Project 3

Project #3

AE/ME 6766

Dr. Seitzman

A Study of Propane-Air C-J detonations

3/9/2014

Marius Popescu

Table of Contents

I. Terms and Symbols…………………………………………………………….1

II. Introduction…………………………………………………………………………1

III. Methods………………………………………………………………………………..1

IV. Results………………………………………………………………………………….2

V. Discussion………………………………………….…………………………….…..Error: Referencesource not found

VI. Conclusion…………………………………………………………………………….10

Marius Popescu Page 1 AE 6766 Spring 2014|Project #3

Terms and Symbols Used

φ=Fuel Equivalence Ratio

K=Kelvin

p=Pressure∈atmospheres (101.325kPa)

X1=initialconditions

X d=conditionsafter shock

X2=conditions after heat release /equilibrium

X=average property

D=shock velocity∈m /s

ρ=density∈kg×m−3

W=molecular weight

γ=specific heat ratio

c p=constant pressure specific heat

cv=constant volume specific heat

R=universal gasconstant

Introduction

This report examines some of the properties of C-J detonations for propane – air mixtures. First using GasEq’s C-J detonation calculation to determine product characteristics from the reactants at various initial conditions concerning equivalence ratios, temperatures, and pressures. This is compared to a simplified mathematical model for temperatures around 2700K. Finally, using the ZND model, the auto-ignition delay is examined with Chemkin using conditions after the initial shock. This will provide insight into what factors play a role in detonations, how accurate models are depending on these conditions, and what affects the bounds of range that C-J detonations are possible.

Methods

Even though Chemkin and GasEq were used in previous projects, different functions were used for this project. GasEq was used for its C-J detonation, which iteratively solves for the final conditions by guessing final pressure and temperature, then solving for equilibrium species concentrations, then getting the density, specific heat ratio, and specific gas constant of products. This information can then be used to calculate D (without pressure) and a new pressure guess. Using the new pressure ratio we get a new Temperature guess. This process is iterated until converged. GasEq is also used for shock


properties. This evaluates properties after a shock front, using conservation equations and constant specific heats. This leads to minor errors after 2000K, but the author compares his results to NASAs for higher temperatures and it does not change appreciably. In addition to using GasEq, the simplified model for D is used, which assumes that the pressure term is neglected for stronger shocks, i.e. higher temperatures; and it is compared to the full C-J iteration for accuracy. Lastly Chemkin is used to determine auto-ignition delays following a shock analysis from GasEq in accordance to the ZND process.

Results

Section 1: Analysis of Various Initial Conditions for C-J Detonation

For the first part of this section pressure was held at 1atm and three different initial temperature cases were ran in GasEq. Equivalence Ratios were varied from 0.0225 (2.2% fuel) and 0.10133 (9.2% fuel).

0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9 2.1 2.3 2.51500

1550

1600

1650

1700

1750

1800

1850

1900

0

1

2

3

4

5

6

300450600300450600

Equvalence Ratio (ϕ)

Shoc

k Ve

locit

y (m

/s)

Mac

h N

o.

Initial Tempera-ture (K)

Figure 1. Results for shock speed. Solid lines are Mach number; dashed are in m/s.


0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9 2.1 2.3 2.51700

1900

2100

2300

2500

2700

2900

3100

300

450

600


Tem

pera

ture

(K) Initial Tempera-

ture (K)

Figure 2. Results for Final Temperature

0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9 2.1 2.3 2.56

8

10

12

14

16

18

20

300450600


Pres

sure

(atm

)


Figure 3. Results for final Pressure


0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9 2.1 2.3 2.51.15

1.2

1.25

1.3

1.35

1.4

300450600300450600


spec

ific r

atio


Figure 4. Results for Final Specific Heat Ratio. Solid lines Reactant specific heat ratio, dashed product specific heat ratio

0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9 2.1 2.3 2.51.621.641.661.68

1.71.721.741.761.78

300

450

600


ρ2/ρ

1


Figure 5. Density Ratio due to changes in reactant temperature

For the second part of this section initial temperature was held at 300 Kelvin and three different initial pressure cases were ran in GasEq. Equivalence Ratios were varied from 0.0225 (2.2% fuel) and 0.10133 (9.2% fuel).


0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9 2.1 2.3 2.51500

1550

1600

1650

1700

1750

1800

1850

1900

0

1

2

3

4

5

6

15201520


Shoc

k Ve

locit

y (m

/s)

Mac

h N

o.

Initial Pressure (atm)

Figure 6. Results for Shock Speed. Dashed lines are Mach Numbers; Solid lines are in m/s

0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9 2.1 2.3 2.51700

1900

2100

2300

2500

2700

2900

3100

1

5

20


Tem

pera

ture

(K)


Figure 7. Results for Final Temperature


0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9 2.1 2.3 2.50

50

100

150

200

250

300

350

400

1520


Pres

sure

(atm

)


Figure 8. Results for final Pressure

0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9 2.1 2.3 2.51.15

1.2

1.25

1.3

1.35

1.4

15201520


spec

ific r

atio


Figure 9. Results for Final Specific Heat Ratio Solid lines Reactant specific heat ratio, dashed product specific heat ratio

0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9 2.1 2.3 2.51.7

1.711.721.731.741.751.761.77

1520


ρ2/ρ

1


Figure 10. Density Ratio for various initial pressures


Section 2: Comparison of GasEq’s C-J Calculation to Simplified Model

1500 1700 1900 2100 2300 2500 2700 2900 3100

-10

-8

-6

-4

-2

0

2

4

6

8

10

Pressure 1 Temp 300Pressure 1 Temp 600Pressure 20 Temp 300Pressure 5 Temp 300Pressure 1 Temp 450

Temperature (K)

% e

rror

Figure 11. % Error in Shock Velocity vs Initial Temperature for every Initial condition

-150 -100 -50 0 50 100

-60

-40

-20

0

20

40

60

80

100

120

140

Difference in Temperature to 2700 K

Abso

lute

Err

or (m

/s)

Figure 12. Absolute Error in Shock Velocity vs. Difference in Temperature from 2700K less than 100K


Section 3: Determination of Auto-Ignition

-1.00E-04 1.04E-18 1.00E-04 2.00E-04 3.00E-04 4.00E-04 5.00E-04 6.00E-040.00E+00

1.00E-03

2.00E-03

3.00E-03

4.00E-03

5.00E-03

6.00E-03

7.00E-03

8.00E-03

0.00E+00

5.00E+02

1.00E+03

1.50E+03

2.00E+03

2.50E+03Auto-Ignition Lean

HOHT

Residence Time (s)

Mol

ar F

racti

on

Tem

pera

ture

(K)

Figure 13. Plug Flow Reactor graph for temperature and H and OH mole fractions showing auto-ignition delays. Initial Conditions: p = 1atm, T = 300K, ϕ = 0.536

-1.00E-04 -8.13E-20 1.00E-04 2.00E-04 3.00E-04 4.00E-04 5.00E-04 6.00E-04-1.00E-04

-8.13E-20

1.00E-04

2.00E-04

3.00E-04

4.00E-04

5.00E-04

6.00E-04

0.00E+00

5.00E+02

1.00E+03

1.50E+03

2.00E+03

2.50E+03

Auto-Ignition Rich

HOHT

Residence Time (s)

Mol

ar F

racti

on

Tem

pera

ture

(K)

Figure 14. Plug Flow Reactor graph for temperature and H and OH mole fractions showing auto-ignition delays. Initial Conditions: p = 1atm, T = 300K, ϕ = 2.413

Table 1. Summary of Rich and Lean Ignition Delays

Rich LeanIgnition Time (ms) 0.1 0.164

Temperature Chemkin (K) 2090 2276Temperature GasEq (K) 1950.6 2131.7

Shock Velocity(m/s) 1673.36 1531.36


Gas Velocity post Shock(m/s) 1436.6 1256.44

Discussion

Section 1

Looking at Figure 1, temperatures effect on shock velocity is to increase it, but Mach number on the other hand is reduced. This effect is due to the products having a higher speed of sound. Since speed of sound is related to specific heat ratio, molecular weight and temperature, some information can be gathered from Figure 2 and 4. Unsurprisingly a rise in reactant temperature results in a rise in product temperature, also closer to φ = 1 raises temperature, although leaning more to the rich side and more so with increasing reactant temperature (this can be explained by propane having a higher specific heat capacity therefore contributing more heat from its initial sensible enthalpy to reactants if initial temperature is higher. Whereas air has a far lower sensible enthalpy compared to products). The specific heat ratio goes down at higher temperatures; however, for instance in the case of initial pressure is 1atm and initial temperature is 300K temperature rises by 31.3% whereas specific heat ratio declines by 3.2%. In addition to this, it is typical for the average molecular weight to decline especially closer to stoichiometry and lean side, this effect is also very slightly increase with higher initial temperatures because of the larger presence of radicals. Lower molecular weight also increases speed of sound. The combined effect of these is that the speed of sound increases more quickly than the shock velocity increases therefore resulting in lower Mach numbers post shock.

Also unsurprisingly, increasing reactant temperatures and pressures result in diminishing returns on rise in product temperature as seen in Figure 2 and 6. Figure 3 shows perhaps a slightly confusing result that higher reactant initial temperatures lower product pressures. One would imagine that higher temperatures result in higher pressures, but because initial pressure was held at 1atm the initial density had to decline with any raise in temperature, thus resulting in a lower final pressure. What is also interesting is that the density ratio between initial and final temperature also declines with increases in reactant temperature. This can be explained by the equation below:

ρ2ρ1

=1+ 1γ2

(1−p1p2

)

There is almost no change in the final specific heat ratio as shown figure 4, but as shown before final pressure decreases with increasing initial temperature therefore this term increase and reduces the value of the ratio. The ratio increases closer to stoichiometry due to the effect that specific heat ratio decreases. This ratio is a linear factor for shock speed.

Pressure affects shock velocity similarly like temperature but even more strongly, that is because while changes in pressures only marginally affect the density ratio as shown in Figure 10, pressure was determine in project 1 to be a significant determinant in final temperature of constant pressure flames; also the term in the square root in the shock velocity can be related to final pressure by

RW 2

T 2=p2ρ2


And final pressure ratio is given by

p2=ρ2ρ1

T2/W 2

T1/W 1

p1

Since the density ratio doesn’t change much and only in an increasing fashion and final temperature only increases with increasing initial pressure, initial pressure increases shock velocity. This effect also explains the curves in figure 8. However, unlike with initial temperature Mach Numbers are not changed much, since temperature only increases slightly with increasing pressure, the same goes for final specific heat capacity. Also pressure very minutely increases the average molecular weight by decreasing the amount of radicals.

Section 2

Looking at Figure 11 this shows the amount of error over the full range of temperatures, just demonstrating that the relative error does not go much beyond 8%. Considering this is the range that C-J detonations are stable this estimate is a good initial guess at. Both Figure 11 and Figure 12 show that when temperatures are close to 2700 K errors are minimal (under 4%). Figure 11 does show an interesting trend that lean mixtures have a more well behaved error across the board, this is perhaps related to fewer variations in final specific heat ratio.

Section 3

Auto Ignition delays for these conditions were around .1 ms suggesting that ignition delays longer than this results in a blowout, probably due to some initiating reaction not producing enough radicals. This can be stated in a sense that the Damköhler number for the reaction rate to produce radicals over the radical species diffusive rate needs to be a certain value to sustain a detonation. This diffusive rate is probably a function of gas velocity and temperature. Since it is likely that temperature increases this diffusion and gas velocity decreases it, looking at the lean vs rich cases, it makes sense that the lean case ignition delay can be longer.

Also comparing the two temperatures shows that Chemkin’s temperature to be higher. Is possible that that is because the reaction hasn’t been allowed to come to equilibrium or that the ZND structure has some errors and does not predict this correctly. In the latter case, ZND is best used for analysis of rates and GasEq for actual final conditions.

Conclusion

Increasing pressure of the reactants is an effective way to increase detonation velocity. Slightly rich mixtures tend to go faster. Anything that increases temperature increases speed as well, but decreases mach number. Using the simplified shock velocity equation it is possible to get close to C-J detonation calculation values and is a good first guess at C-J detonation velocities. Using the ZND model it can be suggested that Damköhler numbers for certain reactions vs shock velocity and temperature can be a way to estimate which C-J detonations are stable and which are not. The ZND model may not be accurate to determine final conditions.


Sources

1. GasEq’s Calculations Document

Documents

Project 3