Radiative Heating in Combustion Chamber of Liquid

Radiative Heating in Combustion Chamber ofLiquid Propellant Rocket Engines*

Takeshi KANDA† and Masaki SATO

Research and Development Directorate, Japan Aerospace Exploration Agency, Kakuda, Miyagi 981–1525, Japan

The effects of radiation heat transfer in rocket engine combustion chambers are studied analytically. The fuel is hydro-gen, methane or ethanol, and the oxidizer is oxygen. Radiative heat fluxes are estimated using empirical equations, andconvective heat flux is estimated using flux on a flat plate, with modification of circumferential length in convergenceto, or divergence from, the throat. The calculated total heat flux including radiation and convection showed reasonableagreement with the flux measured experimentally. The ratio of radiative heat flux to total flux is increased up to 30%in the cylindrical section, whereas it is less than 10% at the throat. The effect of radiation on the total amount of heat trans-ferred to the chamber is remarkable when increasing the length of the cylindrical section and diameter, respectively. It isalso made clear that the conventional estimation method of heat flux based on the pipe flow model can estimate larger heatflux in small chambers, in spite of ignoring the effects of radiation.

Key Words: Radiation, Convection, Heat Flux, Combustion Chamber

Nomenclature

A: cross-section, parametera: parameterB: parameterb: parameterC: constantC1: constantcf: friction coefficientcH : Stanton numberCp: specific heatD: chamber diameterf: function of heat flux

hg: heat transfer coefficientk: factor for multiplying D in order to obtain mean radi-

ation layer thicknessL: lengthM: Mach number_m: mass flow raten: power number of viscosity equationp: pressure

Pr: Prandtl numberQ: heat rateq: heat fluxR: gas constantr: ratioS: parameter

Re: Reynolds numberReu: unit Reynolds number

S: emissivity, surface areas: mean thickness of radiating gas layerT: temperature

U: velocityx: stream-wise distance¡: ratio of boundary layer heights¤: boundary layer thickness¾: parameter in Bartz’s formula£: ratio of specific heats®: viscosityµ: density

Subscriptsam: meanB: Bartz’s formulac: convection, chamber, compressible

CO2: carbon dioxideD: diameterd: divergent sectione: exit, edge of boundary layerg: gas

H2O: steami: incompressiblep: pressurer: radiation, recoveryt: throat

total: totalv: convergent sectionw: wallx: stream-wisey: cylindrical section0: reference condition1: initial condition2: final condition

1. Introduction

The combustion chamber of liquid rocket engines receivesheat by radiation as well as convection. In general, the contri-bution of radiative heat transfer is small, and only convective

© 2016 The Japan Society for Aeronautical and Space Sciences+Received 16 July 2015; final revision received 11 April 2016; acceptedfor publication 22 June 2016.†Corresponding author, [email protected]

Trans. Japan Soc. Aero. Space Sci.Vol. 59, No. 6, pp. 332–339, 2016

332

heat transfer is a major concern in the thermal analysis ofcombustion chambers. The convective heat flux calculatedusing the conventional method often overestimates the fluxmeasured experimentally. However, the effect of radiativeheat transfer was pointed out to be 5 to 35% of the totalamount of heat transferred to the combustion chamber.1) Inrecent studies based on numerical analysis,2,3) it is also re-ported that the local heat flux and total amount of heat trans-ferred are significantly influenced by radiation. For example,radiative heat flux can reach about 10% of the overall heatflux for SSME. Therefore, the accurate estimation of heat fluxincluding the effect of radiation is important in the coolingdesign for combustion chambers. The accurate estimationof the total amount of heat transferred is also essential whendesigning engine systems; especially when an engine adoptsthe expander cycle that utilizes the coolant heated in the com-bustion chamber cooling jacket as the turbine-drive gas.

The present study examines an outline of the effect of ra-diative heat transfer on the local heat flux and the totalamount of transferred heat. In this paper, propellant combi-nations are oxygen/hydrogen, oxygen/methane and oxy-gen/ethanol. The radiative heat flux and convective fluxare estimated analytically using empirical and theoreticalequations. The effects of radiation are discussed in relation-ship to the propellant and the size of chambers. The charac-teristics of heat flux estimated using the conventional methodare also discussed by comparison with the flux estimated bythe present method for further use in the estimation.

2. Calculation Method

2.1. Heat fluxThe heat flux by radiation from steam and carbon dioxide

can be calculated using the formulae of Schack4) as follows,

qr;H2O ¼ 0:0446� 10�6 � S � ðpH2O � sÞ0:6½Tg3 � Tw3� ð1Þ

qr;CO2¼ 0:8729� 10�9 � S � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

pCO2� s3

p � ½Tg3:5 � Tw3:5� ð2Þ

where the constant values are re-calculated based on SI units.pH2O and pCO2

are the partial pressures of steam and carbondioxide in the combustion gas, respectively. Tg is the statictemperature of the combustion gas. Power numbers of tem-perature are specified by the experiments. S represents theemissivity (i.e., the ratio of the radiation constant of actualmaterial to that of black body). The mean thickness of the ra-diating gas layer, s, is expressed as

s ¼ k �D ð3Þk of 0.95 for the gas body shape of infinitely long cylinderand S of 0.65 for the surface of oxidized copper are usedin the present study.

The equations were derived from several experimental re-sults by, for example, Schmidt, Hottel and Mangelsdorf orEckert. In the experiment by Hottel and Mangelsdorf, the ra-diation was collected by a gold-plated concave mirror andthe radiation was measured with a thermocouple. This isone of the ways to measure radiation experimentally, as wellas using a mirror wall.

The formulae were confirmed by comparison with the pre-vious experimental results. The measurements were con-ducted to 1600 K and to several atmospheres. They werelower than conditions in the combustion chamber, respec-tively. The subject of this study is to investigate an outlineof the effect of the radiation. However, the formulae maynot have sufficient accuracy to estimate heat flux in the rocketengine combustion chambers. The applicability of the for-mulae in the combustion chamber will be discussed inChapter 3.

Convective heat flux is calculated using a flow modelalong with a flat plate with modifications on the convergentsection and divergent section near the throat. The convectiveheat flux is

qc ¼ cH � � � u � CpðTr � TwÞ ð4ÞIn another style,

qc ¼ cH_m

ACpðTr � TwÞ ð5Þ

where the wall temperature measured in experiments is usedfor Tw. In case of no measured data of Tw, the presumed tem-perature of 700 K is used in this study. Based on the Rey-nolds analogy, the Stanton number, cH , is calculated usingthe friction coefficient, cf, as follows:

cH � ð1=2Þcf ð6ÞIn the cylindrical section, the friction coefficient is calculatedusing the formula for the turbulent incompressible flow.5)

cf;i � 0:025Rex�1=7 ð7Þ

On the other hand, the coefficients of the compressible flowat the throat and chamber exit are calculated using the formu-la of White.5)

cf;c �0:455

S2 ln20:06

SRex

�e

�w

ffiffiffiffiffiffiTe

Tw

s !

where

S ¼ ðTr=Te � 1Þ1=2sin�1Aþ sin�1B

A ¼ 2a2 � b

ðb2 þ 4a2Þ1=2 and B ¼ b

ðb2 þ 4a2Þ1=2

a ¼ � � 1

2Me

2 Te

Tw

!1=2and b ¼ Tr

Tw� 1

!ð8Þ

Here, a recovery factor of 1.0 is presumed, and the recov-ery temperature, Tr, becomes equal to the total temperature,Ttotal.

The combustion chamber has a convergent section to thethroat and a divergent section to the chamber exit. The massflow rate in the boundary layer is increased as the fluid flowson the wall. Assuming the mass flow rate in the layer is con-served, the boundary layer becomes thicker in the convergentsection than the layer flows in a straight duct. In such a sit-uation, the velocity gradient and temperature gradient to

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the wall, and also the heat flux, are decreasing. The thicknessof the turbulent boundary layer is given as follows5):

� � 0:14x6=7 � Reu�1=7 ð9ÞThe ratio of thickness of the boundary layers at two differentpositions can be expressed as,

ð�2=�1Þ ¼ � � ðx2=x1Þ6=7 ð10ÞDue to conservation of the mass flow rate in the boundarylayer, the thickness of the layer is inversely proportional tothe length of circumference. Therefore, the ratio of thickness,¡, can be redefined as,

� � ðD1=D2Þ � ðx2=x1Þ6=7 ð11ÞThen, Eq. (7) is modified as follows:

cf;2 � 0:025Re2�1=7 ¼ 0:025Re1

�1=7 � ��1=6 ð12ÞThis modification is derived using the relationship for theincompressible gas flow, whereas the gas is compressiblein the divergent section of the chamber. Herein, the modifi-cation is also applied to the flow in the divergent sectionfor simplicity.2.2. Propellant

Three kinds of propellant combustion are examined;oxygen/hydrogen, oxygen/methane, and oxygen/ethanol.The properties of the combustion gases in stagnant conditionare calculated using the CEA code.6) Properties in the cylin-drical section, at the throat and at the chamber exit are calcu-lated under the frozen condition for simplicity in parametriccalculation. Viscosity is presumed to be proportional to statictemperature.5) The subject of the present study is to investi-gate an outline of the effects of radiation. Use of these calcu-lation methods is within the scope of this study.2.3. Amount of heat transferred

The total amount of heat transferred to the chamber wall iscalculated applying a simple integration method using the es-timated values of heat flux. As for the friction coefficient nec-essary for heat flux estimation, there is no general formulaapplicable to both incompressible flow and compressibleflow. Therefore, the friction coefficient in the convergent sec-tion is calculated by interpolation of the coefficient in the cy-lindrical section and that at the throat.

Figure 1 shows a schematic of representative heat flux dis-tribution in a combustion chamber. As shown in this figure, alow heat flux region exists in the vicinity of the injector face

plate. This injector effect with local reduction in the heat fluxis caused by the insufficient progress in combustion. Kuma-kawa et al. derived an empirical relational expression to es-timate the length of the injector effect.7) However, reports donot always refer all of the data necessary for the empirical ex-pression. Thus a specified length of the low heat flux regionis presumed in the present study, based on the experimentalresults reported. The heat flux calculated in the cylinder sec-tion is compared with the flux measured downstream of thisregion. In calculating the total heat transferred, the length is0.05m for a small engine with a throat diameter of 0.1m, and0.1m for a large engine with that of 0.3m, respectively.8) Insmall engines (e.g., LE-5), the length is around 0.05m andthat is around 0.1m in large engines (e.g., LE-7).

3. Results and Discussion

3.1. Calculated and measured heat fluxThe values of heat flux calculated by the present method

are compared with the measured values of past experiments.Table 1 shows the summary of firing test conditions of theexperiments referred in the present paper.7–13) In these firingtests of methane- and ethanol-fueled combustors, it is as-sumed there is no radiation from solid carbon.

The results of comparison between the heat flux calculatedand the flux measured are shown in Fig. 2. Figure 2(a), (b)and (c) show comparisons in the cylindrical section, at thethroat and at the chamber exit, respectively. Each heat flux

Fig. 1. Schematic of heat flux distribution in combustion chamber.

Table 1. Summary of firing test conditions in past experiments.

Diameter ofFuel cylindrical

Diameter at Chamber Combustor Mixture ratio,

section, mmthroat, mm length, mm pressure, MPa O/F

LE-58Þ H2 240 136 309 3.5 6.1Kumakawa et al.7Þ H2 66, 67 23 157, 207 7.12–9.50 4.77–6.86Niino et al.9Þ H2 66 28 236 2.78–3.52 4.27–6.46Elam10Þ H2 144 90.9 379, 481 10.2–16.4 5.24–6.90Hiraiwa et al.11Þ C2H6O 66 25, 33 330 4.0, 7.8 1.6Kumakawa et al.7Þ CH4 66, 67 23 157, 207 3.5–9.6 2.71–5.06Kim and Ju12;13Þ CH4 176 101 342 6.4 3.0


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calculated plotted in these figures is the sum of the radiativeand convective heat flux. The comparisons in the cylindricalsection and at the throat show good agreement, as shown inFig. 2(a) and (b). Figure 2(b) also shows that the presentmodification of the boundary layer thickness in the convec-tive heat flux estimation is effective. The heat flux calculated

with no modification overestimates the experimental flux, asshown in Fig. 3.

As shown in Fig. 2(c), however, it should be noted that theheat flux at the chamber exit can be overestimated. One of thepossible reasons is underestimation of the boundary layerthickness in the divergent section. The modification isderived from the relationship in the incompressible flow.The present modification requires further examination to beapplied to the compressible boundary layer. However, thevalues of heat flux in the divergent section are significantlylower than those near the throat, thus the impact on the totalamount of heat transferred to the chamber would be small.3.2. Effect of radiation on heat flux

Based on the heat flux calculated using the presentmethod, the effect of radiation on heat flux is examined.Figure 4 shows the ratios of radiative heat flux to the totalheat flux for the oxygen/hydrogen combustor. The mixtureratio is O/F = 6 and the chamber pressure is pc ¼ 3 or10MPa. The size of the combustion chamber is varied from

(a) Cylindrical section

(b) Throat

(c) Chamber exit

Fig. 2. Comparison between calculated and measured heat flux.

Fig. 3. Comparison between calculated and measured heat flux at throatwith no modification.

Fig. 4. Ratio of radiative heat flux to total flux in oxygen/hydrogen com-bustion chamber at O=F ¼ 6.


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0.1 to 0.5m in diameter with the contraction ratio of 2.78. Asthe chamber diameter is increased, the ratio of radiative heatflux increases up to 30% in the cylindrical section, whereasthe ratio at the throat is less than 10%.

The ratio of radiative heat flux in the cylindrical sectiontakes a higher value than that at the throat, and it also takesa higher value in the case of lower chamber pressure. Thesecharacteristics can be explained with the equations used inthe present method. Equation (1) of the radiative heat fluxis rewritten as

qr;H2O � 0:0446� 10�6Sðrp � ptotal � k �DÞ0:6½Ttotal3 � Tw3�

1þ � � 1

2M2

!3:6��3��1

ð13Þwhere,

Tw=Ttotal < 1 ð14Þis presumed. Equation (4) of the convective heat flux isrewritten with Eq. (6) as

qc �1

2cf � ptotal

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�

R � Ttotal

sCp

� ðTtotal � TwÞM

1þ � � 1

2M2

! �þ12ð��1Þ

ð15Þ

The radiative heat flux is proportional to the 0.6 power of thechamber diameter and is proportional to the 0.6 power ofthe total pressure, whereas the convective heat flux is propor-tional to total pressure. The ratio of the radiative heat flux tothat of convective heat flux is expressed with Eqs. (13) and(15) asqr;H2O

qc� C1 �D0:6 � Ttotal

2:5

Ptotal0:4

� 1

M� 1

1þ � � 1

2M2

!6:2��7

2 ��1ð Þ

ð16Þ

Here,

C1 ¼ 0:0892� 10�6 � S � ðrp � kÞ0:6cf � Cp

ffiffiffiffiR

�

sð17Þ

The ratio of radiative heat flux becomes larger as the diame-ter increases, and smaller as total pressure increases.

Figure 5 compares the heat flux measured with the fluxcalculated applying the convection only in the cylindricalsection. Many convective heat fluxes calculated are lowerthan those measured. Difference from Fig. 2(a) is causedby the radiation, corresponding to the amount shown inFig. 4. Though the conditions in the combustion chamberare out of the original applicable conditions of Eqs. (1) and(2), there will be no great problem using the equations.The accuracy in estimating heat flux will be improved by in-cluding radiation. It depends on the position of measurement,

chamber geometry and operating conditions of the engine.The effect of radiation on the total amount of transferred heatwill be discussed in Section 3.3.

Figure 6 shows the results for the oxygen/methane com-bustor in the cylindrical section. The combustion gas con-tains carbon dioxide as well as steam. Figure 6 shows notonly the ratio of radiative heat flux to total radiative flux,but also the molecular ratio in the combustion gas for steamand carbon dioxide. In the ratio of radiative heat flux, the ra-diation from carbon dioxide is lower than that from steam.This characteristic is also derived from the equations usedin the present method. From comparison of Eqs. (1) and(2), the constant number for carbon dioxide is smaller thanthat for steam. The exponential number of partial pressurefor carbon dioxide is also smaller than that for steam; besidesthe value of partial pressure for carbon dioxide is smallerthan that for steam as shown in Fig. 6 by the molecular ratio.Therefore, the effect of radiation from carbon dioxide is sig-nificantly lower than that from steam. This implies that theradiation effect in hydrocarbon-fueled combustors is lessthan that in hydrogen-fueled combustors.

Fig. 5. Comparison between measured heat flux and calculated flux byconvection only in the cylindrical section.

Fig. 6. Ratio of radiative heat flux and molecular ratio in cylindrical sec-tion of oxygen/methane combustion chamber at pc ¼ 5MPa.


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3.3. Effect of radiation on total amount of heat trans-ferred

In order to examine the effect of radiative heat transfer onthe engine system design, the total amount of heat transferredto the chamber with and without radiative heat transfer is cal-culated. Figure 7 shows the ratio of those total amounts ofheat for the oxygen/hydrogen combustor. Qtotal denotesthe total amount of heat with radiation. It is calculated by in-tegrating both radiative and convective heat flux. On theother hand, Qc denotes the total amount of heat without ra-diation and with convection only, and is calculated by inte-grating only the convective heat flux. In the value ofQtotal=Qc, the part beyond 1 shows the effect of radiation.The ratio of Qtotal=Qc is plotted as a function of the ratioof Ly=Dt, where Ly denotes the chamber length in the cylin-drical section and Dt denotes the chamber diameter at thethroat. The chamber exit diameter is the same as the chamberdiameter in the cylindrical section. The contraction ratio forthe large chamber with Dt ¼ 0:3m is 2.78, and the ratio forthe small chamber with Dt ¼ 0:1m is 2.89. Figure 7 showsthat the effect of radiation becomes larger as chamber lengthincreases in the cylindrical section and also as the throat di-ameter increases. On the other hand, the effect of radiationbecomes less as chamber pressure increases.

Herein, the ratio, Qtotal=Qc, is expressed in a numericalformula. The radiative heat flux in Eq. (1) is rewritten witha Mach number as follows:

qr � Cr �D0:6 � ptotal0:6 � Ttotal3 � frðMÞ ð18Þwhere,

Cr ¼ 0:0446� 10�6 � S � ðrp � kÞ0:6 ð19Þ

fr ¼ 1 1þ � � 1

2M2

!3:6��3��1

,ð20Þ

In Eq. (19), rp is the ratio of the steam partial pressure to thewhole pressure. On the other hand, convective heat flux inEq. (4) is rewritten as follows:

qc � Cc � ptotal � Ttotal0:5 � fcðMÞ ð21Þ

where,

Cc ¼1

2cf �

ffiffiffiffi�

R

s� Cp ð22Þ

fc ¼ M 1þ � � 1

2M2

! �þ12ð��1Þ

,ð23Þ

From Eqs. (18) and (21), the ratio Qtotal=Qc is expressed asfollows:

Qtotal

Qc

�1þ Cr

Cc

� Ttotal2:5 �Dt

0:6

ptotal0:4

� SyfrðMyÞgðMyÞ þ SvðMtÞgðMtÞ þ SdfrðMeÞgðMeÞSy � fcðMyÞ þ Sv � fcðMtÞ þ Sd � fcðMeÞ

ð24Þwhere, S is surface area, and g is defined as a function of theratio of diameters as follows:

g Mð Þ ¼ ðD=DtÞ0:6

¼ 1

M0:3

2

� þ 11þ � � 1

2M2

!" #0:6ð�þ1Þ2ð��1Þ

ð25Þ

From Eq. (24), it is clarified that ratioQtotal=Qc is propor-tional to 0.6 power of the throat diameter, Dt, and is in-versely proportional to 0.4 power of total pressure, ptotal.

In order to show the effect of each length of the chamber, asample calculation is executed for an oxygen/hydrogen com-bustor with the mixture ratio of O=F ¼ 6. Mach numbers areMy ¼ 0:2,Mt ¼ 1, andMe ¼ 2. The ratio of partial pressureof steam of rp ¼ 0:75, with molecular weight of 14. Othersare presumed as � ¼ 1:2, S ¼ 0:65, k ¼ 0:95, cf ¼ 0:003,R ¼ 594 J0kg¹10K¹1, and Cp ¼ 3560 J0kg¹10K¹1. HereCr ¼ 0:02366� 10�6 and Cc ¼ 0:2402. Under these condi-tions, Eq. (24) is rearranged as follows:

Qtotal

Qc

�1þ 0:0985� 10�6 � Ttotal2:5 �Dt

0:6

ptotal0:4

� 1:98 � Sy þ 0:90 � Sv þ 0:33 � Sd0:33 � Sy þ Sv þ 0:53 � Sd

ð26Þ

In the numerator of the second term, the coefficient of thecylindrical surface area, Sy, is the largest. This means that thecylindrical surface area (i.e., the chamber length in the cylin-drical section) has an intensive effect on Qtotal=Qc. There-fore, the effect of radiation should be taken into accountwhen designing an engine with a long combustion chamber.3.4. Conventional estimation of heat flux

As shown in the previous section, radiation has a signifi-cant effect on the total amount of heat transferred to thechamber, especially for chambers with a long cylindrical sec-tion. In experiments, the heat flux measured was larger thanthe convective heat flux, as shown in Fig. 4. However, therehas not been sufficient attention paid to the effect of radiationin the estimation of heat flux.

One of the reasons considered for this is the fact that mostcombustion chambers do not have a long cylindrical section,

Fig. 7. Ratio of total heat transferred to that by convection only in an oxy-gen/hydrogen combustion chamber at O=F ¼ 6.


337©2016 JSASS

,

,

so the effect of radiation is not prevalent. Another reason isthe fact that the heat flux estimated using the conventionalmethod frequently results in overestimation compared tothe heat flux measured in spite of having ignored the effectof radiation.

In the estimation of heat flux, the gas-side heat transfer co-efficient by Bartz14) is well known and frequently used. Wa-kamatsu showed the results of comparison between the heatflux using the formula of Bartz and that using the formula ofMayer,15) and pointed out that the heat flux using Mayer’sformula agreed well with the experimental values, whereasthat using Bartz’s formula did not.16) There is a differencein the boundary layer model between them. The formula ofMayer adopts a turbulent boundary layer integral methodalong the flow direction of the combustion gas. On the otherhand, the simplified formula of Bartz adopts the pipe flowmodel. As a specific example, Fig. 8 shows the results ofthe comparison between the heat flux calculated using theBartz’s formula and the flux measured in past experiments.The overestimation of the heat flux using Bartz’s formulais also reported in other reports.7,9,11)

Herein, the numerical expression of heat flux using Bartz’sformula is compared with that in the present paper. The heattransfer coefficient of Bartz’s formula is:

hg;B ¼ 0:026

D0:2

�am0:2 � CpamPram0:6

ð�am � uÞ0:8 ð27Þ

The parameters with a subscript of am are evaluated usingthe mean temperature defined as follows:

Tam ¼ ðT þ TwÞ=2 ð28ÞSpecific heat, Cp, is not a function of temperature in a frozenideal gas, so Eq. (27) is rewritten as follows:

hg;B ¼ 0:026

Pr0:6� 1

ReD0:2� ð� � uÞ � Cp � �am

�

!0:2� �am

�

!0:8ð29Þ

where,

ReD ¼ ð� � u �DÞ=� ð30ÞViscosity, ®, is a function of temperature, so its ratio can bereplaced by the temperature ratio as follows:

�=�0 � ðT=T0Þn ð31ÞFor the power number of n, Bartz adopted 0.6. Prandtl num-ber, Pr, is around 0.7 in most gases.5) Therefore, Eq. (29) isrewritten as follows:

hg;B � 0:0644

2� 1

ReD0:2� ð� � uÞ � Cp � " ð32Þ

where

" ¼ 1þ ðTw=T Þ2

!0:2n�0:8

¼ 1

1

2

Tw

Tt1þ � � 1

2M2

!þ 1

2

( )0:8�0:2nð33Þ

On the other hand, the heat transfer coefficient in the presentconvective heat flux in Eq. (4) is expressed as follows, usingthe friction coefficient of the incompressible flow in Eq. (7):

hg;c ¼0:025

2� 1

Rex1=7� D

Dy

!1=6� � u� � � Cp ð34Þ

With the use of Eqs. (32) and (34), the ratio of the heat trans-fer coefficients is expressed as follows:

hg;B

hg;c� 0:0644

0:025� Rex

1=7

ReD0:2� Dy

D

!1=6�" ð35Þ

Finally, Eq. (35) is rewritten as follows:

hg;B

hg;c� 2:576 � 1

� � u �Dy

�

!0:06 x

Dy

!0:14Dy

D

!0:37�" ð36Þ

In the evaluation of ¾, static temperature is used in this paper,whereas Bartz used the total temperature in his formula.

Figure 9 shows the ratio of the heat flux at the throat cal-culated using Eq. (36) for the hydrogen-, methane-, and etha-nol-fueled chambers. Here, a short chamber length of x =

0.05m in the cylindrical section is assumed. The throat diam-eter, Dt, is varied up to 0.3m with the constant contractionratio of 2.78. The heat flux ratio in Fig. 9 is decreased as Dt

increases. However, the heat flux ratio is always larger than1.0. This means it is not possible to calibrate heat flux using ascaled chamber to estimate the heat flux of a larger chamber.Here, it should also be noted that the heat flux calculated inthe denominator is the convective heat flux alone, and the ra-diative heat flux is excluded from the value. Therefore, theresults shown in Fig. 9 imply that it is difficult to distinguishan increase in heat flux due to radiation from the overesti-mated heat flux due to Bartz’s formula.

The overestimation of heat flux is caused by the pipe flowmodel used in the Bartz’s formula. In general, the entranceregion of the pipe flow must have enough length to establish

Fig. 8. Comparison between heat flux calculated using Bartz’s formulaand heat flux measured at the throat.


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a fully developed velocity profile, and that length is severaltens to several hundreds of times the diameter.17) Thoughthe effect of the radiation can be large in heat transfer, therewas a general problem in calculating the heat transfer coeffi-cient in a conventional manner. The formula significantly de-grades accuracy when calculating heat flux. Bartz proposedanother estimation method by integrating the turbulentboundary layer along the flow direction. In that article, headdressed the inadequate application of the pipe flow modelto the rocket combustion chamber and nozzle.18)

4. Conclusion

The effects of radiative heat transfer in rocket engine com-bustion chambers were investigated analytically. The resultsattained in the present study show that careful attention to theeffect of radiation is necessary during the thermal design ofthe chamber. The conclusions are summarized as follows:

(1) Radiative heat flux can be calculated using empiricalequations and convective heat flux can be calculated usingequations of a flat-plate flow model with modifications tothe boundary layer thickness. The heat flux estimated includ-ing radiation and convection shows reasonable agreementwith the heat flux measured.

(2) The effect of radiation on heat flux increases as thechamber diameter increases, and decreases as the chamberpressure increases. The ratio of radiative heat flux to totalheat flux is about 10 to 30% in the cylindrical section anda few percent at the throat.

(3) In the hydrocarbon-fueled combustor, the effect of ra-diation from carbon dioxide is smaller than that from steamcontained in the combustion gas.

(4) The effect of radiation on the total amount of heattransferred to the chamber increases as chamber length inthe cylindrical section increases and chamber diameter in-creases.

(5) In spite of having ignored the effect of radiation, theconvective heat flux estimated using the conventional meth-od with a pipe flow model results in overestimation, espe-cially for small chambers.

Acknowledgments

The authors appreciate the discussions with and help of Mr. T.Hiraiwa, Researcher of JAXA.

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J. R. HulkaAssociate Editor

Fig. 9. Ratio of heat flux by Bartz’s formula to convective heat flux by thepresent method at the throat.O=F ¼ 6 for oxygen/hydrogen, O=F ¼ 3:6 for oxygen/methane, andO=F ¼ 1:6 for oxygen/ethanol.


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Radiative Heating in Combustion Chamber of Liquid