Download ENERGY TRANSFER SYSTEMS - Engineering School Class

ENERGY TRANSFER SYSTEMS AND THEIR DYNAMIC ANALYSIS
Many mechanical energy systems are devoted to transfer of energy between two points: the
source or prime mover (input) and the load (output). For chemical systems there is either a
transformation of chemical species, such as a chemical reaction or a separation of products, such
as a refining distillation column or a centrifuge. There are also mass transfers, such as flow in and
out of a chemical reactor.
In this lab, we will only be concerned with mechanical systems. Hydraulic systems are considered
as a subset of mechanical systems for this discussion.
The mechanical system may be as simple as a single prime mover, an automobile engine
connected to a load, wheels, through gears and fluid couplings. Or it may be a complex system
as shown below:
1 Burner-> 2 Boiler -> 3 Steam Turbine -> 4 Electric Generator -> 5 Transformer -> 6 Electric
Motor -> 7 Hydraulic Pump -> 8 Hydraulic Cylinder -> 9 Load
There are four main types of subsystems:
Energy Conversion
Energy Storage
Energy Transmission
Energy Transformers
Energy Conversion: These systems receive energy from one type of input and convert it to
another form of energy. The most common types of energy are mechanical, fluid and thermo.
Examples of energy conversions are electrical to mechanical, a motor; mechanical to fluid, a
pump. Can you think of others?
Other forms of energy may be involved either as a source or an intermediate step, chemical,
nuclear, magnetic.
When the output of the energy conversion is mechanical energy, it is called a prime-mover. The
input and output of an energy converter can be defined in terms of two quantities of power
(HP,KW)
Electrical:
Current, I (Amp) * Voltage, V (volt) = Power, watts
Mechanical:
Speed, v (m/sec) * Force, F (Kgf) = Power, Kgf-m/sec
Fluid Power:
Flow, Q (m^3/sec) * Pressure, P (Kgf/m^2) = Power, Kgf-m/sec
In every case one of the quantities is a rate or flux quantity while the other is a potential quantity.
The energy conversion performance is analyzed in terms of output quantities as functions of
combinations of input quantities. Centrifugal pumps are common energy converter used in the
chemical process industries. The pump’s capacity are shown by means of a “pump curve” that
shows the pumping head or pressure, the power, efficiency, and suction head.
This curve demonstrates the relationship between the potential, head or pressure and the flux,
flow rate in GPM. Note that the linear increase in flow with power delivered to the pump, yet the
potential decreases in a non-linear manor.
Energy Storage: In this case, energy is stored. Examples of these are:
Electrical
Chemical
Mechanical
Fluid
Capacitor, inductive coil
Fuel cell, battery
Spring (elastic energy), flywheel (kinetic energy), elevated weight (potential)
Accumulator
In some cases, the energy is stored in one form of energy and also transformed to another.
Example: a battery stores the energy as chemical energy and converts it to electrical energy.
Energy Transmission: Seldom can energy be used at the site where it is generated; therefore a
means must be provided to transmit energy between the converter or generator and the load.
Examples:
Electrical:
Mechanical:
Fluids:
Cables or wire
Shafts, couplings
Pipes, tubes
Energy Transformers: These devices neither change the nature of the energy nor the power
transmitted (input power = output power). They do alter one of the quantities, either the flux or
potential. They match the output of one energy converter to the input of the next allowing both of
them to operate at maximum efficiency.
Examples:
Electrical:
Transformers
Mechanical:
Gears
Fluid:
Double bore cylinders
We should also consider that none of these machines are perfect or ideal, there are losses, such
as frictional losses, thermal losses. These losses are actually energy converters, but the
conversion is not useful. It is wasted. This is why we are interested in efficiency.
Other elements that may be grouped under a general category of energy conditioners or
controllers, used to regulate the flow of energy or control the direction where the energy flows,
valves, switches, clutches, pressure regulators, etc.
As systems engineers you are interested in the dynamic behavior of these systems.
INTRODUCTION TO DYNAMIC SYSTEMS
Any control system may be broken down, step by step, into smaller and smaller sub-systems.
While there is no advantage in proceeding to where only the individual parts and pieces are to be
considered, a certain degree of division is very useful. It is especially convenient to consider two
levels of sub-division. The first includes those items in a control loop that may be manufactured,
tested, purchased and perhaps even designed as individual pieces of equipment. We have
chosen to call these items components and examples would include valves, controllers,
transmitters, actuators and even the process itself. For the second level of sub-division, the
components are broken down into their fundamental working parts or sub-assemblies. These we
have elected to call basic elements and examples would include beam assemblies, orifices,
amplifiers, etc. We shall examine several basic elements and derive their transfer functions. Our
intentions are to illustrate the method of developing the mathematical model of a physical device
and to also provide working equations for the more commonly encountered basic elements. It
would be both impractical and unnecessary to describe every basic element that might occur in
process control. Once the techniques of analysis are mastered, they may be employed in the
modeling of any device whose physical principles are understood.
MATHEMATICAL MODELS OF PHYSICAL DEVICES
The mathematical representation of a physical device is accomplished by the application of the
fundamental physical laws such as Newton's Laws, conservation of mass and energy, flow
equations, Ohm's Law, etc. Proper representation requires more and more engineering ability
than mathematical dexterity.
It is always necessary to make simplifying assumptions in applying the basic laws to real
situations. This requires the ability to discriminate between significant and trivial effects and
recognition of the limitations resulting from the simplifications. The compromise must always be
made between exact, completely general, but unwieldy equations and those that are limited and
sometimes approximate, but which readily yield solutions.
THE GENERAL CONCEPT OF IMPEDANCE
A general concept of impedance is often, but not always, helpful in deriving a mathematical
model. When working with a dynamic system, there must be a situation or condition, which is
forcing change. This forcing condition is always some form of potential energy. When a change
takes place; that is, when the dynamic action occurs, a movement or flow that can be generally
described as a flux represents that change. The forms that flux and potential take depend upon
the physical nature of the system. Some familiar physical forms of flux and potential are
described below. Impedance describes the mathematical relationship between flux and potential.
Specifically, it is the ratio of an incremental change in potential to an incremental change in flux.
System
Mechanical
Fluid
Electrical
Magnetic
Thermal
Potential
Force
Pressure
EMF
MMF
Temperature
Flux
Velocity
Flow
Current
Flux
Heat Flow
There are three basic possibilities for this relationship.
1. If the relationship is not time dependent, the device is resistive and the relationship can be
expressed as
d ( potential )
=R
d ( flux )
POTENTIAL
The term R may or may not be a function of flux but in either case it is defined as resistance.
A plot of potential versus flux for a resistive element with linear resistance; that is, R is
independent of flux.
R
1
FLUX
POTENTIAL
The next plot illustrates a non-linear resistance characteristic.
R
1
FLUX
The transfer function of a resistive element is obtained by replacing flux and potential with the
Laplace transform of those variables. However, since only linear equations can be transformed,
an element with non-linear resistance must be approximated by a straight line, whose equation is
transformable. A convenient operating point is selected, such as indicated, and a transfer function
which represents a straight-line approximation is used. If the excursions from this operating point
are kept small, no significant error is introduced. In both cases then, the transfer function will be:
L( potential )
=R
L( flux )
In the non-linear case, the expression is valid only over a small interval about the operating point.
2. If the impedance is such that potential is proportional to the time integral of flux, the device is a
capacitive element. The equation for this relationship is:
potential =
1
( flux )dt
C∫
The coefficient, C, is defined as capacitance and may be either linear or non4inear. Before
transforming the non-linear case must be linearized in a manner similar to the method used to
handle non-linear resistance. Because of the integral relationship, capacitive elements are often
referred to as integral elements or integrators. The transfer function is
L( potential )
1
=
L( flux)
Cs
3. The third basic impedance relationship is inductive and occurs when potential is proportional to
the rate of flux change as given by:
potential = I
d ( flux )
dt
The proportionality constant, I, is defined as inertance and in electrical systems is inductance and
in mechanical systems is either mass or moment of inertia. An element possessing this
characteristic is sometimes called a differentiator. The transfer function is:
L( potential )
= Is
L( flux )
These three impedance functions and their inverses, which are defined as admittances, appear
frequently. Normally, the transfer function of a device is determined without intentionally seeking
any impedance relationships but these quantities naturally appear and it is convenient to have
them defined at the outset.
THE NON-LINEAR REAL WORLD
The linear assumptions concerning systems described in the above discussion do not truly reflect
the real world behavior.
Real world examples of non-linear behavior
In the case of a charging capacitor, there are passive electrical components that exhibit a varying
capacitance as the applied voltage changes, called a varistor.
In the case of a magnetic circuit, the applied magneto motive force, MMF can cause the magnetic
flux to saturate in an iron core. The potential flux relationship will, in addition to non-linear
behavior, will also exhibit hysteretic behavior:
Hysteric behavior is also exhibited in automatic control valves with linkages that wear in splines
and linkages.
For heat transfer applications, the potential is the temperature of the material, while the flux is the
heat flow. For solids this relationship is the simplified relationship
Q = mC p (∆T )
Assuming specific heat Cp does not vary with temperature.
However in the case of a heat exchanger where fluids are involved, as is the case in our lab, the
heat transferred across the tubes takes on a more complex function and is proportional to the hot
and cold fluids flow rates.
Q = UA(∆T )
A is the area and U is the heat transfer coefficient, that varies non-linearly with flow rates.
The use of simple Laplace transforms is not adequate to properly
describe the dynamic behavior many industrial process systems,
including those in our lab.
References:
Notes from ME 551 Class Dr Gomez Washington University
st
Lloyd, S. G., Anderson, G. D. Industrial Process Control 1 edition, Fisher Controls Company,
Marshalltown, IA, pp. 93-95, 1971.