I
P. A. LONGWELL,
H. H. REAMER, N. P. WILBURN,
and 6. H.SAGE
California Institute of Technology, Pasadena, Calif.
Ballistic Piston for Investigating Gas Phase Reactions This compressor and that at the Naval Ordnance Laboratory are the only reported facilities in this country for high temperature, high pressure studies of gases
Dmw, years, interest has arisen in rapid compressional procTHE LAST 10
esses whereby gases can be subjected to extreme conditions not maintainable in steady state equipment. Thus far, mostly pressure-volume-temperature relationships (PVT) or products formed in chemical reactions have been investigated by this method. However, using equipment of the free piston type, it is believed that reaction rates and apparent order can be determined, together with information on reaction mechanism. The ballistic piston apparatus, described here and used since 1954 at the California Institute of Technology, is a free piston device capable of subjecting gas samples to transient high pressures and temperatures. ~
Literature
on. Rapid Compressional Processes
Subject Air-accelerated free piston; argon ionization measured Adiabatic compressor, free piston PVT measurements Volumetric behavior of COZ and Nt Rapid expansion of gas behind free piston PVT measurements below ambient temp: Limited data for NZ Thermodynamic properties of argon NHa formation from adiabatically compressed NZand HZ NO formation by compressing mixtures of Nz, 02,and A2 Methane oxidation under adiabatic compressioneffect of cold wall on chain reaction Patents, adiabatic compressiond processes Manufacture of hydrazine from the elements Products and, probably by crusher gage, maximum pressures
Ref.
(IS) (1, 19)
(11)
(6,16, 16)
(6) (17 ) (18)
(14)
(3)
(7, 8) (9) (3, 16, 18)
Principles of Operation
The equipment (Figure 1) consists essentially of the heavy-walled, hollow cylinder A 3.0 inches in inside diameter within which is located the free piston B. Samples are introduced into the cylinder C through valves D and E and pressure of the air, introduced into space
&. F through valve G, may be measured either by gage H or through the mercuryoil U-tube, I, by means of the pressure balance, H' (Figure 1). A mechanical vacuum pump, J, is connected through valve K to prevent contamination of sample in C from possible slow leakage of the gas from F past the O-ring seals, L and L', which are located in grooves on piston B. T o release the piston, a holding pin is sheared by rotation of the handle M after all valves are closed. The initial pressure in C is a few pounds per square inch absolute, whereas that in F is about 1000, although this may be higher or lower as conditions dictate. When the retainer pin is sheared, piston B accelerates rapidly and compresses the gas in C. This process continues under normal conditions until the piston has entered the bottom closure N and the clearance between the lower face of the piston and the end of the cylinder is from 0.50 to 0.01 inch. Because total travel of the piston is nearly 100 inches, this represents a rather high volumetric compression ratio, and pressures of about 5000 to 100,000 pounds per square inch may be realized at the lower end of the piston travel, depending on the initial conditions. This high pressure causes the piston to accelerate rapidly upward, and after some oscillation, iJ assumes some neutral position where the pressure on the two sides is of the same order. Significant mechanical friction causes the oscillating motion to attenuate rapidly. As a result of the rapid rise of pressure in C, there is a corresponding rapid rise of temperature. Temperatures higher than 10,OOOo F. may be obtained with certain gases for a few milliseconds near the end of the travel of the piston. The sample is withdrawn. from C through valve E after piston movement has ceased, An electrical pulse is obtained from each of the side contacts, P and P' as piston B passes, and the time interval between these pulses is used to determine piston velocity. Contact wires of varying heights also are mounted in holders a t locations Q in the bottom closure N . Electrical pulses obtained from these contact wires are used to determine the position of the piston as a function of time for the last half inch or so of travel. A piezoelectric gage for measuring pres-
sure and an instrument for measuring thermal flux are also mounted in the bottom closure N . Details of Apparatus
Essential features of the equipment are rather simple but care in design is necessary. The cylinder was constructed of two pieces of SAE 1040 steel tubing with a 1.5-inch wall thickness. After the interior surface was bored and honed, the cylinder was 3.000 inches in inside diameter with a variation in diameter of less than 0,001 inch in the unstressed condition. The two cylindrical sections were connected by a sleeve (Figure 1). Neoprene O-rings were used to seal the exterior of the cylindrical section of the cylinders. Buttress threads were provided to attach the cylinders to the sleeve and the axes of the cylinders were held in alignment by short cylindrical sections. The bore of the cylinders was enlarged approximately 0.001 inch over a region of about 0.2 inch from each end to avoid any impingement of one edge of the piston on the lip of the cylinder during its passage down the bore. The release mechanism (Figure 2), consisting of screw U actuated by handle M through an appropriate packing gland, applies force on rod V. The rod is attached with a brass shear pin X to sleeve W provided with hardened steel sleeves to localize shear stress in the brass pin without deforming the soft steel assembly. Operation of handle M applies a shear to the pin, thereby increasing the shear already present from pressure in space F acting on the upper part of the piston. When the pin is sheared, piston B is free to move down cylinder A (Figure 1). Bottom closure, N , of the free pistoncylinder combination is attached to the sleeve Y by the unsupported area seal, Z (Figure 3). A cup is provided in the upper part of the closure N to receive piston B which is shown just entering the cup. This cup permitted a closure with negligible clearance volume and also localized erosion in closure N which was easier to repair than the main barrel A of Figure 1. Piston B has a replaceable head as shown. The cylindrical surface of this piston head between the face and the VOL. 50, NO. 4
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c 3 -
Figure 2. The release mechanism. Manual operation has been without difficulty at pressures in space F from 200 to 1200 pounds per square inch
Figure 1. Essentially, the equipment consistsof a 132-inch heavywalled cylinder and piston
I
Figure 3. Bottom closure N is of Type 4 16 stainless steel i- THERMCCOUPLE
i
JUNC T I 01d
,/
Figure 4. One of four side contact holders placed a t approximately equal intervals along the barrel
Figure 5. Bottom contact holder and contact wire
lower O-ring groove is covered with stellite which resists erosion satisfactorily. The remainder of the piston head is made of Type 302 stainless steel. Two pistons were employed-that shown in Figure 1 weighed approximately 31 pounds, while a shorter piston
604
b
weighed only 3.5 pounds. ’i’alve E was provided in the lower part of closure N to permit withdrawal of samples and to aid in evacuating and introducing the charge before initiating a compression cycle. Before each test, the barrel and piston are lightly coated with
INDUSTRIAL AND EN6lNEERlNO CHEMISTRY
graphite which is a suitable boundary lubricant. ~ns+rumen+a+ion A 0.015-inch diameter lead wire from the side contact holder (Figure 1: P and
S M A L L SCALE ENGINEERING D A T A
P' and Figure 4) projects approximately 0.030 inch into the 3-inch bore. A positive potential of 30 volts is placed on the contact wire which is insulated by a soapstone seal, and when the piston touches the contact wire a negative pulse is transmitted through coaxial cables to an electronic gating circuit of local design which controls Model 5424 and 4 2 4 3 preset counters (Berkeley Instrument Go.) to record time intervals in units of 0.0001 second. A continuous electrical circuit from the piston to ground is provided by a wire coiled in the form of a tapered helix attached to the piston and to the top closure. The side contact holders must be removed from the barrel after each run in order to replace the lead wire. The neoprene unsupported area seal has functioned satisfactorily under this service. A bottom contact holder and contact wire (Figure 5), mounted in the bottom of the chamber at positions such as Q in Figure 1, are subjected to the large transient forces produced by the sample pressure (up to 100,000 pounds per square inch). The seal and electrical insulation are provided by soapstone parts. The portion of the insulation exposed to the chamber is made from soapstone baked at 1700' F. to produce hardness. A bare copper contact wire is inserted in the holder, and its height above the bottom of closure N (Figure 3) is measured to 0.0002 inch with a depth micrometer. Before firing, a negative potential of about 14 volts is supplied to each contact wire by individual sources with internal impedances of 1000 ohms. When the piston touches a contact wire, a positive pulse is generated and these pulses are used to control Berkeley Model 5120 time interval meters. Th? latter instruments record time intervals with a precision of 1 microsecond. The measuring element of the thermal flux meter (Figure 6) consists of a platinum-constantan thermocouple which is soft-soldered to the back of a 0.030-inch diaphragm of Type 302 stainless steel. The front surface of the diaphragm forms a portion of the bottom face of the lower closure since the thermal flux meter is installed in one of the six locations, Q, in Figure 3. The steel diaphragm is supported mechanically by a Micarta insulator, and the thermocouple leads pass through a small hole in this insulator and the supporting stainless steel follower. The cold junction is contained in the thermal flux meter housing. The sample gas reaches high temperature but only for a very short time and, compared to the frequency response of the thermal flux meter, the energy
source may be considered as instantaneous. . If an amount of heat, Q,is transferred instantaneously to one face of an infinite slab of thickness, a, the temperature rise at the opposite face is
if no energy loss is suffered. For large times, this approaches the value of temperature rise for uniform temperature :
For the geometry and physical constants involved here the temperature rise reaches 0.95 that given by Equation 2 in about 40 milliseconds. The diaphragm is not infinite in extent and there is some radial conductive transport. However, with a diameter of 10 times the thickness, this radial conduction starts to affect the thermocouple temperature by an amount greater than 0.5% only after 150 milliseconds and therefore does not interfere with the measurements. The thermocouple voltage is measured with a sensitive galvanometer unit mounted in a Wm. Miller oscillograph Type H. This galvanometer unit yields a deflection of 1 inch for 2.45 microamperes and has a natural frequency of 10 cycles per second. With an external damping resistance of 411 ohms, 1 inch on the record corresponds to 59.2' F. temperature rise of the diaphragm (Figure 7). ' This temperature rise corresponds to an energy transfer of 4.55 B.t.u. per square foot to the lower face of the piston and to the upper face of the lower closure and represents about 14y0 of the maximum kinetic energy of the piston for this case which involved a sample of hydrogen and n-hexane. The closest approach of the piston to the bottom of the chamber has been measured with lead crusher gages (Figure 8). These have taken the form of lead shot for approaches of closer than 0.075 inch, while taller gages have been made for runs where the piston stops at a greater distance from the bottom. The height of the deformed gages was measured to 0.0001 inch. When two gages of different heights have been used on the same run they have agreed in final height within the flatness tolerances of the piston head and bottom closure. I t is believed therefore that they measure the actual closest approach of the piston to within 0.0005 inch in most cases. A tourmaline piezoelectric gage made by Atlantic Research Corp. measures sample pressure as a function of time. The gage, similar in external shape to
0.1 0 TIME SEC.
0.05
Figure 7. meter
0.1'5
Data from the thermal flux
the bottom contact holder (Figure 5), is mounted in one of the holes, Q (Figure 3). The output of the piezoelectric gage is connected to a preamplifier with an input impedance of 900 megohms, and to a network of parallel capacitors. The output impedance of the preamplifier is matched to the characteristic impedance of the coaxial cable which carries the signal to a Hewlett-Packard Model 150-A oscilloscope (Figure 9,A). The two beam traces are a reference line obtained with zero input to the preamplifier, and the pressure trace. A third beam trace, appearing on the negative and consisting of timing marks from a Berkeley Model 5630-15 time standard, was too faint for reproduction. The beam is blanked out momentarily when contact is made with one of the wires in the apparatus and in this manner the pressure- and volume-time scales are related (Figure 9,B). Static calibration of the gage was furnished by the vendor, and dynamic calibration has been done by integration of the pressure-time record and comparison with piston position information. The initial pressure in the air chamber was measured by means of a calibrated Bourdon tube gage having an uncertainty of about 1%; thus, the determination of this variable contributes only a small amount to the uncertainty of predicting the over-all behavior of the system. Temperature of the air in F of Figure 1 is measured by a mercury-inglass thermometer with an uncertainty of 0.5" F. These measurements can easily be refined if greater accuracy is required in determining the state of the air in F; however, this has not been necessary since piston velocities are measured. The glass sample-addition equipment (Figure IO) includes a McLeod gage and a mercury-in-glass manometer used to determine the pressure in chamber C of Figure 1 between the additions of the several components of the mixture to be investigated. The elevation of the mercury in the arms of the mercuryVOL. 50, NO. 4
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Information Obtained and Operating Experience
The experimental information obtained may be summarized as follows:
Figure 8. Lead crusher gages. They are thought to be accurate within 0.0005 inch
in-glass manometer is measured with a cathetometer with an uncertainty of 0.010 inch. Arrangement of Equipment
T h e barrel itself (Figure 11) is mounted upon a framework constructed of relatively heavy steel angle iron. The horizontal members of this framework were purposely made rather flexible so that some movement of the barrel could be permitted without damage to the equipment. The introduction of the sample through the valve D of Figure 1 is made from the landing. The piston is released from the mezzanine above by operation of the handle M of Figure 1.
1. Amount and initial state of the sample placed in the lower chamber are determined from pressure, temperature, and the known chamber volume. Amount and final state of the sample after piston motion has ceased are determined from pressure, total volume, temperature, and the results of an analysis. Leakage can be calculated from this and the original amount of sample. 2. Amount and initial state of the driving air is determined from measurements of its pressure, temperature, and total volume. 3. The times a t which the piston face passes four rather widely separated points during the main portion of its travel and four rather closely spaced points close to the end of its travel are measured for the compression portion of the initial stroke. Piston velocities are obtained from these measurements. 4. The integrated thermal flux to a small area in the bottom closure is measured. 5. The minimum volume of the sample is determined from the height of the lead gage. 6. The pressure of the sample is determined as a function of time for the high pressure portion of the cycle by a piezoelectric gage.
I
I
I
2 TIME
Figure 9.
3 MILLISEC
Pressure-time record
A. Actual record; of time
6. Pressure as a function
and showed little damage. The insulation on the bottom contact holders gradually deteriorated and it was occasionally necessary to repair a contact holder. However, the maintenance was much less than originally expected.
By the end of March 1957, 173 tests had been made. The barrel remained in good condition. Two bottom closures became \Qorn during this period and were replaced. The piston heads with a stellite facing performed satisfactorily
Mathematical Relationships
Many principles of ballistics, such as have been summarized by Corner (Z),
VALVE BLOCK
3CHL
’
’ IJ
1
AIR SUPPLY
“A
BULSS
TRAP
-
I
SAMPLE ADDITION
VACUUN PUMP
SYSTEM
COMPRESSED GAS TANKS
/ / I PRESSURE GAUGE
EAL COMPRESSED A I R FOR PISTON RETURN
Figure 10.
606
Samples are introduced through valves
INDUSTRIAL AND ENGINEERING CHEMISTRY
4
D
and
E
(Figure 1) with this equipment
SMALL SCALE ENGINEERING D A T A are applicable to the analysis of the ballistic piston. In the authors’ experience, however, the relative magnitude of the several quantities involved is so different from those normally encountered in guns as to make a slightly different technique of analysis desirable. If forces associated with the linear acceleration of the gases are neglected and shear ofthe gas along the walls of the piston is not considered, a simple expression describes the motion of the piston :
mad&
-
+ PAVAdmA -k PBvBdms (5)
It was assumed in deriving Equation 5 that all weight changes are negative, corresponding to leakage out of each chamber, and that properties of the gases are uniform throughout each system. Expansion of the differential internal energy terms into partial derivatives and change of all total differentials into rates yields,
sign of the friction term is reversed it applies equally well to an upstroke. Solutions for Simplifled Case
Although solution of Equation G in the general case is difficult, useful solutions can be obtained on the basis of simplifying assumptions. It will be assumed for the sample gas that an ideal solution (10) is formed, that there is no leakage or heat transfer, and that the composition is constant. The equation of state is assumed to be
P(v
In the above equation forces, distances, velocities, and accelerations are taken as positive in the upward direction. Equation 3 takes into account friction associated with movement of the piston in the cylinder. I t is probable that this friction is a function of the velocity and pressure at each end of the piston and should be so treated. Equation 3 is written for a downstroke, and the sign of the friction term must be changed for an upstroke. If the forces of Equation 3 are applied through a distance, dx, an energy equation is obtained,
In deriving Equation 6 it is assumed that com’position of the driving gas remains unchanged and that local equirelationlibrium ( 6 ) is attained-i.e., ships between properties of the system at a point are the same as those for a macroscopic system at equilibrium. The partial internal energies Zkof Equation 6 include internal energies of formation. The chemical reaction taking place in the sample gas may be represented symbolically by
If the rate of reaction rl is expressed as moles of component 1 reacted per unit volume per unit time, the following relationship is valid:
If the process in the gases is considered frictionless, use of the first law of thermodynamics and differential relationships for systems of variable weight gives
= RT
(9)
where the covolume b is constant. For this equation of state the internal energy is independent of pressure and volume. Under the assumptions noted, the comIt pressional process is isentropic. can be shown for an isentropic change of state of a gas having a constant covolume that
T* = ( V * - a ) - R / +
(10)
Equation 10 is expressed in terms of normalized variables. The function 4 is a function of the specific heat and is defined by the relationship:
The equation of state, Equation 9, when expressed in normalized variables and a is neglected with respect to unity, becomes P*(V*
- a) =
T*
(12)
The driving gas is assumed to follow a polytropic expansion path given by the formula P A V A ~= constant
(13)
For the situation as described, Equation 6 is equivalent to the equality:
The relationship of Equation 8 may be substituted in Equation 6 if desired. If more than one reaction is taking place, a term of the form on the right of’Equation 8 must be substituted for each reaction.
Figure 1 1 . Position of the ballistic piston relative to landing and mezzanine
- 6)
Equation 6 may be solved for piston position if frictional forces, thermal transport, gas leakages, thermodynamic properties and equations of state of the driving gas and of the sample gas, and the reaction rates are known. As written it applies to a downstroke; however, if the
If the dependence of friction on V* is known, Equation 14 may be integrated. If the friction is assumed constant, this integration for a downstroke gives
where $ =
1
T* WT*
(16)
Equation 15 may be solved by f k s t obtaining V* and the normalized velocity dV*/dO as numerical functions of T*, and VOL. 50, NO. 4
APRIL 1958
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TIME
PRESSURE
MILLISEC
TWE
Figure 12.
v* dV*
(17)
For the return stroke of the piston the velocity is given by _ = dV*
d8
[__ 2Ag mPPB0
]''2
I (
-_ PbAAPAo _ [(I f p k-I
The integrand of Equation 17 exhibits a singularity when the velocity is zero although the integral is bounded. This difficulty may be overcome by approximating over a small interval around the singularity by A 8 =-
(19)
A series of calculations was made using these relationships to illustrate the behavior to be expected of gas samples in the ballistic piston apparatus and to display the effects of various parameters on the temperature-time and pressuretime histories. These calculations were made for perfect gases having the heat capacities at infinite attenuation of helium, of carbon dioxide, and of a n equimolal mixture of helium and carbon dioxide. The heat capacity of carbon dioxide was obtained from Rossini (72) while a constant value of 2.981 B.t.u./(lb.-mole) (" R.) was used for Cv for helium. A polytropic exponent k of 1.424, a value of frictional force F, of 50 pounds, and the dimensions of the ballistic piston apparatus were used for these calculations.
608
MILLISEC
Effect of several parameters on some relationships
11. Apparent sample temperature-time; on this figure indicate minimum volume
1. Piston position-time;
then calculating times by numerical or graphical integration of the relationship
LE PER SO IN
i l l . Apparent sample temperature-pressure for a position weight of 31 pounds.
The parameters selected for variation were sample composition, initial pressure of the driving air, ratio of initial pressure of the driving air to the initial pressure of the sample, and the weight of the piston (Figure 12). The behavior of the sample is markedly affected bv the heat capacity of the gas.
-
V*)l-k
I ( 1
+ p - Vz")'-k]
(18)
Helium, with a low heat capacity, reaches a high temperature which is maintained for a relatively long time, and the maximum pressure is relatively low. With carbon dioxide, the piston comes close to the bottom-maximum pressure is high, and maximum temperature, heid for a short time, is low. Effect of the initial pressure of the driving air on the lowest position of the piston and on the maximum temperature
INDUSTRIAL AND ENGINEERING CHEMISTRY
Table I.
Circles
is minor (Figure 12,ID and IID) and is the result of (m, - 8')in Equation 15. The maximum pressure and the time dependence of position, temperature, and pressure, however, are markedly affected by changing the initial air pressure. SVhile the change in sign of frictional force F with the reversal of the direction of piston travel makes the decompression nonsymmetric with the compression, the effect is not apparent in the lower portion of the stroke where other forces are much greater. Increasing the ratio of the initial air pressure pressure to the causes the piston to approach the bottom closelJ', and the presare raised sure and 12.IC: 1IIB, and IIc, respectively). 'The shape of the temperature-time relationship is altered drastically by changing the initial pressure ratio (Figure 12,IIC). The upper curve in Figure 12,IIB, is similar to the lower curve in
Data from Two Tests Test N o . _ I _ _
Quantity Initial press., driving air Initial press., sample Initial temp., sample Closest approach of piston Fraction leakage out Weight of piston Height of bottom contacts 1 2 3 4 Time at bottom contacts 1 2 3 4
Cnit Lb./sq. in. Lb./sq. in. ' R. In. Lb. Kn.
132
147
544,O 1.9039 529.5 0.0311 0.023 30.978
843.0 2.1181 531.29 0.0928 0.039 32.0872
0.5121 0.3506 0.2108 0,1046
1.0038 0.7504 0,4887 0.2552
psec. 0 254 491 698
0 296 639 988
SMALL SCALE ENGINEERING D A T A Table
II. Composition" of Samples Test No. 132
Hydrogen n-Hexane Propane Propylene Ethane Ethylene Methane
Initial 0.9508 0.0492
...... .. .
... 9 . .
Final 0.7529 0.0066 0.0008 0.0011 0.0082 0.0021 0.2283
Test No. 147 Nitrogen, 0.4727 0.4639 Oxygen 0.1672 0.1583 Helium 0.3601 0.3601 Nitrogen oxides* 0.0177 a Mole fraction and adjusted t o yield a total of unity. 3. Reported as nitric oxide.
...
Figure 12,IIC, and the lower curve in Figure 12,IIB, is similar to the upper curve of Figure 12,IIC. Minimum volume and maximum temperature are affected only slightly by the weight of the piston (Figure 12,IA and 12 IIA). However, velocity of the piston at a given position and, therefore, rates of change of temperature and pressure, are inversely proportional to the square root of the piston weight. Dependence of sample pressure on temperature is unaffected by weight of the piston. Thus, conditions to which a sample is subjected can be varied widely by appropriately selecting values of operating parameters such as initial sample pressure, initial air pressure, sample composition, and piston weight. Figure 12, for the adiabatic case with no chemical reaction, illustrates trends rather than exact relationships. Temperatures and Pressures for Tests with Chemical Reaction
Test 132 (Tables I and 11) involved the decomposition of n-hexane in the presence of hydrogen. The final composition (Table 11) was derived from an analysis made with a mass spectrometer. The main product of this reaction was methane, although traces of 2- and 3-carbon hydrocarbons were formed. Although the over-all reaction to give the products shown in Table I1 is exothermic, that portion taking place for the short time during which the pressures were high, was endothermic. Calculations which yielded sample temperature and pressure and the position of the piston as functions of time were made for this test. Volumetric properties of the sample were estimated from information presented by Hirschfelder (4) and the assumption of ideal solutions (70); specific heats were obtained from Rossini (72). I t was assumed that the sample was compressed
isentropically to a temperature and pressure a t which chemical reaction commenced, and that the temperature then remained approximately constant until the piston reached the bottom of its travel. The initial portion of the decompression was then taken as isentropic. The calculated temperatures and pressures are shown in Figure 13. The maximum pressure was 31,000 pounds per square inch. The pressure, however, exceeded 10% of the maximum pressure for only one millisecond. The temperature was greater than 2150' R. for about one millisecond and the maximum cooling rate was about 3' R. per microsecond. These pressures and cooling rates, while large, are not the extremes obtainable with this apparatus. Test 147, involved the compression of a mixture of nitrogen, oxygen, and helium (Tables I and 11). Nitric oxide in the amount of 0.0177 mole fraction was formed. A numerical integration of the equations of motion for this test was made neglecting thermal transfer but with the chemical reaction taken into account (Figure 13). A maximum rate of temperature change of 13' R. per microsecond was calculated for this test.
a
0
TIME
MILLISEC.
P W
Analytical Procedure
The quantitative determination of thermodynamic properties or reaction rates from data obtained with the ballistic piston apparatus require extensive computational effort. However, such properties can be obtained with this apparatus for conditions not easily attainable in more conventional equipment. Primary attention is focused upon the lower part of the stroke where the driving air pressure lis low and the sample pressure is relatively high. The effects of piston friction, driving air pressure, and potential energy of the piston are minor. First, the sample pressure is integrated twice with respect to time and substituted in the equation:
a v) w v)
n -2
0
I
MILLISEC.
Figure 13. Calculated relationships for two tests Top. Temperature-time Bottom. Pressure-time Initial isentropic compression Period during reaction and decompres-
-
-----
sion
sample pressures. As a result, the sample pressure and volume, and the piston position and velocity, are known continuously as functions of time. Next, functional forms for d B and h~ - are assumed. These may be functions of sample temperature, pressure, and volume. The internal energy is found by using mBEB
Equation 20 represents two integrations of Equation 3. The term involving ( f F - m p - APA) is relatively small and can ordinarily be approximated by use of average values. Equation 20 is then compared to the known piston . position-time data and the minimum sample volume, and the best fit is obtained by selection of the parameters F, UO, and the pressure calibration. Equation 20, with the sign of friction reversed, is then applied to the decompression process using the measured
-1
TIME
=
mBoEBo
-k
Equation 21 may be derived from Equation 4 and the first law of thermodynamics as applied to a frictionless process in the gas. Solution of Equation 21 requires numerical integration in the general case. The integral involving for a compression-decompression cycle is then compared with the data
i~
VOL. 50, NO. 4
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obtained from the thermal flux meter, and the functional dependence of 8~ is adjusted if necessary to give agreement. Likewise, the leakage rate, hB, is integrated and compared with-measured leakage. If the functional forms of h~or 4~ must be adjusted, Equation 21 12 solve2 again. When a satisfactory solution of Equation 21 is obtained, the internal energy is known as a function of time. The specific internal energy of the sample may then easily be determined as a function of specific volume and pressure. If no chemical reactions or dissociations are taking place, the pressure-volumeinternal energy relationship of the sample gas is thus known for one path; runs under different conditions will give these relationships for other paths. Temperatures corresponding to the internal energies may be calculated as functions of pressure and volume if specific heat data are available. When chemical reactions are studied, the thermodynamic properties of the sample gases must be known. Knowing the pressure and the specific volume as functions of time, temperature and t h e change in internal energy from reaction can be estimated as functions of time. From stoichiometry and AEE for the reaction, the composition is estimated and a set of compositions and temperatures which match the known pressure-volumeinternal energy-time relationship is determined. The final composition calculated is compared to that found by analysis of the gas sample. If these compositions agree, the reaction rate may be determined as a function of time for the test and then may be related to the composition, pressure, and temperature of the sample. If data are obtained for several tests under different conditions it may be possible to determine the functional dependence of reaction rate on temperature end on fugacities of the components present.
-
Conclusion
This equipment has been satisfactory for investigating a number of binary atomic systems such as nitrogen and oxygen, nitrogen and hydrogen, and carbon and hydrogen. The pressuretime and temperature-time conditions imposed upon samples may be varied within rather wide limits by changing parameters such as sample composition and pressure, driving air pressure, and piston weight. Although the quantitative analysis of data obtained with the ballistic piston is difficult, conditions which are not ’attainable otherwise may be imposed on samples with this apparatus. I t is expected that the thermodynamic properties of some nonreactive gases may be determined. The experience with this apparatus has been
61 0
satisfactory from a mechanical standpoint. Maintenance has been minor even though very severe conditions have been imposed. Acknowledgment
The equipment and methods described were developed through the financial support of the Hercules Powder Co. and the Texas Co., represented by Lyman Bonner and duBois Eastman, respectively. L. S. Wood prepared a number of the illustrations and J. B. Olin did the calculations associated with the nitrogen-oxygen system. B. L. Miller assisted in preparing the manuscript which was reviewed by W.N. Lacey.
a A 8
= normalized covolume, b/vg = YAO/YBO = increment in = time, sec.
K
=
thermometric conductivity, sq. ft ./sec.
C
=
summation ofn terms
u
=
specific weight, Ib./cu. ft.
0
= [ ~ ~ v ~ ~ n T * ) ] / T”) ( l n B.t.u./
P
k=l
(1b.- mole)( O R.) T*
#
= k u d T * B.t.u./(lb.-mole) ( R.)
a
= partial differential operator = integration operator
f
Nomenclature
Subscripts
A
A
= cross-sectional area of cylinder,
v*
sq. ft. = thickness of diaphragm, ft. = molal covolume, cu. ft./lb. mole = specific heat (solid), B.t.u./(lb.) ( ” R.1 = molal specific heat a t constant volume, B.t.u./(lb.-mole) ( ” R.1 = differential operator = specific internal energy, B.t.u./ lb. = partial specific internal energy of component k , B.t.u./lb. = change in total internal energy due to reaction of amounts in stoichiometric equation, B.t.u. = exponential function = frictional force, lb. = acceleration due to gravity, ft./ secZ = specific enthalpy, B.t.u./lb. = index of summation = polytropic path exponent, index of summation = natural logarithm = molecular weight of component k = weight, lb. = weight of piston, lb. = total leakage rate, lb./sec. = moles of component k in stoichiometric reaction = index of summation = weight fraction of component k pressure, lb./sq. in. = normalized pressure = P/Po = thermal energy from instantaneous source, B.t.u./sq. ft. = heat absorbed by system for infinitesimal change, B.t.u. = thermal flux into system, B.t.u./ sec. = molal gas constant, B.t.u./(lb. mole) ( O R.) = degrees Rankine = reaction rate, component 1 (1b.mole)/(cu. ft.)(sec.) = temperature, OR. = normalized temperature, T/Ts = velocity, ft./sec. = specific volume, cu. ft./lb. = total volume, cu. ft. = molal volume, cu. ft./lb.-mole = normalized volume = _V/_V, =
X
= distance measured upwards, ft.
a
6
C C, d
E
.Ek A_E, ex@
F g
H
i k
In
M, m
r9
m
Nk n nk
P P* Q ‘I,
5 R OR
r1
T T* U
V
vV
INDUSTRIAL AND ENGINEERING CHEMISTRY
I -
v/vo
B
i k
0 1 2
driving air sample gas index of summation index of summation, component k = initial condition = component 1 = a t bottom of stroke = = = =
Literature Cited
(1) .4blard, J. E., Larson, R. L., U. S. Naval Ordnance Laboratory, NOL 10,526, Sept. 9, 1949. (2) Corner, J., “Theory of the Interior Ballistics of Guns,” Wiley, New York, 1950. (3) Furman, M. S., Tsiklis, D. S., Doklady Akad. Nauk. (S.S.S.R.) 91, 597 (1953). (4) Hirschfelder, J. O., Curtis, C. F., Bird, R. B., “Molecular Theory of Gases and Liquids,” Wiley, New York, 1954. (5) Jacobs, S. J., Ph.D. thesis, University of Amsterdam, June 1953. (6) Kirkwood, J. G., Crawford, B., Jr., J . Phys. Chem. 56, 1048 (1952). (7) Kouzmine, E.,French Patent 860,410 (Sept. 30, 1940). (8) Zbid.,884,219 (April 19, 1943). (9) Zbid.,966,320 (March 1, 1950). (10) Lewis, G. N., J . Am. Chem. Sac. 30, 668 (1908). (11) Price, D., Lalos, G. T., Edwards, P. L., Allgaier, R. S., U. S. Naval Ordnance Laboratory, NAVORD 3990, May 10,1955. (12) Rossini, F. D., others, “Selected Values of Physical and Thermodynamic Properties of Hydrocarbons and Related ComDounds.” Carnegie Press, Pittsburgh; 1953. ’ (13) Ryabinin, Yu. N., Zhur. Eksptl. 2 Teoret. Fiz. 23,461 (1952). (14) Ryabinin, Yu. N., Markevich, A. M., Tamm, I. I., Doklady Akad. Nauk (S.S.S.&95,.111 .) (1954). (15) Seigel, A. E., Ph.D. thesis, University of Amsterdam, January 1952. (16) Seigel, A. E., U. S. Naval Ordnance Laboratory, NAVORD 2692, July I S . 1952.
(17)
(18)
(19)
Ordnance ’Laboratory, NAVORD 2219, Oct. 4, 1951. RECEIVED for review July 29, 1957 ACCEPTEDDecember 26, 1957
