nozzle first decreases and then increases.

The relationship between the area, velocity, and the Mach number for a nozzle is governed by the equation

\(\frac{{dA}}{A} = - \frac{{dV}}{V}\left( {1 - {M^2}} \right)\)

So, from the figure, it is shown that the flow velocity in the convergent section of the nozzle is subsonic flow.

Concept:

According to the continuity equation, we have

ṁ_in = ṁ_out 

_{\(\dot m = \rho \times A \times v = \frac{{A \times velocity}}{{specific\;volume}}\;kg/s\)}

The steady flow energy equation

\(\dot m \times \left[ {{h_1} + \frac{{v_1^2}}{{2000}}} \right] - \dot Q = \dot m \times \left[ {{h_2} + \frac{{v_2^2}}{{2000}}} \right]\)

ṁ = mass flow rate in nozzle, h = specific enthalpy, Q̇ = heat lost from the nozzle, v = steam velocity, A = area of nozzle, ρ = density

Calculation:

Given:

A = 50 cm² = 50 × 10^-4 m², Q̇ = 75 kJ/s

At nozzle inlet:

P₁ = 4 MPa, T₁ = 673 K, v₁ = 60 m/s

At nozzle outlet:

P₂ = 2 MPa, T₂ = 573 K

\(m = \rho \times A \times v = \frac{{A \times velocity}}{{specific\;volume}}\;kg/s\)

\(̇̇ m = \frac{{50 \times {{10}^{ - 4}} \times 60}}{{0.0734310}} = 4.085\;kg/s\)

\(\dot m \times \left[ {{h_1} + \frac{{v_1^2}}{{2000}}} \right] - \dot Q = \dot m \times \left[ {{h_2} + \frac{{v_2^2}}{{2000}}} \right]\)
\(4.085 \times \left[ {1.87 \times 673 + \frac{{{{60}^2}}}{{2000}}} \right] - 75 = \;4.085 \times \left[ {1.87 \times 573 + \frac{{v_2^2}}{{2000}}} \right]\)

v₂ = 581.723 m/s.