Unit 4: Flow in constant area duct - Subjective Questions

Fanno flow is the steady, one-dimensional flow of a compressible fluid through a constant-area duct in which wall friction is present but no heat is transferred.

Assumptions:

• The flow is steady and one-dimensional.
• The duct has a constant cross-sectional area.
• The fluid is a perfect gas with constant specific heats.
• The process is adiabatic, so .
• No shaft work is performed.
• Changes in potential energy are negligible.
• Wall friction is the only irreversibility.

The energy equation gives

Therefore, the stagnation temperature remains constant. Friction increases entropy and reduces stagnation pressure. Both subsonic and supersonic flows tend toward the sonic condition .

Consider steady flow through a differential length of a constant-area duct.

1. Continuity equation:

Since is constant,

2. Momentum equation:
For wall shear stress ,

Using the Fanning friction factor , where ,

3. Energy equation:
Because the flow is adiabatic and has no shaft work,

For a perfect gas,

which gives

4. Equation of state:

Combining these equations and integrating between a state at Mach number and the sonic state produces the standard Fanno-flow length equation.

Combining the continuity, momentum, energy and perfect-gas equations gives a differential relation between Mach number and frictional duct length. Integration from a state having Mach number to the limiting sonic state gives

Here:

• is the duct length required to reach .
• is the hydraulic diameter.
• is the Fanning friction factor.
• is the ratio of specific heats.

Define the Fanno function

For two sections,

If the Darcy friction factor is used, the left side is written as . The friction-factor convention must therefore be checked before using tables or formulas.

The superscript denotes the sonic state on the same Fanno line. For a perfect gas,

Temperature ratio:

Pressure ratio:

Density ratio:

Velocity ratio:

Stagnation-pressure ratio:

The stagnation temperature remains constant:

These relations allow all flow properties to be determined when and are known.

In a subsonic Fanno flow, friction causes the Mach number to increase toward unity.

As duct length increases:

• Mach number: increases toward .
• Velocity: increases.
• Static pressure: decreases.
• Static temperature: decreases because kinetic energy increases while remains constant.
• Density: decreases to satisfy .
• Stagnation temperature: remains constant because the flow is adiabatic.
• Stagnation pressure: decreases because friction is irreversible.
• Entropy: increases and reaches its maximum at .

Thus, friction accelerates a subsonic stream even though it causes a loss of stagnation pressure. The flow becomes choked if the duct is long enough for the exit Mach number to reach unity.

In a supersonic Fanno flow, friction causes the Mach number to decrease toward unity.

As the fluid moves downstream:

• Mach number: decreases toward .
• Velocity: decreases.
• Static pressure: increases.
• Static temperature: increases as kinetic energy is converted into internal energy.
• Density: increases.
• Stagnation temperature: remains constant.
• Stagnation pressure: decreases due to irreversibility.
• Entropy: increases toward its maximum value at the sonic state.

Therefore, wall friction decelerates supersonic flow but accelerates subsonic flow. In both cases, the sonic state is the limiting condition for a constant-area adiabatic duct.

For Fanno flow, the Mach number approaches unity as the frictional duct length increases.

• If , friction causes to increase.
• If , friction causes to decrease.
• The limiting condition is .

The remaining length required to reach the sonic state is

where is the Fanno function. For a duct between states 1 and 2,

Choking: When the available duct length equals , the exit becomes sonic. A longer duct cannot sustain the same inlet state and mass flow rate. The system must respond through a reduction in mass flow, a change in inlet conditions, or the formation of compression waves or shocks.

Hence, is the maximum possible duct length for specified inlet conditions and friction factor.

A Fanno table normally lists , , , , and as functions of and .

Procedure:

1. Use the inlet Mach number to obtain .
2. Calculate the friction parameter of the actual duct:
   
3. Determine the outlet Fanno parameter:
   
4. Read or interpolate the corresponding on the same subsonic or supersonic branch.
5. Determine property ratios using star values, for example:
   
6. Apply the same ratio method for temperature, density, velocity and stagnation pressure.

If , the assumed inlet state and mass flow cannot exist because the duct would require passage beyond the sonic limit.

A Fanno line represents all possible states of steady, adiabatic flow in a constant-area duct for a fixed mass flux and stagnation enthalpy.

On a - diagram:

• The line has separate subsonic and supersonic branches.
• Friction causes entropy to increase in the direction of flow.
• Both branches approach the sonic point .
• The sonic point corresponds to the maximum entropy state on that Fanno line.
• Stagnation temperature and stagnation enthalpy remain constant.
• Stagnation pressure decreases as entropy increases.

A continuous Fanno process cannot pass smoothly from subsonic to supersonic flow through . The sonic state is a limiting or choked state. A normal shock may connect a supersonic state to a subsonic state, but it introduces an additional irreversible change.

Rayleigh flow is steady, one-dimensional flow through a constant-area duct in which heat transfer is present while wall friction is neglected.

The assumptions include perfect-gas behavior, constant area, no shaft work and negligible potential-energy change.

From continuity,

where is the mass flux. Since ,

For frictionless constant-area flow, the momentum equation is

Integration gives

Using ,

Therefore,

This is a straight line in the - plane with slope

It is called the Rayleigh line. Each line corresponds to a fixed mass flux and momentum constant.

For steady, one-dimensional flow in a constant-area duct:

1. Continuity:

Since is constant,

2. Momentum:
With negligible wall friction,

Since ,

3. Energy:
Heat transfer changes stagnation enthalpy:

For a perfect gas,

where

4. Perfect-gas equation:

The momentum relation compared with the sonic state gives

Combining this with continuity and the perfect-gas equation produces the remaining Rayleigh property relations.

For a perfect gas, the Rayleigh-flow property ratios relative to the sonic state are:

Pressure:

Temperature:

Density:

Velocity:

Stagnation temperature:

Stagnation pressure:

These relations form the basis of Rayleigh-flow tables and charts.

When heat is added to a subsonic Rayleigh flow:

• The Mach number increases toward .
• Velocity increases.
• Static pressure decreases.
• Density decreases.
• Stagnation temperature increases.
• Entropy increases.
• Stagnation pressure decreases.

The static temperature does not vary monotonically over the entire subsonic branch. It reaches its maximum when

and then decreases slightly as the flow approaches .

At the sonic state, the entropy and stagnation temperature attain their maximum values on the Rayleigh line. Further heat addition is impossible for the same inlet state and mass flow because the flow is thermally choked.

For supersonic Rayleigh flow, heat addition produces the following changes:

• Mach number decreases toward .
• Velocity decreases.
• Static pressure increases.
• Static temperature increases.
• Density increases.
• Stagnation temperature increases.
• Entropy increases.
• Stagnation pressure decreases.

Thus, heat addition accelerates subsonic flow but decelerates supersonic flow. In both cases, it drives the flow toward the sonic condition. The maximum permissible heat addition is reached when , at which point the flow becomes thermally choked.

Heat rejection moves a Rayleigh flow away from the sonic condition.

For subsonic flow:

• Mach number decreases.
• Velocity decreases.
• Static pressure generally increases.
• Density increases.
• Stagnation temperature decreases.
• Entropy decreases in the direction of the idealized cooled flow.

For supersonic flow:

• Mach number increases.
• Velocity increases toward its high-Mach limiting value.
• Static pressure decreases.
• Density decreases.
• Stagnation temperature decreases.

Therefore, heat addition drives either branch toward , whereas heat rejection drives the subsonic branch toward lower Mach numbers and the supersonic branch toward higher Mach numbers. Actual heat rejection remains irreversible overall when the surroundings are included, even though the fluid entropy may decrease.

For steady Rayleigh flow, the energy equation is

Thermal choking occurs when the final state is sonic, so and .

Therefore, the maximum heat addition per unit mass is

From the Rayleigh relation,

Hence,

Alternatively, if is obtained directly from a table,

This heat limit applies to both subsonic and supersonic inlet states. Additional heat cannot be accommodated without changing the mass flow rate or upstream conditions.

Rayleigh tables list ratios such as , , , , and for different Mach numbers.

Calculation procedure:

1. Use the inlet Mach number to read .
2. Calculate the outlet stagnation temperature from
   
3. Determine
   
4. Use the table to find the corresponding outlet Mach number on the physically appropriate branch.
5. Obtain property ratios using star-state values. For example,
   
   
6. Repeat this ratio method for velocity, density and stagnation pressure.

Interpolation may be required when the calculated ratio falls between tabulated values.

Fanno flow:

• Adiabatic: .
• Wall friction is present.
• Stagnation temperature remains constant.
• Stagnation pressure decreases.
• Duct length and friction factor determine the change of state.
• The governing curve is called the Fanno line.

Rayleigh flow:

• Heat transfer is present.
• Wall friction is neglected.
• Stagnation temperature changes according to .
• Stagnation pressure generally decreases during heat addition toward the sonic state.
• Heat addition or rejection determines the change of state.
• The governing curve is called the Rayleigh line.

Common features:

• Both represent steady, one-dimensional, constant-area compressible flow.
• Both have distinct subsonic and supersonic branches.
• Friction in Fanno flow and heat addition in Rayleigh flow drive both branches toward .
• The sonic state is the maximum-entropy limiting state on the corresponding line.
• Both processes can produce choking.

Frictional choking:

• Occurs in an adiabatic constant-area duct with wall friction.
• The controlling parameter is .
• Subsonic flow accelerates and supersonic flow decelerates toward .
• The maximum allowable length is .
• Stagnation temperature remains constant while stagnation pressure decreases.

Thermal choking:

• Occurs in a frictionless constant-area duct due to heat addition.
• The controlling parameter is the added heat or the change in .
• Both subsonic and supersonic streams approach when heated.
• The maximum allowable heat is
  
• Both stagnation temperature and entropy increase during heat addition.

In either case, once the sonic condition is reached, the same mass flow cannot be maintained if additional frictional length or heat input is imposed.

Solution method:

1. Identify whether friction or heat transfer is the dominant mechanism.
2. Verify the constant-area and one-dimensional assumptions.
3. Calculate the inlet Mach number and stagnation properties.
4. For Fanno flow, evaluate and use
   
5. For Rayleigh flow, use
   
   and the tabulated ratios.
6. Select the correct subsonic or supersonic branch.
7. Determine pressure, temperature, density and velocity from star-property ratios.
8. Check whether the calculated process reaches and is therefore choked.

Limitations:

• Real ducts can have both friction and heat transfer simultaneously.
• Area variations invalidate the constant-area assumption.
• Boundary layers make the flow nonuniform.
• Friction factors and specific heats may vary with temperature.
• Shocks, separation and chemical reactions require additional models.

Fanno and Rayleigh models are therefore idealized but valuable for preliminary analysis of nozzles, combustors, heat exchangers and high-speed gas pipelines.