Applied Thermodynamics Module 4: Compressible Flow Introduction Flows that involve significant changes in density is called compressible flows. They are frequently encountered in devices that involve the flow of gases at very high velocities. Compressible flow combines fluid dynamics and thermodynamics in that both are necessary to the development of the required theoretical background. In this module, we develop the general relations associated with one-dimensional compressible flows for an ideal gas with constant specific heats. Stagnation Properties Before going into details of stagnation properties, let’s first understand the way it comes. In analysis of control volume, two terms are frequently encountered internal energy (푢) and flow energy (푃푣). For convenience these two terms are combined and termed as enthalpy, ℎ = 푢 + 푃푣 (defined per unit mass. Whenever the kinetic and potential energies of the fluid are negligible, as is often the case, the enthalpy represents the total energy of a fluid. For high-speed flows, the potential energy of the fluid is still negligible, but the kinetic energy is not. In the analysis of high-speed flows enthalpy and kinetic energy appear frequently. For convenience these two terms are combined together and termed as stagnation (or total) enthalpy ℎ0, defined per unit mass as (1) When the potential energy of the fluid is negligible, the stagnation enthalpy represents the total energy of a flowing fluid stream per unit mass. Thus it simplifies the thermodynamic analysis of high-speed flows. Throughout this chapter the ordinary enthalpy h is referred to as the static enthalpy, whenever necessary, to distinguish it from the stagnation enthalpy. Stagnation process: A process in which flowing- fluid is brought to rest adiabatically. The properties of a fluid at the stagnation state are called stagnation properties (stagnation temperature, stagnation pressure, stagnation density, etc.). The stagnation state and the stagnation properties are indicated by the subscript 0. Fig. (1) Kinetic energy is converted to stagnation enthalpy represents the enthalpy of a enthalpy during a stagnation process. fluid when it is brought to rest adiabatically. Consider the steady flow of a fluid through a duct where the flow takes place adiabatically and with no shaft or electrical work and no change in potential energy. Energy balance to the system yields If the fluid were brought to a complete stop, then the velocity at state 2 would be zero and above equation becomes or, (2) The stagnation state is called the isentropic stagnation state when the fluid is brought to rest reversibly and adiabatically (i.e., isentropically). The entropy of a fluid remains constant during an isentropic stagnation process. The actual (irreversible) and isentropic stagnation processes are shown on the h-s diagram in Fig. (2). Notice that the stagnation enthalpy (and the stagnation temperature if the fluid is an ideal gas) of the fluid is the same for both cases. The stagnation processes are often approximated to be isentropic, and the isentropic stagnation properties are simply referred to as stagnation properties. stagnation (or total) temperature represents the temperature an ideal gas attains when it is brought to rest adiabatically. When the fluid is approximated as an ideal gas with constant specific heats, its enthalpy can be replaced by 푐푝푇 and above equation of stagnation enthalpy can be Fig.(2) The actual state, actual stagnation written as state, and isentropic stagnation state of a fluid on an h-s diagram. Or, (3) The term V2/2cp corresponds to the temperature rise during stagnation process and is called the dynamic temperature. stagnation (or total) pressure represents the temperature an ideal gas attains when it is brought to rest adiabatically. For ideal gases with constant specific heats, P0 is related to the static pressure of the fluid by (4) From fig. (2) it may be noted that the actual stagnation pressure is lower than the isentropic stagnation pressure because entropy increases during the actual stagnation process as a result of fluid friction. 1 푘 푘 By noting that 휌 = ⁄푣 and using the isentropic relation 푃푣 = 푃0푣0 , the ratio of the stagnation density to static density can be expressed as (5) Speed of Sound and Mach Number Consider a pipe that is filled with a fluid at rest, as shown in Fig. (3). A piston fitted in the pipe is now moved to the right with a constant incremental velocity dV, creating an infinitesimally small pressure wave. The speed at which an infinitesimally small pressure wave travels through the medium is known as speed of sound or sonic speed. The wave front moves to the right through the fluid at the speed of sound c and separates the moving fluid adjacent to the piston from the fluid still at rest. The fluid to the left of the wave front experiences an incremental change in its thermodynamic properties, while the fluid on the right of the wave front maintains its original thermodynamic properties, as shown in Fig. (3). To simplify the analysis, consider a control volume that encloses the wave front and moves with it, as shown in Fig. (4). To an observer traveling with the wave front, the fluid to the right will appear to be moving toward the wave front with a speed of c and the fluid to the left to be moving away from the wave front with a Fig. (3) Propagation of a small speed of c-dV. pressure wave along a duct. Applying mass balance equation to the control volume or, Neglecting higher order terms, above equation reduces to Now, applying energy balance equation to the control volume Fig. (4) Control volume moving with the small pressure In absence of any heat and work interaction and change in potential wave along a duct. energy above equation can be expanded to Which yields where we have neglected the second-order term dV². Since the variation in pressure and temperature are negligibly small and the change of state is so fast that the propagation of sonic wave is not only adiabatic but also very nearly isentropic. Then the second T ds relation reduces to Or, Combining Eqs. (a), (b), and (c) yields the desired expression for the speed of sound as Or, For an ideal gas, in an isentropic process 푃푣 푘 = 푐표푛푠푡푎푛푡 푃 = 푐표푛푠푡푎푛푡 휌푘 where k is the specific heat ratio of gas. Taking logarithm of above equation yields, 푙푛푃 − 푘푙푛휌 = 푙푛푐표푛푠푡푎푛푡 Now differentiating the above equation yields 푑푃 푑휌 − 푘 =0 푃 휌 푑푃 푃 Or, = 푘 푑휌 휌 푑푃 푑푃 For isentropic process = ( ) 푑휌 푑휌 푠 푑푃 푃 ∴ ( ) = 푘 푑휌 푠 휌 푑푃 Since 푐2 = ( ) and 푃 = 푘푅푇 푑휌 푠 푐2 = 푘푅푇 (6) 푈푛푣푒푟푠푎푙 푎푠 푐표푛푠푡푎푛푡 Where, 푅 = 푐ℎ푎푟푎푐푡푒푟푠푡푐 푔푎푠 푐표푛푠푡푎푛푡 = 푀표푙푒푐푢푙푎푟 푤푒ℎ푡 Note: 1. Lower molecular weight and higher value of k gives higher sonic velocity at the same temperature as shown in Fig. (5) 2. For a specified ideal gas R is a constant. Ratio of specific heat k, at most, could be a function of temperature. Hence the speed of sound in a specified ideal gas is a function of temperature alone (Fig. (5)) Mach Number: It is the ratio of the actual velocity of the fluid (or an object in still air) to the speed of sound in the same fluid at the same state. Fig. (5) The speed of sound (7) changes with temperature and varies with the fluid. Note that the Mach number depends on the speed of sound, which depends on the state of the fluid. Therefore, the Mach number of an aircraft cruising at constant velocity in still air may be different at different locations as shown in Fig. (6). Fluid flow regime on the basis of Mach number, the flow is called 1. Sonic when Ma=1 2. Subsonic when Ma<1 3. Supersonic when Ma>1 4. Hypersonic when Ma>>1 (i.e. Ma≥5) and Fig. (6) The Mach number can be 5. Transonic when Ma≅1 different at different temperatures even if the velocity is the same. Variation of Fluid Velocity with Flow Area Consider the mass balance for a steady-flow process: Differentiating and dividing the resultant equation by the mass flow rate, we obtain (8) Now, energy balance equation for steady flow process, without heat and work interaction and without change in potential energy, can be written as 푉2 Or, ℎ + = 푐푛푠푡푎푛푡 2 Differentiating the above expression, 푑ℎ + 푉푑푉 = 0 (9) Second Tds equation can be written as 0 (for isentropic flow) 푇푑푠 = 푑ℎ − 푣푑푃 1 푑ℎ = 푣푑푃 = 푑푃 휌 Here, it is worthwhile to mention that v denotes specific volume and V denotes velocity of flow. Putting the value of dh in equation (9) (10) Combining equations (8) and (10) gives 푑휌 1 Putting the value of = in above equation, yields 푑푃 푐2 (11) Above equation describes the variation of pressure with flow area. Two important points can be noted from above equation. 1. It may be noted that A, 휌, and V are positive quantities. For subsonic flow (Ma<1), the term 1 − 푀푎2 is positive; and thus dA and dP must have the same sign, i.e. P increases as A increases and P decrease as A decreases. Thus, at subsonic velocities, the pressure decreases in converging ducts (subsonic nozzles) and increases in diverging ducts (subsonic diffusers). 2. In supersonic flow (Ma>1), the term 1 − 푀푎2 is negative, and thus dA and dP must have opposite signs, i.e. P decreases as A increases and P increases as A decreases. Thus, at supersonic velocities, the pressure decreases in diverging ducts (supersonic nozzles) and increases in converging ducts (supersonic diffusers). Another important relation for the isentropic flow of a fluid is obtained by substituting 휌푉 = − 푑푃⁄푑푉 from equation (10) into equation (11) (12) This equation governs the shape of a nozzle or a diffuser in subsonic or supersonic isentropic flow.