Experimental Results
Where the gravitational acceleration g=9.81ms2
The formulas that were employed in the writing of this report included the following,
Es1=Es2, where Es=y+V22g Eq. 1
Velocity, V=qy;where q=QB Eq. 2
coefficient of contraction, cc=yx Eq. 3
velocity head=V22g.m Eq. 4
Momentum theory:0.5ρgy12-Fg-0.5ρgy22=ρqV2-ρqV1 Eq. 5
Froude Number; Fr=Vgxy Eq. 6
Conjugate equation; y4=y32×[1+8Fr32-1] Eq. 7
Calculations: Part 1
For detailed calculations refer to the appendix.
Q=7.948×10-4m3s;q=9.947×m2s; V1=0.071ms; V2=0.09ms
Velocity head1=2.57×10-4m; velocity head2=4.13×10-4m
Es1=Es2. Therefore, we have;
0.125y23-0.018y22+6.317×10-7=0,∴y2=0.006 m; cc=y2x=0.0080.01=0.8
Calculations: Part 2
Detailed calculations are to be found in the appendix.
V3=0.711ms; V4=0.343ms;0.5ρgy12-Fg-0.5ρgy22=ρqV2-ρqV1
Therefore, Fg=85.59 Nm; Fh=0.5×1000×9.81×0.141-0.012=84.17 Nm
Calculations: Part 3
Detailed calculations are to be found in the appendix.
Employing the momentum equation,
4905y43-8.033y4+0.0989=0
Thus, y4=0.032 m
Fr=0.719.81×0.014=1.92; Thus, E3=0.039
Using the conjugate equation to counter check;
0.0142×1+8×1.922-1=0.04; And, E3=0.035
Discussion
The experimental values obtained were slightly different from the theoretical values found in literature. For instance, the experimental y2 value obtained was 0.008 m. But, after application of the specific energy equation, Es1=Es2, the calculated value of y2 was 0.006 m. The difference in y2 values could be better understood by expressing the variance as a percentage. That is, 0.008-0.0060.008×100%=25% and this is the difference between the y2 theoretical value and the experimental value.
The slight variation of the values may be due to such reasons;
- Lack of precision and accuracy during the measurement of the y2 values by use of a ruler.
- Difficulty in handling fluids involved in the experiment
- The value of y1might also have been not accurate and as such the error was transferred to the final y2 calculated.
- Frictional forces between the apparatus and the fluid
Given the above reasons, an accurate value of y2 was determined by getting the mean of the calculated values, that is 0.007 m, while that for averaged y4 was 0.031 m. A similar explanation is valid for the variance between the theoretical and experimental value of y4 by utilization of the momentum equation.
The ratio of FgFhtends to near one of the depth ratios. This is an indication that the variations between the force actually acting on the gate may be denoted the hydraulic force on it. The hydrostatic pressure force was determined by measuring the downstream and upstream water depths. The momentum equation was used with the flow rates to evaluate the horizontal force present on the sluice gate. (Baines, 2003)
Conclusion
- The variations between the measured and computed values fell within the expected experimental uncertainty.
- Minor errors like parallax could have contributed to difference the measured values from those found in literature.
- Average values for the measured parameters were used as they were closer to the theoretical values.
- Frictional resistance between the hydraulic structure and the liquid contributed to energy dissipation which affected value of y2
- Inaccuracy and difficulty in taking maintaining steady flow of the fluid are some of the errors that affected the obtained values.
- Pressure distribution in the system decreased with increase in the velocity. (Vanden-Broeck, 1997)
Appendix
Calculations: Part 1
Q=0.01518.89=7.948×10-4m3s
q=7.948×10-40.0799=9.947×10-3m2s
V1=9.947×10-30.141=0.0705ms
V2=9.947×10-30.008=0.099ms
Es1=Es2; Es1=0.141+(9.947×10-4)2/2×9.81=0.141
Es2=y2+(7.948×10-4)2/2×9.81×0.07992×y22=y25.043×10-6/y22
Therefore, 0.141=y2+(7.948×10-4)2/2×9.81×0.07992×y22
Multiply each side of the equation by 2×9.81×0.07992×y22
0.018y22=0.125y23+6.317×10-7=0
Re-arranging the equation and solving, we obtain;
y2=0.006 m, -0.005 m or 0.144 m
y2=0.006 m is taken as the approximate figure since it is closer to the experimental value.
Calculations: Part 2
V3=9.947×10-30.014=0.711ms
V4=9.947×10-30.029=0.343ms
0.5ρgy12-Fg-0.5ρgy22=ρqV2-ρqV1
0.5×1000×9.81×0.1412-Fg-0.5×1000×9.81×0.0082=1000×9.947×10-3×1.24-1000×9.947×10-3×0.071
97.52-Fg-0.31=12.33-0.71, or Fg=85.59 Nm
Fh=0.5×ρ×g×(y1-X)2
Fh=0.5×1000×9.81×(0.141-0.01)2=84.17Nm
Calculations: Part 3
V3=9.947×10-30.014=0.711ms
V4=9.947×10-30.029=0.343ms
0.5×1000×9.81×0.0142+1000×9.947×10-3×0.711=0.5×1000×9.81×y42+1000×(9.947×10-3)2/y4
0.961+7.072=4905y42+0.0989y4
Multiply both sides with y4
8.033y4=4905y43+0.0989
Re-arranging the equation and solving for y4;
y4=-0.046 m, 0.032 m or, 0.014 m
E3=y3+V22g=0.014+0.71122×9.81=0.039
E3=y4+V32g=0.029+0.34322×9.81=0.03
Bibliography
P. G. Baines and J. A. Whitehead, ‘‘on multiple states in single-layer flows,’’ Phys. Fluids 15, 298 (2003)
J. M. Vanden-Broeck, ‘‘Numerical calculations of the free-surface flow under a sluice gate,’’ J. Fluid Mech. 330, 339 (1997)
