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Inversion of the Critical Back Pressure Relation in Isentropic Nozzle Flow

Joseph Majdalani* and Esam M. Abu-Irshaid† University of Tennessee Space Institute, Tullahoma, TN 37388

We consider the one-dimensional isentropic flow equation that relates the nozzle expansion area ratio to the critical back pressure that must not be exceeded in avoidance of a subsonic throat condition. To eliminate guesswork and numerical root solving in deducing the critical back pressure, we apply asymptotic tools to invert this relation analytically. Our perturbation technique is based on the reciprocal of the nozzle area expansion ratio which, in most applications, does not exceed 0.3. A five-term approximation for the critical back pressure ratio with respect to the total pressure is readily obtained. By extending our series approximation to higher orders, we unravel a recursive formula that permits the efficient calculation of the pressure ratio to arbitrary level of precision. Favorable agreement with the numerical solution at several gas compression ratios is demonstrated as the relative error in a three-term approximation is found to be a mere 0.28 percent for a nozzle area ratio of 0.3 and γ = 1.4. The error slightly decreases as the gas compression ratio is increased. Using a newly derived formula for the exit pressure (Majdalani, J. and Abu-Irshaid, E. M., “General Solutions for Some Isentropic Equations in Variable Area Duct Flow,” AIAA Paper 2005-4382), we evaluate the ratio for optimal expansion. In concert with the present solution, we then present an explicit solution for the ratio of critical back pressure and the optimal exit pressure for a given nozzle expansion area ratio. All solutions are numerically verified.

Nomenclature A = local cross sectional area

tA = nozzle throat area p = normalized pressure

ep = normalized exit pressure at optimal expansion bp = normalized critical back pressure

α , β = exponents in p pα βεξ = − given by Eq. (8) ε = perturbation parameter, /t eA A γ = ratio of specific heats λ = constant related to γ via Eq. (4) σ = exponent in 1/

b bp pγ ελ− = ξ = constant related to γ via Eq. (8) Subscripts and Symbols 0 ,1 = leading and first order e = condition in the exit plane n = asymptotic level t = condition at the nozzle throat

= dimensional property *Jack D. Whitfield Professor of High Speed Flows, Department of Mechanical, Aerospace and Biomedical Engineering. Member AIAA. †Graduate student and Research Associate, Department of Mechanical, Aerospace and Biomedical Engineering. Member AIAA.

41st AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit10 - 13 July 2005, Tucson, Arizona

AIAA 2005-4552

Copyright © 2005 by J. Majdalani. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

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I. Introduction OR isentropic flow through a converging-diverging nozzle with throat area, tA , a nonlinear relation connects the nozzle area expansion ratio, /t eA Aε = , to the critical pressure ratio, crit 0/bp p . This well-known expression

takes the form1

11/ 1 1/1

crit crit

0 0

1 211 1

b b t

e

p p Ap p A

γ γγγ

γ γ

−− − − = + +

(1)

According to one-dimensional nozzle theory, we recall that the critical back pressure critbp is the maximum exit pressure that will still induce choked conditions at the throat. Further reductions in the back pressure can no longer affect conditions upstream of the nozzle; they only lead to further delays in oblique shock formation along the diverging sidewall of the nozzle. Shocks will cease when we approach the optimal pressure ep for a given nozzle area expansion ratio. Conversely, when critbp is exceeded, the establishment of sonic conditions at the throat is mitigated. Here 0p represents the total stagnation pressure and /p vc cγ = is the ratio of specific heats. Equation (1) is applicable to a variety of propulsive applications involving variable area duct and nozzle flow. Due to its transcendental nature, however, it is only amenable to inversion using numerical root solving. For this reason, numerical solutions rendering back pressure data versus nozzle area ratios have so far been collected and presented in graphical or tabular format in many textbooks on the subject.1-3 While the optimal pressure ep has been resolved analytically in a recent study by the authors,4 a closed form solution for the back pressure remains at large. It is hence the purpose of this note to provide the necessary detail leading to a full analytical inversion of Eq. (1). This will be accomplished using asymptotic ‘expansions’ that will enable us to express critbp as a direct function of 0p , γ , and most importantly, /t eA A . The advent of a closed form expression for critbp will also enable us to calculate the pressure tolerance of a given nozzle by directly evaluating the ratio of critbp and the optimal expansion pressure ep .

II. Analysis Before applying perturbation theory, we introduce the dimensionless back pressure crit 0/b bp p p= and the small perturbation parameter /t eA Aε = . At the outset, Eq. (1) collapses into

( )1

11/ 1 1/ 1 211 1b bp p

γγ γ γεγ γ

−− −− = + +

(2)

Note that ε is the actual reciprocal of the nozzle area expansion ratio. We make use of the relative size of ε which, according to Sutton,5 varies customarily between 0.04 and 0.3 although in high altitude nozzle applications, can be as small as 0.0025 . Such low values provide an optimal environment for applying perturbation theory and extracting an explicit series approximation for ( , )b bp p ε γ= . In what follows, we sketch the procedural steps needed to arrive at the desired solution. We also recognize that Eq. (2) exhibits two possible roots, specifically, for subsonic and supersonic exit conditions. Since any exit pressure between the optimal ep and critical critbp will trigger subsonic exit conditions, only the subsonic root will be pursued here. As usual, results will be verified both theoretically and numerically. To start, a regular perturbation approximation is invoked with ε at its epicenter. The dimensionless back pressure can be constructed from a series of diminishing terms, namely, 2 1

0 1 2( , , ) ( )n nb np n p p p p Oε γ ε ε ε ε += + + + + +… (3)

where ( , , )bp nε γ represents the solution at the nth order. Equation (2) may be further simplified into

1/b bp pγ ελ− = ; where

111 2

1 1γγλ

γ γ

− −≡ + +

(4)

Upon substitution of Eq. (3) into Eq. (2), one can rearrange and collect terms of the same order. One gets ( )1 2

0 0 1 1 0 2 1 ( ) 0p p k p p k p Oσ σ λ ε ε− +− + − − + = ; 1/σ γ= (5)

Note that Eq. (5) is truncated at the second order. Using symbolic programming, it can be expanded to any desired order needed to reach a user-defined level of precision. Terms of the same order can thus be separated and solved, one-by-one, for the higher order corrections, 1p , 2p , … , np . The total series solution is hence constructed and presented as

F

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( ) ( )1 3 2

21 11 11 2 1 2 1bpγγ γ

γ γγ γγ γ ε γ γ ε−

− −− −= − + − +

( ) ( )( )( )7 3

11 2 1 4

3 4 511 1 13 8 1 2 2 1 2 1 ( )O

γγγ γ

γ γ γγ γ ε γ γ γ γ ε ε−−+

− − −− + − + + + + (6) One can continue to higher orders until a repeatable trend is uncovered. The solution for the dimensionless critical back pressure may hence be expressed as a recurrence formula that provides the solution to the nth order. This useful equation is given by

( )

( ) ( )( )

1

1

0

11

( , , ) ( )1 ! 1

m

nm n

b m mm

im i i

p n Om

σε γ λε ε

σ

−

+

=

=− + −

= +− −

∏∑ (7)

This asymptotic solution may be compared to the numerical solution of Eq. (2) and displayed graphically. These are shown in Fig. 1 at 1.2γ = and 1.4 , respectively. Clearly, the deviation between asymptotics and numerics is quite acceptable over a significant range of area ratios, especially when 4n ≥ . In Fig. 1a, the error in the fourth-order solution (0.5,1.2, 4)bp is a mere 0.5% for an area ratio of 0.5ε = ; this increases to 7.1% for 0.7ε = . Depending on the required precision, more or fewer terms could be retained in Eq. (7). For example, using three terms only, one only incurs an error of 0.28 percent at a relatively large area expansion ratio, 0.3ε = . As stated earlier, since most nozzles have 0.3ε ≤ two or three terms in Eq. (6) will be sufficient for most applications. This is corroborated by Fig. 2 in which the relative error in bp is characterized over the full range of γ and discrete nozzle area ratios. The relative errors show very slight improvement as γ is increased irrespective of whether three or five-term approximations are used. Also note that the error in the three-term approximation at 0.3ε = varies between 2.4 and 3.4% over the full range of γ . The resulting precision may be adequate in many practical applications. In a companion paper,4 an asymptotic expression is found relating the optimal exit pressure (for optimal expansion) to the total pressure, 0p . This is

( )

( ) ( )

1

2 2 10

0

11

( )1 !

m

nm m n

m mme

im i i

pO

p m

α βξ ε ε

β α

−

+

=

=− + −

= +− −

∏∑ ; ( )( )

11 1 1 12 2 21

γγξ γ γ+−≡ − + ; 2 / , 1 1/α γ β γ= − = − − (8)

In view of Eq. (7), one may multiply both recursive formulas to obtain a direct relation between the critical back pressure and its (lower) optimal value. This can be obtained at even orders of ε from the composite product

( )

( ) ( )( )

( )

( ) ( )

1 1

22 2 1crit crit 0

0 00

1 11 1

( )1 ! 1 1 !

m m

n nm m m nb b

be m m m mm me e

i im i i m i i

p p pp O

p p p m m

σ α βλε ξ ε ε

σ β α

− −

+

= =

= =− + − − + −

= = = +− − − −

∏ ∏∑ ∑ (9)

The resulting ratio obtained asymptotically is plotted in Fig. 3 for two values of γ while being compared to the numerical solution.

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.70.5

0.6

0.7

0.8

0.9

1.0

1.1pb

a)

numeric 1 term asymptotic 2 3 4 5

At /A

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.70.5

0.6

0.7

0.8

0.9

1.0

1.1pb

b) At /A

Figure 1. Comparison between numerical and asymptotic solutions of the critical back pressure relation at increasing orders. Results are shown for a) γ = 1.2 and b) γ = 1.4.

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III. Conclusions By inverting the critical back pressure versus nozzle area expansion ratio, we have provided simple and direct analytical means to calculate this important nozzle property. Our asymptotic solutions are extended to arbitrary order to the extent of becoming applicable to very large exit area ratios. By combining the present solution with that for the optimal exit pressure, we have constructed a composite expansion for the ratio of critical and optimal back pressures. Our formulas increase our repertoire of isentropic flow expressions that are widely used in both industry and academe. Their purpose is to circumvent the need to consult with tabulated material and to supplant the writing of individual codes while discretely solving for these properties. As shown here, the presentation of explicit formulas can be combined with those obtained in other studies to help in deriving yet newer expressions in related applications.

Acknowledgments This work was sponsored by the National Science Foundation through Grant No. CMS-0353518, Program

Manager, Dr. Masayoshi Tomizuka.

References 1Anderson, J. D., Modern Compressible Flow with Historical Perspective, 3rd ed., McGraw-Hill, New York, 2003, p. 204. 2Moran, M. J., and Shapiro, H. N., Fundamentals of Engineering Thermodynamics, 5th ed., John Wiley, New York, 2004. 3Çengel, Y. A., and Boles, M. A., Thermodynamics: An Engineering Approach, 4th ed., McGraw-Hill, New York, NY, 2002. 4Majdalani, J., and Abu-Irshaid, E. M., “General Solutions for Some Isentropic Equations in Variable Area Duct Flow,” AIAA Paper 2005-4382, July 2005. 5Sutton, G. P., Rocket Propulsion Elements, 6th ed., John Wiley, New York, 1992.

1.1 1.2 1.3 1.4 1.5 1.6 1.7

10-4

10-2

100

0.70.5

0.3

γ

ε = 0.1

a)1.1 1.2 1.3 1.4 1.5 1.6 1.7

10-4

10-2

100

0.7

0.5

0.3

γb)

Figure 2. Relative error in the critical back pressure formulation using a) three and b) five term approximations.

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.70.5

0.6

0.7

0.8

0.9

1.0

numeric asymptotic ~ ε5

pbe

At /A

γ = 1.4

a)0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

0.5

0.6

0.7

0.8

0.9

1.0

b)

γ 1.2 1.4 5/3

At /A

pbe

Figure 3. Ratio of critical back pressure and optimal exit pressure shown in a) for γ = 1.4 alongside the numerical solution and in b) at three representative values of γ. Here we use the fourth-order solution.