Design of a Centrifugal Pump for an Expander Cycle Rocket Engine

Università degli Studi della Basilicata

SCUOLA DI INGEGNERIA

CORSO DI LAUREA IN INGEGNERIA MECCANICA

Tesi di Laurea in Macchine e Sistemi Energetici

DESIGN OF A CENTRIFUGAL PUMP FOR AN EXPANDER CYCLE ROCKET ENGINE

Relatore: Prof. Ing. Aldo BONFIGLIOLI

Correlatore: Ing. Angelo LETO

Laureando: Antonio CANTIANI Matricola: 41036

ANNO ACCADEMICO 2014/2015

List of contents Chapter 1

1.1 Definitions and fundamentals (2) ...... 1

1.1.2 Thrust ...... 2

1.2 Liquid-fuel rocket engines cycles ...... 3

1.3 Expander cycle engines ...... 5

1.3.1 Closed expander cycle ...... 6

1.3.2 Closed split expander cycle (5) (6) ...... 8

1.3.3 Closed dual expander cycle ...... 8

1.3.4 Open expander cycle ...... 9

1.4 Existing expander cycle systems ...... 10

1.4.1 The RL10 engine ...... 10

1.4.2 The LE-5 engine ...... 12

1.4.3 Vinci ...... 13

1.5 Liquid propellants ...... 15

1.4.3 Liquid oxygen ...... 18

1.4.4 Liquid hydrogen ...... 19

1.4.5 Methane ...... 19

Chapter 2

2.1 Introduction ...... 21

2.2 Pump description ...... 23

2.3 Cavitation ...... 25

2.4 Pump parameters ...... 27

2.5 Pump design methods...... 40

2.5.1 Method 1 ...... 40

2.5.2 Method 2 ...... 41

2.5.3 Method 3 ...... 42

2.6 Volute design ...... 43

2.6.1 Effect of the volute design on efficiency ...... 44

2.6.2 Volute geometry esteem ...... 46

Chapter 3

3.1 Introduction ...... 48

3.2 Impulse turbines ...... 49

3.3 Velocity-compounded impulse turbine ...... 49

3.4 Reaction turbine ...... 50

3.5 Impulse turbines design ...... 50

Chapter 4

4.1 Introduction ...... 55

4.2 CoolProp libraries ...... 55

4.3 MatLab functions: pump.m ...... 59

4.4 MatLab functions: bladesNumber.m ...... 66

4.5 Software test-case ...... 68

Chapter 5

5.1 Turbopump specifics and applications ...... 73

5.2 Conclusions ...... 76

Appendix

Appendix A – MatLab additional functions ...... 77

Appendix B – RL10-3-3A LH2 pump output parameters...... 81

Appendix C– RL10-3-3A LO2 pump output parameters...... 84

Appendix D: Designed methane pump output parameters ...... 86

Introduction

The present work is aimed at designing a centrifugal pump for an expander cycle engine fed system. In particular, the pump has been designed to work with liquid methane. The choice of this type of cryogenic working fluid has been driven by the characteristic required by a rocket propulsion system: the methane has a lower cost and a higher density with respect to the more commonly used hydrogen. The lower cost is obviously a very appreciated feature; in addition to this, the higher density allows the design of more compact stages, which reduces the total weigh and the aerodynamic drag.

The design phase led to the development of a software in MatLab environment. This software aims to be a tool capable of providing a preliminary design of a generic centrifugal pump, given certain input data. The CoolProp libraries have played a key role in the software development. These libraries are an open source tool that, once implemented in MatLab, enabled to easily determine the thermodynamic variables required for the software calculations.

The developed software passed through a validation process, in which we have performed various design simulation based on the known data of the hydrogen and oxygen pump of a RL10A-3-3A; the results have shown an acceptable error margin.

Later, we moved to the design of the methane pump. It has been design with input data that should provide comparable performances to those of the methane pump developed by the Italian company AVIO for the LM10-Mira engine, which is currently under development in collaboration with the Russian KBKhA.

In addition to the various graph of the velocity triangles and 2-Dimensional models of the impeller, a 3-Dimensional model of the pump has been developed through the use of SolidWorks

Chapter 1 Liquid-fuel rocket engines

Since the beginning of the “rocket era” liquid-fuel engines have been the most widely used rocket engines. They passed through a long improvement process, which has led to engines that develop higher thrust, weigh less and are more reliable. The main goal of interest is to increase the payload, cost reduction, reliability improvement and design reusable launch vehicles (1).

1.1 Definitions and fundamentals (2)

The function of rockets engines is to convert the chemical energy provided by the propellant into thrust, through the combustion process.

The total impulse It is defined as the thrust force F (which may be time dependent) integrated over the burning time t, as shown by equation (1.1).

푡 퐼푡 = ∫ 퐹(푡) 푑푡 (1.1) 0

The total impulse is proportional to the energy released by the propellant. For constant thrust force and negligible start and stop transients, equation (1.1) reduces to:

퐼푡 = 퐹 푡 (1.2)

The specific impulse Isp is the total impulse per unit of weight flow rate of propellant. It is an important parameter that describes the performances of the rocket propulsion system. If we denote by 𝑔0 the acceleration of gravity at sea-level and by 푚̇ the mass flow rate of propellant, then equation (1.3) returns the specific impulse.

푡 퐹 푑푡 ∫0 퐼푠푝 = 푡 (1.3) 𝑔0 ∫0 푚̇ 푑푡

This expression gives a time-averaged specific impulse value. For constant thrust and propellant flow, the equation (1.3) can be simplified as follows:

퐼푡 퐼푠푝 = (1.4) 푚푝 𝑔0 where 푚푝 is the total effective propellant mass.

Chapter 1 ______

In the SI system Isp is expressed in seconds, however it does not represent a measure of elapsed time.

The exhaust velocity in the rocket nozzle is not uniform over the entire cross-section. Since the velocity profile is difficult to measure accurately, a uniform axial velocity c is assumed, which allows a one-dimensional description of the problem. This effective exhaust velocity c is the equivalent velocity at which the propellants should be ejected from the vehicle to achieve the engine specific impulse. It is defined as

퐹 푐 = 퐼 𝑔 = (1.5) 푠푝 0 푚̇

It the SI system, the effective exhaust velocity is expressed in meters per seconds. Since the effective exhaust velocity c and the specific impulse 퐼푠푝 only differ by an arbitrary constant 𝑔0, either one or the other can be used to measure the rocket performances.

The mass ratio MR of a vehicle or a particular vehicle stage is defined as the final mass mf

(after the rocket has consumed all usable propellant) divided by the initial mass m0:

푚푓 푀푅 = (1.6) 푚0

1.1.2 Thrust

The thrust is the force produced by a rocket propulsion system acting upon a vehicle. In a simplified way, it is the reaction experienced by its structure due to the ejection of mass at high velocity. This phenomenon is a consequence of the law of conservation of linear momentum: the high-pressure gases generated by the combustion of the propellant are accelerated by the nozzle and ejected at high velocity. The momentum of the combustion products is balanced by a momentum imparted to the vehicle in the opposite direction. In rocket propulsion relatively small masses are involved, which are carried with the vehicle and ejected at high velocities.

The thrust due to the change in momentum is given by:

푑푚 퐹 = 푣 = 푚̇ 푣 (1.7) 푑푡 2 2

This force represents the total propulsion force when the nozzle exit pressure equals the external pressure. To be more precise, also the pressure of the surrounding fluid has an influence upon the thrust.

______1.2 Liquid-fuel rocket engines cycles

Since rockets move through the atmosphere, they experience a pressure gradient, therefore the thrust changes during the flight. Figure (1-1) shows the pressure environment acting on the surface of a rocket combustion chamber and nozzle. The size of the arrows indicates the relative magnitude of the pressure forces. The axial thrust can be determined by integrating the pressure acting on areas that can be projected on a plane normal to the nozzle axis. The forces acting radially do not contribute to the axial thrust because a rocket is typically an axially symmetric structure.

Figure 1- 1 Combustion chamber and nozzle pressure environment Because of the change of external pressure due to the variations of altitude, when the pressure at the nozzle exit differs from the atmosphere pressure, there is an imbalance between the atmospheric pressure p3 and the pressure at the exit of the nozzle p2. Thus, for a rocket propulsion system moving through a homogeneous atmosphere, the total thrust is equal to:

퐹 = 푚̇ 푣2 + (푝2 − 푝3)퐴2 (1.8)

The first term represents the thrust due to the conservation of linear momentum, the second term represent the pressure thrust, due to the difference between the exhaust gas pressure and the external pressure. For the reason that the atmospheric pressure decreases with increasing altitude, the thrust and the specific impulse increase as the vehicle reaches higher altitudes.

1.2 Liquid-fuel rocket engines cycles

In liquid-fuel rocket engines, the propellant can be pressurized directly in the tank (pressure- fed systems) or by the use of a turbopump (pump-fed systems), then combusted in a combustor. For the pump-fed systems, the engine cycles are classified according to the driving method of 3

Chapter 1 ______the turbopump turbine. Depending on how the gas is handled after driving the turbine, liquid rocket engines cycle can be classified as:

 Closed cycle: the gas flows through the main combustion chamber and it is combusted (Figure 1-2 [A]);  Open cycle: the gas is exhausted (Figure 1-2 [B]);

Another possible classification is based on the gas generation method, so the engine can be:

 Gas generator: the gas is generated in an auxiliary chamber (Figure 1-2 [1]);  Expander cycle: the gas is generated by the heat produced by the main combustor (Figure 1-2 [2]);

In this work we are focusing on expander cycle engines.

Figure 1- 2 Rocket engines types

______1.3 Expander cycle engines

1.3 Expander cycle engines

The working principle of an expander cycle engine is to increase the energy of the propellant fluid (and also of the oxidizer fluid in some variants of the cycle) thanks to the heat produced by the combustion chamber. The heat is transferred to the fluid using the cooling jacket placed around the combustion chamber and the nozzle. This high-energy fluid goes into a turbine that uses part of the fluid energy to drive the propellant and oxidizer turbopump. Once it has been discharged from the turbine, it is injected into the combustion chamber and burned in the case of closed expander cycles, or exhausted if the engine works with an open expander cycle.

Generally, cryogenic propellants are used, such as liquid hydrogen (LH2) and liquid oxygen (LOX).

The main elements of an expander cycle engine are:

 Tank system (fuel tank, oxidizer tank);  Turbopump system;  Cooling system;  Injector system;  Main combustion chamber;  Thrust chamber;

The turbopump has the purpose of increasing the fluid pressure to a value able to satisfy the pressure drop in the cooling system and pipes, the fuel expansion in the turbine and the required pressure at the injectors.

There are many possible configurations for the turbopump system, shown in Figure (1-3) (3). The first possibility is to install the turbine and the two pumps on the same shaft (Direct drive). This is a very simple configuration, but the two pumps rotate at the same speed and it may be not desired. Another option is to install one turbine for each pump on separated shafts (Dual shaft): in this way we can independently adjust the oxygen and propellant flow going into the combustion chamber, by acting on a valve placed before the turbine. On the other hand we must install two turbine, which means less reliability and higher costs and weight. At last, we can install only one turbine, but oxygen and propellant pumps on two different shafts (Geared). In this configuration, a gearbox is required to synchronize the rotational speed of the turbopumps. The gearbox makes the system more complex, heavier and less reliable.

Chapter 1 ______

Figure 1- 3 Pump-turbine drive systems

The reliability of the expander cycle engines is provided by the simple configuration and the relatively low structural and thermal load on the turbine side.

The limit of this engine cycle is the inlet turbine temperature, since we have a limited area where we can place the cooling jacket. The problem lies with the square-cubic rule: as the size of the nozzle increases with increasing thrust, the nozzle surface area increases as the square of the diameter, but the volume of propellant that must be heated increases with the cube of the diameter. Therefore, there exists a maximum engine size beyond which there is no longer enough nozzle area to heat the propellant enough to drive the turbine and, therefore, the turbopumps (4).

1.3.1 Closed expander cycle

A closed expander cycle is the simplest form of expander cycle. Since it is a closed cycle, all of the fluid coming from the tank is burned in the combustion chamber and then expelled through the nozzle. A simplified scheme of a closed expander cycle is presented in Figure 1-4.

______1.3 Expander cycle engines

Both fuel and the oxidizer fluids go through a turbopump that increase their pressure. The oxidizer is then directly injected into the combustion chamber. The propellant travels along the combustion chamber and the nozzle to increase its temperature. Usually the combustion chamber is cooled first, because most of the heat is transferred in this part of the engine, then the nozzle is being cooled. This high energy fluid is now delivered to the turbine. The turbine must provide enough power to compress the propellant and oxidizer to the pressure level needed. After that, the propellant goes through some injector and enters the combustion chamber. Figure 1- 4: Closed expander cycle schematic. MFV: Main Fuel Valve, MOV: Main Oxidizer Valve, OTBV:

Oxidizer Turbine Bypass Valve (for regulation)

The pressure that the fuel pump has to generate is obtained by adding the pressure generated by the combustion chamber with the pressure drops that occurs into the cooling system, the injection system and the turbines pressure drop:

ΔP퐹푃 = 푃푐 + ΔP푖푛푗 + ΔP푡 + ΔP푐표표푙 (1.9)

Regarding the oxidizer fluid, the pressure that the oxidizer pump has to generate equals to

ΔP푂푃 = 푃푐 + ΔP푖푛푗 (1.10)

With the oxidation (combustion) of the propellant in the combustion chamber, there is a release of energy and the generation of high-velocity combustion products, accelerated by the nozzle to a supersonic speed in order to provide a high thrust level.

Chapter 1 ______

1.3.2 Closed split expander cycle (5) (6)

To increase the thrust of an expander cycle engine the fuel-flow rate provided to the combustion chamber must be increased. Consequently, the turbopump needs more power, and the turbine should provide it, but the increase of the fuel flow rate (that means a higher flow velocity) results in a decrease of the turbine inlet temperature that limits the power available at the turbine, since the fluid has less time to take energy from the cooling jacket. Because of these considerations, it is clear that the thrust we can achieve from an expander cycle has a defined upper limit. The idea of a split expander cycle (Figure 1-5) is to pump some of the fuel with a first pump stage to a lower pressure and the rest of it to a higher pressure with a second pump stage. The high-pressure flow goes through the cooling jacket and provides the energy needed by the pumps, then it goes into the main injector (as in a normal closed expander cycle). The low-pressure flow goes directly into the main injector. Since not all of the fluid is pumped to the highest pressure, the power requirements of the turbine decrease (the turbopump power requirements decrease by approximately 15 to 25 percent (6)), so we can achieve a higher Figure 1- 5: Closed split expander cycle schematics combustion chamber flow rate, compared to a standard closed expander cycle.

1.3.3 Closed dual expander cycle

In a closed dual expander cycle, both oxygen and propellant are used as cooling fluids. The heated propellant drives the fuel turbine, the heated oxygen drives the oxidizer turbine.

______1.3 Expander cycle engines

It is also possible to combine the split expander cycle with the dual, on one or on both (propellant and oxidizer) sides. In the Figure 1-6 is shown a dual expander cycle with split oxidizer side.

One of the main vulnerability of this cycle is given by the oxygen driving through the cooling jacket. Whenever there is a cooling jacket, there is the potential risk of cracking and leaking. Since on the outside we have a fuel- rich environment, a leak of fuel is not dangerous, but a leak of oxygen may be, on the contrary, very dangerous. This problem adds complexity to the design and results in an increase of costs. Furthermore, oxygen is less efficient as turbine working fluid than the propellant (6).

Similarly to the split expander cycle, the dual expander cycle is currently just a concept and only few of them are being developed. The results show Figure 1- 6: Closed dual expander cycle schematics a decrease of weight to achieve the same thrust level.

1.3.4 Open expander cycle

In an open expander cycle (also known as “bleed” expander cycle) the fuel flow is split after it leaves the turbopump: a part flows through the main injector, the rest drives through the cooling jacket providing the energy the turbine needs. Since the cycle is “open”, this second flow never reach the main combustion chamber, but it is expelled to the outside or directly injected into the lower part of the nozzle, performing a sort of post-combustion (see Figure 1-7). Because part of the fuel bypasses the combustion chamber, the engine works at a lower efficiency.

Chapter 1 ______

Even if we lose efficiency, the power needed by the turbopump to achieve a certain combustion chamber pressure decreases. Consequently, if the same power at the turbopump is kept, a higher combustion chamber pressure can be reached, which results in a higher thrust level.

Figure 1- 7: Open expander cycle schematics.

1.4 Existing expander cycle systems

Due to their advantages, there are many expander cycle engines in use today. These include the RL-10 which powers the Centaur upper stage (NASA), the LE-5 used by the Japanese Aerospace Exploration Agency (JAXA) and the Vinci used by the European Space Agency (ESA).

1.4.1 The RL10 engine

The RL10 liquid rocket engine is a closed expander cycle engine, originally developed by Pratt & Whitney (P&W) in the middle 1950s. This engine is in fact derived from a liquid hydrogen powered turbojet engine, designed as model 304 (7). It has been the pioneer of all expander cycle rocket engines.

______1.4 Existing expander cycle systems

Figure 1- 8 RL-10 schematic Over the years, the LR10 passed through many restyling, but all of them have in common the usage of liquid hydrogen (LH2) as propeller and liquid oxygen (LOX) as oxidizer. The key mechanical features of the basic configuration are:

- Two-stage centrifugal fuel turbopump - Single-stage centrifugal oxidizer pump - Two-stage axial flow turbine on the fuel pump shaft - Reduction gear system to drive the oxidizer pump and the fuel pump with a single turbine, each at the required speed - Tubular stainless steel combustion chamber/primary nozzle (thrust chamber)

The LR10A-1 has been the first liquid hydrogen engine to fly on a rocket, and two of them have been used on the first Atlas/Centaur vehicle AC-1.

The RL10-A4 LOX pump delivers 17,78 kg/s at 5,725 kPa with a design speed of 14300 rpm (1).

The data and performances of the RL-10 engines are presented in the table 1-1.

Chapter 1 ______

On June 1995, the development of a new RL10 engine started to face the grown payload of the new Delta III lunch vehicle. The most distinctive feature of this new engine, known as RL10B- 2, is the translating nozzle extension and an expansion ratio of 285:1.

Parameter RL10B-2 RL10A-4

Vacuum Thrust [kN] 110 99

Chamber Pressure [kPa] 4440 4205

Mixture Ratio 6,0 5,5

Specific Impulse [sec] 466,5 451,0

Fuel Flow Rate [kg/s] 3,45 3,45

Oxidizer Flow Rate [kg/s] 20,64 19,05

Fuel Pump Speed [rpm] 37900 37360

Fuel Pump Discharge pressure [kPa] 10404 9618

Oxidizer Pump Speed [rpm] 15163 14944

Oxidizer Pump Discharge pressure [kPa] 5915 5839

Table 1- 1 The main limits of this cycle are (1):

 While the RL-10 is designed to have restart capability, it is designed for a relatively short life.  The RL-10 has a high parts count, a large gearbox and requires many hours of maintenance.  Limited throttling range.

1.4.2 The LE-5 engine

The LE-5B engine was designed as the second-stage engine of the H-IIA launch vehicle. It was the first LH2/LOX Japanese engine and one of the first expander bleed cycle engine ever produced.

______1.4 Existing expander cycle systems

One of the most interesting feature of this engine is the wide operating throttle range: in addition to 100% rated operations, throttling tests at 60%, 30% and at extremely low levels only using the tank pressure without operating the turbine were tested and verified a stable operating capability. (8)

The LOX pump delivers 19.4 kg/s at a design speed of 17000 rpm (1). The full list of the engine data is presented by Table 1-2.

Parameter LE-5A LE-5B

Vacuum Thrust [kN] 122 137

Chamber Pressure [MPa] 4,0 3,6

Mixture Ratio 5,0 5,0

Specific Impulse [sec] 453 450

Fuel Pump Speed [rpm] 50500 50200

Oxidizer Pump Speed [rpm] 17400 17100

Turbine inlet temperature [K] 600 380

Table 1- 2 Figure 1- 9 LE-5A engine

1.4.3 Vinci

The Vinci was designed to be the upper stage of the Ariane 5. It has a design thrust of 180 kN, a design Isp of 464 s and uses a closed expander cycle engine. It has a dual turbine design, eliminating the need for a gearbox. The two turbopumps are in a direct drive configuration, as shown by the Figure 1-11. The turbines are set in series and a set of two by-pass valves adjust their flow rates. It is intended to be capable of five restarts. The engine characteristics are listed in the Table 1-3.

The peculiarity of this engine is the extended combustion chamber (Figure 1-10), designed to increase the heat transferred to the turbine working fluid. This is one way to improve the performances of an expander cycle, but on the other hand it increases the engine weight.

Chapter 1 ______

Parameter Vinci

Vacuum Thrust [kN] 180

Chamber Pressure [MPa] 6,0

Mixture Ratio 5,8

Specific Impulse [sec] 465

Fuel Pump Speed [rpm] 90000

Oxidizer Pump Speed [rpm] 18000

Fuel flow rate [kg/s] 5,80

Oxygen flow rate [kg/s] 33,70

Fuel pump discharge pressure [MPa] 22,4

Oxidizer pump discharge pressure [MPa] 8,1 Figure 1- 10: Vinci combustion Table 1- 3 chamber.

Figure 1- 11: Vinci schematics

______1.5 Liquid propellants

1.5 Liquid propellants

Propellants characteristics affects engine design and performances and the propellant storage. When choosing a propellant, many factors should be taken into account:

Economic factors. It’s a very important factor. Takes into account the raw material availability and cost, as well as the production process complexity.

Corrosion hazards. Some propellants (such as hydrogen peroxide and nitrogen tetroxide) have to be handled in containers of special materials to avoid corrosion. The corrosion products may make the propellant unsuitable for the designed rocket engine.

Explosion and fire hazards. Some propellants (such as hydrogen peroxide and nitromethane) are unstable and tend to detonate under certain conditions (temperature, pressure, shock). The fire hazard is usually associated with the oxidizers, which may start chemical reactions with a large variety of organic substances.

Health hazards. Many propellants are toxic or poisonous, therefore special precautions have to be taken.

Specific gravity. Higher is the propellant density, lower is the space needed to store it, resulting in a lower structural vehicle mass and aerodynamic drag. In addition, it affects the mass flow rate, resulting in better performances for higher specific gravities. Therefore, specific gravity has an important effect on the engine performances and the maximum flight speed. Figure 1-12 shows the specific gravities of several liquid rocket engines propellants with respect to the temperature.

Heat transfer. High specific heat, high thermal conductivity and high boiling point are desirable for propellants used for thrust chamber cooling.

Vapor pressure. A low vapor pressure permits easier handling of propellants and more effective pump design (it reduces the potential cavitation point). Propellants with high vapor pressure (Liquid hydrogen / liquid oxygen) require special design and low-temperature materials.

Propellant performances. The propellant performances can be compared in terms of specific impulse, effective exhaust velocity, specific propellant consumption or other engine parameters.

Performances parameter of various propellants combination can be found in the table 1-4.

Chapter 1 ______

Figure 1- 12: Specific gravities of several liquid propellants as function of temperature (Reference density 103 kg/m3). For high-performances engines, a good propellant must have a high content of chemical energy to permit a high chamber temperature.

The highest potential specific impulse is achieved using a toxic liquid fluorine oxidizer with hydrogen fuel plus suspended solid particles of beryllium, which gives approximately a 480 s specific impulse at 1000 psia (6,894 MPa) chamber pressure. Due to the toxicity of the oxidizer, there are not rocket engine developed with these propellants.

The most common liquid propellants combinations are Liquid oxygen-Liquid hydrogen (used in Centaur upper stage, the Space Shuttle main engine and other upper stage engines developed in Japan, Russia, Europe and China) and Liquid oxygen-Hydrocarbon. In the following sections we will describe oxygen, hydrogen and methane. The characteristics of these three propellants are shown in Table 1-5.

______1.5 Liquid propellants

Table 1- 4: Theoretical performances of liquid propellants combinations [ref. Sutton]

Chapter 1 ______

Liquid oxygen Liquid hydrogen Methane

Chemical formula O2 H2 CH4

Molecular mass 32.00 2.016 16.03

Melting point [K] 54.4 14.0 90.5

Boiling Point [K] 90.0 20.4 111.6

Heat of vaporization [kJ/kg] 213 446 510

Specific heat [kcal/kg*K] 0.4 (65 K) 1.75 (20.4 K) 0.835 (111.6 K)

Specific gravity 1.14 (90.4 K) 0.071 (20.4 K) 0.424 (111.6 K)

1.23 (77.6 K) 0.076 (14 K)

Viscosity [centipoise] 0.87 (53.7 K) 0.024 (14.3 K) 0.12 (111.6 K)

0.19 (90.4 K) 0.013 (20.4 K) 0.22 (90.5 K)

Vapor pressure [MPa] 0.0052 (88.7 K) 0.2026 (23 K) 0.033 (100 K)

0.87 (30 K) 0.101 (117 K)

Table 1- 5

1.4.3 Liquid oxygen

Liquid oxygen (often abbreviated as LOX) is the most commonly used oxidizer, because of the high performances obtained using it. It has a light blue color (Figure 1-13). At atmospheric pressure boils at 90 K, has a specific gravity of 1.14 and a heat of vaporization of 213 kJ/kg.

It usually does not burn spontaneously with organic matter at ambient pressure, but combustion or explosion may occur when oxygen and organic matter are pressurized. Therefore, to avoid it, the contact materials used for handling and storage must be clean. Liquid oxygen Figure 1- 133: Liquid is a non-corrosive and nontoxic liquid, thus will not cause the oxygen deterioration of container walls.

______1.5 Liquid propellants

Liquid oxygen can be obtained by several raw materials, for example from air. Air is made mostly of oxygen and nitrogen. In order to obtain liquid oxygen, the air is first compressed and cooled to obtain liquid air. Nitrogen gas turns into liquid at -196 °C, oxygen at -183 °C, so to separate oxygen and nitrogen, the liquid air is heated just enough for nitrogen to turn into gas, leaving only the liquid oxygen.

1.4.4 Liquid hydrogen

Liquid hydrogen delivers very high performances when burned with liquid oxygen or fluorine, and it is also an excellent regenerative coolant. Among all the used liquid fuels, liquid hydrogen is the lightest (having a specific gravity of 0.07) and the coldest (having a boiling point of about 20 K). Because of the very low fuel density, big tanks are needed to store a proper fuel quantity, which means a large vehicle volume, with relatively high drag. The extreme low temperature makes the material choice very problematic for tank and piping, because many materials become brittle at these temperatures. In addition, tanks and lines must be isolated to minimize the heat exchange and the consequent hydrogen evaporation. Another consequence of the hydrogen low boiling point is that all common liquids and gases solidify in it. Because of that, all of the lines and tanks must be carefully emptied of air and moisture before introducing the propellant, to avoid that solids particles may obstruct orifices and valves. Furthermore, mixture of liquid hydrogen and solid oxygen or solid air can be explosive.

Liquid hydrogen is manufactured from gaseous hydrogen by successive compression, cooling and expansion processes.

Hydrogen burning with oxygen forms a non-toxic exhaust gas. This combination gives the highest specific impulse for a non-toxic combination.

1.4.5 Methane

Methane (CH4, often called Liquid Natural Gas LNG) is one of the commonly used hydrocarbon fuels. The hydrocarbon fuels usually give good performances and are relatively easy to handle.

Methane is a cryogenic hydrocarbon fuel, it is denser than liquid hydrogen and has a relatively low cost. Its higher density allows the design of more compact stages when compared to

Chapter 1 ______hydrogen, making a lighter vehicle with lower drag, which compensates the lower specific impulse.

There are some concepts for operating a rocket stage using two fuel, initially with methane and then switching during flight to hydrogen, but they are not fully developed yet. (2)

Chapter 2 Propellant pump

2.1 Introduction

The propellant feed system of a liquid rocket engine is responsible for delivering the propellants from the tanks to the combustion chamber, at the required flow rate and pressure conditions. The propellant feed system consists of propellant tanks, feed lines (tubes and ducts), valves and pressurization devices. Depending on how propellants are pressurized and fed into the combustion chamber, the feed systems are classified as pressure-fed or pump-fed systems (1).

Figure 2- 1 shows a pressure-fed system schematics. This system relies on the simple idea of take advantage of the tank pressure to feed propellant into the combustion chamber. It gives the name to this class of rocket engine (pressure- fed rocket engine). They have a very simple design and are very reliable, but are limited to low chamber pressure requirements, because high pressures make the tanks too heavy (for material Figure 2- 1 strength reasons). This results in lower thrust.

The pump-fed systems are used for high-pressure, high performances applications. Any expander cycle engine uses this type of feed system. The main elements are turbine, pump, gearbox (optional), pipe and tubes, housing (Figure 2- 2). The turbine takes the energy from the hot gas flowing around the combustion chamber and drives the pump. If the engine has not a “dual” configuration, a gearbox may be required the drive and synchronize oxygen and fuel pumps. It is one of the main disadvantages of this type of propellant feed system, since it adds weight and failure probability. The propellant pressure is achieved thanks to the pump, which can either be axial or centrifugal.

Chapter 2 ______

Figure 2- 2: Typical geared turbopump assembly used on the RS-27 engine (Delta I and II Launch Vehicles) with liquid oxygen and RP-1 propellants. (Courtesy of The Boeing Company, Rocketdyne Propulsion and Power.) [Reference: (2)] The use of turbopumps enables the engine to operate at high chamber pressure without increasing the vehicle tank weight. The turbopump typically consists of one or more pump stages driven by a turbine. Its basics elements are shown in the Figure 2- 3: pump, turbine, bearings, seals, housing. The pump operates over a wide range of pressures, pumping the fluid from low pressure at the inlet to a very high pressure at the outlet. The pressure at the inlet of the pump is relatively low due to the low tank pressure. Therefore, the cavitation phenomenon must be taken into account.

Since expander cycle engines use a pump-fed system, in the following sections we will describe the main components of this configuration. Specifically, the focus is on the pump and turbine elements and the design methods.

______2.2 Pump description

Figure 2- 3: Turbopump main elements

2.2 Pump description

A pump is a component designed to transfer energy to a fluid, with the purpose of increasing its pressure. The main elements of a pump are: housing, inducer (optional), impeller, diffuser and volute. The impeller blades (rotating element) transfer energy to the fluid, increasing its pressure and its speed. The diffuser (stationary element) converts kinetic energy into pressure. The volute has the main task of collecting the fluid and guiding it to the discharge. Also a little pressure rise takes place within the volute. The fluid enters the pump at a low pressure, coming from the supply tank. The pump inlet pressure is usually minimized to reduce the tank size and weight. To prevent cavitation phenomena, an inducer is placed before the impeller. The inducer is usually an axial-flow type impeller that increases the fluid pressure to permit normal operation of the main impeller. The inducer needs to add sufficient energy to the fluid to suppress the cavitation of the fluid passing through the impeller. It should be pointed out that not all designs require an inducer. According to the flow direction at the exit of the impeller, the pump can be axial or centrifugal.

Centrifugal pumps. Almost all rocket propellant pumps are of this type. Centrifugal pumps are characterized by high heads and low flow rates. They can handle high-pressure requirements efficiently as well as economically in terms of weight and size. The impeller is basically a rotating wheel with radial vanes. The flow enters axially respect to the impeller and leaves in

Chapter 2 ______radial direction with increased pressure. Figure 2- 4 shows the main elements of a centrifugal pump. The centrifugal impellers can be either cantilevered or shrouded. The shrouded (or covered) impeller is preferred because it maintains a tighter clearance, which reduces the leakage. The shroud, however, adds mass, and thus puts higher stresses on the part. The centrifugal impeller tip speeds are limited by the material strength (610 m/sec with titanium for LH2 and 274 m/sec with Inconel 718 for LOX) (Humble, Henry and Larson, 1995). The maximum head produced by a single stage is limited by the maximum allowable tip speeds. Thus, low-density fluids such as LH2 require the use of multiple stages for high-pressure applications. The basic construction of a multistage pump is similar to that of a single stage pump, except that proper channeling of the fluid between stages is added. At the impeller outlet, the flow goes through the diffuser, and then it is guided by the volute to the discharge.

Figure 2- 4 Axial pumps. This type of pump (see Figure 2- 5) can be represented as an impeller in a tube. It is generally used for high flow - low heads requirements. The flow enters and leaves the impeller in axial direction, experiencing a pressure gain thanks to the rotational movement of the impeller. The axial pumps are less efficient than the centrifugal pumps (1), so it is necessary

______2.3 Cavitation to use a larger number of axial pump stages to generate a pressure rise equivalent to that of a centrifugal pump.

Figure 2- 5: Axial pump schematics

2.3 Cavitation

The phase change of any liquid substance is influenced by temperature and pressure. If, keeping a constant pressure, we increase the temperature of a liquid, we will reach one maximum point, commonly called boiling point. This is the temperature at which the vapor pressure of the liquid reaches the pressure acting on it. The vapor pressure is defined as the relative pressure exerted by vapor in thermodynamic equilibrium with its condensed phase (solid or liquid) at a given temperature in a closed system and it is strongly dependent on temperature. In particular, an increase of temperature determines an increase of vapor pressure (Figure 2- 6).

It is well known that the boiling point changes with the pressure: higher pressure means higher boiling point, while lower pressure means lower boiling point.

Chapter 2 ______

Any impeller causes a pressure drop at the inlet. According to what we have said previously if the pressure of the liquid is lower than the vapor pressure at the working temperature, it may cause a phase Figure 2- 6 change of the fluid, which means, for liquid propellants, the formation of vapor bubble. This phenomenon is called cavitation. The bubbles usually form around the blades, since the lowest pressure points are typically located on the blades’ surfaces. This phenomenon is one of the main reasons of performances degradation in liquid rocket engines (3).

Figure 2- 7: Cavitation phenomenon (left) and effects (right). The first effect of cavitation is the erosion of the pump surface (Figure 2- 7). The erosion is due to the instability of the vapor bubbles: when the pressure of an infinitesimal volume drops below the liquid vapor pressure, a bubble appears; when this bubble travels with the flow, it will reach a point where the pressure is above the vapor pressure, so it will collapse, generating very high- pressure waves. This pressure waves recur with high frequency, due to the rapid creation and collapse of the bubbles. Thus, the blades’ surfaces are subjected to a mechanism of fatigue which causes the erosion.

Another effect, due to the same reason of the erosion effect, are vibrations, which may cause problems to the engine.

______2.4 Pump parameters

The last effect considered is the instability of the volumetric flow, due to the increase of volume caused by the bubbles formation. This phenomenon may cause non-optimal combustion and reduce the thrust.

Figure 2- 8: Fuel pump inducer impeller of the Space Shuttle main engine lowpressure fuel turbopump. It has a diameter about 10 in., a nominal hydrogen flow of148.6 lbm/sec, a suction pressure of 30 psi, a discharge pressure of 280 psi at 15,765 rpm,an efficiency of 77%, and a suction specific speed of 39,000 when tested with water. (Courtesy of The Boeing Company, Rocketdyne Propulsion and Power.) [Reference: (2)] As we will see in the section 2.4, the cavitation phenomenon is related to the tank pressure. Thus to avoid it we should increase the tank pressure, and consequently the vehicle weight. One method to avoid cavitation without increasing the tank pressure is to place an inducer (Figure 2- 8) before the impeller inlet. The inducer is a special pump impeller, usually connected to the same shaft and rotating at the same speed of the main impeller. Inducers are basically axial flow pumps with a spiral impeller, and they usually operate under slightly cavitation conditions (at their inlet). The inducer head is typically very low (2 to 10% of the total pump head (2)), since it has to be just large enough to suppress cavitation in the main pump. This allows a smaller and lighter main pump.

2.4 Pump parameters

In this section we are going to define some important parameters needed during the design process of a rocket propellant pump.

풌품 Required pump mass flow - 풎̇ [ ] 풔 The required mass flow:

푚̇ = 휌 ∗ 푣 ∗ 퐴 (2.1)

where v is the flow speed, A is the cross sectional area and 휌 is the fluid density is determined by the rocket design to achieve a given thrust, effective exhaust velocity and mixture ratio. In

Chapter 2 ______addition to the flow required by the combustion chamber, if part of the flow bypasses the pump, it has also to be accounted for.

푚3 The product v*A is defined as volumetric flow rate Q [ ] and can be determined by one of the 푠 two relations presents in Equation (2.2).

푚̇ 푄 = 푣 ∗ 퐴 = (2.2) 휌

Mixture ratio - 휶 The mass flow (as well as the volumetric flow rate) represent the total flow of propellant, so it is important to know the mixture ratio between fuel and oxidizer. The mixture ratio is defined as the oxidizer mass flow divided by the fuel mass flow, as shown by equation (2.3).

푚̇표 훼 = (2.3) 푚푓̇

For RP-1/LOX it usually ranges between 2.2 and 3, for Hydrogen/LOX has values between 5 and 7.

Required pump discharge pressure - 풑풅 [푷풂] It is determined from the chamber pressure and the hydraulic losses in valves, tube, pipes, cooling jacket and injectors.

푝푑 = 푝푠 + (Δ푝)푝푢푚푝 = 푝1 + (Δ푝)푑푟표푝푠 (2.4)

The eq. (2.4) shows that the pump discharge pressure (pd) equals the propellant pump suction pressure (ps), see Eq. (2.5), plus the pressure rise across the pump (Δ푝)푝푢푚푝; for the pressure balance, the required pump discharge pressure equals the chamber pressure (p1) plus all the pressure drops that occur downstream of the pump: valves, tube and pipes, cooling jacket, injectors.

The suction pressure depends on the propellant tank pressure, and can be estimated using the Bernoulli relation:

2 푝푠 = 푝푡푎푛푘 − (0.5 ∗ 휌 ∗ 푐1 ) ∗ 퐶푑 (2.5) where Cd is the friction coefficient in the suction channel and valves (3).

To avoid cavitation phenomena the suction pressure must be higher than the vapor pressure in the pump inlet section.

______2.4 Pump parameters

Manometric head - H [m] The Manometric head represent the pressure increase generated by the pump between discharge and suction. It can be expressed as

푝푑−푝푠 ∆퐻 = + 푌 + 푌 (2.6) 휌 ∗ 𝑔 푠 푑 where Ys and Yd are the friction losses in the channels before and after the pump

Net positive suction head – NPSH [m]

The NPSH can be divided into two type: the required NPSH (NPSHr) and the available NPSH

(NPSHa). Both are pressure measurement expressed in meters.

The NPSHr is the limit value of the head at the pump inlet that allows to avoid cavitation phenomena (see section 2.1.1). It is necessary to have this pressure at the inlet because every pump generates a pressure drop at the entrance of the impeller vanes. Therefore, all pump systems must maintain a positive suction pressure to overcome this pressure drop. The NPSHr depends upon the pump design and the working-fluid.

The NPSHa measures the absolute pressure at the suction. It combines the effects of tank pressure, the elevation of the propellant level with respect to the pump inlet and the friction losses in the line between tank and pump:

푁푃푆퐻푎 = 퐻푡푎푛푘 ± 퐻푒푙푒푣푎푡푖표푛 − 퐻푓푟푖푐푡푖표푛 − 퐻푣푎푝표푟 (2.7)

To avoid pump cavitation NPSHa>NPSHr.

shows the NPSHa and some other head references.

Chapter 2 ______

Figure 2- 9 Definition of pump net positive suction Let us make some consideration in order to determine the NPSH from a theoretical point of view. By applying the generalized Bernoulli’s equation between the tank and the pump inlet (which is the point of minimum pressure) we obtain:

푝 − 푝 푐2 −퐿 = 1 퐴 + 1 + 𝑔(푧 − 푧 ) (2.8) 푤 휌 2 1 퐴

In writing Eq. (2.8) we have neglected the fluid velocity in the tank and we have used Lw to denote the head loss that occurs in the line between tank and pump.

Calling Helev = g(z1-za) the fluid level – pump inlet head, we can rearrange the Equation (2.8) as:

푐2 푝 = 푝 − 훾퐿 − 휌 1 − 훾퐻 (2.9) 1 퐴 푤 2 푒푙푒푣 where γ = ρ*g is the specific weight, defined as weight per unit of volume [N/m3].

If we want to be more accurate, we should consider that around the impeller blades there are some areas where pressure is lower than p1. To account for this, we can subtract a ∆p to the p1, obtaining equation (2.10).

푐2 푝 = 푝 − 훾퐿 − 휌 1 − 훾퐻 − Δ푝 (2.10) 1 퐴 푤 2 푒푙푒푣

To avoid cavitation, the pressure at the pump inlet p1 must be higher than the vapor pressure pv of the fluid at the fluid temperature.

We can rearrange the equation (2.10) into equation (2.11) by separating the system characteristic (NPSHa) from the pump characteristic (NPSHr)

푝 − 푝 휌푐2 Δ푝 푁푃푆퐻 = 퐴 푣 − 퐿 − 퐻 > 1 + = 푁푃푆퐻 (2.11) 푎 훾 푤 푒푙푒푣 2훾 훾 푟

Suction specific speed – S

The suction specific speed can be defined by the equation (2.12), where Nr is the number of revolutions per seconds, or by the equation (2.13), where ω is the angular velocity expressed in radians per seconds.

푁 √푄 푟 (2.12) 푆 = 3/4 (𝑔 ∗ 푁푃푆퐻푟)

______2.4 Pump parameters

휔√푄 (2.13) 푆 = 3/4 (𝑔 ∗ 푁푃푆퐻푟)

When using the number of revolutions per second, the suction specific speed ranges between 1.8 and 2.4 for axial turbopumps, and between 2.5 and 3 for centrifugal turbopumps (3).

When using the angular velocity S ranges between 14 and 24.

In both equations (2.12) and (2.13) NPSHr is the required net positive suction head, therefore knowing the suction specific speed, the volumetric flow and the impeller speed, we can determine the NPSHr.

Specific speed – Ns The pump specific speed is a characteristic value typically defined at the point of maximum efficiency, which is usually the design point (4). It is defined as:

휔√푄 푁 = (2.14) 푠 (gΔ퐻)3/4 where ω is the pump rotating speed (radians per second), Q is the pump flow rate (m3/s) and ∆H is the pump head (m).

Ns is a function of the design configuration and does not vary significantly for a series of geometrically similar impellers (having the same angles and proportions), or for a particular impeller operating at any speed.

Knowing Ns and the volumetric flow rate Q, we can estimate the overall efficiency of a Figure 2- 10: Turbopump diagram specific pump using the turbopump diagram (Figure 2- 10).

Relationships between the specific speed and some non-dimensional coefficients

The specific speed Ns (Eq. 2.14) can be expressed as a function of the head coefficient ψ and the flow coefficient φ, referred to the outlet section.

The volumetric flow rate is expressed from a cross section multiplied by the absolute speed meridian component cm2 (Eq. 2.15).

푄 = 휋퐷2푏2푐2푚 (2.15)

Chapter 2 ______

The discharge flow coefficient is defined as:

푐2푚 휑2 = (2.16) 푢2

We can replace 푐푚2 in the Eq. (2.15) with a function of 휑 and 푢2:

푄 = 휑휋퐷2푏2푢2 (2.17)

Replacing Equation (2.17) into Equation (2.14) we obtain

휔√휑휋퐷 푏 푢 푁 = 2 2 2 (2.18) 푠 (𝑔Δ퐻)3/4

The angular velocity ω can be written as:

2 ∗ 푢 휔 = 2 (2.19) 퐷2

Substituting the (2.19) in the (2.18) we obtain Equation (2.20).

3/4 2√휋퐷 푏 푢3/2 푏 푢2 2 2 2 1/2 2 2 1/2 (2.20) 푁푠 = 3/4 휙 = 2√휋√ ( ) 휑 퐷2 (𝑔Δ퐻) 퐷2 𝑔Δ퐻

By definition, the head coefficient equals to:

𝑔Δ퐻 휓 = 2 (2.21) 푢2

Substituting the (2.21) into the (2.20) we obtain:

푏 휑1/2 2 (2.22) 푁푠 = 2√휋√ 3/4 퐷2 휓

푏 Equation (2.22) express the specific speed as a function of the ratio 2 and the flow and head 퐷2 coefficients.

For pumps typically used for liquid rocket engines, the values of flow coefficients are:

- Inlet flow coefficient (푢푒= impeller eye velocity): 푐 1푚 (2.23) 휑1 = = 0.08 ÷ 0.4 푢푒

- Discharge flow coefficient (푢2=impeller tip speed, usually between 182.88 and 609.6 [m/s])

푐2푚 휑2 = = 0.05 ÷ 0.3 (2.24) 푢2

______2.4 Pump parameters

The discharge meridian velocity 푐푚2 should be approximately 1.5 times higher than the inlet meridian velocity 푐푚1.

There is a relation between the head and flow coefficients and the pump specific speed, shown by Figure 2- 11.

Figure 2- 11: Flow and head coefficients vs specific speed

Specific diameter – Ds [m]

The specific diameter is a value that depends on the specific speed Ns and it is necessary to determine the impeller diameter. Impeller diameter and specific diameter are related by the equation (2.25).

1/4 퐷2 ∗ (𝑔 ∗ Δ퐻) 퐷푠 = (2.25) √푄

The value of Ds can be read by the Cordier diagram (figure 2-12).

This diagram was determined by Cordier in 1950s, by empirical analysis of turbomachines, trying to correlate Ns, Ds and η. He found out that turbomachines with good efficiencies tend to group along a defined curve, when plotted with their Ns vs Ds values.

Chapter 2 ______

He further found that machines with low efficiency were found far away from the ones with good efficiency in the diagram (5).

The curve on the diagram represent the value

of Ds optimized to have a pump efficiency of

90%. The value of Ds far from the curve represent machines with lower efficiency.

Figure 2- 12: Cordier diagram

Impeller diameter – D2 [m]

Knowing the specific diameter we can determine the impeller diameter D2 by the Equation (2.25) as:

퐷 √푄 퐷 = 푠 (2.26) 2 (g ∗ Δ퐻)1/4

If the impeller diameter is not acceptable, we must change the specific diameter, so the input parameter of the Cordier diagram Ns.

An alternative method to determine the impeller diameter passing through the determination of the impeller tip speed. The tangential velocity (u2) can be determined by the equation (2.27).

𝑔 ∗ 퐻 푢 = √ (2.27) 2 휓

Where ψ is the head coefficient that take into account the impeller losses. Typical values of ψ for centrifugal pumps are 0.4 – 0.7 (6), in particular 0.6 – 0.7 for pumps which works with low density fluids (LH2). For centrifugal pumps ψ is determined by equation (2.21).

Knowing u2, the impeller diameter is defined by the equation (2.28).

______2.4 Pump parameters

2 ∗ 푢 퐷 = 2 (2.28) 2 휔 The impeller speed is limited by the design material strength to about 610 m/s. With titanium (lower density than steel) and machined cantilevered impellers a speed of over 655 m/s (2). For cast impellers this limiting value is lower than for machined impellers. Consequently, once we have determined the impeller diameter and velocity, we have to verify if the stress acting on the impeller is lower than the material admissible stress.

Impeller-inlet velocity triangle Since at the impeller inlet the pipe is axial, the flow enters with a negligible tangential component of velocity. Therefore, as first approach, the inlet absolute velocity C1 can be considered, completely axial (i.e. α1 = 90°) and equals to 푄 푐1 = (2.29) 퐴1 where A1 is the inlet cross sectional area.

Figure 2- 13: Centrifugal pump blade detail (left), Inlet velocity triangle (right)

The dragging speed u1 is orthogonal to the pump axis and directed in accordance to the impeller angular velocity. It can be determined by the shaft rotational speed by the equation:

푢1 = 2휋푁푟퐷1 (2.30)

Defining the angle α1 as the angle between the absolute velocity and the tangential direction and β1 as the angle between the relative speed and the tangential direction, taking into account the velocity addition formula, we can determine the relative velocity magnitude as

푐1 푤1 = (2.31) sin (훽1)

푐1 where 훽1 = 푎푟푐푡푎푛 ( ) 푢1

A sample inlet velocity triangle is shown by figure 2-13. Obviously, the absolute speed can be also determined by

Chapter 2 ______

푐1 = 푢1 ∗ tan (훽1) (2.32)

If, for any reason, a device that gives to the flow a whirl is installed before the impeller inlet, the α1 angle is not equal to 90° any longer. The figure (2-14) shows the inlet triangles for “pre- rotation” α1<90° and “counter-rotation” α1>90°. It can be seen that the approach flow angle β1 increase with pre-rotation and decrease with counter-rotation, keeping constant the dragging speed and the axial component of the absolute velocity. The picture also shows the incidence angle i1, defined as the difference between the blade angle β1B and the flow angle β1. If the incidence is zero the blade has only a displacement effect on the flow. This situation is called shockless entry. A leading edge different from zero will generate circulation around the leading edge, resulting in lower pump efficiency.

Figure 2- 14: Inlet velocity triangle with pre-rotation and counter-rotation

Impeller-outlet velocity triangle For the impeller outlet triangle two conditions must apply simultaneously: 1. The work, determined by the Euler equation, must be equal to the theoretical head required by the machine (equation 2.33); 2. The meridian component of the absolute velocity must guarantee the discharge of the mass flow. 𝑔Δ퐻 퐿 = 푐2푢2 cos(훼2) − 푐1푢1 cos(훼1) = 푐2푢푢2 − 푐1푢푢1 = (2.33) 휂푦

Assuming α1=90° and introducing the head coefficient ψ (2.21), equation (2.33) becomes: 𝑔Δ퐻 푢 휓 2 (2.34) 푐2 cos(훼2) = = 푢2휂푦 휂푦 Therefore, the first condition calculates the tangential component of the impeller outlet absolute velocity:

푢 휓 2 (2.35) 푐2푢 = 휂푦

______2.4 Pump parameters

The second condition, under the assumption of uniform speed on the cross section at the impeller outlet, immediately defines the meridian component of the absolute velocity (2.36), directly from the definition of flow coefficient (2.16)

(2.36) 푐2푚 = 휑 ∗ 푢2 Being known the absolute speed components, it is possible to determine the absolute speed magnitude (2.37)

2 2 (2.37) 푐2 = √푐2푚 + 푐2푢

The angle β2 can be determined by:

푐2푚 훽2 = 푎푟푐푡푎푛 ( ) (2.38) 푢2 − 푐2푢

The tangent of α2 equals to:

푐2푚 휑 ∗ 푢2 휑 tan (훼2) = = = (2.39) 푐2푢 휓 ∗ 푢2 휓 휂푦 휂푦 Therefore

푐 휑 푚2 (2.40) 훼2 = arctan ( ) = 푎푟푐푡푎푛 ( ) 푐푢2 휓 휂푦 It easy to see the validity of equations (2.38), (2.39) and (2.40) from figure 2-15.

Figure 2- 15: Impeller outlet velocity triangle To complete the outlet velocity triangle the only remaining data necessary is the relative velocity, which can be determined by the absolute and the tangential velocity (2.41)

2 2 (2.41) 푤2 = √(푐2 + 푢2) − [2푢2푐2 cos(훼2)]

It is now possible to determine the two impeller outlet components:

Chapter 2 ______

- The meridian component w2m coincides with c2m (2.42) 푤2푚 = 푐2푚 = 푤2sin (훼2)

- The tangential component w2u (2.43) 푤2푢 = 푤2cos (훼2) or (2.44) 푤2푢 = 푢2 − 푐2푢 = 푢2 − 푐2cos (훼2)

Another relation to determine the β2 angle is:

푤2푚 β2 = 푎푟푐푡푎푛 ( ) (2.45) 푤2푢

Since the relative speed w2 is inclined at the angle β2, it can be written as

푢2 − 푐2푢 푤2 = (2.46) cos (훽2)

There are basically three configurations of impeller blades: backward blades, radial blades, forward blades (figure 2-16). The impeller-outlet triangle angles change in these three configurations.

Figure 2- 16: Blades configurations Forward blades are directed forward respect to the impeller rotating velocity. This turns in a high absolute velocity, which means that basically all of the energy transferred to the fluid is kinetic energy.

Backward blades are directed to the opposite direction respect to the impeller rotating velocity. In this case we have a small absolute velocity, so almost all of the energy transferred is

______2.4 Pump parameters transformed in pressure. Backward-blades impeller are more often used, because they have a higher efficiency and prevent boundary layer separation.

Blade numbers - z The number of blades (z) needed to have an efficient impeller can be estimated by a simple empirical relation (3)

푅푠 푧 = 2 ∗ 푘 ∗ ∗ sin (훽푚) (2.47) 푙푏

Where lb is the blades length expressed in meters (2.48), Rs is the mean radius expressed in meters (2.49), βm is the average angle (2.50) and k is a coefficient that depends on the pump architecture (k=6.5 for centrifugal pumps; k=4.5 for axial pumps).

퐷 − 퐷 푙 = 2 1 (2.48) 푏 2

퐷 + 퐷 (2.49) 푅 = 2 1 푠 4

훽 + 훽 (2.50) 훽 = 1 2 푚 2

Total pump efficiency - 휼풑 The total efficiency of a pump can be seen as the product of three efficiencies (2.51):

휂푝 = 휂푦휂푣휂푇푟 (2.51)

- 휂푦 hydraulic efficiency. The hydraulic efficiency depends on the specific speed and on the machine architecture. It also depends on the Reynolds number and on the roughness of the impeller surface. [Typical values between 0.7 and 0.96].

- 휂푣 volumetric efficiency. The volumetric efficiency is defined as the ratio of the outlet flow rate delivered by the pump and the theoretical discharge flow rate produced by the pump (i.e. the inlet flow rate). It gives an esteem of the flow loss due to leakage of the fluid through the pump. [Typical values between 0.85 and 0.97]

- 휂푇푟 transmission efficiency. The transmission efficiency considers the losses between the pump and the turbine, due to bearings, gears and any other moving part. [Typical values between 0.88 and 0.97]

Chapter 2 ______

2.5 Pump design methods

2.5.1 Method 1

The input parameter is the number of rotation per minute Nr [rpm]. The choice of Nr depends on the propellant density.

푟푎푑 The first step is calculate the angular velocity ω[ ], as shown by equation (2.52). 푠

2 ∗ 휋 ∗ 푛 ω = (2.52) 60 Then we calculate the volumetric flow rate Q and the head H.

It is now possible to calculate the specific speed Ns, consequently it is possible to read on the

Cordier diagram the specific diameter Ds that makes the pump work with a good efficiency.

From the specific diameter we can calculate the outlet diameter D2

퐷 √푄 퐷 = 푠 (2.53) 2 (g ∗ Δ퐻)1/4

Knowing the outlet diameter the impeller tip speed is equal to

퐷 ∗ 푁 푢 = 2 0 (2.54) 2 2 The power absorbed [W] by the pumps is determined by equations (2.55) and (2.56) respectively for oxidizer and fuel pump.

(푚̇ )표푥 ∗ [(푝푑)표푥 − (푝푠)표푥] (푃푎)표푥 = (2.55) 휌표푥 ∗ 휂표푥

(푚̇ )푓 ∗ [(푝푑)푓 − (푝푠)푓] (푃푎)푓 = (2.56) 휌푓 ∗ 휂푓

Knowing the suction specific speed S we can determine the required NPSHr [m] by the equation (2.57), which directly comes from eq. (2.13).

1 푁푃푆퐻푟 = 푆 1.33 (2.57) (𝑔 ∗ ) ω ∗ 푄0.5

- S = 1.8 ~ 2.3 Axial turbopumps; - S = 2.5 ~ 3 Radial turbopumps;

While the NPSHa [m] is equals to

______2.5 Pump design methods

푝푠 − 푝푣 푁푃푆퐻 = (2.58) 푎 휌 ∗ 𝑔

2.5.2 Method 2

The input parameters are: volumetric flow Q, Manometric head H, specific speed Ns, suction specific speed S.

The first step is to calculate the specific angular velocity ω

3/4 푁푠 ∗ (𝑔 ∗ 퐻) ω = (2.59) √푄

The impeller tip speed u2 [m/s] is obtained by the pump coefficient ψ

𝑔 ∗ 퐻 푢 = √ (2.60 2 휓

The cavitation verification is the same as the previous method. In this case

1 푁푃푆퐻푟 = 푆 1.33 (2.61) (𝑔 ∗ 0.5) 푁0푠 ∗ 푄

Where S = 15~24

푁 ∗ √푄 0푠 (2.62) 푆 = 3/4 (𝑔 ∗ 푁푃푆퐻푟)

The suction diameter (in meters) can be calculated by equation (2.63).

1/3 8푄⁄ 휋 (2.63) 퐷1 = ( 2 ) 휑푖푛 ∗ 푁0푠 ∗ (1 − 휈 )

Where

- φin= flow coefficient, generally equals to 0.1 for the suction section - ν = shaft diameter – suction diameter ratio, usually between 0.20 and 0.35 The powers can be calculated by [W]

푚̇ 표푥 ∗ 𝑔 ∗ 퐻표푥 (푃푎)표푥 = (2.64) 휂표푥

푚̇ 푓 ∗ 𝑔 ∗ 퐻푓 (푃푎)푓 = (2.65) 휂푓

Respectively for oxidizer and fuel pump.

Chapter 2 ______

2.5.3 Method 3

The input parameter are: volumetric flow Q, manometric head H, specific speed Ns, suction channel diameter D.

The angular velocity ω [rad/s] is equal to

3/4 푁푠 ∗ (𝑔 ∗ 퐻) 휔 = (2.66) √푄

The absolute speed at the inlet section of the pump c1 [m/s] is 4 ∗ 푄 푐 = (2.67) 1 휋 ∗ 퐷2

Knowing the specific speed Ns it is possible to read the non-dimensional coefficient ku2 from the empirical diagram in the Figure 2-17.

Thanks to this parameter is it possible to determine the impeller tip speed u2 [m/s]

푢2 = 푘푢2 ∗ √2 ∗ 𝑔 ∗ 퐻푑 (2.68) where Hd = discharge head

The impeller discharge diameter D2 [m] can be determined by the tip speed and the specific angular velocity

2 ∗ 푢2 퐷2 = (2.69) 푁0푠

Figure 2- 17: D1 = Impeller inlet diameter [m], D1=Shaft diameter [m], b2 = blades width [m]

______2.6 Volute design

To determine the blade outlet angle we prescribe:

- ηv = volumetric efficiency (between 0.90 and 0.97)

- ψ2 = adimensional coefficient which consider blades volume, generally equal to 0.95

The discharge speed c2 [m/s] is equal to 푄 푐2 = (2.70) 휋 ∗ 퐷2 ∗ 푏2 ∗ 휓2 ∗ 휂푣

In conclusion the outlet blades β2 angle can be determined by equation (2.71).

푢2 − 푣2푡 훽2 = 퐴푟푐퐶표푡 ( ) (2.71) 푐2 where

𝑔 ∗ 퐻푑 푣2푡 = (2.72) 푢2 ∗ 휂푦

ηy = hydraulic efficiency (0.7~0.96 for rocket engines turbopumps).

2.6 Volute design

There are several possible volute shape design. The most common are:

- Spiral volute: is the first type of volute used on centrifugal pumps. Volute casing of constant velocity of this type are simple to design and more economical to produce because of the open areas around the impeller periphery. They can be used on large as well as small pumps of all specific speeds with good efficiency. - Double volute: consists of two opposed single volute. The total throat area is the same as the one that would be used on a comparable single volute. The hydraulic performance is comparable to the single volute design. Tests indicate that a double volute pump will be less efficient at the best efficiency point with respect to a single spiral volute, but more efficient on both sides of the best efficiency point respect a single spiral volute. Double volute should not be used in low flow pumps, because the small cross section makes difficult manufacturing and cleaning. - Concentric volute: several experimental studies have been carried out on this volute type. The results have revealed that concentric volutes improve the hydraulic performances of low head or low specific speed pumps. Specifically, pump efficiency is improved for pumps with specific speed under 600 m/s. (7)

Chapter 2 ______

2.6.1 Effect of the volute design on efficiency

One of the main challenges for a pump designer is to design an efficient and durable pump. Mainly two effects have to be taken into account: the force load on the impeller and the hydraulic efficiency. Too high forces may cause premature failure of bearings or other components. Low efficiency turns into higher costs in terms of power needed to drive the pump.

Small changes in volute design have been shown to affect one or both force load on the impeller and hydraulic efficiency (8).

Experimental studies have been carried on different volute and impeller design (9). Three different volute types (Spiral, Double and Concentric figure 2-18) have been tested on a four vanes impeller and a five vanes impeller.

Figure 2- 18: Volute geometry types

The result shown a strong dependence on the volute geometry by the lateral impeller force magnitude and direction (figure 2-19).

The hydraulic performances are presented for the four-vane impeller spiral volute combination. To facilitate simple and direct comparison Figure 2- 19: Force magnitude and force orientation for different of the efficiency between all capacities in the three volute types

______2.6 Volute design volute-impeller combinations, all efficiencies were normalized by the efficiency of the four- vane impeller–spiral volute combination at the design flow, which corresponds to the BEP (Best

Efficiency Point) and hence is symbolized by η4SV,BEP ≡ ηn. Therefore, normalized efficiency

η/ηn > 1 represent an increase in hydraulic efficiency respect to the four-vane/spiral combination. η/ηn < 1 represent a decrease in hydraulic efficiency.

Figure 2-20 shows the normalized head coefficient

ψ/ψn and the normalized

efficiency η/ηn versus the

normalized capacity φ/φn for the four vane impeller operating in each of the three volutes. The concentric volute develops a

higher head for φ/φn< 0.4 respect to the baseline spiral volute. The head developed by Figure 2- 20: Head coefficient and efficiency vs flow coefficient for the three volute types with a four vanes impeller the concentric volute is consistently lower than that of the spiral volute for φ/φn> 0.6 and, consequently, a reduced efficiency at the same flow range is observed. The peak of normalized efficiency for the concentric volute is approximately 0.95 and occurs at φ/φn ≈ 0.95.

For low normalized flow coefficients, the double volute has a constant normalized head and it is therefore the most stable configuration. The peak normalized efficiency for the double volute is approximately 0.99 and occurs at φ/φn ≈ 0.8. For φ/φn > 0.8 the efficiency and the head characteristic for the double volute drop rapidly Figure 2- 21: Head coefficient and efficiency vs flow coefficient for the three volute types with a five vanes impeller

Chapter 2 ______compared to the spiral and concentric volute. This phenomenon is due to the higher volute losses: increasing the wet surface (because of the two tongues) results in higher incidence losses compared to the single volute. Figure 2-21 shows the normalized head coefficient and normalized efficiency versus the normalized flow coefficient for the five-vane impeller in each of the three volute configuration. Compared to the four-vane impeller the shut-off head coefficient decreases in the five-vane for each volute configuration.

From the two charts we can see how the four-vane impeller in each volute type is stable (has a negative slope) over a wider flow range than the corresponding characteristic for the five-vane impeller in each volute type. At high flow rates we can see an higher head produced by the five vanes impeller.

The table 2-1 shows the normalized head coefficient and the normalized efficiency for each impeller type in each volute at the normalized flow coefficient corresponding to the BEP of each configuration.

Table 2- 1

2.6.2 Volute geometry esteem

The purpose of the volute is to collect and slow down the flow coming from the impeller outlet. A simple sizing criterion of sizing consists of considering the outlet radial flow coming from the impeller and a spiral volute with a development of 360°.

Typically the inlet volute radius is obtained by increasing by 5 mm the impeller radius. The volute radius will be increased every 90°. The increment is controlled by a parameter (kr), which may take different values (kr = 1, 2, 3, … , N).

For example, kr = 2 means doubling the radius every 90°.

______2.6 Volute design

The steps that leads, as first approach, to the volute design are:

rv0 = rimp + 0.005

rv90 = rv0*kr

rv180 = rv90*kr

rv270 = rv180*kr

rv360 = rv270*kr

The meridian velocity at the output section of the volute can be obtained from:

푄 푐푚푣,표푢푡 = 2 (2.73) 휋 ∗ 푟푣360

The flow, passing through the volute, not only experiences a velocity decrease, but also a pressure increment. Therefore the pressure at the output section of the volute will be greater than the impeller discharge pressure (푝퐷):

2 2 푝푣,표푢푡 = (0.5 ∗ 휌 ∗ 푐2푚 + 푝퐷) − 0.5 ∗ 휌 ∗ 푐푚푣,표푢푡 (2.74)

Consequently, the pressure increment is:

Δ푝푣 = 푝푣,표푢푡 − 푝퐷 (2.75)

Chapter 3 Propellant turbines

3.1 Introduction

Turbines provide shaft power to the propellant pumps, taking the energy from the expansion of a high-pressure, high-temperature gas (coming from the cooling jackets) to a lower pressure and temperature. Turbines can be divided into two major types: impulse and reaction turbines. Impulse turbines can be either single or multi stage. Reaction turbines are usually multi-stage. A sample turbine is presented in Figure 3-1.

Figure 3- 1: Gas turbine general elements

______3.3 Velocity-compounded impulse turbine

3.2 Impulse turbines

An impulse turbine consists of a nozzle and a single rotor disk to which are attached the turbine blades. The nozzle is a stationary element which delivers the gas to the turbine blades. In the nozzle the gas pressure is converted into kinetic energy, with a consequent drop of pressure. The maximum gas velocity is achieved at the nozzle-exit/blades-inlet. The gas flowing through the turbine blades transfers its kinetic energy to the rotor as rotational mechanical energy. Ideally, the turbine rotor works in a constant pressure environment (neglecting the friction effects). Figure 3-2 shows the nozzle and the rotor of a single stage impulse turbine.

Figure 3- 2: Single stage impulse turbine. The graphs show the pressure drop and the velocity trend through the machine

3.3 Velocity-compounded impulse turbine

The velocity compounded impulse turbine can be seen as multi-stage impulse turbine, even if technically it is considered a single stage turbine because it involves only one pressure drop. Like the previous type, a nozzle directs the flow through the first rotor blades. When the flow leaves the first set of rotor blades, the flow changes its direction. A set of stationary blades (stator) changes again the flow direction and directs it into the second set of rotating blades, where the working fluid exchanges further energy with the machine. Other stages may be placed after the second rotor blades, in sets of stationary and Figure 3- 3: Single stage, two rotor, velocity compounded impulse turbine moving elements. Ideally the entire pressure drop occurs in the stationary nozzle. The flow velocity is ideally constant through the stator and drops through each rotor. Stator, rotor, pressure and velocity diagrams are presented in Figure 3-3.

Chapter 3 ______

3.4 Reaction turbine

Unlike the impulse turbines, in reaction turbines the working fluid experiences a pressure drop while it passes through the rotating blades, as shown in Figure 3-4. The energy transferred to the machine is due to a change of the working fluid momentum. In a pure reaction turbine the driving force is totally given by the expansion of the gas, which is usually roughly equally split between rotor and stator.

Figure 3- 4: Reaction turbine schematics and pressure - velocity diagrams.

3.5 Impulse turbines design

For rocket engine application the impulse turbines are preferred, both for their simplicity and light weight. The single stage impulse turbines are used for low-power application and they work at high rotational speeds. For higher performances, single stage, two rotor velocity compounded impulse turbine are used, they at lower rotational speed, since the kinetic energy is taken by two rotors. (1)

Turbine nozzles The turbine nozzles of most rocket engines have a convergent-divergent profile. As first approach we can consider an isentropic nozzle expansion, which gives an ideal velocity at the rotor inlet C0.

0 0 퐶1푖푑 = √2 ∗ (ℎ0 − ℎ1푠) = √2 ∗ 퐶푝 ∗ (푇0 − 푇1푠) (3.1)

Taking into account the isentropic law of expansion and the ideal gas equation, we can rewrite the (3.1) as:

______3.5 Impulse turbines design

훾−1 푝 훾 0 1 (3.2) 퐶1푖푑 = √2 ∗ 퐶푝 ∗ 푇0 [ 1 − ( 0) ] 푝0

This velocity has to be reduced in order to obtain the actual rotor inlet velocity C1. In order to do so, we can use a nozzle velocity coefficient Kn (usually between 0.89 and 0.98 (1)), defined as:

퐶1 퐾푛 = (3.3) 퐶1푖푑

Therefore, the real rotor inlet velocity equals to:

훾−1 푝 훾 0 1 (3.4) 퐶1 = 퐾푛 ∗ 퐶1푖푑 = 퐾푛 ∗ √2 ∗ 퐶푝 ∗ 푇0 [ 1 − ( 0) ] 푝0

The nozzle expansion is shown on an enthalpy-entropy diagram in Figure 3-5.

Figure 3- 5 The efficiency of a turbine nozzle is defined as the real kinetic energy at the nozzle exit divided by the kinetic energy at the nozzle exit of an isentropic expansion, which equals to the nozzle velocity coefficient squared, as shown by equation (3.5). It ranges from 0.80 to 0.96 (1).

2 퐶1⁄ 2𝑔 2 휂푛 = 2 = 퐾푛 (3.5) 퐶1푖푑⁄ 2𝑔

While passing through the nozzle, the flow changes its direction from nearly axial to a direction forming an α1 angle with the plane of rotation at the nozzle exit. Theoretically a better efficiency

Chapter 3 ______is obtained with a small exit angle, but with the decrease of it, the flow deflection increases causing higher frictional losses. The design values of α1 range from 15° to 30° (1).

Rotor blades The function of the rotor blades is to transform the gas flow kinetic energy into work. Ideally there should be no change of gas pressure, temperature or enthalpy in rotor blades, but in fact some expansion of the gas usually occurs. Furthermore, due to the frictional losses, the velocity at the exit of the rotor blades is lower than the velocity at the rotor blades inlet (W2

푊2 퐾푏 = (3.6) 푊1

Practical values of Kb vary from 0.8 to 0.9 (1).

Figure 3-6 shows the nozzle and the rotor blades of a typical single stage impulse turbine, with the corrisponding velocity diagrams.

Figure 3- 6: Single stage impulse turbine nozzles, rotor blades and velocity diagrams. C1=Nozzle exit absolute velocity, V1=Nozzle exit relative velocity, U=dragging speed, C2=Rotor outlet absolute velocity, V2=Rotor outlet relative velocity, dm=mean diameter. Assuming as reference the rotor blades mean diameter, the dragging speed u can be determined knowing the number of revolution per minutes (Nr) by:

휋 ∗ 푑 ∗ 푁 푢 = 푚 푟 (3.7) 60

______3.5 Impulse turbines design

The work obtained by a single stage of rotor blades, can be determined from the Euler equation, and is equals to:

(퐿)표푏푡 = 푢 ∗ [퐶1 ∗ cos(훼1) + 퐶2 ∗ cos (훼2)] (3.8)

Using some geometrical considerations we can rewrite Eq. (3.8) as:

(퐿)표푏푡 = 푢 ∗ (1 + 퐾푏)[퐶1 ∗ cos(훼1) − 푢] (3.9)

The blade efficiency is defined as the work obtained divided by the kinetic energy at the rotor blades input:

(퐿)표푏푡 휂푏 = 2 퐶1 ⁄2𝑔 (3.10)

The power developed by the turbine can be determined as:

푃푊 = 휂푏 ∗ (퐿)표푏푡 ∗ 푚̇ (3.11) where 푚̇ is the mass flow passing through the turbine.

Single stage, two rotor velocity compounded impulse turbine The velocity diagrams of a two rotor velocity compounded impulse turbine are shown in Figure 3-7. The first part of this turbine works in the same way as a single rotor impulse turbine, so it is described by the equations we have just seen above. After the first rotor blades, the flow goes into the stator blades, where the absolute velocity of the flow passes from C2 to C3. Then the flow passes through the second rotor blades, which work with the same principles as the first one. The total work transferred to the blades of a two-rotor turbine is the sum of the work transferred in each rotor. For a two-rotor impulse turbine it is equal to:

(퐿)표푏푡 = 푢 ∗ [퐶1 cos(훼1) + 퐶2cos (훼2)+퐶3 cos(훼3) + 퐶4cos (훼4)] (3.12)

As in the single rotor turbines, the exit velocity from any row of blades is lower than the inlet velocity because of frictional losses. It can be assumed that the blades velocity coefficient has the same values for any row of blades (Equation (3.12)).

푊2 퐶3 푊4 (3.13) 퐾푏 = = = 푊1 퐶2 푊3

Chapter 3 ______

Figure 3- 7

Chapter 4 Software development

4.1 Introduction

In this chapter we will discuss the software developed to achieve a preliminary design of the turbopump. The code has been written in MatLab language considering the theoretical background described in the previous chapters. To determine properties of the working fluids the CoolProp libraries have been used (version 5.1.1, exported in MatLab).

4.2 CoolProp libraries

CoolProp is an open-source database, which is intended to determine the fluid proprieties for a particular thermodynamic condition. It has been validated against the most accurate data available from relevant references. Many project are currently using the CoolProp libraries, and several are the organizations and university that supports CoolProp (1).

The libraries can be used in several environments, such as: Python, Octave, C#, Java, MatLab, FORTRAN and Microsoft Excel.

A wide variety of fluids are supported, which vary from pure and pseudo-pure fluids to many pre-set mixture. Table (4-1) shows some of the fluids available, the full list is available on the CoolProp official website. The list of predefined mixtures that can be obtained on MatLab with the command line: CoolProp.get_global_param_string('predefined_mixtures') is reported in Table 4.

LIST OF FLUIDS

Name Formula Name Formula Name Formula

1-Butene C4H8 Helium He Propylene C3H6

Acetone C3H6O1 Hydrogen H2 R134a C2F4H2

Air - Methane CH4 SulfurDioxide SO2

Benzene C6H6 Methanol CH4O Toluene C7H8

Chapter 4 ______

Carbon CO2 Nitrogen N2 Water H2O dioxide

Ethane C2H6 NitrousOxide N2O n-Heptane C7H16

Ethanol C2H6O1 Oxygen O2 n-Octane C8H18

Table 4- 1: List of part of the fluids available in CoolProp The most common function used with the CoolProp libraries are the PropsSI function and the PhaseSI function, which are part of the High-level interface. In both cases, SI stands for Système International d'Unités, meaning that the input parameters must be provided in SI units and the function returns the data in SI units.

The PropsSI function works for pure fluids, pseudo-pure fluids and mixtures (does not work for humid air, which has a separate command). The syntax is as follow:

CoolProp.PropsSI(‘Output’, ‘Input1’, Input1 Value, ‘Input2’, Input2 Value, ‘Fluid’)

The ‘Output’ and the various ‘Input’ are strings of characters which define a particular thermodynamic property. The complete list of the input strings can be found on CoolProp official website (2), Table (4-2) shows some of them.

PARAMETER INPUT/OUTPUT DESCRIPTION

‘D’ / ‘DMASS’ / Dmass I/O Mass density [kg/m^3]

‘H’ / ‘HMASS’ / ‘Hmass’ I/O Mass specific enthalpy [J/kg]

‘P’ I/O Pressure [Pa]

‘Q’ I/O Mass vapour quality [mol/mol]

‘S’ / ‘SMASS’ / ‘Smass’ I/O Mass specific entropy [J/kg/K]

‘T’ I/O Temperature [K]

______4.2 CoolProp libraries

‘A’ / ‘SPEED_OF_SOUND’ / O Speed of sound [m/s] ‘speed_of_sound’

O Mass specific, constant ‘C’ / ‘CPMASS’ / ‘Cpmass’ pressure specific heat [J/kg/K]

O Mass specific, constant ‘0’ / ‘CVMASS’ / ‘Cvmass’ pressure specific heat [J/kg/K]

‘V’ / ‘VISCOSITY’ / O Viscosity [Pa*s] ‘viscosity’

Table 4- 2: High level interface main input and output parameters

An example command could be:

% Command string (input) CoolProp.PropsSI(‘T’, ‘P’, 101325, ‘Q’, 0, ‘Water’) % Q=0 means only vapor

% Result (output) ans = 373.1243

For pure and pseudo-pure fluids, two thermodynamic points are required to fix the fluid state. The equation of state that CoolProp uses are based on Temperature and Density as state variables, so [T, D] will always be the inputs which returns the fastest results. If other inputs are used, the function will be slower. For example, if the inputs are Pressure and Temperature the calculation will be 3 to 10 times slower. If neither Temperature nor density are used (for example if we use Pressure, Enthalpy) the calculation will be much slower. In this case, if speed is an issue, it is possible to avoid the evaluation of the full equation of state using a tabular interpolation method. These methods are basically two: Bicubic interpolation and Tabular Taylor Series Extrapolation (TTSE). The concept behind this method is very simple: first a table is created, then an interpolation is performed in order to find the value needed.

The PhaseSI function is useful to determine the phase of a fluid at a given state point. To obtain the fluid phase, it is necessary to give as inputs two thermodynamic parameters and the fluid name. The syntax is as follow:

Chapter 4 ______

CoolProp.PhaseSI(‘Input1’, Input1 Value, ‘Input2’, Input2 Value, ‘Fluid’)

A sample code can be:

% Command string (input) CoolProp.PhaseSI(‘P’, 101325, ‘T’, 374, ‘Water’)

% Result (output) ans = gas

The phase index can be obtained also using the PropsSI function:

% Command string (input) CoolProp.PropsSI(‘Phase’, ‘P’, 101325, ‘T’, 374, ‘Water’)

% Result (output) ans = 5

Each index corresponds to a phase. Table (4-3) shows the index-phase correlation.

Index Phase

0 Liquid

1 Supercritical (p>pc, T>Tc)

2 Supercritical gas (p

3 Supercritical liquid (p>pc, T

4 At the critical point

5 gas

6 Two phase

7 Unknown phase Table 4- 3: Phase index for PropsSI function Figure 4- 1: Water phase diagram

______4.3 MatLab functions: pump.m

The function we have seen until now are part of the High level interface. For more advanced use of the CoolProp libraries, it can be useful to use the Low-level interface. This interface allow the user to access to lower-level parts of the CoolProp code. This approach is useful especially because it is much faster (the high level interface internally calls the low-level interface). Furthermore, the low-level interface only operates using numerical values, which is much faster with respect to the use of strings. Anyway, the low-level interface is much more complex to use with respect to the high-level one.

For our concern the High-level interface is the best option.

4.3 MatLab functions: pump.m

The first MatLab function developed has been the pump.m function, which basically calculates all the required parameter necessary to design a fuel or oxidizer pump. The function code is shown below.

1 function pump() 2 disp('*************************************************'); 3 disp('***** Pump preliminary design for LRE *****'); 4 disp('*************************************************'); 5 disp('--> Input parameters'); 6 T = input('Suction temperature [K] = '); 7 PS = input('Suction pressure [Pa] = '); 8 PD = input('Discharge pressure [Pa] = '); 9 fl = input('Mass flow [kg/s] = '); 10 Ns = input('Specific speed = '); 11 H = input('Height [m] = '); 12 s = input('Suction specific speed = '); 13 ETAp = input('Total efficiency = '); 14 ETAy = input('Hydraulic efficiency = '); 15 sb = input('Blades thickness [m] = '); 16 kx = input('Eye diameter / Tip diameter ratio = '); 17 fluid = input('Working fluid = ', 's'); 18 19 whitebg('k'); %plot background color 20 set(gcf, 'color', 'k'); 21 set(gcf, 'InvertHardCopy', 'off'); 22 23 %Inlet density and vapor pressure 24 rho = CoolProp.PropsSI('D', 'T', T, 'P', PS, fluid); %density [kg/m^3] 25 pv=CoolProp.PropsSI('P', 'T', T, 'Q', 0, fluid); %inlet vapor pressure [Pa] 26 inlet_state = CoolProp.PhaseSI('P', PS, 'D', rho, fluid) 27 28 Q = fl/rho; %volumetric flow [m^3/s] 29 30 g = (6.67*10^-11*5.97*10^24)/(6378*10^3+H*10^3)^2; %gravity acceleration 31 32 Hn = (PD-PS)/(rho*g); %nominal head 33 HS = (PS-pv)/(rho*g); %suction head

Chapter 4 ______

34 HD = (PD-PS-pv)/(rho*g); %discharge head 35 36 Hdt = HS*0.8; %required head 37 38 PW = Q*(PD-PS)/ETAp; %power 39 omega = Ns*((g*HD)^0.75)/(Q^0.5); %angular velocity 40 torque = PW/omega; %torque 41 42 rps = omega/(2*pi); %revolutions per second 43 rpm = rps*60; %revolution per minute 44 45 %cavitatio verification 46 NPSHr = (( (omega*Q^0.5)/s)^(4/3))/g; 47 Hsc = HS-NPSHr; %anticavitation margin 48 49 %outlet velocity triangle 50 u2 = sqrt((g*HD)/(0.4/((omega*Q^0.5)/((g*HD)^0.75))^0.25)); 51 C2u = (u2*(0.4/((omega*Q^0.5)/((g*HD)^0.75))^0.25))/ETAy; 52 C2m = 53 0.1715*sqrt(((omega*Q^0.5)/(g*HD)^0.75))*(sqrt((g*HD)/(0.4/((omega*Q^0.5)/((g*HD)^0.75))^0. 54 25))); 55 56 betaTip = atan(C2m/(u2-C2u)); 57 betaTipDEG = atan(C2m/(u2-C2u))*180/pi; %tipical values from 15 to 35 58 alpha2 = atan(C2m/C2u)*180/pi; 59 60 C2 = sqrt(C2m^2+C2u^2); 61 62 W2 = sqrt((C2^2+u2^2)-(2*u2*C2*cos(atan(C2m/C2u)))); 63 W2m = W2*sin(atan(C2m/(u2-C2u))); 64 W2u = W2*cos(atan(C2m/(u2-C2u))); 65 66 Dt2 = 2*(u2/omega); %tip diameter 67 68 %Head and Flow coefficient 69 PSId = (g*HD)/u2^2; %discharge head coefficient 70 PHId = C2m/u2; %discharge flow coefficient 71 72 %plot outlet velocity triangle 73 p1 = [0, 0; u2, 0; 0, 0]; 74 p2 = [u2, 0; (-W2u+u2), W2m; C2u, C2m]; 75 arrow3(p1, p2, '_b'); 76 text((u2-W2u/2)+10, W2m/2, 'W2'); 77 text(u2/2, 1, 'u2'); 78 text(C2u/2-25, C2m/2, 'C2'); 79 text(u2/6, C2m/10, '\alpha', 'FontSize', 12); 80 text(u2-u2/6, C2m/10, '\beta', 'FontSize', 12); 81 title('Outlet velocity triangle'); 82 text(0, C2m, datestr(clock)); 83 xlabel('[m/s]'); 84 ylabel('[m/s]'); 85 86 %suction section 87 D1s = kx*Dt2; %suction diameter 88 Dsh1s = (1-kx)*D1s; %shaft diameter 89 PHIs = Q/(rps*D1s^3); %suction flow coefficient 90 C1m = PHIs*(2*pi*0.5*D1s*rps); 91 CmRatio = C2m/C1m;

______4.3 MatLab functions: pump.m

92 D1ms = (D1s+Dsh1s)/2; %suction mean diameter 93 94 %suction mean diameter triangle 95 u1ms = (2*pi*0.5*D1ms*rps); 96 C1ums = (u1ms*(0.4/((omega*Q^0.5)/((g*HD)^0.75))^0.25))/ETAy; 97 beta1ms = atan(C1m/(u1ms-C1ums)); 98 beta1msDEG = beta1ms*180/pi; %typically from 15° to 50° 99 W1ms = (u1ms-C1ums)/cos(beta1ms); 100 101 102 %suction eye triangle 103 u1e = omega*0.5*D1s; 104 C1ue = (u1e*(0.4/((omega*Q^0.5)/((g*HD)^0.75))^0.25))/ETAy; 105 C1e = sqrt(C1m^2+C1ue^2); 106 107 beta1e = atan(C1m/(u1e-C1ue)); 108 beta1eDEG = beta1e*180/pi; 109 alpha1e = atan(C1m/C1ue)*180/pi; 110 111 W1e = (u1e-C1ue)/cos(beta1e); 112 W1ue = W1e*cos(beta1e); 113 114 %plot inlet velocity triangle (mean diameter) 115 figure 116 set(gcf, 'color', 'k'); 117 set(gcf, 'InvertHardCopy', 'off'); 118 p1 = [0, 0; u1e, 0; 0, 0]; 119 p2 = [u1e, 0; u1e-W1ue, C1m; C1ue, C1m]; 120 arrow3(p1, p2, '_g'); 121 text(u1e-20, C1m/2+2, 'W1e'); 122 text(u1e/2, 0.5, 'u1e'); 123 text(C1ue/2-5, C1m/2+2, 'C1e'); 124 text(u1e/6, C1m/10, '\alpha', 'FontSize', 12); 125 text(u1e-u1e/6, C1m/10, '\beta', 'FontSize', 12); 126 title('Inlet-eye velocity triangle'); 127 text(0, C1m, datestr(clock)); 128 xlabel('[m/s]'); 129 ylabel('[m/s]'); 130 131 %blades number optimal value 132 [Zopt, B2] = bladesNumberV3(C2m, C2u, u2, Dt2, fl, Q, PD, PS, ETAy, sb, betaTipDEG, 133 beta1msDEG, beta1eDEG, Dsh1s, fluid, rho); %blades number parametrical analisys 134 135 %inlet blades 136 kz1 = ((((pi*D1ms)/Zopt)-(sb/sin(atan(C1m/(u1ms-C1ums)))))/((pi*D1ms)/Zopt))^2; 137 %contraction factor impeller inlet 138 B1ms = Q/(pi*D1ms*C1m*kz1); 139 140 %volute design 141 kr = sqrt(2); %valute radius coefficient 142 rv0 = (Dt2+0.005)/2; %volute starting radius = impeller diam + 5mm 143 rv90 = rv0 * kr; 144 rv180 = rv90 * kr; 145 rv270 = rv180 * kr; 146 rv360 = rv270 * kr; 147 148 Avolu = pi*rv360^2; %volute discharge section 149

Chapter 4 ______

150 Cvolu = Q/Avolu; %Volute discharge velocity 151 152 Pvolut = (0.5*rho*C2m^2+PD)-0.5*rho*Cvolu^2; %Volute discharge pressure 153 154 DeltaPvolut = Pvolut - PD; %Volute pressure increment 155 156 %material strength 157 stress = rho * u2^2 /(1E6) %[MPa] 158 159 %save data to file 160 fileID = fopen('Pump propeties.txt', 'w'); 161 162 fprintf(fileID, '%-40s\r\n', datestr(clock)); 163 164 fprintf(fileID, '************************************************************\r\n'); 165 fprintf(fileID, '************************************************************\r\n'); 166 fprintf(fileID, '******** ********\r\n'); 167 fprintf(fileID, '******** Pump properties ********\r\n'); 168 fprintf(fileID, '******** ********\r\n'); 169 fprintf(fileID, '************************************************************\r\n'); 170 fprintf(fileID, '************************************************************\r\n\r\n'); 171 fprintf(fileID, '------Input Parameters------\r\n'); 172 fprintf(fileID, '%-40s %12s\r\n', 'Fluid = ', fluid); 173 fprintf(fileID, '%-40s %12.4f\r\n', 'Suction temperature T [K] = ', T); 174 fprintf(fileID, '%-40s %12.4f\r\n', 'Scution Pressure PS [Pa] = ', PS); 175 fprintf(fileID, '%-40s %12.4f\r\n', 'Mass flow fl [kg/s] = ', fl); 176 fprintf(fileID, '%-40s %12.4f\r\n', 'Discharge pressure PD [Pa] = ', PD); 177 fprintf(fileID, '%-40s %12.4f\r\n', 'Specific speed Ns = ', Ns); 178 fprintf(fileID, '%-40s %12.4f\r\n', 'Suction speed s = ', s); 179 fprintf(fileID, '%-40s %12.4f\r\n', 'Height H [m] = ', H); 180 fprintf(fileID, '%-40s %12.4f\r\n', 'Efficiency ETAp = ', ETAp); 181 fprintf(fileID, '%-40s %12.4f\r\n', 'Hydraulic efficiency ETAy = ', ETAy); 182 fprintf(fileID, '\r\n\r\n------Results ------183 \r\n'); 184 fprintf(fileID, '%-40s %12.4f\r\n', 'Density rho [kg/m^3] = ', rho); 185 fprintf(fileID, '%-40s %12.4f\r\n', 'Vapor pressure pv [Pa] = ', pv); 186 fprintf(fileID, '%-40s %12.4f\r\n', 'Volumetric flow Q [m^3/s] = ', Q); 187 fprintf(fileID, '%-40s %12.4f\r\n', 'Gravity acceleration g [m/s^2] = ', g); 188 fprintf(fileID, '%-40s %12.4f\r\n', 'Nominal Head Hn [m] = ', Hn); 189 fprintf(fileID, '%-40s %12.4f\r\n', 'Suction head HS [m] = ', HS); 190 fprintf(fileID, '%-40s %12.4f\r\n', 'Dischagre head HD [m] = ', HD); 191 fprintf(fileID, '%-40s %12.4f\r\n', 'Minimum required head Hdt [m] = ', Hdt); 192 fprintf(fileID, '%-40s %12.4f\r\n', 'Power PW [W] = ', PW); 193 fprintf(fileID, '%-40s %12.4f\r\n', 'Angular velocity omega [rad/s] = ', omega); 194 fprintf(fileID, '%-40s %12.4f\r\n', 'torque [N/m] = ', torque); 195 fprintf(fileID, '%-40s %12.4f\r\n', 'Revolution per second rps [1/s] = ', rps); 196 fprintf(fileID, '%-40s %12.4f\r\n', 'Revolution per minute rpm [1/min] = ', rpm); 197 fprintf(fileID, '%-40s %12.4f\r\n', 'NPSHr [m] = ', NPSHr); 198 fprintf(fileID, '%-40s %12.4f\r\n', 'Anticavitation margin Hsc [m] = ', Hsc); 199 fprintf(fileID, '%-40s %12.4f\r\n', 'Absolute velocity (outlet) C2 [m/s] = ', C2); 200 fprintf(fileID, '%-40s %12.4f\r\n', 'Dragging speed (outlet) u2 [m/s] = ', u2); 201 fprintf(fileID, '%-40s %12.4f\r\n', 'Relative velocty (outlet) W2 [m/s] = ', W2); 202 fprintf(fileID, '%-40s %12.4f', 'Beta angle impeller tip betaTip [deg] = ', betaTipDEG); 203 if betaTipDEG >= 15 && betaTipDEG <= 35 204 fprintf(fileID, ' *OK*\r\n'); 205 else 206 fprintf(fileID, ' *****ERROR*****\r\n'); 207 end

______4.3 MatLab functions: pump.m

208 fprintf(fileID, '%-40s %12.4f\r\n', 'Alfa angle impeller tip alpha2 [deg] = ', alpha2); 209 fprintf(fileID, '%-40s %12.4f\r\n', 'Tip diameter Dt2 [m] = ', Dt2); 210 fprintf(fileID, '%-40s %12.4f\r\n', 'Head coefficient PSId = ', PSId); 211 fprintf(fileID, '%-40s %12.4f\r\n', 'Flow coefficient PHId = ', PHId); 212 213 fprintf(fileID, '%-40s %12.4f\r\n', 'Suction diameter D1s = ', D1s); 214 fprintf(fileID, '%-40s %12.4f\r\n', 'Suction shaft diameter Dsh1s = ', Dsh1s); 215 fprintf(fileID, '%-40s %12.4f\r\n', 'Suction flow coefficient PHIs = ', PHIs); 216 fprintf(fileID, '%-40s %12.4f\r\n', 'Suction absolute vel (meridional) C1m = ', C1m); 217 fprintf(fileID, '%-40s %12.4f', 'Meridional components ratio C2m/C1m = ', CmRatio); 218 if CmRatio <= 1.6 219 fprintf(fileID, ' *OK*\r\n'); 220 else 221 fprintf(fileID, ' *****ERROR*****\r\n'); 222 end 223 fprintf(fileID, '%-40s %12.4f\r\n', 'Suction mean diameter D1ms = ', D1ms); 224 fprintf(fileID, '%-40s %12.4f\r\n', 'Suction-mean dragging speed u1ms = ', u1ms); 225 fprintf(fileID, '%-40s %12.4f\r\n', 'Suction-mean abs. vel. (tang.) C1ums = ', C1ums); 226 fprintf(fileID, '%-40s %12.4f\r\n', 'Suction-mean relative velocity W1ms = ', W1ms); 227 fprintf(fileID, '%-40s %12.4f', 'Suction-mean beta angle beta1ms [deg] = ', beta1msDEG); 228 if beta1msDEG >= 15 && beta1msDEG <= 50 229 fprintf(fileID, ' *OK*\r\n'); 230 else 231 fprintf(fileID, ' *****ERROR*****\r\n'); 232 end 233 fprintf(fileID, '%-40s %12.4f\r\n', 'Suction eye dragging speed u1e = ', u1e); 234 fprintf(fileID, '%-40s %12.4f\r\n', 'Suction eye absolute velocity C1e = ', C1e); 235 fprintf(fileID, '%-40s %12.4f\r\n', 'Suction eye relative velocity W1e = ', W1e); 236 fprintf(fileID, '%-40s %12.4f\r\n', 'Suction eye beta angle beta1e [deg] = ', beta1eDEG); 237 fprintf(fileID, '%-40s %12.4f\r\n', 'Suction eye alpha angle alpha1e = ', alpha1e); 238 fprintf(fileID, '%-40s %12.4f\r\n', 'Contraction factor impeller inlet kz1 = ', kz1); 239 fprintf(fileID, '%-40s %12.4f\r\n', 'Suction blades width B1ms = ', B1ms); 240 fprintf(fileID, '\r\n%-40s\r\n', '-VOLUTE DESIGN-'); 241 fprintf(fileID, '%-40s %12.4f\r\n', 'Volute increment coefficient = ', kr); 242 fprintf(fileID, '%-40s %12.4f\r\n', 'Volute starting radius = ', rv0); 243 fprintf(fileID, '%-40s %12.4f\r\n', 'Volute 90° radius = ', rv90); 244 fprintf(fileID, '%-40s %12.4f\r\n', 'Volute 180° radius = ', rv180); 245 fprintf(fileID, '%-40s %12.4f\r\n', 'Volute 270° radius = ', rv270); 246 fprintf(fileID, '%-40s %12.4f\r\n', 'Volute 360° radius (volute discharge) = ', rv360); 247 fprintf(fileID, '%-40s %12.4f\r\n', 'Volute discharge pressure = ', Pvolut); 248 fprintf(fileID, '%-40s %12.4f\r\n', 'Volute pressure increment = ', DeltaPvolut); 249 fclose(fileID); 250 251 %plot impeller 2D 252 figure 253 rot = 360/Zopt; 254 ii = 0; 255 for ii=0:rot:359 256 257 impellerPlot(D1s/2*1000 , Dt2/2*1000, betaTip, ii); 258 end 259 figure 260 impellerPlotDx((D1s/2-Dsh1s/2)*1000, B2*1000, Dsh1s*1000, Dt2*1000); 261 262 end

Chapter 4 ______

The first part of the code, from line 6 to line 17, asks the input parameters necessary to run the function. These are: suction temperature, suction pressure, discharge pressure, mass flow, specific speed, height, suction specific speed, total efficiency, hydraulic efficiency, blades thickness, eye diameter / tip diameter ratio and the working fluid. After that, the program starts to calculate the necessary data. The inlet density and the inlet vapor pressure (line 23 to 25) are determined by the use of the CoolProp libraries. Also a first check about cavitation is performed through the CoolProp PhaseSI function (line 26) that returns a console feedback about the inlet fluid state, which can be one of those list in table 4.3.

Once the density is known, it is possible to determine the volumetric flow rate (line 28) using equation (2.2). The successive line calculates the gravity acceleration as a function of the input height.

The nominal head, suction head and discharge head are calculated using equation (2.6), while the required head Hdt, as a first guess, is determined to be the 80% of the suction head (line 36). This value of the required head gives us a first idea of the anti-cavitation margin that should be obtained.

The power needed by the pump (line 38) is calculated using equation (2.56), the angular velocity using equation (2.14). It is now possible to determine the required torque dividing the power by the angular velocity (line 40). Other necessary data are the revolutions per second and the revolutions per minutes, which are directly related to the angular velocity (line 42 and 43).

The required NPSH results by rearranging equation (2.13). We can now determine the anticavitation margin Hsc (line 47). This values should be compare with the Hdt determined in line 36, which is an esteem of the anti-cavitation margin based on known data.

The next step is to determine the velocity triangles. In all the variables related to the velocity triangles the subscript “2” stands for the outlet section, while “u” stands for tangential component and “m” for meridian component. The outlet velocity triangle is determined first.

The starting point is the dragging speed u2 (line 50), which can be calculated using equation (2.27). It is now possible to determine the tangential component of the absolute velocity using equation (2.35) and its meridian component by equation (2.36). Being known the two components of the absolute velocity, it is possible to determine the absolute velocity magnitude (line 60). In addition we can determine the alpha and beta angles of the velocity triangle by simple geometrical considerations, that results in equations (2.38) and (2.40). The only

______4.3 MatLab functions: pump.m unknown component is now the relative velocity, which can be calculated as the magnitude by equation (2.41) and as component-wise using equations (2.42) and (2.43).

The impeller outlet diameter is determined by the dragging speed u2 and the angular velocity. The adimensional coefficients φ and ψ are determined using equations (2.16) and (2.21) respectively.

The lines 72 to 84 simply plot the outlet velocity triangle thanks to the function arrow3.m (3). Some text strings are added to plot the vector names (W2, u2 and C2), the angles (α and β), the axis units, the title and the date and time when it has been plotted.

The suction diameter is calculated reducing the discharge diameter of a certain value, given by the parameter kx. The next calculation are the same as for the outlet triangle, but in this case the variables are determined first for the suction mean diameter and then for the suction eye. Also the plot follow the same logic used for the outlet velocity triangle.

The next code lines use the function bladesNumber.m to determine the optimal blades number and the outlet blades width. This function will be examined in section 4.4.

Knowing the optimal blades number we can determine the impeller inlet contraction factor kz1.

The volute design exactly follows what we have described in section 2.6.2: first a value is assigned to the kr parameter, then the radius at 0, 90°, 180°, 270° and 360° is determined. The volute discharge velocity is determined using equation (2.73), the output pressure by equation (2.74) and the pressure increment using equation (2.75).

After that, we estimate the stress to which the blades are subjected: if the impeller material is not strong enough to tolerate this stress we may change material or decrease the pump pressure increment adding an additional pump stage.The lines from 166 to 258 have the task of create a .txt file, called Pump properties, in which the input parameters and the results are saved. Since we know the typical range of the design variables for our application (see chapter 2), this file also plots the text “*OK*” if the value is between the typical range or Figure 4- 2: Pump properties.txt example

Chapter 4 ______

“*****ERROR*****” if the value exceeds the boundaries. A sample part of this file is shown in fig. 4-2.

The last part of the function prints the frontal section and the right section of the designed impeller. To plot these two figures other two functions have been designed. However we are not going to explain them since they just use the previously calculated data, with some geometrical considerations. These functions are listed in Appendix A.

The working scheme of the pump.m function is explained in Figure 4-3.

Figure 4- 3: Matlab functions working structure.

4.4 MatLab functions: bladesNumber.m

The function bladesNumber.m is internally called by the function pump.m (see section 4.3), which gives to it the necessary input data. The first part of the code performs a parametrical analysis by calculating the variables for impellers with 3, 5, 7 and 9 blades, then the function determines the number of blades which returns the outlet density value which is closest to the inlet density.

______4.4 MatLab functions: bladesNumber.m

The second part of the function follows a different approach by determining the optimal number of blades using the empirical relation (Eq. (2.45)). All the calculated data are saved in a .txt file, named blades.

%blades number parametrical analisys %sb blades thickness function [Zopt, B2b] = bladesNumberV3(C2m, C2u, u2, Dt2, fl, Q, PD, PS, ETAy, sb, betaTipDEG, beta1msDEG, beta1eDEG, Dsh1s, fluid, opt)

%parametrical analisys i=1; for ZN=3:2:9

%contraction factor (impeller discharge) [typical values 0.85 - 0.95] KZ2(i) = ((((pi*Dt2)/ZN)-(sb/sin(atan(C2m/(u2-C2u)))))/((pi*Dt2)/ZN))^2; ETApp(i) = ETAy*KZ2(i)*0.9; PW(i) = (Q*(PD-PS))/ETApp(i); B2(i) = (Q*KZ2(i))/(((KZ2(i))^0.5)*pi*Dt2*C2m); %blades height Q2(i) = C2m*((pi*Dt2*B2(i))-(ZN*(sb/sin(atan(C2m/(u2-C2u))))*B2(i))*KZ2(i)); rho2(i) = (fl*KZ2(i))/Q2(i); T2(i) = CoolProp.PropsSI('T', 'D', rho2(i), 'P', PD, fluid);

i=i+1; end

[v, index] = min(abs(opt-rho2)); %closest value to optimal density %Zopt = index+(index+1); %optimal blades number

%approximate blades number "b" betaMedDEG = mean([betaTipDEG, beta1msDEG, beta1eDEG]); betaMedRAD = betaMedDEG * pi/180;

ZNa = 2*6.5*(0.5*((Dt2/2)+(Dsh1s/2))/((Dt2/2)-(Dsh1s/2)))*betaMedRAD; ZNb = round(ZNa); KZ2b = ((((pi*Dt2)/ZNb)-(sb/sin(atan(C2m/(u2-C2u)))))/((pi*Dt2)/ZNb))^2; ETAppb = ETAy*KZ2b*0.9; B2b = (Q*KZ2b)/((KZ2b^0.5)*pi*Dt2*C2m); Q2b = C2m*((pi*Dt2*B2b)-(ZNb*(sb/sin(atan(C2m/(u2-C2u))))*B2b)*KZ2b); rho2b = (fl*KZ2b)/(Q2b); PWb = (Q2b*(PD-PS))/ETAppb; T2b = CoolProp.PropsSI('T', 'D', rho2b, 'P', PD, fluid);

Zopt = ZNb;

fileID = fopen('blades.txt', 'w');

fprintf(fileID, '%-40s\r\n', datestr(clock));

fprintf(fileID, '******************************************************************\r\n'); fprintf(fileID, '******************************************************************\r\n'); fprintf(fileID, '*********** ***********\r\n'); fprintf(fileID, '*********** Blades number parametrical analisy ***********\r\n'); fprintf(fileID, '*********** ***********\r\n');

Chapter 4 ______

fprintf(fileID, '******************************************************************\r\n'); fprintf(fileID, '******************************************************************\r\n\r\n');

fprintf(fileID,'%12s %12s %12s %12s %12s\r\n', ' ','3 blades','5 blades','7 blades','9 blades'); fprintf(fileID,'%12s %12.4f %12.4f %12.4f %12.4f\r\n', 'KZ2', KZ2); fprintf(fileID,'%12s %12.4f %12.4f %12.4f %12.4f\r\n', 'ETApp', ETApp); fprintf(fileID,'%12s %12.2f %12.2f %12.2f %12.2f\r\n', 'PW [W]', PW); fprintf(fileID,'%12s %12.4f %12.4f %12.4f %12.4f\r\n', 'B2 [m]', B2); fprintf(fileID,'%12s %12.4f %12.4f %12.4f %12.4f\r\n', 'Q2 [m^3/s]', Q2); fprintf(fileID,'%12s %12.4f %12.4f %12.4f %12.4f\r\n', 'rho2[kg/m^3]', rho2); fprintf(fileID,'%12s %12.4f %12.4f %12.4f %12.4f\r\n', 'T2 [K]', T2);

fprintf(fileID,'\r\n------\r\n'); fprintf(fileID,' approximate blades number ("solution b") \r\n'); fprintf(fileID,'------\r\n\r\n'); fprintf(fileID,'%14s %12.4f\r\n', 'ZNa = ', ZNa); fprintf(fileID,'%14s %12i\r\n', 'ZNb = ', ZNb); fprintf(fileID,'%14s %12.4f\r\n', 'KZ2b = ', KZ2b); fprintf(fileID,'%14s %12.4f\r\n', 'ETAppb = ', ETAppb); fprintf(fileID,'%14s %12.2f\r\n', 'PWb [W] = ', PWb); fprintf(fileID,'%14s %12.4f\r\n', 'B2b [m] = ', B2b); fprintf(fileID,'%14s %12.4f\r\n', 'Q2b [m^3/s] = ', Q2b); fprintf(fileID,'%14s %12.4f\r\n', 'rho2b [kg/m^3] = ', rho2b); fprintf(fileID,'%14s %12.4f\r\n', 'T2b [K] = ', T2b); fclose(fileID); end

4.5 Software test-case

In order to test the reliability of the MatLab functions that we have developed, we have considered the first stage of a RL10-3-3A expander cycle rocket engine, whose pumps data are available in the literature (4).

The input data for the hydrogen pump are:

Fluid H2 Suction specific speed 24.0

Suction temperature [K] 21.44 Height [m] 350.0

Mass flow [kg/s] 2.7946 Pump efficiency 0.5854

Suction pressure [Pa] 184779.0 Hydraulic efficiency 0.75

Discharge pressure [Pa] 3694000.0 Blades thickness [m] 0.002

Specific speed 0.2020 Inlet/outlet diam. ratio 0.55

______4.5 Software test-case

With this values we obtain a 5 blades impeller with a 3.1 mm outlet blades width and 93.9 mm suction diameter (Fig. 4-4). Table 4-4 shows the engine known data compared with the results coming from the developed software. As we can see in the “relative error” column, the data gap is very tight, which proves the reliability of the software that we have developed.

Variable RL10-3-3A (LH2) Pump.m outputs Relative error

Speed [rpm] 33A3 31494 31430 0.203%

Torque [Nm] 72.93 73.07 0.192%

Power [W] 240404.41 242701.9 0.956%

Inlet density [kg/m3] 69.472 69.659 0.269%

3 Outlet density [kg/m ] 68.639 68.9 0.380%

Outlet temperature [K] 26.47 26.36 0.416% 4 Efficiency 0.5854 0.5801 0.905% Table 4- 4: LH2 pump orginal data and software results

The data obtained by the blades number parametrical analysis (Appendix B) shows how we would obtain a higher pump efficiency with a lower blades number. On the other hand the mass flow is being increased and so the discharge density. In this case we have chosen a lower efficiency level in order to achieve a definite discharge density, which is required by the cooling jacket and the injector and combustion chamber design. In the Fig. 4-5 we can see the impeller inlet (eye) and the impeller outlet velocity triangles. The full list of the pump data can be read in Appendix B.

Figure 4- 4: RL10-3-3A LH2 Impeller frontal wiev (left) and right wiev (right). 69

Chapter 4 ______

Figure 4- 5: RL10-3-3A LH2 Impeller inlet and outlet velocity triangles In order to offer a realistic image of the pump, a 3-dimensional model of the impeller have been designed using Solidworks. The result is shown in Figure 4-6.

Figure 4- 6: RL10-3-3A LH2 pump impeller.

In order to have and additional confirmation of the reliability of the software that we have developed, also the oxidizer pump of the RL10-3-3A engine has been tested In this case the input data are:

______4.5 Software test-case

Fluid O2 Suction specific speed 20.0

Suction temperature [K] 97.05 Height [m] 350.0

Mass flow [kg/s] 13.953 Pump efficiency 0.6408

Suction pressure [Pa] 275514.5 Hydraulic efficiency 0.75

Discharge pressure [Pa] 4282333.8 Blades thickness [m] 0.001

Specific speed 0.33 Inlet/outlet diam. ratio 0.60

As the previous simulation, also the LO2 pump designed with our software gives virtually identical results to the real pump; the two are compared in (Table 4-5). In this case it must be pointed out that the approximate blades number (“solution b”) returns 8 as optimal blades number. The results shown in Table 4-5 are for a 3-blades solution, since with this solution the values are the closest to the real ones. This is just a design choice, since with this solution we obtain a lower efficiency, but more suitable values of temperature and density. The same set of graph and images of the LH2 pump are presented (Figure 4-7, Figure 4-8 and Figure 4-9), the full set of data can be found in Appendix C.

Variable RL10-3-3A (LO2) Pump.m outputs Relative error

Speed [rpm] 33A3 12598 12614 0.127%

Torque [Nm] 59.78 59.68 0.167%

Power [W] 78825.35 78693.1 0.168%

Inlet density [kg/m3] 1106.4 1106.6 0.000%

3 Outlet density [kg/m ] 1105.6 1105.2 0.000%

Outlet temperature [K] 99.22 99.36 0.141% 4 Efficiency 0.6408 0.6420 0.187% Table 4- 5: LO2 pump original data and software results.

Chapter 4 ______

Figure 4- 7: RL10-3-3A LO2 Impeller frontal wiev (left) and right wiev (right).

Figure 4- 8: RL10-3-3A LO2 Impeller inlet and outlet velocity triangles

Figure 4- 9: RL10-3-3A LO2 pump impeller.

Chapter 5 Methane pump development

5.1 Turbopump specifics and applications

The final stage of this work consists in designing a turbopump for an expander cycle liquid rocket engine powered with liquid methane (LCH4) as propellant and liquid oxygen (LO2) as oxidizer. This fuel oxidizer combination has already been used for the LM10-MIRA developed by the collaboration of the Italian company AVIO and the Russian KBKHA. This engine is the first liquid methane-powered vehicle to fly (1). The turbopump designed for the LM10 is characterized by a low volumetric flow pump with a high pressure rise, achieved with only one centrifugal stage (2).

The pump we are going to design has roughly the same specifics of the LM10 pump, which makes it suitable for an engine of the same thrust class (10 tons). In particular the used input data that have been used are:

Fluid CH4 Suction specific speed 20

Suction temperature [K] 110 Height [m] 350.0

Mass flow [kg/s] 3.95 Pump efficiency 0.62

Suction pressure [Pa] 135000 Hydraulic efficiency 0.80

Discharge pressure [Pa] 14200000 Blades thickness [m] 0.001

Specific speed 0.19 Inlet/outlet diam. ratio 0.52

Running the pump.m function with these values we obtain a 6 blades impeller with a 96.8 mm outlet diameter, which is designed to work at a rotational speed of 45965 rpm. The power needed by the pump equals 208.7 kW, the pump efficiency is 0.6267 and we achieve an outlet temperature of 122 K. The full set of data can be found in Appendix D.

Using the blade-number parametrical analysis (blades.txt file), it is possible to see how decreasing the number of blades we obtain a more efficient pump, with a lower discharge temperature. On the other hand, by doing so, we increase the fluid density, which may be a

Chapter 5 ______problem for the cooling system. For this reason, as first trial, we consider optimal the number of blades obtained using the empirical equation (2.47) (“solution b” in the blades.txt file).

The velocity triangles are shown in Figure 5-1, while Figure 5-2 shows the frontal and right view. The impeller 3-D model of the impeller is shown in Figure 5-3.

Figure 5- 1: Inlet eye and outlet velocity triangle for the LCH4 pump impeller

Figure 5- 2: Frontal and right view of the LCH4 pump impeller

The volute has been designed with kr = 1.25 and a starting radius of 0.0509 m. Using these two parameters we obtain: 90° radius = 63.6 mm, 180° radius = 79.6 mm, 270° radius = 99.4 mm,

______5.1 Turbopump specifics and applications discharge radius = 124.3 mm. With this configuration the volute discharge pressure equals 14.26 MPa, meaning a pressure increment through the volute of 0.0645 MPa. The 3D model of the designed volute is shown in Figure 5-4 and Figure 5-5.

Figure 5- 4 Figure 5- 3

Figure 5- 5 75

Chapter 5 ______

5.2 Conclusions

The developed software is very useful to give a first indication about the dimensions and the specifics of a generic pump. Since the software has been validated with hydrogen and oxygen, we can say that it works properly for a wide range of fluids density. In addition with the parametrical study of the impeller blades it is possible to choose the number of blades according to the design requirements, since changing the blades number we change the pump efficiency, the outlet temperature and the outlet density.

The CoolProp libraries seems to work properly, and we can say that they are a reliable source which can be used to determine the fluid properties in different conditions.

Not surprisingly, we have found that the chosen working fluid and the input parameters radically change the pump characteristics. Figure 5-6 shows the difference between the designed methane impeller (green), the LR10 oxygen impeller (gold) and the LR10 hydrogen impeller (red). As we can see the methane impeller is much smaller than the other two. This is a desirable property since it reduces the pump dimensions, thus allowing a more compact engine design, which results in lower weight and drag.

Figure 5- 6

Appendix

Appendix A – MatLab additional functions function impellerPlot(r1, r2, rsh, beta2, rot) whitebg('k') set(gcf, 'color', 'k') set(gcf, 'InvertHardCopy', 'off'); ang([0,0], r2, [0, 2*pi], 'w-'); hold on ang([0,0], r1, [0, 2*pi], 'w-'); text(-r2, r2+r2/10, datestr(clock)); i=0; for teta = 0:0.01:2*pi i=i+1; x2(i) = r2*cos(teta); y2(i) = r2*sin(teta); end j=0; i=0; ii=0; for ra=20:1:120 i=i+1; delta(i) = inter(ra, x2, y2, beta2); r(i) = ra; end c = delta(delta~=0); pos=find(delta==min(c)); rFin=r(pos);

j=0; for teta = 0:0.01:pi j=j+1; xa(j) = -rFin+rFin*cos(teta); ya(j) = rFin*sin(teta); end

[a, b] = intersections(x2, y2, xa, ya);

for j=1:(pi/0.01) if xa(j)>a xf(j) = xa(j); yf(j) = ya(j); end end h = plot(xf,yf, 'b'); xlabel('[mm]'); ylabel('[mm]'); 77

Appendix ______

rotate(h, [0,0,1], rot); %chiamare la funzione con ciclo per far ruotare

for teta = 0:0.01:2*pi i=i+1; xsh(i) = rsh*cos(teta); ysh(i) = rsh*sin(teta); end area(xsh,ysh, 'ShowBaseLine', 'off', 'FaceColor', 'w');

daspect([1,1,1]); end

function impellerPlotDx(B1, B2, Dshaft, Dt2)

Rshaft = Dshaft/2; Rt2 = Dt2 / 2; whitebg('k') set(gcf, 'color', 'k') set(gcf, 'InvertHardCopy', 'off'); plot([-10, Rt2+Rt2/10],[0,0], 'color', 'w', 'LineWidth', 1); hold on plot([0, 0], [-(Rt2+Rt2/10); Rt2+Rt2/10], 'color', 'w', 'LineWidth', 1); text(2*B2, Rt2+Rt2/10, datestr(clock));

%superior half plot([0, B2], [Rt2, Rt2], 'b'); i=0; r = Rt2-(B1+Rshaft); for teta = 0:0.01:pi/2 i=i+1; x1(i) = r-r*cos(teta)+B2; y1(i) = -r*sin(teta)+Rt2; end plot(x1, y1, 'b');

plot([r+B2, r+B2], [(B1+Rshaft), 0], 'b'); rr = Rt2-Rshaft; i=0; for teta = 0:0.01:pi/2 i=i+1; x2(i) = rr-rr*cos(teta); y2(i) = -rr*sin(teta)+Rt2; if x2(i)<(r+B2) xx(i) = x2(i); yy(i) = y2(i); 78

______Appendix A – MatLab additional functions

end end plot(xx, yy, 'b--'); %area(xx, yy);

%bottom half plot([0, B2], [-Rt2, -Rt2], 'b'); i=0; r = Rt2-(B1+Rshaft); for teta = 0:0.01:pi/2 i=i+1; x1(i) = r-r*cos(teta)+B2; y1(i) = +r*sin(teta)-Rt2; end plot(x1, y1, 'b'); plot([r+B2, r+B2], [-(B1+Rshaft), 0], 'b'); rr = Rt2-Rshaft; i=0; for teta = 0:0.01:pi/2 i=i+1; x2(i) = rr-rr*cos(teta); y2(i) = rr*sin(teta)-Rt2; if x2(i)<(r+B2) xxb(i) = x2(i); yyb(i) = y2(i); end end plot(xxb, yyb, 'b--');

%Shaft i=0; for teta = 0:0.01:pi i=i+1; x(i) = Rshaft*sin(teta)+r+B2; y(i) = Rshaft*cos(teta); end plot(x, y, 'b'); xlabel('[mm]'); ylabel('[mm]'); daspect([1,1,1]); hold off end

function delta = inter(ra, x2, y2, beta2)

j=0; for teta = 0:0.01:pi j=j+1; 79

Appendix ______

xa(j) = -ra+ra*cos(teta); ya(j) = ra*sin(teta); end %plot(xa,ya);

[a, b] = intersections(x2, y2, xa, ya); if a~=0 [xai, xaiPos] = min(abs(a-xa)); xa(xaiPos); betaA=abs(atan((ya(xaiPos)-ya(xaiPos-1))/(xa(xaiPos)-xa(xaiPos-1))));

delta = abs(beta2-betaA); else delta = 0; end end

______Appendix B – RL10-3-3A LH2 pump output parameters

Appendix B – RL10-3-3A LH2 pump output parameters

Appendix ______

______Appendix B – RL10-3-3A LH2 pump output parameters

Appendix ______

Appendix C– RL10-3-3A LO2 pump output parameters

______Appendix C– RL10-3-3A LO2 pump output parameters

Appendix ______

Appendix D: Designed methane pump output parameters

______Appendix D: Designed methane pump output parameters

Appendix ______

References

1. Strain, William S. Design of an oxygen turbopump for a dual expander rocket engine. 2008.

2. Sutton, George P. Rocket propulsion elements (7th edition).

3. Huang, Dieter K. Huzel - David H. Modern engineering for design of liquid propellant rocket engines.

4. Sexton, Gabriel L. Johnson - Thomas D. Johnson sexton cycle rocket engine patent.

5. Limerick, Charles D. METHOD AND APPARATUS FOR INCREASING COMBUSTION CHAMBER PRESSURE IN A HIGH PRESSURE EXPANDER CYCLE ROCKET ENGINE. 5,410,874 USA, 02 05 1995.

6. NASA contractor report 187217 - Advanced engine study program. Masters, D.E. Galler, T.F. Denman, R.A. Shied, J.R. Black. 1993.

7. SANTIAGO, JORGE R. Evolution of the RL10 liquid rocket engine for a new upperstage application.

8. Mitsubishi Heavy Industries Technical Review Vol. 48 No. 4. ONGA, M. ATSUMI - K. YOSHIKAWA - A. OGAWARA - T. 2011.

9. Cannon, James L. Liquid Propulsion: Propellant Feed System Design. Encyclopedia of Aerospace Engineering, Volume 2 Propulsion & Power, Wiley Publishers.

10. Leto, A. Turbopump design methods for liquid rocket engine.

11. Huang, Dieter K. Huzel - David H. MOdern engineerinf for design of liquid-propellant rocket engines. 1992.

12. Wright, Terry. Fluid Machinery: Performance, Analysis, and Design.

13. Cooper, I. J. Karassik - J. P. Messina - P. Pump Handbook.

14. Centrifugal pumps design and application - 2nd edition. Ross, V. S. Lobanoff - R. R. 1992.

15. Effect of relative impeller-to-volute position on hydraulic efficiency and static radial force distribution in a circular volute centrifugal pump. D. O. Baun, R. D. Flack, L. Kostner. s.l. : ASME Journal of Fluids Engineering, 2000.

16. Effect of volute design and number of impeller blades on laterla impeller forces and hydraulic performances. D. O. Baun, R. D. Flack. 2003.

References ______

17. huang, Huzel and. Design of liquid propellants rocket engines.

18. [Online] www.coolprop.org.

19. [Online] http://www.coolprop.org/coolprop/HighLevelAPI.html#parameter-table.

21. Binder, Michael P. A transient model of the RL10A-3-3A rocket engine.

22. http://www.caeconference.com/proceedings/abstract/abstract_08/avio_cira_cfx.html. [Online]

23. FUel turbopump development for the LM10-MIRA LOx-LNG expander cycle enginein the frame of the LYRA program. M. Rudnykh, G. Cagiano, A.G. Kravchenko, S.V. Chembartsev, A. De Lillis, E. D'Anversa.