MATLAB SIMULINK MODEL FOR THE
DOUBLE ACTING PNEUMATIC CYLINDER
REPORT
Presented to Dr. Mitty C. Plummer,
Professor in Engineering Technology,
University of north Texas in partial fulfillment
of the requirements for special problem
MSET 5900.703, Spring 2003
By
Sunitha Vegerla
Denton, Texas
May, 2003
ACKNOWLEDGEMENTS
I would like to extend my very sincere thanks to my advisor, Dr. Mitty C. Plummer, for
introducing me to the area of cryogenic heat engines, for the wealth of knowledge he shared with
me, and for the great enthusiasm with which he directed all the time.
This work would not have been possible without the continuous help and support from Dr. Igor
N. Kudryavtsev in the Matlab design.
Finally, I would like to thank Department of Engineering Technology, UNT for the computer
support.
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Contents
 Objective
 Introduction
 Pneumatic Systems
 The double acting Pneumatic Cylinder
 Significance of Parameters used in the Mathematical Model
 Step1: During First Cycle - Filling
o Physical Assumptions
o Parameters used for Calculations
 Step2: During Second Cycle – Expansion
o Physical Assumptions
o Parameters used for Calculations
 Mathematical Model of Piston Dynamics
 Matlab - Simulink Design
 References
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Objective
Develop a Matlab-Simulink model of the proposed dual expansion engine for the next liquid
nitrogen automobile.
Introduction
Pneumatic Systems
The pneumatics is termed as compressible fluid mechanics since it deals with the use of
compressed or pressurized gas as a source of power. Pneumatics differs from the theory of
aerodynamics in the same way that mechanics of materials differs from the theory of elasticitythat is, pneumatics is concerned with the macroscopic point of view. From this point of view a
volume containing gas is characterized by single values of pressure and temperature which are
presumed constant throughout the volume1.
A basic result of the compressibility is the in a fixed volume of gas either temperature of mass
(or both) must change before pressure changes. Therefore, when a pneumatic volume under a
spring loaded piston has been vented, opening a valve does not move the piston immediately.
Rather, sufficient gas must flow into the volume to increase the pressure until the force
overcomes the spring preload.
Another result of compressibility is the energy storage capacity of gases. It is the capacity that
has placed much pneumatic equipment on board space vehicles of all types, where energy
storage is vital, as is package size and weight.
To increases the load on a pneumatic actuator (motor) either gas must be added to the volume or
the motor must stroke, decreasing the volume of the enclosed mass of gas. This means the load
changes are difficult to absorb by means of a piston held in place by gas pressure.
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The major application of pneumatics is in the area of cryogenic tank pressurization, in which the
use of pneumatics in low-temperature environments is highly favored by the very low
temperatures. 2
The double acting pneumatic cylinder
Figure (1)
Direction of movement of piston is considered to be one directional, i.e., from left to right as
shown in the figure (1).
The working and exhaust cavities are denoted as 1st cavity and 2nd cavity respectively
including all the indexes associated with them.
The points A, B, C, and D are piston position detectors that control opening and closing of
electromagnetic valves at the manifolds. The sequence of operation is shown in table 1
below.
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TABLE 1
Pm1
Pm1′
Pm2′
Pm2
A
0
X
X
0
B
X
X
X
0
C
X
X
X
0
D
X
X
X
X
Where
Pm1′ is the inlet manifold for the 1-st cavity
Pm1 is the -----------------------Pm2′ is the exhaust manifold for the 2-nd cavity
Pm2 is the ---------------------------S1 and S2 are the squares of the piston
ts is the width of the piston,
D is the diameter of the cylinder,
Dd1 and Dd2 are the diameters of the rod (from left and right sides accordingly),
xA, xB, xC and xD are the correspondent distances for the piston position measured from zero,
F is the force of load (constant for the first approximation),
x is the position of the piston (variable, dependent on time), and
dx/dt is the velocity of piston motion during operation.
In the first cycle (step1), the inlet valve is open for the gas to enter into the working cavity. In the
second cycle (step2), the inlet valve is closed for the gas to expand in the working region.
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Significance of Parameters used in the Mathematical Model
x – Coordinate of the piston (m)
v – Velocity of the piston (m/sec)
t – Time of operation (sec)
M – Mass of piston (kg)
FL – Load force (N)
FFR – Frictional force (N)
S – Cross-sectional area of the piston (m2)
Sv – Cross-sectional area of the input valve (m2)
vg - Velocity of gas entering the cylinder (m/sec)
P1 – Pressure in working cavity (Pa)
V1 – Volume of gas in working cavity (m3)
m1 – Mass of gas in working cavity (kg)
ρ1 – Density of gas in working cavity (kg/m3)
T1 – Temperature in working cavity (K)
P2 – Pressure in exhaust cavity (Pa)
Pm – Pressure of gas in the intake manifold (Pa)
Variables are indicated with blue color.
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First Cycle (Step 1)
The inlet valve is open and hence the pressure is taken as the atmospheric pressure. The
following assumptions are considered on the basis of ideal gas behavior.
Physical Assumptions
Consider the ideal gas as a working fluid, which possesses constant temperature T1 during the
working cycle. We assume constant pressure at the manifold Pm and exhaust cavity P2 for the
first approximation.
Parameters for Calculations
Length of the cylinder
L= 0.3 m
Density of Nitrogen gas at n.c.
ρ0 = 1.2506 kg/m3
Cross-sectional area of the piston
S = 3.5256∙10-3 m2
(diameter 0.067 m)
Cross-sectional area of the intake valve
Sv = 3.1416∙10-4 m2
(diameter 0.02 m)
Mass of piston with all moving parts
M = 2 kg
Load force
FL = 400 N
Frictional force
FFR = 10 N
Gas constant3 (for nitrogen)
R = 296.8 J/(kg∙K)
Pressure of gas in the intake manifold
Pm = 0.5 MPa
Temperature of gas in working cavity
T1 = 295 K
Pressure of gas in exhaust cavity
P2 = 0.101325 MPa
Position of piston detector location A
XA= 0.01 m
Position of piston detector location B
XB = 0.02 m
Position of piston detector location C
XC = 0.28 m
Position of piston detector location D
XD = 0.29 m
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Mathematical Model of Piston Dynamics
The change in position of the piston with respect to time depends on the mass of the piston (M),
load force (FL), frictional forces (FFR), pressure in the working cavity (P1), and on the pressure in
the exhaust cavity (P2). This can be represented with the equation (1) as following:
d 2x
M 2  ( P1  P 2)  S  ( FL  FFR )
dt
(1)
Applying Ideal gas law for the working cavity results in equation (2)
P1V1  m1RT1 ,
(2)
where R is a gas constant.
The change in mass of gas with respect to time in working cavity depends on the density of the
gas,
1 , velocity of the gas, vg , and cross sectional area of the valve in the working cavity, Sv ;
which can be represented using equation (3)
dm1
 1v g Sv .
dt
(3)
Bernoulli’s equation 3 for gas flow through the input valve is represented using equation (4)
( Pm  P1 ) 
1v g 2
2
.
(4)
The volume of the gas in the working cavity is calculated from the piston position and cross
sectional area of the piston, which is shown in equation V1  x  S .
From equation (4), the velocity of gas can be written as,
vg 
2( Pm  P1 )
1
.
(5)
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Substituting the formula for velocity, vg in equation (4) results dm1/dt as,
2( Pm  P1 )
dm1
,
 1Sv
dt
1
(6)
dm1
 Sv 2 1 ( Pm  P1 ) .
dt
(7)
or
From the equation (2) one can obtain
m1 
P1V1
R T1
or
1 
P1
.
R T1
(8)
Substituting the given formulas for m1 and 1 in equation (7), one can obtain
d  P1V1 
P

  Sv 2 1 ( Pm  P1 ) .
dt  RT1 
R T1
(9)
Substituting the equation for V1 results in,
d  P1 xS 
P

  S v 2 1 ( Pm  P1 ) .
dt  RT1 
R T1
(10)
Taking out the constants S, R, and T1 from the derivative results in,
S
d
( P1  x)  v 2 RT1P1 ( Pm  P1 ) .
dt
S
Using product rule of the differentiation, above equation is further simplified as;
x
dP1
dx S v
 P1

dt
dt S
2 RT1 P1 ( Pm  P1 ) .
(11)
Hence dP1/dt can be written as;
dP1 1 S v
 {
dt
x S
2RT1 P1 ( Pm  P1 )  P1
dx
}.
dt
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As a result, we have a system of two differential equations
d 2 x 1
 [( P  P2 ) S  FL  FFR ],

 dt 2 M 1

 dP1  1 { S v 2 RT P ( P  P )  P dx
1 1
m
1
1
 dt
x S
dt

(12)
In order to solve the equations (12) numerically it is necessary to add the initial coordinate,
velocity of piston as well as initial pressure in the working cavity:
x (0)=0.01 m, v (0) = 0,
P1 (0)=0.2 MPa.
Second Cycle (Step2)
Second cycle is the expansion cycle, in which both the inlet and exhaust valves are closed,
leaving the conditions in close approximation to polytropic expansion.
Physical Assumptions
In reality, the compressor conditions are neither purely isothermal nor isentropic but
somewhere in between. This is accounted for calculating the compression work by using the
isentropic equation with n, where n (1< n <1.4) is the polytropic constant.
Parameters for Calculations
Pressure of gas in the intake manifold
Pm = 0.5 MPa
Pressure of gas in exhaust cavity
P2 = 0.101325 Mpa

As the name indicates temperature is constant.

For an ideal gas under adiabatic frictionless (isentropic) conditions, specific heat ratio, k, is approximately 1.4 for
diatomic gases (O2, N2. Etc.)
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Mathematical Equations for the Expansion Cycle
Polytropic Expansion is given by the following relationship
P1V1n  P1BV1nB  Const
(1)
General Expressions for the Double Acting Pneumatic Cylinder
The following are the set of equations based on the position of piston.
0, x  xA

d 2x  1
  [( P1  P2 ) S  FL  FFR ], xA  x  xD
dt 2  M
0, x  xD

dP1 1 S v
 [
dt
x S
P1  P1B (
2RT1 P1 ( Pm  P1 )  P1
dx
], x  xB
dt
xB n
) , x  xB
x
References:
1. Blaine W. Anderson, “The Analysis and Design of Pneumatic Systems”, John Wiley &
Sons, Inc. 1967, page No. 1-16.
2. Ron Darby, “Chemical Engineering Fluid Dynamics”, Marcel Dekker, Inc. 1996, page.
No. 106.
3. David R. Lide, “Handbook of Chemistry and Physics”, 71st Edition, 1990-1991, The
Chemical Rubber Co. Page 1-1.
4. Simulink Dynamic System Simulation for Matlab, Using Simulink Version 4,
Mathworks, 1990-2000.
5. Sample results
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