Factors Affecting Characteristic Length of the Combustion Chamber of Liquid Propellant Rocket Engines

Optimum characteristic length of the combustion chamber of liquid rocket engine is very important to get higher energy from the liquid propellants. Characteristic length is defined by the time required for complete burning of fuel. Combustion reactions are very fast and combustion is evaporation dependent. This paper proposes fuel droplet evaporation model for liquid propellant rocket engine and discusses the factors which can affect the required size of characteristic length of the combustion chamber based on proposed model. The analysis is performed for low temperature combustion chamber. A computer code based on proposed model is generated, which solve analytical equations to calculate combustion chamber characteristic length under various input conditions. The analysis shows that characteristic length is affected by combustion chamber temperature, pressure, fuel droplet diameter, chamber diameter, mass flow rate of propellants and relative velocity of the droplet in the combustion chamber.

V cc is volume of the chamber up to throat. Characteristic length has normally value between 0.8 and 3 meter. It can be higher in some conditions. It is even higher for some

Factors Affecting Characteristic Length of the Combustion Chamber of Liquid Propellant Rocket Engines
monopropellant [1]. Normally characteristic length for a new combustion chamber is selected on the basis of experiments or data available in literature of successful combustion chambers. Values of characteristic lengths for some propellants are shown in Table 1 [2][3]. Mulkey et. al. [4] reported decrease in characteristic velocity with increase in characteristic length without insulation due to heat losses through copper wall of the chamber.
Jo et. al. [5]  propellants which is misguiding. Table 1  Characteristic length is given as follow [7].  In spray combustion, combustion is controlled by fuel droplet evaporation rate. The speed of evaporation depends on chamber temperature, chamber pressure, initial fuel droplet temperature, size of fuel droplet, type of propellants, relative velocity of fuel droplet, ambient gas transport properties and turbulent intensity [8].

SINGLE DROPLET EVAPORATION
Combustion takes place in gaseous phase, therefore evaporation of fuel is very important process for the performance of liquid fueled combustion systems, such as gas turbines, automobiles and propulsion systems. A lot of work has been done on the evaporation of spray and droplet evaporation [9][10][11][12][13].
In liquid combustion systems, the spray is dense and droplets are close to one another. Modeling or experimental study of such spray is difficult; therefore, modeling of single droplet is performed, which gives a very good picture of actual problem [14]. The modeling of evaporation of single droplet is simplification of real complex spray evaporation. Single fuel droplet is studied in past and will be studied in future in quiescent state, gas moving around the droplet and turbulent condition around the droplet.

Stagnant Condition or no Relative Movement: At stagnant
condition there is no movement of the drop relative to ambient gas. Heat transfer takes place due to conduction, radiation and natural convection in such condition. Most of the work for spray or droplet evaporation is done at stagnant condition. The work of Godsave in 1950s [15] led to the quasi steady state model, which is also called d 2 law.

Factors Affecting Characteristic Length of the Combustion Chamber of Liquid Propellant Rocket Engines
Convective Conditions: When there is relative motion between droplet and ambient gases then force convection come into play its role and then heat transfer take place through conduction radiation and force convection.
Relative movement helps to increase heat transfer and mass transfer so, increasing evaporation rate. Convection effects on droplet evaporation were studied by Froessling [16]. Applying Froessling correlation.
Turbulent Environment: Combustion of spray is turbulent. The relative movement of hot gas around droplet affects the spray/droplets in different ways. Faeth [17] reports three different types of effects of turbulence on spray combustion. He reports droplet distribution over wide area, change of turbulence properties and change of transport properties due to motion of droplet.
Theoretical and experimental work on droplet evaporation in turbulent condition is limited. Work dealing with turbulent droplet evaporation at high temperature and pressure is almost unavailable and most of the work done is at standard temperature and pressure [18]. In Maisel and Sherwood [19] were the first who started work on effects of ambient turbulence condition on mass transfer.
The entire researcher reported increase in mass transfer rate except [20]. Later on Gokalp et. al. [21], introduced vaporization Damkuhler number. Gokalp et. al. [21] reported that mass transfer is more for vaporization Damkuhler number from 0.0001-0.1. Mass transfer decreases with increase in vaporization Damkuhler number and it will not affect mass transfer if vaporization Damkuhler number is one or more than one. The findings of Gokalp at. el. [21] were verified by Wu et. al. [22][23].
According to Wu at. el. [22], for 0.0001 Dav 0.1 Hiromitsu and Kawaguchi [24] tested Damkuhler vaporization correlation of Gokalp et. al. [21] for ambient temperature from 323-423 K and they found that correlation of Gokalp et. al. [21] does not work for temperature above atmospheric.
A correlation was proposed by Birouk and Gokalp [25] for turbulent effects on evaporation of droplet with zero convection mean velocity. The correlation is given below.
The classical droplet evaporation rate is given by [9] Putting in Equation (6) Vapor pressure,P fs at T s is calculated by using Antoine correlation [26], which is given Equation (14).
Where, A, B and C are constant depending on type of fuel.
Curvature of liquid droplets affects vapor pressure.
The rate of change of mass of the droplet or evaporation rate is: Putting Equation (23) in Equation (10).   For stagnant droplet with no evaporation Nu 0 is equal to 2 and Sh 0 with no relative movement of the droplet with ambient is also 2.

Application on Combustion Chamber of Liquid Rocket Engine
Velocity of hot gas inside combustion chamber is given by: Solution procedure is shown in Fig. 1. Fig. 1 shows that heat up phase is ended when change in surface temperature of the fuel droplet is equal to or less than 10 -6 and droplet life is finished when diameter of fuel droplet is equal to or less than 10 -9 . Computer Code starts with initial, given surface temperature of the droplet.
Reference temperature is calculated using this initial temperature. Corresponding required values of density, viscosity, thermal conductivity, enthalpy, entropy, specific heat capacity at constant pressure and volume, pressure and etc. at that reference temperature is calculated.

Factors Affecting Characteristic Length of the Combustion Chamber of Liquid Propellant Rocket Engines
Thermodynamic values for liquid oxygen is taken from data file. Vapor pressure at the droplet surface is

RESULTS AND DISCUSSION
The developed model is applied on low temperature combustion chamber with different input conditions. The output results showing the effects of input condition on characteristic length are shown. Fig. 7 shows increase in droplet evaporation time and characteristic length with increase in droplet velocity up to a point and then start

Factors Affecting Characteristic Length of the Combustion Chamber of Liquid Propellant Rocket Engines
decreasing. Fuel droplet evaporation rate increases with increase in relative velocity not actual so as droplet velocity increases, it reduces relative velocity between droplet and hot gas. Relative velocity is zero at peak point in Fig. 7. Reduction in relative velocity reduces convection coefficient, so required time increases.  Fig. 9 shows that effects of the combustion chamber diameter on characteristic length and relative velocity at fixed mass flow rate. When diameter of chamber increases for a given input condition then more area is available for hot gas to flow due to which velocity of hot gas decreases. Decrease in hot gas velocity causes decrease in relative velocity. Reynolds number, turbulent intensity and convection coefficient decreases. Decrease in turbulent intensity reduces heat and mass transfer, which consequently decreases rate of fuel droplet evaporation as long as relative velocity get zero.