Class 11 mathematical modeling of pneumatic and hydraulic systems

ICE401: PROCESS INSTRUMENTATION

AND CONTROL

Class 11

Mathematical Modeling of Pneumatic and Hydraulic Systems

Dr. S. Meenatchisundaram

Email: [email protected]

Process Instrumentation and Control (ICE 401)

Dr. S.Meenatchisundaram, MIT, Manipal, Jan – May 2015

Pneumatic System:



• Pneumatic system uses compressible fluid as working

medium and it is usually air.

• In pneumatic systems, compressibility effects of gas cannot

be neglected and hence dynamic equations are obtained using

conservation of mass.

• In pneumatic systems, change in fluid inertia energy and the

fluid’s internal thermal energy are assumed negligible.

• In pneumatic system, the mass and volume flow rates are not

readily interchangeable.

• Pneumatic devices involve the flow of gas or air, through

connected pipe lines and pressure vessels.

Pneumatic System:



• Hence, the variables of pneumatic system are mass flow rate,

qm, and pressure P.

• The mass flow rate is a through variable and it is analogous

to current. The pressure variable is across variable and is

analogous to voltage.

• The two basic elements of a pneumatic system are the

resistance and capacitance.

• The gas flow resistance, R is defined as the rate of change in

gas pressure difference for a change in gas flow rate.

( )( )

2/

/ sec

Changein gas pressure difference N mR

Changein gas flowrate Kg=

Pneumatic System:



• Pneumatic capacitance is defined for a pressure vessel and

depends on the type of expansion process involved.

• The capacitance of a pressure vessel may be defined as the

ratio of change in gas stored for a change in gas pressure.

( )

( )2/

Changein gas stored KgC

Changein gas pressure N m=

Pneumatic System:



Pros and Cons of Pneumatic systems:

• Advantages

– The air or gas used is non inflammable and so it offerssafety from fire hazards.

– The air or gas has negligible viscosity, compared to highviscosity of hydraulic fluids.

– No return pipelines are required and since air can be letout at the end of work cycle.

• Disadvantage

– The response is slower than that of hydraulic systemsbecause of the compressibility of the working fluid.

Pneumatic System:



Applications of Pneumatic systems:

• Guided Missiles

• Aircraft systems

• Automation of production machines

• Automatic controllers

• Many more……

Pneumatic System:



Pi = air pressure of the source at steady state (newton/m2)

P0 = air pressure in the vessel at steady state (newton/m2)

∆ Pi = small change in air pressure of the source from its steady state

∆ P0 = small change in air pressure of the vessel from its steady state

Pneumatic System:



• Rate of gas storage in vessel = rate of gas inflow

• Applying Laplace and rearranging the terms, we get

0 0id P P PP

Cdt R R

∆ ∆ − ∆∆= =

0 ( ) 1

( ) ( 1)i

P s

P s RCs

∆=

∆ +

Pneumatic System:



Consider the system shown below:

Pneumatic System:



Let

Steady-state value of input air pressure

Increase in the pressure of air-source

Steady-state value of pressure inside the bellows

P = Increase in pressure inside the bellow

Steady state value of air flow rate

qm = Increase in air flow rate

A = Area of each flat surface of the bellows

R = Resistance of the restriction

C = Capacitance of the bellows.

x = Displacement of the movable surface of the bellows

iP =

ip =

P =

mQ =

Pneumatic System:



• Let the pressure of air source be increased from its steady

state value by an amount Pi.

• This results in an increase in air flow by qm and increase in

the pressure inside the bellows by p.

• Due to increase in pressure, there will be a displacement of

the movable surface of the bellows, by an amount x.

• Here, the terms Pi, qm, p and x are all functions of time, t and

therefore can be expressed as Pi(t), qm(t) p(t) and x(t) .

• The force exerted on the movable surface of the bellows is

proportional to increase in pressure inside the bellows,

i.e, Fb ∝ p(t)

Pneumatic System:



• Force exerted on the movable surface of the bellow,

Fb =A p(t) (8.1)

• The force opposing the movement of the flat surface of

bellow walls is proportional to the displacement, i.e.,

F0 ∝ x(t) (8.2)

• Force opposing the motion,

F0 =K x(t) (8.3)

• Where, K is a constant representing the stiffness of the

bellows. At steady state the above two forces are balanced,

Fb = F0 → A p(t) =K x(t) (8.4)

Pneumatic System:



• The resistance R can be given as,

• Rearranging, (8.5)

• The Capacitance C can be given as,

• Rearranging,

(8.6)

( ) ( )Difference between Change in pressure

Change in air flow rate ( )

i

m

p t p tR

q t

−= =

( ) ( )( ) i

m

p t p tq t

R

−=

( )Change in air flow rate

( )Rate of change of pressure

mq tC

dp tdt

= =

( )( )

m

dp tq t C

dt=

Pneumatic System:



• Equating eqn. 8.5 and 8.6,

(8.7)

• Rearranging, (8.8)

• From the eqn. 8.4,

(8.9)

• Differentiating eqn. 8.9 yields,

(8.10)

• Substituting eqn. 8.10 into 8.8,

( ) ( )( ) ip t p tdp tC

dt R

−=

( )( ) ( )

i

dp tRC p t p t

dt+ =

( ) ( )K

p t x tA

=

( ) ( )dp t K dx t

dt A dt=

Pneumatic System:



(8.11)

• Taking Laplace

(8.12)

• Rearranging,

(8.13)

• Where,

( )( ) ( )i

K dx t KRC x t p t

A dt A+ =

( ) ( ) ( )i

K KRC sX s X s P s

A A+ =

( )

( ) 1 1i

A AX s K K

P s RCs sτ= =

+ +

RCτ =

Hydraulic Systems:



• They are used in hydraulic feedback systems and in combined

electro-mechanical-hydraulic systems.

• In hydraulic devices, power is transmitted through the action of

fluid flow under pressure and the fluid is incompressible.

• The fluid used are petroleum based oils and non – inflammable

synthetic oils.

• Hydraulic devices used in control systems are generally

classified as hydraulic motors and hydraulic linear actuators.

• The output of a hydraulic motor is rotary motion.

• The hydraulic motor is physically smaller in size than an

electric motor for the same output.

Hydraulic Systems:



• Applications are power steering and brakes in automobiles,

steering mechanisms in large ships, control of machine tools

etc.

Advantages:

• Hydraulic fluid acts as a lubricant and coolant.

Comparatively small sized hydraulic actuators can develop

large forces or torques.

• Hydraulic actuators can be operated under continuous,

intermittent reversing and stalled conditions without damage

• Hydraulic actuators have a higher speed of response. They

offer fast starts, stops and speed reversals.

Hydraulic Systems:



• With availability of both linear and rotary actuators, the

design has become more flexible.

• Because of low leakages in hydraulic actuators, when loads

are applied the drop in speed will be small.

• Hydraulic components are more rugged than their

counterparts.

Disadvantages:

• Hydraulic power is not readily available compared to electric

power.

• Inherent problems of leaks and sealing them against foreign

particles.

Hydraulic Systems:



• Operating noise

• Costs more compared to electrical system

• Fire and explosion hazards exist

• Hydraulic lines are not flexible as electric cables

• Highly non linear response

Hydraulic Systems:



Hydraulic Systems:



Let

• qp = Rate at which the oil flows from the pump

• qm = Oil flow rate through the motor

• qi = Leakage flow rate

• qc = Compressibility flow rate

• x = Input stroke length

• θ = Output angular displacement of motor

• P = Pressure drop across motor

Hydraulic Systems:



• The rate at which the oil flow from the pump is proportional to

stroke angle, i.e., qp ∝ x.

• Oil flow rate from the pump, qp = Kp x

• Where Kp is a constant and is equal to the ratio of rate of oil

flow to unit stroke angle.

• The rate of oil flow through the motor is proportional to motor

speed, i.e.,

• Therefore, Oil flow rate through motor

• Where, Km = Motor displacement constant.

m

dq

dt

θ∝

m m

dq K

dt

θ=

Hydraulic Systems:



• All the oil from the pump does not flow through the motor in

the proper channels.

• Due to back pressure in the motor, a portion of the ideal flow

from the pump leaks back past the pistons of motor and

pump.

• The back pressure is the pressure that is built up by the

hydraulic flow to overcome the resistance to free movement

offered by load on motor shaft.

• It is usually assumed that the leakage flow is proportional to

motor pressure, i.e. qi ∝ P

Hydraulic Systems:



• Therefore, Leakage flow rate, qi= Ki P

• Where Ki = constant.

• The back pressure built up by the motor not only causes

leakage flow in the motor and pump but also causes the oil in

the lines to compress.

• Volume compressibility flow is essentially proportional to

pressure and therefore the rate of flow is proportional to the

rate of change of pressure, i.e.,

→ Try the rest.c

dPq

dt∝

References:

• Modern Control Engineering, 5th Edition, by Katsuhiko Ogata.

• Advanced Control Systems Engineering, Ronald Burns

• Control Systems, Nagoor Kani.

• A course in Electrical, Electronic Measurements and

Instrumentation, A.K. Sawhney.



Class 11 mathematical modeling of pneumatic and hydraulic systems

Engineering

Transcript of Class 11 mathematical modeling of pneumatic and hydraulic systems