User:Milton Beychok/Sandbox: Difference between revisions

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==Velocity of gases exiting the rocket engine nozzle exhaust==


The purpose of a propelling nozzle in a rocket engine is to expand and accelerate the [[combustion]] product gases from burning [[propellant]]s so that the gases are exhausted from the nozzle exit at [[supersonic]] [[Velocity|velocities]]. Most usually, rocket propelling nozzles are of the [[De Laval nozzle|de Laval]] type as depicted in the adjacent Figure 1. 
The analysis of gas flow exit velocity from de Laval nozzles involves a number of concepts and assumptions:
* For simplicity, the combustion gas is assumed to be an [[ideal gas]].
* The gas flow is isentropic (i.e., at constant [[entropy (thermodynamics)|entropy]]), [[Sliding friction|frictionless]], and [[adiabatic]] (i.e., there is little or no [[heat]] gained or lost)
* The gas flow is constant (i.e., steady) during the period of the propellant burn.
* The gas flow is along a straight line from gas inlet to exhaust gas exit (i.e., along the nozzle's axis of symmetry)
* The gas flow behavior is that of a [[compressible fluid]] (i.e., gas at high velocity).
As the combustion gas enters the rocket nozzle, it is traveling at [[subsonic]] velocities. As the throat contracts down the gas is forced to accelerate by the [[venturi effect]] until at the nozzle throat, where the cross-sectional area is the smallest, the linear velocity becomes sonic (i.e., attains the [[speed of sound]]). From the throat the cross-sectional area then increases, the gas expands and the linear velocity becomes progressively more supersonic.
The linear velocity of the exiting exhaust gases can be calculated using the following equation <ref name=Nakka>[http://members.aol.com/ricnakk/th_nozz.html Richard Nakka's Equation 12]</ref><ref name=Braeuning>[http://www.braeunig.us/space/propuls.htm#intro Robert Braeuning's Equation 2.22]</ref><ref>{{cite book|author=Sutton, George P.|title=Rocket Propulsion Elements: An Introduction to the Engineering of Rockets|edition=6th Edition|publisher=Wiley-Interscience|year=1992|isbn=0471529389|page=636}}</ref>
:<math>V_e = \sqrt{\;\frac{T\;R}{M}\cdot\frac{2\;k}{k-1}\cdot\bigg[ 1-(P_e/P)^{(k-1)/k}\bigg]} </math>
{| border="0" cellpadding="2"
|-
|align=right|where:
|&nbsp;
|-
!align=right|<math>V_e</math>
|align=left|=&nbsp; Exhaust velocity at nozzle exit, m/s
|-
!align=right|<math>T</math>
|align=left|=&nbsp; absolute [[temperature]] of inlet gas, K
|-
!align=right|<math>R</math>
|align=left|=&nbsp; [[gas constant|Universal gas law constant]] = 8314.5 J/(kmol·K)
|-
!align=right|<math>M</math>
|align=left|=&nbsp; the gas [[molecular mass]], kg/kmol&nbsp; &nbsp; (also known as the molecular weight)
|-
!align=right|<math>k</math>
|align=left|=&nbsp; <math>c_p/c_v</math> = [[Specific heat ratio|isentropic expansion factor]]
|-
!align=right|<math>c_p</math>
|align=left|=&nbsp; [[specific heat]] of the gas at constant pressure
|-
!align=right|<math>c_v</math>
|align=left|=&nbsp; specific heat of the gas at constant volume
|-
!align=right|<math>P_e</math>
|align=left|=&nbsp; [[pressure|absolute pressure]] of exhaust gas at nozzle exit, [[pascal (unit)|Pa]]
|-
!align=right|<math>P</math>
|align=left|=&nbsp; absolute pressure of inlet gas, Pa
|}
Some typical values of the exhaust gas velocity '''''V<sub>e</sub>''''' for rocket engines burning various propellants are:
* 1.7 to 2.9&nbsp;km/s (3800 to 6500&nbsp;mi/h) for liquid [[monopropellant]]s
* 2.9 to 4.5&nbsp;km/s (6500 to 10100&nbsp;mi/h) for liquid [[bipropellant]]s
* 2.1 to 3.2&nbsp;km/s (4700 to 7200&nbsp;mi/h) for [[solid rocket|solid propellant]]s
As a note of interest, '''''V<sub>e</sub>''''' is sometimes referred to as the ''ideal exhaust gas velocity'' because it based on the assumption that the exhaust gas behaves as an ideal gas.
As an example calculation using the above equation, assume that the  propellant combustion gases are: at an absolute pressure entering the nozzle of '''''P''''' = 7.0 MPa and exit the rocket exhaust at an absolute pressure of '''''P<sub>e</sub>''''' = 0.1 MPa; at an absolute temperature of '''''T''''' = 3500 K; with an isentropic expansion factor of '''''k''''' = 1.22 and a molecular mass of '''''M''''' = 22&nbsp;kg/kmol. Using those values in the above equation yields an exhaust velocity '''''V<sub>e</sub>''''' = 2802 m/s or 2.80&nbsp;km/s which is consistent with above typical values.
The technical literature can be very confusing because many authors fail to explain whether they are using the universal gas law constant ''R'' which applies to any ideal gas or whether they are using the gas law constant ''R''<sub>s</sub> which only applies to a specific individual gas.  The relationship between the two constants is ''R''<sub>s</sub> = ''R''/''M''.

Revision as of 10:21, 22 March 2010

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