Design Problem Fuel Economy for Tractor Trailer Trucks Brad Peirson EGR 365 Fluid Mechanics Instructor Prof Fleischmann School of Engineering Padnos College of Engineering and Computing Grand Valley State University July 25 2007 1 Purpose The purpose of this analysis is to determine the fuel efficiency of a tractor trailer truck given in given wind conditions using the aerodynamic properties of the truck 2 Background The 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 These probabilities are represented graphically in Figure 1 Figure 1 Truck Heading and Wind Direction Probability For the purposes of angle calculations North is considered the zero reference The angle 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 relative velocity This is accomplished using the relationship shown in Figure 2 In Figure 2 V is the speed of the truck V W is the speed of the wind and Vr is the relative velocity vector Because the truck is heading south both the wind angle and the yaw angle are included 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 is done by applying the law of cosines to Figure 2 This equation is given in equation 1 1 Figure 2 Relative Air Speed Vector Diagram Table 1 Truck Configuration Information Truck Velocity Average Wind Speed Density of Air Projected Frontal Area Weight of the Truck Driveline Efficiency D Rear End Gear Ratio Density of Fuel f Tire Radius R Tire Type Trailer Gap Weight W 2 55 mph 5 15 mph 0 00237 slug f t3 114 5f t2 53 780 lbf 0 85 4 17 6 95 lbm gal 21 25 in 7 5lbf b 0 a 1000lbf 13 6 smooth sides 70 in 53 780 lbf Once the relative velocity is known the yaw angle needs to be calculated in order to determine the drag coefficient CD for the truck The yaw angle is found by applying the law of sines equation 2 to Figure 2 Once is known the drag coefficient is found from Figure 3 i1 h Vr V 2 V W 2 2 V V W cos 2 VW Vr sin sin 1 2 Figure 3 Drag Coefficient vs Yaw Angle for Various Truck Configurations Given the drag coefficient and the relative velocity the aerodynamic drag can be found using equation 3 The aerodynamic drag changes for each angle as the drag coefficient changes 1 DA AVr2 CD 3 2 The probabilities for the wind coming from a given compass direction are given in Appendix A Because the wind does not come from a single direction the aerodynamic drag must be averaged to account for the direction of the wind This is done with equation 4 D A P DA 3 4 Where P is the probability of the wind blowing from a given direction The next step in the analysis is to determine the rolling resistance that is the force required to overcome the 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 can be input into equation 6 to determine the total drag on the truck W nI DT D A Dr W sin 2 A g R 6 Where is the grade angle zero in this analysis n is the number of tires an 18 wheel truck trailer I is the rotational moment of inertia for the wheel assembly and A is the acceleration of the truck zero in this analysis The braking horsepower BHP is then found using equation 7 BHP DT V pacc D 7 Where pacc is the power taken from the engine by accessories zero in this analysis The engine speed is then found using equation 8 rev 60s V RearEndRatio T ransmissionRatio 8 R 2 rad min The braking horsepower and the engine speed are used together to determine the brake specific fuel consumption BSF C from Figure 4 Once the brake specific fuel consumption has been determined the fuel usage in gallons per mile can be determined using equation 9 RP M BSF C GP M BHP f V 9 The reciprocal of equation 9 gives the fuel economy of the truck in miles per gallon 3 Results The equations provided in section 2 were input into Excel The complete Excel data sheet is provided in Appendix A Sample calculations verifying Excel s results are provided in Appendix 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 1994 wind conditions the truck will use 5 22 gallons of fuel per mile 4 Figure 4 Fuel Map Table 2 Analysis Results D A lbf Dr lbf DT lbf BHP Hp Engine RPM lbm BSF C BHP hour GP M MPG 5 689 85 403 35 1093 2 188 63 1813 94 0 388 0 191 5 22 A Excel Data Sheet B Sample Calculation C Graded Rough Draft