Adiabatic conditions refer to conditions under which overall heat transfer across the boundary between the thermodynamic system and the surroundings is absent. Examples of processes proceeding under adiabatic conditions and applied in engineering are expansion and compression of gas in a piston-type machine, the flow of a fluid medium in heat-insulated pipes, channels and nozzles, throttling and setting of turbomachines and distribution of acoustic and shock waves. The flow of a viscous liquid or gas through a heat-insulated channel is often referred to as adiabatic.
A reversible adiabatic process in the thermodynamic system (see Thermodynamics) working only in expansion is described by the differential equation:
i.e., the process is also isentropic (ds = 0, s = const). Equation (1) implies the following relations:
where h = u + pv is the enthalpy, cp and cv are the heat capacities at a constant pressure and volume, respectively. γ is referred to as an isentropic exponent (or adiabatic exponent, which is less strict). If γ = const the system states are described by an adiabatic (Poisson) equation
Equation (3) is used for irreversible adiabatic processes too. In the case of gas expansion, an actual adiabatic exponent γ' is within 1 < γ < γ and in the case of gas compression, γ' > γ. For solids and liquids, the values of γ are extremely high and change appreciably with the temperature. For instance, for water at 0°C γ = 3602000 and at 100°C γ = 22300. For perfect gases, (∂ln p/∂ln ν)T = −1, such that γ = cp/cv. For a monatomic gas, ; for atomic gas, 1.40; for triatomic and polyatomic gases, 1.29. The variation of g with temperature and pressure for gases is small except near the saturation curve and when the gas is in a dissociated and ionized state.
For constant γ, the work for system expansion in an adiabatic process is given by:
for a perfect gas.
Useful work, that is the difference between the work of expansion and the work of pushing the working medium, is given by:
where wf is the friction work (wf ≥ 0).
Adiabatic throttling is the adiabatic limit irreversible process for reducing the fluid pressure. Under adiabatic throttling, h = const. The variation of the system thermal state in adiabatic throttling is described by the equation:
where μJT is the differential throttling (Joule-Thompson) coefficient. Variation of temperature in throttling is:
For a perfect gas, μJT = 0 and T2 = T1. For imperfect fluids, μJT may be negative or positive, and may change from negative to positive depending on p and T. The locus of this change in sign of μTR forms, the p - T diagram, an “inversion curve” of a given substance. The curves of adiabatic gas expansion and compression plotted in p - v coordinates are shown in Figure 1.
An adiabatic flow of fluid in pipes and channels is described by the energy equation:
where u is the flow velocity, z the height above the ground level, and w' the useful work input or output. Equation (9) holds for both frictionless flows and flows with friction. In the particular case of a reversible adiabatic flow of incompressible fluid (v = const, dh = v dp) and in the absence of useful work (dw' = 0), an integral of Eq. (3) has the form:
and is known as Bernoulli’s Equation or integral. ρu2/2 is the dynamic pressure and ρgz the hydrostatic pressure. The sum p0 = p + ρu2/2 is called the stagnation pressure, p0. The stagnation pressure can be measured by the Pitot Tube and the difference p0 - p by Pitot-Prandtl tube (which has a static reference tapping on the probe). For an adiabatic flow of comprehensive gas that performs no useful work (variation in gz is commonly neglected), from Eq. (9) we have:
h0 is called the stagnation enthalpy. Propagation of small disturbances in an elastic medium, i.e., acoustic waves, can be assumed adiabatic and isoentropic, which leads to the Laplace equation for the velocity of sound:
In the case of a perfect gas , where Ri is the gas constant for the specific substance. The finite Velocity of Sound is responsible for the critical regime of gas flow in the channel under which the local sound velocity and the maximum flow rate of gas (for given initial parameters) are achieved in the channel throat.
The increase in sound velocity with temperature causes large disturbances to propagate in gaseous media as shock waves travel, in relation to the undisturbed medium, with a supersonic velocity. This is because of the elevation in temperature in the compressed region. The shock wave is an irreversible adiabatic process of substance compression, accompanied by entropy growth. The variations of specific volume and pressure for perfect gas during the passage of the shock wave are related by the shock adiabatic equation (Hugoniot adiabat):
This equation makes it possible, using the Clapeyron equation, to determine the temperature variation.
There is much current interest in the adiabatic processes of high-velocity viscous fluid flow over an insulated wall. Friction brings about an elevation of wall temperature to a value at which there is a balance between heat generation and heat release to the external flow. This temperature is referred to as the Adiabatic Wall Temperature (see separate item in this topic) and characterizes an aerodynamic heating of the surface in the flow.