3 Design Approach(Reviewed Rearanged)

8/10/2019 3 Design Approach(Reviewed Rearanged)

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3. Design methodology

3.1 Introduction

In this chapter, firstly the design strategy is defined by specifying theanalysis priorities and by specifying some variables such as the type of

the propellants to be used, the engine cycle, etc, besides other

considerations. Then, relations that can be used to quantify the engine

requirements for mass and pressure, besides relations and other

considerations which are necessary for pumps selection, turbine

preliminary selection, turbine optimization and turbo-pump

configuration all are discussed in a step by step manner. Broadlyspeaking, it can be said that the content of this chapter is a projection

to the design flow chart in verbal and mathematical expressions.

3.2Design flow chart

Figure (3-1.A) is a proposed basic flow chart that can further be

developed to a more detailed chart, figure (3-1.B), addressing the

design sequences. Figure (3-1.A) can be read in the following simpleparagraphs:

The specific thrust required by the missile when analyzed can point out

the design goals of the pumping system which in turn give the

specifications of each pump.

The specifications of the turbine performance and the arrangement of

the different components of the turbo-pump can then be decided on.

Analysis of engine requirements, analysis of the available energy in the

turbine drive gas, the hydrodynamic constraints, the mechanical

constraints, optimization among the numerous design requirements


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and constraints, all are samples of the details waiting to be addressed.

So, the basic flow chart only highlights the way to the more detailed

chart.

Missile requirements

(Thrust and specific impulse)

Design goals of the pumping system

Fuel pump specifications

Turbine specifications Turbo-pump arrangementTurbine shaft power and speed

Oxidizer pump specifications

Figure (3-1 .A): Basic flow chart.


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Figure (3-1.B):Detailed flow chart
http://c/Users/gcc/Documents/%D9%8A%20%D9%8A/%D8%B7%D9%84%D8%A7%D8%A8%20%D8%AF%D8%B1%D8%A7%D8%B3%D8%A7%D8%AA%20%D8%B9%D9%84%D9%8A%D8%A7/KARARY/JaafarKhalifa/30Sep2012/Thesis/chart_updated.xlsmhttp://c/Users/gcc/Documents/%D9%8A%20%D9%8A/%D8%B7%D9%84%D8%A7%D8%A8%20%D8%AF%D8%B1%D8%A7%D8%B3%D8%A7%D8%AA%20%D8%B9%D9%84%D9%8A%D8%A7/KARARY/JaafarKhalifa/30Sep2012/Thesis/chart_updated.xlsmhttp://c/Users/gcc/Documents/%D9%8A%20%D9%8A/%D8%B7%D9%84%D8%A7%D8%A8%20%D8%AF%D8%B1%D8%A7%D8%B3%D8%A7%D8%AA%20%D8%B9%D9%84%D9%8A%D8%A7/KARARY/JaafarKhalifa/30Sep2012/Thesis/chart_updated.xlsm


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3.3 Design Strategy

The design strategy was based on the following considerations:

Because pump configuration is based on the requirementsderived from the engine system, the engine requirements will be

analyzed first.

Engine thrust at sea level and chamber pressure, as known

parameters, are assumed to be the key parameters for starting

the work.

Pumps parametric analysis will be performed before the turbine

analysis.

Each component of the system will be optimized separately and

then after the optimum arrangement of the components is

selected, a new optimization process for the whole system will be

conducted.

Readydeveloped technology will be under focus while conducting

the design specially when selecting the turbine. The gas generator cycle is assumed to be the working cycle.

Due to its ease of application, the regenerative cooling is chosen

to be the working cooling system.

The thesis will not go deeply in the cooling system analysis in

order to keep the way forward to the main subject.

The propellant combination is kerosene and red fuming nitric acid

for their availability and ease of handling and storing.

Some data will initially be inputs to the program depending on the

common literature, then, more realistic values when predicted

will be final inputs to the program.


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3.4. The engine requirements for flow and pressure

3.4.1 Flow requirements

On the basics of mass conservation, the mass flow rate of exhaust gases

of the engine is the same as the propellantsmass flow rate delivered by

the pumps to the engine combustion chamber. Equation 3.1 below, the

simple thrust equation, relates the engine thrust, to the mass flowrate,

,of the propellant:

3.1The ideal exhaust gas velocity, ,of equation 3.1 can be expressed inequation 3.2 as follows [14]:

3.2

The actual exhaust velocity, , is about 0.85 to 0.98 of the theoreticalone [14].An average correction factor of 0.92 is going to be assumed inthis thesis.

With the aid of combustion charts, equations (3.1) and (3.2) can be

solved for a given engine (specifications of which are known such as:

chamber pressure, thrust, propellant type) resulting in good data

about mass flow rate of the propellant required to be injected by the

pumping system into the engine combustion chamber.

Combustion charts are available for different pairs of propellant [10].

Figures (A.1), (A.2), (A.3) and (A.4) of appendix (A) show the variations


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of combustion parameters with chamber pressure for Nitric Acid and

Kerosene (Liquid oxygen and Kerosene, Liquid oxygen and Methane are

also shown in appendix A). These charts can be considered as sources of

data for driving relationship between each parameter and the chamberpressure as the independent variable [10].

Equations (3.3), (3.4), (3.5), and (3.6) below are deduced from

combustion charts of Nitric acid and Kerosene by making use of curve-

fitting procedure. Each equation relates one of the combustion

parameters to the chamber pressure. They can be used in solving

equation (3.2).

Equation of the mixture ratio, : 3.3Equation of the chamber temperature, : 3.4Equation of the average molecular weight, : 3.5Equation of the specific heat ratio, : 3.6It is worth mentioning that:

The mixture ratio calculated by equation (3.3) is the optimum

mixture ratio corresponding to an exit pressure of 1 atm.


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The chamber temperature calculated by equation (3.4) is the

adiabatic flame temperature which also corresponds to an exit

pressure of 1 atm.

The molecular weight calculated by equation (3.5) corresponds to

a modified mixture ratio.

The specific heat ratio calculated by equation (3.6) corresponds to

a modified mixture ratio.

The concept of the mixture ratio can be used to estimate the mass flow

rate of each propellant injected into the combustion chamber. It is

explained in equation (3.7) below:

3.7Of course these are not the actual pumps flow rates as long as the gas

generator is the working cycle. The mass flow rates required by the

turbine have to be estimated, which, when added to the mass flow

rates required by the combustion chamber of the main engine, give the

exact mass flow rates delivered by the pumps. This is expressed in

equation (3.8) below:

(3.8a) (3.8b)Initially, the percentage of the propellant bled off the main flow and

sent to the gas generator will be assumed. Typical values of such a

percentage are 1 to 7 [3, 1]. This will result in a suitable mass flow rate or say , injected into the gas generator and then expandedin the turbine. By assuming a suitable value for the mixture ratio of the

gas generator, the mass flow rate of each propellant ( and )can be estimated. A more realistic value of the propellant flow rate


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injected into the gas generator will be estimated later in section

(3.6.2.4) of this chapter, using relations addressing the turbine

requirements for the drive power.

3.4.2 Pressure requirements

Regarding the pressure, it should be put in mind that there is the

injection requirement for pressure difference , besidesconsiderable friction losses in the flow lines mainly the cooling jacket

expressed as pressure drops,

. Hence; the pumps should deliver

propellants at a discharge pressure

considerably higher than the

pressure of the main combustion chamber . This is expressed inequation (3.9) below: (3.9)The term disappears when dealing with the oxidizer line providedthat the coolant is the fuel and the piping losses are negligible.

Therefore, for a given chamber pressure of a rocket engine, the

discharge pressure for each pump can be obtained if other pressure

drops could be estimated. Other pressure drops can be estimated as

follows:

Pressure drop due to injection

For injection pressure drops, equation (3.10) below can be used [3]:

3.10This implies that:


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Or:

, which implies that: For rocket engines, the injection velocity is between 30 to 45 m / s [6].

So, if suitable injection parameters are selected based on common

literature, the pressure drop due to injection can be predicted. Also,

considering the injector, its number of orifices

is countable such

that:

3.11Pressure drop in the flow passage

Flow lines include the piping and cooling jacket. The following empirical

correlations can be used in estimating pressure drop (

) in the

cooling system and pipes assuming smooth tubing: (For turbulent flow), [6] 3.12a (For laminar flow), [19] 3.12bThe fluid velocity is the variable through which the pressure dropcaused by frictional losses can be controlled. The fluid velocity can be

controlled through the dimensions such as the hydraulic passage whendealing with the cooling requirements. So, the hydraulic passage should

be the one that expected to satisfy the cooling requirements when

dealing with the thermal considerations, section (3.8). It is clear that to


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obtain the pressure drop caused by frictional losses of the flow,

preliminary data about the engine and piping dimensions are required.

Preliminary estimations of engine dimensions

The assumption of a simple De Laval configuration with a convergent

half angle of 30 and a divergent half angle of 15 is quiet enough to

help in estimating the engine dimensions for the purpose of pressure

drop and heat transfer calculations [6].

Equations of ideal rocket parameters [3]will be used and they are

summarized in the following three paragraphs:

a.Equations defining the parameters of nozzle throat

The nozzle throat area is such that:

3.13

All parameters are the same as those of equation (3.2).

The gas pressure at the nozzle throat is such that:

3.14

The gas temperature at the nozzle throat is such that:

3.15


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b.Equations defining parameters of nozzle exit

To find the nozzle exit area, the Mach number ( ) at that area should

first be found. It is given by the perfect gas expansion law as follows:

3.16

The nozzle exit area is such that:

3.17

c. Characteristic parameters of engine chamber

To estimate the chamber dimensions, the following two characteristic

parameters are defined:

Contraction ratio ( ):

It is the ratio of the chamber diameter to the throat diameter. It defines

the optimum diameter to be given to the combustion chamber so thatreduced losses due to flow velocity of gases within the chamber, and in

the same time the least usable face area for the injection, both are

satisfied. It is expressed as:


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3.18

Typical values for ( ): 1.2 to 4 according to the engine size [6]. It

decreases with increasing engine thrust.

Characteristic chamber length ( ):

It is defined as the length that a chamber of the same volume ( )

would have if it were a straight tube (no convergent section). It defines

the minimum chamber length that permits sufficient time for the

combustion to complete. It is expressed as:

3.19

Typical values for ( )[6]: 2 to 3 m for nitric acid and hydrocarbons

combinations.

Now, by assuming suitable values for the contraction ratio,

, and the

characteristic length, , the previous set of equations yield good dataabout the chamber dimensions. Then, the pressure drop caused by thecooling mission of the coolant is predictable for a suitable coolant

velocity.

3.5 Pump parameters

Now, for each pump, the design speed and the configuration are going

to be selected based on optimizing each pump for its propellant in such

a way that the discharge pressure, flow rate, inlet conditions (NPSH)

and operating range must all be satisfied as they are pumping


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requirements deduced from the engine system. The power required by

each pump besides its efficiency is also going to be estimated.

3.5.1 Design speed

A suitable shaft speed based on the suction conditions available at the

pump inlet has to be estimated. Equation 3.20below [6] represents a

practical formula which relates the shaft speed (N) to the lowest

required suction pressure, above vapor pressure of the liquid beingpumped. It is an empirical formula derived for pumps employed in

rocket engines by Russian engineers.

- - 3.20Where, is the minimum required suction pressure, is the vaporpressure of the liquid being pumped, is the specific density, isThe parameter is a constant having a value lying between 13 and 17[6].

Equation 3 .20 can be rearranged to read: 3.21Equation 3.21 can be used to estimate the maximum shaft speed

(rpm) for afixed suction pressure.

3.5.2 Configuration

An important selection criterion which reflects the difference in

characteristics of pump geometry is the specific speed () which is afunction of the shaft speed (N), volumetric flow (Q), and pump head

(H). It is expressed in equation 3.22 below:


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3.22The pump head, , is a function of the required discharge pressure,

,

the available inlet pressure, , and the propellant density, . This isexpressed in equation 3.23 below: 3.23

Specific speed ranging from 0.2 to 0.4 indicates centrifugal pumps,

whereas that ranging from 0.6 to 0.8 indicates mixed flow pumps and itis above 2.5 for pure axial flow pumps[3].

Further speed and configuration optimization is possible, if required,

but it dictates elevating the head available at the pump inlet.

3.5.3 Size of the pumping element

The size of the pumping element can be obtained by making use of the

concept of the head coefficient () which is the ratio of discharge head, to the kinetic head of the blade tip as in equation 3.24 below: 3.24Empirical correlations relating the head coefficient to the specific speed

are available. So, it is a function of the pump type and in the same time

it establishes the diameter of the required pumping element and

number of stages to develop the required pump head for a given shaftspeed. Equation 2.25 below is an empirical correlation that can be used

to estimate the head coefficient[22?]:


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(For specific speed less than unity) 3.25a

(For specific speed greater than unity) 3.25b

It is mentioned that the head requirements for high-density fluids such

as RP-1 (refined petroleum grade 1 or rocket propellant 1) can be

generated with a single stage centrifugal pump, with the impeller

diameter well within aluminum and nickel-base alloy steel structural

limits [1].

3.5.4 Efficiency and power

The pump overall efficiency, , can be estimated using Anderson

correlation [22] with some care to its limitations. It is presented in

equation 3.26:

3.26

is the specific speed in U.S terms which is expressed as:

In SI units the equation could be rewritten as:

3.27

Anderson correlation has the following limitations:


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a.X is a correction factor for pumps with specific speed greater

than unity.

b. It assumes that the fluid being pumped is water. Therefore,

another correction factor may be required especially when thehead - flow Reynolds number ( ) is relatively low. Typical

values of which do not impose a correction factor are

values greater than 2E5 [22?].

We have to check for the two correction factors. If the value of sp is

less than unity then the value of X in Anderson correlation is unity. If

the value of our is greater than 2E5 then also no correction

factor is required. is expressed as follows [22]:

3.28

Shaft power of each pump can be estimated as follows:

Oxidizer pump shaft power ( ) is:

3.29Fuel pump shaft power ( ) is:

3.30


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3.5.5 Optimization scenario

According to equation (3.27), the pump efficiency can further be

optimized dramatically by increasing the rotational speed. Maximum

pump efficiency can generally be developed in the specific speed rangeof 0.6 to 0.8 approximately [3].

The decision to increase the pump rotational speed to achieve a better

optimum performance is not an easy decision. The pump rotational

speed strongly affects all other design parameters of the turbo-pump

components and the component arrangement. Chart of figure (3-2)

describes briefly the effect of increasing the pump rotational speed on

other turbo-pump design parameters.


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Figure 3.2 Effect of increasing pump speed on other design parameters


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3.6 Turbine parameters

3.6.1 Preliminary selection

The previous analysis on the pump section has quantified the powerrequired by the feed pumps which in turn indicates how much power

should be produced by the turbine. If the two pumps are driven by a

single turbine, figure (3.3), then turbine shaft power( ),is:

3.31

Figure 3.3 Two pumps driven by a single turbine [12].

If each pump is driven by a separate turbine, figure (3.4), then the

shaft power of each turbine will be equal to the shaft power of its


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driven pump, assuming no power consumers onboard the vehicle such

as electric generators and hydraulic pumps actuators. In such a case,

the turbine shaft power can be expressed by equation 3.31 below:

3.32

Figure 3.4 Each pump is driven by a separate turbine

On the bases of the quantity of power produced, gas turbines arecategorized to different classes.

The gas turbine to be dealt with is categorized as a micro gas turbine.

Micro gas turbines utilize technology already developed which agrees

with the strategy of the research. Radial-inflow is a widely used design

in micro turbines and specifically the mixed-flow type. The greatest

advantage of radial-inflow design is that the power produced by a

single stage is equivalent to that of two or more stages of the axial-

flow design [8]. It is worth mentioning that one of its common

applications is that it is used to power helicopters.

As any gas turbine, the overall performance of a radial-inflow gas


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turbine depends upon the following three variables:

The available energy content per kilogram of drive gas.

The blade tangential velocity (U).

The number of turbine stages.

The drive gas available energy is always expressed in terms of the gas

theoretical velocity, , sometimes named the spouting velocity. This

quantity of energy is predictable since the following working

conditions are specified:

The working propellant combination and their mixture ratio. The engine cycle which may indicate the turbine pressure ratio.

The spouting velocity is expressed in terms of the combustion

parameters of the drive gas and the working pressure ratio as:

3.33

(subscript 1 indicates the entrance of turbine )

Combustion parameters of the gas generator can be deduced from gas

generator analysis which will follow in section 3.6 of this chapter.

Dumping the turbine exhaust overboard or sending it to the nozzle

downstream implies that the turbine discharge pressure is our

selection. So, if the turbine inlet total pressure is , then, the

pressure ratio is quantifiable and conditionally it is such that: This is made so in order to permit sending the turbine discharge to a


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choked nozzle so that a stable turbine performance is insured during

vehicle flight.

If the turbine discharge pressure, ( ), is decided, then:

3.34In this manner, the spouting velocity ( ) can be estimated.

The blade tangential velocity (U) is an important parameter in turbo-

machinery in general but when considering gas turbines, it has a

special significance. It directly addresses the material stress as one ofthe most important mechanical constraints at the design stage [12].

The maximum centrifugal stress will be expressed in terms of

maximum blade tangential velocity, U. For titanium rotors which are

the prevailing technology nowadays, U varies between 457 to 549 m/s

according to numerous references [8].

The selection of blade diameter is constrained by the blade height-to-

diameter ratio (around 2.2) [8] and the inertia considerations. Also it is

constrained by the shape uniformity of the turbo-pump. It should not

be greater than three times the pump diameter in the case of a direct

turbine-pump coupling [12].

The blade tangential velocity (U) can be assumed and so can the blade

diameter, both within the mentioned constraints. Consequently, the

turbine rotational speed (N) can be deduced.

Alternatively, for a given rotational speed, the blade tangential

velocity (U) can be deduced by assuming a suitable blade diameter

also within the mentioned constraints.

The turbine velocity ratio can now be pointed out. It is a useful


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quantity in studying the turbine performance. It can be used to make a

preliminary selection of the turbine configuration that best suits the

efficiency requirement. Increasing the velocity ratio will improve the

efficiency but there are the mechanical constraints and also the

selected turbo-pump configuration may become a constraint. If the

resulting efficiency is extremely low, some of the expansion has to

take place in the rotor blades (increasing the degree of reaction) or

even an additional stage can be used.

The experimentally obtained values for this ratio lie between 0.68 and

0.73 for maximum efficiency according to S.M Yahia [2] and also

Courtesy Institution of Mechanical Engineers reported values withinthe same range [8]. It is expected that it is difficult to attain this range

in our case due to the relatively high spouting velocity resulting from

the high pressure ratio. It is mentioned that rocket turbines have

velocity ratio always less than 0.4 when the gas generator is the

working cycle [6, 12].

Up to here, the turbine velocity ratio (

) and consequently the

turbine type, size and rotational speed can all be pointed out but may

not be for the optimum operational conditions.

It should be remembered that the design selection is to be made to

maximize the turbine efficiency and minimize the turbine weight both

within the mentioned constraints.

3.6.2 Optimization scenarios

After a preliminary turbine selection is made, many scenarios are

possible to satisfy the goals of turbine optimization. For example, the


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pressure ratio can be maximized to minimize mass flow rate, the

velocity ratio can be maximized to maximize turbine efficiency for a

given speed, the velocity ratio can be maximized to maximize turbine

efficiency for a given blade diameter, etc.

3.6.2.1 Mass flow rate minimization

The gas generator cycle is an open cycle in which a considerable

amount of energy will be lost in sending the exhaust overboard and

this may negatively affect the engine specific impulse. To minimize the

energy lost in the turbine exhaust, the mass flow rate of the turbine

working fluid has to be minimized. According to equation 3.35 below,this can be achieved by maximizing the turbine pressure ratio {12}:

(concept of turbine overall efficiency) 3.35

Where, is the turbine weight flow rate and is the specific heat of

turbine working fluid.

The maximum total pressure at turbine inlet where is the

static pressure of the gas generator. The static pressure of the gasgenerator is a function of the discharge pressures of the pumps. It is

less than the discharge pressure of, say, the oxidizer pump by a value

of such that is pressure drop due to injection into the gas

generator. The same injection characteristics used for the main


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chamber are going to be assumed so that the static pressure of the gas

generator is almost the same as the static pressure of the main

chamber.

The turbine discharge pressure will be chosen to be slightly greater

than the atmospheric pressure by a quantity just sufficient to have the

nozzle being choked [12, 14].

Accordingly,

[

]

3.36

So, the pressure ratio of equation 3.35 when substituted by themaximum pressure ratio []of equation 3.36, the correspondingweight flow rate of the turbine will be the minimum one for a given

working fluid, turbine shaft power and turbine efficiency.

The net effect of maximizing the pressure ratio is not entirely positive.

Chart of figure 3.5 describes the effect of pressure ratio maximization

on other design parameters.


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Figure 3.5 Effect of maximizing the turbine pressure ratio on other design

parameters


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3.6.2.2 Efficiency maximization

Efficiency can be maximized by maximizing the blade tangential speed

(U) whether by increasing the blade diameter for a given rotational

speed (N) or increasing the rotational speed for a given blade

diameter. Chart of figure (3-6) describes the effect of tangential speed

maximization on other design parameters (for a given rotational speed

N).

Figure 3.6 effect of increasing turbine tip speed for a given constant rotational

speed


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Turbine efficiency can further be maximized by increasing the number

of stages after the upper limit of the tangential velocity is reached but

this will increase the weight. Therefore, a tradeoff between weight

and efficiency is also available. For missions of short burning time, theefficiency can be sacrificed.

3.6.2.3 Turbine inlet temperature maximization

Increasing the turbine inlet temperature can increase the efficiency

through decreasing the turbine weight flow rate but it has negative

effects on other parameters as shown in figure (3-7).

Figure 3.7 effect of raising the turbine inlet temperature on other design

parameters


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3.6.2.4 The maximum allowable turbine flow rate

The turbo-pump efficiency should be adequate for the engine to meet

its requirements. It seems that there is a maximum allowable turbine

flow rate beyond which the turbo-pump is considered to be infeasible

from a payload standpoint. The maximum allowable turbine weight

flow rate according to NASA criteria [12] is:

3.37

The turbine specific impulse is very small compared to the main

chamber specific impulse so that for the purpose of simplification it

can be neglected it in equation 3.36 to read:

3.38

The specific impulse is the thrust per weight flow rate of the

propellants. So, the engine specific impulse can be estimated by the

following expression:

is the total weight flow rate through the pumps.

The chamber specific thrust can be estimated by the expression:

3.39


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3.40

The maximum allowable turbine flow rate corresponds to the

minimum allowable turbo-pump efficiency. The turbine weight flow

rate and the turbine efficiency can be related to each other by

equations :

3.41

Equation 3.41 assumes that each pump is driven by a separate turbine.

If the two pumps are driven by a single turbine, the equation

becomes:

3.42

Or alternatively,

3.43

The maximum allowable turbine weight flow rate of equations (3.37)


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or (3.38) when substituted in equations (3.41), (3.42) or (3.43), the

resulting turbine efficiency is the minimum allowable efficiency.

So, up to this point, the minimum allowable turbo-pump efficiency (or

turbine efficiency for a given pump efficiency) can be obtained.

After the turbine is optimized, the obtained turbo-pump efficiency is

going to be compared to the minimum allowable one. If it is found to

be critical, then we have to improve it by increasing the turbineefficiency. The turbine efficiency can further be improved by adding a

new stage provided that all other means of efficiency improvement

are exhausted and the efficiency has no priority over weight

minimization.

3.6.3 Turbine stage efficiency

For an inward- flow gas turbine of the pure impulse type with an axial

exit, the stage efficiency can be expressed as follows [8]:

() 3.44

3.45

such that is the nozzle angle at turbine inlet.


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Now, a relationship between the pressure ratio and turbine rotational

speed can be established using equations (3.44) and (3.45).

() 3.46

3.7 Combustion parameters of the gas generator

A fuel rich mixture will be used to suppress the combustion

temperature of the gas generator. G. Sutton stated that typical

values of suppressed gas temperature are in the range of ( 900 to

1350 K ) and for practical reasons the turbine inlet temperature is

always kept within the range of (900 to 950 K ), [3].

It can be deduced from the above expression that the range (900 to

1350 K ) is for the combustion environment inside the gas generator

chamber whereas the range (900 to 950 K ) is for the gas approaching

the rotor after being throttled at the fixed nozzles of the turbine.

Assuming a Nitric acid-Kerosene mixture ratio of 0.5 seems to be

reasonable and the corresponding combustion temperature is

expected to be around 940k [3] according to table (2-4). The turbine

rotor inlet temperature will be less than this value since the

combustion gas is going to be expanded at the turbine nozzles

With aid of table (2-4), extrapolation on the combustion chart

[appendix (A)] beyond the plotted curves can show the combustion

parameters.


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3.8 Guarding against thermal failure

As mentioned in points of design strategy, the heat transfercomputation will be simplified. The problem to be dealt with is a

steady state heat transfer problem in which heat will be transferred

firstly across a gaseous film, then across the chamber wall and lastly

across a liquid coolant film in a series manner, figure (3.8).

Figure 3-8: Schematic diagram of heat transfer across the walls of a

combustion chamber.


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Assuming a one dimension heat transfer problem, the following three

equations can be used to model the heat flux ( ) for the three cases

respectively:

3.47

3.48

3.49

The following empirical correlations can be used to predict some

unknown parameters of the above equations:

3.50

Where Reynolds number is calculated with the diameter of the

chamber.

3.51

Where and represent Prandtl and Reynolds numbers

within the coolant whereas represents Prandtl number for


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conditions near the wall.

The mathematical expressions for the dimensionless numbers involved

in equations (3.50) and (3.51) are as follows:

(Nusselt number)

(Prandtl number)

(Reynolds number)

It can be shown that D in this case is twice the thickness separating the

outer and the inner wall of the chamber.

Having data about the inlet and outlet temperatures and ofthe coolant and its viscosity, the heat flow rate absorbed by thecoolant can be calculated. The coolant inlet temperature is expected

to be slightly greater than the ambient temperature say by 5C,

whereas the coolant outlet temperature will be assumed to be less

than its boiling temperature say by 10% to avoid abrupt pressure

elevation inside the pumping line.

Since the engine dimensions are known, the surface heat flux can beestimated. The total surface area () of the nozzle and chamberacross which heat is transferred can be calculated from the relation

[6]:

3.52

The above empirical correlations for estimating the gas film coefficient


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and coolant film coefficient are samples of a wide variety of such

correlations suggested by specialists. Unfortunately the results

obtained by those empirical correlations do not agree with each other

or with the results obtained experimentally as stated by Huzel and

Huang [14] (of NASA) who also decided that the complete analysis of

the chamber cooling system is a specialized field due to the complex

interrelations of the chamber design. Therefore, this issue will be

skipped by now and assigned to other researchers or will be

considered in separate papers in the future.


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