Exhaust velocity

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Exhaust Velocity

The exhaust velocity of a rocket engine is the relative velocity with which propellant is ejected from its nozzle. The exhaust velocity is determined by the properties of the propellant and the properties of the rocket nozzle.

Relation to Thrust

The exhaust velocity, ve, is related to the thrust by the equation:

<math>F_{T} = {dp \over dt} = m_{e} v_{e}</math>

FT is the rocket engine thrust
<math>{dp \over dt}</math> is the time derivative of the rocket’s momentum
me is the reaction mass of the rocket engine

The exhaust velocity is related to the specific impulse by the equation:

<math>I_{sp} = { F_{T} \over m_{e} g_{s} } = { v_{e} \over g_{s} }</math>

Isp is the specific impulse
gs is the acceleration due to gravity at the Earth’s surface

Relation to Gas Properties

By approximating the exhaust jet as a perfect gas (both ideal and isentropic), the maximum possible exhaust velocity of a de Laval nozzle (a type of convergent-divergent nozzle) can be estimated using the equation:

<math>v_{e} = \sqrt{{2 W \gamma R T_{o} \over \gamma - 1}( 1 - ( {P_{f} \over P_{o}} )^{\gamma - 1 \over \gamma})}</math>

W is the mean molecular weight of the exhaust
<math>\gamma</math> is the ratio of specific heats of the exhaust
R is the universal gas constant
To is the operating temperature of the rocket’s reaction chamber
Po is the operating pressure of the rocket’s reaction chamber
Pf is the exhaust pressure

No propellants produce an exhaust stream that is a perfect gas. Fortunately, the behaviors of real gases often make them slightly more effective propellants than their perfect gas approximations. That makes this approximation conservative for many propellants.

The following rule of thumb generally holds true for thermal rockets (rockets whose exhaust jet is not accelerated by an electromagnetic field):

<math>v_{e} = (250 {m \over s}) \sqrt {T_{o} \over W } {+ \over -} 10%</math>

Most real gases will have maximum possible exhaust velocities that fit in this range.

Given an exhaust stream that consists of a mixture of two gases, the mean molecular weight is:

<math> W = {W_{1} m_{1} + W_{2} m_{2} \over m_{1} + m_{2}}</math>

W1 and W2 are the molecular weights of the two components
m1 and m2 are the mass flow rates of the two components

m1 + m2 is the rocket engine’s reaction mass, me.

Relation to Nozzle Properties

The equations for ve given above are for maximum possible values. Reaching those values depends on the design of the rocket nozzle.

The effect of nozzle shape on a perfect gas exhaust stream tends to be determined by Mach number as well as the properties of the gas.

The values given by the approximations above tend to be supersonic, even at low operating temperatures. (The speed of sound in the rocket exhaust decreases with temperature; thus, at low temperatures the speed of sound is low as well.) However, accelerating from rest in the reaction chamber to supersonic values in the exhaust stream can be problematic if the nozzle is not shaped correctly.

The basic relationship between Mach number, Ma, and nozzle shape is:

<math>{dP \over dA} = {P \over A} \cdot {\gamma Ma^{2} \over {1 - Ma^{2}} } </math>

<math>dP \over dA</math> is the change in pressure per change in nozzle cross sectional area.
P is the pressure at some section of the nozzle
A is the cross sectional area at that section of the nozzle

For subsonic flow through the nozzle, Ma < 1, and therefore <math>{dP \over dA} > 0 </math>. This means that the pressure in a nozzle section will increase directly with its cross sectional area. Since the gas is accelerated from regions of high pressure toward regions of low pressure, the best nozzle shape for accelerating a subsonic flow is by tapering down the nozzle. This is the same principle as using a tapered fire hose nozzle or putting your thumb over a garden hose in order to increase the speed and range of the spray. Decreasing the area increases the flow speed.

For supersonic flows, Ma > 1, and therefore <math>{dP \over dA} < 0 </math>. This means that the pressure in a nozzle section will decrease with greater area, causing a supersonic flow to be accelerated by an expanding nozzle, not a tapered one. Decreasing the area decreases the flow speed for a supersonic nozzle. To accelerate a supersonic exhaust stream, you need to allow it to expand, not taper it down.

While you can slow down a supersonic flow to less than the speed of sound by tapering the nozzle (thereby ruining its performance in a rocket nozzle), you can never accelerate a subsonic flow to high supersonic speeds just by tapering the nozzle. As soon as the flow is accelerated past the speed of sound, the tapered nozzle begins to decelerate it. This gives you a flow that is “choked” – its velocity is restricted to the speed of sound no matter how much pressure you drive it with or how much reaction mass you try to shove through it.

One solution to this is to use a convergent-divergent nozzle, also known as a de Laval nozzle. The nozzle section leading out of the reaction chamber is tapered to accelerate subsonic flows (i.e., it converges), and sized to allow the gas to accelerate to the speed of sound. At the speed of sound, the flow begins to choke. Just past the point where the flow chokes, the nozzle then expands (i.e., it diverges), allowing the choked flow to begin accelerating again, this time to supersonic speeds. Thus, a de Laval nozzle allows the flow velocity to pass the speed of sound relatively smoothly, without too much restriction.

Since gas under sufficient pressure can be made to accelerate to the speed of sound at an abrupt restriction in a pipe, too, some rocket engines use a cylindrical reaction chamber connected directly to a divergent nozzle without trying to taper the nozzle first. This design is less efficient than the de Laval nozzle since it requires higher operating pressure, but is lighter and easier to build at large scales.

Note that the convergent-divergent nozzle is more efficient than a simple divergent nozzle because it lowers the operating pressure needed, not because it “pre-accelerates” the exhaust gas. The flow in both types of nozzles chokes at the same velocity - the speed of sound - and any prior acceleration is irrelevant, no matter how great.

A supersonic nozzle also has a critical ratio of exhaust pressure to reaction pressure, below which no increase in operating pressure or decrease in exhaust pressure will improve its exhaust velocity.

<math>{P_{f} \over P_{o}} = {2 \over \gamma + 1}^{\gamma \over \gamma - 1}</math>

The critical exhaust pressure for a given operating pressure is always greater than that of vacuum. Further, since the pressure of the exhaust stream decreases as the nozzle diverges, there is no advantage in extending the divergent section of the nozzle past the point at which the critical exhaust pressure is reached. The critical pressure ratio determines the optimum size of the nozzle.

For a rocket operating in an atmosphere whose pressure is greater than its critical exhaust pressure (e.g., Earth’s surface pressure is greater than the critical exhaust pressure of many rocket engines), its exhaust velocity will decrease if the reaction pressure is not increased to compensate. However, increasing the reaction pressure beyond the critical reaction pressure defined by the critical pressure ratio will no longer improve the exhaust velocity.

Because of their higher operating pressures, simple divergent nozzles are better suited for operation in high pressure environments, such as the Earth’s surface. De Laval nozzles are best suited for low pressure environments, such as outer space.

Part of the art of rocket engine design lies in selecting rocket nozzle parameters that allow the engine to approach its maximum theoretical exhaust velocity as closely as possible.

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