Thermodynamics II

🧊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.

Key Concepts and Definitions

  • Compressible fluid flow involves fluids with significant density changes due to variations in pressure and temperature
  • Mach number (MM) represents the ratio of the fluid's velocity to the local speed of sound and characterizes the compressibility of the flow
    • Subsonic flow: M<1M < 1
    • Sonic flow: M=1M = 1
    • Supersonic flow: M>1M > 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: ρt+(ρV)=0\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \vec{V}) = 0
  • Momentum equation represents the conservation of momentum: ρDVDt=p+τ+ρg\rho \frac{D\vec{V}}{Dt} = -\nabla p + \nabla \cdot \tau + \rho \vec{g}
  • Energy equation describes the conservation of energy: ρDeDt=qp(V)+ϕ\rho \frac{De}{Dt} = -\nabla \cdot \vec{q} - p(\nabla \cdot \vec{V}) + \phi
  • Equation of state relates pressure, density, and temperature for a given fluid (ideal gas law: p=ρRTp = \rho RT)
  • Speed of sound equation: a=γpρa = \sqrt{\frac{\gamma p}{\rho}}, where γ\gamma 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\frac{\partial}{\partial t} = 0)
  • Unsteady flow occurs when fluid properties at a point vary with time (t0\frac{\partial}{\partial t} \neq 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<1M < 1)
    • Transonic (M1M \approx 1)
    • Supersonic (M>1M > 1)
    • Hypersonic (M>>1M >> 1, typically M>5M > 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
  • Numerical methods for solving complex compressible flow problems (finite difference, finite volume, finite element)
    • Computational Fluid Dynamics (CFD) simulations
  • Experimental techniques for measuring compressible flow properties (pressure probes, temperature sensors, Schlieren imaging, Particle Image Velocimetry)

Real-World Examples

  • Supersonic aircraft (Concorde, SR-71 Blackbird) and their aerodynamic design challenges
  • Jet engines in commercial and military aviation (turbofans, turbojets, ramjets, scramjets)
  • Rocket engines for space launch vehicles (SpaceX Falcon 9, NASA Space Shuttle)
  • High-speed wind tunnels for testing aircraft, vehicles, and components (NASA Glenn Research Center, AEDC)
  • Supersonic inlets and nozzles in jet engines and rocket engines (Bell X-1, North American X-15)
  • Shock waves generated by explosions, blast waves, and supersonic projectiles
  • Shock wave phenomena in medical applications (extracorporeal shock wave lithotripsy for kidney stone treatment)
  • Natural gas pipelines and compressor stations for long-distance transport
  • High-pressure compressed air systems in industrial applications (pneumatic tools, air brakes)
  • Supersonic flow in valve and regulator designs for pressure control in various industries


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© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.