development and testing of paraffin-based hybrid...

(c)2001 American Institute of Aeronautics & Astronautics or Published with Permission of Author(s) and/or Author(s)' Sponsoring Organization.

VI MA A _ A01 -34608

AIAA 2001-4503DEVELOPMENT AND TESTING OFPARAFFIN-BASED HYBRID ROCKETFUELS

M. A. Karabeyoglu, B. J. Cantwell, D. AltmanDepartment of Aeronautics and AstronauticsStanford UniversityStanford, California 94305

37th AIAA/ASME/SAE/ASEEJoint Propulsion Conference and Exhibit

July 8-11,2001/Salt Lake City, UtahFor permission to copy or to republish, contact the copyright owner named on the first page.

For AIAA-held copyright, write to AIAA Permissions Department,1801 Alexander Bell Drive, Suite 500, Reston, VA, 20191-4344.

(c)2001 American Institute of Aeronautics & Astronautics or Published with Permission of Author(s) and/or Author(s)' Sponsoring Organization.AIAA 2001-4503

DEVELOPMENT AND TESTING OF PARAFFIN-BASED HYBRID ROCKET FUELS

M. A. Karabeyoglu*, B. J. Cantwellf, D. Altaian*Department of Aeronautics and Astronautics

Stanford UniversityStanford, California 94305

Abstract

The classical hybrid combustion theory is generalized to solid fuels that form a liquid layer on their burning surface.For several classes of liquefying fuels, the layer is hydrodynamically unstable leading to substantial dropletentrainment from the melt layer into the gas stream. The susceptibility of a given fuel to this shear driven instabilityincreases with decreasing viscosity and surface tension of the melt layer. The entrainment mass transfer, which actsin addition to the conventional gasification mechanism, is not affected by the blocking phenomenon induced byblowing from the surface. For practical oxidizer flux levels encountered in hybrid rocket applications, dropletentrainment can dominate direct gasification. Such liquefying fuels can exhibit greatly increased surface regressionrates compared to classical materials such as HTPB. One application of the theory is to solid cryogenic hybrids,which utilize frozen materials for the solid propellant. The theory successfully predicts why high regression rates areobserved in tests of cryogenic solid pentane, solid pentane and solid oxygen. In addition, the theory explains thedependence of the burning rates of other tested cryogenic materials on the physical properties of the liquid layer.The theory also leads to the conclusion that certain non-cryogenic materials such as paraffin and PE waxes will alsoexhibit high regression rates. This important result is confirmed by lab scale tests performed at Stanford Universityon a paraffin-based fuel.

1) Nomenclature

a: Absorption coefficient, m'1aent: Entrainment coefficientB, Bg: Blowing parameter and evaporation blowing

parameterb: Regression rate parameterC: Specific heat, kJ/kg-secCm, CB2' Blowing correction coefficientsCH, CHo: Stanton number with and without blowing

Amplification parameterSkin friction coefficientHeat transfer correction factor for surfaceroughnessPort diameterFroude numberViscous solution for the gas phase Orr-Sommerfeld equationInstantaneous mass flux, time averaged massflux, kg/m2-secBody force per unit mass

h:

hm,he:A/i:

i :I:k:

c:

F:

p

Fr:

G , G :

m,:

0/F:

P0:

Melt layer thickness, mmEffective heat of gasification, kJ/kgEffective heats, kJ/kgEnthalpy difference between the flame andthe surface, kJUnitary complex numberGas phase velocity profile integralWave number, cm"1

Latent heat of melting and vaporization,kJ/kgEntrainment component of mass flux fromfuel surface, kg/m2-secMass flow rate of liquid flowing throughmelt layer per unit width, kg/m-secOxidizer to fuel ratioDynamic pressure in the port, Pa

Nondimensional gas phase normal stressperturbationRadiative and convective heat flux at thesurface, kJ/m2-sec

* Post-doctoral fellow, Member AIAAf Professor, Member AIAA* Consulting Professor, Fellow AIAA, President, Space Propulsion Group Inc.


AIAA 2001-4503

Qw: Total heat flux at the surface, kJ/m2-secR: Ratio of thermal to radiative thicknessRe: Liquid layer Reynolds numberRe _ : Gas phase Reynolds number

6

Rhv, Rhe: Ratio of effective heat of gasifications forentrainment and vaporization

r, r: Instantaneous and time-averaged regressionrate, mm/sec

rcl: Regression rate predicted by classicaltheory, mm/sec

Ta: Initial fuel temperature, KTg : Average gas phase temperature, KTm,Tv: Melting and vaporization temperatures, KU0 (y): Mean axial velocity of the liquid, m/secUl, U°: Mean axial velocity of the liquid at the

interface with and without blowing, m/secUg (f): Mean velocity of the gas flow, m/secU°: Gas velocity at the center of the port, m/secV,: Mean normal velocity of the liquid, m/secWe: Weber numberXe: Entrainment parameter, N1/2

y: Distance from the interface, mz: Axial distance along the port, ma: Nondimensional wave numbera, /?: Dynamic pressure and thickness exponentsA: Gas phase stress perturbation parameter8m : Liquid layer momentum thickness0(>0: y-component of nondimensional liquid

stream function7: Viscosity exponent77: Surface waveformK : Thermal diffusivity, mVsecILL : Viscosity, milliPa-secn: Surface tension exponentp: Density, kg/m3

<3: Surface tension, milliN/mtg : Mean shear stress exerted by the gas flow on

the liquid-gas interfaceT g : Nondimensional gas phase shear stress

perturbation\l/: Nondimensional thickness parameter\lfg (f): Inviscid solution for the gas phase Orr-

Sommerfeld equation£: Normal curvilinear coordinate in the gas

phaseSubscripts:ent: Entrainmentg: Gas/: Liquido: Oxidizer

s: Solidv: VaporizationSuperscripts:•: Nondimensional quantity

2) Introduction

Despite their safety and cost advantages oversolid and liquid systems, conventional hybrid rocketspossess one very significant shortcoming, very low fuelregression rates. For many practical applications, thisleads to a complex, multi-port grain design such as thecommonly used wagon wheel configuration. The multi-port design presents serious shortcomings, whichsignificantly degrade the overall performance,reliability and cost effectiveness of a hybrid propulsionsystem. Some of the disadvantages of a multi-portconfiguration can be listed as:

• Large sliver fractions, which may in practice leavelarge quantities of fuel unburned.

• Fairly low volumetric loading of the fuel in thecasing leading to low thrust densities.

• Fuel grain integrity problems, especially towardsthe end of the burn when the web thicknessbetween ports becomes vulnerable to structuraldisintegration. A costly solution for this problem isto use web supports. This method, however, willcompromise the system performance and will addweight and complexity.

• Difficult and expensive manufacturing of the fuelgrain

• Requirement for a pre-combustion chamber ormultiple injectors.

• Potentially non-uniform burning from port to port.• Increased risk of instabilities related to multi-port

dynamic interactions or due to the existence of alarge pre-combustion chamber.

Fundamentally, the limit on regression rate forconventional hybrid fuels is set by the physicalphenomena of heat and mass transfer from therelatively remote flame zone to the fuel surface1. Heattransfer to the fuel surface is further reduced by thewell-known "Blocking Effect" which is induced by theradial blowing of a large quantity of gas from the fuelsurface. As a consequence, the regression rates ofmodern hybrids that utilize polymers as the fuel aremuch lower than conventional solid rocket burningrates.

So far many techniques have been suggestedor tried to increase the regression rates of hybrids.Unfortunately, all of these methods suffer importantshortcomings. Some of these techniques are listed here.


AIAA 2001-4503

• Use of fuel with low effective heat of gasification.This method yields only a modest improvementsince, as revealed in the classical hybrid theory1'24,the exponent of the heat of gasification is a smallnumber (approximately 0.32). The weakdependency of the regression rate on the heat ofgasification is due to the "Blocking Effect"described earlier.

• Insertion of screens or other mechanical devices inthe port to increase the turbulence level and thusthe heat transfer rates. This method complicates thedesign significantly and increases the likelihood offailure. In addition, this approach may lead tononuniform burning along the port.

• Add swirl to the incoming oxidizer flow to increasethe effective mass flux and thus improve the heattransfer rate. This method also complicates thehybrid design, especially for large scale motors,and requires heavy injectors or vanes.

• Addition of oxidizing agents or self decomposingmaterials in the hybrid fuel. This well knowntechnique reverts to a quasi-solid design andeliminating the inherent safety characteristic ofhybrid rockets. In addition, the regression ratebecomes sensitive to chamber pressure.

• Addition of metal additives. This is also a commontechnique that improves the fuel mass burning rate.The improvement is small and there are severalshortcomings such as the increased vulnerability toinstabilities due to the pressure dependentregression rate and increased environmentalimpact. This is a separate consideration from theaddition of metal additives to increase fuel density,which is a useful improvement.

• Increasing the roughness of the burning surface byadding dispersed solid particles in the fuel thatwould burn at a different rate compared to thematrix material. This technique can only give alimited improvement and large solid particlesinjected in the gas stream might reduce theefficiency of the system. Fuel grain manufacturingcosts would also increase.

We believe that a natural and effectivemethodology to increase the regression rate is tocarefully formulate a hybrid fuel that will generate masstransfer by mechanical means in addition to the masstransfer by direct gasification from the fuel surface. Formaterials forming a low viscosity melt layer on theirburning surfaces, the mechanical mass transfer will takeplace by the entrainment of liquid droplets into the gasstream. An obvious class of entraining substancesincludes liquids or gases at standard conditions, whichare frozen to form solids (i.e. solid cryogenic hybrids).Typical members of this class range from H2, a deepcrypgen, to liquid amines and hydrocarbons. However

it is clear that the same internal ballistic behavior can beexperienced by materials that are solids at standardconditions if they form a low viscosity melt layer at thecombustion surface.

In fact, increased regression rates by severalhundred percent have been observed with solidcryogenic hybrids by researchers at the Air ForceResearch Laboratory (AFRL)2'3'4 and ORBITEC5'6. TheAFRL program was concentrated on burning severalfrozen organic liquids including normal pentane withgaseous oxygen in a small test motor. Fig. 1 shows theburning rate behavior of pentane and several otherhydrocarbons tested by AFRL in the space-timeaveraged regression rate, r , time averaged oxidizermass flux G0, plane. Note that for the presented set ofdata, the averaged oxidizer mass flux is defined basedon the average port radius. The regression rate law for atypical conventional hybrid fuel, HTPB, is alsoincluded in Fig. 1 for comparison purposes. A typicalregression rate expression for an HTPB-O2 hybrid7,r = 0.025G^65 mm/sec where G0 is in kg/m2-sec (grainlength = 15 cm), is used in the calculations. As shownin Fig. 1, the measured regression rates for cryogenicpentane are 3-4 times larger than the reportedregression rates for the conventional fuel, HTPB, underthe same oxidizer mass flux condition.

Similarly, ORBITEC conducted tests usingsolidified kerosene and methane with gaseous oxygenand solidified oxygen with gaseous methane (i.e. areverse hybrid configuration) with small AFRL scalemotors. In the process of developing a productiontechnique for frozen methane fuel grains, ORBITECalso performed a limited number of tests with a lowmolecular weight paraffin wax. The oxidizer massfluxes corresponding to the measured regression ratesfor the paraffin tests were not reported. Similar to theAFRL results, solid kerosene and methane showed veryhigh burning rates relative to the polymeric classicalhybrids.

We may conclude from these variousexperiments that the regression rates for cryogenicpropellants are many times higher than for conventionalhybrids. This observation cannot be explained by alower heat of vaporization, which is insufficient toaccount for the observed regression rate increase.Recall that in the classical regression rate expression1'24,the heat of vaporization is incorporated in the blowingparameter, B, raised to a power 0.32. We haveestimated a maximum B value of 13.5 for the pentane-O2 system using the expressions given in Ref. 1, whichwould only cause a minor increase in the regression rate(30-50 %) over the conventional systems rather than the


AIAA 2001-4503

observed increases of 300-400%. A differentmechanism is clearly indicated.

The regression rate model developed for theseliquefying propellants, in addition to the classicalgasification, is based on a mass transfer mechanisminvolving the entrainment of liquid droplets from themelt layer. As will be demonstrated, droplet formationis due to liquid layer instabilities, which result from thehigh velocity gas flow in the port. The purpose of thispaper is to explore the development of the liquid layerat the burning surface and to determine the conditionsfor the onset of droplet entrainment so that itsoccurrence can either be invoked or avoided. Aschematic of the presumed model for entrainment isshown in Fig. 2.

3) Development of the Liquid Layer HybridCombustion Theory

In this section we will describe thedevelopment of a theory that can be used to explain thephenomena observed in cryogenic hybrids and that willextend the applicability of the hybrid combustiontheory to fuels that may form a reasonably thick liquidlayer during their combustion. Due to the complexity ofthe problem, we will perform the development in threestages. In the first stage, we will investigate therequirements for the formation of a melt layer on thefuel grain. In the second stage, we will consider thelinear stability of a melt layer under the strong shear ofa gas flow. The linear stability model discussed in thefollowing paragraphs includes the effect of the injectionof liquid at the liquid-solid interface due to theregression of the fuel slab. Later in this stage the linearstability results will be linked to the entrainment ofliquid droplets with use of some experimental resultsand some semi-empirical relations developed in thenuclear engineering literature. Eventually in the finalstage, classical theory will be extended to the case ofliquid droplet entrainment. It is shown that the primaryeffect of the entrainment mass transfer is to increase theregression rate of the fuel without materially increasingthe thermochemically defined blowing parameter, B.Note that the thermochemical blowing parameter canonly be altered, to some degree, by changing thethermochemical properties of the propellants or thecombustion parameters such as the local flame O/Fratio.

A) Liquid Layer Thickness Estimation

In this section the thermal analysis to estimatethe film thickness formed on a burning slab under thecombined heating by convection and radiation issummarized. The thickness of the liquid layer can bedetermined by the energy transfer relations in the solid

and liquid phases. The schematic of the thermal modelfor the slab configuration is shown in Fig. 3. For thepurposes of this paper, we are solely interested in thesteady-state regression of the fuel slab. Namely, thevelocity of the liquid-gas interface and the solid-liquidinterface are assumed to be equal and constant. This, ofcourse, implies that the thickness of the melt layer isalso constant. For the sake of simplicity we furtherassume that the thermophysical properties of thematerial both in the liquid phase and also in the solidphase are uniform. The effect of convection in theliquid layer is also ignored because of the small meltlayer thicknesses for which the Reynolds numbers arerelatively small (a couple of hundreds) and thetemperature gradients are fairly large.

In our analysis we have included thepossibility of the penetration of the thermal radiationinto the fuel slab. Several simplifying assumptions areintroduced in the radiation treatment. First, the radiativeflux field is assumed to be one-dimensional. Since thetemperature levels in the slab are small, the contributionof radiation emitted by internal material to the radiativeintensity is negligible. The absorbing character of boththe liquid and also the solid material are assumed tobehave like a gray body, namely the absorptioncoefficient is independent of the frequency of theradiation. Note that the absorption characteristics fortypical hydrocarbons used in cryogenic hybridapplications, such as pentane, are reasonably flat in themedium IR range of the spectrum8'9. Even if this is notthe case, the use of a constant absorption coefficientobtained by averaging over the band of the spectrum ofthe incoming radiation can be justified11. Furthermorethe non-collimated effects of the incoming radiation tothe fuel slab surface are neglected.

Under these simplifying assumptions thethermal analysis yields the following expression for thethickness of the melt layer10.

(la)

where the characteristic thermal thickness in the liquidphase is defined as 8l = Klps/rpl. The thicknessparameter, y, which depends mainly on thethermophysical properties and radiative characteristicsof the fuel and the nature of the heat transfer to the fuelsurface can be found as a solution of the followingnonlinear equation.


AIAA 2001-4503

Here the following definitions of the effectiveheating parameters are introduced for convenience.

hm=Lm+Cs(Tm-Ta) (2a)v-Tm) (2b)

(2c)

The form of effective heat of gasification, hv , given inEq. 2c is slightly different than the expressionscommonly encountered in the literature, since no heat isrequired to vaporize the material transported by meansof entrainment .

Note that Ta, Tm, Tv are the initial, meltingand vaporization temperatures of the fuel, respectively.<2r and Qw are the radiative and total heat fluxes to thefuel surface, Q and Cs are the average specific heats ofthe liquid and solid and Lm and Lv are the latent heatsfor melting and vaporization. Also note that r and rv

are the total and vaporization components of theregression rate. Another important parameter thatappears in the thickness expression is thenondimensional radiation parameter, Rt, which isdefined as the ratio of the thermal thickness to theradiative thickness in the liquid phase.

Ri = 8iai (3)

Note that a{ is the average gray body absorptioncoefficient in the liquid phase.

An interesting observation is that, solidabsorptivity does not appear in Eqs. la or Ib and themelt layer thickness is independent of the radiativecharacteristics of the solid material. This can bededuced from the fact that the only contribution of thesolid-state temperature distribution on the thickness isthrough the heat transfer from the solid at the liquid-solid interface. It can be shown that the solid thermalprofile in the neighborhood of the interface and thus theheat flux at the interface (from the solid side) isindependent of the distribution of the radiative heatingin the solid.

An explicit solution for the algebraic nonlinearequation, Eq. Ib, for the general case could not beobtained. We focus on the following two limiting casesof practical interest.

1) R{ » 1: The absorption of the radiation in the liquidlayer is very large. In this extreme case of an opaqueliquid layer, all the radiative heat is absorbed at the

liquid-gas interface. The thickness can be solvedexplicitly as

(4)

Note that all the thermophysical properties of the fuelmaterial are lumped in the logarithmic term and theQr/Qc ratio§ does not affect the thickness. This case isimportant for propellants that are loaded with stronglyabsorbing materials such as carbon black.2)/?j «1: In this other extreme, the absorption of theradiation in the liquid phase is small. Here the thicknessof the thermal layer in the liquid is much smaller thanthe radiative thickness in the liquid and as aconsequence all the radiative flux is absorbed in thesolid. The effect of the entrainment mass transfer on thethickness is also included in this formulation. Unlikethe other extreme, in this case the film thicknessdepends on the ratio of the radiative heat flux to theconvective heat flux and it can be expressed as

h = 5 In 1 +hm-hv(Qr/Qw)

(5)

Note that the dependence of the thickness onrv/r comes from the effective heat of gasification, hv

that appears explicitly in this formula and is defined inEq. 2c. Note that an important common property of theregression rate expressions Eq. 4 and Eq. 5 is that themelt layer thickness is proportional to the characteristicthermal length of the liquid, <5/5 and thus inverselyproportional to the regression rate (h^\/r\

As indicated by Eq. Ib, some knowledge ofthe order of magnitude of the liquid absoptivityparameter is essential for a reliable estimation of themelt layer thickness. We have calculated the averageabsorptivity of n-pentane using the published NearInfrared Region (NIR) and Middle Infrared Region(MIR) absorption spectra for this material8'9. In thecalculations, we have assumed a continuous spectraldistribution (according to Planck's Law) for theintensity of the impinging radiation from thecombustion gases at an effective radiation temperatureof 1800 K11. At this temperature the radiation peaktakes place at a wave number of 6211 cm'1 and most ofthe heat transfer is in the Near Infrared Region of thespectrum. The estimated average absorption coefficient

Note that Qr=

Qw l + Qwhere Qc is the

convective heat flux to the fuel surface.


AIAA 2001-4503

for pentane is ^=1.62 mm~l which results infy = 0.13/r (i.e. r is in mm/sec). For pure pentane atregression rate values larger than 1 mm/sec, which aretypical for fast burning liquefying propellants, theliquid absoptivity can be ignored. Thus for all pentanemelt layer thickness calculations presented in thispaper, the approximate relation, Eq. 5, is used insteadof the exact nonlinear relation, Eq. Ib.

One of the important parameters that affect thethickness of the melt layer is the ambient temperatureof the fuel slab. Fig. 4 shows the plots of the melt layerthickness versus the regression rate for various ambienttemperatures for a pentane fuel slab for Qr/Qc =0.1and rv/r = 0.5. The material properties used in thecalculations are given in Table 1. Note that an increasein the ambient temperature of the fuel slab results in anincrease in the melt layer thickness. The maximum filmthickness for a given regression rate is attained whenthe ambient temperature is equal to the meltingtemperature.

A plot of the variation of the film thicknesswith the relative radiative heat transfer, Qr/Qc , for apentane system is shown in Fig. 5. The figure revealsthe interesting phenomenon that at a critical value ofthe radiation factor (around 0.3 for pentane with 10 %entrainment), the steady-state film thickness growsindefinitely. This fact can be seen from Eq. 5, namely atthe critical value of this factor the numerator of theargument of the logarithm diminishes. In physicalterms, during this critical operation, the energy balanceat the liquid-solid interface requires zero conductiveheat transfer from the liquid side. This conditionimposes a zero temperature gradient on the liquid sideof the interface, which can only be achieved with aninfinite thickness of the liquid layer. However thegrowth to large thicknesses can only be achieved in along transient period, which cannot be estimated withour steady-state model. Fig. 5 also shows that the effectof entrainment mass transfer is to reduce the thicknessat a given radiation factor and to increase the criticalvalue of the radiation factor. This effect is generatedfrom the entrainment regression rate dependence of theeffective heat of gasification (Eq. 2c).

B) Linear Stability Investigation

The instability of the liquid film formed on thesurface of a hybrid fuel grain is essential for thepossibility of entrainment from the liquid interface.Although the stability of liquid layers are extensivelystudied in the past12'13'14'15, the behavior of films understrong blowing conditions and relatively high liquid

Reynolds numbers, which are encountered in hybrids, isnot explored. In this section, we will summarize thelinear stability investigation of a liquid layer withstrong blowing under the effect of the strong shearforce generated by the gas flow in the port. Details ofthe stability investigation are presented in Ref. 10.

The schematic of the liquid layer stabilitymodel is shown in Fig. 6. For simplicity, we ignore theeffect of density variations and all chemical reactions inthe gas phase on the stability. In the calculations wemodify and adapt the linearized gas phase equationsderived by Benjamin16 for the incompressible flow overa wavy wall. Since our predictions for the Reynoldsnumbers for the hybrid liquid films are typically on theorder of a couple of hundreds, both the small Reynoldsnumber and also the high Reynolds numberapproximations12'13 as the solution techniques for thefilm stability equation introduced in the literature arenot satisfactory. For that reason we develop newsolution techniques for the liquid layer aspect of theproblem.

The body force, g, is taken in the normaldirection in order to make direct comparison with theprevious stability work reported in the literature13. Notethat, for a spin stabilized rocket, the centrifugalacceleration will generate a body force in the normaldirection acting outwards as indicated in Fig. 6. Theeffect of axial acceleration of a missile on the stabilityis not addressed in this paper. Since the melt layerthicknesses are very small for typical hybrid fuels, it isfair to state that for moderate accelerations, the effect ofthe axial body force on the stability is small comparedto the dominant effect of the shear forces generated bythe gas flow in the port.

As stated earlier, in our analysis we ignore thevariations of the fluid properties across the thickness ofthe layer. We believe that the effect of these variationsis small on the stability behavior of the film. We furtherassume that liquid flow in the melt layer is laminar innature. This is a reasonable approximation for mosthybrid melt layers, which are typically characterizedwith Reynolds numbers less than 300.

For a burning fuel slab, the base flow in themelt layer will be steady state only with respect to acoordinate frame moving with the burning surface. Inthis coordinate frame there will be a constant non-zeronormal velocity component in the liquid layer which isinduced by the density difference between the liquidand solid phases. The mass balance requires that thisvertical velocity is

Vi = rps/Pl. (6)


AIAA 2001-4503

For the described flow field, the velocitycomponent of the base flow parallel to the burningsurface can be written as

h (7)

where the velocity at the liquid-gas interface, Uf andthe liquid phase blowing parameter (or regression rateparameter), b are define as

U? = rgh/jLil and b-VihPi = P* h

Vl Pi 8m '

Note that the momentum thickness for theliquid layer is defined as 8m = ju //p /r. Here rg is themean shear stress exerted on the liquid surface by thegas and fa is the liquid viscosity. For typical operatingconditions of hybrids the liquid blowing parameter fallsin the range of 0.1 -1.0.

In the limit of zero liquid blowing (r = 0), theapplication of the L*Hospital's rule yields the followinglinear velocity profile of the standard Couette flow asexpected.

(8)

According to Eq. 7 the liquid velocity at thesurface is

(9)

In Fig. 7 the plot of the velocity profilesaccording to Eq.7 and Eq. 8 are shown for a typicalblowing parameter value of 0.4. It is apparent that theexact velocity profile is quite different from the simpleCouette flow profile. However it is observed that forvalues of b smaller than 0.8, the exact solution is fairlyclose to a linear variation from the zero wall value tothe maximum velocity predicted by Eq. 9. Note that thefirst order effect of the injection is to reduce the surfacevelocity of the liquid. For that reason the followingcorrected linear profile is a good approximation forsmall blowing parameters (i.e. See Fig. 7)

eb -\ ybeb h

(10)

The corrected linear velocity profile given by Eq. 10will be used in the linear stability investigations as anapproximation for the base flow. This simplification isessential for the analytic development of an explicitstability expression for the film layer.

After perturbing the liquid flow field aroundthe base flow presented in the previous paragraphs andassuming a general harmonic disturbance for the liquid-gas interface waveform, we obtain the followingnondimensional Orr-Sommerfeld equation for the y-component of the perturbation stream function, 0.

IV - 2a2<t>" + a4(t> - b((t>'" - a V) =

(11)

Here a is the nondimensional wave number and c isthe nondimenional amplification parameter and Re isthe liquid Reynolds number. Note that in Eq. 1.1 wehave used the approximate mean velocity expressiongiven by Eq. 10, which simplifies to U0-y in thenondimensional form. The extra term multiplied by bis due to the vertical motion of the liquid.

Next we consider the boundary conditions forthe Orr-Sommerfeld equation. The first two of theboundary conditions can be obtained from the velocityrequirements at the solid-liquid interface. The physicalstatement of the requirements are 1) The parallelcomponent of the velocity at solid wall must be zerodue to the no slip condition. 2) The regression rate ofthe slab is steady. These can be written in terms of thestream function formulation as

0(0) = 0 and (12)

Another requirement that needs to be satisfiedby the solution is the kinematic boundary condition atthe liquid-gas interface. In the linear form this can bewritten as

(13)

There are two other constraints that limit thepossibility of solutions, which come from the dynamicconditions at the liquid-gas interface. These are theshear and the normal force balances at the liquidsurface. The shear force balance in the linear formdictates

(14)

In this relation fg is the nondimensional gas phaseshear stress perturbation on the liquid interface. Notethat the gas shear perturbation should be determined bysolving the gas phase perturbation equations, which willbe discussed later. For present purposes we consider rg

as a known input for the liquid layer stability problem.


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yieldsFinally the normal force balance at the surface ^(y) = — J sinh[a(y - j)]e

- c)f (1) - "(1) - W(l) -

Fr 7] 1-c(15)

Here /^ is the nondimensional gas phasepressure perturbation on the liquid surface, We is theWeber number and Fr is the Froude number where <Jis the surface tension.

and Fr=gh (16)

Eqs. 12, 13, 14 and 15 set five conditions onthe fourth order differential equation, Eq. 11. Thus thisis an overposed boundary value problem. The correctapproach is to consider the Orr-Sommerfeld equationwith those five conditions as an eigenvalue problem,the eigenvalue being the amplification parameter, c.The direct numerical solution of this eigenvalueproblem is computationally expensive, since it requiresmany iterations (each iteration is a finite differencesolution itself) for every selection of the parameterssuch as the Reynolds number, Froude number or theWeber number that might effect the stability of thelayer. For that reason it is desirable to developanalytical solutions.

We have obtained solutions for the presentedstability problem by using two independent techniques.We first developed a power series solution that can onlybe applied to films with Reynolds numbers less thanunity. For the power series solution, we modifiedCraik's technique13, which was originally applied to asimpler case of a layer with no blowing. Since in hybridapplications the film layer Reynolds numbers can besignificantly larger than unity, we also formulated anexact solution for the liquid stability problem definedabove.For the purposes of this paper, we will only present theexact solution developed in great detail in Ref. 10.

(17)n=l

The four independent particular solutions of the Orr-Sommerfeld equation are

^=e^ (18a)#2=*"°* (18b)

1 V;inh[a^-^L-(^

y.

y

a Jy.

+

(18c)

(18d)where

a - c] ~ f /, . 1'- and B = -ib /(aRe)1/ V '(aRe)

These boundary conditions with the analyticalexpressions for the independent solutions can now beused to solve the eigenvalue problem for the eigenvaluec. It is important to note that the selection of y0 - 0significantly simplifies the solution since the first twoof the boundary conditions require q = c2 = 0 and dropout of the system. In this case the system is reduced tothree equations for three unknowns, c3 , c4 , c. Note thateven the reduced system involves difficult integrals andit is nonlinear in c. Thus the solution requiresnumerical integration and some iterative procedure onc . In order to perform the calculations, we developed aMATHEMATICA program that automates the solutionprocedure and iteratively solves the eigenvalue c, forany given selection of parameters. The time requiredfor the solution at a specified point of the parameterspace is less than a second with a modern desktopcomputer. Thus the analytic solution developed for thefilm stability problem is apparently very efficient interms of the computation time over the numericalsimulation of the Orr-Somerfeld equation or the Navier-Stokes equations.

The linear stability problem discussed in theprevious paragraphs requires the gas phase response asthe input. Here we will describe the gas phaseperturbation treatment that was first developed byBenjamin16 and later adapted to the thin film stabilityproblem by Craik13. Benjamin investigated the flowfield generated by a shearing gas flow on a wavysurface and presented explicit formulas for the shearand normal stresses disturbances induced by the shearflow on the surface. The following fundamentalsimplifications were introduced in the treatment.• The primary gas flow is parallel and in the

direction of the surface waves.• In the case of a turbulent gas flow, the interactions

with the turbulence and the wave motion areneglected. In other words, the mean velocitydistribution for the turbulent case is disturbed by


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the waves in the same fashion as the velocitydistribution in an equivalent laminar flow.

• The complete solution for the Orr-Sommerfeldequation derived for the gas phase is expressed as asum of the inviscid solution, V^g(C) and a viscoussolution confined to a small thickness next to thesurface, /g(C)» where £ is the curvilinearcoordinate that is perpendicular to the wavysurface. The viscous solution, which rapidlydiminishes as one moves away from the surface, isrequired to satisfy the wall boundary conditionsimposed on the fourth order Orr-Sommerfeldequation. The region where fg(£) is significantcompared to the inviscid solution is named as the"wall friction layer", not to be confused with theviscous sublayer of a turbulent boundary layer.

The explicit solution for the shear stress andpressure perturbations presented by Benjamin is revisedand reformulated by Craik in the context of the liquidlayer stability problem formulated earlier in this paper.We have discovered that the simplifying assumption,A «1, that was used by both Benjamin and Craiksignificantly restricts the applicability of the gas phasestress perturbation expressions given in Refs. 13 and16. This assumption on A is particularly invalid forhybrid rocket applications, typically characterized byhigh gas velocities and small disturbance wave lengths.We followed Benjamin's original derivation to refinethe set of gas phase stress perturbation formulas byrelaxing the restricting assumption on A. We foundthat the nondimensional normal and the tangentialstress perturbations Pg and fg which are valid for anyvalue of A can be approximated by the followingequations.

77 1 + 1.288 e'W6A Cf(19a)

A =(19b)(19c)

Here Ug is the mean boundary layer velocity profile ofthe gas flow, U° is the gas velocity at the center of theport and \JL , p are the average viscosity and the

density of the gas in the port, respectively. The realparts of stress parameters correspond to the componentsof stress that are in phase with the wave displacement,whereas the imaginary parts of the stress parametersrepresent the stress components, which are in phasewith the wave slope.

For the sake of simplicity, the following formof the gas phase velocity profile will be used inevaluating the integral, /.

for £<dp/2h (20a)

(20b)

Note that for hybrid rocket applications, the boundariesof the integral in Eq. 19d need to be from 0 to dp/h.

Finally, it can be shown that with use of theshear balance at the liquid gas interface, the liquid layerReynolds number can be written in terms of the gasflow parameters and the properties of the liquid.

Re = Re^c f^-' P,

2eb-lbeb (21)

Here Reg is the gas phase Reynolds number,z is the axial distance in the port, cf is the skin frictioncoefficient.

In the following paragraphs we present theresults for the liquid layer stability problem that areobtained with the application of the solution techniquesdiscussed in the previous section. For all of thecalculations that will be conducted in this paper, thefollowing geometrical and gas phase properties will beused.

dp -2 cm, = 14 cm= 6.5 = 8 kg/ m

Liquid properties for various materials are listed inTable 1. The solid densities of all the cryogenicpropellants are assumed to be equal to the density ofsolid pentane, 850 kg/m3. The solid density of the waxis taken to be 930 kg/m3. The gas phase blowingparameter, #, used in the calculations for all the fuelsis 7.


AIA A 2001-4503

One of the most interesting results of thislinear theory is that even at very small film thicknessesthere exists a finite range of amplified wave numbers,namely the layer is unstable over a finite range of wavenumbers. The instabilities of that type were firstdiscovered for thin water films in a wind tunnel byCraik13 and they are called the "slow waves". These aregenerated by the interaction of the gas phase shearstresses acting on the liquid surface with the slope ofthe liquid layer surface. Fig. 8 shows a typical form ofthe dimensional amplification rate of the interfacedisturbance, -(fil/h2pl\lm(caR&)9 as a function ofthe dimensional wave number, k = a h, calculatedwith use of the linear stability theory outlined in thispaper. For this calculation, the liquid is water, liquidReynolds number is 50, film thickness is 0.3 mm,regression rate is zero and the Froude numbercorresponding to the body force of 9.81 m/sec2 is 9.458.As indicated in the figure, there is a positiveamplification domain, which lies between two cutoffwave numbers. The amplification takes a maximumvalue at a wave number between these two cutoff wavenumbers. This is the most amplified wave number andthe corresponding wavelength is expected to be theobserved size of the disturbance in the actual flowsystem. As the body force diminishes the first cutoffpoint moves towards zero.

Intuitively, the most important parameter thatlinks the linear stability results to the entrainment rateof the liquid from the surface is the amplification rate ofthe disturbances. Fig. 9 shows the effect of the liquidReynolds number on the amplification curves. This plotis for a 0.3 mm thick pentane film. The liquid Reynoldsnumber is adjusted by changing the port mass flux from42.7 kg/m2-sec to 92.2 kg/m2-sec. The regression rateparameter is 0.55 and the body force is assumed to bezero. Note that as the liquid layer Reynolds number(which is directly proportional to the gas streamdynamic pressure) increases, the amplification rateincreases. The Reynolds number also increases the mostamplified wave number, meaning that at higher gasflow velocities the expected wavelengths of theinstabilities are smaller. This latter result is in goodagreement with the experimental findings for the scaleof waves formed on the surface of a thin film17.

The effect of the liquid injection on the filmstability is shown in Fig. 10. The amplification rate as afunction of the wave number for three values ofregression rate parameter, b, is calculated with use ofthe exact solution. This case is for a 0.15 mm pentanefilm with a liquid Reynolds number of 50 and a Froudenumber of zero. Fig. 10 shows that the normal liquidinjection has a slight stabilizing effect on the film. This

conclusion is also confirmed by the power seriessolution. For a situation for which the mass flux is keptconstant rather than the liquid Reynolds number, theeffect of the regression rate parameter will be moredominant. This is due to the fact that, according to Eq.52, an increase in the regression rate parameterdecreases the liquid Reynolds number for a given portmass flux value.

Another very important result that came out ofthe stability investigation is that both the surfacetension and also the viscosity of the liquid have astabilizing effect on the liquid film. It turns out that thisfeature of the thin film instabilities plays a major role indetermining which propellant is likely to sustaininstabilities and potentially entrain droplets into the gasstream. In order to demonstrate the effect of materialproperties on the stability, the amplification curves forvarious liquids at a fixed film thickness of 0.15 mm areshown in Fig. 11. For all the liquids the regression rateis 1 mm/sec and the port mass flux is 80 kg/m2-sec.Note that liquids with higher viscosities such asisopropanol and HFI are more stable than the liquidswith small viscosity values. The stability curve for onegrade paraffin wax, which is solid under ambientconditions, is included in the figure. It is predicted bythe theory that this particular grade of wax would forman unstable melt layer promoting entrainment of liquiddroplets into the gas stream.

As a final note of this section we like to statethat the linear stability theory presented in this papercan also be applied.to classical hybrid propellants thatform a liquid layer on their burning surfaces. One suchpropellant is HDPE (high-density polyethylene)polymer, which has been tested extensively as a hybridfuel. In Ref.10 it is shown that the melt layer of HDPEis 4 orders of magnitude more viscous than pentane.The linear theory suggests that the melt layers of thesehighly viscous polymeric fuels will be extremely stableat the port mass flux levels typically encountered inhybrid motors.

C) Liquid Entrainment Relationships

The linear stability treatment outlined in theprevious section revealed that the melt layers typicallyencountered in hybrid applications could developinterfacial instabilities. Even though the existence ofthese linear instabilities is a necessary condition, it isnot sufficient for the entrainment process to occur.Before entrainment of droplets can take place, theinfinitesimal harmonic disturbances growing at theliquid-gas interface must grow into a non-harmonicwaveform with a finite amplitude. The process ofgrowth to a finite amplitude and eventually entrainmentof droplets is a highly nonlinear phenomenon, which is

10


AIAA 2001-4503

extremely difficult to model. We bypass this difficultbut critical step by using some emprical relations thatare introduced in the existing literature for theentrainment from thin liquid films under strong shearforces exerted by a gas flow. As stated in the literature,one plausible mechanism for droplet entrainment fromthese thin films at large Reynolds numbers is throughthe formation of nonlinear "Roll Waves". Asschematically indicated in Fig. 2, the droplets are drawnfrom the tips of these nonlinear waves by the stressesexerted by the local gas flow.

The most relevant experimental work onentrainment is reported by Gater and L'Ecuyer in anearly paper17. In this study which was motivated toaddress the liquid injection requirements for filmcooling applications, the entrainment rates from thinfilms of various liquids (including some hydrocarbonssuch as RP-1, methanol) under strong gas flow weremeasured. The experiments were performed in a windtunnel and some tests were executed with hot gas flow.The entrainment relationship suggested by Gater andL'Ecuyer, which is slightly modified in Ref. 10 can bewritten as

™ent = PSem =1-41 \V~\Xe -2109) m{. (22)

Here is rent the entrainment portion of theregression rate. The entrainment parameter, Xe and theliquid flow rate per unit width in the melt layer, m/, canbe written as

general empirical expression (for operation sufficientlyremote from the critical conditions for entrainment) forthe entrainment rate of liquid droplets in terms of therelevant properties of the hybrid motor.

(24)

The experimental data and the linear stabilitytheory suggest that the dynamic pressure exponent isexpected to be in the range of 1-1.5. For example Gaterand L'Ecuyer17 scaling for large mass fluxes indicatesthat a is approximately 1.5 and ft is equal to 2. Theviscosity and surface tension exponents are bothpredicted to be 1 . The series of entrainment relationsgiven by Nigmatulin et a/18 suggest a weakerdependence on both the dynamic pressure and also thethickness (a = l and /? = !). The scaling with respectto the material properties is not estimated accurately bythe entrainment relations given by Nigmatulin, sincemainly water data was considered in the study. Webelieve that under the conditions of hybrid rocketapplications, the melt layer viscosity plays a moreimportant role compared to the surface tension inestablishing the entrainment mass transfer rate (i.e.

A very useful empirical relation for the criticalconditions for the onset of entrainment given by Ref. 20is rearranged here in terms of the practical conditions inthe motor.

pO.5—a

(23) Cf Pi(25)

Note that Pg is the dynamic pressure of the

gas flow (Pg = G2/Pg) anc* Tg is the gas streamaverage temperature. The temperature ratio is includedas a correction for the density variation in the gas.

In the nuclear reactor field there has beensignificant research on the entrainment masstransfer18'19, since it has importance in the emergencycooling of the reactors. Various models for entrainmentare developed at different regimes of operation. Ingeneral, the experimental findings of those studies alsoconfirm the results previously mentioned. However theform of dependence of the entrainment mass transfer onthe parameters is somewhat different in every study.Thus it is fair to state that the proper form of scaling forthe entrainment still needs to be resolved.

In the light of experimental findings and theresults of the linear theory, we suggest the following

Note that this relation only holds for a laminar film(Re<300). Unlike the entrainment relations this formulahas the correct scaling with the material properties ofthe liquid. Fig. 12 shows the plot of the entrainmentboundary for various liquid materials in the mass fluxfilm thickness plane (See Table 1 for the materialproperties used in calculations). Note that for eachliquid, the entrainment takes place in the region abovethe curve.

D) Modification of the Classical Theory forEntrainment

It has been shown in the previous sections thata liquid layer can be formed on the fuel grain and thisliquid layer can be unstable over a reasonable range ofparameters typical of hybrid operation. It is alsoindicated that the instabilities in the liquid layer mayinduce the entrainment of liquid droplets into the gas

11


AIAA 2001-4503

stream. The final phase of the development involves themodification of the classical hybrid boundary layercombustion theory for the possibility of entrainmentmass transfer from the fuel grain.

The formation of the liquid layer instabilitiesand entrainment of liquid droplets require three majormodifications in the classical hybrid combustion theory.

• The ratio of the enthalpy difference to the effectiveheat of gasification (&h/hv) that appears in thethermal blowing parameter expression is altered.The effective heat of gasification is reduced sincethe evaporation energy required for the fuel masstransfer from the surface is partly avoided by themechanical entrainment of the liquid. Whereas, theenthalpy difference between the flame and thesurface is also reduced, since some of the reactantsare now in liquid phase. It is estimated that, thereduction in the effective heat of gasification ismore dominant than the change in the enthalpydifference. Thus as a first approximation weassume that the reduction in the flame enthalpy isnegligible.

• The blocking factor, CH/CHo, that modifies theconvective heat flux to the surface is also altereddue to the presence of the two-phase flow. As afirst approximation we ignore the effect of theliquid droplets on the momentum and energytransfer. Under this assumption the blocking factorcan be expressed as a function of evaporationblowing parameter.

O =f(Bg)

The evaporation blowing parameter, Bg9 includesonly the gaseous phase mass transfer from the fuelsurface. We assume that evaporation of thedroplets released from the liquid surface into thegas stream does not take place beneath the flamesheet. This assumption is consistent with the flamesheet approximation, which is a standard one inhybrid combustion modeling. Moreover, it is areasonable approximation for typical hybridoperating conditions that are characterized by highblowing rates and thus low residence time ofdroplets between the liquid surface and thediffusion flame. However, we believe that a morerigorous treatment of the two-phase flow in thehybrid boundary layer will be an important step inthe further improvement of the liquefying hybridtheory.

• The ripples formed on the liquid layer surfaceincrease the surface roughness and the heat transferfrom the flame front to the surface.

In general, the total regression rate of a hybridmotor can be written as a sum of the evaporationregression rate that is generated by the vaporization ofthe liquid into the gas stream and the entrainmentregression rate, which is related to the mass transfermechanically extracted from the liquid surface.

= * + r (26)

For an arbitrary combination of theentrainment and evaporative mass transfer the energybalance at the liquid gas interface is

Pfwhere

hvand he

(27)

(28)

Here the nondimensional energy parametersfor entrainment (/?^) and vaporization (R^v) areintroduced because the material that is extractedthrough the entrainment mechanism possesses differentheating histories (i.e. no heat of vaporization is requiredfor entrainment). We postulate that the effective heatingin the liquid phase required for fuel material which isgoing through the entrainment mass transfer mechanismreduces linearly as the vaporization component of theregression rate decreases. This extra complexity, whichwas ignored in the thickness estimation arguments, isintroduced to capture the asymptotic behavior ofregression rate in the entrainment-dominated region ofoperation. We would like to note that the effect ofreduced liquid phase heating on the thickness atmoderate entrainment conditions is very small since thethickness presents a logarithmic dependence on theenergy terms.

The roughness parameter, Fr, is introduced inthe energy equation to account for the increased heattransfer by wrinkling of the liquid surface. It has beenargued by Gater and L'Ecuyer17 that the surfaceroughness decreases with increasing dynamic pressureof the gas flow. This argument is based on the observedreduction in the scale of the interface disturbances withincreasing dynamic pressure. Note that thisphenomenon is captured by the linear theory which

12


AIAA 2001-4503

shows decreasing wave length with increasing gas massflux. The empirical formula for the roughnesscorrection parameter suggested by Gater and L'Ecuyercan be expressed in terms of the operational parametersof the motor as

R1=1 + -14.1 p 0.4

0.2 (29)

In this formula the units of mass flux and gas densityare kg/m2-sec and kg/m3, respectively.

We developed a new curve fit expression forthe blowing correction, CH/CHo , which is a reasonableapproximation for the analytical expression given byMarxman21 for a wide B range of 0-14 (See Fig. 13).

CHO 2 +

Here the coefficients are defined as

^ 2 A T -CR1 = —————PPTT anu c#2 =

\0.75 (30)

1.25£a75

0'752 + 1.255

It is not acceptable to adapt one of the forms that arecommonly used in the hybrid literature1

__f\ X-O

(i.e. CH/CHo =B ) in this study in which accuracyat low B values is essential. Classical expressionspredict unrealistically large blocking factors (evenlarger than one) for B values close to zero. Note that Bis the classical blowing parameter and rcl is theclassical regression rate of the system in the case of noentrainment. The classical regression rate can be writtenas

rcl='

0.2

Pf(31)

Following the arguments in the proceedingsection (i.e. Eq. 24) the entrainment regression rate canbe expressed in terms of the mass flux in the port andthe total regression rate.

(32)

Here we used the definition of the dynamic pressure( p d = G2/pg) and Eq. 4 or Eq. 5 for the film thicknessexpression. The entrainment parameter, aent, is afunction of the properties of the selected propellant and

average gas density (p.) in the chamber. Foro

simplicity, we will assume that aent is constant for agiven propellant.

Eqs. 26, 27, 28, 29 and 32 together with theblowing correction expression, Eq. 30, form a nonlinearset of algebraic equations which can be solved for agiven propellant combination to obtain the totalregression rate as a function of the axial location andlocal mass flux.

E) Application to Cryogenic Hybrid Rockets

For pentane as the frozen fuel, the regressionrates are calculated iteratively from this nonlinear set ofequations as a function of the mean mass flux in theport. Fig. 14 shows both the predicted regression ratesand also the reduced instantaneous regression rate datafor several pentane runs10. In the calculations a = 1.5,£ = 1.5, ^=810-14(m8-5sec°5/kg3), 4/4=0.15,z = 0.075 (m), p^=4(kg/m3), and ^=6.510'5 (N-sec/m2) numerical values are used. The estimatedblowing parameter value of 13.5 for the pentane-oxygen system is used in the calculations. The figureshows that the high regression rates observed in thepentane tests can be explained in the context of theliquid layer combustion theory. Most of the increase inthe burning rate comes from the entrainment ofdroplets; reduced effective heat of gasification andreduced blocking correction. The roughness effect,however, accounts for a smaller increase in regressionrates (i.e. increase in the surface heat transfer rangesfrom 30 % at low mass fluxes down to 5 % at thehigher flux levels).

It is useful to obtain an upper limit for theregression rate of a hybrid fuel forming a liquid layer.In the extreme case the vaporization from the liquidsurface is negligible, namely all the mass is transferredto the gas flow in the form of droplets. This upperbound can be quantitatively determined upon theconsideration of Eq. 27. The maximum regression rateoccurs when no vaporization at the surface takes place.Under the assumptions of our theory, for this extremecase, the blocking factor becomes unity. The maximumregression rate normalized with the classical value ofthe regression rate, for which no entrainment from thesurface is allowed, becomes

1 (33)

For pentane fuel the maximum regression rateis estimated to be approximately 23 times the classical

13


AIAA 2001-4503

regression rate. Note that for pentane calculations theblowing parameter value of 13.5 and Rhe value of 0.23are used. This large upper limit for the amplificationexplains the experimentally observed regression ratesthat are determined to be 2-5 times larger than theexpected (classical) values.

As shown in Table 1 and Fig. 1 somecryogenic hydrocarbons such as pentane, acetone and2,2,5 tmh burn fast whereas some others such asisopropanol and HFI burn considerably more slowly.This observation can also be explained within thecontext of the liquid layer hybrid theory described inthis paper. Fig. 12 shows the entrainment onsetboundary for various propellants generated with use ofEq. 25. The parameter values of pg=4(kg/m3),cf =0.004 and jng =6.5 10~5 (N-sec/m2) are used inthe calculations. It is clear that isopropanol and HFI,which are much more viscous compared to the othermaterials, require large thresholds for entrainment. Inorder to reinforce this finding, we have used the scalinglaw, Eq. 24, to estimate the entrainment rates for givenmass flux and film thickness for various materials. Theentrainment rates are then reduced to entrainmentparameters by normalizing with the entrainment rate forthe reference propellant pentane. Calculations areperformed with use of the viscosity and surface tensionexponents of 1 and 0.7, respectively. The relativeentrainment parameters are shown in Table 1. Clearly,the entrainment parameters of the viscous propellants(i.e. isopropanol, HFI) are one order of magnitudesmaller than the other propellants. This finding revealsviscosity as a determining parameter controlling theburning rate of a liquefying hybrid propellant. In fact,viscosity of the imbedded propellant can be adjusted totailor the regression rate of a hybrid rocket during thecourse of its operation. Moreover it can be stated thatoxygen, which is an interesting alternative solidcryogenic hybrid propellant, is also predicted to be afast burning material due to its low viscosity. Thisresult is confirmed by the tests performed byORBITEC6.

Another important practical issue for theprediction of the performance of a propellant is itssusceptivity to bulk heating by radiation penetratedthrough the slab. Specifically, it can be stated that thematerials, which have small radiative absorptivities intheir solid phases, potentially burn in an uncontrolledfashion. This is due to the possibility of sloughing ofthe propellant that is internally heated. In order toinvestigate this, we have estimated the outer webtemperature as a function of the remaining webthickness for a typical pentane run10. It can be shownthat, for pentane tests, the outer web temperature

changes very little until the last portion of the run afterwhich it increases rapidly. Note that in the last third ofthe run the chamber pressure drops suddenly for thesame test. This observation supports the possibility ofsloughing due to bulk heating of the propellant. Theextent of this uncontrolled operation mode over thewhole burn time is expected to be more significant forpropellants with smaller absorptivities of radiation. Infact for a propellant to be practical its absorptivity mustbe increased over a minimum value by possibly addingmaterials that are good absorbers of radiation (i.e.carbon black).

4) Paraffin Based Hybrid Rocket Fuel Test Results

Even though the liquid layer combustiontheory is first developed to explain the burning behaviorof solid cryogenic propellants, it is applicable to anymaterial that would form a liquid layer on its burningsurface.

In fact we have applied the theory to formulatea noncryogenic fast burning paraffin-based hybridrocket fuel, SP-1. The entrainment onset boundary andthe relative entrainment rate for the paraffin-based fuelare shown in Fig. 12 and Table 1, respectively.Surprisingly, the plot and table indicate that thepredicted entrainment rate for this formulation iscomparable to that of a low molecular weight cryogenicmaterial such as pentane.

Experiments conducted with gaseous oxygenat Stanford University confirm the theory predictionson the paraffin-based fuel22. The burning rate of the fuelformulation SP-1 as a function of mass flux isapproximately 3 times faster than the burning rates ofHTPB as indicated by Fig. 15. Note that the averageoxidizer mass flux is calculated based on the averageport diameter.

This new fuel formulation delivers very highburning rates without compromising any of theadvantages of the hybrid fuels. We believe thatparaffin-based propellants may very well play a centralrole in the development of the hybrid rocket technologyinto operational propulsion systems.

Some of the key characteristics of theexperimental setup used in the tests can summarized asfollows (i.e. See Ref. 10 for a detailed description of thefacility).

• Oxidizer is gaseous oxygen.• Motor OD is 3 1/4".• Initial fuel port diameter used in the tests ranges

from 0.5" to 1.2".• Oxidizer flow rate is set by a choked sonic orifice.• Motor is ignited with a methane/Gox torch.

14


AIAA 2001-4503

• Sonic orifice upstream pressure is measured to beused in the oxygen mass flow calculations. Thechamber pressure is measured at the fore end of themotor.

• ATJ graphite is the material used for the nozzle andcombustion chamber insulation.

The liquid layer hybrid combustion theory alsocovers the classical propellants that form a liquid layer.The reason that the classical polymeric propellants suchHDPE do not burn fast (even though they might form aliquid layer) is their highly viscous melt layers. Forexample, the melt layer viscosity of HDPE ispredicted10 to be 4 orders of magnitude larger than themelt viscosity of pentane or wax. Naturally, at thesesuper high viscosities, even at very high mass fluxlevels, the onset of entrainment is not achieved.

5) Conclusions

• In this paper the hybrid diffusion limited theory isgeneralized to hybrid fuels that burn by forming aliquid layer. In the first step of our development,we determined the conditions required to form amelt layer with a thickness sufficient to produceentrainment. It is shown that the melt layerthickness can be a few hundred microns thick fortypical hybrid operating conditions. Next we haveshowed that these thin films with strong blowingcould be unstable under hybrid operatingconditions. Results reported in the literature areused to infer that the instabilities will result in theentrainment of droplets into the gas stream.

• It is determined that both the surface tension andthe melt layer viscosity play a critical role indetermining the entrainment characteristics of amaterial. It can be stated that materials with lowviscosity and low surface tension will generatemore entrainment. This fact explains why some ofthe classical fuels that form a melt layer (i.e.HDPE) do not generate high regression rates.

• The extra mass transfer mechanism can potentiallyincrease the regression rates an order of magnitudelarger than the estimated rates from the classicaltheory. This is mainly due to the reduced effectiveheat of gasification, decreased blocking factor inthe hybrid boundary layer and higher heat transferas a result of the increased surface roughness.

• The cryogenic hybrid rocket tests revealedregression rates 2-5 times larger than the expectedrates estimated from the classical theory for lowviscosity propellants such as pentane. The extendedtheory developed here successfully predicts thesevery high regression rates observed in thecryogenic tests. The liquid layer theory also

explains the slow burning rates of high viscositypropellants.

• The application of the theory is not limited to solidcryogenic hybrid fuels. In fact we used our modelto formulate a paraffin-based fuel which wouldexhibit high regression rates comparable topentane. Tests at Stanford on a laboratory scalemotor confirmed the predicted high regressionrates for this fuel. The new paraffin-basedmaterials comprise a class of fast burning non-cryogenic hybrid fuels with a wide spectrum ofproperties. This provides the opportunity to satisfya broad range of mission requirements for the nextgeneration of hybrid rockets.

6) Acknowledgments

This work was partially supported by theNASA Ames research center and the Joint Institute forAeronautics and Acoustics under grants NCC2-55 andNCC2-1172. Additional, funds were provided byStanford University.

7) References

^arxman, G. A., Wooldridge, C. E., and Muzzy, R.J. "Fundamentals of Hybrid Boundary Combustion",Progress in Astronautics and Aeronautics, Vol.15, 1964p485.

2Larson, C. W., Pfeil, K. L., DeRose, M. E. andCarrie, P. G., "High Pressure Combustion of CryogenicSolid Fuels for Hybrid Rockets", AIAA paper No. 96-2594, AIAA/SAE/ASME/ASEE 32 nd Joint PropulsionConference and Exhibit, July 1996.

3Larson, C. W., DeRose, M. E., Pfeil, K. L. andCarrie, P. G. "High Pressure Combustion of CryogenicHybrid Fuels in a Lab-Scale Burner", Proceedings ofthe 1996 JANNAF Joint Propulsion Conference,Albuquerque, NM, December 9-13, 1996, published byCPIA, John Hopkins University.

4DeRose, M. E., Pfeil, K. L., Carrie, P. G., andLarson, C. W. "Tube Burner Studies of CryogenicSolid Combustion", AIAA paper No. 97-3076,AIAA/SAE/ASME/ASEE 33 rd Joint PropulsionConference and Exhibit, July 1997.

5Gramer, D., Rice, E., Knuth, W. and Clair, C. St.,"Experimental Investigation of a Metallized CryogenicHybrid Rocket Engine", AIAA paper No. 98-3509,AIAA/SAE/ASME/ASEE 34th Joint PropulsionConference and Exhibit, July 1998.

6Clair, C. St., Rice, E., Knuth, W., and Gramer, D.,"Advanced Cryogenic Solid Hybrid Rocket EngineDevelopments: Concept and Testing", AIAA paper No.98-3508, AIAA/SAE/ASME/ASEE 34th JointPropulsion Conference and Exhibit, July 1998.

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AIAA 2001-4503

7Estey, P., Altman, D. and McFarlane, J., "AnEvaluation of Scaling Effects for Hybrid RocketMotors", AIAA paper No. 91-2517, 1991.

8Lang, L. "Absorption Spectra in the InfraredRegion", London Butterworths, 1974.

9Bentley, F. F., Smithson, L. D. and Rozek, A. L."Infrared Spectra and Characteristic Frequencies -700-300 cm -1", Interscience Publishers, 1968.

10Karabeyoglu, M. A., "Transient Combustion inHybrid Rockets", Ph. D. Thesis, Stanford University,August 1998.

"Seigel, R. and Homel, J. R. "Thermal RadiationHeat Transfer", Hemisplane Publishing Coorporation,1992.

12Schlichting, H., "Boundary Layer Theory", McGrawHill, 1955.

13Craik, A. D. D., "Wind Generated Waves in ThinLiquid Films", Journal of Fluid Mechanics, vol. 26,part 2, pp. 369-392, 1966.

14Chang, I-Dee and Russel, P. E., "Stability of a LiquidLayer Adjacent to a High-Speed Gas Stream", ThePhysics of Fluids, vol. 8, no. 6, 1965.

15Nayfeh, A. H., Saric, W. S., 'Non-Linear Kelvin-Helmholtz Instability", Journal of Fluid Mechanics,vol. 46, part 2, pp. 209-231, 1971.

16Benjamin, T. B., "Shearing Flow over a WavyBoundary", Journal of Fluid Mechanics, vol. 6, pp.161-205, 1959.

17Gater, R. A. and L'Ecuyer, M. R. L., "AFundamental Investigation of the Phenomena thatCharacterize Liquid Film Cooling", InternationalJournal of Heat and Mass Transfer Vol. 13, pp 1925-1939,1970.

18Nigmatulin, R., Nigmatulin, B., Khodzaev, Y. A.and Kroshilin, V., "Entrainment and Deposition Ratesin a Dispersed-Film Flow", International Journal ofMultiphase Flow Vol. 22, pp. 19-30, 1996.

19Ishii, M. and Grolmes, M. A., "Inception Criteriafor Droplet Entrainment in Two Phase Concurrent FilmFlow", AlCh Journal, vol. 21, no. 2, pp. 308-318, 1975.

20Nigmatulin, B. I., Klebanonov, L. A., Kroshilin, A.E., "Heat transfer crisis for process steam-liquiddispersed annular flows under non-stationaryconditions", High Temp. Thermal Phys. No. 18, 1242-1251, 1980.

2IMarxman, G. A. "Combustion in the TurbulentBoundary Layer on a Vaporizing Surface", TenthSymposium on Combustion, 1965, p 1337-1349.

22Karabeyoglu, M. A., Altman, D., Cantwell, B. J.,Patent Pending, "High Regression Rate Hybrid RocketFuels"

23Dauber,T. E., Danner, R. T., "Physical andThermodynamic Properties of Pure Chemicals, DataCompilation", Taylor and Francis, 1997.

24Altman, D. and Humble, R. "Hybrid RocketPropulsion Systems" in Space Propulsion Analysis andDesign, McGraw Hi l l , 1995, p382.

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8) Tables and Figures

Propellant

Molecular Weight(g/mol)Heat of Formation(kJ/mol)Surface Tension(mN/m)Viscosity(mPa-sec)Density-liquid phase(kg/m3)Melting Temperature(K)Boiling Temperature(K)Heat of Fusion(kJ/kg)Heat of Vaporization(kJ/kg)Thermal Conductivity-liquid phase (W/mK)Specific Heat-liquid phase (kJ/kg-K)Specific Heat-solid phase (kJ/kg-K)Entrainment Parameter1,Relative to PentaneBurning RateCharacteristic

PentaneQH1272.15

-146.8

14.3

0.46

688.4

143.3

309.6

116.7

357.8

0.14

2.06

1.10

1

High

AcetoneC3H6058.08

-217.2

19.2

0.51

835.2

178.45

329.44

98.0

513.0

0.17

2.07

1.33

0.73

High

2,2,5 tmh^9"20

128.26

-254.0

11.3

0.42

719.8

167.4

397.2

48.3

261.8

0.12

2.00

-

1.29

High

Oxygen02

32.0

0.0

13.2

0.33

1226.9

54.4

90.2

13.7

190.58

0.17

1.67

0.83

1.48

High

HFI

69.11

-231.7

15.6

2.5

785.0

181

350

96.3

540.2

0.137

2.03

1.09

0.17

Low

IsopropanolC3H8060.10

-272.4

16.4

5.0

808.9

183.3

355.4

89.4

663.4

0.14

2.28

1.10

0.08

Low

Table 1: Material properties of the propellants used in the calculations23. Liquid properties other than surfacetension are evaluated at a mean temperature between the melting and vaporization temperatures. Surfacetension is evaluated at the boiling temperature. See Ref. 10 for the estimation of properties for the mixtureHFI.

1 Calculations are performed with the viscosity and surface tension exponents of 1 and 0.7, respectively.

17


AIAA 2001-4503

2.5-

2-

1.5

*cc

0.5

o-f*V

o

PentaneAcetone2,2,5 tmhHFIIsopropanol

Curve Fit forPentane/Gox system

HTPB/Gox System

30 40 50 60 70Oxidizer Mass Flux,

80 90 100 110

Figure 1: Space-time averaged regression rates are plotted versus the average oxidizer mass fluxes forvarious materials tested by AFRL. Curve fit expression for pentane data is r- 0.123 Go

063 (i.e. r is inmm/sec and G0is in kg/m2-sec). For comparison purposes the burning law of a classical propellant, HTPB,is included in the plot.

peUe

ReactingCg^Droplets Djffusjon

Ran

Rollraves

Figure 2: Schematic of the entrainmentmechanism. Figure 3: Schematic of the thermal model used

in the melt layer thickness estimation.

18


AIAA 2001-4503

0.5

0.4 0.6 0.8 1 1.2Regression Rate, mm/sec

1.4 1.6

Figure 4: Effect of regression rate and initial slab temperature on the melt layer thickness for pentane.

0.05J-0.1 0.2 0.3 0.4

Radiation Level, Qr/Qc

Figure 5: Effect of radiative heat transfer on the melt layer thickness of a pentane slab for variousvaporization rates are plotted. The plots are for regression rate of 1 mm/sec and initial slab temperature of77 K.

19


AIAA 2001-4503

Wave formMean VelocityProf He

Figure 6: The schematic of the stability model.

5

o

0.9 -

0.8-

0.7

i 0.6i

! 0.5

0.3

0.2

0.1

Approximate velocity profile

No blowing velocity profile

0.1 0.2 0.3 0.4 0.5 0.6 0.7Nondimensional Velocity

0.8 0.9 1

Figure 7: The exact velocity profile for the regression rate parameter of b = 0.4 the linear approximationgiven by Eqn.10 and also the velocity profile with no blowing are plotted.

20


AIAA 2001-4503

jo

-2

Re-50We-0.116Fr=9,458

b=0

Maximum amplification point

> First cutoff pointSecond cutoff point

8 10Wave Number, 1/cm

12 14 16 18

Figure 8: Amplification rate of a surface disturbance versus the wave number. This plot is for a water filmwith a thickness of 0.3 mm and film Reynolds number of 50. The Froude number is 9.458 and theregression rate parameter is zero.

45

40

35-

30-8

•52£ 2 5 -

[ 20

15

10

10 15 20 25 30 35Wave Number, 1/cm

40 45 50 55

Figure 9: Amplification rate of a surface disturbance versus the wave number for various film Reynoldsnumbers. This plot is for a pentane film with a thickness of 0.3 mm. Reynolds number is altered bychanging the gas mass flux in the port. The body force and the regression rate is taken to be zero for thiscalculation.

21


AIAA 2001-4503

10 15 20 25Wave Number, 1/cm

30

Figure 10: The effect of regression rate on the stability of the film is illustrated. This plot is for a pentanefilm with a thickness of 0.15 mm and a liquid Reynolds number of 50. The body force is taken to be zero.

35

30

25

15

10

Fr=0

Wax

10 15Wave Number, 1/cm

20 25

Figure 11: Amplification curves for various liquids for a port gas mass flux of 80 kg/m2-sec and aregression rate of 1 mm/sec. A film thickness of 0.15 mm is used in these calculations. The body force isassumed to be zero.

22


AIAA 2001-4503

0.05

00 110 120 130 140 150 160 170 180 190 200

Figure 12: Entrainment onset boundaries in the thickness-mass flux plane for various propellants areplotted. Eq. 25 is used in the calculations.

1

0.9

0.8 • \ )

£0.7

CD 0.5

0)0.4c

0.3

0.2

0.1

\\ o

—— Marxman- - Curvefito Experimental Data

0 5 10Blowing parameter, Bg

15

Figure 13: Blowing corrections according to Marxman's formula and our curve fit expression, Eq. 30, areplotted as a function of the blowing parameter. The experimental points are reported in Ref. 21.

23


AIAA 2001-4503

x Run 100o Run 101* Run 102+ Run 103

100 150Mass Flux, kgAn2-sec

200 250

Figure 14: Estimates of instantaneous regression rates and corresponding mass fluxes evaluated at midpoint of the port for 4 different pentane tests is plotted. Liquid layer theory predictions for the vaporization,entrainment and total regression rates are included in the plot.

: 1.5

O) 1<D

DC

0.5

i————i—————i————i————r

o Wax/GOX Data, Stanford University

10 20 30 40 50 60 70 80 90Oxidizer Mass Flux, kg/m2-sec

100 110 120

Figure 15: Space-time averaged regression rates are plotted versus the average oxidizer mass fluxes for theparaffin-based fuel (SP-1) tested at Stanford University. Curve fit expression for SP-1 data isr = 0.091 G°'69 (i.e. r is in mm/sec and G0is in kg/m2-sec). For comparison purposes the burning law of aclassical propellant, HTPB, is included in the plot.

24

development and testing of paraffin-based hybrid...

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