ROTARYROCKET7

This is the report me and my teammates have made for v-notch weirs and rectangular weirs, just wanted to ask someone who knew about them and see what they think about the report





























CALIBRATION OF

SHARP-CRESTED WEIRS









































HYDRAULICS LAB #2

PROFESSOR KHANVILBARDI

OCTOBER 22, 2005

GROUP 2B



JESSICA BLACK

JASON RODRIGUEZ

VALENTINA PONOCHOVNAYA

MITZA ZOBENICA



TABLE OF CONTENTS





ABSTRACT…………………………………………………………….….2



INTRODUCTION……………………………………………………….…3



Background on Weirs………………………………………………3



Experiment Objectives……………………………………………..4



THEORETICAL EQUATIONS…………………………………………...4



PROCEDURE……………………………………………………………...6



DISCUSSION OF RESULTS….…………………………………………..6



CONCLUSION…………………………………..……………………..…10



LITERATURE CITED…………………………………………………….10



APPENDIX………………………………………………………………..11







ABSTRACT





Experiments were conducted using the open channel of a “hydraulic bench” and two sharp-crested weirs of different geometry to determine discharge coefficients (Cd). A rectangular weir with a width of 0.03m and a V-notch weir with V angle: θ = 90บ were used for these experiments.



The Cd for the rectangular weir was found to be 0.867, with a standard deviation of 0.123. The Cd developed for the V-notch weir was 1.115, with a standard deviation of 0.312. Flowrates were then calculated with the average Cd terms above to determine an average error between the actual and calculated flowrates. The error found using the Cd for the rectangular weir was found to be 9.6%, while it was found to be 19.8% for the V-notch weir. Both the high standard deviation and error found for the V-notch weir could either point to a potential error in data collection methods or just to normal variation of empirical tests.



Flowrate (Q) over a weir is a function of the relative width of the weir and the head (H) above the edge of the weir. Though a basic theoretical relationship between Q and H can be derived, the empirically determined Cd term is necessary to account for effects on actual flow over a weir.



INTRODUCTION



Background on Weirs

Weirs are man-made structures constructed normal to the direction of flow in open channels. Weirs are generally classified as “sharp-crested” or “broad-crested” and serve a number of purposes. Broad-crested weirs act as spillways for streams and reservoirs by discharging excess water levels. Sharp-crested weirs provide good discharge measurements for small flow rates, and are commonly used for this purpose (as in this lab). The flow over a weir depends on both the geometry of the weir and the head on the weir.



In order for a weir to be considered sharp-crested, it must be constructed with a sharp upstream corner such that the water passing over the weir touches only a line. By contrast water will flow over the surface of a broad-crested weir . These weirs typically have a ratio of crest length to upstream head greater than 1.5 to 3 . There is a third type of weir known as “weirs not-sharp crested”. These weirs, though too thin to be considered as broad-crested, are thick enough for water to still flow along a surface in its passing . It should be noted that the weirs used in this experiment are correctly noted as sharp-crested in that the bottom nappe (or edge of the overflow volume as it passes a weir) only touches the sharp, upstream edge of the weir before springing over the weir’s far edge. Figure 1 shows a weir cross-section typical of the ones used in this experiment.

Figure 1: Section view of sharp-crested weir typical of both weirs used in experiment



Experiment Objectives

This report details an experiment to determine flowrates and an empirical discharge coefficient for two different types of sharp-crested weirs:

1. A rectangular weir

2. A V-notch weir



Using the boundary conditions of open channels, a theoretical equation for discharge (Q) over weirs can be developed using the Bernoulli equation (to be discussed in more detail later). However, actual flow over a weir is affected by both viscous effects and the convergence of streamlines downstream of the plane of the weir (similar to the streamlines of water being poured from a pitcher held high in the air). Therefore, a discharge coefficient (Cd) is found using empirical methods to account for these effects . The experimental data collected were used to determine and calibrate a Cd appropriate to each of the weirs used.





THEORETICAL EQUATIONS



Discharge can be found theoretically for weirs using Bernoulli’s equation between a point upstream of the weir - where the approach velocity is negligible and the pressure is atmospheric – and a point taken as the flow reaches the plane of the weir (see Figure 2).



Figure 2: Rectangular weir discharge parameters





Bernoulli’s equation between 1 and 2 can be expressed as:



H = (H – h) + V2/2g (EQ 1)



This equation can be simplified to yield velocity as:



V = (2gh)1/2 (EQ 2)



Integrating the velocity over the width of the weir notch (B) and the differential head yields a theoretical discharge (Q).



Q = ∫ (2gh)1/2*B dh (EQ 3)

Evaluating from zero to H yields:



Q = 2/3*B*(2g)1/2*H3/2 (EQ 4)



In order to approximate actual flow conditions the discharge coefficient is introduced:



Q = 2/3*Cd*B*(2g)1/2*H3/2 (EQ 5)





The theoretical expression is modified for the V-notch weir to account for the fact that the notch width (B) varies with h. Notch width is expressed then in terms of similar triangles and notch angle θ (see Figure 3).

Figure 3: V-notch weir discharge parameters





The expression to be integrated then becomes:



Q = ∫ (2gh)1/2*(H – h)*2tan(θ/2) dh (EQ 6)



Evaluating between zero and H yields:



Q = 8/15*(2g)1/2*tan(θ/2)*H5/2 (EQ 7)



Applying the empirical discharge coefficient for actual flow yields:



Q = 8/15*Cd*(2g)1/2*tan(θ/2)*H5/2 (EQ 8)







PROCEDURE



To find the discharge coefficients for each weir, five sets of data, based on five different flowrates, were taken recording time, head above weir notch, and volume of overflow. For each flowrate examined, set with the bench’s pump control valve, time was recorded for 3 liters of water to flow out of the channel. An actual flowrate (Qact) was computed by dividing the 3 L volume by the time recorded. Using this value, a discharge coefficient (Cd) was computed for each flowrate examined. These values were averaged to determine a single Cd to be used for the flowrate equation appropriate to each weir.



This experiment was conducting using the apparatus found on the mobile “hydraulics bench” owned by the City College Hydraulics Laboratory. The bench employs a centrifugal pump to provide flow into a rectangular open channel, and later recycle the water through the system. A stilling baffle was used close to the channel’s inlet nozzle to provide smooth flow in the channel (by mitigating the effects of water flow exiting the inlet and striking the back and sides of the channel). Each weir plate was locked into place normal to the direction of the channel’s flow.



Critical to this experiment was the measured head (H) from the bottom of the weir’s notch to the top of the water level, prior to the effects of any drawdown close to the plane of the weir. This measurement was performed with a Vernier hook and point gauge. At the time each weir was placed, the coarse adjustment screw was set to a water level just imminent to flow over the weir edge, and set as zero. The fine adjustment gauge was then used to measure the head above that point for the subsequent flows examined. All measurements were taken only after the flow was determined visually as stable. Care was taken when measuring volume to measure from the same portion of the bench’s scale to ensure consistency if the data.







DISCUSSION OF RESULTS



The average coefficient of discharge terms computed for the rectangular and V-notch weirs used are listed below in Table 1:



Table 1: Cd computed for each weir using experimental data















These coefficients are only useful as adjustment factors if there exists a direct relationship between Q and H. This relationship can be expressed as:



Q = kHn (EQ 9)



Where k is a constant, and n is an exponent based on the weir’s geometry. Strong evidence of this exponential relationship was found by plotting both Q1/n vs. H, and log Q vs. log H in Figures 4 through 7. (Note: all tabulated data and computations can be found in the appendix)



























Figure 4: Q1/n vs. H, Rectangular Weir Figure 5: Q1/n vs. H, V-notch Weir



























Figure 6: Log Q vs. Log H, Rectangular Weir Figure 7: Log Q vs. Log H, V-notch Weir



Figures 4-7 demonstrate exponential relationship of Q and H. The R2 values confirm the correlation of the data collected, and verify the performance of the discharge equations.







The most interesting analysis comes from plotting flowrates calculated by using the average Cd and actual flowrates against head (see Figures 8 and 9), and comparing the error between the actual Q and computed Q values.

























Figures 8 & 9: Variation of actual Q with Q calculated with average Cd



Errors between actual and computed flowrates are presented in Table 4. The trend in both graphs and table show an increase in error as the head increases. For both cases, the calculated Q predicts a higher value than the actual Q for greater values of H.





Table 2: Error between actual Q and computed Q for both weirs





































Examining Cd vs. H can explain some of this variation. Figures 10 and 11 show the tendency for the coefficient of discharge (as computed for each data set) to vary with head. Given that this experiment is to determine an average discharge coefficient, an average will provide greater error for some calculated Q values than others. This begs the question: how much error is acceptable?

























Figures 10 & 11: Variation of Cd with H for rectangular and V-notch weirs



It should be noted that typically a V-notch weir will yield more accurate flowrates than rectangular weirs . It was initially thought that the Cd developed for the V-notch weir would be a more sensitive predictor and have less error, which is not the case here. Though perhaps the higher error could be a function of the V-notch geometry, there may be a simple physical explanation. Volume measurements were taken at the same gauge levels to provide consistency in the data, but it was observed that jumps in the volumetric gauge levels occurred while recording the time for the V-notch weir. It was thought at the time that there might have been some “sloshing” in the collection tank, which may be responsible for some error in the data collection. It may have been more effective to record the 3 liters over time at a higher point on the volumetric scale (i.e. between 5 and 8 liters instead of 0 and 3 liters).









CONCLUSION



The experiment performed gives insight into the use of discharge coefficients in computing flowrate over weirs. Because values of Cd vary with changing H and the increase in error for greater values of H, 2 ways of increasing the predictive accuracy of Cd may be considered for future experiments:

1. Imposing a lower and upper limit on the magnitudes of flowrates examined/expected.

2. Collecting a greater quantity of data for a more accurate average discharge coefficient value.



Although there is a degree of error inherent in empirically developing a general Cd to be used for a specific weir and its geometry, it still provides a good predictive tool over limited ranges. For the Hydraulic Engineer the question is, what degree of accuracy is necessary? For example: if these computations are to determine how much energy needs to be dissipated downstream of a weir, the coefficients found in this lab would provide a higher predicted flowrate, and a more conservative design.











LITERATURE CITED



1. L. Urquhart, Civil Engineers Handbook, (McGraw-Hill Book Company, 1959)

2. N. Hwang and R. Houghtalen, Fundamentals of Hydraulic Engineering, (Prentice Hall 3rd Edition, 1996)

3. X. Chanson, The Hydraulics of Open Channel Flow (Publisher, City, Year)

4. C. Crowe, D. Elger, and J. Roberson, Engineering Fluid Mechanics, (John Wiley and Sons, Inc. 7th Edition, 2001)

5. K. Edwards, V-Notch (Triangular) Weirs, available at http://www.lmnoeng.com/Weirs/vweir.htm







APPENDIX

A – Experimental Data





















B – Computed Actual Flowrates and Coefficients of Discharge























C – Computed Excel Values

Rob x-7

Get Brett Weir I said

drftk1d

when things go wrong, blame it on the engineer

94touring

As an ex-engineer, I don't care.

teknics

my dads an engineer. but not that kind.



kevin.

Cheers!

Haha, that is what came to my mind as well.



Who is JESSICA BLACK btw?

l8t apex

So is the point to determine a viable error acceptance or was it determine that there is one which in itself is a dynamic constant in each model.

What the objective? To confirm the variable or to reduce it by the average.

Juct curious.