Nozzles are profiled ducts for speeding up a liquid or a gas to a specified velocity in a preset direction. Nozzles are used in the rocket and aircraft engineering to produce jet propulsion, in intensive shattering and spraying technologies, in jet devices and ejectors and in gas dynamic lasers and gas turbines (see Gas turbine). Nozzles are the basic components of wind tunnels (see Wind Tunnels), and they allow the design of systems of intensive (jet) cooling, such as in electrical engineering.

Nozzles are extremely diverse in geometry (Figure 1): round (axisymmetric) and two-dimensional, annular and tray, with oblique and right-angle exit sections, etc. The diversity of nozzle contours enables one to obtain a high degree of outflow uniformity both in absolute value and the divergence angle of the velocity vector, which is of prime importance for increasing jet propulsion and for modeling flow around aircraft and rockets.

**Figure 1b. Conical nozzles with pin shaped plug (b) Aerospike nozzles with pin shaped plug, (c) Tray nozzles, (d) Tray nozzles with backward flow.**

Axisymmetric and two-dimensional nozzles of the simplest shape are smoothly converging and then diverging ducts (see Figure 3). Known as Laval nozzles, they were named after a Swedish engineer who was the first to design them in 1889 for generating supersonic water vapor jets to rotate an impeller in a steam turbine.

We shall first discuss the principal laws governing an adiabatic flow of an ideal gas in the Laval nozzle. In order to describe the gas state we make use of the Mendeleev-Clapeyron equation

and the adiabatic equation p/r^{γ} = const, where p, ρ, and T are the gas pressure, density, and temperature, R the universal gas constant,
the gas molecular mass, and γ the adiabatic exponent which, for an ideal gas, is determined as the ratio of heat capacities at a constant pressure and volume γ = c_{p}/c_{v} with c_{p} – c_{v} =
. The sound velocity is determined as

Let the nozzle axis be rectilinear and horizontal and the deflection of the stream velocity vector from the axis be negligible. In each section x = const (Figure 3) the gas velocity, pressure, density, and temperature take constant values (a one-dimensional flow model).

Then the mass, momentum, and energy conservation laws take the form

for a nonfrictional and non-heat-conducting flow). Here v(x) is the velocity, S(x) the cross section area of the nozzle, and u = ∫c_{v}dT = c_{v}T the internal gas energy related to enthalpy by h = ∫c_{v}dT = c_{v}dT = (c_{v} +
)T = u + p/ρ. The integral of Eq. (1) is the constant gas flow rate
= ρvS = const. The integral of Eq. (2) is known as the Bernoulli integral. For the adiabatic process it is

where p_{0} and ρ_{0} are the values for the gas brought to rest, i.e., the "stagnation" values. The integral of the energy conservation equation v^{2}/2 + h = h_{0} = const makes it possible to introduce the concept of an overall enthalpy of gas flow h0 which corresponds to the total energy of the gas brought to rest (the stagnation enthalpy). Accordingly, T_{0} = h_{0}/c_{p}, or the total gas temperature is linked with p_{0} and ρ_{0} by the adiabatic equation
and by the equation describing the state of the ideal gas a p_{0} =
.

Correlating gas parameters in different nozzle sections, we can derive from the energy conservation law

Here Ma = v/a is the dimensionless ratio of the flow velocity to the sound velocity referred to as the Mach number (see Mach Number). Since

then

Thus, pressure and density monotonically decrease with increasing local value of the Mach number

Table 1 presents the relative values of gas temperature, pressure, and density for various Mach numbers Ma and γ = 1.4 (the value of γ for air). As is seen from Table 1, at moderate Mach numbers the temperature drops approximately as Ma, the pressure drops as Ma^{3.5}, and the density drops as Ma^{2.5}. As Ma increases, the exponents grow tending to 2, 7 and 5 for γ = 1.4.

In contrast to pressure and density, the temperatures of gases and gaseous mixtures cannot diminish indefinitely when they are accelerated in the nozzle. This is because condensation, i.e., a transition of the gas molecules to a condensed state occurs. Even for dry air, where drops of water cannot be formed, liquid oxygen droplets show up in the flow at a temperature ~ 90 K and nitrogen begins to condense at T = 77 K. To avoid this the air in the nozzle inlet is heated to a temperature T_{0} that depends on the required value of the Mach number.

Figure 4 graphs the dependence of the minimum temperature T_{0} and pressure p_{0} on the Ma number of the air stream, which during expansion, does not bring about oxygen condensation. The dependence on pressure p is due to the change of the gas condensation (saturation) temperature with pressure along the saturation curve (see Vapor Pressure). An account for this phenomenon requires a joint (solution) of the set of equations

The laws governing the contour or the cross section area S(x) of the nozzle can be derived by combining Eqs. (1) and (2). Eliminating density and pressure yields an equation for the Laval nozzle

In a subsonic flow (Ma < 1) an increase in velocity (dv > 0) can be achieved only by diminishing of the cross section area dS < 0. Conversely, expanding the duct, we cause the gas flow velocity to decrease. However, after the sound velocity in the flow (M > 1) has been attained, the law of variation of the cross section area is reversed. The minimum of function S(v) corresponds to the nozzle throat S* at which Ma = 1 (Figure 5).

**Figure 4. Temperature T _{0} to which air entering a nozzle must be preheated in order to prevent condensation of oxygen at the throat.**

The one-dimensional inviscid gas model is insufficient for the accurate determination of the nozzle contour, i.e., find the law of variation of the cross section area S(x). In addition, in order to provide a highly uniform flow at the nozzle exit section, the angles and radii of the convergent and divergent section of the nozzle must be carefully selected to match one another. There exists an easy technique of shaping the nozzle subsonic section. It is based on using a conical flow with an inclination angle a of the wall (Figure 6). As a rule, α < 24°. The radii r_{0} and r* must also be specified as initial data for the inlet section and a throat. In the entrance section a cylindrical prechamber smoothly matches the conical section. Here the law of profiling is

where z = π/2/x/r_{00} tan α.

The length of this section is determined by the range of variation of the cylindrical coordinate x

After the conical section the flow must be changed into a cylindrical one, the length of the third section being

and the contour being described by the function r(x) = r* + x tan α

where z = πx tan α/2r*.

The minimum admissible ratio of the inlet section radius to the throat radius n = r_{0}/r* = 2.015 yields the contour of the subsonic section without using a conical section. However, a low value of contraction ratio n does not guarantee suppresion of flow fluctuations at the nozzle outlet.

A universal contour can also be recommended for the supersonic section with a conical flow with an angle α < 15°. In this case the law of profiling is

Numerical calculations demonstrate that the nozzle so designed is advantageous within a wide range of gas composition and with an adiabatic exponent in the range 1.13 < γ < 1.67.

In engineering, great emphasis is placed on problems related to particle acceleration in the nozzle and, in particular, finding an optimized profile or, similarly, optimum distribution throughout the nozzle length of gas dynamic parameters (gas velocity or pressure).

The one-dimensional equation of motion of a solid and accelerating particle, motion in an ideal gas flow is as follows

Here v_{p}, ρ_{p}, and d_{p} are the velocity, density, and diameter of a particle, C_{d} the drag coefficient of a particle,
, λ = v/a_{cr}.

The dimensionless gas velocity λ depends on the nozzle geometry which allows us to consider it as a parameter which may be varied with the aim of designing nozzles with high performance characteristics.

It is required to find the gas velocity distribution which ensures the particle speeds up from
to the required value of
in the minimum length (x – x_{0}). This variational problem consists in seeking the minimum of the function

Assuming C_{d} = const and introducing new variables
and
reduces the solution of this problem to taking an integral

For γ = n — 1/n – 2, where n = 3,4,5, ... the solution can be obtained in an analytical form

Speeding up the particle to the specified velocity in the minimum length obviously requires the maximum aerodynamic force on the particle. This force, in the model postulated in which C_{d} = const, depends only on the particle and gas velocities, which follows from the equation of motion (Eq. (6)). It is not difficult to separate out two limiting cases when a particle with the velocity
does not at all interact with the gas flow (dv_{p}/dx = 0). The first is observed in equilibrium flow of particles and gas λ =
, and the second occurs at the maximum gas velocity
when its density diminishes and becomes zero. Within the range,
< λ <
, there is a certain
-depending value of λ* at which a particle acceleration
is maximum. In the optimized nozzle it must be maximum throughout the particle path. Fulfilling these conditions yields Eq. (7).

We now consider the operation of this optimized nozzle under off-design regimes. All the factors leading to off-design regimes, except for an isentropic exponent, are incorporated in a generalized complex m = C_{d}ρ_{0}/r_{p}d_{p}. It has a dimension which is the reciprocal of length and as a scale factor it is involved in a dimensionless coordinate of solution (7). Optimization is achieved only for a given, quite specific, value m = m*. In this case the particle possesses, in each nozzle section, the velocity for which the flow parameters provide the maximum acceleration. We denote this velocity by
.

If the equality m = m* does not hold, for instance, due to deviation of particle size or gas density in the prechamber from the estimated values, this nozzle is no longer optimized. For m > m* particle velocities are higher than and lie within the range < < λ. For m < m* the particle begins to lag behind < and, failing to gather velocity at the initial portion of the path, it will steadily deviate from the optimum speadup parameters. Thus, a polydisperse flow is characterized by a broadened range of particle velocities.

We note one more specific feature of the solution obtained. Equation (8) relates those instantaneous dimensionless velocities of gas and particles which make possible the most rapid speedup of particles and implies that holds for each . In other words, an optimized nozzle is supersonic everywhere, while actual nozzles have a subsonic section and a supersonic section . Theoretically the subsonic section can be made arbitrarily short by using steep walls and changing to a multinozzle structure (an assembly of nozzles). In actual fact, this can be advisable, for instance, for avoiding particle overheating by the gas flow since the particle temperature rise is predominantly in the subsonic section.

The profile of the optional nozzle is closely linked with the distribution of particle velocity in the acceleration of a polydisperse flow. If relatively low particle velocities that correspond to the entrance section of the nozzle are needed, the particle velocity spectrum in the polydisperse flow will be strongly blurred due to a fairly broad range of particle velocities and high gradients of their variation along the flow. To equalize the velocities, a constant gas velocity section is needed in which the velocities of particles of various size will be equalized and approach the gas velocity. Here, the Re numbers of the particles, become smaller because the difference in velocity between the gas and the particles is small. C_{d} can no longer be taken as constant and should be calculated from:

which allows for the dependence of the drag coefficient on Re (see Drag Coefficient).

The particle motion equation is written as

or, after transformations,

This equation has an analytical solution

where

The solution allows us to find the length of the section with constant parameters of the gas flow at which the particle is speeded up from velocity η_{0} to velocity η.

The physics of the flow of a medium in a nozzle is substantially more complex if viscosity becomes significant, if the gas is chemically reactive, electromagnetic fields appear, or the concentration of the disperse phase is high.