Flow of Compressible Fluids

Definition A compressible flow is a flow in which the fluid density ρ varies significantly within the flowfield. Therefore, ρ(x, y, z) must now be treated as a field variable rather than simply a constant.

The main difference between the flow behavior of incompressible and compressible fluids, and between the equations that govern them, is the effect of variable density, e.g., the dependence of density upon pressure and temperature. At low velocities (relative to the speed of sound), relative changes in pressure and associated effects are often small and the assumption of incompressible flow with a constant (average) density may be reasonable. It is when the gas velocity approaches the speed at which a pressure change propagates (i.e., the speed of sound) that the effects of compressibility become the most significant.

Definition Many of the relations developed for incompressible (i.e. low speed) flows must be revisited and modified. For example, the Bernoulli equation is no longer valid,

Definition The following table compares the variables and equations which come into play in the two cases

Flow of Compressible Fluids Some Thermodynamic Concepts

Basic Equations Momentum Equation (Momentum conservation) Continuity Equation (Mass conservation) Total Energy Balance, Mechanical Energy Balance (Energy Conservation) State Equation

Internal Energy and enthalpy The law of conservation of energy involves the concept of internal energy, which is the sum of the energies of all the molecules of a system In fluid mechanics we employ the specific internal energy, denoted by u, which is defined for each point in the flowfield. A related quantity is the specific enthalpy, denoted by h, and related to the other variables by h = u + pv The units of u and h are (velocity) 2, or m 2 /s 2 in SI units.

Internal Energy and enthalpy For a calorically perfect gas, which is an excellent model for air at moderate temperatures, both u and h are directly proportional to the temperature. Therefore we have u = c v T h = c p T where c v and c p are specific heats at constant volume and constant pressure, respectively. h − u = pv = (c p − c v )T

Internal Energy and enthalpy comparing to the equation of state, we see that c p − c v = R Defining the ratio of specific heats, γ ≡ c p /c v Notes: k = γ; R = Gas constant / Molecular weight

Thermodynamics Concepts

Thermodynamic Relationships

Continuity equation Total-energy balance Mechanical energy balance

Stagnation temperature From Total Energy Balance

Isothermal Condition The isothermal (constant temperature) condition may be approximated, for example, in a long pipeline in which the residence time of the gas is long enough that there is plenty of time to reach thermal equilibrium with the surroundings. Under these conditions, for an ideal gas,

Isentropic

Adiabatic/ isentropic condition The adiabatic condition occurs, for example, when the residence time of the fluid is short as for flow through a short pipe, valve, orifice, etc. and/or for well-insulated boundaries. When friction loss is small, the system can also be described as locally isentropic. It can readily be shown that an ideal gas under isentropic conditions obeys the relationship

Speed of sound (acoustical velocity) Sound is a small-amplitude compression pressure wave, and the speed of sound is the velocity at which this wave will travel through a medium. An expression for the speed of sound can be derived as follows.

Speed of sound The conservation of mass principle applied to the flow through the wave reduces to

Speed of sound Likewise, a momentum balance on the fluid ‘‘passing through’’ the wave is which becomes, in terms of the parameters

Speed of sound Eliminating V and solving for c 2 gives For an ideal gas, it reduces to

Mach number Mach number (Ma) is defined as the ratio of u, the speed of the fluid to c, the speed of sound in the fluid under conditions of flow M<1 Subsonic flow M=1 Transonic flow M>1 Supersonic flow

Mach Number Relationship

Example An object is immersed in a air-flow with static pressure of 200 kPa absolute, a static temperature of 20 o C and a velocity of 250 m/s. What are the pressure and temperature at the stagnation point?

Example2 An airflow at Ma= 0.75 passes through a conduit with a cross section area of 50 cm 2. The total absolute pressure is 360 kPa and the total temperature is 10 o C. Calculate the mass flow rate through the conduit.

Process of Compressible Flow Isentropic flow Adiabatic friction flow Isothermal friction flow

Isentropic expansion The cross-sectional area of the conduit must change The stagnation temperature does not change in the conduit

Isentropic compressible flow

Example1 (isentropic flow) Suppose we are designing a supersonic wind tunnel to operate with air at a Mach number of 3. If the throat area is 10 cm 2, what must be the cross-sectional area of the test section to be? Assume k = 1.4 for air.

Example2 (Isentropic Flow) Air enters a convergent-divergent nozzle at a temperature of K and a pressure of 20 atm. The throat area is half that of the discharge of the divergent section (a) Assuming the Mach number in the throat is 0.8, what are the value of p, T, v, ρ and G at the throat? (b) What are the value of p*,T*,u*,G* (c) What is the maximum Mach number at the discharge of the divergent area? Given γ = 1.4 and MW = 29

Adiabatic friction flow Adiabatic friction flow through a pipe of constant cross section This process is irreversible, and the entropy of the gas increases Since Q=0, the stagnation temperature is constant throughout the conduit

Adiabatic Compressible Flow Maximum Conduit Length

Example (Adiabatic Flow) Air flows from a reservoir through an isentropic nozzle into a long, straight pipe. The pressure and temperature in the reservoir are 20 atm and K, respectively, and the Mach number at the entrance of the pipe is (a) What is the value of fL max /r H ? (b) What are the pressure, temperature, density, linear velocity, and mass velocity when L b = L max ? (c) What is the mass velocity when fL max /r H = 400?

Isothermal friction flow Isothermal friction flow through a pipe of constant cross-sectional area Flow of heat through the pipe wall sufficient to keep the temperature constant The stagnation temperature changes during the process, since T is constant

Isothermal Friction Flow

Example Air at 1.7 atm gauge and 15 o C enters a horizontal 75-mm steel pipe that is 70 m long. The flow rate of the entering air q is m 3 /s. Assuming isothermal flow, what is the pressure at the discharge end of the pipe?

Practical Example