Isentropic nozzle flow describes the movement of a gas or fluid through a narrowing opening without an increase or decrease in
entropy
Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynam ...
.
Overview
Whenever a gas is forced through a tube, the gaseous molecules are deflected by the tube's walls. If the speed of the gas is much less than the
speed of sound, the gas density will remain constant and the velocity of the flow will increase. However, as the speed of the flow approaches the speed of sound,
compressibility
In thermodynamics and fluid mechanics, the compressibility (also known as the coefficient of compressibility or, if the temperature is held constant, the isothermal compressibility) is a measure of the instantaneous relative volume change of a f ...
effects on the gas are to be considered. The
density
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...
of the gas becomes position dependent. While considering flow through a tube, if the flow is very gradually compressed (i.e. area decreases) and then gradually expanded (i.e. area increases), the flow conditions are restored (i.e. return to its initial position). So, such a process is a reversible process. According to the
Second Law of Thermodynamics
The second law of thermodynamics is a physical law based on universal experience concerning heat and energy interconversions. One simple statement of the law is that heat always moves from hotter objects to colder objects (or "downhill"), unles ...
, whenever there is a reversible and adiabatic flow, constant value of entropy is maintained. Engineers classify this type of flow as an ''
isentropic
In thermodynamics, an isentropic process is an idealized thermodynamic process that is both adiabatic and reversible. The work transfers of the system are frictionless, and there is no net transfer of heat or matter. Such an idealized process ...
'' flow of fluids. Isentropic is the combination of the Greek word "iso" (which means - same) and entropy.
When the change in flow variables is small and gradual, isentropic flows occur. The generation of
sound waves
In physics, sound is a vibration that propagates as an acoustic wave, through a transmission medium such as a gas, liquid or solid.
In human physiology and psychology, sound is the ''reception'' of such waves and their ''perception'' by the ...
is an isentropic process. A
supersonic flow that is turned while there is an increase in flow area is also isentropic. Since there is an increase in area, therefore we call this an ''isentropic expansion''. If a supersonic flow is turned abruptly and the flow area decreases, the flow is irreversible due to the generation of
shock waves
In physics, a shock wave (also spelled shockwave), or shock, is a type of propagating disturbance that moves faster than the local speed of sound in the medium. Like an ordinary wave, a shock wave carries energy and can propagate through a med ...
. The isentropic relations are no longer valid and the flow is governed by the
oblique or normal shock relations.
Stagnation properties
![Enthalpy–entropy diagram illustrating the definition of stagnation state](https://upload.wikimedia.org/wikipedia/commons/f/f3/Enthalpy%E2%80%93entropy_diagram_illustrating_the_definition_of_stagnation_state.jpg)
In fluid dynamics, a
stagnation point
In fluid dynamics, a stagnation point is a point in a flow field where the local velocity of the fluid is zero.Clancy, L.J. (1975), ''Aerodynamics'', Pitman Publishing Limited, London. A plentiful, albeit surprising, example of such points seem ...
is a point in a flow field where the local velocity of the fluid is zero. The isentropic stagnation state is the state a flowing fluid would attain if it underwent a reversible
adiabatic deceleration to zero velocity. There are both ''actual'' and the ''isentropic'' stagnation states for a typical gas or vapor. Sometimes it is advantageous to make a distinction between the actual and the isentropic stagnation states. The actual stagnation state is the state achieved after an actual deceleration to zero velocity (as at the nose of a body placed in a fluid stream), and there may be irreversibility associated with the deceleration process. Therefore, the term "stagnation property" is sometimes reserved for the properties associated with the actual state, and the term total property is used for the isentropic stagnation state. The
enthalpy
Enthalpy , a property of a thermodynamic system, is the sum of the system's internal energy and the product of its pressure and volume. It is a state function used in many measurements in chemical, biological, and physical systems at a constant ...
is the same for both the actual and isentropic stagnation states (assuming that the actual process is adiabatic). Therefore, for an
ideal gas
An ideal gas is a theoretical gas composed of many randomly moving point particles that are not subject to interparticle interactions. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is a ...
, the actual
stagnation temperature In thermodynamics and fluid mechanics, stagnation temperature is the temperature at a stagnation point in a fluid flow. At a stagnation point the speed of the fluid is zero and all of the kinetic energy has been converted to internal energy and is a ...
is the same as the isentropic stagnation temperature. However, the actual stagnation pressure may be less than the isentropic stagnation pressure. For this reason the term "total pressure" (meaning isentropic stagnation pressure) has particular meaning compared to the actual stagnation pressure.
Flow analysis
The isentropic efficiency is
.
The variation of fluid density for
compressible flow
Compressible flow (or gas dynamics) is the branch of fluid mechanics that deals with flows having significant changes in fluid density. While all flows are compressible, flows are usually treated as being incompressible when the Mach number (the r ...
s requires attention to density and other fluid property relationships. The fluid
equation of state
In physics, chemistry, and thermodynamics, an equation of state is a thermodynamic equation relating state variables, which describe the state of matter under a given set of physical conditions, such as pressure, volume, temperature, or intern ...
, often unimportant for incompressible flows, is vital in the analysis of compressible flows. Also, temperature variations for compressible flows are usually significant and thus the energy equation is important. Curious phenomena can occur with compressible flows.
* For simplicity, the gas is assumed to be an ideal gas.
* The gas flow is isentropic.
* The gas flow is constant.
* The gas flow is along a straight line from gas inlet to exhaust gas exit.
* The gas flow behavior is compressible.
There are numerous applications where a steady, uniform, isentropic flow is a good approximation to the flow in conduits. These include the flow through a
jet engine, through the nozzle of a rocket, from a broken gas line, and past the blades of a turbine.
Mach number = M Velocity = V
Universal gas constant = R Pressure = p
Specific heat ratio = k Temperature = T
* = Sonic conditions Density =
Area = A Molar mass =
Energy equation for the steady flow:
To model such situations, consider the control volume in the changing area of the conduit of Fig. The continuity equation between two sections an infinitesimal distance dx apart is
If only the first-order terms in a differential quantity are retained, continuity takes the form
The energy equation is:
![Steady, uniform, isentropic flow through a conduit](https://upload.wikimedia.org/wikipedia/commons/c/c1/Steady%2C_uniform%2C_isentropic_flow_through_a_conduit.jpg)
This simplifies to, neglecting higher-order terms,:
Assuming an isentropic flow, the energy equation becomes:
Substitute from the continuity equation to obtain
or, in terms of the
Mach number:
This equation applies to a steady, uniform, isentropic flow.
There are several observations that can be made from an analysis of Eq. (9.26).
They are:
* For a subsonic flow in an expanding conduit (M <1 and dA>0), the flow is decelerating (dV <0).
* For a subsonic flow in a converging conduit (M <1 and dA <0), the flow is accelerating (dV >0).
* For a supersonic flow in an expanding conduit (M >1 and dA >0), the flow is accelerating (dV >0).
* For a supersonic flow in a converging conduit (M >1 and dA <0), the flow is decelerating (dV <0).
* At a throat where dA =0, either M =1 or dV =0 (the flow could be accelerating through M =1, or it may reach a velocity such that dV =0).
Supersonic flow
![Fig, A supersonic nozzle](https://upload.wikimedia.org/wikipedia/commons/7/77/Fig%2C_A_supersonic_nozzle.jpeg)
A nozzle for a supersonic flow must increase in area in the flow direction, and a diffuser must decrease in area, opposite to a nozzle and diffuser for a subsonic flow. So, for a supersonic flow to develop from a reservoir where the velocity is zero, the subsonic flow must first accelerate through a converging area to a throat, followed by continued acceleration through an enlarging area.
The nozzles on a rocket designed to place satellites in orbit are constructed using such converging-diverging geometry. The energy and continuity equations can take on particularly helpful forms for the steady, uniform, isentropic flow through the nozzle. Apply the energy equation with Q_ W_S 0 between the reservoir and some location in the nozzle to obtain
Any quantity with a zero subscript refers to a stagnation point where the velocity is zero, such as in the reservoir. Using several thermodynamic relations equations can be put in the forms:
If the above equations are applied at the throat (the critical area signified by an
Asterisk (*) superscript, where M =1), the energy equation takes the forms
The critical area is often referenced even though a throat does not exist. For air with k =1.4, the equations above provide
T* = 0.833333·T
o
p* = 0.528282·p
o
ρ* = 0.633938·ρ
o
The mass flux through the nozzle is of interest and is given by:
With the use of Eq. (9.28), the mass flux, after applying some algebra, can be
Expressed as
If the critical area is selected where M =1, this takes the form
which, when combined with previous it provides:
Converging nozzle
![A Converging Nozzle](https://upload.wikimedia.org/wikipedia/commons/8/83/A_Converging_Nozzle.jpg)
Consider a converging nozzle connecting a reservoir with a receiver. If the reservoir pressure is held constant and the receiver pressure reduced, the Mach number at the exit of the nozzle will increase until Me=1 is reached, indicated by the left curve in figure 2. After Me =1 is reached at the nozzle exit for
, the condition of choked flow occurs and the velocity throughout the nozzle cannot change with further decreases in
. This is due to the fact that pressure changes downstream of the exit cannot travel upstream to cause changes in the flow conditions.
The right curve of figure 2. represents the case when the reservoir pressure is increased and the receiver pressure is held constant. When
, the condition of choked flow also occurs; but Eq indicates that the mass flux will continue to increase as
is increased. This is the case when a gas line ruptures.
![The pressure variation in the nozzle](https://upload.wikimedia.org/wikipedia/commons/7/79/The_pressure_variation_in_the_nozzle.jpg)
It is interesting that the exit pressure
is able to be greater than the receiver pressure
. Nature allows this by providing the streamlines of a gas the ability to make a sudden change of direction at the exit and expand to a much greater area resulting in a reduction of the pressure from
to
. The case of a converging-diverging nozzle allows a supersonic flow to occur, providing the receiver pressure is sufficiently low. This is shown in figure 3 assuming a constant reservoir pressure with a decreasing receiver pressure. If the receiver pressure is equal to the reservoir pressure, no flow occurs, represented by curve
. If pr is slightly less than p_0, the flow is subsonic throughout, with a minimum pressure at the throat, represented by curve B. As the pressure is reduced still further, a pressure is reached that result in M =1 at the throat with subsonic flow throughout the remainder of the nozzle.
There is another receiver pressure substantially below that of curve C that also results in isentropic flow throughout the nozzle, represented by curve D; after the throat the flow is supersonic. Pressures in the receiver in between those of curve C and curve D result in non-isentropic flow (a shock wave occurs in the flow). If pr is below that of curve D, the exit pressure pe is greater than pr. Once again, for receiver pressures below that of curve C, the mass flux remains constant since the conditions at the throat remain unchanged. It may appear that the supersonic flow will tend to separate from the nozzle, but just the opposite is true. A supersonic flow can turn very sharp angles, since nature provides expansion fans that do not exist in subsonic flows. To avoid separation in subsonic nozzles, the expansion angle should not exceed 10°. For larger angles, vanes are used so that the angle between the vanes does not exceed 10°.
See also
*
de Laval nozzle
A de Laval nozzle (or convergent-divergent nozzle, CD nozzle or con-di nozzle) is a tube which is pinched in the middle, making a carefully balanced, asymmetric hourglass shape. It is used to accelerate a compressible fluid to supersonic speeds ...
*
Fanno flow
Fanno flow is the Adiabatic process, adiabatic flow through a constant area duct where the effect of friction is considered. Compressible flow, Compressibility effects often come into consideration, although the Fanno flow model certainly also appl ...
*
Supersonic gas separation
References
* Colbert, Elton J.
Isentropic Flow Through Nozzles'.
University of Nevada, Reno
The University of Nevada, Reno (Nevada, the University of Nevada, or UNR) is a public land-grant research university in Reno, Nevada. It is the state's flagship public university and primary land grant institution. It was founded on October 12 ...
. 3 May 2001. Accessed 15 July 2014.
* Benson, Tom.
Isentropic Flow. ''NASA.gov''. National Aeronautics and Space Administration. 21 June 2014. Accessed 15 July 2014.
* Bar-Meir, Genick.
Isenotropic Flow. {{Webarchive, url=https://web.archive.org/web/20150226104124/http://www.potto.org/gasDynamics/node75.php , date=2015-02-26 ''Potto.org''. Potto Project. 21 November 2007. Accessed 15 July 2014.
Thermodynamic processes
Thermodynamic entropy