Compressible flows in fluid mechanics in chemical engineering
- 2. 9.1. Introduction • All previous chapters have been concerned with “low-speed’’ or “incompressible’’ flow, i.e., where the fluid velocity is much less than its speed of sound. • When a fluid moves at speeds comparable to its speed of sound, density changes become significant and the flow is termed compressible. • Such flows are difficult to obtain in liquids, since high pressures of order 1000 atm are needed to generate sonic velocities. • In gases, however, a pressure ratio of only 2:1 will likely cause sonic flow. • Thus compressible gas flow is quite common, and this subject is often called gas dynamics.
- 3. • The two most important and distinctive effects of compressibility on flow are: – Choking, wherein the duct flow rate is sharply limited by the sonic condition, and – Shock waves, which are nearly discontinuous property changes in a supersonic flow. • The proper criterion for a nearly incompressible flow was a small Mach number where V is the flow velocity and a is the speed of sound of the fluid. • Under small-Mach number conditions, changes in fluid density are everywhere small in the flow field. an incompressible flow requires only a momentum and continuity analysis.
- 4. • For compressible flows which have Mach numbers greater than about 0.3, the density change is significant and it follows from equation of state that the temperature and pressure changes are also substantial. • Large temperature changes imply that the energy equation can no longer be neglected. • Therefore the work is doubled from two basic equations to four: – Continuity equation – Momentum equation – Energy equation – Equation of state • to be solved simultaneously for four unknowns: pressure, density, temperature, and flow velocity (p, ρ, T, V). • What assumption you can make for simplification ?
- 5. • Depending upon the magnitude of Ma, the following rough classifications of Ma are commonly used external high-speed aerodynamics: – Ma < 0.3: incompressible flow, where density effects are negligible. – 0.3 < Ma < 0.8: subsonic flow, where density effects are important but no shock waves appear. – 0.8 < Ma < 1.2: transonic flow, where shock waves first appear, dividing subsonic and supersonic regions of the flow. – 1.2 < Ma < 3.0: supersonic flow, where shock waves are present but there are no subsonic regions. – 3.0 < Ma: hypersonic flow, where shock waves and other flow changes are especially strong. • For internal (duct) flows, the most important question is simply whether the flow is subsonic (Ma < 1) or supersonic (Ma > 1), because the effect of area changes reverses.
- 6. Specific heat ratio: In addition to geometry and Mach number, compressible- flow calculations also depend upon a second dimensionless parameter, the specific-heat ratio of the gas: K slowly decreases with T. Variations in k have only a slight effect upon compressible flow k = 1.4 for air.
- 7. • Fluid equations of state: The most elementary treatments are confined to the perfect gas with constant specific heats: • For all real gases, cp, cv , and k vary with temperature but only moderately; for example, cp of air increases 30 percent as temperature increases from 0 to 5000°F. therefore it is quite reasonable to assume constant specific heats. • The changes in the internal energy û and enthalpy h of a perfect gas are computed for constant specific heats as: • For variable specific heats one must integrate internal energy and enthalpy.
- 8. • Isentropic process: For most of compressible problems for simplification, we take zero change in entropy. • From the first and second laws of thermodynamics for a pure substance: • Introducing dh = cpdT for a perfect gas and solving for ds, • Substitute ρT = p/R from the perfect-gas law and obtain: • If cp is variable, the gas tables will be needed, but for constant cp we obtain the analytic results: For Isentropic flow of a perfect gas
- 9. Properties of Common Gases at 1 atm and 20°C
- 11. 9.2 Speed of sound • The so-called speed of sound is the rate of propagation of a pressure pulse of infinitesimal strength through a still fluid. • It is a thermodynamic property of a fluid. • Let us analyze it by considering a pulse of finite strength as show here:
- 12. • The continuity equation is thus, • the thickness of pressure waves in gases is of order 10-7 m at atmospheric pressure. Thus we can safely neglect friction and apply the one-dimensional momentum equation across the wave: • Using the continuity equation result: • For infinitesimal strength Δρ 0 and we get:
- 13. • For adiabatic or isentropic process the relationship with be: • For perfect gas: • The speed of sound increases as the square root of the absolute temperature. • For air, with k = 1.4, the speed of sound will be ?
- 14. 9.3 Adiabatic and isentropic steady flow • Consider high-speed flow of a gas past an insulated wall as shown in figure. • There is no shaft work delivered to any part of the fluid. Therefore every stream-tube in the flow satisfies the steady-flow energy equation: The wall in figure could be either the surface of an immersed body or the wall of a duct.
- 15. • Neglecting the terms gz, and taking q and wv equal to zero and for the flow out side the boundary layer, we get: • The constant in the above equation is equal to the maximum enthalpy which the fluid would achieve if brought to rest adiabatically. It is called the stagnation enthalpy h0 of the flow. Thus: • For a perfect gas h = cpT, and equation becomes: • An alternate interpretation of equation occurs when the enthalpy and temperature drop to (absolute) zero, so that the velocity achieves a maximum value: For steady state adiabatic flow
- 16. • To get the adiabatic equation in dimension less form we divide it by cpT to obtain: • Also from the perfect gas law: • This gives: • And: • Since a α T1/2, • Note that these equations for adiabatic flow conditions and hold even in presence of frictional losses or shock wave.
- 17. • If the flow is also isentropic, then for a perfect gas the pressure and density ratios can be computed as: • The quantities p0 and ρ0 are the isentropic stagnation pressure and density, respectively, i.e., the pressure and density which the flow would achieve if brought isentropically to rest. • In an adiabatic non-isentropic flow p0 and ρ0 retain their local meaning, but they vary throughout the flow as the entropy changes due to friction or shock waves. • The quantities h0, T0, and a0 are constant in an adiabatic non-isentropic flow.
- 19. • The isentropic assumptions are effective, but are they realistic? • Differentiating the adiabatic equation. • Also from the thermodynamic law for isentropic process: • Combining: • Which is exactly the Bernoulli relation, for steady frictionless flow with negligible gravity terms. • Thus we see that the isentropic-flow assumption is equivalent to use of the Bernoulli or streamline form of the frictionless momentum equation.
- 20. • The stagnation values (a0, T0, p0, ρ0) are useful reference conditions in a compressible flow. • But the more useful conditions are where the flow is sonic, Ma = 1.0. These sonic, or critical, properties are denoted by asterisks: p*, *, a*, and T*. So for when Ma 1.0; for k 1.4. • In all isentropic flow, all critical properties are constant; in adiabatic non-isentropic flow, a* and T* are constant, but p* and * may vary. • The critical velocity V* equals the sonic sound speed a* by definition and is often used as a reference velocity in isentropic or adiabatic flow:
- 23. 9.4 Isentropic flow with area changes • For steady one-dimensional flow the equation of continuity is: • The differential form will be: • Momentum: • Sound speed: • Eliminate dp and dρ between Eqs., relation between velocity change and area change in isentropic duct flow:
- 24. What about the sonic point Ma 1?
- 25. Since infinite acceleration is physically impossible, the Eq. indicates that dV can be finite only when dA = 0, that is, a minimum area (throat) or a maximum area (bulge). The throat or converging diverging section can smoothly accelerate a subsonic flow through sonic to supersonic flow, as in Fig. (a). The bulge section fails; the bulge Mach number moves away from a sonic condition rather than toward it.
- 26. • Equate the mass flow at any section to the mass flow under sonic conditions (which may not actually occur in the duct): • Both the terms on the right are functions only of Mach number for isentropic flow: • Combining Eqs. • For k = 1.4
- 27. It shows that the minimum area which can occur in a given isentropic duct flow is the sonic, or critical, throat area. In many flows a critical sonic throat is not actually present, and the flow in the duct is either entirely subsonic or, more rarely, entirely supersonic.
- 28. • For given stagnation conditions, the maximum possible mass flow passes through a duct when its throat is at the critical or sonic condition. Since • The duct is then said to be choked and can carry no additional mass flow unless the throat is widened. • If the throat is constricted further, the mass flow through the duct must decrease. • The maximum flow is: • For k = 1.4 • For isentropic flow through a duct, the maximum mass flow possible is proportional to the throat area and stagnation pressure and inversely proportional to the square root of the stagnation temperature.
- 30. 9.5 Normal shock wave • A common irreversibility occurring in supersonic internal or external flows is the normal-shock wave • Such shock waves are very thin (a few micrometers thick) and approximate a discontinuous change in flow properties. • The analysis is identical to that of a fixed pressure wave.
- 31. • Using all our basic one-dimensional steady-flow relations: • Note that area term is canceled since A1 = A2, which is justified even in a variable duct section because of the thinness of the wave. • Here we have five equations for five unknowns (p, V, ρ, h, T). • Because of the velocity-squared term, two solutions are found, and the correct one is determined from the second law of thermodynamics, which requires that s2 > s1.
- 32. • The following equations were determined by making use of above equations: • The change in enthalpy (Rankine-Hugoniot relation). • Introducing perfect gas law: • Also, the actual change in entropy across the shock can be computed from the perfect gas relation:
- 33. Assuming a given wave strength p2/p1, we can compute the density ratio and the entropy change and list them as follows for k = 1.4: It is seen here that the entropy change is negative if the pressure decreases across the shock, which violates the second law. Thus a rarefaction shock is impossible in a perfect gas. It is also seen that weak-shock waves (p2/p1 = 2.0) are very nearly isentropic.
- 34. Mach-Number Relations can also be obtained from these relations:
- 38. Problems from McCabe & Smith