Laboratory 4 Diverging Channel Brad Peirson Laboratory Group Nick O Brien and Ryan Lillibridge EGR 365 Fluid Mechanics Instructor Prof Fleischmann School of Engineering Padnos College of Engineering and Computing Grand Valley State University June 5 2007 1 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 equation Based on these results regions of flow in which Bernoulli s equation cannot be applied will be identified 2 Background Figure 1 Schematic of Converging Diverging Channel Conservation of Mass V1 A1 V2 A2 Bernoulli s Equation p1s 21 V12 p2s 12 V22 pt Bernoulli applies to steady flow with no losses and no net energy transfer Table 1 Variables for Bernoulli and Conservation of Mass Variable V1 A1 V2 A2 p1s p2s pt Definition Velocity at the throat Area at the throat Velocity at the scanned position Area at the scanned position Static pressure at the throat Static pressure at the scanned position Total pressure 1 Combine Bernoulli s and Conservation of Mass to obtain full derivation and sample 2 p2s p1s A 1 calculation in Appendix A 1 A pt ps 1 2 3 Results Pressure differences were measured every 2 cm beginning at the top of the channel The complete set of data is shown in appendix B Figure 2 shows the plot of the pressure ratio against the area ratio for both the theoretical model above and the experimental results Figure 2 Pressure Ratio vs Area Ratio Note that in figure 2 the theoretical area ratio is a mirror image of itself about the point where the ratio equals 1 This was an added manipulation to the data points in order to plot them using Excel otherwise the data before and after 1 would lie directly on top of one another The same was done to the experimental data in order to prevent it from backing up on itself in the Excel plot 1 The experimental data correlates to the model up to the point where A A2 1 the throat 4 Discussion Conclusions The experimental results follow the theoretical model up to the throat of the channel At this point there is a large discrepancy between the two sets of data as shown in figure 2 This is largely due to the flow in this section of the channel At the beginning of the channel the flow is steady It remains steady as the channel converges into the throat though the velocity 2 increases As the flow exits the channel the velocity decreases again but there is also the introduction of turbulent flow The slower moving fluid has a chance to mix itself around Because the flow in this section is no longer steady Bernoulli s equation no longer applies This also makes the theoretical model invalid in this section as it is based on Bernoulli 3 A Equation Derivation Question 1 and Sample Calculations B Excel Data Sheet C Question 3 Response Experimentalists often present data in non dimensional form Give one reason why this would be convenient Developing non dimensional models for data makes them universal Scientists in the United States can use the US Customary unit system in their calculations while scientists in Europe can use the SI system Given that the model is inherently dimensionless both parties should obtain the same results regardless of the unit system used This makes it extremely convenient for researchers willing to reproduce experiments who may not have access to tools that will provide them with the identical unit system originally used D Question 4 Response Where is Bernoulli s Equation Valid Where is it not valid Could Bernoulli be modified for use in these regions The experimental results follow the theoretical model up to the throat of the channel At this point there is a large discrepancy between the two sets of data as shown in figure 2 This is largely due to the flow in this section of the channel At the beginning of the channel the flow is steady It remains steady as the channel converges into the throat though the velocity increases As the flow exits the channel the velocity decreases again but there is also the introduction of turbulent flow The slower moving fluid has a chance to mix itself around Because the flow in this section is no longer steady Bernoulli s equation no longer applies This also makes the theoretical model invalid in this section as it is based on Bernoulli The model could be modified to account for this turbulent region however Bernoulli could not Bernouli s equation is based on the assumption of steady flow with no losses In order to account for the losses due to unsteady flow it would be necessary to use Euler s Equation and derive a new model without the assumption of steady flow