Fundamentals of Hydrology - Speech or Presentation Example

Task Carry out an investigation to establish the relationship between depth of flow and discharge rate of water over both a “V” and a rectangular notch weir. You are to compare the discharge results against the given theoretical formulae. a). Determine the theoretical flow rate for each reading. For the V-notch weir: The formula used in the computation of the theoretical flow rate is: Q = 8/15 (2g) tan  .H5/2 where: θ = 45 Source Height Theoretical Flow H Q (m3/s) Q (l/s) Al-Dsri - Alhri 0.0161 0.00013 0.12585 0.0205 0.00023 0.23024 0.0226 0.00029 0.29382 0.0291 0.00055 0.55276 0.0381 0.00108 1.08422 Brbry - Gry 0.0168 0.00014 0.13998 0.0195 0.0002 0.20318 0.0241 0.00035 0.34502 0.0283 0.00052 0.51555 0.0342 0.00083 0.82769 Hon - Zag 0.0152 0.00011 0.10900 0.0181 0.00017 0.16866 0.0228 0.0003 0.30036 0.0295 0.00057 0.57195 0.0398 0.00121 1.20923 zop - kos 0.0148 0.0001 0.10197 0.0173 0.00015 0.15063 0.0233 0.00032 0.31710 0.0352 0.00089 0.88953 0.0387 0.00113 1.12741 For the rectangular weir: The formula used in the computation of the theoretical flow rate is: Q = 2/3B (2g) H3/2 where: B = 0.030 m; g=9.81 m s2 Source Height Theoretical Flow H Q (m3/s) Q (l/s) Al-Dsri - Alhri 0.0161 0.00018 0.18097 0.0205 0.00026 0.26002 0.0226 0.00030 0.30098 0.0291 0.00044 0.43976 0.0381 0.00066 0.65882 Brbry - Gry 0.0168 0.00019 0.19290 0.0195 0.00024 0.24123 0.0241 0.00033 0.33144 0.0283 0.00042 0.42175 0.0342 0.00056 0.56030 Hon - Zag 0.0152 0.00017 0.16601 0.0181 0.00022 0.21572 0.0228 0.00030 0.30499 0.0295 0.00045 0.44886 0.0398 0.00070 0.70340 zop - kos 0.0148 0.00016 0.15950 0.0173 0.00020 0.20158 0.0233 0.00032 0.31507 0.0352 0.00059 0.58505 0.0387 0.00067 0.67444 b). Plot a graph of Q vertical against H horizontal for both observed and theoretical values of flow rate and comment on these graphs. For the V-notch weir: Figures 1 and 2 present a plot of the theoretical flow rate and the observed flow rate, respectively, against the depth of flow, when the flow rate is in liters/sec. Similarly, Figures 3 and 4 show the same data when the flow rate is expressed in cubic meters per second. Figure 1. Plot of the theoretical flow rate (l/s) against depth of flow (m) Figure 2. Plot of the observed flow rate (l/s) against depth of flow (m) It may be observed from Figures 1 and 2 that the plot of flow rate against depth of flow is a smoother curve with the theoretical flow rate in l/s, as compared to the curve with the observed flow rate. The same observation was noted in Figures 3 and 4 when the flow rates are expressed in m3/s. The curves in Figures 1 and 3 are smoother and the flow rates tend to increase as the depth of flow increases. The curves in Figures 2 and 4 have slight outliers from the typical pattern of the curve. However, like in Figures 1 and 3, the flow rates also tend to increase with the depth of flow. Figure 3. Plot of the theoretical flow rate (m3/s) against depth of flow (m) Figure 4. Plot of the observed flow rate (m3/s) against depth of flow (m) For the rectangulat weir: Figures 5 and 6 present a plot of the theoretical flow rate and the observed flow rate, respectively, against the depth of flow, when the flow rate is in liters/sec. Similarly, Figures 7 and 8 show the same data when the flow rate is expressed in cubic meters per second. Figure 5. Plot of the theoretical flow rate (l/s) against depth of flow (m) It will be noted that Figure 6 is exactly the same as Figure 2, since the same observed values were used for both the rectangular weir and the V-notch weir. It was observed from Figures 5 and 6 that the plot of flow rate against depth of flow in the rectangular weir is a smoother curve with the theoretical flow rate in l/s, as compared to the curve with the observed flow rate, and that the flow rates tend to increase with the depth of flow. The same observation was noted in Figures 7 and 8 when the flow rates are expressed in m3/s. The curves in Figures 7 and 8 are smoother and the flow rates tend to increase as the depth of flow increases. The curves in Figures 6 and 8 have slight outliers from the typical pattern of the curve. Figure 6. Plot of the observed flow rate (l/s) against depth of flow (m) Figure 7. Plot of the theoretical flow rate (m3/s) against depth of flow (m) Figure 8. Plot of the observed flow rate (m3/s) against depth of flow (m) c). Plot a graph of log Q vertical against log H horizontal, and obtain the gradient of the best straight line of fit (estimated by eye). Comment on this value compared to the theoretical value expected. For the V-notch weir: Figure 9 presents the plot of the theoretical flow rate in liters per second against the depth of flow. As shown the plot almost defines a straight line, with a slope of the gradient line at about 49. Figure 9. Plot of the log of theoretical flow rate (l/s) against the log of the depth of flow (m) Figure 10 presents the plot of the observed flow rate in liters per second against the depth of flow. The best straight line of fit labelled as the gradient line was drawn. The slope of the gradient line is approximately 52. Figure 10. Plot of the log of observed flow rate (l/s) against the log of the depth of flow (m) Figure 11 presents the plot of the theoretical flow rate in cubic meters per second against the depth of flow. As shown the plot almost defines a straight line, with a slope of the gradient line at about 20. Figure 11. Plot of the log of theoretical flow rate (m3/s) against the log of depth of flow (m) Figure 12 presents the plot of the observed flow rate in cubic meters per second against the depth of flow. The best straight line of fit labelled as the gradient line was drawn. The slope of the gradient line is approximately more than 20. Figure 12. Plot of the log of observed flow rate (m3/s) against the log of depth of flow (m) For the rectangulat weir: Figure 13. Plot of the log of theoretical flow rate (l/s) against the log of the depth of flow (m) Figure 13 presents the plot of the theoretical flow rate in liters per second against the depth of flow. As shown the plot almost defines a straight line, with a slope of the gradient line at about 48. Figure 14. Plot of the log of observed flow rate (l/s) against the log of the depth of flow (m) Figure 14 presents the plot of the observed flow rate in liters per second against the depth of flow. The best straight line of fit labelled as the gradient line was drawn. The slope of the gradient line is approximately 47. Figure 15. Plot of the log of theoretical flow rate (m3/s) against the log of depth of flow (m) Figure 15 presents the plot of the theoretical flow rate in cubic meters per second against the depth of flow. As shown the plot almost defines a straight line, with a slope of the gradient line at about 14. Figure 16. Plot of the log of observed flow rate (m3/s) against the log of depth of flow (m) Figure 16 presents the plot of the observed flow rate in cubic meters per second against the depth of flow. The best straight line of fit labelled as the gradient line was drawn. The slope of the gradient line is approximately 24. Based on the plots, it may be concluded that flow rate in both V-notch and rectangular weirs tend to increase as the weir head increases, which suggests a linear relationship between flow rate and weir head. d). Define the term “coefficient of discharge” and calculate the values indicated by the data. The coefficient of discharge (c) may as defined in King and Wisler (2008) is the ratio between the observed flow rate or discharge (Qo) and the theoretical flow rate (Qt). By formula: c = Qo Qt For the V-notch weir: Following are the data used in the computation (Qo and Qt) and the calculated values of the coefficient of discharge for the V-notch weir. Source Theoretical Flow Observed Flow Coefficient of Discharge Q (m3/s) Q (l/s) Q (m3/s) Q (l/s) c (Q in m3/s) c (Q in l/s) Al-Dsri - Alhri 0.00013 0.12585 6.5E-05 0.065 0.51663 0.51663 0.00023 0.23024 0.000124 0.12438 0.5402 0.5402 0.00029 0.29382 0.00016 0.16000 0.54456 0.54456 0.00055 0.55276 0.000275 0.27473 0.49701 0.49701 0.00108 1.08422 0.000562 0.56180 0.51816 0.51816 Brbry - Gry 0.00014 0.13998 7.18E-05 0.07179 0.51283 0.51283 0.0002 0.20318 0.000108 0.10834 0.53322 0.53322 0.00035 0.34502 0.000175 0.17452 0.50582 0.50582 0.00052 0.51555 0.000284 0.28409 0.55105 0.55105 0.00083 0.82769 0.00042 0.42017 0.50764 0.50764 Hon - Zag 0.00011 0.10900 5.71E-05 0.05708 0.52367 0.52367 0.00017 0.16866 9.07E-05 0.09074 0.53805 0.53805 0.0003 0.30036 0.000165 0.16502 0.5494 0.5494 0.00057 0.57195 0.000302 0.30211 0.52822 0.52822 0.00121 1.20923 0.000658 0.65789 0.54406 0.54406 zop - kos 0.0001 0.10197 5.55E-05 0.05549 0.54424 0.54424 0.00015 0.15063 8.18E-05 0.08177 0.54282 0.54282 0.00032 0.31710 0.000167 0.16722 0.52736 0.52736 0.00089 0.88953 0.000474 0.47393 0.53279 0.53279 0.00113 1.12741 0.000613 0.61350 0.54417 0.54417 It was observed from the calculated values of the coefficient of discharge for a 45 V-notch weir is smaller than the standard values specified in Davie (2002) which is supposed to be greater than 0.575 and less than 0.580. For the rectangulat weir: Source Theoretical Flow Observed Flow Coefficient of Discharge Q (m3/s) Q (l/s) Q (m3/s) Q (l/s) c (Q in m3/s) c (Q in l/s) Al-Dsri - Alhri 0.00018 0.18097 6.5E-05 0.065 0.35927 0.35927 0.00026 0.26002 0.000124 0.12438 0.47834 0.47834 0.00030 0.30098 0.00016 0.16000 0.53159 0.53159 0.00044 0.43976 0.000275 0.27473 0.62471 0.62471 0.00066 0.65882 0.000562 0.56180 0.85273 0.85273 Brbry - Gry 0.00019 0.19290 7.18E-05 0.07179 0.37214 0.37214 0.00024 0.24123 0.000108 0.10834 0.44912 0.44912 0.00033 0.33144 0.000175 0.17452 0.52655 0.52655 0.00042 0.42175 0.000284 0.28409 0.67359 0.67359 0.00056 0.56030 0.00042 0.42017 0.7499 0.7499 Hon - Zag 0.00017 0.16601 5.71E-05 0.05708 0.34381 0.34381 0.00022 0.21572 9.07E-05 0.09074 0.42065 0.42065 0.00030 0.30499 0.000165 0.16502 0.54106 0.54106 0.00045 0.44886 0.000302 0.30211 0.67307 0.67307 0.00070 0.70340 0.000658 0.65789 0.9353 0.9353 zop - kos 0.00016 0.15950 5.55E-05 0.05549 0.34791 0.34791 0.00020 0.20158 8.18E-05 0.08177 0.40563 0.40563 0.00032 0.31507 0.000167 0.16722 0.53074 0.53074 0.00059 0.58505 0.000474 0.47393 0.81007 0.81007 0.00067 0.67444 0.000613 0.61350 0.90963 0.90963 Typical values of the coefficient of discharge for a rectangular range from 0.57 to 0.62 (Australian Standards AS 3778.4.1-1991). Hence, only one value of the coefficient fall within the standard range (value was rendered in bold font). The rest of the values either fall below or above the range. Task 2. Using the flow visualisation apparatus investigate the effect on the fluid flow of various weirs or obstructions placed in the channel. Comment on the characteristics (weir shape, type and strength of hydraulic jump, critical and sub-critical flow) of flow over weirs being illustrated in each case relating your observations to theoretical expectations It was observed that an increase in channel slope increases the flow velocity in both the V-notch and rectangular weirs and this resulted in the reduction of the depth of flow. Such reduction in the depth of flaw is called hydraulic drop by Charlton (2008). It was also observed in both the V-notch and the rectangular weirs that at the base, a breaking wave occurred. In theory, this is termed as the hydraulic jump, and it is the breaking waves which gives the observer the cue for such a transition where the flow changes back to subcritical. Subramanya (2009) explained that theoretically, “a hydraulic jump occurs when a supercritical stream meets a subcritical stream of sufficient depth” (p. 248). References Australian Standards (1991). AS 3778.4.1-1991. Retrieved, November 11, 2010, from: http://services.eng.uts.edu.au/~phuoc/docs/fluid/ WeirFlowMeasurement.pdf Charlton, R. (2008). Fundamentals of fluvial geomorphology. Oxon, UK: Routledge. Davie, T. (2002). Fundamentals of hydrology (2nd ed.). Oxon, UK: Routledge.. King, H. W. & Wisler, C. O. (2008). Hydraulics. Charlston, SC: BiblioBazaar. Subramanya, K. (2009). Flow in open channels (3rd ed.). New Delhi, IND: Tata McGraw-Hill Publishing. Read More