[American Institute of Aeronautics and Astronautics 42nd AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit - Sacramento, California ()] 42nd AIAA/ASME/SAE/ASEE Joint Propulsion

One Dimensional Hydrocarbon Fueled Scramjet Model with Droplet Evaporation and Finite Rate Chemistry

K. Srinivasan* and Nayan Dubey†

Defence Research and Development Laboratory, Hyderabad, India, 500058

The quasi-one-dimensional equations of fluid motion are coupled with the equations for finite-rate chemistry to model high speed (supersonic) combustor flow fields. A numerical calculation method has been incorporated to predict the evaporation percentage of the transversely injected fuel spray with downstream distance x from the injector(s) by tracking a single droplet from each injector. The effects of the droplet heat up period and forced convection on the evaporation process have been taken into account. This helps the proposed model to free itself from the huge responsibility of assuming appropriate values for the mixing length of the fuel droplet, which is usually the case. The variation in the evaporation percentage with the downstream distance for varying air velocities, pressures, temperatures etc have been taken care of. The model also incorporates the effects of area change, friction, mass injection, heat transfer through the walls of the combustor on the combustor performance. The model is clearly oversimplified compared to the complex three dimensional turbulent mixing and induced shock train processes known to exist in supersonic combustors. Nevertheless, it includes sufficient physics to provide a rapid interpretation of the global features of the supersonic combustors. Accordingly, it may be useful in the early conceptual design studies of the supersonic combustor. The simulations done, as a part of this work, are for Jet-A fuel, C12H23, assuming it to be a single component fuel.

Nomenclature A = combustor cross-sectional area, m2

dA = projected area of the droplet

yB = mass transfer number

DC = drag coefficient

fC = friction coefficient

HC = Stanton number

pc = specific heat capacity at constant pressure, J/(kg – K)

D = hydraulic diameter, m

bodyFr

= body force, N

dFr

= net downward force, N

dragFr

= drag force, N

g = gravitational acceleration constant h = enthalpy per unit mass, J/kg, heat transfer coefficient W/(m2 – K) K = thermal conductivity, W/(m – K) L = latent heat of fuel vaporization, J/kg M = Mach number m& = mass flow rate, kg/s * Scientist, Hypersonic Propulsion Division, e-mail: s† Scientist, Hypersonic Propulsion Division, e-mail: n

American Institute of

42nd AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit9 - 12 July 2006, Sacramento, California

AIAA 2006-5036

Copyright © 2006 by K. Srinivasan and Nayan Dubey. Published by the American Instit

[email protected][email protected]

Aeronautics and Astronautics

1

ute of Aeronautics and Astronautics, Inc., with permission.

A = molecular weight of aM ir, kg/kmol

FM = molecular weight of fuel, kg/kmol

Fm& = rate of droplet evaporation, kg/s MW = molecular weight, kg/kmol

MW = mixture molecular weight, kg/kmol Nu = Nusselt number p = pressure, N/m2

Pr , m

W

R t, J/ (kg – K)

Sh

= Prandtl number

wP = wetted perimeter

transQ = heat transferred from air,

eQ = heat used for fuel evaporation

Q& = heat transfer rate, J/s = Universal gas constan

Re = Reynolds number

sr = droplet radius, m = Sherwood number

T = temperature, K t

ty between the droplet and the gas stream u

v in the y – direction

= time, s , m/s U = velocity

Ur

R = relative veloci= droplet velocity in the x – direction d

V = volume, m3

= droplet velocity dx = axial coordinate, m Y = mass fraction

x = control volume∆ increment of x ε = ratio of gas injection velocity to free stream velocity γ = ratio of specific heats µ = viscosity, kg/(m-s) ρ = density, kg/m3

ω& = molar production rate, kmol/(s – m3)

ubscripts

ies added for mass/fuel injection

r

nation conditions

itions

SA = air

ded ad = specaw = adiabatic wall crit = critical conditions d = droplet

el or vapouF = liquid fuG = gas

pecies i = ith s0 = total or stagr = reference condition s = droplet surface w = wall

nd∞ = free stream co

American Institute of Aeronautics and Astronautics

2

Superscripts r unit area

e

I. Introduction The combustion process of a Scramjet is one of the research in the development of a

hypersonic vehicle. The fuel evaporation characteri ent mixing process of fuel in a supersonic air stre

ion, and turbulence is now feasible using contemporary methods. Det

ing efficiency, mixing length and fuel inje

the fuel injected. Thus there must be some rati

re, area of the combustion chamber etc. Unlike the

sis for the sake of analytical and numerical simplicity. Fol

II. Governing Flow Equations for the Model The governing ow equation h tions: 1) Quasi-one-dimensional flo h e x- direction only,

`` = pe``` = per unit volum

primary areas of stics, the turbul

am, the shock wave – boundary layer interactions, the residence and reaction times of the fuel-air mixture that determines the supersonic combustor size, the burning characteristics of the fuel-air mixture in an environment of high speed and low pressure flow, the high enthalpy flow contributing to the real gas effects such as species dissociation of gas, the chemical reaction of combusting media etc. are few of the many complexities that are present in the Scramjet system analysis. The Scramjet system analysis is, therefore, complicated because it requires the understanding of many such phenomena

An analysis of such a process based on solution of the Navier-Stokes including the effects of finite rate chemical kinetics, viscosity, thermal and mass diffus

ailed analyses of the Scramjet concept are being pursued by many CFD researchers, and their work is well-summarized by White et. al1. However, these approaches are very costly, time consuming and subject to limitations of available turbulence modeling and are unsuitable for conducting preliminary design studies. A method that could quickly predict the overall performance of a scramjet combustor would have great practical utility for the preliminary design process, as well as for rapid interpretation of experimental data. Few quasi-one dimensional combustor models, that include finite rate chemistry, have been developed2-3.

However the major drawback with the existing models is the assumption of the mixing length and the mixing efficiency2-3. Studies show that the fuel volume fraction, fuel mixing and burn

ctor locations are one of the most critical design parameters for a supersonic combustor. The errors in the assumed values for mixing length, fuel injector location and fuel mixing efficiency were shown to have a large impact on the vehicle thrust and also in optimizing the vehicle design2.

Thus the performance of the Scramjet and the vehicle, in turn, is very sensitive to the errors in the assumed values for parameters like the mixing length and the mixing efficiency of

onale for making the choice of these parameters. One, therefore, needs to incorporate the effects of fuel vaporization on the overall combustion process. Many droplet combustion models exist in literature4. Studies have shown that different droplet model show different characteristics which will ultimately influence the accuracy of the analysis of a spray combustion system. A model similar to Droplet Evaporation Model (DEM) 4 has been used in this paper along with a set of quasi-one-dimensional equations of fluid motion coupled with the equations for finite-rate chemistry to model high speed (supersonic) combustor flow fields. Although better models have been reported to exist for droplet combustion4, a model comparable to DEM is used in this work because of its relative simplicity as compared to others and its extensive applications in the spray combustion calculations. The present model DEM can be used to predict the evaporation percentage of the transversely injected fuel spray with downstream distance x from the injector(s) by tracking a single droplet from each injector. The effects of the droplet heat up period and forced convection on the evaporation process have been taken into account. No droplet burning is assumed. The model presented in this paper also incorporates the effects of area change, friction, mass injection, heat transfer through the walls of the combustor on the combustor performance.

The mathematical model presented here allows one to quantify the rates of fuel vaporization and combustion under the conditions of continuously changing pressure, temperatu

previous work, the work presented here does not assume the value of the mixing length and the value comes out from the conditions prevailing in the combustion chamber.

The process of fuel vaporization and combustion are simulated and quantified for Jet-A fuel (C12H23). Jet-A is assumed to be a single component fuel in the coming analy

lowing are the controlling parameters in the simulations done: fuel droplet initial diameter, fuel droplet initial temperature.

fl s for t e engine model are based on the following assumpw, w ere all variables (including area) are functions in th

2) Steady- state flow, 3) Changes in stream properties are continuous, and


3

4) The gas is thermally perfect, i.e. it obeys Boyle’s and Charles’ laws and has a specific heat which varies only with

The continuity equation under the above assumptions can be written in logarithmic differential form as5:

temperature and composition.

A. Equation of Continuity

dxdA

AdxdU

Udxd

dxmd

m1111

++=ρ

ρ&

& (1)

he above equation (1) incorporates the geometry effects (area variation) afuel etc. The area variation is prescribed by the user.

ated from the differential change in the droplet diameter. The remainin

urth

The quasi – one dimensional momentum equation in differential form5 can be expressed as:

T nd the effects of the mass injection of

It is to be noted that only the fuel which evaporates in the given length, dx, is considered to be the rate of mass added to the control volume ( md & ) and is evalu

g fuel which travels through the length dx without evaporating is assumed to be neither entering nor leaving the control volume. It is assumed f er that the evaporated fuel is mixed perfectly with the main stream at the downstream boundary of the control volume.

B. Momentum Equation

0)1(21 222 − mdMCMdUMdp f &εγγγ2 2 =+++

dxmDdxUdxp & (2)

where ε is the ratio of the velocity of gas injection in the x-direction over the velocity of the flow field

∞

=UUiε (3)

Equation (3) accounts for the angled injection (ε = 0 for transverse injection). The friction factor, Cf, accounts for the frictional shearing stress acting on the combustion chamber wall, the drag of stationary bodies immersed in the

stream within the control-surface boundaries etc. The value of Cf is user-specified and is based on the experimental results and /or some analytical methods. The friction coefficient used in simulations to follow is calculated using Eckert’s reference-temperature method assuming fully turbulent flow 6, 7. The viscosity is evaluated at the reference temperature by using Sutherland’s three parameter law for air 8. The hydraulic diameter of a rectangular engine cross-section can be found from:

wPD=

A4 (4)

C. Equation of State For a thermally perfect gas, the equation of state can be written in differential form as3:

dxMWddTddp 1111 ρ

MWdxTdxdxp−+=

ρ (5)

Equation (5) is useful in the sense that that it allows the inclusion of the chemical changes caused by mass injection and combustion. The mixture molecular weight can be written as:


4

∑=

=

= Ni

i i

i

MWY

MW

1

1 (6)

The differential form of (6) is as follows:

⎟⎟⎠

⎞⎜⎜⎝

⎛−= ∑

=

=

Ni

i

i

i dxdY

MWMW

dxMWd

1

2 1 (7)

D. Species Conservation Equation The species conservation equation is based on derivations given by Turns8. The equations are listed here without any proofs. Conservation of species i can be formulated as (refer fig.1):

[ ] [ ] addediixixxi mVmAmAm ,&&&& +′′′=′′−′′ ∆+ (8)

Neglecting molecular diffusion in the flow direction, Equation (8) can be re-formulated as:

[ ] [ ] addediixixxi mxAmAmYAmY ,&&&& +∆′′′=′′−′′ ∆+ (9)

In differential form, equation (9) can be expressed as:

[ ]

dxmd

Am

dxAmYd

Aaddedi

ii ,11 &

&&

+′′′=′′

(10)

Equation (10) can be re-written as:

dxmd

mY

dxmUMW

dxdY iaddediiii &

&&

&−+= ,1

ρmd &ω

(11)

where,

iii MWm ω&& =′′′ (12)

The significance of equation (11) is that it shows the change in mass fraction of species, i is a function of the rate of production/destruction of species i, the amount of species i added, and the total amount of mass added. The net rate of species i, iω& , is evaluated from the reaction mechanism proposed by Kundu et. al.9, for the combustion of Jet-A fuel with air. The net rate equations are manually formed from the individual reaction rates.

E. Energy Equation Based on the formulation of Turns8, the quasi-one-dimensional, steady energy equation for the finite

control volume can be written as:

(13) [ ] [ ] xPQmhhAmhAm w

added

Ni

iiixoxxo ∆′′−⎥⎦

⎤⎢⎣

⎡=′′−′′ ∑

=

=∆+

&&&&1


5

2

Uhho +=2

(14)

Axial heat conduction, axial species diffusion, radiation, and work done on the control volume have in neglected in equation (13). Figure 2 explains the derivation of equation (13). The equation can be written in differential form as:

[ ]

wadded

Ni

iii

o PQdx

mhd

dxhAmd ′′−

⎥⎦

⎤⎢⎣

⎡

=′′

∑=

= &

&& 1

(15)

Using the chain rule, equation (15) can be written as:

dxdUU

dxmmdh

AmPQ

dx

mhd

mdxdh owadded

Ni

iii

−−′′′′

−⎥⎦

⎤⎢⎣

⎡

=∑=

=

&

&

&

&&

&

11 (16)

The enthalpy of the flow field can be written in terms of the species enthalpy as:

∑ (17) =

=

=Ni

iiiYhh

1

The species enthalpy ‘ ’ is available in standard thermodynamic database for all species of interest as a polynomial in temperature. Here it is assumed that the flow field is thermally perfect so that the enthalpy is not a function of flow field pressure. The derivative of enthalpy as a function of axial distance is given by,

ih

dxdTcY

dxdYh

dxdh

pii

Ni

i

ii +⎥⎦

⎤⎢⎣⎡=∑

=

=1 (18)

The second term on the right hand side of equation (16), i.e. the heat transfer term, is calculated from the definition of the Stanton number

( )wawH hhU

QC−′′

=ρ

& (19)

From the Reynolds analogy (i.e. analogy between momentum transfer and heat transfer), the Stanton number can be related to the friction coefficient as

32

Pr2

fH

CC = (20)

where, Pr is assumed to have a constant value of 0.713. Substituting equation (20) in equation (19), the second term in equation (16) can be written as


6

( )

D

TTcCAmPQ wawpfw

32

Pr

2 −=

′′′′&

& (21)

The adiabatic wall temperature can be evaluated from the following relation,

( ) ( )⎥⎦

⎤⎢⎣

⎡⎥⎦⎤

⎢⎣⎡ −

+= 232

*

21Pr1 MTTaw

γ (22)

where *Pr is Prandtl number evaluated at reference temperature. In the present analysis, *Pr is also assumed to have a constant value of 0.713. After some substitutions and rearrangements, the one-dimensional, steady, differential energy equation is:

( )

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡−−

−−⎟

⎠⎞

⎜⎝⎛+−= ∑ ∑

=

=

=

= dxdUU

dxmd

mh

D

TTcCdxmdh

mdxdYh

cdxdT Ni

i

owawpfNi

i added

ii

ii

p 1 32

1 Pr

211 &

&

&

&) (23)

[ ]⎭⎬⎫

⎩⎨⎧

−= ∑=i

addedpiipp cmm

cc1

1ˆ &&

=Ni

(24)

Further, after some algebraic manipulations an equation for derivative of velocity can be obtained as,

( )

( )⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡ −−+

−⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛+−+

⎟⎠⎞

⎜⎝⎛−−+

+−=

∑ ∑=

=

DC

h

TTcM

dxMWd

MWdxmdh

mdxdYh

hdxmd

mhhM

dxdA

AdxdU

fwawp

Ni

i added

ii

ii

o

2

Prˆ

11ˆ1ˆ11

11

32

2

1

2

γ

ελ

α

&

&

&

& (25)

⎟⎟⎠

⎞⎜⎜⎝

⎛+−⎟

⎠⎞

⎜⎝⎛=

hUM

U ˆ11 22

ˆ

γα (26)

(27) Tch pˆ=

III. Governing Equations for Droplet Evaporation Above were mentioned the equations used to model the flow and species conservation in the combustor. Those

equations were coupled with the finite rate chemistry equations. The equations mentioned in this section were used to model the droplet. These equations when solved simultaneously with the above equations gave a good picture of the various phenomena happening in the combustor. These equations relieved the user from assigning an appropriate value to the mixing length, which was a major drawback in the earlier existing models. A single droplet of a given initial diameter was tracked and the analysis was done. The equations used to model the droplet are mentioned.

A. Heat transfer analysis The following assumptions have been made.

I. The fuel considered here possess well defined boiling point.


7

II. Strong recirculation exists within the droplet and hence the intra droplet gradients are neglected. III. Lewis number is unity.

The heat transferred to a droplet moving in a hot air-flow can be expressed as

(28) )(4 2sstrans TThrQ −= ∞π

Assuming steady-state evaporation, heat transfer coefficient for a droplet is

y

y

G

s

BB

KrhNu

)1(ln.2..2 +

== (29)

The effect of forced convection on heat transfer can be accounted for, using the following empirical correlation10

320Re PrRe3.01/ Adhh +==

11 (30)

Thus, the heat transfer coefficient for a vaporizing droplet with forced convection can be expressed as

y

yAd

s

G

BB

rKh

)1ln(.)PrRe3.01( 3

121 +

+= (31)

For numerical calculation, By is defined as the mass transfer number by

Fs

Fsy Y

YB−

=1

(32)

where

1

11−

∞ ⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−+=

F

A

FsFs M

MPPY (33)

Thus, the heat transferred from the hot air to a vaporizing droplet is expressed as

y

yAdsGstrans B

BTTKrQ

)1ln().PrRe3.01()(4 3

112

++−= ∞π (34)

B. Heat used for Droplet Evaporation Latent heat consumed for droplet evaporation is

LmQ Fe .&= (35)

For a spherical droplet, the evaporation of a stagnant droplet can be written as10


8

)1ln(4 yp

GsF B

CKrm

G

+= π& (36)

If the forced convection is taken into account, then

31

21

0Re PrRe3.01)/( AdShSh +== (37)

Assuming the Lewis number is unity, we get

)1ln().PrRe3.01(4 31

21

yAdp

GsF B

CKrm

G

++= π& (38)

and the latent heat of vaporization of the fuel can be written as

n

crit

s

TTAL ⎟⎟

⎠

⎞⎜⎜⎝

⎛−= 1 (39)

Therefore, we get

n

crit

syAd

p

Gse T

TABCKrQ

G

⎟⎟⎠

⎞⎜⎜⎝

⎛−++= 1)1ln()PrRe3.01(4 3

121

π (40)

C. Analysis of Evaporation Process The steps involved in a fuel droplet evaporation process are discussed by Zhu et al10. Based on this, we may write the energy equation for a droplet as

dt

dTCrQQ sFpFsetrans ,

3

34 ρπ=− (41)

FpF

etrans

s

s

CQQ

rdtdT

,3 .

.4

3ρπ

−= (42)

with in each time increment ∆t, equation (38) is still valid and

⎟⎠⎞

⎜⎝⎛−= FsF r

dtdm ρπ 3

34

& (43)

Combining equation (38) and (43), we get

)PrRe3.01)(1ln( 31

21

AdysFp

Gs BrC

Kdtdr

G

++−=ρ

(44)


9

The time step can be converted to the incremental step in the x-direction using following relation:

dudxdt /= (45)

For better accuracy, and are evaluated at the following reference temperature and composition using the “one – third rule” of Sparrow and Gregg,

GKGpC

( ssr TTTT −+= ∞31 ) (46)

( )sFFsFrF YYYY ,,,, 3−+= ∞

1 (47)

Assuming = 0, ∞,FY

sFrF YY ,, 32

= (48)

rFrA ,, YY 1−= (49)

( ) ( )YTKYTKK .. FvrvFrArAG ,, += (50)

( ) ( )YTCYTCC .. rFrFvPrArAPGP ,,,,, += (51)

D. Droplet Trajectory Equation The trajectories of the droplets can be tracked by applying a Lagrangian-based analysis to the droplets. The following assumptions were made:

1) Droplets are always spherical; 2) No droplet break-up occurs; 3) Lift, virtual mass and Basset Forces which takes into account the acceleration history of the droplet are

neglected. These assumptions reduce the droplet momentum equation to include only the effects of the drag and body

forces. The general momentum equations for a single droplet injected along the positive y-direction, transversely into a downward flowing air stream in the positive x-direction, as shown in the figure 3, is described by

bodydragd FFFrrr

+= (52)

where the net force that drives the droplet motion is balanced by the drag force opposing its motion, and the field forces acting on the droplet. The aerodynamic drag force is given by

dFr

DdRRAdrag CAUUFrrr

ρ21

−= (53)

The body force, resulting from an equivalent volume of air that buoys the droplet, includes the gravitational and buoyancy forces. It is given by


10

( ) gVF dAdbodyrr

ρρ −= (54)

Substituting equations (53) and (54) in equation (52), we get

( ) DdRAdAd

dd CAUuudt

duVr

−−= ρρ21

(55)

( ) ( ) gVCAUvvdt

V ddADdRAdAd

dd ρρρρ −+−−=dv r

21

2

(56)

(57) sd rA π=

3

3 sd rV π=4

(58)

dudt

=dx

(59)

dvdt

=dy

(60)

The drag coefficient of the droplet depends on the droplet Reynolds number and is given by

(61)

(24/Red) (1 + Red2/3/6) Red <= 1000

CD = 0.424 Red > 1000

where Red is the droplet Reynolds number and is defined as follows

A

sRAd

rU

µ

ρr

2Re = (62)

in which rs is the droplet radius and µA, ρA are the air viscosity and density at the combustion gas temperature. By solving equations for droplet, we can get the droplet velocity in the x and y direction. The x and y position of the droplet can be calculated by numerically integrating equations (59) and (60).

E. Physical Properties The properties of air can easily be obtained from handbooks and for this reason the physical properties of fuel and fuel vapor are stated here. The fuel used in the present analysis is Jet-A (C12H23). The Critical temperature for the fuel is assumed to be 737 K. Its normal boiling point is 529 K and the molecular weight is 167 kg/kmol. The various other physical properties of the fuel are stated as a function of temperature.


11

1. Fuel liquid density (g/cm3)

n

crit

sTT

F AB⎟⎟⎠

⎞⎜⎜⎝

⎛−−

=1

ρ (63)

where, A = 0.29292, B = 0.26661, n = 0.298.

2. Latent heat of vaporization of fuel liquid (kJ/mol)

n

crit

s

TTAL ⎟⎟

⎠

⎞⎜⎜⎝

⎛−= 1 (64)

where A = 73.509, n = 0.347

3. Specific heat at constant pressure of fuel liquid (J/(mol – K))

(65) 32, sssFp DTCTBTAC +++=

where A= 142.238, B= 1.5261, C= -0.0034477, D = 0.0000032968. 4. Specific heat at constant pressure of fuel vapor (J/(mol – K))

(66) 432, rrrrFvp ETDTCTBTAC ++++=

where A= -128.032, B= 1.4622, C= -0.00086193, D= 0.00000018462, E= 0.00000000000036227. 5. Thermal conductivity of fuel vapour (W/(m – K))

(67) 2rrFv CTBTAK ++=

where A= -0.01184, B= 0.00061839, C= 0.000000025082. 6. Vapor pressure (mm Hg)

21010 loglog sss

svap ETDTTC

TBAP ++++= (68)

)(TPP svapFs = (69)

where A= -50.5512, B= -2705.3, C= 28.273, D= -0.045702, E= 0.000020443.

IV. Solution Methodology To solve the present set of governing differential equations, numerical integration was done. The present set of

governing differential equations formed a stiff set of non-linear ordinary differential equations (ODEs). The ODEs generated were “stiff” as a result of the wide disparity between the concentrations of molecular and radicals species and also because of varying time scales for different physical and chemical phenomena. Out of the numerous techniques available for the numerical integration of stiff ODE in the literature, Gear’s11 method based on backward differencing scheme was found to be the most suitable method. A problem specific code was developed to solve the species conservation equations using Gear’s method (because stiffness of the ODEs is mainly due to the chemical


12

production/destruction terms) and the remaining equations were solved using Runge – Kutta fourth order method12. The Jet – A reaction mechanism proposed by Kundu et al., was used in this study to model the chemical kinetics of the combustion of Jet-A in air. The reaction mechanism consists of 17 species and 23 reactions. The ambient air composition was assumed to be 78 % nitrogen, 21 % oxygen, and 1 % argon by volume. The values for the molecular weight, heat of formation etc. for various species was obtained from the standard references available. The governing equations for the droplet characteristics were solved separately to obtain droplet diameter, droplet temperature, x – component velocity, y – component velocity, x and y coordinates for droplet penetration. The mass flow rate of fuel and mass flow rate derivative are then calculated from the change in the droplet diameter injected from all the injectors. The substitution of the mass addition and chemical information in the species conservation equation gives the change in mass fraction of all the species which can then be substituted into the Equation (7) to solve for the change in mixture molecular weight. The cross-sectional area profile is prescribed by the user. The friction coefficients etc. are known quantities or at least can be calculated. Thus, the velocity derivative derived in Equation (25) can be calculated at a particular x location. With the knowledge of velocity derivative, the density derivative may be found from the continuity equation [Equation (1)]. The pressure derivative can be calculated from the momentum equation [Equation (2)]. The temperature derivative is then found from the equation of state [Equation (5)]. The derivatives of all the variables except mass fraction are then integrated using the Runge – Kutta fourth order method. The mass fraction derivative is integrated numerically by using Gear’s method.

V. Results and Discussion The cross-sectional area variation of the combustor is given in Fig. 4. The program was run for staged fuel injection. The injection points were located at 0.117m. 0.467m, 0.567m, 1.067m and 1.230m along the length of the combustor and the mass fraction of the total fuel injected in each injector is 0.45, 0.10, 0.10, 0.25 and 0.10. The total length of the combustor was 1.890m. The fuel was injected transversely (only evaporated fuel enters the control volume and it is assumed to enter in the transverse direction) (although the code has the provision for simulating the angled injection). The mass flow rate of air is 9.9 kg/s and a fuel equivalence ratio is 1.0. The inlet Mach number of air and static temperature are 1.9 and 1505 K respectively. The heat transfer through the combustor walls were neglected in the present simulation although the given model can account for that. The results obtained for various flow conditions, droplet diameters and droplet initial temperature were plotted. The benchmark conditions for the droplet were that its initial diameter is 20µm, injection velocity of fuel in transverse direction is 50 m/s and its initial temperature is 300K. The results for other droplet conditions are compared with this. The static temperature profile of the main flow field is shown in Fig. 5. The trend shown in the figure indicates the tug of war between the expansion phenomenon occurring due to the diverging duct and the effects of heat addition due to combustion. For the first 0.090m, the temperature rises due to friction and then there is a dip in temperature. This is attributed to the fact that in that zone expansion starts taking place, while fuel injection hasn’t yet occurred. At 0.117m, the first fuel injection occurs, there is a further dip in temperature which is because the droplet initial temperature is lower than the main flow temperature and hence it extracts some heat from the main flow which results in decrease in temperature. This is followed by a gradual rise in temperature. This gradual rise in temperature occurs primarily because of the heat addition due to combustion of the evaporated droplet. In this region, heat addition is more influential than the expansion phenomenon. Further beyond the distance of 0.990m from the inlet, the expansion process increases and hence there is a temperature fall. Besides, the injected fuel is almost exhausted and hence no significant heat addition occurs. Finally after the fuel injection at 1.067m and 1.230m, there is some heat release and the temperature rises and then again falls due to expansion. This suggests that by a clever distribution of fuel between various injector locations, one can get a desired temperature profile. Based on the similar lines of reasoning, one can validate the Mach number (Fig.6) and the pressure plots (Fig. 7). The pressure plots looks almost like a mirror image of the Mach number plot. Fig. 8 shows the droplet trajectories. It can be seen from the figure that the droplet penetration is maximum for the first droplet and is lower for the droplet injected from second and third injection ports. The penetration length increases again for the last two injected droplets. This is attributed to the temperature profile. At lower main flow field temperatures, the time taken by droplet to evaporate is more and so it has more time to penetrate into the core flow whereas for the higher temperatures (as for second and third droplets) the time available to the droplets is less and hence they get evaporated before they can penetrate much. The distance traveled by the droplet along the length of the combustor, before it evaporates completely, can be seen from Fig. 9 and Fig. 10. One observation that can be made from Fig. 10 is that the droplet wet bulb temperatures is varying along the combustor length. This is attributed


13

to the fact that the pressure is varying along the length. Since, the boiling point of the droplet increases /decreases with the increase/decrease in the pressure, the wet bulb temperature is higher for the second and third droplet and is lower for the rest of the cases. Studies were also conducted on the effect of droplet initial temperature and droplet initial diameter on the various parameters that have been discussed earlier. Fig. 11-14 shows the effect of droplet initial temperature on the static temperature, static pressure, Mach number of the main flow and also its effect on droplet penetration length. The temperature of the main flow field is shown to rise little, which is because the heat absorbed by the newly introduced droplet to attain the wet bulb temperature and to evaporate decreases with the increase in the droplet initial temperature and hence the temperature of the main flow field is higher for the droplet with higher initial temperature. Based on this reasoning, the effect of droplet initial temperature on other parameters can be understood. The droplet initial temperature has a very little effect on the droplet wet bulb temperature (Fig. 16). In reality, it should not affect the wet bulb temperature, but a small change in the wet bulb is observed which is attributed to the fact that the pressure along the combustor length is different for the three cases of different droplet initial temperature. Thus, due to this change in pressure the droplet wet bulb temperature is getting affected, which is an indirect effect of the droplet initial temperature. The change in the droplet initial diameter has a predominant influence on the various flow parameters and finally shows its influence in terms of the lower thrust output for the droplets of larger diameters. Thrust calculations made from this model showed that the thrust output decreases from 5121N to 4350N when the droplet size is increased from 20µm to 40µm. From the Fig.20, it can be seen that the model shows that the larger size droplets are not completely evaporated even after they reach the end of the combustor, which should be the case. Thus, the model is able to follow the physics of the problem and the results predicted by it seem to be formidable.

VI. Conclusion A mathematical model has been developed and simulated in order to understand the various physical and

chemical processes occurring in the (supersonic) combustor. A set of quasi-one- dimensional equations of fluid motion have been coupled with the equations of finite rate chemical kinetics for the same. The single droplet evaporation model has been used to incorporate the effects of the fuel vaporization on the combustion process. The proposed model has a clear physical meaning with a definite advantage, besides being simple to use. Mathematical dependencies between the caloric parameters and thermal parameters have been provided for describing the thermodynamic behavior of the gas under the flow conditions of the Scramjet combustor. The model is simulated for Jet-A fuel (C12H23) assuming that Jet-A is a single component fuel. The model gives an opportunity to investigate the interactions between the combustion-related parameters and combustion process and engine performance with respect to the engine operating conditions and the parameters of fuel injection and evaporation. The model assists in better understanding of various ongoing physical and chemical processes that occur during the combustion in the Scramjet combustor. The process of fuel vaporization and combustion are simulated and quantified with respect to the following controlling parameters: fuel droplet initial diameter, fuel droplet initial temperature and the mass distribution between different injectors located at different locations. The results show very strong influence of the droplet diameter and droplet initial temperature on the vaporization and combustion processes, with respect to the duration of these processes.

References 1White, M.E., Drummond, J.P., and Kumar, A., “Evolution and Application of CFD technique for Scramjet Engine

Analysis,” Journal of Propulsion, Vol.3, No. 5, Sept-Oct. 1987, pp.423-438. 2Starkey, R.P., and Lewis, M.J., “Sensitivity of Hydrocarbon Combustion modeling for hypersonic Missile Design,” Journal

of Propulsion and Power, Vol.19, No.1, January-February, 2003, pp. 89-97. 3O’Brien, T.F., Starkey, R.P., and Lewis, M.J., “Quasi-One-Dimensional High Speed Engine Model with Finite-Rate

Chemistry,” Journal of Propulsion and Power, Vol.17, No.6, Nov- Dec, 2001, pp. 1366-1374. 4Huang, J.S., and Chiu, H.H., “Comparison of Droplet Combustion Models in non-premixed spray combustion,” 38th AIAA

/ASME/SAE Propulsion Conference and Exhibit, Indianapolis, Indiana. AIAA Paper 2002-4181 5Shapiro, A., “The Dynamics and thermodynamics of Compressible Fluid Flow,” Vol.1, Ronald, New York, 1953, pp. 219-

260. 6Eckert, E.R.G., “Engineering Relations for Friction and Heat Transfer to Surfaces in High Velocity Flow,” Journal of

Aeronautical Sciences, Vol.22, 1955, pp.585-587. 7Eckert, E.R.G., “Engineering Relations for Heat Transfer and friction in High Velocity Laminar and Turbulent Boundary-

Layer Flow over Surfaces with Constant Pressure and Temperature,” Transactions of the ASME, Vol.78 , No.6, 1956, p.1273-1283.

8 Turns, S.R., An Introduction to Combustion, 1st edition, McGraw-Hill, New York, 1996, pp.151-207.


14

9Kundu, K.P., Penko, P.F. and Yang, S.L., “Reduced Reaction Mechanisms for Numerical Calculations in Combustion of Hydrocarbon Fuels,” AIAA Paper 98-0803, Jan. 1998.

10Zhu, J.Y., and Chin, J.S., “Characteristics and Evaporation History of Fuel Spray Injected into Cross-flowing Airstreams,” Journal of Propulsion, Vol. , No.3, May-June, 1987, pp. 227-234.

11Gear, C. W., “Numerical Initial Value Problems in Ordinary Differential Equations”, Prentice Hall, New Jersey, 1971. 12Press, W.H., Teukolsky, S.A., Vetterling, W.T., and Flannery, B.P., “Numerical Recipes in C++: The Art of Scientific

Computing,” Second Edition, Cambridge University Press, Cambridge, pp. 715-716.


15

LIST OF FIGURES

1. Quasi-one-dimensional control volume for species conservation derivation. 2. Quasi-one-dimensional control volume for energy derivation. 3. Free body diagram of the Droplet. 4. Schematic of the Scramjet Combustor. 5. Static temperature distribution along the length of the combustor. 6. Mach number distribution along the length of the combustor. 7. Static pressure distribution along the length of the combustor. 8. Droplet trajectory. 9. Droplet evaporation history along the length of the combustor. 10. Droplet surface temperature variation along the length of the combustor. 11. Effect of Droplet initial temperature on Static temperature distribution of the flow. 12. Effect of Droplet initial temperature on Mach number distribution of the flow. 13. Effect of Droplet initial temperature on Static pressure distribution of the flow. 14. Effect of Droplet initial temperature on Droplet trajectory. 15. Effect of Droplet initial temperature on Droplet evaporation history. 16. Effect of Droplet initial temperature on Droplet surface temperature variation. 17. Effect of Droplet initial diameter on Static temperature distribution of the flow. 18. Effect of Droplet initial diameter on Mach number distribution of the flow. 19. Effect of Droplet initial diameter on Static pressure distribution of the flow. 20. Effect of Droplet initial diameter on Droplet trajectory. 21. Effect of Droplet initial diameter on Droplet evaporation history. 22. Effect of Droplet initial diameter on Droplet surface temperature variation.


16

Figure 1

added

Ni

iiimh ⎥⎦

⎤⎢⎣

⎡∑=

=1

&

xPQ w ∆′′&

[ ] xxohAm ∆+′′&[ ]xohAm ′′&

∆x

Figure 2

gVforcebuoyant dAρ,

dtUdmmomentumdroplet d

d

r

,

directionyinforcedrag −and

forcenalgravitatio

y

x

directionxinforcedrag − Air flow

Figure 3


17

Figure 4

270

144.0

90 90 94.5

90 180 720 900

NOT TO SCALE

Figure 5

Figure 6


18

Figure 7

Figure 8

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.5

1

1.5

2x 10

-5

Length of the combustor, m

Dro

plet

dia

met

er, m

First dropSecond dropThird dropFourth dropFifth drop

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2300

350

400

450

500

550


Dro

plet

tem

pera

ture

, C

Figure 9

Figure 10

First drop Second drop Third drop Fourth drop Fifth drop


19

Figure 11

Figure 12

Figure 13

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

1

2

3

4

5

6x 10-3

Length of combustor, m

Pen

etra

tion,

m

Red: 350 K Black: 300 K Blue: 260 K

Figure 14


20

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

1

2x 10-5

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

1

2x 10-5

Dro

plet

dia

met

er, m

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

1

2x 10-5


Droplet Temperature =300 K

Droplet Temperature = 260 K


Figure 15

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2300

400

500

600

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2200

400

600

Dro

plet

tem

pera

ture

, K

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2300

400

500

600


Droplet Temperature =300 K



Figure 16


21

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 21400

1600

1800

2000

2200

2400

2600


Tem

pera

ture

, K

Droplet diameter = 20 micronsDroplet diameter = 30 micronsDroplet diameter = 40 microns

Figure 17

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 21.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2


Mac

h nu

mbe

r


Figure 18

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 22

3

4

5

6

7

8

9

10

11x 104


Pre

ssur

e, N

/m2


Figure 19

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

Length of combustor, m

Pen

etra

tion,

m

Figure 20 Red – 40 microns droplet, Blue – 30 microns droplet,

Black – 20 microns droplet


22

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

1

2x 10-5

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

1

2

3x 10-5

Dro

plet

dia

met

er, m

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

2

4x 10-5


Droplet diameter =20 microns

Droplet diameter = 30 microns


Figure 21

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2300

400

500

600

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2300

400

500

600

Dro

plet

tem

pera

ture

, K

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2300

400

500


Droplet diameter =20 microns



Figure 22


23

Documents

[American Institute of Aeronautics and Astronautics 42nd AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit - Sacramento, California ()] 42nd AIAA/ASME/SAE/ASEE Joint Propulsion