Design ProblemFuel Economy for Tractor-Trailer TrucksBrad PeirsonEGR 365 – Fluid MechanicsInstructor: Prof. FleischmannSchool of EngineeringPadnos College of Engineering and ComputingGrand Valley State UniversityJuly 25, 20071 PurposeThe purpose of this analysis is to determine the fuel efficiency of a tractor-trailer truck givenin given wind conditions using the aerodynamic properties of the truck.2 BackgroundThe analysis was performed on a southbound truck using wind statistics for 1994 at night.The probability of the wind coming from a given direction is provided in Appendix A. Theseprobabilities are represented graphically in Figure 1.Figure 1: Truck Heading and Wind Direction ProbabilityFor the purposes of angle calculations, North is considered the zero reference. The angleφ proceeds clockwise from the North (East = 90◦, South = 180◦etc.).In order to determine the drag on the truck it is necessary to determine its relativevelocity. This is accomplished using the relationship shown in Figure 2. In Figure 2 V isthe speed of the truck, V W is the speed of the wind, and Vris the relative velocity vector.Because the truck is heading south, both the wind angle (φ) and the yaw angle (ψ) areincluded angles in the vector triangle.Table 1 lists the given information for the configuration of the truck.The first step in the analysis is to determine the relative velocity of the truck. This isdone by applying the law of cosines to Figure 2. This equation is given in equation 1.1Figure 2: Relative Air Speed Vector DiagramTable 1: Truck Configuration InformationTruck Velocity 55 mphAverage Wind Speed 5.15 mphDensity of Air 0.00237slugft3Projected Frontal Area 114.5ft2Weight of the Truck 53,780 lbfDriveline Efficiency, ηD0.85Rear End Gear Ratio 4.17Density of Fuel, γf6.95lbmgalTire Radius, R 21.25 inTire Type a =7.5lbf1000lbf, b = 0Trailer 13’6”, smooth sidesGap 70 inWeight, W 53,780 lbf2Once the relative velocity is known, the yaw angle (ψ) needs to be calculated in order todetermine the drag coefficient (CD) for the truck. The yaw angle is found by applying thelaw of sines (equation 2) to Figure 2. Once ψ is known the drag coefficient is found fromFigure 3.Vr=hV2+ V W2− 2(V )(V W ) cos(φ)i12(1)Vrsin φ=V Wsin ψ(2)Figure 3: Drag Coefficient vs. Yaw Angle (ψ) for Various Truck ConfigurationsGiven the drag coefficient and the relative velocity the aerodynamic drag can be foundusing equation 3. The aerodynamic drag changes for each angle φ as the drag coefficientchanges.DA=12ρAV2rCD(3)The probabilities for the wind coming from a given compass direction are given in Ap-pendix A. Because the wind does not come from a single direction, the aerodynamic dragmust be averaged to account for the direction of the wind. This is done with equation 4.¯DA= ΣP (φ)DA(4)3Where P (φ) is the probability of the wind blowing from a give n direction φ. The next stepin the analysis is to determine the rolling resistance, that is the force required to overcomethe forces acting in the truck itself. This is done using equation 5.Dr= (a + bV )W (5)Once both the average aerodynamic drag and the rolling resistance are known they canbe input into equation 6 to determine the total drag on the truck.DT=¯DA+ Dr+ W sin δ + Wg+nIR2!A (6)Where δ is the grade angle (ze ro in this analysis), n is the number of tires (an 18 wheeltruck/trailer), I is the rotational moment of inertia for the wheel assembly and A is theacceleration of the truck (zero in this analysis). The braking horsepower (BHP ) is thenfound using equation 7.BHP =DTVηD+ pacc(7)Where paccis the power taken from the engine by accessories (zero in this analysis). Theengine speed is then found us ing equation 8.RP M =VRrev2πrad60smin(RearEndRatio)(T ransmissionRatio) (8)The braking horsepower and the engine speed are used together to determine the brakespecific fuel consumption, BSF C, from Figure 4.Once the brake specific fuel consumption has been determined the fuel usage in gallonsper mile can be determined using equation 9.GP M = BHP"BSF CγfV#(9)The reciprocal of equation 9 gives the fuel economy of the truck in miles per gallon.3 ResultsThe equations provided in section 2 were input into Excel. The complete Excel data sheetis provided in App endix A. Sample calculations verifying Excel’s results are provided inAppendix B. Table 2 shows the results of the primary calculations in this analysis.The results of the analysis show that for the given truck configuration in night, July 1994wind conditions the truck will use 5.22 gallons of fuel per mile.4Figure 4: Fuel MapTable 2: Analysis Results¯DA(lbf) 689.85Dr(lbf) 403.35DT(lbf) 1093.2BHP (Hp) 188.63Engine RPM 1813.94BSF ClbmBHP hour0.388GP M 0.191MP G 5.225A Excel Data SheetB Sample CalculationC Graded Rough