Trans. JSASS Aerospace Tech. JapanVol. 8, No. ists27, pp. Pa_7-Pa_11, 2010

Original Paper

Copyright© 2010 by the Japan Society for Aeronautical and Space Sciences and ISTS. All rights reserved.

Pa_7

The Effect of Fuel Grain Size on the Combustion Characteristics in the Primary Combustion Chamber of Staged Combustion Hybrid Rocket

By Harunori NAGATA, Kenta HASHIBA, Hiroya SAKAI, Tsuyoshi TOTANI and Masashi WAKITA

Division of Mechanical and Space Engineering, Hokkaido University, Sapporo, Japan

(Received June 15th, 2009)

To clarify the fuel gasification characteristics in a primary combustion chamber of a staged combustion hybrid rocket, the effect of fuel grain size on the regression rate of a grain was investigated experimentally. The grain size distribution inthe combustion region achieved a steady state in 30 seconds burning duration. Examining fuel size distributions and fuel consumption rate at steady states, we obtained a history of fuel size and the regression rate of a grain in the combustion region. Regression rate increases with decreasing grain size. With a constant oxidizer flow rate, the regression rate is a function of grain size and independent to the initial grain size. After an initial transient the grain size decreases followingthe classical d-square law in droplet combustion: The square of the grain size decreases linearly with time. Although why the regression history of a grain in the combustion region follows the d-square law is not clear, this result is useful to esti-mate the fuel gasification rate of a staged combustion hybrid rocket.

Key Words: Hybrid Rocket, Solid Fuel, D-square Law.

Nomenclature

F(x) : Cumulative size distribution function f(x) : Size distribution function Kb : Evaporation constant [mm2/s] N : Number of grains in the combustion

region ns : Fuel number nl : Another definition of fuel number n : Number flow rate of fuel grains [1/s]

r(x) : Regression rate of a grain [mm/s] Seq : Equivalent burning surface area [mm2]V : Volume of the combustion region [mm3]x : Characteristic length of a grain [mm]

: Lifetime of a grain [s] : Volumetric occupancy of grains

Subscripts0 : initial l : l-th layer in the combustion region

1. Introduction

In recent space development activities, more and more group, particularly in universities, are developing small satel-lites. Because there is no rocket specialized for a launch of small satellites less than 100 kg, most small satellites are launched as a ‘piggyback’, using excess capacity on larger launch vehicles. In this system, piggyback satellites likely have no choice of an orbit because the primary satellites have the initiative. When a mission requires a small satellite to se-lect a particular orbit, a thruster for orbit transfer is necessary. Because piggyback satellites are required to be safe not to damage primary satellites, the thruster should not carry any

explosives or hazardous materials. The authors have proposed staged combustion hybrid rock-

ets for a kick motor of small to micro scale satellites1, 2). Vir-tues of hybrid rockets, throttling and restart abilities and com-pactness, are right qualifications for small kick motors. Con-ventional hybrid rockets, however have a drawback that the throttling causes O/F shift, resulting in the loss of specific impulse. To overcome this drawback, staged combustion hy-brid rockets use primary and secondary combustion chambers. The primary combustion chamber doubles as a fuel container, storing fuel grains. Fuel-rich combustion gas from the primary combustion chamber flows into the secondary combustion chamber, in which the gas reacts with the secondary oxidizer to reach the optimum O/F. A previous study shows that the fuel gasification rate depends on the primary oxidizer flow rate and is independent on the secondary oxidizer flow rate. Consequently, the primary oxidizer acts as a throttle of the fuel gas supply. As a result, the rocket can change the thrust with keeping the optimum O/F.

It is necessary to clarify the fuel gasification characteristics in the primary combustion chamber for an appropriate design of the rocket. To derive a fuel gasification equation, the effect of fuel grain size on the combustion characteristics was stu-died experimentally. In the primary chamber only a portion of granular fuel near the bottom burns. New fuels go down into the combustion region as fuel grains diminish. In the steady state, the distribution of the fuel grain size according to the distance from the bottom does not change with time. This steady fuel grain size distribution is important because it gives the fuel regression history of each single fuel grain in the combustion region.

Trans. JSASS Aerospace Tech. Japan Vol. 8, No. ists27 (2010)

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2. Theoretical Background

Figure 1 shows the basic concept of the primary combustion chamber. The primary oxidizer gas flows into the chamber from the bottom and the primary combustion gas flows out through four pipes on the sidewall. Fuel grains gasify in the gasification layer ranging from the bottom to the height of the pipes. New fuel grains get into the gasification layer from the upper side as the gasification of the grains in the layer proceeds. In the steady state, the distribution of the fuel size according to the distance from the bottom does not change with time. Fuel grains in more advanced gasification stages distribute near the bottom of the layer. The combustion gas flows upward from bottom with O/F decreasing and gas flow density increasing, according to the fuel addition along a streamline. Consequently, gas composition, temperature, O/F,and gas flow density vary with height. Because the regression rate of a fuel depends on the local gas composition, tempera-ture, O/F and gas flow density, it is possible to express the regression rate as a function of fuel grain size, which is a function of height.

Suppose that the characteristic length of a fuel grain is x[m], the regression rate of a grain is r(x), the normalized size distribution function in the combustion region is f(x), and the number of grains in the region is N. Because the number of grains whose characteristic length becomes less than x in unit time is Nf(x)r(x), the following equation is obtained in the steady state:

0))()((dx

xrxNfd , Cxrxf )()( (1)

where C [1/s] is a constant. Using normalization condition; 0

01)(

xdxxf (2)

The grain lifetime [s], which is the lifetime of a fuel grain in the combustion region, and the total fuel flow rate mf [kg/s] are functions of fuel grain ingression/burnout rate n [1/s]:

nN

Cdx

xrx 1

)(10

0 (3)

00

00)()()()(

xx

f dxxsCNdxxfNxrxsm (4)

where x0, , s(x) are initial characteristic length, density, and surface area of fuel grains, respectively. Accordingly, one can obtain the regression rate as a function of characteristic length from the steady state distribution function of the characteristic length in the gasification layer.

3. Experimental Apparatus and Procedure

In the experiments, fuel grains are column-shaped unsatu-rated polyester resins whose diameter and length are the same with each other. Two cases of grain size are employed: One is S-grain of 8 mm in diameter and length. The other is L-grain of 10 mm in diameter and length. The volume of S-grain is about 50% less than the volume of L-grain.

Figure 2 shows a schematic diagram of the chamber for combustion experiments. The water-cooled chamber dupli-cates a primary chamber of a staged combustion hybrid rocket

motor. The chamber is made of stainless steel, with inner di-ameter of 50 mm and height of 260 mm. Gas oxygen with a constant flow rate of 0.85 g/s flows into the chamber through an injector plate at the bottom. There are many ejection holes of 0.7 mm in diameter on the injector plate. Alumina balls of 4 mm in diameter fill the bottom of the chamber up to 40 mm from the injector plate. Fuel grains pile on the top of alumina balls. Accordingly, the top face of alumina balls is the bottom face of the combustion region. There are four exhaust outlets on the sidewall, being at right angles to each other, at 30 mm from the top of alumina balls. The top face of the combustion region is at the level of the exhaust outlets. A nichrome wire at the top of grains ignites the fuel. To avoid the combustion ash preventing the supply of unburned fuel grains into the com-bustion region, a piston at the top of the chamber presses fuel grains.

Experimental procedure is in the followings: Supply elec-tricity to a nichrome wire at the top of fuel grains for about 30 s, followed by the oxidizer supply to start burning. Simulta-neously, pressurize the piston room with a nitrogen gas and start supplying cooling water. After some burning duration, stop supplying oxidizer and supply nitrogen gas simulta-neously for extinction. After removing unburned fuel grains over the combustion region, pick up ten fuel grains at a time from the top of the combustion region and measure the height of the top of fuel grains above the bottom of the combustion region. This value is defined to be the characteristic height of the layer of ten fuel grains. Measure mass of a fuel grain one by one. The characteristic length x [m] of a grain is defined to be the diameter of a cylinder of the same mass, with equal length and diameter.

Combustion region

Oxidizer (O2)

Primarycombustion gas

Fuel grains

InjectorAlumina balls

Nichrome wirePiston

Cooling water

High-pressure N2

Fig. 2. Schematic of the combustion chamber.

Combustion region

Oxidizer

Primarycombustion gas

Fuel grains

Fig. 1. Basic concept of primary combustion chamber of staged combustion hybrid rocket.

H. NAGATA et al.: Effect of Fuel Grain Size on Combustion Characteristics in Staged Combustion Hybrid Rocket

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4. Result and Discussion

4.1. Fuel size distribution The characteristic length of fuel grains in the same layer spread because the boundary between layers is not clear. Open circles in Fig. 3 show relationship between characteristic length and fuel number ns. How to define the fuel number is in the followings: Arrange layers of ten grains in the order of ascending characteristic height of layers. Then arrange grains in each layer in the order of ascending characteristic length of grains. Another method to number fuel grains is much simple; arrange all grains in the combustion region in the order of ascending characteristic length of grains. Solid circles in Fig. 3 show results by this method. Distributions of open circles and solid circles agree well with each other, showing a clear

tendency that the characteristic length of a fuel grain decreases as a grain approaches the bottom.

Figure 4 shows distributions of characteristic length for each instant of time with Fuel L and S for Fig. 4 (a) and Fig. 4 (b), respectively. Each plot represents the mean value of each layer. As the figure shows the distribution varies little after 30 seconds, indicating that the distribution was at a steady state after 30 seconds. We defined an average distribution among those of 30 to 40 seconds as a steady state distribution. Figure 5 shows steady state distributions for L-grain and S-grain cas-es. Horizontal and vertical axes are fuel number and characte-ristic length normalized by the number of grains in the com-bustion region and the initial grain size, respectively. What is interesting is that these two profiles agree well with each other, indicating the presence of a common regression formula as a function of characteristic length. 4.2. Fuel flow rate

Figure 6 shows fuel consumption histories for both fuels. After the fuel size distribution in the combustion region reaches steady state, fuel consumption rates keep constant. Slopes of linear lines give fuel gasification rates of L-grain and S-grain cases to be 1.86 g/s and 2.59 g/s, respectively. The fuel gasification rate of S-grain is about 1.39 times larger than that of L-grain. If a fuel regression rate is independent on a grain size, a fuel gasification rate increases proportionally with the fuel surface area. Because steady state distributions of characteristic length for S-grain and L-grain are virtually similar with each other, as Sec. 4.4 shows, the equivalent sur-face area Seq the following equation gives is applicable to compare the total surface area of fuel grains in the combustion

50 100 150

2

4

6

8

0

Unmodified fuel numberModified fuel number

Fuel number

Char

acte

ristic

leng

th [m

m]

Fig. 3. Characteristic length vs. fuel number.

20 40 60 80

2

4

6

8

10

0Fuel number

Char

acte

ristic

leng

th [m

m]

25 s30 s35 s40 s

(a) L-grain

50 100 150

2

4

6

8

0

Char

acte

ristic

leng

th [m

m]

Fuel number

25 s30 s35 s40 s

(b) S-grain

Fig. 4. Distributions of characteristic length.

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

0

x/x 0

ns /N

: L-grain: S-grain

Fig. 5. Steady state distribution of characteristic length.

30 35 4030

40

50

60

70

80S-grain:y = 2.59x + const

L-grain:y = 1.86x + const

Burning duration [s]

Fuel

con

sum

ptio

n [g

]

Fig. 6. Fuel consumption histories.

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zone.

0/ xVSeq (5)

where V, , and x0 are the volume of the combustion region [m3], the volumetric occupancy of fuel grains in the combus-tion region, and the initial characteristic length of fuel grains [m]. In the steady state, Seq for S-grain is 1.21 times larger than that for L-grain. This value is smaller than the ratio of fuel gasification rates (= 1.39), indicating that the regression rate increases with decreasing grain size.4.3. Regression history

The diminishing rate of the number of grains is constant at a steady state, yielding the following equation:

tnnN s (6)

where N, ns, n , and t are the total number of grains in the combustion region, fuel number of a grain, number flow rate of grains [1/s], and the time from when the grain entered into the combustion region, respectively. For simplicity, the fol-lowing equation introduces another definition of fuel number:

sl nNn (7)

Figure 7 shows correlations between square of characteristic length and fuel number ln for S-grain and L-grain. After an initial transient, the square of characteristic length decreases linearly with the fuel number ln :

constnKx l2 (8)

Note that the fuel number ln of a grain is proportional to the time from when the grain entered into the combustion region.

Accordingly, Fig. 7 shows that after an initial transient the characteristic length of a grain decreases following the clas-sical d-square law in a droplet combustion: The square of the characteristic length of a grain decreases linearly with time3).Although why the regression history of a grain in the combus-tion region follows the d-square law is not clear, this result is useful to estimate the fuel gasification rate of a staged com-bustion hybrid rocket.

Mean masses of a single L-grain and S-grain are 0.885 [g] and 0.435 [g], respectively. Accordingly, using fuel gasifica-tion rates Fig. 6 shows, 5.95 and 2.10 grains diminishes every one-second in S-grain and L-grain cases, respectively. Mul-tiplying these values with K (absolute values of slopes Fig. 7 shows) yields ‘evaporation constants’ Kb:

nKKb (9)

The evaporation constant for L-grain and S-grain are 3.08 and 3.04 [mm2/s], respectively. Virtually the same value of eva-poration constants in both cases show that the regression rate of a grain in the combustion region is independent with the initial size of the grain. 4.4. Distribution function

Employing the mean value of 3.06 [mm2/s] as the evapora-tion constant common to both sizes, Eq. (9) gives slopes in Fig. 7 to be -1.46 and -0.514 for L-grain and S-grain, respectively. By assigning 0 and x0

2 to x and const in Eq. (8), nl gives the total number of grains N in the combustion region, the initial transient region being eliminated. The results are 68.5 for L-grain and 125 for S-grain. The following equation gives the cumulative distribution function of the grain size:

Nn

NnxF ls 1)( (10)

Eliminating nl by Eq. (8), “const” being x02, gives

'1)(

220

NKxx

xF . (11)

Because NK’ = x02,

2

020

2

20

2201)(

xx

xx

xxx

xF (12)

Figure 8 shows a comparison between experimental results and the model, showing a good agreement commonly to both cases of L-grain and S-grain.

20 40 60 80

20

40

60

80

100

0

y = -1.47x + const

x2 [mm

2 ]

Fuel number(a) L-grain

50 100 150

20

40

60

0

y = -0.510x + const

Fuel number

x2 [mm

2 ]

(b) S-grain

Fig. 7. Square of characteristic length vs. fuel number.

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

0

: L-grain: S-grain

x/x0

Dis

tribu

tion

func

tion

Fig. 8. Distribution function vs. fuel number.

H. NAGATA et al.: Effect of Fuel Grain Size on Combustion Characteristics in Staged Combustion Hybrid Rocket

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4.5. Regression rateFrom Eq. (1), the regression rate is a function of the size

distribution function f (x):

xfCxr (13)

Using Eq. (3),

xfNnxr (14)

The size distribution function f (x) is the first derivative of the cumulative size distribution function with respect to x:

20

20

2 2)()(x

xxx

dxd

dxxdFxf (15)

Substituting Eq. (15) into Eq. (14) and eliminating n by Eq. (9) gives

xK

xKNxx

xKxKN

xKxr bbb

222202

0

20

20 . (16)

Experimentally, the following equation gives the regression rate rl+1/2 of a grain whose characteristic length is xl+1/2:

ll

ll

ll

lll FF

xxNn

fNnr

xxx

1

12/1

12/1 ,

2 (17)

where l is the number of layer and Fl is the cumulative size distribution function of l-layer. Figure 9 shows the result. The solid line in the figure represents Eq. (15). In both L-grain and S-grain cases, regression rate increases with decreasing grain size. Experimental results lie close to the solid line, showing that the regression rate follows the d-square law and is inde-pendent to the initial grain size.

5. Conclusion To clarify the fuel gasification characteristics in a primary

combustion chamber of a staged combustion hybrid rocket, the effect of fuel grain size on the regression rate of a grain was investigated experimentally. The grain size distribution in the combustion region achieved a steady state in 30 seconds burning duration. Examining fuel size distributions and fuel consumption rate at steady states, we obtained a history of fuel size and the regression rate of a grain in the combustion region. Regression rate increases with decreasing grain size. With a constant oxidizer flow rate, the regression rate is a function of

grain size and independent to the initial grain size. After an initial transient the grain size decreases following the classical d-square law in droplet combustion: The square of the grain size decreases linearly with time. Although why the regression history of a grain in the combustion region follows the d-square law is not clear, this result is useful to estimate the fuel gasification rate of a staged combustion hybrid rocket.

Acknowledgments

This research was partially supported by the Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Scientific Research (B), 21360410, 2009.

References

1) Satori, S., Nagata, H., Aoki, Y., Akiba, R., Kudo, I. and Kubota, I.: Preliminary Study for Staged Combustion Hybrid Rocket, Proceed-ings of the 22nd International Symposium on Space Technology and Science, 2000, 1, pp.116-120.

2) Akiba, R., Aoki, Y., Kayuta, S., Fujii, A., Nagata, H. and Satori, S.: Leading Studies of the Staged Combustion Hybrid Rocket, Journal of the Japan Society for Aeronautical and Space Sciences (in Japa-nese), 51, No. 591 (2003), pp. 141-150.

3) For example, Turns, R. S., An Introduction to Combustion,McGraw-Hill, 1996, pp. 89.

2 4 6 8 10

0.2

0.4

0.6

0.8

1

0

: L-grain: S-grain

Grain size [mm]

Regr

essi

on ra

te [m

m/s

]

Fig. 9. Regression rate vs. grain size.