CC BY
14КРИОГЕННЫЕ ТРАНСПОРТНЫЕ СРЕДСТВА CRIOGENIC VEHICLES
COMPUTER SIMULATION OF PNEUMATIC ENGINE OPERATION
I. N. Kudryavtsev , A. V. Kramskoy, A.I.Pyatak , M.C.Plummer
Htii Members of International Editorial Board
УДК 621.541; 532.51; 519.683
Kharkov National Automobile and Highway University 25 Petrovskogo st., Kharkov, 61002, Ukraine Ph.: +38 057 707 36 72; e-mail: kudr@khadi.kharkov.ua
* University of North Texas 310679, Denton, Texas, 76203, USA
Physicist diploma in 1987 and he got Ph.D. in Physics in 1990 at Kharkov State University.
Associate Professor of the Faculty of Physics, Director of Laboratory for Cryogenic and Pneumatic Devices at Kharkov National Automobile & Highway University (Ukraine).
His research activity is focused on investigation of thermal and kinetic properties of solid, liquid and gaseous nitrogen, superconductivity, development and optimization of the pneumatic engines, heat exchangers and cryogenic pumps, computational physics. He has more than 90 scientific publications.
Igor N. Kudryavtsev
Ms.D. in Engineering (2001).
Post-graduate student of the Automobile Department, Kharkov National Automobile & Highway University (Ukraine).
Present scientific interests: computing gaseous dynamics and aerodynamics processes; control systems; heat and mass transfer; computer modeling and simulation. He published more than 10 scientific papers.
Alexander V. Kramskoy
Physicist diploma in 1965, he got his Doctorate of Physical and Mathematical Science in 1993.
Professor of the Faculty of Physics, Mechanical Department, Kharkov National Automobile & Highway University (Ukraine).
His research interests include investigation of physical properties of liquid, gaseous, solid and plasma states of matter, power engineering, ecologically clean transport systems and alternative energy sources. He has more than 100 scientific publications
Alexander I. Pyatak
Nuclear engineering diploma at Texas A&M University (1967), Ph.D. in Nuclear Engineering (1970).
Associate Professor of Engineering Technology Department, University of North Texas (USA).
His research interests include ecologically clean transport systems and power, cryogenic vehicles, design and development of heat exchangers and pneumatic systems, applications of nuclear power, vibrations. He published more than 60 scientific papers.
Mitty Charles Plummer
A mathematical model for the double acting pneumatic cylinder operation is proposed, which allows to calculate both the dynamic characteristics of piston motion and flow gas parameters without using any fitting procedures. The corresponding computer code in MATLAB-SIMULINK software is developed and numerical simulation of the double acting cylinder operation has been accomplished. The PV diagram for the working cycle is calculated and analysis of the working process is presented. The approach proposed allows calculation of a wide set of thermodynamic and operational parameters for various pneumatic cylinders and can be used for development of the highly efficient pneumatic engine intended for vehicle propulsion.
Introduction
At the present time, a new direction in designing automobiles using cryogenic technologies and pneumatic power-plants is being developed [1-3]. Pneumatic engines possessing high efficiency and correspondingly low consumption of compressed gas are necessary for the development of non-polluting cryogenic automobiles that run on liquid nitrogen as well as for the pneumatic vehicles operating on compressed air. The piston expansion machines based on pneumatic cylinders most closely correspond to this criterion [4]. Recent developments in pneumatic servosystems and innovative pneumatic components [5, 6] show important advances, which are expected in vehicle applications also.
Design optimization for a pneumatic engine of a given set of characteristics is possible as a result of mathematical modeling of working cycles. Therefore, the development of an adequate mathematical model is a reasonable scientific and technical approach.
In this work the mathematical model of the double acting pneumatic cylinder, which can be used independently or in a group of pistons in the pneumatic engine of a vehicle, is proposed. It is necessary to note, that the given design is actually equivalent to a two-cylinder pneumatic engine with a consecutive operating mode for the cylinders and a drive with crank-shaft mechanism.
The purpose of the mathematical model developed is to determine the basic dynamic parameters; namely, gas pressure in the cylinders, position and speed of the piston in time, cycle frequency, and calculation of the operational characteristics (power, efficiency, specific work, gas consumption, etc.) of the pneumatic engine being considered.
Development of mathematical model
The schematic of a pneumatic cylinder is represented on Fig. 1. We will first consider a stroke of the piston from left to right. The movement
of the piston from point A to point B corresponds to the process of filling of cavity 1 by compressed gas from manifold Pm1 and exhaust of gas from cavity 2 to manifold Pm2. At point B the supply from Pm1 is cut off by closing the inlet valve. The movement of the piston between points B and D corresponds to the process of expansion of the working gas in cavity 1. At point D the exhaust valve is closed. Next, the reverse motion of the piston takes place and points A, D and B, C are functionally interchanged.
For a description of the dynamics of the piston movement between the points A and D, it is necessary to determine the parameters of the working gas state. The equation of the piston motion generally can be written as
d x
M-T = p1S1 - p2 S2 - F, dt
(1)
where p2 are the pressures in the working and exhaust cavities correspondingly; S1, S2 are the useful areas of the piston for cavity 1 and 2; F is the resistance force, which consists of the force of friction Ffr and loading force FL; M is the mass of piston with all moving parts (rods, crank-shaft mechanism, etc.). In Eq. 1 the quantities p1 and p2 are the unknowns. Now, we derive the equations for the pressures that correspond to every stage of motion measured from the right edge of the piston.
Suppose that all thermal energy dQmi admitted with the gas is changed to internal energy dUj
Fig. 1. The schematic of the double acting pneumatic cylinder
Статья поступила в редакцию 13.01.2005 г. The artisle has entered in publishing office 13.01.2005
and the work of the gas expansion dWj and write the equation of the energy balance according to the first law of thermodynamics
dQmi = dUi + dWx. (2)
Assuming, that pressure in the system of the receiver-manifold does not vary during filling of the working cavity, we use the relation dQmi = dH . In this case the quantity of thermal energy, which has arrived from the inlet (Pm1) to the cavity 1, is equal to the product of the mass of gas dmm1
and the specific enthalpy (dQmi = hmi dmmi) and the change of the gas internal energy dU1 and work dW1 made by it are equal accordingly dU1 = d (u1m1) and dW1 = pldVl. Therefore, Eq. 2 can be written in the following form:
hmi dmmi = u1dm1 + m1du1 + p1dV1, (3)
where u1 is the specific internal energy of gas in the cavity 1; V1 is a volume of cavity 1; mass of gas entering the cavity 1 ; the quantities with index m relate to the manifold or pipeline.
We can express in Eq. 3 the values of enthalpy and internal energy through the product of temperature and heat capacity at constants pressure cp and volume cV, according to equation
CpTm1dmm1 = CVT1dm1 + CvmidTi + p^.
(4)
The equation of state of the real gas in the working cavity is written as
pV = zmRT, (5)
where R is the gas constant, and compressibility factor z determines the extent of non-ideality of the working fluid.
Numerical calculations of the compressibility factor z for nitrogen, accomplished in the approach
Compressibility factor vs temperature and pressure
1
0.99 0.98 0.97 0.96
Pressure, MPa
4 200
220
280
260
240
Temperature, K
proposed in Ref. [7] for the real gases, showed that in the pressure range considered (P = 0.1...3.5 MPa) and temperature range T > 210 K the assumption that z = 1 is highly accurate, and the gas can be treated as ideal (see Fig. 2).
Substituting in Eq. 4 the value m1dT1, taken from Eq. 5 with the approximation z = 1, and us-
c
ing the common notations — = k and c - cV = R,
cv
where k is the adiabatic exponent, after simple transformations one can obtain the following expression
kRTmi dmmx = Vdp1 + kp\dV\. ( 6)
We can then replace the mass of gas dmm , entering the volume V1 during time dt in Eq. 6, by the corresponding value of the consumption function G1 (defined as dmmi = G\dt) and then express the equation relative to the pressure
kGlRTmidt dv
dp, =-1--крл-.
1 v v
и
к
= const
к
Pi
к P.
mj
П
300
0.995 0.99 0.985 0.98 0.975 0.97 0.965 0.96 0.955 0.95 0.945
2к ' Pmj \ Pi
к - i vPmi Pi /
и =
From the adiabatic equation
ч1/к
Pi
Pm1
Pi Pm
Pi =P
m1
ч1/к
Pi Pm
Fig. 2. Compressibility factor z calculated for nitrogen in the real gas approach
(7)
We can also determine the function G1 that describes the gas consumption. For this purpose we can find the parameters from Bernoulli's equation
2 k -1 p
of the adiabatically braked stream (parameters of braking). In this case we will put the velocity of a stream equal to zero over the area of the inlet valve into Bernoulli's equation. Then we have
2 k -1 p\ k -1 Pm\ (8)
Now one can find the velocity of gas entering the first cavity from Eq. 8
(9)
one can find the value of gas density p1 in the working cavity
(10)
By substituting expression 10 into formula 9 we have
2
The gas consumption function can be defined as G1 = u^a^, where a1 is the area of the cross section of the inlet valve; is a coefficient of gas consumption through the inlet valve. Let us substitute the velocity of gas, determined by formula 11, into the expression for the gas consumption. Then we obtain
1 -
/ \ Л.
Рщ
V
k -1
(12)
If we substitute the gas density in the first cavity from Eq. 10 to 12 we can derive
g =p
щ
/ k Pi
Pm1
V 1 /
2k RT k-1 RTmi
k-1
1 -
/ \ Pi
Pm1 V 1
= «1^1
2k p 2 RT
k - ^ m1 m1
2/k
P1
Pm1 V 1
k+1
/ \ P1
Pm1 V 1
= «1^1 P
m1
г k+1 1
2/k ( \
2k 1 P1 P1
k -1 RTm1 Pm1 V 1 Pm1 V 1
As a result, the function G1 of the gas consumption from the unlimited volume (manifold) can be determined by the formula [8-11]
G1 = P
2k 1
m1 I
Ik -1 RT„
m1
/ \ k / \
P1 P1
Pm V 1 У Pm V 1 /
k+1
k
' (13)
stant parameter, n. In reality, it follows from formula 13, that the consumption G1 is a function of the pressure ratio, in which the numerator is the pressure of the medium into which gas flows, and the denominator is the pressure of the medium from which this gas moves.
We will present formula 13 for gas flow from a pipe in a more convenient form [9]
G =
JRTm
k+1
Л/k _ k
(14)
2k
where 9(a) = Vc2/k -a k ; 0l = p/pm; K =
V k -1
In order to find the maximum of the gas consumption factor 9(a), let us set its derivative to
2 2/k-1 k +1 1/k n zero, which can be written as 7a---— a = 0,
k k
from which one can obtain the critical ratio of pressures
G =
k +1
k k-1
(15)
For example, for the adiabatic exponent k = 1.4 we have a* = 0.5282.
Fig. 3 shows the consumption factor 9(a) calculated for k = 1.4. We must note, that a branch of 9(a), shown by a dotted line, is invalid. As shown by experiment, in this range the consumption function is a maximum [8, 11] (a continuous horizontal line on Fig. 3). Function 91(a) in the given figure corresponds to the process of the gas being exhausted from a cavity of constant volume. The corresponding formula for 91(a) will be presented below.
фН
where p and Tmi are the gas pressure and temperature in manifold Pm1.
Note that the losses of gas pressure in the pipeline and local resistances are taken into account by introducing the coefficient of consumption ^ [9, 11], which besides takes into account the compression of flow stream during exhaust, the speed of the gas as it approach the aperture and other factors. More often, this coefficient of consumption is determined experimentally or with the help of approximations.
When the flow of gas occurs within a short span of a pipe at high velocity, the exhaust process is considered to be adiabatic, and it is possible to use formula 13. The process of compression of the gas in a working cavity is described by Eq. 6, which we can solve simultaneously with Eq. 13. As a result it is possible to determine the pressure p1 in cavity 1 as a function of time. We must note that this process cannot be described by means of elementary polytropic process with a con-
0.3
0.2
0.1
\ Ф1 (g)
л
/ / f / / / \
/ / / /
0.1
0.3
0.5
0.7
0.9
1.1
G
Fig. 3. The consumption factors for filling the cavity of constant volume by the compressed gas 9(a) and for the exhaust 91(a) at k = 1.4
It is necessary to distinguish between two regimes of the flow; subcritical, when the consumption function G1 is determined by the formula 13, and supercritical, at which the maximum critical consumption of gas G* is obtained after substitution of the critical ratio of pressures from Eq. 15 into Eq. 14 [9]
2k
V(k - №
(16)
mi
where 9(0*) = 0.2588 for k = 1.4.
If we substitute the value of the critical gas consumption G* into Eq. 7 instead of G1, we can obtain the equation that describes the process during the supercritical mode in a cavity of changing volume
dp1 =-— dt - kp1 —-1.
V v
(17)
The analysis of Eqs. 7, 13, 16 and 17 shows, that the process of change of gas state in a cavity being filled, both for variable and for constant volume does not coincide with any one of the elementary thermodynamic processes, which occur with a constant polytropic exponent. The process can be described using the variable polytropic exponent n, which, in the beginning of the process, equals the adiabatic exponent, and then monoton-ically decreases [8]
n = 1 +
I (k -1)
(18)
where o0 = p0 / p ; p0 is a pressure in the filling cavity at the start time.
At 0 = 1, i. e. at the end of process, the value of polytropic exponent asymptotically approaches the isothermal exponent n = 1.
Change of the gas state by polytropic process (with a constant polytropic exponent) is possible in a working cavity when there is a constant mass of gas. Some examples include the internal combustion engines after closure of the inlet valve, in flap-valve pneumatic motors, where the chamber is completely isolated by plates from the inlet and outlet ports, and in compression-driven devices and accumulators, when there is expansion of the compressed gas. In the case of variable gas quantity in a cavity, it is necessary to investigate the process by the energy balance stated in equation 6.
Thus, we can obtain the differential equation for determination of the gas pressure during the filling of cavity 1 in a general form, by substituting the value of gas consumption G1 from expression 14 into Eq. 7
—P1 dt
m , s kp1 dV1
9(G1)--dV-
(19)
V V
Let's consider the expansion stage of the pneumatic cylinder operation; the process of gas expan-
sion during the movement of piston from point B to point D (see Fig. 1). We will describe this polytropic process with a parameter 1 < n < 1.4, that allows us to take into account, to a first approximation, the possible processes of heat exchange in the pneumatic cylinder.
Then the pressure in cavity 1 can be determined by the following expression
P1 = P1„
' xB - ts^
v x - ts /
и
T
k p
= const
и
m2 2
k P,
m^2
k
p2
k - 1 Pm2 k - 1 P2
Um2 =
с \
2k p2 p,2
Jk -1 P2 pm2 ,
In accordance with the adiabatic equation
P2
pm2
/ \1/k p2_ pm2
P 2 =P
m2
Г V k
pm 2
(20)
where xB is the corresponding distance traveled by the piston to arrive at position B (see Fig. 1); x is the current piston position; ts is a thickness of the piston; p, is the gas pressure at the beginning of the expansion stage.
Also consider the process of gas outflow from cavity 2. The first law of thermodynamics can be applied here, but in this equation it is necessary to put a minus sign in the left side as there is a removal of gas
-dQm2 = dU2 + dW2. (21)
Accordingly, we change the index "1" (concerning to the first cavity), to an index "2" for the second cavity. As a result, we obtain the following expression
-kRTm2 dmmi = V2 dp2 + kp2 dV2. (22)
Let's determine the gas consumption function. For this purpose, from Bernoulli's equation
,2
k -1 p
we will find the parameters of the adiabatically braked stream. Also in this case we will put the velocity of the gas stream inside the second cavity equal to zero. Then we have
,2
(23)
where um2 is the velocity of the gas exhaust to the
outlet pipe.
From Eq. 23 we can find the gas stream velocity, exhausting to the manifold Pm2
(24)
one can obtain the value of the gas density in the second cavity
(25)
Let's substitute the gas density from expression 25 into 24. Then we have
Um2 = ,
2k
k - 1
RT
k-1
1 -
Pm2
P2
(26)
The gas consumption function for the second cavity we can define as G2 =um pm a2^2, where a2 is the area of the cross-section of outlet valve; p,2 is a coefficient of gas consumption through the outlet valve. If we substitute the velocity from Eq. 26 to the expression of the gas consumption function, then we obtain
G2 =Pm2 «2^2
2k
k -1
RT2
(27)
41/k
«2^2,
G2 =P
= «2M JTl P22RT2
2k
k - Г
RT
k-1
1-
m2 \ P2 ,
2/k
P2
k+1
P2
(28)
= «2^2 P2,
2k 1
k -1RT
2/k
k+1
P2
V /
m2 \ P2 ,
At the supercritical regime of the exhaust from the limited volume, the gas consumption is variable, as the value of p2 is varying, though the consumption function in this case is constant. Having substituted dmm2 = G2dt and G2 from Eq. 28 into Eq. 22, we can obtain the equation for determining pressure in the exhaust cavity, which is connected to pipeline Pm2
dP2 = k«2 KPl4RT2
Ф
1
kP2 dx xE - x dt '
(29)
P2
Pm2
T
n—1
T
(30)
Then we can obtain the following equation for determining the gas pressure in cavity 2 during exhaust from the limited volume [9]
dP2 dt
3n-1
k^2«2KP2 2n
Ф
1
S2 (xE - x)P,
n-
2n
kP2 dx xE - x dt
. (31)
m2
As a result it is necessary to solve the next equation for determining the dynamic characteristics of the piston motion in the pneumatic cylinder
0, x = xA d2 x = PA - P2 S2 - F
dt2
M
0, x = xD
xA < x < xD.
(32)
Let's substitute the value of the gas density in the second cavity, determined from Eq. 25, into expression 27. Thus we have
Thus, the total set of the equations, describing the dynamics of the pneumatic cylinder can be presented as
dP1
dt ~ V1 Ф(С ) V1 dt '
if x < xB;
xB - ts Y f > B
p1 = pb - , if x > xB;
x - ts
V У
dp, = k^2«2Kp22n JkT, dt
S2 (xE - x)P
mo
n-1— Ф
2n m2
kp2 dx ; xE - x dt
0, x = xA
d_x = P1- P2 S2 - F dt2 M
0, x = xD
xA < x < xD
(33)
dt S2 (xE - x)
where xE is the length of the pneumatic cylinder (Fig. 1); x is the current piston position; a 2 = p2 / pm2.
The temperature T2 in the Eq. 29 can be expressed through the pressure p2 from the poly-tropic equation, accounting for the processes of heat exchange in an exhaust cavity of the pneumatic cylinder
For the numerical solution of the derived differential equations 33 it is necessary to add the initial parameters, describe the physical sizes and operating conditions of the pneumatic cylinder.
Modeling of the operation
A computer program in MATLAB-SIMULINK software has been developed [12] for calculation of the dynamic parameters of the pneumatic cylinder by using derived Eqs. 33 (see interface at Fig. 4).
The following parameters and initial conditions were chosen for the numerical simulation:
- working fluid-nitrogen, R = 296.8 J/(kgK);
- adiabatic k = 1.4 and polytropic n = 1.25 exponents;
- initial pressure in cavity 1 p1) = 0.255 MPa
- initial pressure in cavity 2 p2) = 0.113 MPa
- pressure in the 1st manifold pmi = 3.5 MPa
- pressure in the 2nd manifold pm2 = 0.113 MPa;
- gas temperature in the 1st manifold T = = 300 K;
n
00 СП
H
о о -1 ю S <
> >
2 ¡Z и» и»
К) К)
о о
о о
1Л 1Л
Simulation of the biateral pneumatic cylinder operation
Help
Input parameters for the 1st cavity
Input parameters for the 2nd cavity
Pneumatic cylinder
32.1 1
48.17
1.5
19.33
899.2
0.01939
0.03878
25.79
247.6
300.7
53.07
0.5247
0.9701
0.509
0.3257
0.8962
1.389
71.63
3.438
14.54
340
365.7
392.5
333.1
342.2
Clearance volume V0, cm3 Filling volume V1: cm3 Ratio l/yv0
Volume expansion ratio Working volume, cm3
Time for one piston motion, s Total cycle time, s Frequency on shaft, Hz
Net specific work, kJ/kg Sp. expansion work, kJ/kg Sp. compression work, kJ/kg
Thermal efficiency Mechanical efficiency Effective efficiency Effectiveness Isothermicity
Mass of gas entered to cyl., g Gas flow rate G, g/s Gas consumption, m3/min Consumption ge, kg/kW h
28
32.11
4.179
87.19
0.07143
0.02
0.02
3.5
0.289
3.336
0.1572
0.15
0.3118
0.113
Piston stroke (xD-xA), cm Active piston square, cm2 Ratio (xD-xA)/D Ratio (xD-xA)/S, 1/m Ratio d^xD-xA) Diameter of bore d1t m Diameter of bore d2, m
Pm1 pressure, MPa Pi initial, MPa P, in point S, MPa P, in point D, MPa P2 initial, MPa P2 final, MPa Pm2 pressure, MPa
Resultant time dependences
343.9| Work for one piston motion J
1 12| Mechanical work on rod, J
115.5| Mechanical work total, J
655.4| Total heat brought, J
490.4| Quantity of heat on stage I, J
1 651 Quantity of heat on stage II J
9.2911 Heat input on stage II, kW
Initial temp, in 1st cavity, K Temperature in point B, K Average T, at stage I, K Temperature in point D, K Average T, in 1st cavity, K
0.09197
1.389
1.413
0.09907
334.6
m-i0, g
m^ final, g
m2o, g
m2 final, g Average T2 in 2nd cavity, K
Plot 3D temperature in 2nd cavity
Plot 3D temperature in 1st cavity (method 2)
IIH4SIIM
17.74
5.776
35.65
51.91
292
5.624
Power, kW Power on rod, kW Torque, N-m
Work on stage I, J Work on stage II, J Ratio Ai/A
Fig. 4. Interface of the computer program for calculation the dynamical and operational characteristics of the pneumatic cylinder
© 2005 Научно-технический центр «TATA»
© 2005 Scientific Technical Centre «TATA:
- gas temperature in the 2nd manifold Tm2 = = 270 K;
- loading force FL = 400 N;
- constant component of the friction force Ffr= =10 N;
- coefficient of friction b = 0.15 [11];
- gas consumption coefficient in cavity 1 m1 = = 0.7 [11];
- gas consumption coefficient in cavity 2 m1 = = 0.7 [11];
- diameters of input and output valves d1 = = 0.02 m, d2 = 0.02 m;
- diameter of the piston D = 0.067 m;
- rod diameters in the 1st and 2nd cavities Dd1 = 0.02 m, Dd2 = 0.02 m;
- initial piston speed u0 = 0,
- mass of piston M = 2 kg,
-distances xA = 0.01m, xB = 0.02 m, xC = = 0.28 m, xD = 0.29 m,
- piston stroke xE = 0.3 m.
The results of numerical modeling of the gas pressure in the 1st and 2nd cavities of the pneumatic cylinder as functions of the piston position and time are presented in Fig. 5.
Now, let's analyze the physical processes, which occur within the cavities of the pneumatic cylinder, in accordance with the numerical results obtained [13].
At the opening of input valve in pipe Pmi, gas at pressure of 3.5 MPa enters the cavity 1, also the outlet valve of cavity 2 is open and the piston remains motionless until the force arising from
Pressure in 1st cavity
0.1 0.2 0.3 Piston stroke, m
Pressure in 1st cavity
0.4
8 £ CL
CL
5
CD
ffi 0.2
Fig. 5.
0.01 0.02 0.03 0
Time of operation, s The dynamics of pressure changes in the cavities
the pressure in cavity 1 exceeds the force of friction and force of useful resistance. During this period there is filling of the constant volume of cavity 1 , and the piston remains motionless at point A. On the graph of pressure as a function of the piston motion this process is displayed by a practically vertical segment near the value x = 0.01 m.
The following stage of the pneumatic cylinder operation is the continuing process of filling of cavity 1 , for an already moving piston from point A up to B. During this process the outlet valve of cavity 2 is open and exhaust of gas from cavity 2 takes place. The pressure in cavity 1 , which was established after filling the cavity while the piston was motionless, falls to 3.2 MPa. The pressure in cavity 2 rises slightly (see Fig. 5).
After the piston passes point B, the outlet valve of the exhaust cavity remains open, the inlet valve of the working cavity is closed, and the piston continues to move to the right under the action of pressure forces in the first cavity. During this period the pressure in cavity 1 decreases, but there is some increase of pressure in cavity 2. The pressure increase can be explained by the fact that the flow capacity of the outlet valve is insufficient to dissipate all of the pressure formed by the movement of piston. In Fig. 5 of pressure dependence vs position of piston in cavity 1 , the process of passing point B and closing of the inlet valve corresponds to the break of the pressure curve.
Further movement of the piston there corresponds to the gas expansion with the closed inlet valve of cavity 1 and the open outlet valve in cavity 2 (piston position between points B and D). During this period gas pressure decreases by poly-tropic expansion of the gas in cavity 1 . In cavity 2 the pressure rises slightly to 0.255 MPa, which precisely equals the initial pressure in cavity 1 .
The final period of the pneumatic cylinder operation is the termination of the piston movement at point D and closing of the outlet valve.
It can be seen from Fig. 5, that the process under consideration has both a non-stationary and essentially nonlinear character caused by
Pressure in 2nd cavity
0.1 0.2 0.3 Piston stroke, m
Pressure in 2nd cavity
0.01 0.02 Time of operation, s of pneumatic cylinder
0.03
the big differences of pressures in cavities and manifolds.
The final pressure in cavity 1 (0.113 MPa) accurately coincides with the pressure in the exhaust pipeline Pm2 and with the initial pressure in cavity 2 as well. This supports the self-consistency of the calculation approach used for the description of cyclic operation of the double acting pneumatic cylinder.
Note that the mathematical model developed here allows us to calculate the time dependences of position and velocity of the piston, and frequency of operation for pneumatic cylinder. In Fig. 6 the calculated PV diagram is presented for the process under consideration, using the pressure dependences obtained for piston motion in the cavities 1 and 2.
Pressure, MPa
3.5
0.8 1.0 Volume, m310-3
Fig. 6. PV diagram of the process
during filling and exhaust of the working gas. We plan to accomplish this generalization in following stages of the investigation.
It is necessary to point out that for finding the temperature and entropy of gas in the filled cavity it is necessary to fix the value of polytropic exponent. As specified above, in this process the polytropic exponent is a variable value and can be determined, for example, from expression 13. In a more general approach for numerical calculation of the polytropic exponent, it is necessary to solve the equations of gas dynamics (even in a one-dimensional form).
Conclusions
The mathematical model developed in this paper allows one to accomplish the numerical simulation of the working process and to determine the main dynamic characteristics of the double acting pneumatic cylinder. By using the results of the numerical calculations, the analysis of the particulars of changes of gas pressure in cavities can be accomplished. The subsequent stages of pneumatic cylinder operation can also be studied to calculate the main operational characteristics. In the next stage of research the authors plan to use the mathematical model developed and the corresponding program in MATLAB-SIMULINK software for optimization and for determination of the most effective operating mode of the pneumatic cylinder for a pneumatic (cryogenic) engine for vehicle propulsion.
Nomenclature
A — specific useful work, kJ/kg;
a
sp
^ a2 — cross sections of inlet and outlet valves, m2
Within the framework of the model developed and by using the given initial parameters we have also made numerical calculations of the specific work, power, frequency of operation and specific nitrogen consumption in the pneumatic cylinder, which are important parameters for the development of the vehicle:
Asp = 185.7 kJ/kg;
N = 19.3 kW;
v = 1620 rot/min;
Gn2 = 19.4 kg/(kW h).
The results obtained agree with the data available in literature [4, 14] for pneumatic power systems.
Note, that the theoretical approach suggested in the present paper allows one to take into account the heat exchange between the working fluid and the pneumatic cylinder (environment) only at the process of gas expansion in cavity 1 on the section B-D in the model of polytropic expansion with constant polytropic exponent. It seems useful to take into consideration more details of the phenomenon of heat exchange, including processes
b — coefficient of friction;
cv — heat capacity at constant volume, J/(kgK); cp — heat capacity at constant pressure, J/(kgK); D — diameter of piston, m;
Dd1, Dd2 — rod diameters in 1st and 2nd cavities, m; d1, d2 — diameters of input and output valves, m; F, Fl, Ffr — resistance, loading and friction forces, N;
9(0) — gas consumption factor;
G1, G2 — gas consumption functions for cavities
1, 2, kg/s;
GN2 — specific gas consumption, kg/(kWh); Hm1 — enthalpy of gas entering the cavity 1, J; hm1 — specific enthalpy, J/kg; k — adiabatic exponent; M — mass of piston, kg; m1 — mass of gas entering the cavity 1 , kg; Mi, M2 — coefficients for gas consumption in cavities 1, 2;
N—working power of cylinder, kW; n — polytropic exponent; v — frequency of operation, rot/min; P1, P2 — pressures in cavities 1, 2, MPa;
ГШ1' Fm2
MPa;
— pressures in 1st and 2nd manifolds,
p1o, p2o — initial pressures in cavities 1, 2, MPa; Q — thermal energy entering the cavity 1, J; Qm2 — thermal energy, which is removed from cavity 2, J;
R — gas constant, J/(kgK); « p1, P2 — gas densities in 1st and 2nd cavities, i kg/m3;
* pm1, pm2 — gas densities in 1st and 2nd pipelines, | kg/m3;
^ S1, S2 — useful areas of piston for cavities 1 and 1 2, m2;
u
^ a0, a1, a 2, a* — ratios of pressures; 1 T , Tm2 — gas temperatures in 1st and 2nd maniS folds, K;
g T1, T2 — gas temperatures in cavities 1, 2, K;
® U1, U2 — internal energies of gas in cavities 1, 2, J;
u1 — specific internal energy of gas in the cavity 1, J/kg; 3
V1, V2 — volumes of cavities 1, 2, m3; u1 — velocity of gas entering the 1st cavity, m/s; um2 — velocity of gas exhaust from the 2nd cavity, m/s;
W1, W2 — works for gas expansion in cavities 1, 2, J;
x — current piston position, m;
xA, xB, xC, xD — distances (see Fig. 1), m;
xE — piston stroke, m;
z — compressibility factor.
References
1. Plummer M. C., Koehler C. P., Flanders D. R., Reidy R. F., Ordonez C. A. Cryogenic Heat Engine Experiment // Advances in Cryogenic Engineering. 1998. Vol. 43. P. 1245.
2. Williams J., Knowlen C., Mattick A. T., Hertzberg A. Frost-free cryogenic heat exchanger for automotive propulsion // Proc. of 33rd AIAA/ ASEE Joint Propulsion Conference & Exhibition, Seattle, USA, 1997.
3. Turenko A. N., Pyatak A. I., Kudryavt-sev I. N. et al. Ecologically clean cryogenic transport: modern state of problem // Bulletin of the
Q.
I
CL
С I T
I
LP С С
e
Kharkov National Automobile and Highway University (Ukr). 2000. Vol. 12-13. P. 42-47.
4. Turenko A.N., Pyatak A.I., Kudryavtsev I.N. et al. Pneumatic power plants for ecologically clean transport vehicles // Automobile Transport (Ukr). 2001. Vol. 7-8. P. 193-197.
5. Belforte G. New developments and new trends in pneumatics // Proc. 6th Triennal International Symposium on Fluid Control, Measurement and Visualization, Sherbrooke (Qc), Canada, 2000.
6. Stoll K. New developments in pneumatics // Proc. of The 5th Int. Conf. on Fluid Power Transmission and Control (ICFP). Hangzhou, China, 2001.
7. Kudryavtsev I. N., Pyatak A. I., Kudr-yash A. P., Marinin V. S., Bondarenko S. I. Ther-modynamic properties of solid, liquid and gaseous nitrogen // Ukr. J. of Physics. 2002. Vol. 47(8). P. 784-790.
8. Gertz E. V. Dynamics of pneumatic systems of machines. Moscow: Mashinostroyeniye, 1985.
9. Gertz E. V., Kreinin G. V. Calculation of pneumodrive. Moscow: Mashinostroyeniye, 1975.
10. Loytsyanskij L. G. Mechanics of liquid and gas. Moscow: Nauka, 1973.
11. Metlyuk N. F., Avtushko V. P. Dynamics of pneumatic and hydraulic drives of automobiles. Moscow: Mashinostroyeniye, 1980.
12. Kudryavtsev I. N., Kramskoy A. V., Pyatak A. I. Program for calculation and visualization of dynamical and operational characteristics for double-acting pneumatic cylinder // Copyright of Ukraine. No. 10560 from 15 July 2004.
13. Turenko A. N., Bogomolov V. A., Kudryavtsev I. N., Kramskoy A. V., Pyatak A. I., Plummer M. C. Mathematical model of the pneumatic cylinder with the bilateral drive // Automobile Transport (Ukr). 2002. Vol. 10. P. 10-16.
14. Plummer M. C., Ordonez C. A., Reidy R. F. A review of liquid nitrogen propelled vehicle programs in the United States of America // Bulletin of the Kharkov National Automobile and Highway University (Ukr). 2000. Vol. 12-13. P. 47-52.
