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Cavitation avoidance in nitrous oxide rocket engines using the efficient

TRANSIENT 1D downwind prediction

RUGESCU Radu Dan

1, a *

, BOGOI Alina

2,b

and CONSTANTINESCU Emil Cristian

2,c

1

ADDA Ltd, 18 Bancila St, Bucharest 060144, Romania

2

UPB, 1 Polizu St, Bucharest 011061, Romania

a

rugescu@yahoo.com,

b

bogoi_alina@yahoo.com,

c

constantinescu_ce@yahoo.com

*corresponding author

Keywords: cavitation, nitrous oxide, rocket engines, space propulsion, dual phase flow

Abstract. Nitrous oxide has recently entered violently the arena of space propulsion and gained

interest, due to its high energy and gasification potential and despite its low oxygen content as an

oxidizing chemical and its instability over some 600 C. However, its physical and chemical

instability soon proved to be a potential hazard and led to a renewed interest in the study of its

behavior as a fluid. In the present contribution computer simulation of the liquid phase flow of the

nitrous oxide under high pressure is used to predict and avoid the cavitation into the feeding line

tract of rocket engines, specifically of the compound rocket engines feeding line. The method

involves a substantially simplified 1-D description of the fluid motion with sufficiently accurate

determination of cavitation risk where the feeding duct suffers blunt variations of the cross area or

steep turns and corners involving sensible static pressure variations of the fluid. A means of

avoiding dangerous behaviors of the nitrous oxide is thus developed that could increase safety

margins during the handling of this quite unpredictable oxidizer for the compound, combined or

hybrid rocket engines.

Introduction

Although nitrous oxide (N

2

O), first discovered in 1793 by the English scientist Joseph Priestly, is

known as a convenient curing means in medicine [1, 2] and often used for more than two hundred

years as an anesthetic [3, 4], with recent extension in the area of maternal care [5], little was known

in fact on its challenging behavior and detonation potential, disastrously demonstrated in the 2007

accident at Scaled Composites Co. on an county airport near Los Angeles, US [6]. The fact that no

specific cause of the accident could have been found during the official investigation [7] leaves

open the question of safety procedures with N

2

O handling. Since no hazard have been ever found to

develop with N

2

O in liquid phase, a series of potential causes were considered as potential hazards

for N

2

O handling, including contamination with organic soot, local overheating due to diverse

causes, leading to a local vaporization and formation of bubbles, in fact a sort of static cavitation,

the sudden opening of valves that produces a pressure transient with bubble formation, local

catalytic effects due to the presence on unsuitable materials in the plumbing etc. [8, 9] .One of the

most subtle causes which are suspected for the apparently irregular detonation of this substance is

the sudden dynamic cavitation that could develop into the transfer feed line during the liquid flow

for tank feeling procedure [10].

Cavitation is commonly associated to the rotating machines with blades, where a reduced

pressure would develop around the convex part of their profile and leads to local vaporization due to

the liquid instant boiling [11-14]. The same effect is producing however around any sudden

variation of the area/shape of the duct where the instable liquid is flowing and the main target of the

contribution of the authors is to identify this potential hazard along the feed line of the experimental

compound motor MEC-80 developed by ADDA.

Applied Mechanics and Materials Vol. 656 (2014) pp 95-100 Submitted: 2014-08-12

© (2014) Trans Tech Publications, Switzerland Accepted: 2014-08-15

doi:10.4028/www.scientific.net/AMM.656.95 Online: 2014-10-27

All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of Trans

Tech Publications, www.ttp.net. (ID: 84.117.82.164-13/08/15,16:13:24)

Euler type transient equations

In order to focus the research on the transient variation of the static pressure in the fluid as a

response to variations in the cross area and shape of the duct the viscous effects in the fluid are

considered insignificant and the fluid is assessed as an ideal, incompressible, non-viscous and non-

vaporizing liquid. However, the fast transients produced at flow startup and shut down are important

and are thus considered into the computational model that has the known 3-D form

0

=

⋅

∇

V,

p

dt

d∇−=

ρ

1

F

V, (1)

used directly in the form (2) for the one-directional motion,

,0

1

,

ln

=

∂

∂

+

∂

∂

+

∂

∂

−=

∂

∂

y

p

x

u

u

t

u

dx

Ad

u

x

u

ρ

(2)

or in the generalized [17] Chorin‘s 2-D pseudo-compressibility form,

0

,0

,0

1

2

2

2

=

∂

∂

+

∂

∂

+

∂

∂

+

∂

∂

+

∂

∂

=

∂

∂

+

∂

∂

+

∂

∂

+

∂

∂

+

∂

∂

=

∂

∂

+

∂

∂

+

∂

∂

y

p

y

v

v

x

v

u

t

v

t

pv

x

p

y

u

v

x

u

u

t

u

t

pu

y

v

x

u

t

p

β

β

β

(3)

both for negligible field forces F=0. The latter also transforms, for the one-directional case

applicable to slender duct flows, into the 1-D basic equations of mass and impulse conservation of

the dependent state variables

speed

and

pressure

,

,0

ln

,0

ln1

2

=+

∂

∂

+

∂

∂

+

∂

∂

=−

∂

∂

dx

Ad

u

x

p

x

u

u

t

u

dx

Ad

u

t

p

β

(4)

where the physical conditions are given as

.

const

=

ρ

,

givenxA

=

)( .

According to recommendations in [16] the constant value for β should be around u

~

. The

cavitation and the subsequent two phase flow that accompanies cavitation is still considered as

ineffective, in order to arrive at the conditions for local vaporization first. The main problem in this

case is the accommodation of the one-directional motion concept with local variations in the cross

section size of the duct and with the blunt turns that appear at duct corners.

The two-point boundary value problem of the controlled duct

When the application is focused on the computation of pressure distribution along a given pipe

with a given profile of the cross area and given pressure values at both ends of the open duct, the

problem resides in the computation of the velocity and pressure distribution in time along the whole

duct, in compliance with the given and constant values of the pressure at the two end openings.

96 Monitoring, Controlling and Architecture of Cyber Physical Systems

Although the inviscid-incompressible flow into the duct is governed by that simple description in

(2) the flow rate and local velocities that accomplish this flow rate are not known ab initio. The

problem enters the area of the two-point boundary value problems and the initial (t=0) distribution

of velocity and pressure values along the pipe must first be computed.

The specific given data of the problem are:

a) the size and profile of the duct in the form of the cross area distribution A(x) along the pipe;

b) the given values of the pressure at both ends of the pipe p

1

and p

N

;

c) the given value of the liquid constant density ρ.

Nothing else is known of the flow.

As far as the liquid is incompressible, at every instant in time the flow rate, that means both

volume and mass flow rates, are absolute constants along the pipe and this gives the possibility to

build the velocity distribution along the pipe when the flow rate is known. But it isn’t. Following

the physical phenomenon, when a control valve is opened at the downwind end of the pipe a

continuous flow of liquid starts to be displaced along the pipe due to the pressure head and this

mass of liquid gradually accelerates until the pressure values at both ends equal the imposed

pressure values (b) of the system.

In order to model this type of flow a numerical scheme is conceived based on the finite, constant

difference computational mesh as informally presented in Fig. 1. A finite difference algorithm is

then attached to this mesh in order to approximate the continuous flow with the discontinuous

replacer that will be forced to reproduce the real flow with the desired accuracy, by transforming the

partial derivatives equations (2) or (4) into a finite difference form convenient for direct

computations from node to node in the mesh.

Fig. 1. Equi-positioned computational mesh into the real {t-x} plane.

The given problem is permeable to an indirectly implicit, finite difference algorithm that replaces

the continuous equations (4) with their working form (5), according to Fig. 1,

,

,

2

AB

AB

A

A

AB

AB

AB

AB

A

AD

AD

AB

AB

A

A

AD

AD

xx

AA

A

u

xx

pp

xx

uu

u

tt

uu

xx

AA

A

u

tt

pp

−

−

−=

−

−

+

−

−

+

−

−

−

−

=

−

−

β

(5)

v

B

v

E

v

C

v

F

v

A

v

D

O x

elapsed time

position along the duct

t

D

C B A

E F

Applied Mechanics and Materials Vol. 656 97

where the formulae are to be gradually applied along the initial row of nodes established on the

abscise axis. Equations (5) are only written for the initial time value (t=0) in the given form.

However, this computational working form has the disadvantage of simultaneously involving too

many increments in the dependent variables and this is why the more useful form is the transcription

obtained by extending the equations from the form (5) into

.

1

,

,

2

2

A

B

AB

A

B

AB

A

A

A

D

AD

AB

AB

A

A

AD

AD

AB

AB

A

A

AB

AB

xx

pp

xx

AA

A

v

tt

uu

xx

AA

A

u

tt

pp

xx

AA

A

u

xx

uu

−

−

−

−

−

=

−

−

−

−

=

−

−

−

−

−=

−

−

ρ

β

(6)

For a quasi-viscous flow the pressure increment along the abscise axis could be rebuilt from the

function Ø

n+1

through the relationship [18], as suggested by [16]

1211

2

+++

∇

∆

=

nnn

eR

t

p

ϕϕ

. (7)

In order to compute Ø

n+1

the Poisson equation is used [19]

t

u

n

∆

⋅

∇

=∇

+

*

12

ϕ

. (8)

For the inviscid case however the pressure gradient along the direction of motion is only related

to the variation in cross area values of the duct, as seen from the second equation in (6). The

following computational steps are then used.

Stage I. A steady state value of the flow rate Q

st

≡A

m

v

0

is first obtained from the global static

pressure difference (p

1

-p

N

) and the mean cross area of the duct A

m

, using the Bernoulli equation of

energy in integral form,

ρ

)(2

1N

mst

pp

AQ −

=, (7)

where the assumption was made that the entrance in the pipe is at rest (entrance from a large tank).

In the absence of viscosity no frictional losses are present and at a constant diameter of the pipe the

pressure does not vary along the duct. The first numerical value of the fluid velocity at the entrance

of the tube is thus given by

11,0

/AQv

st

=, (8)

Extending equations (6) to the entire computational mesh the following solving system appears

.

1

,,

,1,

,1,

,1,

,1,

,

2

,

,,1

,,1

,

,

,

,

2

,,1

,,1

,1,

,1,

,

,

,1,

,1,

jiji

jiji

jiji

jiji

ji

ji

jiji

jiji

jiB

jiB

ji

ji

jiji

jiji

jiji

jiji

ji

ji

jiji

jiji

xx

pp

xx

AA

A

v

tt

uu

xx

AA

A

u

tt

pp

xx

AA

A

u

xx

uu

−

−

−

−

−

=

−

−

−

−

=

−

−

−

−

−=

−

−

+

+

+

+

+

+

+

+

+

+

+

+

ρ

β

(9)

98 Monitoring, Controlling and Architecture of Cyber Physical Systems

Stage II. Starting from a given distribution of pressures along the pipe and successively applying

the difference algorithm in (9) a new distribution of pressures and velocities develop along the pipe

within a given time frame of the computation, up to the steady state. This values end in a different

final pressure at the end of the pipe and a linearized correction is made to obtain the desired fit of

the given end-value pressure. The process is repeated to the desired accuracy.

Numerical results for a given configuration

The following configuration given in Fig. 2 and resulting from the design of the feed system of

the compound rocket engine MEC-80JMIC was subjected to the method here presented.

Fig. 2. The design configuration of the duct.

The variation of the pressure along the given pipe as resulting after 27 time steps is shown in Fig.

3. It reveals the presence of pressure raises and drops at the stages where cross area variations of the

duct are visible.

Fig. 3. Smoothed computed pressure distribution along the pipe and experiments.

These variations have to be compared with the admissible variations of the liquid phase of the

fluid in order to establish the limits of vaporization at the given temperature. This aspect is specific

to the fluid and for the present application is characteristic to the nitrous oxide subjected to tests.

Conclusion

The transient numerical method allows predicting the pressure distribution along a given length pipe

fast and easy. The results are in good agreement with the experimental measurements in the set.

O

p

0.4

0.2

0.1

1.0

0.9

0.8

0.7

0.6

0.5

0.3

100%

experimental data

x %

Applied Mechanics and Materials Vol. 656 99

References

[1] Amort, Dr. Robert, The Physiological Action of Nitrous Oxide, New York Medical Journal,

August, 1870.

[2] I. O’Sullivan, and J. Benger, Nitrous oxide in emergency medicine, Emerg. Med. J., May 2003;

20(3), 214–217.

[3] Johnson, Dr. George, A Lecture on the Physiology of Coma and Anaesthesia, Medical Times

and Gazette, April 3, 1869.

[4] Ruben, H., Nitrous oxide analgesia in dentistry. Its use during 15 years in Denmark, Br. Dent.

J., 1972 Mar 7, 132(5), 195–196.

[5] Rosen, M. A., Nitrous oxide for relief of labor pain: a systematic review, Am. J. Obstet.

Gynecol. May 2002, 186(5 Suppl Nature), 110-126.

[6] Abdollah Tami, and Stuart Silversteine, Test site explosion kills three, Los Angeles Times, 27

July 2007.

[7] ***, Inspection Report No. 0950625 at Scaled Composites, LLC, OSHA Report, 2007.

[8] B. Berger, Is nitrous oxide safe? Swiss Propulsion Laboratory, 2007.

[9] *** Scaled Composites accident - Mojave Desert, California, Observations and comments on

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accident/, 2014.

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of a nitrous oxide/propane rocket engine, 37th Joint AIAA/ASME/SAE/ASEE Propulsion

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[11] S. Khurana, Navtej and Hardeep Singh, Effect of cavitation on hydraulic turbines - A review,

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[12] Escaler, Xavier, Egusquiza, Ed., Farhat, Mohamed, Avellan, Francois, and Coussirat, Miguel,

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[13] Knapp, R.T., Daili, J.W., and Hammit, F. G., Cavitation, New York, Mc Graw Hill Book

Company, 1970, 578p.

[14] Terentiev, A., Kirschner, I., and Uhlman, J., The Hydrodynamics of Cavitating Flows,

Backbone Publishing Company, 2011, 598pp.

[15] Peyret, Roger (ed), Handbook of Computational Fluid Mechanics, Academic Press, 25 Mar.

1996, 467 pag.

[16] Turkel, E., Fiterman, A., and B. Van Leer, Preconditioning and the Limit to the Incompressible

Flow Equations, ICASE Report No. 93-42, Langley Research Center, Hampton Virginia, USA

1993.

[17] Chorin, A. J., A Numerical Method for Solving Incompressible Viscous Flow Problems,

Journal of Computational Physics, 2 (1967), pp. 12-26.

[18] Lee, Long, A class of high-resolution algorithms for incompressible flows, Department of

Mathematics, University of Wyoming, Laramie, WY 82071, USA, 2010.

[19] D. Brown, R. Cortez, and M. L. Minion. Accurate projection methods for the incompressible

Navier-Stokes equations, J. Comput. Phys.,

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100 Monitoring, Controlling and Architecture of Cyber Physical Systems