Chapter Six/Isentropic Flow in Converging Nozzles ... Chapter Six/ / Isentropic Flow in Converging...

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Transcript of Chapter Six/Isentropic Flow in Converging Nozzles ... Chapter Six/ / Isentropic Flow in Converging...

  • UOT

    Mechanical Department / Aeronautical Branch

    Gas Dynamics

    Chapter Six//Isentropic Flow in Converging Nozzles ----------------------------------------------------------------------------------------------------------------------------- ---------------

    1-6 ch.6

    Prepared by A.A. Hussaini 2013-2014

    Chapter Six/Isentropic Flow in Converging Nozzles

    6.1 performance of Converging Nozzle

    Two types of nozzles are considered: a converging-only nozzle and a

    converging–diverging nozzle. A assume a fluid stored in a large reservoir, at

    and , is to be discharge through a converging nozzle into an extremely large

    receiver where the back pressure can be regulated. We can neglect frictional

    effects, as they are very small in a converging

    section.

    If the receiver (back) pressure is set at ,

    no flow results. Once the receiver pressure is

    lowered below , air will flow from the

    supply tank. Since the supply tank has a large

    cross section relative to the nozzle outlet area,

    the velocities in the tank may be neglected.

    Thus and (stagnation

    properties). There is no shaft work and we assume no heat transfer and no friction

    losses, i.e. the flow is isentropic.

    We identify section 2 as the nozzle outlet. Then from energy equation

    And for perfect gas where specific heats are assumed constant

    It is important to recognize that the receiver pressure is controlling the flow. The

    velocity will increase and the pressure will decrease as we progress through the

  • UOT

    Mechanical Department / Aeronautical Branch

    Gas Dynamics

    Chapter Six//Isentropic Flow in Converging Nozzles ----------------------------------------------------------------------------------------------------------------------------- ---------------

    2-6 ch.6

    Prepared by A.A. Hussaini 2013-2014

    nozzle until the pressure at the nozzle outlet equals that of the receiver. This will

    always be true as long as the nozzle outlet can “sense” the receiver pressure.

    Example: Let us assume

    For receiver

    For reservoir

    for isentropic flow

    From isentropic table corresponding to ⁄

    and ⁄

    ( )

    √ √ ⁄

    Figure 6.2 shows this process on a T –s

    diagram as an isentropic expansion. If the

    pressure in the receiver were lowered further,

    the air would expand to this lower pressure and

    the Mach number and velocity would increase.

    Assume that the receiver pressure is lowered to

    . Show that

    This gives:

    ( )

  • UOT

    Mechanical Department / Aeronautical Branch

    Gas Dynamics

    Chapter Six//Isentropic Flow in Converging Nozzles ----------------------------------------------------------------------------------------------------------------------------- ---------------

    3-6 ch.6

    Prepared by A.A. Hussaini 2013-2014

    √ √ ⁄

    and are critical properties

    Notice that the air velocity coming out of the nozzle is exactly sonic. The

    velocity of signal waves is equal to the velocity of sound relative to the fluid into

    which the wave is propagating. If the fluid at cross section is moving at sonic

    velocity, the absolute velocity of signal wave at this section is zero and it cannot

    travel past this cross section.

    If we now drop the receiver pressure below this critical pressure ( ),

    see figure (6.3), the nozzle has no way of adjusting to these conditions. That’s

    because fluid velocity will become supersonic and signal waves (sonic velocity)

    are unable to propagate from the back pressure region to the reservoir.

    Assume that the nozzle outlet pressure could continue to drop along with the

    receiver. This would mean that ⁄ , which corresponds to a

    supersonic velocity (point 4).We know that if the flow is to go supersonic, the area

    must reach a minimum and then increase. Thus for a converging-only nozzle, the

    flow is governed by the receiver pressure until sonic velocity is reached at the

  • UOT

    Mechanical Department / Aeronautical Branch

    Gas Dynamics

    Chapter Six//Isentropic Flow in Converging Nozzles ----------------------------------------------------------------------------------------------------------------------------- ---------------

    4-6 ch.6

    Prepared by A.A. Hussaini 2013-2014

    nozzle outlet and further reduction of the receiver pressure will have no effect on

    the flow conditions inside the nozzle. Under these conditions, the nozzle is said to

    be choked and the nozzle outlet pressure remains at the critical pressure.

    Expansion to the receiver pressure takes place outside the nozzle (points 5 and 6).

    The analysis above assumes that conditions within the supply tank remain

    constant. One should realize that the choked flow rate can change if, for example,

    the supply pressure or temperature is changed or the size of the throat (exit hole) is

    changed.

    The pressure ratio below which the nozzle is chocked can be calculated for

    isentropic flow through the nozzle. For perfect gas with constant specific heats,

    (

    )

    ( )⁄

    (

    ( ) )

    ( )⁄

    Example 6.1Air is allowed to flow from a large reservoir through a convergent

    nozzle with an exit area of . The reservoir is large enough so that

    negligible changes in reservoir pressure and temperature occur as fluid is

    exhausted through the nozzle. Assume isentropic, steady flow in the nozzle, with

    and . Assume also that air behaves as a perfect gas

    with constant specific heats, . Determine the mass flow through the nozzle

    for back pressures , and .

    At and the critical pressure ratio is 0.5283; therefore for all back

    pressures below;

  • UOT

    Mechanical Department / Aeronautical Branch

    Gas Dynamics

    Chapter Six//Isentropic Flow in Converging Nozzles ----------------------------------------------------------------------------------------------------------------------------- ---------------

    5-6 ch.6

    Prepared by A.A. Hussaini 2013-2014

    The nozzle is choked. Under these conditions, the Mach number at the exit plane is

    unit and the pressure at exit plane is and the temperature at exit plane

    The nozzle is chocked for back pressures of and the mass

    flow rate is;

    ̇

    For back pressures of the nozzle is not choked

    and the exit plane pressure equals to back pressure;

    From isentropic table at , , , and

    ̇

    Example 6.2 Nitrogen is stored in a tank in volume at a pressure of

    and a temperature of . The gas is discharge through a converging nozzle

    with an exit area of . For back pressure of , find the time for the

    tank pressure to drop to . Assume isentropic nozzle flow with nitrogen

    behaves as a perfect gas with and ⁄ . Assume quasi-

    steady flow through the nozzle with the steady flow equation applicable at each

    instant of time assume also that is constant too

  • UOT

    Mechanical Department / Aeronautical Branch

    Gas Dynamics

    Chapter Six//Isentropic Flow in Converging Nozzles ----------------------------------------------------------------------------------------------------------------------------- ---------------

    6-6 ch.6

    Prepared by A.A. Hussaini 2013-2014

    Solution; As the reservoir pressure drops from to , the ratio

    ⁄ ⁄ and ⁄ ⁄ remains below

    critical pressure ratio ( ) and .

    ̇

    ̇ ( )

    From conservation of mass

    ∬ ( ̂)

    The mass inside the tank at any time is m;

    ∬ ( ̂)

    The mass coming out of tank exit at any time

    (

    )

  • UOT

    Mechanical Department / Aeronautical Branch