Problem Solving One of te primary tasks for engineers is often solving problems. It is wat tey are, or sould be, good at. Solving engineering problems requires more tan just learning new terms, ideas and concepts, and using rules and pysical laws. To find out wat engineering is really about, we must learn ow to apply tese concepts and rules to real or ypotetical situations. Experience as sown tat tis kind of learning cannot occur witout practice. Tis means spending more time working problems tan reading te text. It is common in engineering education to talk about te matematics problem i.e. te weakness in matematics of students entering university engineering education. Certainly te lack of fluency in specific matematical tecniques is an obvious aspect of tis problem, but te more serious aspect may be te lack of understanding of problem solving processes. In fact, problem solving is more tan merely substituting numbers in a matematical formula. We sould begin by studying te ideas, te concepts, and teir relationsips first. Ten we sould attempt te problems as a way to find out weter or not we understand te subject. Problem Solving Process In order to build a model and solve te problem, te following steps may be followed: 1. Carefully identify wat is given in te problem. Tis is an important step because it makes tings clear before attacking te problem. Read te entire problem carefully. You may need to briefly write down te question using symbols, make lists or tables of known and unknown information, and draw a diagram of te pysical situation.. Carefully identify te objective of te problem. Tis could be te most important step, because it becomes te foundation for all te rest of te steps. Wat is te problem asking you to solve or find? Sometimes te objective of te problem is clear; some oter times you need to lists te unknown information in order to identify te objective. 3. Decide wic matematical tool best suits te problem. Wat are te possible pats of solution to be followed? Wat are te processes involved? Wat are te relationsips involved? And, determine wic pat and process promise te greatest likeliood of success. Tis step will elp you build a collection of analytical metods, many of tem will work to solve te 1
problem, and owever, many oters may not work. Also, one metod may produce fewer equations to be solved tan anoter, or it may require less matematics tan oter metods. 4. Write equations and develop a model. After you decide on te metod, document te process very well by writing te equations to actually start solving te problem. 5. Attempt a solution. Present detailed solution before putting real numbers into equations. Paper-and pencil, calculator, and computer tools are all available to pursue te solution. 6. Verify te solution. Is it realistic, expected, or not at all? Ask yourself weter te solution you ave obtained makes sense. Does te magnitude of te answer seem realistic? You may want to rework te problem via an alternative metod to test te validity of your original answer. By doing so, you develop your perception about te most efficient metods to solve te problem. In real life, any design is cecked by several independent means. Acquiring te abit of verifying te answer will elp you as a student and as an engineer in te future. 7. Finally, if te solution is realistic, present te solution to your professor, boss, or team members. If not, ten return to step 3 and continue troug te process again. It is important to know tat altoug te above steps ave been organized to apply to engineering foundations types of problems, te problems to be solved during one s career will vary in complexity and magnitude. Working Example As an example of te general guidelines for problem solving, let us work a sample problem. Consider a tank tat is used to store a liquid. Liquid can be let into te tank troug an inlet pipe at te top, and it discarges from te tank troug an orifice near te base. Suc a situation occurs frequently in mecanical and cemical engineering applications. Consider two cases for te flow troug te orifice: laminar and turbulent. Wat is te rate of outflow from te tank if te eigt of te liquid is 0.5 m and te discarge coefficient is 0.7.
3 Solution By reading te problem carefully and drawing a diagram similar to Figure 1, we will ave covered steps 1 and. Figure 1 A liquid storage system. Te basis of tis approac to model building is tat te equations wic constitute a model are not arbitrary matematical entities, but ave a consistent pysical basis. Tere are certain types of equation wic describe different aspects of a model. Knowledge of tis elps to ensure tat all equations are written down. For step 3 we recall te relationsip between te volumetric rate of outflow Q measured in (m 3 /s) and te eigt of te liquid measured in (m). Q = Cd (1) were C d is a constant of proportionality called te discarge coefficient. Te discarge coefficient is constant tat depend on te sape of te orifice. It could be a sarp-edged, a sort flus-mounted tube, or a rounded orifice. In tis example, we ave assumed tat te volumetric flow rate of te liquid is proportional to te eigt only, owever, in practice te flow rate depends on te pressure drop across te orifice, orifice cross-sectional area, and fluid density. Te dependent variable Q is a function of te independent variable. Terefore, te input function is te ead, te function rule is multiply te input by C d, and te resulting output is te flow rate Q. A grap of Q against is sown in Figure. Q
4 Q Q = C d Figure A grap of Q and relationsip for a laminar flow. If te flow troug te orifice is turbulent ten a different functional relationsip will exist between Q and A grap of Q against is sown in Figure 3. Q Q = Cd () Q = Cd Figure 3 A grap of Q against Now in accord wit step 5, we substitute te values and te units for te algebraic symbols. For laminar flow For turbulent flow 3 Q = 0.7 0.5 = 0.35 m /s
5 Q = 0.7 3 0.5 = 0.49 m /s Te solution looks realistic! According to step 6, we may want to rework te problem via an alternative metod to test te validity of te answer.
6 System of Units A unit is a particular pysical quantity, defined and adopted by convention, wit wic oter particular quantities of te same kind are compared to express teir value. A pysical quantity is a quantity tat can be used in te matematical equations of science and engineering. Te value of a pysical quantity is te quantitative expression of a particular pysical quantity as te product of a number and a unit, te number being its numerical value. Tus, te numerical value of a particular pysical quantity depends on te unit in wic it is expressed. Wen making measurements, it is customary to record bot te quantity (ow muc) and te unit (of wat). Science and tecnology depend largely on te unit of measurement. For example, te value of te eigt of building is = 10 m. Here is te pysical quantity, its value expressed in te unit meter, unit symbol m, is 10 m, and its numerical value wen expressed in meters is 10. Te SI System of Measurement A system of units is a class of units defined by composition from a base set of units, suc tat every instance of te class is standard unit for a pysical dimension and every pysical dimension as an associated unit. A measurement of any pysical quantity must be expressed as a number followed by a unit. A unit is a standard by wic a dimension can be expressed numerically. Te units for te fundamental dimensions are called te fundamental or base units. Wile carrying out engineering calculations, tere are several systems of base units tat are available. However, tey may be broken into two main groups. First, te International System of Units (also called SI, from te Frenc Système International des Unités ) introduced by Griorgi in 1901, including te meterkilogram-second-ampere (MKSA) subsystem representing te four fundamental dimensions lengt, mass, time, and electric current, respectively. Second is te centimeter-gram-second (CGS) system. Te units for oter dimensions are called secondary or derived units and are based on te above fundamental units. Te International System of Units as seven base units, several derived units wit special names, and many derived units wit compound names. Te seven base units are te building blocks from wic te derived units are constructed. Eac base unit is defined by a very precise measurement standard tat gives te exact value of te unit. Te base units are not related to one oter, no do tey depend on eac oter for teir definition. Te complete SI system involves units and oter recommendations, one of wic is tat multiple and submultiples of te MKSA units be set in steps of 10 3 or 10-3. Te base units SI units and abbreviations are listed in Table 1. Te SI as tremendous advantages over previous systems because it uses a
7 unique unit name for eac pysical quantity and it assigns a unique symbol for eac name. Te SI is te standard system used in today s scientific literature. Table 1 Te Seven Fundamental SI Units Quantity Unit Abbreviation Lengt Mass Time Electric current Temperature Luminous intensity Matter meter kilogram second ampere kelvin candela mole m kg s A K cd mol Te SI derived units are formed from te previously defined SI base units. Table lists many of te SI derived units used in electric and electronic circuits. Table SI Derived Units Quantity Symbol Unit Unit Symbol Angle θ radian rad Capacitance C farad F Conductance G siemens S Electric carge Q coulomb C Electromotive force E volt V Energy, work W joule J Force F newton N Frequency f ertz Hz Inductance L enry H Power P watt W Resistance R om Ω Pressure p pascal Pa Magnetic Flux φ weber Wb Magnetic Induction B tesla T Ligt Flux L lumen lm Te SI uses te decimal system to relate larger and smaller units to te basic units, and employs prefixes to signify te various powers of 10. A list of prefixes and teir symbols is given in Table 3. Tese prefixes are very important in engineering studies and are wort memorizing.
8 Table 3 SI Prefixes Metric Symbol Metric Prefix Value Power of Ten T One Trillion Tera 10 1 G One Billion Giga 10 9 M One Million Mega 10 6 K One Tousand Kilo 10 3 m One Tousandt Milli 10-3 µ One Milliont Micro 10-6 n One Billiont Nano 10-9 p One Trilliont Pico 10-1 Dimensional Analysis Dimensional analysis is a tecnique used in cecking an equation by establising te same dimension formula on eac side of te equation, tat is Left and side = Rigt and side Numbers wit no units attaced to tem are dimensionless. A pysical requirement is tat dimensional omogeneity old tat is bot sides of an equation ave te same dimension. Example 1 An airliner, initially at rest, undergoes a constant acceleration of m/s down te runway for 30 s before it lifts off. How far does it travel down te runway before taking off? Prove te dimension of te distance. Solution Te distance te airplane travels is given by 1 x = x 0 + v 0 t + at Were x 0 is te initial distance te airliner traveled before t = 0. Te direction of acceleration is along te positive x axis. Assume x 0 = 0, v o = 0, a = m/s, and t = 30 s.
9 Solution Using te values given above, te distance is 1 x = 0 + 0 + m/s (30 s) = 900 m Hence te distance at te left and side of te equation as te same dimension of te rigt and side tat is meter (m). Exercise 1 Wic of te following are dimensionally correct (ave dimensional omogeneity)? mv F = r F = mgl 1 mg = mv p = mv w = mg Were m = mass, r = distance, g = gravitational acceleration, l = lengt, t = time, v = velocity, F = force, and p = pressure. Exercise Te viscous drag force of a fluid is found from te following formula F = 6πηrv Were r is te radius of te spere, η is te viscosity of te fluid, and v is te speed of te object wit respect to te fluid. Find te dimension of η.