Grand Valley State University The Padnos School of Engineering CONVERGING DIVERGING CHANNEL EGR 365 FLUID MECHANICS Brad Vander Veen June 3 2003 Lab Partners Julie Watjer PURPOSE The purpose of this lab is to measure the static pressure variation through an enclosed converging diverging channel nozzle at a given flow rate and to compare those measurements to the predictions obtained from Bernoulli s equations Based on these results regions of flow in which Bernoulli s equation can and cannot be applied will be identified THEORY Consider the diagram of the flow area and pitot static tube below in Figure 1 Figure 1 Diagram of Flow Area and Pitot Static Tube Writing Bernoulli s equation at any two points 1 and 2 yields 1 2 1 2 p1s v1 p 2 s v 2 pt 2 2 1 where ps is the static pressure v is the velocity and pt is the total pressure Knowing conservation of mass to be true v1 A1 v 2 A2 2 where v is the velocity and A is the cross sectional area Combining these two equations gives us a measure of pressure in non dimensional form v 2 2 A2 2 1 2 1 2 2 0 5 v1 v1 A1 p 2 p1 3 where p1 v1 and A1 are properties at the throat and p2 v2 and A2 are all properties at some other point in the flow and is the density of the fluid In this experiment a pitot static tube will be used for measuring the dynamic head which is pt p s 0 5 v 2 4 Combining Equation 3 and 4 yields 2 p 2 s p1s A2 1 2 p1t p1s A1 5 APPARATUS ITEM Airflow Bench Pitot Static Tube Manometer Meterstick PROCEDURE 1 Assemble the experimental setup as seen in Figure 1 2 Measure the depth of the flow channel 3 Turn on the Airflow Bench 4 Start at the top and move down at increments of 1cm until the bottom is reached At each distance conduct the following a measure the width of the channel b record static pressure using the manometer c record total pressure using the manometer RESULTS distance from top m width m area m 2 static pressure mbar total pressure mbar 0 0 01 0 02 0 03 0 04 0 05 0 06 0 07 0 08 0 09 0 1 0 11 0 12 0 13 0 14 0 15 0 16 0 17 0 18 0 19 0 2 0 21 0 22 0 23 0 24 0 25 0 26 0 27 0 28 0 29 0 3 0 075 0 07 0 065 0 061 0 056 0 052 0 047 0 044 0 044 0 044 0 044 0 044 0 045 0 046 0 048 0 05 0 052 0 054 0 056 0 057 0 058 0 06 0 061 0 063 0 065 0 066 0 068 0 07 0 071 0 072 0 074 0 00360 0 00336 0 00312 0 00293 0 00269 0 00250 0 00226 0 00211 0 00211 0 00211 0 00211 0 00211 0 00216 0 00221 0 00230 0 00240 0 00250 0 00259 0 00269 0 00274 0 00278 0 00288 0 00293 0 00302 0 00312 0 00317 0 00326 0 00336 0 00341 0 00346 0 00355 11 4 10 8 9 8 9 2 8 2 7 5 6 4 4 3 8 3 4 3 4 3 4 3 6 4 4 6 5 2 5 6 6 6 6 7 7 4 7 6 8 8 2 8 4 8 6 8 8 9 9 2 9 4 9 6 15 6 15 6 15 6 15 6 15 6 15 6 15 6 15 6 15 8 15 8 15 8 15 8 15 8 15 8 15 8 15 8 15 8 15 8 15 8 15 8 15 6 15 6 15 6 15 6 15 6 15 6 15 6 15 6 15 6 15 6 15 6 Table 2 Recorded Results throat in yellow ANALYSIS In Figure 3 below the experimental normalized pressure is plotted versus the known cross sectional area ratio On the same plot the theoretical values for normalized pressure are plotted The experimental normalized pressure was calculated using the left side of Equation 5 while the theoretical was calculated using the right side Norm alized Pressure vs Area Ratio 0 700 Norm alized Pressure 0 600 0 500 0 400 Theoretical Actual 0 300 0 200 0 100 0 000 0 59 0 72 0 94 1 00 0 98 0 88 0 79 0 73 0 68 0 63 0 59 A1 A2 Figure 4 Plot of Theoretical Normalized Pressure vs Actual Normalized Pressure Note that before the throat the theoretical normalized pressure seems to match the actual normalized pressure quite well However after the throat the two sets seem to diverge from each other This can be explained because the theoretical values do not assume any losses throughout the flow channel As measurements were taken at points further down in the flow channel the losses accumulated and were very apparent in the results Even though the beginning and end of the channel had the same area the pressure at the end did not return to where it started This was due to the losses In Figure 5 below the dimensional parameters of the flow can be seen Pressure vs Area 18 16 Pressure m bar 14 12 10 P t 8 P s 6 4 2 0 0 0036 0 0029 0 0023 0 0021 0 0022 0 0024 0 0027 0 0029 0 0031 0 0034 0 0036 Area m 2 Figure 5 Actual Pressure vs Cross Sectional Area The velocity of the fluid in the channel can also be plotted as seen below in Figure 6 These velocity values were calculated using Equation 4 Air Speed vs Area 50 00 Air Speed m s 45 00 40 00 35 00 Air Speed 30 00 25 00 20 00 0 0036 0 0027 0 0021 0 0022 0 0025 0 0028 0 0031 0 0034 Area m 2 Figure 6 Air Speed vs Cross Sectional Area CONCLUSION In this lab measurements of pressure were taken throughout a flow channel Theoretical data showed that in areas where the channel narrowed the static pressure would decrease and the fluid velocity would increase These predictions were confirmed by actual results found in this lab Also discovered in this lab was that Bernoulli s equation holds well for flow through a channel before in converges but after the channel diverges the losses are significant and Bernoulli s equation becomes erroneous