3 Design Approach(Reviewed)

8/10/2019 3 Design Approach(Reviewed)

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3. Design methodology

3.1 Introduction

In this chapter, firstly the design strategy is defined by specifying the

analysis priorities and by specifying some variables such as the type of

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

considerations. Then the factors which affect the turbo-pump system

design besides pumps selection, turbine preliminary selection, turbine

optimization and turbo-pump configuration all are discussed in a step

by step manner and the discussion is supported by the relevant

mathematical relations. Broadly speaking, it can be said that the

content of this chapter is a projection to the design flow chart in verbal

and mathematical expressions.


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3.2Design flow chart Design approach

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 simple

paragraphs:

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

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.


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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
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3.3 Design Strategy

The design strategy was based on the following considerations:

Factors which affect the design of turbo-pump system would bediscussed first and then the parametric analysis will be conducted.

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.

Engine thrust at sea level and chamber pressure, as known

parameters, are assumed to be the key parameters for starting

the work.

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 thecommon literature, then, more realistic values when predicted

will be final inputs to the program.


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3.4 Factors affecting the turbo-pump system design

The design of the turbo-pump system is highly dependent on thefollowing factors which mainly characterize therocket engine itself:-

The engine requirements for flow and pressure.

The engine cycle (power cycle).

The engine throttling requirements.

The types of propellants used.

The propellant inlet conditions.

The engine dimensions The chamber material.

3.4.1 The engine requirements for flow and pressure

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, shows that the engine thrust, as one of thevehicle requirements is a function of the mass flow rate, ,of thepropellant . 3.1Where, is the ideal exhaust gas velocity, is the pressure of theexhaust gases at the nozzle exit,

is the pressure of the ambient

atmosphere, and is the area of the nozzle exit.

Also, for a specified engine and pair of propellants the pressure of the

combustion chamber can be increased by increasing the mass flow rate
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of the propellants(3)

. Increasing the mass flow rate leads to a better

engine performance since the exhaust velocity is a performancecriterion of rocket engines

(12). This fact is explained by equation 3.2

blow:

3.2

Where, Kis the specific heat ratio, R' is the universal gas constant,

is the combustion temperature, M is the average molecular weightof the exhaust gases, is the combustion chamber pressure, and isthe pressure at the nozzle exit.

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

solved for a given engine (specifications of which are known suchas: chamber pressure, thrust, propellant type) resulting in good

data about mass flow rates required to be delivered by the pumps

to 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 variationsof 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


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data for driving relationship between each parameter and the chamber

pressure 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.

3.3

3.4

3.5 3.6It is worth mentioning that:

The mixture ratio calculated by equation (3.3) is the optimummixture ratio corresponding to an exit pressure of 1 atm.

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.

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

considerable friction losses in the flow lines including the cooling jacket

expressed as pressure drops, and that there is the injection


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requirement for pressure difference, the pumps should deliver

propellants at a discharge pressure considerably higher than the

pressure of the combustion chamber.

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 in the flow passage

Flow lines include the piping and cooling jacket. The following empiricalcorrelations can be used in estimating pressure drop (p) in the cooling

system and pipes assuming smooth tubing:

(For turbulent flow) (6) 3.7 (For laminar flow)

(19)

3.8

Such that:is the fluid mean velocity in the pipe is the passage length, is thehydraulic diameter and is the Reynolds number ( ) basedon the hydraulic diameter and coolant dynamic viscosity .


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It is clear that to obtain the pressure drop caused by frictional losses of

the flow, preliminary data about the engine and piping dimensions are

required.

It is clear that the fluid velocity, is the variable through which we cancontrol the pressure drop resulting from the frictional losses. The fluidvelocitycan be controlled through the dimensions such as the hydraulic

passage when dealing with the cooling requirements. The hydraulic

passage will be the one that expected to satisfy the cooling

requirements when dealing with the thermal considerations, section

(3.7).

Pressure drop due to injection

For injection pressure drops, equation (2.9) below can be used:

3.9Or:

Where,

is the mass flow rate injected through the injection area,

is the total cross-section area of the orifices (the total injectionarea), is the discharge coefficient, is the pressure drop acrossthe injection orifices, is the injection velocity.


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It is clear that for a specified injection diameter , the number oforifices is countable: 3.10For rocket engines, the injection velocity is between 30 to 45 m/ s

(6).

Pressure drop due to injection is now predictable if suitable injection

parameters are selected.

3.3.2 The engine cycle

The engine cycle affects the turbo-pump design in the following ways:

The pump flow rate and discharge pressure are either maximized

or minimized according to the type of the engine cycle, see table

(3.1), reference [14].

The turbine flow rate, available energy and operatingtemperature are all affected by the engine cycle; table (3-1)

explains in details these effects.

For example, the gas generator cycle maximizes the pump flow rate by

adding the mass flow rate of the gas generator to the mass flow rate of

the main combustion chamber whereas the pump discharge pressure

remains unaffected by the presence of the gas chamber.

Table (3-1)Comments on engine cycles from turbo-pump components

view points.

Cycle Characterizing Results


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features

The gas

generator

Turbine exhaust is

discharged

outboard or to the

main nozzle

Maximized pump-required

flow rate.

Minimized turbine required

flow rate

Turbine operating

temperature can be

maximized to the material

limit.

Pump is

discharging

directly to theengine chamber

Minimized pump discharge

pressure

Staged

combustion

No outboard

discharge.

Maximized turbine operating

temperature

Turbine exhaust is

discharged to themain chamber.

Minimized pump flow rate

The pump is not

discharging

directly to the

main chamber,

Maximized pump discharge

pressure

Theexpander

Fuel is vaporizedby the heat

extracted from the

engine in the

cooling process

Limited turbine availableenergy

Limited pump discharge

pressure


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3.3.3 The engine throttling requirements

Rockets can be throttled by controlling the propellant mass rate during

the mission. The engine throttling requirements define the range of

flow and discharge pressure that the turbo-pump must deliver with

stable operation.

3.3.4 The types of propellants used

Although both the propellants used in any liquid rocket engines are

physically in the liquid phase, still the pump and turbine selection is

affected by the type of the propellant used. That is due to the fact that

propellants are found in wide density ranges and wide thermal

properties.

The variations in density lead to different pump head rise requirements

and large differences in volumetric flow. For example, lower density

propellants require a much higher head rise to develop the same

discharge pressure. Equation (3.11) below relates the pressure rise tothe head rise and the fluid density : 3.11


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Higher head rise implies higher tip speed of the pump impeller.

Impeller tip speed () is related to the pump head () by the relation(3)

:

3.12For centrifugal pumps, has values between 0.9 and 1.1 for different

designs but for most pumps, = 1.0, reference [3].

Propellant density directly affects the pump inlet conditions (NPSH of

equation 2.13) and the power required by the pump to deliver a certain

head (equation 3.27).

Also, different propellants when combusted results in different

combustion parameters. The quantity of the available energy of the

drive gas is strongly related to the combustion parameters. In this way,

the turbine design is affected by the difference in propellant

combinations.

Appendix B shows some common propellants used for liquid rocketengines and their properties affecting the turbo-pump design.

3.3.5 The propellant inlet conditions

By the propellant inlet conditions is meant the suction head at the

pump inlet. A criterion for metering this inlet condition is the net

positive suction head (NPSH). Mathematically this can be expressed as:

3.13
http://en.wikipedia.org/wiki/Bipropellant_rockethttp://en.wikipedia.org/wiki/Bipropellant_rocket


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Where, is the sum of the tank pressure and the propellant static

pressure at the pump inlet and is propellant vapor pressure

(provided that the tank is initially pressurized to suppress cavitation).

The net positive suction head (NPSH) dictates the pump's suction

performance requirements. The pump suction performance

requirement is its ability to operate at the available NPSH without

harmful cavitation. A pump with low (NPSH) will suffer bad suction

performance and cavitation is more likely to occur.

A useful parameter for defining the range of operation in which a

pump will experience a stable operation without cavitation is the

suction specific speed, , which is defined as [1, 3] :

3.14

Where, is the design speed in radians per second and is the

volumetric flow rate.

is the required suction head at pump's best efficiency point. It

is defined as the limit value of the head at the pump inlet (above vapor

pressure); above this value cavitation in the impeller will not occur. To

avoid cavitation, (NPSHr) should always be less than the (NPSH). Here

NPSHr is assumed to be as 80% of the NPSH.

It is usually indicated by a head loss of 2 to3% in a pump test when

increasing the throttling in the suction side [3]. It is a function of the

impeller and pump design quality.


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For a given volume flow rate and pump inlet conditions, there is a

certain range for the maximum pump speed within which safe

operation is ensured.

3.3.6 Engine dimensions and chamber material

Engine dimensions are involved in equations 3.7 and 3.8 when

estimating pressure drop due to hydraulic losses in engine jacket. Also

engine dimensions and chamber material are involved in computations

of heat transfer rate, section (3.7). There is a maximum limiting value

for the heat flux that should ensure a wall temperature well below the

melting point of the chamber material.

In this way, as sufficient cooling of the chamber wall is one of the

functions of the pumping system, we can say that the pumping systemis affected by the engine dimensions and chamber material. Should a

designed pumping system not satisfy the cooling requirement, a

redesign process has to be performed for either the engine or the

pumping system but it is easier to do so for the cooling system. Using

the oxidizer as a coolant, or both the fuel and the oxidizer as coolants,

the helicoidally cooling system instead of the jacket cooling system, all

are suggested options to solve the cooling problem should narrowingthe jacket (increasing the coolant velocity) not solve the problem.

3.3.7 Preliminary estimations of engine dimensions


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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:

3.15

All parameters are as previously defined in equation (3.2).

The gas pressure at the nozzle throat is such that:

3.16

The gas temperature at the nozzle throat is such that:

3.17

b.Equations defining parameters of nozzle exit


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To find the nozzle exit area, we should first find the Mach number ( )

at that area. It is given by the perfect gas expansion law as follows:

3.18

The nozzle exit area is such that:

3.19

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 that

reduced 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:

3.20


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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:

3.21

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 data

about the chamber dimensions. Then, the pressure drop caused by the

cooling mission of the coolant is predictable for a suitable coolant

velocity.

3.4 Pump selection

Firstly, we have to estimate a suitable shaft speed based on the suctionconditions available at the pump inlet. Equation 3.22

below [6]

represents a practical formula which relates the shaft speed (N) to the

lowest required suction pressure above vapor pressure of the liquid


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being pumped. It is an empirical formula derived for pumps employed

in rocket engines by Russian engineers.

3.22

Where, is the minimum required suction pressure, is the vapor

pressure of the liquid being pumped, is the specific density, is

the maximum speed in revolutions per minute and is a constant

having a value lying between 13 and 17 [6].

Equation 2.22 can be rewritten as:

3.23

Equation 3.23 can be used to estimate the maximum shaft speed(rpm) for afixed suction pressure.

An important selection criterion which reflects the difference in

characteristics of pump geometry, is the specific speed ( ) expressed

in equation 2.24. It is a function of the shaft speed (N or ), volumetric

flow (Q), and pump head (H).

2-412-21

3.24


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Centrifugal pumps have specific speed typically ranging from 0.2 to 0.4

whereas mixed flow pumps have specific speed ranging from 0.6 to 0.8,

and it is above 2.5 for pure axial flow pumps(3)

.

Now, for each pump, the design speed and the configuration can be

selected based on optimizing the pump for its propellant in such a way

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

operating range must all be satisfied as pumping requirements deduced

from the engine system.

Further speed and configuration optimization is possible, if required,but it dictates elevating the head available at the pump inlet.

The pump head, , which is a function of the required discharge

pressure, , the available inlet pressure, , and the propellant

density, , [ ], are major factors in selecting our

pump configuration as seen in equation 3.24.

The head coefficient ( ) is the ratio of discharge head, to the kinetic

head of the blade tip as in equation 3.25 below:

3.25

It is a function of the pump type and it establishes the diameter of the

required pumping element and number of stages to develop the

required pump head for a given shaft speed. Empirical correlations

relating the head coefficient to the specific speed are available.


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Equation 2.26 below can be used to estimate the head coefficient

[22?]:

(For specific speed less than unity) 3.26

(For specific speed greater than unity)

Now, the size of the pumping element can be obtained by making use

of the previous relations while caring for the material strength.

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].

The pump overall efficiency, , can be estimated using Anderson

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

equation 3.27:

3.27

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


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In SI units the equation could be rewritten as:

3.28

Limitations of Anderson correlation are:

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

3.29

is the kinematic viscosity.

According to equation (3.28), the pump efficiency can be obtained, and,

also can further be optimized dramatically by increasing the rotational

speed. Maximum pump efficiency can generally be developed in the

specific speed range of 0.6 to 0.8 approximately [3].


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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 theturbo-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:

Figure 3.2 effect of increasing pump speed on other design parameters


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Shaft power of each pump can be estimated as follows:

Oxidizer pump shaft power ( ) is:

3.30

Fuel pump shaft power ( ) is:

3.31

3.5 Turbine selection

3.5.1 Preliminary selection


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The previous analysis on the pump section has quantified the power

required 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:

3.32

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

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 3.33 below:

3.33


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Figure 3.4 Each pump is driven by a separate turbine

On the bases of the quantity of power produced, gas turbines are

categorized 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 gasturbine depends upon the following three variables:

The available energy content per kilogram of drive gas.

The blade tangential velocity (U).


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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:

3.34

(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 achoked nozzle so that a stable turbine performance is insured duringvehicle flight.


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

3.35

In 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 of

the 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

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

preliminary selection of the turbine configuration that best suits the


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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 we are going to have some of the

expansion taking 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 within

the 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 fromthe 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 theturbine 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.

3.5.2 Optimization scenarios

After a preliminary turbine selection is made, many scenarios are

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

can maximize the pressure ratio to minimize mass flow rate, maximize

the velocity ratio to maximize turbine efficiency for a given speed,


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maximize the velocity ratio to maximize turbine efficiency for a given

blade diameter, etc.

3.5.3 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 we have to minimize the mass

flow rate of the turbine working fluid. According to equation 3.36

below, this can be achieved by maximizing the turbine pressure ratio

{12}:

(concept of turbine overall efficiency) 3.36

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 gas

generator is a function of the discharge pressures of the pumps. It isless 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

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


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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,

[] 3.37So, the pressure ratio

of equation 3.36 when substituted by the

maximum pressure ratio []of equation 3.37, the correspondingweight flow rate of the turbine will be the minimum one for a givenworking 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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3.5.4 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.

3.5.5 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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3.5.6 The maximum allowable turbine flow rate

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

3.38

Where:

F = engine thrust

= engine specific impulse

= chamber specific impulse

= turbine discharge specific impulse

The turbine specific impulse is very small compared to the main

chamber specific impulse so that for the purpose of simplification we

can neglect it in equation 3.38 to read:

3.39

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:


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= turbine fuel weight flow rate

= pump weight flow rate

= turbo-pump efficiency

Equation 3.40 assumes that each pump is driven by a separate turbine.

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

becomes:

3.43

Or alternatively,

3.44

The maximum allowable turbine weight flow rate of equations (3.38)

or (3.39) when substituted in equations (3.42), (3.43) or (3.44), 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 turbine


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efficiency. 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.

3.5.7 Turbine stage efficiency

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

exit [8]:

()

3.45

3.46

Where, is the turbine stage efficiency is the whirl velocity of

the gas approaching the blades and is the adiabatic head.

such that is the nozzle angle at turbine inlet.

Now, a relationship between the pressure ratio and turbine rotational

speed can be established using equations (3.45) and (3.46).

() 3.473.6 Gas generator analysis

A fuel rich mixture will be used to suppress the combustion


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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.

The following points have to be considered when designing the gas

generator and when analyzing its thermodynamics [12]:

The basic design parameters for the gas generator are assumed

to be the same as those of the main combustion chamber.

The total throat area of the turbine nozzles are going to be

assumed as the equivalent throat area of the gas generator

combustion chamber.

When calculating the characteristic length, the volume between

the injection unit and the turbine nozzle throats is to be used.


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Analysis of gas generator products showed the following

combustion parameters: Specific heat ratio , heat

capacity , average molecular weight

and the resulting spouting velocity at the maximum possible

pressure ratio is

3.7 Guarding against thermal failure

As mentioned in points of design strategy, the heat transfer

computation will be simplified. The problem to be dealt with is a

steady state heat transfer problem.

In our case, 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).


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

combustion chamber.

Assuming a one dimension heat transfer problem, the following three

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

respectively:

3.47

3.48

3.49


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Where, and are the wall temperatures on the gas side and

the coolant side respectively, is the average temperature of the

coolant, is the wall material conductivity and is the wall thickness.

Both the gas film coefficient, , and coolant film coefficient, will be

computed using empirical correlations shown in equations (2.50) and

(2.51) respectively(9)

.

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

conditions near the wall.

The mathematical expressions for the dimensionless numbers involved

in equations (3.39) and (3.40) are as follows:


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(Nusselt number)

(Prandtl number)

(Reynolds number)

Where h is the convective coefficient, D is a characteristic length (the

hydraulic diameter), k is the thermal conductivity of the fluid, is the

constant pressure specific heat, and is the fluid dynamic viscosity. Itcan 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 of

the coolant and its viscosity, the heat flow rate, ( ), absorbed by the

coolant 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 be

estimated. The total surface area ( ) of the nozzle and chamber

across which heat is transferred can be calculated from the relation

[6]:


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