ENERGY TRANSFER SYSTEMS AND THEIR DYNAMIC ANALYSIS


 Beverly Woods
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1 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: Burner> 2 Boiler > 3 Steam Turbine > 4 Electric Generator > 5 Transformer > 6 Electric Motor > 7 Hydraulic Pump > 8 Hydraulic Cylinder > 9 Load
2 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 primemover. 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, Kgfm/sec Fluid Power: Flow, Q (m^3/sec) * Pressure, P (Kgf/m^2) = Power, Kgfm/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 nonlinear manor.
3 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 ( 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 subsystems. 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 subdivision. 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 subdivision, the components are broken down into their fundamental working parts or subassemblies. 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.
4 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 Potential Flux Mechanical Force Velocity Fluid Pressure Flow Electrical EMF Current Magnetic MMF Flux Thermal Temperature Heat Flow There are three basic possibilities for this relationship.. If the relationship is not time dependent, the device is resistive and the relationship can be expressed as d( d( = R 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. POTENTIAL R FLUX
5 The next plot illustrates a nonlinear resistance characteristic. POTENTIAL FLUX R 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 nonlinear 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 straightline 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: = R In the nonlinear 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 = ( dt C The coefficient, C, is defined as capacitance and may be either linear or non4inear. Before transforming the nonlinear case must be linearized in a manner similar to the method used to handle nonlinear resistance. Because of the integral relationship, capacitive elements are often referred to as integral elements or integrators. The transfer function is = Cs 3. The third basic impedance relationship is inductive and occurs when potential is proportional to the rate of flux change as given by: d( potential = I dt
6 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: = Is 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 NONLINEAR REAL WORLD The linear assumptions concerning systems described in the above discussion do not truly reflect the real world behavior. Real world examples of nonlinear 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 nonlinear behavior, will also exhibit hysteretic behavior: Hysteric behavior is also exhibited in automatic control valves with linkages that wear in splines and linkages.
7 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 nonlinearly 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 55 Class Dr Gomez Washington University Lloyd, S. G., Anderson, G. D. Industrial Process Control st edition, Fisher Controls Company, Marshalltown, IA, pp , 97.
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