Fluid flow in a duct of varying cross-section
The objective of the experiment was to highlight the relationship between pressure, fluid velocity and static head in a pipe of varying cross section. The arrangement that was used is called a Venturi, this is because the pipe (which had constant diameter) narrowed with a short contraction to a narrow throat and was followed by a long diffuser back to the pipes original diameter. Results See figures 1, 2 and 3. Discussion During the experiment as the water travelled through the narrower neck of the pipe the pressure was reduced.
This was because as the water travelled through the neck the velocity of the water increased as the cross sectional area decreased. The increase in velocity increased the force with which the water travelled through the pipe. The increase in force was grater than the reduction in cross sectional area, which was why a pressure drop occurred. The greatest change in static head occurred between tap 3 and 4 (as can be seen in figure 1). The static head was increased significantly for all three flow rates due to the drop in pressure.
The results obtained from experiment (which can be seen in figure 2) are actually very different to the results obtained from theory. The results obtained from theory are for the ideal static head which was very small whilst with the experiment the variations in static head values were a lot bigger. The change in total head was similar to the change in static head. The total head values obtained were practically identical for all three flow rates, the only significant difference in the total head occurred at the neck of the pipe.
This was because the decrease in pressure was different for each flow rate because the water was travelling at different speeds. Conclusions In conclusion the results showed that as the water travelled through the pipe the static head was uniform until it reached the neck of the pipe. When the water flowed through the neck the static head increased due to the decrease in pressure. The results also showed that as the water flowed through the neck of the pipe the velocity increased because of the reduction in area, which reduced the pressure.
Here is the raw data obtained from the experiment for flow rates of 46, 34 and 24 litres / minute. Table 1 Tap flow rate (Q) / l/min Distance (x) /mm Diameter (d) /mm Height (h) /cm Velocity (v) / mms-1 1 46 Specimen calculations; The velocity (v) was calculated from the continuity equation Q = v A , where A = the cross sectional area of the pipe. i. e Table 1 tap 4: v = Q / A = 46 / ? r2 = 46 / 2 = 0. 0180 mms-1 When plotting (hn – h1 ) versus x graph, (hn – h1) was calculated by subtracting the first value for height from the rest of the values for height. i. e. Table 1, tap 4: (hn – h1 ) = (h4 – h1 ) = 46.
7 – 140. 6 = -93. 9 mm The calculations for the second graph involved multiplying (hn – h1 ) by 2g (19. 6) and dividing by v12 i. e Table 1 tap 4: 2g(hn – h1 ) = 19. 6(46. 7 – 140. 6) v12 0. 001214 = -90507. 9 mm The calculation for the third graph was similar to the second graph. The formula is basically the same except (Vn / V1)2 -1 is added to the equation. i. e. Table 1 tap 4: 2g(hn – h1 )+(Vn / V1)2 -1 = 19. 6(46. 7 – 140. 6)+(0. 074705/0. 034842)2 -1 v12 0. 001214 = -90507. 9 + 3. 5972 = -90504. 3 mm Error estimation Error in flow rate measurements = + 1 litre / min.
Mean value of flow rate = 35 litre / min Therefore the percentage error in the flow rate = ? Q x 100 Q = 1 x 100 35 = 2. 9% Error in pressure readings = + 1 mm Value of h at the neck (with flow rate of 46 l/min) = 46. 7cm or 467 mm Therefore the percentage error = ? h x 100 h = 1 x100 467 = 0. 2 % Figures Figure 1 = (hn – h1 ) versus x which, represents the change in static head. Figure 2 = 2g(hn – h1 ) versus x, which represents the change in v12 normalised static head. Figure 3 = 2g(hn – h1 ) + (Vn / V1)2 -1 versus x, which represents the . v12 variation of normalised total . head along the duct.