🧊Thermodynamics II Unit 11 – Compressible Fluid Flow
Compressible fluid flow is a crucial area in thermodynamics, focusing on fluids that experience significant density changes due to pressure and temperature variations. This unit covers key concepts like Mach number, stagnation properties, and isentropic flow, as well as fundamental equations for continuity, momentum, and energy.
The study explores different types of compressible flow, including internal and external flows, and various flow regimes from subsonic to hypersonic. It also delves into shock waves, expansion waves, and their applications in nozzles, diffusers, and engineering systems like jet engines and rocket propulsion.
Compressible fluid flow involves fluids with significant density changes due to variations in pressure and temperature
Mach number (M) represents the ratio of the fluid's velocity to the local speed of sound and characterizes the compressibility of the flow
Subsonic flow: M<1
Sonic flow: M=1
Supersonic flow: M>1
Stagnation properties refer to the fluid properties (pressure, temperature, and density) that would be achieved if the fluid were brought to rest isentropically
Critical properties are the fluid properties at which the Mach number equals 1 (sonic conditions)
Isentropic flow assumes no heat transfer or friction, and the entropy remains constant throughout the flow
Fanno flow is an adiabatic flow with friction, leading to entropy increase along the flow path
Rayleigh flow is a frictionless flow with heat transfer, causing entropy to change along the flow path
Fundamental Equations
Continuity equation expresses the conservation of mass in a compressible flow: ∂t∂ρ+∇⋅(ρV)=0
Momentum equation represents the conservation of momentum: ρDtDV=−∇p+∇⋅τ+ρg
Energy equation describes the conservation of energy: ρDtDe=−∇⋅q−p(∇⋅V)+ϕ
Equation of state relates pressure, density, and temperature for a given fluid (ideal gas law: p=ρRT)
Speed of sound equation: a=ργp, where γ is the specific heat ratio
Isentropic relations for pressure, temperature, and density ratios as functions of Mach number
Fanno and Rayleigh flow equations relate fluid properties along the flow path
Types of Compressible Flow
Internal flow occurs within a confined space, such as pipes, ducts, or nozzles
Characterized by the presence of solid boundaries that guide the flow
External flow occurs around objects immersed in a fluid, such as airfoils, vehicles, or buildings
Characterized by the absence of solid boundaries confining the flow
Steady flow implies that fluid properties at any point do not change with time (∂t∂=0)
Unsteady flow occurs when fluid properties at a point vary with time (∂t∂=0)
One-dimensional flow assumes that fluid properties vary only in the flow direction, neglecting variations in other directions
Two-dimensional and three-dimensional flows consider variations in fluid properties in multiple spatial dimensions
Compressible flow regimes:
Subsonic (M<1)
Transonic (M≈1)
Supersonic (M>1)
Hypersonic (M>>1, typically M>5)
Shock Waves and Expansion Waves
Shock waves are thin regions of abrupt changes in fluid properties (pressure, density, temperature, and velocity) that occur when a supersonic flow encounters an obstacle or a sudden change in flow direction
Normal shock waves are perpendicular to the flow direction
Oblique shock waves form at an angle to the flow direction
Shock wave properties can be calculated using normal and oblique shock relations, which relate the fluid properties across the shock
Expansion waves occur when a supersonic flow encounters a sudden expansion or change in flow direction, leading to a decrease in pressure, density, and temperature, and an increase in velocity
Prandtl-Meyer expansion fans are a series of infinitesimal expansion waves that turn the flow gradually, maintaining isentropic conditions
Shock-expansion theory combines shock wave and expansion wave analysis to predict the flow field around supersonic objects (wedges, cones)
Shock-boundary layer interaction can lead to flow separation, increased drag, and heat transfer rates
Nozzles and Diffusers
Nozzles are devices used to accelerate a compressible fluid by converting pressure energy into kinetic energy
Converging nozzles accelerate subsonic flow and decelerate supersonic flow
Diverging nozzles accelerate supersonic flow and decelerate subsonic flow
Converging-diverging (CD) nozzles can accelerate a fluid from subsonic to supersonic velocities
Nozzle flow regimes depend on the pressure ratio between the inlet and outlet:
Subsonic flow throughout the nozzle
Choked flow at the throat (sonic conditions)
Supersonic flow in the diverging section
Over-expanded or under-expanded flow at the exit
Diffusers are devices used to decelerate a compressible fluid and increase its pressure
Subsonic diffusers have a gradually increasing cross-sectional area
Supersonic diffusers often employ a combination of shock waves and area changes to decelerate the flow efficiently
Nozzle and diffuser performance is characterized by parameters such as thrust, mass flow rate, and efficiency
Applications in Engineering
Jet engines (turbojets, turbofans) rely on compressible flow principles for propulsion
Inlet, compressor, combustion chamber, turbine, and nozzle components
Rocket engines use converging-diverging nozzles to accelerate the exhaust gases to supersonic velocities
Supersonic wind tunnels employ nozzles and diffusers to generate and control high-speed flow for aerodynamic testing
Compressible flow analysis is essential in the design of high-speed vehicles (aircraft, spacecraft, missiles)
Gas pipelines and distribution systems involve compressible flow in the transport of natural gas and other compressible fluids
Compressed air systems, such as pneumatic tools and air brakes, utilize compressible flow principles
Refrigeration and air conditioning systems involve the compression and expansion of refrigerants, which are modeled as compressible fluids
Shock wave phenomena are relevant in explosion and blast wave analysis, as well as in medical applications (lithotripsy)
Problem-Solving Techniques
One-dimensional isentropic flow analysis using isentropic relations and tables
Normal and oblique shock wave calculations using shock relations and tables
Prandtl-Meyer expansion fan calculations using Prandtl-Meyer function and tables
Fanno flow analysis using Fanno flow equations and tables
Rayleigh flow analysis using Rayleigh flow equations and tables
Nozzle and diffuser design calculations, including area ratios, pressure ratios, and mass flow rates
Shock-expansion theory for analyzing supersonic flow over wedges and cones