16 CHAPTER Thermodynamics of High-Speed Gas Flow

Steady Flow of a Liquid through an Adiabatic Duct 16-1 Steady Flow of a Liquid through an Adiabatic Duct (fig.16-1)

Three States of a Fluid on an h-s Diagram 16-2 Three States of a Fluid on an h-s Diagram The actual state, actual stagnation state, and isentropic stagnation state of a fluid on an h-s diagram (Fig.16-3)

16-3 Completely Arresting the Flow of an Ideal Gas can Raise its Temperature (Fig.16-5) The temperature of an ideal gas flowing at a velocity V rises by V2/(2CP) when it is brought to a complete stop STOP

Propagation of a Small Pressure Wave Along a Duct 16-4 Propagation of a Small Pressure Wave Along a Duct

Control Volume Moving With the Small Pressure Wave Along a Duct 16-5 Control Volume Moving With the Small Pressure Wave Along a Duct (Fig. 16-8)

The Velocity of Sound Changes With Temperature 16-6 The Velocity of Sound Changes With Temperature (Fig. 16-9)

16-7 The Mach Number Can Vary With Different Temperatures Even With Equivalent Velocities (Fig. 16-10)

Throat: The Smallest Flow Area of a Nozzle 16-8 Throat: The Smallest Flow Area of a Nozzle

16-9 Supersonic Velocities Cannot be Obtained by Attaching a Converging Section We cannot obtain supersonic velocities by attaching a a converging section to a converging nozzle. Doing so will only move the sonic cross section farther downstream

16-10 Variation of Flow Properties in Subsonic and Supersonic Nozzles and Diffusers (Fig. 16-17)

16-11 Effect of Back Pressure on the Pressure Distribution Along a Converging Nozzle (Fig. 16-20)

16-12 Effect of Back Pressure of a Converging Nozzle on Mass Flow Rate and Exit Pressure The effect of back pressure Pb on the mass flow rate m and the exist pressure Pe of a converging nozzle (Fig. 16-21)

16-13 Variation of the Mass Flow Rate Through a Nozzle with Inlet Stagnation Properties (Fig. 16-22)

16-14 The Effect of Back Pressure on the Flow Through a Converging-Diverging Nozzle (Fig.16-26)

The h-s Diagram for Flow Across a Normal Shock 16-15 The h-s Diagram for Flow Across a Normal Shock (Fig. 16-29)

Entropy Change Across the Normal Shock 16-16 Entropy Change Across the Normal Shock (Fig. 16-31)

Isentropic and Actual (Irreversible) Flow in a Nozzle 16-17 Isentropic and Actual (Irreversible) Flow in a Nozzle Isentropic and actual flow in a nozzle between the same inlet state and the exit pressure (Fig. 16-33)

16-18 Schematic and h-s Diagram for the Definition of the Diffuser Efficiency (Fig. 16-35)

The h-s Diagram for the Isentropic Expansion of Steam in a Nozzle 16-19 The h-s Diagram for the Isentropic Expansion of Steam in a Nozzle (Fig. 16-37)

Schematic and h-s Diagram for Example 16-14 16-20 Schematic and h-s Diagram for Example 16-14 (Fig.16-38)

16-21 Chapter Summary In this chapter the thermodynamic aspects of high-speed fluid flow are examined. For high-speed flows, it is convenient to combine the enthalpy and the kinetic energy of the fluid into a single term called stagnation (or total) enthalpy h0, defined as (kJ/kg) The properties of a fluid at the stagnation state are called stagnation properties and are indicated by the subscript zero.

16-22 Chapter Summary The stagnation temperature of an ideal gas with constant specific heats is which represents the temperature an ideal gas will attain when it is brought to rest adiabatically.

16-23 Chapter Summary The (isentropic) stagnation properties of an ideal gas are related to the static properties of the fluid by

16-24 Chapter Summary When stagnation enthalpies are used, the conservation of energy equation for a single-stream, steady-flow device can be expressed as where h01 and h02 are the stagnation enthalpies at states 1 and 2, respectively.

16-25 Chapter Summary The velocity at which an infinitesimally small pressure wave travels through a medium is the velocity of sound (or the sonic velocity). It is expressed as

16-26 Chapter Summary For an ideal gas the velocity of sound becomes

16-27 Chapter Summary The Mach number is the ratio of the actual velocity of the fluid to the velocity of sound at the same state:

16-28 Chapter Summary The flow is called sonic when M = 1, subsonic when M < 1, supersonic when M > 1, hypersonic when M >> 1, and transonic when M 1. = ~

16-29 Chapter Summary The nozzles whose flow area decreases in the flow direction are called converging nozzles. Nozzles whose flow area first decreases and then increases are called converging-diverging nozzles. The location of the smallest flow area of a nozzle is called the throat.

16-30 Chapter Summary The highest velocity to which a fluid can be accelerated in a convergent nozzle is the sonic velocity. Accelerating a fluid to supersonic velocities is only possible in converging-diverging nozzles. In all supersonic converging-diverging nozzles, the flow velocity at the throat is the velocity of sound.

16-31 Chapter Summary The ratios of the stagnation to static properties for ideal gases with con-stant specific heats can be expressed in terms of the Mach number as

16-32 Chapter Summary When M = 1, the resulting static-to-stagnation property ratios for the temperature, pressure, and density are called critical ratios and are denoted by the superscript asterisk:

16-33 Chapter Summary The pressure outside the exit plane of a nozzle is called the back pressure. For all back pressures lower than P*, the pressure at the exit plane of the converging nozzle is equal to P*, the Mach number at the exit plane is unity, and the mass flow rate is the maximum (or choked) flow rate.

16-34 Chapter Summary Under steady-flow conditions, the mass flow rate through the nozzle is constant and can be expressed as

16-35 Chapter Summary The variation of flow area A through the nozzle relative to the throat area A* for the same mass flow rate and stagnation properties of a particular ideal gas is

16-36 Chapter Summary The parameter M* is defined as the ratio of the local velocity to the velocity of sound at the throat (M = 1): It can also be expressed as

16-37 Chapter Summary In some range of back pressure, the fluid that achieved a sonic velocity at the throat of a converging-diverging nozzle and is accelerating to supersonic velocities in the diverging section experiences a normal shock, which causes a sudden rise in pressure and temperature and a sudden drop in velocity to subsonic levels. Flow through the shock is highly irreversible, and thus it cannot be approximated as isentropic.

16-38 Chapter Summary The properties of an ideal gas with constant specific heats before (subscript x) and after (subscript y) a shock are related by

16-39 Chapter Summary The entropy change across the shock is obtained by applying the entropy-change equation for an ideal gas across the shock:

Chapter Summary Actual kinetic energy at nozzle exit 16-40 Chapter Summary The deviation of actual nozzles from isentropic ones is expressed in terms of the nozzle efficiency N, nozzle velocity coefficient CV, and the coefficient of discharge CD, which are defined as where h01 is the stagnation enthalpy of the fluid at the nozzle inlet, h2 is the actual enthalpy at the nozzle exit, and h2s is the exit enthalpy under isentropic conditions for the same exit pressure. Actual kinetic energy at nozzle exit Kinetic energy at nozzle exit for isentropic flow from the same inlet state to the same exit pressure Actual velocity at nozzle exit Velocity at nozzle exit for isentropic flow from the same inlet state to the same exit pressure Actual mass flow rate through nozzle Mass flow rate through nozzle for isentropic flow from the same inlet state to the same exit pressure =

16-41 Chapter Summary The performance of a diffuser is expressed in terms of the diffuser efficiency D the pressure recovery factor FP, and the pressure rise coefficient CPR. They are defined as Actual stagnation pressure at diffuser exit Isentropic stagnation pressure Fp = = Cpr Actual pressure rise Isentropic pressure rise = =

16-42 Chapter Summary Steam often deviates considerably from ideal-gas behavior, and no simple property relations are available for it. Thus it is often necessary to use steam tables instead of ideal-gas relations. The critical-pressure ratio of steam is often taken to be 0.546, which corresponds to a specific heat ratio of k = 1.3 for superheated steam.

16-43 Chapter Summary At high velocities, steam does not start condensing when it encounters the saturation line, and it exists as a supersaturated substance. Supersaturation states are nonequilibrium (or metastable) states, and care should be exercised in dealing with them.