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A modification of the composite propellant erosive burning model of lenoir and robillard
C O M B U S T I O N A N D F L A M E 24, 365-368 (1975)
365
A Modification of the Composite Propellant Erosive Burning Model of Lenoir and RobiUard MERRILL K. KING Atlantic Research Corporation, Alexandria, Virginia 22314
The Lenoir and Robillard model for the erosive burning of composite propellants is shown to suffer from use of an incorrect additivity law for coupling of the effects of heat feedback from the propellant flame and feedback from the core gas. A revised expression, employing a physically realistic additivity concept is derived and shown to fit data from an independent source at least as well as the original expression.
Introduction Erosive burning refers to the augmentation of "normal" burning rate of a solid propellant by action of a flow of gas parallel to the burning surface. Several empirical relations and models appear in the literature to describe the total burning rate, which is the sum of the normal and erosive rates. Of several proposed models for the erosive burning o f composite propellants, the most widely accepted and used is that of Lenoir and RobiUard [ 1 ]. These investigators assumed that the total burning rate, r, is the sum of the normal nonerosive burning rate, r0, which is a function of pressure alone, and a second erosive rate, r e , which depends on the combustion gas crossflow. The second rate arises from convective heat transfer from the "core" gas to the propellant surface. That is, they assumed: r = ro + r e.
(1)
Lenoir and RobiUard used two empirical relationships to describe the heat transfer from the core gas under cross-flow conditions, with transpiration: h = ho exp (-/3ppr/G),
(2)
ho = 0.0288 G Cp Re "°'2 Pr "°'667,
(3)
h = heat transfer coefficient with transpiration;
ho = heat transfer coefficient in the absence of transpiration; /3 = dimensionless constant; t~e = combustion gas heat capacity; = Reynold's number (length) of cross-flow; Pr = Prandtl Number; G = mass flux of cross-flow; pp = propellant density. They then assumed that the "erosive rate," r e , is proportional to h, thus arriving at: r = a p n + k ho exp (-/3ppr/G),
(4)
where k is a proportionality constant. By grouping certain insensitive quantities into a consant, or, and taking the Prandtl number equal to unity, they arrived at the following expression for the burning rate: r = a P n + a G °'a L -°'2 e x p ( - / 3 p p r / G ) ,
(5)
where L is the distance from the head end of the propellant grain, by which Re is defined, and a and/3 are constants, which are adjusted to best fit a set of data for a given propellant formulation. Theoretical Analysis It is the additivity concept expressed by Eq. (1) that is questioned in this communication. This equation entails an assumption that the pressuredependent "base" (nonerosive conditions) rate,
Copyright © 1975 by the Combustion Institute Published by American Elsevier Publishing Company, Inc.
366
MERRILL K. KING
ro, is unaffected by an increase in total rate at a given pressure, an assumption inconsistent with accepte d physical pictures of the solid propellant combustion process. This can be shown by the following simplified thermal balance burning rate analysis, in which the heat flux required to raise the propellant from its bulk temperature to the surface temperature is equated to the sum of fluxes being supplied by the propellant gas flame and the free-stream feedbacks. Let us examine a propellant burning at a given pressure under erosive and n0nerosive conditions. For the nonerosive case, the heat balance is written as:
ropsCp (L- r? = k(rl.- rs)laXe,,,
Since the velocity of propellant decomposition products normal to the burning surface is directly proportional to the burning rate, Eq. (8) may be rewritten as:
AX[ =CI(rDIFF + ~KIN)r.
For a bimolecular fuel-oxidizer gas phase reaction (generally assumed) rK1N is a function of temperature and pressure alone, independent of cross-flow. According to the granular diffusion flame theory (Sutherland [2] ), the diffusion time may be expressed as:
(6)
while for the erosive case, it is written as:
rPsCps(Ts-Ti)" =k(Tf-Ts)/AXfe + h(Tcore-Ts), (7) where
(9)
TDIFF = b 2/2D,
(I0)
b = Fuel vapor pocket characteristic dimension, D = Diffusivity of fuel vapor in oxidizer plus products, where
Ps = propellant density; Cps = propellant heat capacity; Ts = propellant surface temperature; Ti= propellant bulk temperature; k = effective thermal conductivity of gas from heat release zone back to the surface; AXf = effective standoff distance of average heat release from gas-phase combustion reactions (n and e refer to nonerosive and erosive cases, respectively); Tore = core gas temperature (mainstream gas temperature) The effective flame standoff distance, AXf may be expressed as:
AXf = Ugas,n (rDIFF + rXiN),
(8)
where
Ugas,n = velocity of propellant decomposition products normal to the burning surface; rDIFF = time associated with mixing of fuel
and oxidizer gases leaving propellant surface; r K iN = time associated with reaction of mixed fuel and oxidizer gases.
b =6OFslPFv)1/3 Sin ~/~,
. (11)
PFS = solid fuel density ; PFV = fuel vapor density; S = linear dimension of typical solid fuel pocket; n = number of fuel vapor pockets per solid fuel pocket. Obviously, the diffusivity, the two densities, and the linear dimension of a typical solid fuel pocket (S) appearing in Eqs. (10) and (11) are independent of the cross-flow. The parameter "n" might depend on the cross-flow only if the cross-flow caused turbulence to penetrate sufficiently close to the propellant surface. However, in typical erosive burning situations the laminar boundary layer associated with the cross-flow is on the order of 100 microns or more thick, while the flame zone must be on the order of 20 to 50 microns from the surface in order to supply sufficient heat feedback to the unburned propellant to gasify it at typical observed rates. Accordingly, on the basis of the granular diffusion flame model, TDIFF is independent of the cross-flow and the effective flame standoff distance is accordingly proportional to the propellant burning rate. Therefore:
EROSIVE BURNING OF COMPOSITE PROPELLANTS
k/axre =
(to~r).
(12)
Combination of Eqs. (6, 7, 12) yields, after rearrangement: r = ro2/r + h (Tcore-Ts)/PsCps(Ts - Ti) = ro2/r + re
(13) (The second equality follows from the definition of r e in the Lenoir and Robillard paper.) Thus, it may be seen that the additivity concept used by Lenoir and Robillard is basically incorrect. In physical terms, they have failed to account for the fact that increased burning rate, caused by erosive feedback, at constant pressure, results in the propellant flame being pushed further from the surface, decreasing its heat feedback rate and thus decreasing th.e propellant burning rate part of the way back toward the base rate. For completeness, it should be noted that use of solid propellant combustion models other than the granular diffusion flame model may result in some dependency of rDIFF and thus AXf on cross-flow velocity (use of a columnar diffusi6n flame model will yield such a dependency for example). However, in order for the Lenoir and Robillard expression to be obtained, this dependency of AX.f on G would have to be such as to yield an exact reverse dependency of (rDIFF + rKIN) on the total burning rate "r", a most unlikely result. It should also be pointed out that Lenoir and Robillard do not postulate any dependence of the combustion process on the cross-flow, assuming only that the cross-flow affects heat transfer from the coreflow. Therefore, on the basis of their assumed mechanism, rDIFF should be independent of "r" and hence AXf should be proportional to "r". Substitution of the second term on the righthand side of Eq. (5) (the Lenoir and Robillard rate expression) for r e, and aP n for ro into Eq. (13) yields the following final expression for burning rate as a function of pressure and cross-flow mass flux:
r = (aen)2/r + o~G°'s exp [-[Jrpp/G]/L°'2.(14) Comparison with Data Both the original Lenoir and Roblllard rate expression [Eq. (5)] and the modified expression
367
[Eq. (14)] have been used to analyze a set of erosive burning rate data generated by Marklund and Lake [3] in a device in which the exhaust of a main propellant charge is passed over a sample propellant tablet in a blast-tube. The data analyzed is that for a formulation designated as Propellant A by Marklund and Lake. The "base" burning rate (nonerosive conditions) for this formulation is given by ro = 0.0898P "3ss withP in atm and ro in cm/sec. Under their test conditions (fixed core gas temperature, fixed blast-tube cross-sectional area, and fLxed length, L), Eqs. (5) and (14) become, respectively: r = 0.0898P "3ss + a'P °'s V°'s exp [-3'r/PV],(15)
r =(O.0898P'3Ss)2/r + ot"P °'8 V °'8 exp [-fY'r/PV], (16) with r in cm/sec, P in atmospheres, and V(crossflow velocity) in m/sec. Regression analyses were performed for both of these expressions with the Marklund and Lake data plotted in Fig. 1. These regression analyses resulted in values o f d and B' for Eq. (15) of 0.000648 and 5820, respectively, 3.0
I
1
I
I
~ V : 214 m/sec] - O V = I03 m/sec~ Data ~ V = 49 m/see J
2.0
E
I
1.0
,,
'
~
.#"
O:
I
I
J
/&
/
'
A
q "Pl
.I
z_ o.6
49
0.4q
m/To I
LENOIR AND ROBILLARD MODIFIED L&R
0.3
_
} I1111
0.2 20
40
60 80 100 PRESSURE (atm)
200
Fig. 1. Comparison of erosive burning rate data of Marklund and Lake [3] with predictions by best-fit, Lenoir and RobiUard [1], and modified L & R expressions.
368 and values o f a " and fl" for Eq. (16) of 0.000668 and 2900, respectively. These values were then used to generate "theoretical" curves for each of the three cross-flow velocities tested. As may be seen from Fig. 1, the two expressions give nearly identical results, both fitting the data reasonably well. Thus, it appears that neither expression is clearly superior to the other in fitting the data: however, the modified expression is on firmer theoretical ground and accordingly is to be preferred.
MERRILL K. KING References 1. Lenoir, J. M., and Robillatd, G., A Mathematical Method to Predict the Effects of Erosive Burning in Solid-Propellant Rockets, Sixth Symposium {InternationaO on Combustion, Reinhold Publishing Company, NY, 1957, pp. 663-667. 2. Sutherland, G. S., The Mechanism of Combustion of an Ammonium Perchlorate-Polyester-Resin Composite Solid Propellant, PhD Thesis, Princeton University, 1956. 3. Marklund, T., and Lake A., Experimental Investigation of Propellant Erosion, ARSJournal, 173-178 (Feb. 1960). Received 2 August 1974; revised 6 January 1975
365
A Modification of the Composite Propellant Erosive Burning Model of Lenoir and RobiUard MERRILL K. KING Atlantic Research Corporation, Alexandria, Virginia 22314
The Lenoir and Robillard model for the erosive burning of composite propellants is shown to suffer from use of an incorrect additivity law for coupling of the effects of heat feedback from the propellant flame and feedback from the core gas. A revised expression, employing a physically realistic additivity concept is derived and shown to fit data from an independent source at least as well as the original expression.
Introduction Erosive burning refers to the augmentation of "normal" burning rate of a solid propellant by action of a flow of gas parallel to the burning surface. Several empirical relations and models appear in the literature to describe the total burning rate, which is the sum of the normal and erosive rates. Of several proposed models for the erosive burning o f composite propellants, the most widely accepted and used is that of Lenoir and RobiUard [ 1 ]. These investigators assumed that the total burning rate, r, is the sum of the normal nonerosive burning rate, r0, which is a function of pressure alone, and a second erosive rate, r e , which depends on the combustion gas crossflow. The second rate arises from convective heat transfer from the "core" gas to the propellant surface. That is, they assumed: r = ro + r e.
(1)
Lenoir and RobiUard used two empirical relationships to describe the heat transfer from the core gas under cross-flow conditions, with transpiration: h = ho exp (-/3ppr/G),
(2)
ho = 0.0288 G Cp Re "°'2 Pr "°'667,
(3)
h = heat transfer coefficient with transpiration;
ho = heat transfer coefficient in the absence of transpiration; /3 = dimensionless constant; t~e = combustion gas heat capacity; = Reynold's number (length) of cross-flow; Pr = Prandtl Number; G = mass flux of cross-flow; pp = propellant density. They then assumed that the "erosive rate," r e , is proportional to h, thus arriving at: r = a p n + k ho exp (-/3ppr/G),
(4)
where k is a proportionality constant. By grouping certain insensitive quantities into a consant, or, and taking the Prandtl number equal to unity, they arrived at the following expression for the burning rate: r = a P n + a G °'a L -°'2 e x p ( - / 3 p p r / G ) ,
(5)
where L is the distance from the head end of the propellant grain, by which Re is defined, and a and/3 are constants, which are adjusted to best fit a set of data for a given propellant formulation. Theoretical Analysis It is the additivity concept expressed by Eq. (1) that is questioned in this communication. This equation entails an assumption that the pressuredependent "base" (nonerosive conditions) rate,
Copyright © 1975 by the Combustion Institute Published by American Elsevier Publishing Company, Inc.
366
MERRILL K. KING
ro, is unaffected by an increase in total rate at a given pressure, an assumption inconsistent with accepte d physical pictures of the solid propellant combustion process. This can be shown by the following simplified thermal balance burning rate analysis, in which the heat flux required to raise the propellant from its bulk temperature to the surface temperature is equated to the sum of fluxes being supplied by the propellant gas flame and the free-stream feedbacks. Let us examine a propellant burning at a given pressure under erosive and n0nerosive conditions. For the nonerosive case, the heat balance is written as:
ropsCp (L- r? = k(rl.- rs)laXe,,,
Since the velocity of propellant decomposition products normal to the burning surface is directly proportional to the burning rate, Eq. (8) may be rewritten as:
AX[ =CI(rDIFF + ~KIN)r.
For a bimolecular fuel-oxidizer gas phase reaction (generally assumed) rK1N is a function of temperature and pressure alone, independent of cross-flow. According to the granular diffusion flame theory (Sutherland [2] ), the diffusion time may be expressed as:
(6)
while for the erosive case, it is written as:
rPsCps(Ts-Ti)" =k(Tf-Ts)/AXfe + h(Tcore-Ts), (7) where
(9)
TDIFF = b 2/2D,
(I0)
b = Fuel vapor pocket characteristic dimension, D = Diffusivity of fuel vapor in oxidizer plus products, where
Ps = propellant density; Cps = propellant heat capacity; Ts = propellant surface temperature; Ti= propellant bulk temperature; k = effective thermal conductivity of gas from heat release zone back to the surface; AXf = effective standoff distance of average heat release from gas-phase combustion reactions (n and e refer to nonerosive and erosive cases, respectively); Tore = core gas temperature (mainstream gas temperature) The effective flame standoff distance, AXf may be expressed as:
AXf = Ugas,n (rDIFF + rXiN),
(8)
where
Ugas,n = velocity of propellant decomposition products normal to the burning surface; rDIFF = time associated with mixing of fuel
and oxidizer gases leaving propellant surface; r K iN = time associated with reaction of mixed fuel and oxidizer gases.
b =6OFslPFv)1/3 Sin ~/~,
. (11)
PFS = solid fuel density ; PFV = fuel vapor density; S = linear dimension of typical solid fuel pocket; n = number of fuel vapor pockets per solid fuel pocket. Obviously, the diffusivity, the two densities, and the linear dimension of a typical solid fuel pocket (S) appearing in Eqs. (10) and (11) are independent of the cross-flow. The parameter "n" might depend on the cross-flow only if the cross-flow caused turbulence to penetrate sufficiently close to the propellant surface. However, in typical erosive burning situations the laminar boundary layer associated with the cross-flow is on the order of 100 microns or more thick, while the flame zone must be on the order of 20 to 50 microns from the surface in order to supply sufficient heat feedback to the unburned propellant to gasify it at typical observed rates. Accordingly, on the basis of the granular diffusion flame model, TDIFF is independent of the cross-flow and the effective flame standoff distance is accordingly proportional to the propellant burning rate. Therefore:
EROSIVE BURNING OF COMPOSITE PROPELLANTS
k/axre =
(to~r).
(12)
Combination of Eqs. (6, 7, 12) yields, after rearrangement: r = ro2/r + h (Tcore-Ts)/PsCps(Ts - Ti) = ro2/r + re
(13) (The second equality follows from the definition of r e in the Lenoir and Robillard paper.) Thus, it may be seen that the additivity concept used by Lenoir and Robillard is basically incorrect. In physical terms, they have failed to account for the fact that increased burning rate, caused by erosive feedback, at constant pressure, results in the propellant flame being pushed further from the surface, decreasing its heat feedback rate and thus decreasing th.e propellant burning rate part of the way back toward the base rate. For completeness, it should be noted that use of solid propellant combustion models other than the granular diffusion flame model may result in some dependency of rDIFF and thus AXf on cross-flow velocity (use of a columnar diffusi6n flame model will yield such a dependency for example). However, in order for the Lenoir and Robillard expression to be obtained, this dependency of AX.f on G would have to be such as to yield an exact reverse dependency of (rDIFF + rKIN) on the total burning rate "r", a most unlikely result. It should also be pointed out that Lenoir and Robillard do not postulate any dependence of the combustion process on the cross-flow, assuming only that the cross-flow affects heat transfer from the coreflow. Therefore, on the basis of their assumed mechanism, rDIFF should be independent of "r" and hence AXf should be proportional to "r". Substitution of the second term on the righthand side of Eq. (5) (the Lenoir and Robillard rate expression) for r e, and aP n for ro into Eq. (13) yields the following final expression for burning rate as a function of pressure and cross-flow mass flux:
r = (aen)2/r + o~G°'s exp [-[Jrpp/G]/L°'2.(14) Comparison with Data Both the original Lenoir and Roblllard rate expression [Eq. (5)] and the modified expression
367
[Eq. (14)] have been used to analyze a set of erosive burning rate data generated by Marklund and Lake [3] in a device in which the exhaust of a main propellant charge is passed over a sample propellant tablet in a blast-tube. The data analyzed is that for a formulation designated as Propellant A by Marklund and Lake. The "base" burning rate (nonerosive conditions) for this formulation is given by ro = 0.0898P "3ss withP in atm and ro in cm/sec. Under their test conditions (fixed core gas temperature, fixed blast-tube cross-sectional area, and fLxed length, L), Eqs. (5) and (14) become, respectively: r = 0.0898P "3ss + a'P °'s V°'s exp [-3'r/PV],(15)
r =(O.0898P'3Ss)2/r + ot"P °'8 V °'8 exp [-fY'r/PV], (16) with r in cm/sec, P in atmospheres, and V(crossflow velocity) in m/sec. Regression analyses were performed for both of these expressions with the Marklund and Lake data plotted in Fig. 1. These regression analyses resulted in values o f d and B' for Eq. (15) of 0.000648 and 5820, respectively, 3.0
I
1
I
I
~ V : 214 m/sec] - O V = I03 m/sec~ Data ~ V = 49 m/see J
2.0
E
I
1.0
,,
'
~
.#"
O:
I
I
J
/&
/
'
A
q "Pl
.I
z_ o.6
49
0.4q
m/To I
LENOIR AND ROBILLARD MODIFIED L&R
0.3
_
} I1111
0.2 20
40
60 80 100 PRESSURE (atm)
200
Fig. 1. Comparison of erosive burning rate data of Marklund and Lake [3] with predictions by best-fit, Lenoir and RobiUard [1], and modified L & R expressions.
368 and values o f a " and fl" for Eq. (16) of 0.000668 and 2900, respectively. These values were then used to generate "theoretical" curves for each of the three cross-flow velocities tested. As may be seen from Fig. 1, the two expressions give nearly identical results, both fitting the data reasonably well. Thus, it appears that neither expression is clearly superior to the other in fitting the data: however, the modified expression is on firmer theoretical ground and accordingly is to be preferred.
MERRILL K. KING References 1. Lenoir, J. M., and Robillatd, G., A Mathematical Method to Predict the Effects of Erosive Burning in Solid-Propellant Rockets, Sixth Symposium {InternationaO on Combustion, Reinhold Publishing Company, NY, 1957, pp. 663-667. 2. Sutherland, G. S., The Mechanism of Combustion of an Ammonium Perchlorate-Polyester-Resin Composite Solid Propellant, PhD Thesis, Princeton University, 1956. 3. Marklund, T., and Lake A., Experimental Investigation of Propellant Erosion, ARSJournal, 173-178 (Feb. 1960). Received 2 August 1974; revised 6 January 1975