Conditions at chemical equilibrium 67

Part II: Parametric Studies

4.3 Variation of Equivalence Ratio 51

4.4.4 Conditions at chemical equilibrium 67

and

which corresponds to reactions 2, 17 and the reverse step of reaction 19

t

in table 3.6.3a.

The first reaction is in agreement with Mikolatis[loJ's branched chain cycle of reactions 2, 3 and 4 in the fast-rise stage. The importance of the last two reactions is not exactly certain.

The low temperature data again do not fit into the above result. The presence of a transition as in the start-up period may best explain the departure. However, the possible role of H0 2 and H2 O2 cannot be overlooked either. In the lowest temperature case, the change in mole fraction for all radicals is approximately one percent of that at To = 1500 K except for the above two. Some indication is given in figure 4.4.6. For example, LlX for H2 O 2 is -9 for the lowest temperature case, implying a depletion. The previous analysis has shown that an excessive amount of H20 2 is created during the start-up stage at low tem- perature (c.f. section 4.4.2). This means that a large amount of H202 is rapidly destroyed during fast-rise which may in part contribute to the departure in behavior.

vehicles with hypersonic inlet condition at the diffuser. In this case, the kinetic energy of the inlet flow is large enough such that a large change in temperature can be achieved by a small fractional change in flow speed.

The reason for reduced heat release at higher initial temperatures is the increased survivability of high energy radicals. In this respect, higher combustor inlet temperature may still be tolerated if the energy can be recovered through radical recombination in the downstream components. This aspect is discussed in chapter 4 where the chemical kinetic in a two-dimensional expansion nozzle is analyzed.

Equilibrium Initial Temperature To [K]

Condition 850 900 950

Heat Release !::::..H [Jjg] 2794.90 2755.33 2714.25 Temperature T [K] 2586.22 2603.80 2620.96 Density p [kgjm³] 0.05629 0.05579 0.05531 Sound Speed a [mjs] 1061.11 1065.94 1070.72 Composition YN 4.12 ^X 10-⁷ 4.79 ^X 10-⁷ 5.55 ^X 10-⁷

Y N2 7.31 X 10-¹ 7.30 X 10-¹ 7.30x10-¹ Y NO 7.10 X 10-³ 7.47 X 10-³ 7.85x10-³ Y N02 1.30 X 10-⁶ 1.37 X 10-⁶ 1.44 x 10-⁶ Yo 2.15 ^X 10-³ 2.38xlO-³ 2.64 ^X 10-³ Y0 2 1.47x10-² 1.53 X 10-² 1.60 X 10-² YOH 1.27 ^X 10-² 1.34 ^X 10-² 1.42 ^X 10-² YH 4.02 X 10-⁴ 4.43 X 10-⁴ 4.86 X 10-⁴ Y H2 2.94 X 10-³ 3.09 X 10-³ 3.23 X 10-³ YH0 2 2.14 X 10-⁶ 2.31 X 10-⁶ 2.48 X 10-⁶ YH 20 2.17 X 10-¹ 2.15x10-¹ 2.12x10-¹ Y H202 2.69 X 10-⁷ 2.83 X 10-⁷ 2.96x10-⁷ YAr 1.29 X 10-² 1.29 ^X 10-² 1.29 x 10-²

TABLE 4.4.2a Equilibrium conditions for lower initial temperatures.

Equilibrium Initial Temperature To [K]

Condition 1000 1500 2000 2500

Heat Release 6.H [Jig] 2671.65 2172.68 1571.46 901.84

Temperature T [K] 2637.72 2786.62 2908.96 3013.03

Density p [kg 1m³] 0.05484 0.05069 0.04724 0.04428

Sound Speed a [mls] 1075.46 1121.03 1164.42 1206.64

Composition _{Y N} 6.39 X 10-⁷ 2.09 X 10-⁶ 5.08x 10-⁶ 1.03 X 10-⁵ Y N2 7.30x10-¹ 7.28 X 10-¹ 7.26 X 10-¹ 7.25x10-¹ YNO 8.22xlO-³ 1.21 X 10-² 1.58 X 10-² 1.93 X 10-² Y N02 1.52 X 10-⁶ 2.26 X 10-⁶ 2.94 X 10-⁶ 3.51 X 10-⁶ Yo 2.90 X 10-³ 6.47 X 10-³ 1.17x10-² 1.84 X 10-² Y0 2 1.67 X 10-² 2.32 X 10-² 2.89 X 10-² 3.35 X 10-² YOH 1.50 X 10-² 2.30 X 10-² 3.09 X 10-² 3.78 X 10-² Y H 5.32 X 10-⁴ 1.12x10-³ 1.94 X 10-³ 2.96 X 10-³ YH2 3.38 X 10-³ 4.80 X 10-³ 6.07 X 10-³ 7.12x10-³ Y H02 2.65x 10-⁶ 4.50 X 10-⁶ 6.39 X 10-⁶ 8.05x 10-⁶ YH 20 2.10 X 10-¹ 1.88 X 10-¹ 1.65x10-¹ 1.43x10-¹ YH202 3.09xlO-⁷ 4.27 X 10-⁷ 5.09 X 10-⁷ 5.52 X 10-⁷ YAr 1.29 X 10-² 1.29 X 10-² 1.29 X 10-² 1.29 X 10-² TABLE 4.4.2b Equilibrium conditions for higher initial temperatures.

0-<l>

I <J ...

+-' I

<J

0-<l>

I

<J ...

+-' I

<J

1.0f-

.8 f-

.6

-

.4

-

.2 -

.0 .0

1.2

-

1.0 ^-.f?

f'l

_'II

_"

I'

!

//

.8 // ^I' /:

,: /1 I'

.6

~

^I, I: ,:

~i II I:

"

.4

-n

"

-11

.2

HI f4 hi

~'

Hi

i

.0 ^II .00

TO=850 K

I

.2

+

inflexion pOint start-up time

I I

.4 .6

t [s]

-

-

-

-

I

.8 1.0

FIG.4.4.1a Heat release for initial temperature equals 850 J(.

/' I

../'

I I

I I

.01 .02

t [sJ

I

I

-

-

TO= 900 K _ - TO= 950 K

TO=1000 K TO=1500 K

- TO=2000 K -

To=2500 K _-

.03 .04

FIG.4.4.1b Heat release for 6 initial temperatures.

1.0

.8

TO= 900 TO= 950 TO=1000

I I

/ / I I I

---:.-:--::.-:::=

---

K -- __ + inflexion point

K __ ,,"---. / - - + S t ar -up t t· Ime// -'

K """ ./

/ / ./

l .I

i

./ / --- ."

g

.6 _I ^I

I

,I

i i

i

I

i

/

I <J

"

I

I I I I I I

<J .4 I ^I

0' Q)

I <J

"

I

<J

I I I I I I I

.2

.0 L-~~~~/L-~L-J--L-L~~~ I __ L-L-~~J--L-L~~~ __ L-L-~~-L-L~~

.0000 .0005 .0010 .0015

t [s]

.0020 .0025 .0030

FIG.4.4.2a Heat release at initial start-up for intermediate initial temperatures.

.4

.3

.2

.1

TO=1500 K TO=2000 K TO=2500 K

+ inflexion point

- - + start-up time

-- --

,,-

_ /

.0 ^~~~ __ ^~~~DL-L~ _ _ ~~ _ _ L--L~ _ _ ~~ _ _ L--L~ _ _ ~~~

.00000 .00002 .00004 .00006 .00008 .00010

t [s]

FIG.4.4.2b Heat release at initial start-up for high initial temperatures.

lu 5

4

3

2

\

A

_\

\

\

o , '.

.,

'. ., ,

_.,

-.- , ^., _'

_..

/0-.--.--.-_. __

~~~~~:::::>.

/A ~-_---:-::::--::::--::;::--~--~-

1 .-A" 0---

o

900 1300

0

+

TO= 900 K OH /\ TO= 950 K

H ⁰ TO=1000 K

H02 /'::,. TO=1500 K

H₂O 0 TO=2000 K

*

TO=2500 K

---::::~==---E}---:~_~?~~:~=

1700 T [K]

-'--==-"=-='-==-~

2100 2500

FIG.4.4.3a Concentration for five species at start-up time for six initial temperatures.

Ix 5

0 + TO= 900 K OH /\ TO= 950 K

4 H ⁰ TO=1000 K

H0_{2 )} t:::. TO=1500 K H2O 0 TO=2000 K

*

TO=2500 K 3 ' / \

,

&

2

1

o

^{L-~ __ ~ __ - L _ _ ~ _ _ L _ _ _ L-~ _ _ _ L _ _ _ L _ _ ~ _ _ L_~ _ _ ~ _ _ _ L _ _ ~~}

900 1300 1700

T [K]

2100 2500

FIG.4.4.3h Mole fraction for five species at start-up time for six initial temperatures.

Ix 1

-.-.-.-

.-

^...

A-B--.--.----~~--~:?:=~==·

----

---

o

^~~

__

^~

__

^{_ L _ _ ~ _ _ L _ _ _ ~~ _ _ _ L _ _ ~ _ _ ~ _ _ L_~ _ _ ~ _ _ _ L _ _ ~~}

900 1300 1700

T [K]

2100 2500

FIG.4.4.4a Concentration for five species at inflexion for six initial temperatures.

0

+

^{TO= 900 K}

OH /\ TO= 950 ^K

4 H ⁰ TO=1000 ^K

H02 ) !:::. TO=1500 ^K H2O ⁰ TO=2000 ^K

*

^{TO=2500 K}

3

2

1 ;;_~~~_~~~~:,:~_:,:: ^.. ~::;:~;:;;.~;;;;3-"'."."::'''''''~-=;;;-;;';;-;';;;-:--~-::':.-=-:~==.-~---B---_---_-

I, --- . ~-::.~..:..-:..:.':":"..:....~~~

o

^{L-~ __ - L _ _ _ L _ _ ~ _ _ L _ _ _ L-~ _ _ - L _ _ ~ _ _ ~ _ _ L-~ _ _ ~ _ _ _ L _ _ ~~}

900 1300 1700

T [K]

2100 2500

FIG.4.4.4b Mole fraction for five species at inflexion for six initial temperatures.

I~

I~

5

0

+

TO= 900 K

OH /\ T= 950 K

4 H 0 TO=1000 K

H₂O ^/:). TO=1500 K D TO=2000 K

*

TO=2500 K 3

2

---E}---_____________ _

o

^~~

__

^~

__

^~

__

^{L_~ _ _ ~ _ _ ~ _ _ L_~ _ _ _ L _ _ ~ _ _ L_~ _ _ _ L _ _ ~~}

900

4 r--

3 r--

2 r--

1300 1700

T [K] ²¹⁰⁰

FIG.4.4.5a Change in concentration for four species during fast-rise.

I I

0 + TO= 900 K

OH /\ TO= 950 K

H 0 TO=1000 K

(H0₂ not shown) /:). TO=1500 K

-- H₂O D TO=2000 K

*

To=2500 K 2500

-

-

-

t-°A-o-_o __ _

-.-._.-.- 1 FA-0

_ __________ E} __________________ ~

)0: _ - - - - -

~~--- =

o

^{L - - L _ _ L - - L _ _ L-I-L __ L - - L _ _ L_I~ __ L_~ __ ~I __ ~~ __ ~~}

900 1300 1700 2100 2500

T [K]

FIG.4.4.5b Change in mole fraction for four species during fast-rise.

I~

*

_... ^~

(f)

~ Q) OJ

... 0

75 10

5 ^f-

o

^I-

H0₂ <> TO= 850 K

H₂0₂

+

TO= 900 K 1\ TO= 950 K

-5 ^I- 0 TO=1000 K ^-

I::,. TO=1500 K 0 TO=2000 K

j

*

TO=2500 K

<>

-10 ^{I I I I I}

600 1000 1400 1800 2200 2600

T [K]

FIG.4.4.6 Change in mole fraction for H02 and H202 during fast-rise.

-5

-10

<> TO= 850 K + TO= 900 K

1\ TO= 950 K TO$1000 ^K~

0 TO=1000 K

I::,. TO=1500 K 0 TO=2000 K

*

TO=2500 K

---

---

~

T0>1000 K

non-linear fits (start-up time) -- non-linear fits (inflexion point)

-15 ^L-~ __ _ L _ _ - L _ _ ~ _ _ L_~ _ _ ~ _ _ _ L _ _ J _ _ _ ~ _ _ ~~ _ _ ~ _ _ _ L _ _ ~~

.0004 .0006 .0008

l/T [K- l ]

.0010 .0012

FIG.4.4.7 Fits of start-up time and inflexion point with initial temperature.

4.5 Variation of Pressure

Five combustor pressures are chosen for this study:

{

0.1 aim 0.3aim P

=

0.5aim 0.7 aim 1.0 aim

The range chosen is not very large but is believed to be sufficient for combustor performance study. In short, the range represents one logarithmic decade. This is due to the nature of the rate equation. * Detailed reason will become clear as the results are presented.

As in the previous analyses, the heat release during combustion is examined for each pressure. The evolution of normalized heat release is plotted in figure 4.5.1a with a linear time scale and in figure 4.5.1b with a logarithmic time scale. A quantity i* that equals 1 sec is used in the second plot for dimensional purpose. The similarity and spacing between the curves in the second plot suggest why those pressures are chosen.

The overshoot which was first observed in the equivalence ratio study is again obvious form the figures. An explanation can be drawn from the fast and slow formation and destruction of certain radicals when compared to the rate of heat release though H20 formation. Examples of such radicals are OH and NO. The evolution of their normalized mass fractions is plotted in figure 4.5.2a while that of H20 is shown in figure 4.5.2b. In this example, OH is considered the fast species and NO is considered the slow one since OH approaches equilibrium much faster than H₂0 whereas NO approaches much slower.

Note that the rise and fall in the normalized plots also indicate formation and destruction respectively since the initial mass fractions of those species are zero. The energy initially produced through the formation of H₂0 is locked up in the fast formation of OH. As the formation of H₂0 continues, the rapid destruction of such radicals results in an excessive heat release. The excess heat is to be absorbed eventually by the slow forming radical NO which results in a 'bump' during the time when excess heat is present. In general, any species falling in the fast category has the same function as OH and similarly for the slow ones.

* It involves the product of concentrations which are proportional to pressure.