Fluid Mechanics: Fundamentals th and Applications, 4 edition Yunus A. Cengel, John M. Cimbala Lecture slides by Mehmet Kanoglu ©McGraw-Hill Education. All rights reserved. Authorized only for instructor use in the classroom. No reproduction or further distribution permitted without the prior written consent of McGraw-Hill Education. Chapter 10 COMPRESSIBLE FLOW ©McGraw-Hill Education. All rights reserved. Authorized only for instructor use in the classroom. No reproduction or further distribution permitted without the prior written consent of McGraw-Hill Education. © Settles, G.S. Dynamics Lab,Gas Penn State with permission Used University. High-speed color schlieren image of the bursting of a toy balloon overfilled with compressed air. This 1-microsecond exposure captures the shattered balloon skin and reveals the bubble of compressed air inside beginning to expand. The balloon burst also drives a weak spherical shock wave, visible here as a circle surrounding the balloon. The silhouette of the photographer's hand on the air valve can be seen at center right. ©McGraw-Hill Education. Objectives • Appreciate the consequences of compressibility in gas flow • Understand why a nozzle must have a diverging section to accelerate a gas to supersonic speeds • Predict the occurrence of shocks and calculate property changes across a shock wave • Understand the effects of friction and heat transfer on compressible flows ©McGraw-Hill Education. 12–1 ■ STAGNATION PROPERTIES Stagnation (or total) enthalpy V 2 hh kJ/kg 0 2 Static enthalpy: the ordinary enthalpy h Energy balance (with no heat or work interaction, no change in potential energy) 22 VV12 hh hh01 02 1222 (a) © Corbis RF; (b) Photo courtesy of United Technologies Corporation/Pratt & Whitney. Used by permission. All rights reserved. Aircraft and jet engines involve high speeds, Steady flow of a fluid through an and thus the kinetic energy term should adiabatic duct. always be considered when analyzing them. ©McGraw-Hill Education. If the fluid were brought to a complete stop, the energy balance becomes V 2 h1 h h Stagnation enthalpy: The enthalpy of a 12 2 02 fluid when it is brought to rest adiabatically. During a stagnation process, the kinetic energy of a fluid is converted to enthalpy, which results in an increase in the fluid temperature and pressure. The properties of a fluid at the stagnation state are called stagnation properties (stagnation temperature, stagnation pressure, stagnation density, etc.). The stagnation state is indicated by the subscript 0. ©McGraw-Hill Education. Isentropic stagnation state: When the stagnation process is reversible as well as adiabatic (i.e., isentropic). The stagnation processes are often approximated to be isentropic, and the isentropic stagnation properties are simply referred to as stagnation properties. When the fluid is approximated as an ideal gas with constant specific heats V 2 V 2 cpp T0 c T TT0 2 2cp Stagnation (or total) temperature T0 : It represents the temperature an ideal gas attains when it is brought to rest adiabatically. The actual state, actual stagnation state, and isentropic stagnation state 2 Dynamic temperature V /2cp : of a fluid on an h-s diagram. corresponds to the temperature rise during such a process. ©McGraw-Hill Education. Stagnation pressure P0 :The pressure a fluid attains when brought to rest isentropically. kk/1 PT00 PT Stagnation density 0 1/k 1 1/V 00T kk T PPvv 00 When stagnation enthalpies are used, the energy balance for a single-stream, steady-flow device The temperature of an ideal ĖĖin out gas flowing at a velocity V 2 rises by V /2cp when it is qin w in h 01 gz 1 q out w out h 02 gz 2 brought to a complete stop. When the fluid is approximated as an ideal gas with constant specific heats qin q out w in w out cp T 02 T 01 g z 2 z 1 ©McGraw-Hill Education. 12–2 ■ ONE-DIMENSIONAL ISENTROPIC FLOW cP / speed of s sound c k P / T c kRT for an k the specific heat ratio of the gas ideal gas R the specific gas constant V Ma Mach c number During fluid flow through many devices such as nozzles, diffusers, and turbine blade passages, flow quantities vary primarily in the flow direction only, and the flow can be approximated as one- dimensional isentropic flow with good accuracy. ©McGraw-Hill Education. EXAMPLE: Gas Flow through a Converging– Diverging Duct Variation of normalized fluid properties and cross-sectional area along a duct as the pressure drops from 1400 to 200 kPa. ©McGraw-Hill Education. Variation of fluid properties in flow direction in the duct described in Example 12–2 for m = 3 kg/s = constant P, kPa T, K V, m/s ρ, kg/m3 c, m/s A, cm2 Ma 1400 473 0 15.7 339.4 ∞ 0 1200 457 164.5 13.9 333.6 13.1 0.493 1000 439 240.7 12.1 326.9 10.3 0.736 800 417 306.6 10.1 318.8 9.64 0.962 767* 413 317.2 9.82 317.2 9.63 1.000 600 391 371.4 8.12 308.7 10.0 1.203 400 357 441.9 5.93 295.0 11.5 1.498 200 306 530.9 3.46 272.9 16.3 1.946 * 767 kPa is the critical pressure where the local Mach number is unity. Note that as the pressure decreases, the temperature and speed of sound decrease while the fluid velocity and Mach number increase in the flow direction. The density decreases slowly at first and rapidly later as the fluid velocity increases. ©McGraw-Hill Education. We note from Example that the flow area decreases with decreasing pressure up to a critical-pressure value (Ma = 1), and then it begins to increase with further reductions in pressure. The Mach number is unity at the location of smallest flow area, called the throat. The velocity of the fluid keeps increasing after passing the throat although the flow area increases rapidly in that region. This increase in velocity past the throat is due to the rapid decrease in the fluid density. The flow area of the duct considered in this example first decreases and then increases. The cross section of a nozzle at Such ducts are called converging– the smallest flow area is called diverging nozzles. the throat. These nozzles are used to accelerate gases to supersonic speeds and should not be confused with Venturi nozzles, which are used strictly for incompressible flow. ©McGraw-Hill Education. Variation of Fluid Velocity with Flow Area In this section, the relations for the variation of static-to-stagnation property ratios with the Mach number for pressure, temperature, and density are provided. m AV constant d dA dV 0 AV dP V dV 0 dA dP1 d 2 A V dP dA dP 2 2 1 Ma AV Derivation of the differential form This relation describes the variation of of the energy equation for steady pressure with flow area. isentropic flow. ©McGraw-Hill Education. At subsonic velocities, the pressure decreases in converging ducts (subsonic nozzles) and increases in diverging ducts (subsonic diffusers). At supersonic velocities, the pressure decreases in diverging ducts (supersonic nozzles) and increases in converging ducts (supersonic diffusers). dA dV This equation governs the shape of a 1 Ma 2 nozzle or a diffuser in subsonic or AV supersonic isentropic flow. dA For subsonic flow Ma 1 , 0 dV dA For supersonic flow Ma 1 , 0 dV dA For sonic flow Ma 1 , 0 dV The proper shape of a nozzle depends on the highest velocity desired relative to the sonic velocity. To accelerate a fluid, we must use a converging nozzle at subsonic velocities and a diverging nozzle at supersonic velocities. To accelerate a fluid to supersonic velocities, we must use a converging–diverging nozzle. ©McGraw-Hill Education. We cannot obtain supersonic velocities by attaching a converging section to a converging nozzle. Doing so will only move the sonic cross section farther downstream and decrease the mass flow rate. ©McGraw-Hill Education. Variation of flow properties in subsonic and supersonic nozzles and diffusers. ©McGraw-Hill Education. Property Relations for Isentropic Flow of Ideal Gases The relations between the static properties and stagnation properties of an ideal gas with constant specific heats T0 k 1 2 1 Ma T 2 kk/1 P0 k 1 2 1 Ma P 2 1/k 1 0 k 1 2 1 Ma 2 T *2 Tk1 Critical kk/1 ratios P*2 (Ma = 1) Pk0 1 1/k 1 When Ma = 1, the properties at the nozzle *2 t throat become the critical properties. 0 k 1 ©McGraw-Hill Education. The critical-pressure, critical-temperature, and critical-density ratios for isentropic flow of some ideal gases Superheated Hot products of Monatomic steam, combustion, Air, gases, k = 1.3 k = 1.33 k = 1.4 k = 1.667 푃∗ 0.5457 0.5404 0.5283 0.4871 푃0 푇∗ 0.8696 0.8584 0.8333 0.7499 푇0 휌∗ 0.6276 0.6295 0.6340 0.6495 휌0 ©McGraw-Hill Education. 12–3 ■ ISENTROPIC FLOW THROUGH NOZZLES Converging or converging–diverging nozzles are found in steam and gas turbines and aircraft and spacecraft propulsion systems. In this section we consider the effects of back pressure (i.e., the pressure applied at the nozzle discharge region) on the exit velocity, the mass flow rate, and the pressure distribution along the nozzle. Converging Nozzles Mass flow rate through a nozzle AMa P00 k / RT m 2 kk1 / 2 1 1k 1 Ma /2 Maximum mass flow rate kk1 / 2 1 The effect of back pressure on the k 2 m A* P pressure distribution along a max 0 converging nozzle.