A Multicourse Effort for Instilling Systematic Engineering Problem Solving Skills Through the Use of a Mathematic Computer Aided Environment

Transcription

1 A Multicoure Effort for Intilling Sytematic Engineering Problem Solving Skill Through the Ue of a Mathematic Computer Aided Environment Rogelio Luck 1 and B K. Hodge Abtract Thi paper decribe a coordinated, multicoure effort in the Mechanical Engineering (ME) Department at Miiippi State Univerity (MSU) to inculcate diciplined/ytematic engineering problem olving kill through the ue of mathematical workheet uch a thoe found in Mathcad. The advantage of thee workheet i that, with proper guidance, tudent can learn a ytematic methodology that allow for a better undertanding on how to approach and olve engineering problem a compared to uing pencil, paper, and graphing calculator. For ME tudent at MSU, thee workheet are required in three coure: Engineering Analyi, Sytem Dynamic, and Energy Sytem Deign. A ignificant advantage to equencing coure in thi manner i that tudent pend more time (and effort) in engineering function (formulation, verification, and validation) than in the arithmetic function (primarily accomplihed by Mathcad). Detail, example, and aement of effectivene are dicued in the paper. Keyword: Problem olving, computer aided environment, and engineering approach. INTRODUCTION One important objective in engineering education i to inculcate in the tudent a ytematic and practical approach to problem olving. Traditionally, thi ha been largely accomplihed through the ue of homework following guideline for problem formulation and olution uing engineering paper and handheld calculator. However, the widepread availability of laptop and uer-friendly mathematical computer olver i diplacing the ue of calculator in favor of workheet from mathematical CAD program uch a Mathcad and Maple. The feature available in thee program include automatic recalculation, copy, pating, and line editing of equation and text, ymbolic proceing, automatic handling of engineering unit, and intantaneou graphic capabilitie. Student can pend more time and effort in engineering function uch a problem formulation, olution, verification, and validation rather than recalculating reult and plot each time a mitake i found or a parameter i perturbed. On the other hand, without careful guidance, the ue of thee workheet can quickly degenerate into carele and pointle ue of equation olver, diplaying the olution uing a mimatched et of unit, and generating the wrong algebraic olution. Alo, often, the oftware itelf i not fully debugged and diplay erroneou reult. Thu, the tudent mut be carefully coached on the correct ue of thee mathematical workheet. Thi i achieved by raiing the expectation on the problem formulation/olving documentation procedure plu an added emphai on verification and validation (V&V) of the reult. A benefit of V&V i that tudent pend more time rethinking the entire olution proce in term of the phyical meaning of the reult and the mathematical well-poedne. The genei of the evolution of the author realization of the pedagogical impact of mathematical computer olver on engineering education i documented in Hodge and Taylor (1998), Hodge (5), and Hodge and Luck (6). 1 Department of Mechanical Engineering, Miiippi State Univerity, MS 955 Miiippi State, MS 3976, luck@me.mtate.edu. 9 ASEE Southeat Section Conference

2 At the ME department at MSU, thee workheet are ued in three coure: Engineering Analyi (EA), Sytem Dynamic (SD), and Energy Sytem Deign (ESD). Diciplined, ytematic problem olving kill are inculcated through the ue of trictly enforced homework format in EA and SD. The homework format conit of clearly formulated Given, Find, Solution, and Verification/Validation ection. Picture and ketche are introduced into the workheet by copying and pating from computer generated ketche or drawing, by browing the internet, or by canning. Student are required to carefully read the problem tatement and ummarize thi information in the Given ection of the homework. The variable to be found or the deign requirement are to be lited in the Find ection. It i intereting that the Given and Find ection of the homework format are in congruence with the Given and Find formulation in Mathcad for olution of algebraic and differential equation. Thi help reinforce the importance of decribing the problem in a well-poed manner. The Verification/Validation ection i conidered of equal importance a the Solution ection and i addreed and graded accordingly. Student are required to how that their anwer are reaonable by checking that the unit are conitent, the magnitude are reaonable, and the model/equation behave a expected. The phyical unit proceing a well a the D and 3D plotting feature of Mathcad are heavily ued in thee ection. Finally, the ESD coure take the previouly acquired problem olving kill one tep further into engineering deign cenario. The following ection will decribe the philoophy behind the coure and detail and example of typical homework. ENGINEERING ANALYSIS (EA) COURSE Philoophy behind the Engineering Analyi (EA) coure: A main objective behind the EA coure i to conolidate the mathematical kill acquired during the frehman and ophomore year while emphaizing their ue in poing and olving engineering problem. Thi i partly accomplihed through the proce of learning how to ue Mathcad. Student are introduced to Mathcad functionality by demontrating how to define function, perform differentiation and integration, implify ytem of algebraic and trigonometric expreion, perform erie analyi, find optimal olution, fit equation to data, and olve differential equation. They are aked to verify and interpret the reult uing unit, magnitude, and graphical comparion. Thi proce of verification and interpretation allow them to focu on the definition and meaning of the mathematical expreion ued to decribe engineering problem. Example 1 Figure 1 illutrate the typical homework format and how a Mathcad workheet i ued in a homework et to review baic concept of integration and differentiation, and, more importantly, to introduce the proce of verification. The ame philoophy of verifying the reult in order to review key mathematical concept i ued in everal homework et covering topic uch a Taylor Serie and truncation error, root finding, maximization/minimization, and polynomial fitting. Example Figure examine a homework problem where tudent practice how to etup a problem in matrix form and verify the olution. Given air flowrate in and out of a retaurant and carbon dioxide concentration in the incoming air a well a carbon dioxide generation by moker and by the kitchen grill, tudent are aked to find the equilibrium concentration of carbon dioxide in each room in the retaurant. Thi time, the verification i baed on an important engineering concept: conervation of ma. The idea here i to ue baic undertanding of equilibrium, compatibility, and energy and ma conervation equation to verify the olution. Typically, tudent are aked to verify the conitency of the unit of the reult of the calculation. Thi i not done in thi example a all parameter are defined uing their repective unit and Mathcad will not proceed with a calculation unle all unit are conitent. 9 ASEE Southeat Section Conference

3 Problem 1. Given : The function fx ():= e in( x) co () x + 1. Find : Ue Mathcad to π a) Calculate the derivative of f(x) when x:= 4 b) Calculate the integral of f(x) in the interval x π Ue mathcad graphic capabilitie to verify the olution. Solution : Anwer a) Anwer b) d dx f( x) π =.4 f( x) dx = 3.14 Verification : The validation conit in approximating the derivative by the approximate lope and the area (integral) by the area of a box of height equal to the mean value in the interval of interet. a) f( z) f π z, π 4 The trace function wa ued to collect the following data from the plot on the left: point 1 x1 :=.737 y1 :=.45 point : x :=.847 y :=.4 lope at x= π y y1 Slope := Slope =.45 4 x x1 Thu the lope left of -.45 i quite cloe to the exact value of -.4 Figure 1. Workheet for Example 1. 9 ASEE Southeat Section Conference

4 b) qr := π π 1 1. area1 := area := f( z) qr 1 area1 1 area z, qr, area1 1, area 1 Note that the area (X) bounded on the top by the red curve and on the bottom by y= cancel out the area (X) bounded on the top by y= and the bottom by the red curve. The area of the blue box i roughly equal to the area f(z). ApproxArea := 1 π ApproxArea = 3.14 Thu, the anwer appear reaonable a it i quite cloe to the graphical approximation. Figure 1. Concluded. 9 ASEE Southeat Section Conference

5 Given: The following chematic for Carbon Monoxide Ditribution in a Retaurant Smoker Load (3 mg/hr) Q b =1 m 3 /hr c b = mg/m 3 Q a =5 m 3 /hr c a = mg/m 3 Entrance 1 3 m 3 /hr Non- Smoking 15 m 3 /hr Kitchen 6 m 3 /hr 4 5 Q d =15 m 3 /hr 3 m 3 /hr Q c = m 3 /hr Smoking 3 Smoker Load (1 mg/hr) Grill Load (3 mg/hr) Monoxide Ga Ditribution in a Retaurant Find: a) teady tate carbon monoxide concentration in each room b) percent contribution in ection 4 from moker, grill, and outide air Solution: Labelling the volumetric flowrate and the concentration of carbon monoxide. E 1 := 3 m3 Q hr a := 5 m3 Q hr b := 1 m3 Q hr c := m3 Q hr d := 15 m3 hr E := 15 m3 C hr a := mg m 3 C b := mg m 3 E 3 := 3 m3 S hr 1 := 3 mg S hr 3 := 1 mg S hr 5 := 3 mg E hr 4 := 6 m3 hr The following equation repreent the balance of the carbon dioxide flow in each room. Q a C a + S 1 + E 1 C C 1 Q a C 1 = Q b C b + Q a C 1 + E 1 C 1 C + E 3 C 3 C Q c C Q d C + E C 4 C = Q c C + E 3 C C 3 Q c C 3 + S 3 = Q d C + E C C 4 Q d C 4 + E 4 C 5 C 4 = Q d C 4 + E 4 C 4 C 5 + S 5 Q d C 5 = Figure. Workheet for Example. 9 ASEE Southeat Section Conference

6 Solution uing the invere matrix method: E 1 Q a Q a + E 1 E 1 E 3 E 1 Q c Q d E F := Q c + E 3 E 3 Q c R := Q d + E E Q d E 4 E 4 Q d + E 4 E 4 Q d E 3 E Q a C a S 1 Q b C b S 3 S 5 C 1 C C 3 := F 1 R C 4 C 5 Anwer a) C 1 C C 3 = C 4 C mg m 3 b) percentage contribution in Room #4 coeffinvere := F 1 Anwer b) coeffinvere S 3, 1 + coeffinvere S 3, 3 moker := moker = 14. % C 4 coeffinvere S 3, 4 5 grill := grill = % C 4 coeffinvere Q 3, a C a + coeffinvere Q 3, 1 b C b intake := intake = 3.19 % C 4 Verification: Conervation of ma mean that the total amount of carbon monoxide entering the building mut equal the amount leaving. Becaue we are writting thi equation independently of the previou derivation, thee anwer eem to be reaonable a hown below. Q a C a + S 1 + Q b C b + S 3 + S 5 Q d C 5 Q c C 3 = kg The carbon monoxide contribution from each area are reaonable in that they add up to 1 %. moker + grill + intake = 1 % Figure. Concluded. 9 ASEE Southeat Section Conference

7 Example 3 Figure 3 illutrate a homework problem where tudent practice olving ytem of firt-order differential equation uing the built-in differential equation olver from Mathcad and verify the olution uing teady-tate information a well a conervation of energy. Problem 1. Given : Conider the water heater problem hown below. The thermotat etting for turning the heater off i 13 F. Aume a water flowrate demand of gpm tarting 1 minute after turning on the heater and the ame thermotat etting for both water heater. Find : plot the temperature of the water leaving each water heater until teady tate temperature are reached. Solution : Example of a houehold water heater : KfromF ( T) := ( T 3) ( Convertion function from degree F to K ) FfromK ( T) := ( T 73) ( Convertion function from degree K to F ) kj := 1 J cal C p := 1 ( Cp of water ) ρ := 1 kg ( Denity of water ) gm K m 3 V1 := 5 gal V := 5gal ( Volume of water tank 1 & ) T in := KfromF ( 5) K T in = 83 K ( Inlet water temperature ) T a := KfromF ( 65) K T a = K ( Ambient temperature ) Figure 3. Workheet for Example 3. 9 ASEE Southeat Section Conference

8 Qt ():= gal if t > 1min ( Flowrate : Start the flow after 1 min ) min otherwie Qgen1( T) := 66W if T < KfromF( 13) K ( Heater ON if the temp. i below 13F ) otherwie Qgen( T) := 44W if T < KfromF( 13) K otherwie Uing an energy balance to olve for the time rate of change (Slope) of the temperature : Slope( t, T) := 1 ρ C ρ C p V1 p ( T in T a ) Qt ( ) ρ C p T K T a Qt ( ) + Qgen1 T K K 1 ρ C ρ C p V p ( T K T a ) Qt ( ) ρ C p T ( K T 1 a ) Qt ( ) + Qgen( T K 1 ) K KfromF( 5) Tinit := KfromF( 5) K tinit := tfin := 364 N := 1 Sol rkfixed Tinit tinit tfin,,, N, Slope := T K 1 := Sol 1 T := Sol t := Sol Anwer : FfromK T 1 1 FfromK( T ) t 6 Figure 3. Continued. 9 ASEE Southeat Section Conference

9 Verification : The tank reach a temperature of 13 degree F (thermotat etting) in 89 min (5 gal tank) and 67 min (5 gal tank). The energy tored in the water mut be equal to the energy delivered by the electric heating element: 5 gal tank: 5 gal tank: ( KfromF( 13) KfromF5 ) K C p ρ 5gal = kj ( KfromF( 13) KfromF5 ) K C p ρ 5gal = kj 66W 89 min = kj 44W 67 min = kj The imulation appear to follow energy conervation ince the energy tored in the tank i equal to the energy delive by the heater. The teady tate part of the numerical imulation can be checked by auming that all derivative are zero and olving the reulting et of algebraic equation for the teady olution: ( initial gue for the algebraic olver) T := Given 6 7 Slope( 6 5, T) Setting all derivative (lope) to zero. 7.5 T := FfromKFind ( ( T) ) T = 87.5 Figure 3. Concluded. Note the ue of the Given and Find (Mathcad) command above in order to find the teady-tate olution of the differential equation. SYSTEM DYNAMICS (SD) COURSE Philoophy behind the Sytem Dynamic coure: In the Sytem Dynamic coure tudent learn to model and obtain the tranient and frequency repone of mechanical, hydraulic, electric, and thermal ytem and interpret the reult baed on the ytem parameter. They learn to poe and olve tate variable equation uing analytical method a well a numerical olver and how to combine the tate variable equation into higher-order differential equation. Verification of the homework olution i baed on the proce learned in the Engineering Analyi coure. Example 4 Figure 4 illutrate a homework et in the Sytem Dynamic coure that make extenive ue of Mathcad ymbolic command to implify the algebra and obtain analytical olution for differential equation. Given the tep repone 9 ASEE Southeat Section Conference

10 of a pring-damper ytem tudent are aked to find the time contant of the ytem and ue thi information along with the teady tate information to determine the pring and damping contant for the ytem. Problem 3: Given: The unit tep repone of a firt order ytem ytem i hown below (thi i jut a pring connected in parallel to a damper through a male plate and a force pulling on the plate). The unit for the y-axi and x-axi are mm and econd, repectively. Find: a) Write down the olution for the STEP repone of the ytem in term of the time contant, the initial value and the teady tate value. Ue only ymbol, not number for thi part. b) Ue the equation for part (a) and the value of time when y = -1mm to find the time contant. Repeat the ame proce when y= -3.3 mm and find the time contant. Explain which of the two choice of y i le enitive to error when reading the graph? Which of the two value of time contant i more reliable? Clearly explain your anwer. c) Take the derivative of the equation from part (a) and how how to find the time contant baed on the initial lope and the teady tate value. d) Write down the differential equation of the ytem and ue thi equation along with the teady tate information from the plot below to find the pring contant auming that the tep force i of magnitude 5. e) Find the value of the damping b. Figure 4. Workheet for Example 4. 9 ASEE Southeat Section Conference

11 Solution: a) The differential equation of the a 1t order ytem ubject to a tep input i: τ dx() t + xt () x dt Φ() t The Laplace tranform of thi equation i: τ ( X x ) + X x rearranging, τ X + X τ x + x Solving for X() and applying partial fraction expanion: X τ x + x ( τ + 1) A + B τ + 1 Solving for A and B: A B τ x + x ( τ + 1) τ x + x ( τ + 1) ubtitute, A x 1 ( τ + 1) ubtitute, B τ x τ τ x So X x + τ x x τ + 1 x + x x 1 + τ Applying the invere Laplace Tranform: e xt () x + x x t τ Figure 4. Continued. 9 ASEE Southeat Section Conference

12 b) Uing the Mathcad Trace tool, we find t=.555 for y= -1.11mm, x = -3.5mm, and x()=mm. Subtituting thee value in the equation of part a) give:.56 τ 1 := 1.11 mm 3.5 mm + ( + 3.5) mm e τ 1 olve, τ 1 float, 3.76 τ 1 =.76 Uing the Mathcad Trace tool, we find t=.355 for y= mm. Putting thee value in the equation of part a) give:.36 τ := mm 3.5 mm + ( + 3.5) mm e τ olve, τ.71 float, 3 τ =.71 The firt choice uing the value of time when y=-1mm i the le enitive to error when reading the graph becaue around y= -1mm the lope of the curve i higher o it i eaier to determine the correponding time value. Thu it i the firt value of the time contant who i the more reliable ince it i calculated with the value of t and y that are the le enitive to error. c) The derivative of the equation from part a) i: dx() t dt x x τ t τ e ubtituting t=: dx dt x x e τ x τ x olving fot τ τ x x dx dt Figure 4. Continued. 9 ASEE Southeat Section Conference

13 d) The differential equation of the ytem, auming that the tep force i magnitude 5N, i: b dx() t + kxt () ft () 5N dt At teady tate x = x and the derivative term dx(t)/dt i zero. Thi yield 5N x := 3.5mm kx 5N or k := k = 1.49 N x mm e) Since τ b k the damping coefficient b i: b := τ 1 k b = 1.9 N mm Verification: The plot below how that the reult from part a) through e) are conitent becaue they reproduce the plot given. x := mm τ := b k t τ xt () x ( x x ) 1 d := + e x' () t := dt xt () x' = m Figure 4. Continued. 9 ASEE Southeat Section Conference

14 3 τ xt () mm 1 x' t+ x 1 mm t, t The blue traight line interect the teady tate line at a time equal to the equation from part (c) which can be written a: τ. Thi verifie dx x τ dt + x or xt x' t + x at time t τ Figure 4. Concluded. ENERGY SYSTEMS DESIGN (ESD) COURSE Philoophy behind the Energy Sytem Deign coure In the Energy Sytem Deign coure tudent learn to analyze and deign energy ytem component (erie and parallel piping ytem, piping network, and heat exchanger), to elect and confirm the appropriatene of pump, and to model and undertand the operation of energy ytem. The evolution of the Energy Sytem Deign coure wa delineated in Hodge (1998). Hodge and Taylor (1999) i a textbook baed on the material covered in the coure. Example 4 wa taken from a typical homework aignment in thi coure. Example 5 A ytem to pump oil between two reervoir i illutrated in Figure 5. The pipe, which i 3 ft long, i made of commercial teel and i to handle.4 cf. The oil ha a denity of 56.1lbm/ft 3 and a vicoity of.576 lbm/ft-ec. The pump/motor efficiency i 67 percent. Electricity cot $.5 per kwh, and the demand charge i $1. per kw per month. A a function of diameter, determine the power required to pump the oil from the lower reervoir to the upper reervoir. Select a diameter, and defend your diameter election. A chematic of the ytem i preented in Figure 5. 9 ASEE Southeat Section Conference

15 B 15 ft L, D, ε A Q The energy equation for the ytem can be written a Figure 5. Oil Pumping Sytem Schematic. P γ A V A + g + Z A = P γ B VB + g + Z B + f L D V g + ( K ent + K elbow + K exit V ) g W g c g (1) The minor loe are the entrance, the elbow, and the exit. Since A and B are located at free urface of reervoir open to the atmophere, P A = P B and V A = V B =. The energy equation thu reduce to the form: W Z B Z A L + ( f + K D ent + K elbow + K exit V ) g = () The Mathcad workheet containing the problem olution i included a Figure 6. The power input to the pump i required a a function of the pipe diameter. The pipe diameter will be varied, and the pump increae in head and the power required calculated for each diameter. Diameter from 1.5 to 9 inche, in increment of.5 inche, are precribed by D i = (1 +.5 i), where i i the Mathcad range variable. The calculation of the increae in head of the pump, W, required for a given pipe diameter i a Category I problem. The pump increae in head, W, cab be calculated directly for each pipe diameter. The power required by the pump i WQρ PowerR = (3) η Graphical repreentation of variable of interet are provided in the workheet. For the relatively mall diameter range conidered, the power required by the pump change by two order of magnitude (from 46 hp to 5.9 hp). The figure provide graphic evidence of the almot D 5 dependence that power required and preure drop have on diameter when major loe are dominant in piping ytem. An examination of the power required graph provide ome inight into diameter election. The power required ha a diminihing return relationhip a the diameter i increaed. At about D = 3 inche (the knee in the curve) further increae in diameter yield little decreae in either the pump power required or the pump increae in head. Thu, a curory examination eliminate diameter near 1 inch (too much power) or near 6 inche (too large a pipe) and indicate a diameter of 3 inche to be a rational choice. 9 ASEE Southeat Section Conference

16 Economic metric, uch a the minimum preent worth, are often ued to determine the deired configuration of a ytem. The preent worth depend on the ytem operating cot a well a the initial ytem cot. Since the hour per week of ue i not provided, a two-hift operation (8 hour/week) i aumed. Additionally, the expected lifetime of the ytem i taken a 1 year and an interet rate of 7 percent i reaonable. The total ytem operating cot conit of the energy cot and the demand cot. The total ytem operating cot per year a a function of pipe diameter i depicted in the workheet. For very large pipe diameter, the yearly operating cot aymptotically approach a mall number ($5), depending on the pipe diameter. The initial ytem cot are compoed of the pipe purchae/intallation cot and the pump cot. Pipe and pump cot are taken from Hodge and Taylor (1999). The total initial cot of the ytem i the um of the pipe and pump cot. The initial cot of the ytem are dependent only on the pipe diameter and not on the hour per week of operation. The initial ytem cot a a function of pipe diameter i preented in the workheet. The ytem initial cot are dependent motly on the pipe diameter and length ince the pump cot i very mall compared to the pipe cot except at the maller diameter. With the operating cot per year and the total initial cot known, the preent worth for a n-year operating period i n (1 + int) 1 Pr eentw = CotTot + CotInit (4) n int (1 + int) where int i the interet rate (7 percent) and n (1 year) i the number of year conidered. The preent worth for the ytem a function of pipe diameter i hown in the workheet. A minimum preent worth exit ince operating cot decreae a diameter increae and total initial cot increae a diameter increae. The minimum preent worth correpond to a pipe diameter of 3.5 inche. Hence, the curory indication of about 3 inche from power required i amazingly accurate. The recommended pipe diameter i 3.5 inche for thi ytem. The reult of both the imple analyi examining only the behavior of the increae in power (or pump increae in preure) and the more complex preent worth economic analyi are eentially the ame. Verification of the olution i provided by oberving that the value of the increae in head of the pump, W, aymptotically approache the elevation difference of 15 feet. Thi i appropriate a frictional loe decreae a D 5 and for large pipe diameter become very mall. Conider the fluid velocity a a function of pipe diameter. At a diameter of 3 inche, the fluid velocity i about 8 ft/ec, and at a pipe diameter of 3.5 inche, the fluid velocity i near 6 ft/ec. A general rule of thumb in piping ytem i that economic contraint dictate that the velocity hould not exceed 1 ft/ec. Thi olution i very congruent with the rule of thumb thu providing additional verification for the olution. 9 ASEE Southeat Section Conference

17 ORIGIN 1 Set origin for counter to 1 from the default value of. Input the pipe geometry: Diameter in feet Length in feet Roughne in feet: i := L := 3 ft ε :=.15 ft D i := ( i) in Input the ytem boundary (initial and end) condition: Preure in pi: Elevation in feet: P a P b := lbf in Input the lo coefficient: K factor Z a Z b := 15. ft Equivalent length K := 1.5 C := 6 Input the fluid propertie: Denity in lbm/ft 3 Vicoity in lbm/ft- ρ := 56.1 lb ft 3 µ :=.576 Input the flow rate in cf: Q :=.4 ft3 ec lb ft ec Define contant and adjut unit for conitency: g := ft ec g c := ft lb lbf ec Define the function for Reynold number, fully-rough friction factor, and friction factor: Re( q, D) := 4 ρ q π D µ fq (, D, ε) := log Re( q, D) 64 Re( q, D) ε 3.7 D otherwie 1.11 if Re( q, D) > 3 f T ( D, ε) := log.386 ε 3.7 D 1.11 Figure 6. Workheet for Example 5. 9 ASEE Southeat Section Conference

18 The generalized energy equation i: P b P a W i := ρ + g ( Z b Z a ) g c + 8 π g c Q 4 D i fq, D i, ε L D i + K + Cf T D i, ε Power imparted to fluid: Power := W Q ρ Pump mechanical efficiency. η :=.67 Power required: PowerR := Power η W i ft lbf lb PowerR 3 i kw D i in Conider the cot of running the pump for 8 hour per week (two hift). wk := 7 day month := 4.33 wk HW := 8 hr wk Energy Cot: The total electricity cot i compoed of an energy cot (kwh) and a demand cot (kw). The energy cot i the product of the kwh ued per year and the kwh rate: CotE := PowerR HW 5 wk.5 kw hr The demand cot i the product of the demand (kw) per billing period time the number of billing period in a year (uually 1) time the demand rate per billing period. CotD := PowerR 1 month The total electricity cot i the um. CTot := CotE + CotD 1. kw month D i in Figure 6. Continued. 9 ASEE Southeat Section Conference

19 CTot Initial Cot: The initial cot are compoed of the pipe cot (purchae and intallation) and the pump cot (purchae and intallation). Pipe pumpcot (etimated uing information in Hodge and Taylor, 1999). L D CotP := D CotPump := 6 ft in in The total cot i the um of the pipe and pump cot. CotInit := CotP + CotPump D in Power 13 hp CotInit D in Figure 6. Continued. 9 ASEE Southeat Section Conference

20 Conider the preent worth for 1-year of operation at 7 percent. n := 1 Number of year conidered int :=.7 Interet rate ( 1 + int) n 1 PreW := CTot int ( 1 + int) n + CotInit Preent worth for n year at int interet rate PreW Fluid velocity calculation. Area :=.5 π D Q V := Area D in V ft ec D in Figure 6. Concluded. 9 ASEE Southeat Section Conference

21 ASSESSMENT AND CONCLUSIONS The purpoe of thi paper i to dicu a proce for inculcating a ytematic and practical problem olving approach in engineering tudent. In all the example explored in thi paper, the ame proce i ued. The treatment of all the example problem are identical and emphaize the three tep: (1) formulate a well-poed ytem of equation, () utilize uer-friendly mathematical computer olver to do the arithmetic, and (3) verify the reult. In thi paper, the arithmetic ha been accomplihed by uing Mathcad. Other computational oftware ytem (Mathematic, Matlab,.) offer the ame capability, albeit in different format, but with the ame reult. Anecdotally, tudent appreciate the attention to problem olution uing the three-tep unified approach. The ue of Mathcad relieve the tudent from aimilating different numerical technique ( procedure ) to olve a ytem of equation. The net reult i that more involved and more realitic problem can be aigned. With le time pent on arithmetic, more time i available for tudent to engage i higher-level ynthei and undertanding. Example illutrating a unified approach to olution of engineering problem in three required coure in ME at MSU have been preented and dicued. The approach offer advantage in providing tudent with capability to olve more real world problem while concentrating on the engineering apect of the problem. REFERENCES Hodge, B. K., 5, A Unified Approach to Piping Sytem Problem, Proceeding of the 5 ASEE Annual Meeting and Exhibition, Seion 3666, Portland, Oregon, June. Alo, Proceeding of the 5 ASEE Southeatern Section Annual Meeting, Chattanooga, TN, April. Hodge, B. K., 1998, The Evolution of a Required Energy Sytem Deign Coure, Proceeding of the 1998 ASME International Mechanical Engineering Congre and Expoition, Anaheim, CA, Paper 98-WA/DE-9, Nov. Hodge, B. K., and Luck, R., 6, Uing Computational Root Solver: A New Paradigm for Problem Solution?, Proceeding of the 6 ASEE Annual Meeting and Exhibition, Seion 3666, Chicago, IL, June. Alo, Proceeding of the 6 ASEE Southeatern Section Annual Meeting, Tucalooa, AL, April. Hodge, B. K., and Taylor, R. P., 1999, Analyi and Deign of Energy Sytem, 3 rd ed., Prentice-Hall, Inc., Upper Saddle River, NJ. Hodge, B. K., and Taylor, R. P., 1998, The Impact of MathCad in an Energy Sytem Deign Coure, Proceeding of the 1998 ASEE Southeatern Section Annual Meeting, pp , Orlando, FL, April. Alo, Proceeding of the 1998 ASEE Annual Meeting and Exhibition, Seion 666, Seattle, WA, June. Rogelio Luck received the B.S. degree from Texa Tech Univerity in 1984, and the M.S. and Ph.D. degree from Penn State Univ., Univerity Park, in 1987 and 1989, repectively, all in Mechanical Engineering. In 1989 he joined the faculty at the Mechanical Engineering Dept. at Miiippi State Univerity. Hi current reearch interet i in the area of imulation, optimization and control of building cooling heating and power. He ha publihed in the area of building cooling heating and power, uncertainty analyi, invere heat tranfer, radiation heat tranfer, applied math, theoretical and applied control ytem, piezoelectric enor, electrical power ytem generation and ditribution, and redundant meaurement ytem. B. K. Hodge i Profeor of Mechanical Engineering at Miiippi State Univerity (MSU) where he erve a the TVA Profeor of Energy Sytem and the Environment and i a Gile Ditinguihed Profeor and a Griham Mater Teacher. He i a Fellow of the American Society for Engineering Education and the American Society of Mechanical Engineer and an Aociate Fellow of the American Intitute of Aeronautic and Atronautic. 9 ASEE Southeat Section Conference