Conceptual Study of Throttled Operation of Rocket Engine ...

Conceptual Study of Throttled Operation of Rocket Engine Turbopump�

By Takeshi KANDA

Japan Aerospace Exploration Agency, Kakuda Space Center, Kakuda, Japan

(Received May 2nd, 2012)

Throttled operation of turbopumps of liquid rocket engines is studied analytically. Two types of pressure drops in the

propellant injector are examined, that is, the drop of gas injection and that of liquid injection. Circulation is examined in

relation to throttling to operate the pump in the vicinity of the design condition. An enthalpy increase at the pump

entrance is derived analytically in the circulation system. With the derived analytical relationships, throttled operating

conditions are examined for imaginary LH2 and LOX turbopumps with a LH2/LOX property calculation code. The cir-

culation causes gasification at the entrance of the high-pressure LH2 pump. This does not occur in the mid-pressure LH2

pump or the LOX pump. In the liquid propellant injection system, the unstable region becomes narrower than in the gas

propellant injection. The ratio of the turbine flow rate to the pump flow rate decreases in line with throttling. Throttling

does not degrade the engine specific impulse from the viewpoint of the turbine bleed ratio.

Key Words: Throttling, Rocket Engine, Turbopump, Injection, Circulation

Nomenclature

A: cross-section

C: constant

c0: gas speed at turbine nozzle exit

D: diameter

F: force

g: gravitational acceleration

H: head

h: enthalpy

_mm: mass flow rate

N: rotational speed

p: pressure_QQ: volume flow rate

R: gas constant

r: throttling rate

T : temperature

u: speed

W: power

�: theoretical pump flow coefficient

�: pump flow coefficient

�: ratio of specific heats

�: efficiency

�: density

�: theoretical pump head coefficient

: pump head coefficient

Subscripts

a: turbine entrance

b: turbine exit

c: combustion chamber, throttling by circulation only

e: exit

im: impeller

in: inducer

inj: injection

mn: minimum throttling rate

p: pump

pl: plumbing

r: return flow

s: specific, saturation

t: throat, turbine, total

0: rated design condition

1: pump entrance before mixing

2: pump entrance after mixing in circulation system

3: circulation flow before mixing

4: pump exit

1. Introduction

Liquid rocket engines usually operate according to their

design conditions. Their turbopumps also operate according

to the design conditions, that is, they discharge a design flow

rate of propellant at the design pressure and design rota-

tional speed. Propellant injectors, cooling jacket and valves

also operate at the design flow rate, temperature and pres-

sure except at start and stop transient operations. On the

other hand, throttling of the rocket engines is required in

some operations, for example, powered descent, orbit-to-

orbit transfer, hazard avoidance and hovering. Throttling

can condition acceleration and velocity of a vehicle, espe-

cially for the single-stage-to-orbit vehicle (SSTO), and helps

in optimization of the vehicle trajectory. Throttling has been

investigated and several engines were tested in deep and

shallow throttled conditions.1–14) Throttling is also neces-

sary for the rocket-based-combined-cycle engine (RBCC)

to change its operation to the ramjet mode.15)

In the rocket engine, thrust is proportional to the combus-

tion chamber pressure. Under a specified mixture ratio of

propellants, the chamber pressure is proportional to a pro-� 2013 The Japan Society for Aeronautical and Space Sciences�Presented at the Annual Meeting of JSASS, April 13, 2012

Trans. Japan Soc. Aero. Space Sci.

Vol. 56, No. 1, pp. 49–60, 2013

pellant mass flow rate. In throttling of the rocket engine,

both the mass flow rate and the pressure have to change

simultaneously and proportionally. However, in general,

the flow rate of a pump is proportional to its rotational

speed, while its head is proportional to the square of the

speed. To throttle an engine, therefore, shift of the flow

and head coefficients of the pump is required. The shift of

the coefficients may cause pump operation to be unstable,

where the slope between the flow and head coefficients is

positive.3,11,13) As a result of the shift of the coefficients,

pump efficiency will become lower. To keep operation of

the pump around its design specific speed, a circulation sys-

tem is integrated to the pump for the throttled operation.12,13)

As a result of circulation, the pump operates in the stable

region and its efficiency is kept high. However, temperature

increases at the entrance due to the return flow with higher

enthalpy, and it degrades pump suction performance. It is a

key problem in application of circulation to a cryogenic

pump.

In the present paper, relationships on the turbopump-

related throttling operation of the rocket engine are derived

first. The head required for a pump depends on the injected

propellant condition for the combustion chamber, that is,

gas or liquid. The relationships are derived for these two

cases. Relationships for the circulation and for turbine oper-

ation are also derived. Based on the relationships, operating

conditions are calculated for imaginary LOX and LH2 tur-

bopumps. Based on the calculated results, the turbopump

characteristics in the throttling operation, for example,

temperature of the mixed flow at the entrance of the pump,

stable operating region of the pump and turbine flow rate,

are discussed.

There are several kinds of propellants for the rocket

engine, for example, kerosene and methane. When they

are used for the regenerative cooling, they are injected into

the combustion chamber in gas flow with low density. Pump

power characteristics and the enthalpy increase in the circu-

lation system are similar to those of dense propellant of

LOX, whereas injection-related characteristics are similar

to those of H2 gas. The present results are applied to the

throttling characteristics of these propellants.

2. Relationships in Throttling

The thrust of the rocket engine is proportional to the pro-

pellant flow rates and pressure in the combustion chamber.

To throttle a pump, its operating condition is changed due

to the relationship between the flow and head coefficients.

Otherwise, a flow rate and head of the pump is changed

due to circulation. The throttling of the rocket engine is

proportional to thrust.

r ¼F

F0

�pc

pc0¼

_mmp

_mmp0

ð1Þ

The throttling rate and the ratio of the pressure in the

combustion chamber are also proportional to the flow rate

discharged from the pump in the rocket engine when the

mixture ratio is specified. The definitions of the flow and

head coefficients of the pump are

� ¼_QQp

Ae � up¼

_mmp

�

1

Ae � �Dim � ðN=60Þð2Þ

¼�Hp

up2=g� � ¼ g ��Hp

f�Dim � ðN=60Þg2: ð3Þ

The unit of the rotational speed, N, is rpm. Ae,Dim and up are

the discharge area, impeller diameter and impeller speed at

its diameter, respectively. The change of the pump operating

condition in the �– relationship is equal to change of the

specific speed of the pump. Its definition is

Ns ¼N � _QQp

1=2

�Hp3=4

¼ 60 �Ae

1=2 � g3=4

�Dim

��1=2

3=4: ð4Þ

When the circulation pump system shown in Fig. 1 is

adopted, the relationship between the flow rates is

r �_mmp4

_mmp0

¼_mmp1

_mmp0

¼_mmp2 � _mmp3

_mmp0

: ð5Þ

At the design operation, the flow rate into the pump, _mmp1, is

equal to that discharged from the pump, _mmp4, and they are

equal to the design pump flow rate, _mmp0. There is no circu-

lation flow rate, _mmp3, in the design condition here.

There are two kinds of pressure drops in the propellant

injector of the combustion chamber. One is the drop propor-

tional to the propellant mass flow rate. It appears when gas

propellant, for example, hydrogen gas, is injected. Another

is the drop proportional to the square of the propellant mass

flow rate. It appears when liquid propellant, for example,

liquid oxygen, is injected. Strictly speaking, there is no dis-

tinction between gas and liquid in the supercritical condi-

tion. Herein, low and high density conditions correspond

to gas and liquid conditions, respectively, from the view-

point of the pressure drop in the injector. In both cases, gen-

erally, the pressure drop in the injector should be larger than

15 to 20% of the combustion chamber pressure in order

to avoid combustion instability. A large pressure drop is

required for liquid propellant injectors in the design condi-

tion in order to keep the suitable pressure drop even in the

throttled condition. In the following, the pump discharge

Fig. 1. Pump with circulation.

50 Trans. Japan Soc. Aero. Space Sci. Vol. 56, No. 1

pressure and the throttling condition are examined for each

type of injector pressure drop.

Herein, a mixture ratio is presumed to be fixed at the

design condition. The throttling due to the change of the

mixture ratio is a topic for another study and out of the scope

of this study. A turbine is operated in the coolant bleed cycle

with heated hydrogen after the regenerative cooling.

2.1. Gas injection

2.1.1. Pump characteristics

In the gas propellant injection, the pressure drop is pro-

portional to the propellant flow rate and the combustion

chamber pressure. Assuming that gas in the injector mani-

fold is in the stagnation condition and changes isentropi-

cally, the mass flow rate is written as

_mm ¼ A � Pt

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2�

ð� � 1ÞR � TtPc

Pt

� �2�

�Pc

Pt

� ��þ1�

24

35

vuuut¼ A � ð1þ CinjÞPc

�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2�

ð� � 1ÞR � Tt1

ð1þ CinjÞ

� �2�

�1

ð1þ CinjÞ

� ��þ1�

24

35

vuuut :

ð6Þ

Here, the ratio of the injector pressure drop to the chamber

pressure, Cinj, is

Cinj ¼�pinj

pc: ð7Þ

Cinj should be held to 0.15 to 0.2 for stable operation. As

shown in Eq. (6), when the ratio of Pc=Pt is fixed, the mass

flow rate is proportional to Pt, differing from the relation-

ship in the liquid injection. Based on Eq. (6), the injector

pressure drop, �Pinj, is written as

�Pinj ¼ Cinj � Pc

¼Cinj

ð1þ CinjÞ

�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið� � 1ÞR � Tt

2�1

ð1þ CinjÞ

� �2�

�1

ð1þ CinjÞ

� ��þ1�

24

35

vuuuuut�_mminj

Ainj

:

ð8Þ

When Cinj and Tt are fixed during throttling, the injector

pressure drop is proportional to the mass flow rate.

When the other pressure drops in the engine are also

proportional to the combustion chamber pressure, the ratio

of pressure increase of �pp=�pp0 in the pump is approxi-

mately proportional to the throttling rate of r.

r ��pp

�pp0ð9Þ

The change of the head in throttling is written based on

Eq. (3).

�Hp

�Hp0

¼ð�pp=�pÞð�pp=�pÞ0

��pp

�pp0�

0

N2

N02

ð10Þ

Density of the incompressible pump fluid is almost constant.

Equation (10) is rewritten for the rotational speed based on

Eq. (9) as

N

N0

�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0

�pp

�pp0

s�

ffiffiffiffiffiffiffi 0

s�ffiffir

p: ð11Þ

The flow rate in the pump in the throttled condition is

written based on Eq. (2).

_QQp2

_QQp0

¼ð _mmp2=�p2Þð _mmp2=�p2Þ0

�_mmp2

_mmp0

��

�0

N

N0

ð12Þ

Equation (12) is rewritten based on Eq. (11) as

_mmp2

_mmp0

��

�0


s�ffiffir

p: ð13Þ

When circulation is adopted, with Eq. (5), the circulated

return flow rate is

_mmp3

_mmp0

¼_mmp2

_mmp0

�_mmp4

_mmp0

��

�0


s�ffiffir

p� r: ð14Þ

Based on Eq. (14), the ratio of circulation in throttling is

defined as

_mmp3

_mmp3c

�

�

�0


r�

ffiffir

p

1�ffiffir

p : ð15Þ

The subscript c represents the throttled condition due to cir-

culation only, that is, the condition where the flow and head

coefficients in throttling are equal to those of the design ones.

The flow rate of the circulation is

_mmp3 ¼ Ar �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2�p3 � r ��pp0

p: ð16Þ

Ar is a cross-section of the circulation return valve. In the

throttled condition, the flow rate in the pump is equal to

or larger than the discharged flow rate.

_mmp4

_mmp0

�_mmp2

_mmp0

ð17Þ

Based on Eqs. (5) and (13), Eq. (17) is rewritten as

r ��

�0

� �2 0

: ð18Þ

When there is no circulation, the sign in Eq. (18) becomes

equal.

There is a specified relationship between and � in a

pump, and is often expressed with a quadratic equation

of �. In the throttled pump with circulation only, the flow

and head coefficients are the same as those at the design con-

dition, and the mass flow rate and rotational speed are a

function of the throttling rate. In the pump with no circula-

tion, when a head coefficient is specified to a flow coeffi-

cient, then throttling rate, rotational speed and pump dis-

charge pressure are derived from Eqs. (18), (11) and (10),

respectively.

Jan. 2013 T. KANDA: Conceptual Study of Throttled Operation of Rocket Engine Turbopump 51

2.1.2. Enthalpy and temperature at pump entrance

Temperature at the pump entrance increases by circula-

tion. Enthalpy of the mixed flow is

ht;p2 ¼_mmp1 � ht;p1 þ _mmp3 � ht;p3

_mmp1 þ _mmp3

¼�0

�

ffiffiffiffiffiffiffi

0

s�ffiffir

p� ht;p1 þ 1�

�0

�

ffiffiffiffiffiffiffi

0

s�ffiffir

p !

� ht;p3

ð19Þ

where ht is the total enthalpy. Power required for the

pump is

Wp ¼ _mmp2ðht;p4 � ht;p2Þ ¼1

�p_QQp ��pp: ð20Þ

The pump efficiency, �p, is a function of � and its estimation

method is explained in the next section. From Eq. (20),

enthalpy at the pump exit is

ht;p4 ¼ ht;p3 �1

�p

�p

�p2þ ht;p2: ð21Þ

Equation (21) is put into Eq. (19), then based on Eq. (9),

ht;p2 � ht;p1 þ�

�0


s ffiffir

p� r

!1

�p

�pp0

�p2: ð22Þ

The temperature of the mixed flow is calculated using

Eq. (22).

2.1.3. Turbine characteristics

In the present study, an impulse turbine is presumed. The

power produced by the turbine is

Wt ¼ �t � _mmtðha � hbÞ

¼ �t � _mmt � Cpt � Ta 1�pb

pa

� ��1�

8<:

9=; ð23Þ

where _mmt, Cpt, and Ta are flow rate, specific heat at constant

pressure and total temperature of the turbine driving gas at

the entrance, respectively. Subscripts a and b indicate the

entrance and the exit of the turbine, respectively. The

efficiency of the turbine, �t, is a function of the ratio of

the turbine blade rotational speed, ut, to the driving gas

speed at the turbine nozzle exit, c0. The ratio is

ut

c0¼

�Dt

N

60ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2�

� � 1RTa 1�

pb

pa

� ��1�

8<:

9=;

vuuutð24Þ

where Dt is the diameter of the turbine. The efficiency is

often expressed by a function of the speed ratio.

The turbine gas flow rate is specified in the turbine nozzle

with the choking condition and the rate is proportional to

pressure at the entrance. When the entrance temperature

and the pressure ratio do not change from the design values,

the ratio of the speed ratios is written based on Eqs. (11)

and (24),

ðut=c0Þðut=c0Þ0

¼N

N0

�


s�ffiffir

p: ð25Þ

Changes of the turbine and pump powers by throttling are

Wt

Wt0

¼�t

�t0�

_mmt

_mmt0

ð26Þ

Wp

Wp0

��p0

�p�

_mm2

_mm20

��pp

�pp0: ð27Þ

The power of the turbine is equal to that of the pump. The

required turbine flow rate under throttling is derived using

Eqs. (9) and (13),

_mmt

_mmt0

��t0

�t��p0

�p��

�0


s� r

ffiffir

p: ð28Þ

Using Eqs. (5) and (28), ratio of the turbine gas flow rate to

the pump discharge rate is

_mmt

_mmp4

¼_mmt

_mmt0

�_mmt0

_mmp0

�_mmp0

_mmp4

��t0

�t

�p0

�p

�

�0


s�ffiffir

p�_mmt0

_mmp0

: ð29Þ

Here,

_mmt0

_mmp0

¼�pp0

�p0 � �20�

1

�t0 � Cpt � Ta 1�pb

pa

� ��1�

8<:

9=;: ð30Þ

2.2. Liquid injection

2.2.1. Pump characteristics

Pressure increase required for the pump is

�pp ¼ pc þ�pinj þ�ppl ð31Þ

where �ppl is the pressure drop in plumbing between the

pump exit and the injector, and this drop is presumed to

be proportional to pc.

�ppl ¼ Cpl � pc ð32Þ

The pressure drop in the propellant injector of �pinj is

proportional to the square of the propellant mass flow rate,

that is, the square of the combustion chamber pressure.

�pinj ¼ Kinj � pc2 ð33Þ

In throttling of the rocket engine, the propellant mass flow

rate is proportional to the combustion chamber pressure

shown in Eq. (2). Under a fixed density condition, the mass

flow rate is written as

Pc / _mminj ¼ Ainj

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2� ��Pinj

pð34Þ

or

�Pinj ¼1

2�

_mminj

Ainj

� �2

/ Pc2: ð35Þ

When the most throttled condition is designated with a

subscript ofmn, the ratio of the pressure drop to the chamber

pressure is


�pinj;mn

pc;mn¼

Kinj � ðpc0 � rmnÞ2

pc0 � rmn¼ Kinj � pc0 � rmn¼ Cinj;mn: ð36Þ

Therefore,

Kinj ¼Cinj;mn

rmn � pc0: ð37Þ

Cinj should be held to 15 to 20% even in the minimum throt-

tling rate. Using Eq. (37), Eq. (33) can be expressed in

another form as

�pinj ¼Cinj;mn

rmn � pc0� pc2 ¼

Cinj;mn

rmn� r2 � pc0: ð38Þ

The pressure increase of the pump of Eq. (31) is rewritten

with Cinj;mn as

�pp ¼ r � pc0 þ ðCinj;mn=rmnÞ � r2 � pc0 þ Cpl � r � pc0

¼ pc0 � ð1þ CplÞ � r þ ðCinj;mn=rmnÞ � r2�

ð39Þand

�pp

�pp0¼

ð1þ CplÞ � r þ ðCinj;mn=rmnÞ � r2

ð1þ CplÞ þ ðCinj;mn=rmnÞ: ð40Þ

From Eqs. (10) and (40), the ratio of the rotational speeds is

N

N0

�


s�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1þ CplÞ � r þ ðCinj;mn=rmnÞ � r2

ð1þ CplÞ þ ðCinj;mn=rmnÞ

s: ð41Þ

As is in Eq. (13), the pump mass flow ratio is

_mmp2

_mmp0

��

�0�N

N0

��

�0


s�



s: ð42Þ

The circulation flow rate is

_mmp3

_mmp0

¼_mmp2

_mmp0

�_mmp4

_mmp0

��

�0


s�



s� r:

ð43Þ

The circulated return mass flow ratio to that in the circula-

tion-only throttling is

_mmp3

_mmp3c

�

�

�0


r�



s� r



s� r

:

ð44Þ

In the throttled condition, the flow rate in the pump is

equal to or larger than the discharge flow rate. From Eq. (42),

r �

�

�0

� �2 0

ð1þ CplÞ

1þ Cpl þ 1��

�0

� �2 0

( )Cinj;mn

rmn

� � : ð45Þ

When there is no circulation, the sign is equal.

2.2.2. Enthalpy and temperature at pump entrance

The enthalpy of the mixed flow at the pump entrance is

ht;p2 ¼_mmp1 � ht;p1 þ _mmp3 � ht;p3

_mmp1 þ _mmp3

�r � ht;p1 þ

�

�0


r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1þ CplÞ � r þ ðCinj;mn=rmnÞ � r2


s� r

!� ht;p3

�

�0


r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1þ CplÞ � r þ ðCinj;mn=rmnÞ � r2


s : ð46Þ

Equation (21) is substituted into ht;p3 in Eq. (46), and is rewritten based on Eq. (40) as

ht;p2 � ht;p1 þ1

�p

�pp0

�2

fð1þ CplÞ þ ðCinj;mn=rmnÞ � rgfð1þ CplÞ þ ðCinj;mn=rmnÞg

�

�0


s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1þ CplÞ � r þ ðCinj;mn=rmnÞ � r2


s� r

!: ð47Þ

2.2.3. Turbine characteristics

From Eqs. (24) and (41), the ratio of the turbine speed ra-

tios is

ðut=c0Þðut=c0Þ0

¼N

N0

�


s�


1þ Cpl þ ðCinj;mn=rmnÞ

s: ð48Þ

As in Eq. (28), the turbine mass flow rate is rewritten based

on Eqs. (40) and (42) as

_mmt

_mmt0

��t0

�t��p0

�p��

�0


s

�ð1þ CplÞ � r þ ðCinj;mn=rmnÞ � r2


� �32

: ð49Þ

Based on Eqs. (5), (30) and (49), the ratio of the turbine flow

rate to the pump discharge flow rate is


_mmt

_mmp4

¼_mmt

_mmt0

�_mmt0

_mmp0

�_mmp0

_mmp4

¼

�t0

�t

�p0

�p

�

�0


r�

ð1þ CplÞr þ ðCinj=rmnÞr2

ð1þ CplÞ þ ðCinj=rmnÞ

� �32

r�_mmt0

_mmp0

:

ð50Þ

The ratio of the turbine flow rate to that of the pump at the

design condition is as shown in Eq. (30).

3. Results and Discussion

Operating conditions of imaginary, throttled LOX and

LH2 pumps are calculated with the relationships for a

1,000 kN-level, high-pressure LOX/LH2 rocket engine

and a 100 kN-level, mid-pressure engine, respectively.

Hydrogen is often injected into the combustion chamber

in low-density after the regenerative cooling. Oxygen is

not usually used for cooling and injected in high density.

Herein, the gas propellant injection is applied to a LH2 sup-

ply system, whereas the liquid propellant injection is applied

to a LOX supply system.

The head coefficient of the imaginary pump is expressed

with a quadratic equation of the flow coefficient. Efficiency

of the pump is calculated based on the following procedure.

The theoretical head coefficient is expressed as16)

� ¼ 1�� C: ð51Þ

Here, � and � are theoretical head and flow coefficients,

respectively. C is a constant. The =� of an actual pump

reasonably corresponds to � .16) Thus, the pump efficiency

is presumed to have the following relationship in the present

conceptual study.

�¼ 1� � � C ð52Þ

At the design point, the efficiency of �0 is presumed to be a

maximum, specified value. The coefficient of C in Eq. (52)

is specified with the design point conditions.

C ¼1�

0

�0�0

ð53Þ

A two-stage impulse turbine is presumed for the turbo-

pumps. The turbine efficiency is expressed with a quadratic

equation of the speed ratio and becomes largest at the ratio

of 0.25.

Properties of oxygen and hydrogen are calculated with a

code for LOX/LH2 rocket engines.17) Total pressure of the

mixed flow, pt2, is presumed to be equal to pt1.

Presumed specifications of LH2 turbopumps are listed in

Table 1 and those of LOX turbopumps are listed in Table 2.

The LOX turbopumps are designed at the minimum throt-

tled condition of 10%. As shown in Eq. (39), the pressure

drop in the injector becomes larger at the design condition

as the minimum throttling ability is set to be smaller, and

the pump discharge pressure becomes much higher in a deep

throttling engine. The features and problems in throttling

with the liquid propellant injection are made clear in such

a deeply throttled engine. The design pump discharge pres-

sure is approximately 40MPa under the combustion cham-

ber pressure of 15MPa in the high-pressure engine, listed

in Table 2. CECE has a throttling ability of 10 : 1 and the

high delta pressure injector for LOX.4)

In the examination of the LH2 supply system, the pump

discharge pressure is set to 20MPa for the 1,000 kN-level

engine and 5MPa for the 100 kN-level engine referring to

the LE-5B and LE-X engines.18,19) In the examination of

the LOX supply system, the ratio of the pressure drop in

plumbing to the chamber pressure, Cpl, is set to be

0.15.18–20) The ratio of the injector pressure drop to the

chamber pressure, Cinj, is 0.15.

Figure 2 shows head coefficient and pump efficiency of

LH2 imaginary pumps, and Fig. 3 shows those of LOX

pumps.

Table 1. Specifications of LH2 turbopumps at design condition.

1,000 kN-level, high-pressure engine

Combustion chamber pressure, MPa 15

H2 pump

Inflow temperature, Tt1, K 21

Inflow pressure, Pt1, MPa 0.3

Flow rate, kg�s�1 40

Efficiency 0.8

�P0, MPa 19.7

Entrance diameter, m 0.18

Impeller diameter, m 0.24

Rotational speed, rpm 50,000

Specific speed, m, m3/m, rpm 132

H2 turbine

Rotor diameter, m 0.22

Temperature at entrance, K 500

Pressure ratio 8

Efficiency 0.5

100 kN-level, mid pressure engine

Combustion chamber pressure, MPa 3.5

H2 pump



Flow rate, kg�s�1 3.5

Efficiency 0.8

�P0, MPa 4.7





H2 turbine



Pressure ratio 5

Efficiency 0.5


3.1. Pump characteristics

Figure 4 shows operating conditions of the LH2 pumps in

line with throttling for the no-circulation and the full-circu-

lation conditions. Figure 5 shows those of the LOX pumps.

In the full-circulation condition, the flow and head coeffi-

cients are fixed at the design values during throttling. The ro-

tational speed and the pump discharge flow rate decreases in

line with the throttling rate. In the no-circulation condition,

the flow coefficient also decreases. The head coefficient does

not change greatly during throttling. Unstable in the figures

is shown as the region of the positive slope in �– relation-

ship. The normalized features of the two LH2 pumps are al-

most the same except for the unstable operating area even

though the head coefficients and efficiencies are different

as shown in Fig. 2. This similarity is also seen in LOX pumps.

In the LH2 pumps, the rotational speed shows convex

change in line with throttling. As shown in Eq. (11), the ro-

tational speed is proportional to the square roots of the throt-

tling rate and the head coefficient. The head coefficient is

fixed at the design value in the full-circulation condition

Table 2. Specifications of LOX turbopumps at design condition. Mini-

mum throttling is 10%.

1,000 kN-level, high-pressure engine

Combustion chamber pressure, MPa 15

Injector pressure drop ratio, Cinj 0.15

Plumbing pressure drop ratio, Cpl 0.15

O2 pump




Efficiency 0.8

�P0, MPa 39.5





O2 turbine



Pressure ratio 2.5

Efficiency 0.5

100 kN-level, mid-pressure engine

Combustion chamber pressure, MPa 3.5

Injector pressure drop ratio, Cinj 0.15

Plumbing pressure drop ratio, Cpl 0.15

O2 pump




Efficiency 0.8

�P0, MPa 9.0





O2 turbine



Pressure ratio 2.5

Efficiency 0.5

(a)

(b)

Fig. 2. Head coefficient and efficiency of LH2 pumps.

(a) High-pressure engine. (b) Mid-pressure engine.

(a)

(b)

Fig. 3. Head coefficient and efficiency of LOX pumps designed for 10%

throttling.



and does not change greatly in the no-circulation condition

as shown in Fig. 2. So the rotational speed is approximately

proportional to the square root of the throttling rate. In the

LOX pumps, the rotational speed ratio changes more line-

arly proportional to the throttling rate. As the design mini-

mum throttling rate, rm, becomes lower in the liquid propel-

lant injection system, the term Cinj;mn=rmn becomes larger in

Eq. (41) and the rotational speed changes almost linearly to

the throttling rate.

According to Eq. (14), in the no-circulation condition of

_mmp3 ¼ 0, the flow coefficient is approximately proportional

to the square root of the throttling rate in the gas propellant

injection system. In the liquid propellant injection system,

the flow coefficient changes more slowly in line with throt-

tling until approximately r ¼ 0:3. When there is no circula-

tion, Eq. (43) can be rewritten as

�

�0�

ffiffir

p


r�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1þ CplumbÞ þ ðCinj;mn=rmnÞ � rð1þ CplumbÞ þ ðCinj;mn=rmnÞ

s ð54Þ

whereas Eq. (14) of the gas injection is rewritten as

�

�0�

ffiffir

p,, ffiffiffiffiffiffiffi

0

s: ð55Þ

Since the flow coefficient changes more slowly in line with

the throttling rate under the liquid propellant injection sys-

tem in the shallow throttling region, the unstable operating

limit appears lower in the throttling rate than in the gas pro-

pellant injection system. This feature becomes clearer as the

design minimum throttling rate becomes lower in the liquid

propellant injection system. The liquid propellant injection

system has wider stable operating region in throttling than

the gas injection system from the viewpoint of the change

of the flow coefficient.

In the gas propellant injection system with full circula-

tion, the ratio of the circulation flow rate to the pump dis-

charge rate reaches the largest value of 0.25 at r ¼ 0:25 in

the full-circulation condition, as derived by differentiating

Eq. (14). In the liquid injection system, the maximum value

becomes smaller at a smaller throttling rate.

The pumps can operate between no circulation and the

full circulation. Figure 6 shows the 25% throttled condition

(a)

(b)

Fig. 4. Rotational speed, flow rates and coefficients of LH2 pumps in

throttling.


(a)

(b)

Fig. 5. Rotational speed, flow rates and coefficients of LOX pumps de-

signed for 10% throttling in throttling.



for the LH2 high-pressure pump, and Fig. 7 shows the 10%

condition for the LOX high-pressure pump. �=�0 ¼ 1 indi-

cates full-circulation operation, whereas the left end of

the lines indicates no-circulation operation. In the high-

pressure engine, the operating region of the LH2 pump is

limited not only by the unstable region, but also by the

vaporization region. The region is due to the return flow

of high-enthalpy fluid at the pump entrance. In the mid-

pressure LH2 pump, the increase of enthalpy is smaller,

and this limitation does not appear. The vaporization due

to circulation is discussed in next section. The rotational

speed does not change greatly due to circulation since

0= is almost constant in the throttling region. The pump

efficiency becomes lower for a lower circulation ratio due

to its separation from the design operating condition. At

the same time, however, the pump flow rate becomes small-

er for a lower circulation rate.

3.2. Temperature of mixed flow and suction perfor-

mance

In throttling, the increase of temperature of the mixed

flow at the entrance of a cryogenic pump is a problem for

suction performance in the circulation system. Figure 8

shows temperatures at the entrance and the exit of the

full-circulation LH2 pumps, and Fig. 9 shows temperatures

at the entrance and the exit of the full-circulation, high-pres-

sure LOX pump.

In the high-pressure LH2 pump, the mixed flow at the

entrance is at the saturation temperature under the pump

entrance pressure. This is caused by a large increase of

enthalpy in the low-density hydrogen pump. In the LOX

pump, however, the temperatures at the entrance and exit

do not change greatly from the temperature at the design

point at r ¼ 1, even though the pump discharge pressure

is high. This is caused by large density of oxygen and a

small increase of enthalpy in the pump even though the

pump discharge pressure is very high. In the mid-pressure

LH2 pump, the increase of enthalpy is smaller and the tem-

perature does not reach the saturation temperature. Temper-

Fig. 6. 25% operation of LH2 pump of high-pressure engine.

�=�0 ¼ 1 indicates the full-circulation operation and the left end of the

lines indicates the no-circulation operation.

Fig. 7. 10% throttling operating condition of LOX pump of high-pressure

engine.

�=�0 ¼ 1 indicates the full-circulation operation and the left end of the

lines indicates the no-circulation operation.

(a)

(b)

Fig. 8. Temperature at the entrance/exit of LH2 pumps with the full-cir-

culation throttling system.


Fig. 9. Temperature at the entrance/exit of high-pressure LOX pump de-

signed for 10% throttling with the full-circulation throttling system.


ature at the exit shows a strange change. This is caused by a

quick change of properties around the critical pressure. It

may be true or it may be due to the accuracy of the property

calculation code. The effect of circulation on the suction

performance depends on properties of the pump fluid.

Figure 10 shows the net positive suction head (NPSH)

and the cavitation parameter under the full-circulation oper-

ation of the mid-pressure LH2 pump. Herein, their defini-

tions are

NPSH ¼Pt2 � Ps

�2 � gð56Þ

CP ¼P2 � Ps

ð1=2Þ�2 � ui2: ð57Þ

Here, ui is the tip speed of the pump inducer, and Ps is sat-

uration pressure at Tt2. NPSH decreases in line with throt-

tling in the shallow throttling region. In the full-circulation

condition, the saturation pressure increases with an increase

of the temperature of the mixed flow. This decreases NPSH.

As for the cavitation parameter, the rotational speed and the

dynamic pressure become smaller in line with throttling.

This causes the cavitation parameter to increase in line with

throttling.

Figure 11 shows NPSH and the cavitation parameter of

the mid-pressure LH2 pump at the 25% throttling condition.

The point at �=�0 ¼ 1 indicates operation at full circulation,

and the left end of the lines indicates operation with no cir-

culation. Due to the decrease of the circulation flow ratio,

both NPSH and the cavitation parameter increase. However,

the decrease of the ratio causes a shift of the �– operating

condition and makes pump operation unstable. For a LOX/

LH2 engine, for example, shifting the mixture ratio is a

method of producing deeply throttled operation and avoid-

ing unstable operation and gasification.

Figure 12 shows NPSH and the cavitation parameter of

the LOX pump of the high-pressure engine. Due to an in-

crease of the saturation pressure by an increase of tempera-

ture of the mixed flow, NPSH slightly decreases toward

r ¼ 0:3. Even though this feature appears clear in the

high-pressure engine pump, it is a small change. The cavita-

tion parameter increases in line with throttling, as does that

of the LH2 pump.

Figure 13 shows NPSH and the cavitation parameter at

the 10% throttling condition. The point at �=�0 ¼ 1 indi-

cates operation at full circulation and the left end of the lines

indicates operation with no circulation. At 10% throttling,

the unstable region appears in the small flow coefficient

region. The region does not appear in 25% throttling. This

slow appearance of the unstable region in throttling is caused

by the slow decrease of the flow coefficient in line with the

throttling rate in the liquid propellant injection system, as

explained in 3.1. Circulations are effective in a LOX pump

when deeply throttled operation is required, because gasifi-

cation only occurs to a small extent as a result of circulation.

When propellant with large density, for example, a kind

of kerosene or alcohol, is used for cooling and injected into

the combustion chamber in a gas condition, the pump sys-

tem with the propellant has features similar to the LH2 pump

system in operation because of gas injection into the com-

bustion chamber, and features similar to the LOX pump in

temperature and suction performance at the pump entrance

because of high density of the propellant in the pump. The

pump discharge pressure does not become so high and tem-

Fig. 13. NPSH and cavitation parameter of the high-pressure LOX pump

designed for 10% throttling.

Fig. 10. NPSH and cavitation parameter of the full-circulation LH2 pump

of the mid-pressure engine. Fig. 11. NPSH and CP of LH2 pump at 25% throttled condition of the

mid-pressure engine.

Fig. 12. NPSH and cavitation parameter of the high-pressure, full-circu-

lation LOX pump designed for 10% throttling.


perature of the mixed flow increases little in the circulation

system.

3.3. Turbine characteristics

Figure 14 shows operating conditions of the LH2 tur-

bines, and Fig. 15 shows those of the LOX turbines de-

signed at the minimum throttling rate of 10%. Though the

LOX pump discharge pressure of the high-pressure engine

is high, the ratio of _mmt= _mmp4 is less than 0.07. The ratio is

within the scope of the turbine bleed ratio. The deeply throt-

tled operation is possible in the LOX supply system.

The turbine efficiency shows little difference due to the

ratio of circulation. The efficiency depends on the speed ra-

tio in the present study. The rotational speed shown in

Eq. (11) does not change greatly as a result of circulation,

since the pump 0= does not change greatly even in the

no-circulation system. Thus, the efficiency does not change

greatly due to the ratio of circulation. In line with throttling,

the turbine efficiency decreases with a decrease of the rota-

tional speed.

The ratio of the turbine gas flow rate to that discharged

from the pump, _mmt= _mmp4, does not become 0 due to the de-

crease of the throttling rate in the full-circulation system.

In the gas injection system, the ratio is a function of the tur-

bine efficiency and the throttling rate as is shown in

Eq. (29). The efficiency is a quadratic function of the speed

ratio and this ratio is the function of the square root of the

throttling rate as is shown in Eq. (25). Thus, the efficiency

is almost linear to the square root of the rate in the deeply

throttled condition. Thus, _mmt= _mmp4 of Eq. (29) becomes a val-

ue in the throttling rate of 0. In the liquid injection system,

when the throttling rate is small, the efficiency is almost pro-

portional to the square root of the throttling rate and _mmt= _mmp4

of Eq. (50) also becomes a value in the throttling of 0. In the

no-circulation system, the flow coefficient in Eqs. (29) and

(50) decreases in line with throttling, and _mmt= _mmp4 becomes

0 for the throttling rate of 0.

Since the pump flow rate becomes larger as a results of an

increase of the circulation rate, the turbine flow ratio of

_mmt= _mmp4 also becomes larger in the full-circulation condition

(shown by a broken line) than that in the no-circulation con-

dition (shown by a solid line). However, the ratios of both

the full- and no-circulation systems decrease in line with

throttling in both the LH2 and LOX turbines. The turbine

gas does not contribute to thrust production greatly and an

increase of the turbine gas flow rate degrades the engine spe-

cific impulse in the bleed cycle. The decrease in _mmt= _mmp4 in

line with throttling indicates that throttling does not degrade

the engine specific impulse from the viewpoint of turbine

operation. Attention should be paid to the fact that the pump

and turbine operating characteristics are modeled here. For

example, when the pump efficiency is much smaller than

the value used here, _mmt= _mmp4 becomes much larger as seen

in Eqs. (29) and (50). Then, the specific impulse may be de-

graded.

Figure 16 shows 25% operation of the LH2 turbine of the

high-pressure engine, and Fig. 17 shows 10% operation of

the LOX turbine of the high-pressure engine. The circulation

ratio decreases with a decrease of �=�0, and the left end of

the lines shows operation with no circulation. Though the

flow coefficient has no direct relationship to the turbine op-

eration, the ratio of the coefficients is used here for compar-

ison with figures of the throttled pump operating conditions.

As mentioned above, the efficiency does not change

greatly due to circulation. The pump efficiency becomes

(a)

(b)

Fig. 14. LH2 turbine operation in throttling.


(a)

(b)

Fig. 15. LOX turbine operating conditions designed for 10% throttling.



lower with a decrease of the circulation ratio due to off-de-

sign operation, but the pump flow rate also becomes smaller.

As a result, the required turbine power, that is, the turbine

flow ratio of _mmt= _mmp4, becomes smaller with a decrease of

the circulation ratio.

4. Concluding Remarks

A conceptual study of the throttling of rocket engine tur-

bopumps was conducted. Relationships between the throt-

tling rate, flow rates, pressure and temperature of the pump

and turbine were derived. The relationships under the gas

propellant injection were applied to an imaginary LH2 tur-

bopump system, whereas those under the liquid injection

were applied to a LOX turbopump system.

In the LOX pump of the liquid propellant injection sys-

tem, the pump unstable region was limited to the deep throt-

tling region, compared to the region of the gas injection sys-

tem for the LH2 pump. This was caused by the slow change

of the flow coefficient in line with throttling in the liquid in-

jection system.

In the high-pressure LH2 pump with the circulation sys-

tem, gasification occurred at the entrance of the pump due

to low density of hydrogen and the large enthalpy increase

in the pump. In the LOX pump or the mid-pressure LH2

pump, temperature at the entrance changed slightly because

of a small increase of enthalpy.

In both the gas injection and the liquid injection cases, the

ratio of the turbine flow rate to that discharged from the

pump decreased in line with throttling. Throttling did not

degrade the engine specific impulse from the viewpoint of

the turbine operation in throttling.

Acknowledgments

The author is grateful to Mr. Tomoyuki Hashimoto of JAXA for

advice and discussion.

References

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Fig. 17. 10% throttled operation of LOX turbine of high-pressure engine

designed for 10% throttling condition.


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