Flow over Sharp-Crested Weirs - Research Paper Example

EXPERIMENT 2: FLOW OVER SHARP-CRESTED WEIRS EXPERIMENT 3: FLOW THROUGH AN ORIFICE Name Professor Institution Course Date Experiment 2 - Flow over sharp-crested weirs Aims The performance of the flow over sharp crested weirs experiment was aimed at achieving the following aims; To determine the fundamental properties of a two sharp crested weir To obtain the discharge coefficient in both rectangular and rectangular weirs. To perform the examination of the nature and action of pressure forces around as well as on the rectangular weirs Theory Measurement of fluid flow has been historically been performed with the use weir device (John, 2000). Sharp crested consist of a top structure that is flat and short on the upper edge. This is close to bevel which is found on the lower side. In this types of weirs, the fluid flowing changes into an overflowing water sheet which is known as a nappe. In a sharp-crested weir, the action of water clinging on the lower side of the stream is experienced in the cases where the discharge in the plate of the weir is regarded as unpredictable. The sharp crest weirs are perceived as standing objections that are placed perpendicularly to the direction of flow of water and therefore the water passage on top of the weir is seen. The lower stream edge of weir is made to incline at an angle of 45° or 60°. The weirs that contain sharp crests are not only used as devices for measurement of flow channels that are open but they are also employed in the use in very simple spillway forms. This means that the spillway form is determined in a manner that it conforms to the flow nape shape through a weir that is sharp crested. The classification of the sharp crested weir can be done in accordance with several factors in their geometry into three categories. The first category is the rectangular top sharp crested weir, the second category is the triangular or the V weir and the third category is the one that consists of special geometry of weirs such as the parabolic circular or trapezoidal weirs. In order to achieve the result that is accurate and exact in the measurement of discharge, the measurement of the discharge should be done during the aeration time of the weir. The means that pressure experienced on both the lower and the upper sides are expected to match with that of the atmospheric condition. The differing aspects of both the non-aerated and the aerated weirs become much clearer when the pressure analysis on both the upper and lower sides are done. In a flow that is non-aerated, the clinging of the fluid to the weir happens on the lower side and the level of the water height stays at lowers than for a general case analysis. The action of water clinging on the lower part of the weir implies the occurrence of a non-aerated flow. Whenever this takes place the head of water that is below the mark level is not taken into consideration. The discharge expression is also considered very critical in the analysis of the flow through the sharp crested weirs. The rectangular theoretical discharge for weir that are sharp crested is obtained through assumption of parallel, frictionless and horizontal flow that does not experience any losses. Figure 1: Some of the types of the sharp crested weirs The theoretical profile of fluid flow over sharp crested weir is dependent on springing of nappe of the weir plate in a clear manner as demonstrated by the diagram below; Figure 2: sectional view of water flow over a sharp-crested weir. The Bernoulli expression is applied at several points of the weir profile such as indicated above and this is expressed as; P1/ρg + V12/2g + z1 = p2/ρg + V12/2g + z2................................................................(I) The Bernoullis exprerssion in equation (I) assumes a different form when an atmopheric stage assumption is made and so th expression obtained is as follows 0 + 0 + H = 0 + V12/2g + (H – y)……………………………………….………... (II) Since; p1= 0, p2 = 0 and V1 = 0 Following taking into consideration of a strip with a narrow, where thickness of the strip is denoted by dy and its length is denoted by L the velocity at the second point denoted as pint 2 is obtained as; V2 = √2gy……………………………………………..……..………………..… (III) When the strip thickness as well as its length is taken into consideration and the flow discharge is introduced, the expression takes a different form and becomes; Q = VA = VLdy = (√2gy)Ldy………………………………………………….. (IV) Where Q is the discharge In order to come up with the overall discharge, expression (IV) which represents the elemental discharge is subjected to integration. Therefore integrating the elemental discharge expression results in the overall discharge expressed as; Q = 2/3 (√2g) LH3/2…………………………..………………………….………. (V) When the overall discharge expression has been obtained, another aspect of the discharge is introduced into the expression and this is the aspect of the coefficient of discharge. The expression therefore takes another form and is now expressed as; Q = Cd 2/3 (√2g) LH2/3........................................................................................... (VI) Where; Cd is the representation that denotes the coefficient of discharge. For the case of V-shaped notch type of the crested weir the losses on the fluid flow through it are determined by taking into account the coefficient of discharge among other factors as is shown through the use of the equation below; Q = Cd 8/15(√2g) tan θ /2H5/2………………………………………………….. (VII) Where θ is the v angle of notch Apparatus This experiment involving the analysis of various types of sharp crested weirs as well as the fluid flow over sharp crested weirs employed the use the following apparatus; An hydraulic bench A stop watch Basic weir apparatus Point and hook gauge Method This experimental performance employed the method for use that involved taking into consideration the Flow over sharp-crested weirs. The experimental method consisted of carrying out the operations in a sequential manner that included several steps carried out in a particular manner and order. The experiment commenced with the installation of the experimental setup. This was put in place through the involvement of making the flume bed to be maintained at a horizontal level. The location of the datum was then performed on the top side of the sharp crested weir. This was followed by the determination of the head of water in the sharp crested weir before locating the zero mark again for the v-notch type o sharp crested weir. The surge tank valve was then released and the pump flow control valve value was closed and the pump and the pump were made to start. Pouring of the fluid which was water into the weir up to the top level was done. This was so as to maintain the level of water at a constant flow channel new position. The process of topping up the weir was carefully done performed in order to avoid flooding beyond the desired level. The steady flow of the whole circuit was then developed and maintained as the experimental process continued. Water head levels were then measured at a water head mark level that was beyond the top part of the weir. The values obtained from this measurement were used in the experimental data evaluation and analysis. This was then followed by observation and recording of the measurements of the flow rate with the use of a stop watch and a graduated measuring cylinder. The water level was then measurement was then taken in the approach channel using a gauge at a particular given position. This was followed by decreasing the water level in the channel of approach to about half of its initial water level where the differences between the water levels in the channel were calculated and recorded at each reduction with the use of a gauge. The v notch sharp crested weir was then connected to the flow stream where the angle of the notch that was used was 45o. The whole procedure was then performed again and the appropriate values optimally calculated and recoded for analysis. Results Following the carrying out of the final experimental steps, the data values were collected, recorded and presented in a tabulated forma as shown as shown below; Dimensions Rectangular weir V-notch weir Length, L(m) 0.03 m Angle of V-notch 90o Datum (m) 0.065 Datum (m) 0.04 Results on both Rectangular and V-notch weir Results of total flow rate and losses in the weirs For the rectangular weir For the V-notch weir Flow rate (m3/s) Q = V/t/1000 = 0.04/30.19/1000 Q= 1.32 x 10-3 m3/s Q = V/t/1000 (m3/s) = 0.02/40.35/1000 Q= 4.957x 10-4 m3/s Coefficient of discharge Q = Cd ()LH3/2 Cd = = 0.592 Q = Cd () tanLH5/2 = = 0.6340 Cd 0.332 0.753 Discussion For the rectangular type of the sharp crested fluid flow weir, it was obtained from the evaluation of the results collected the discharge was maintained at an almost constant level. This suggested the rectangular type of the sharp crested weirs allowed for passage of larger volumes of fluids though it. For the v-notch weir it was seen that the coefficient of discharge was not maintained at a constant value in which case the value of the mean discharge coefficient value was obtained and recoded as 0.7762. The analysis and evaluation of the result data values obtained from the experiment also took into consideration the errors that were encounter. This was computed as shown; For an error of +0.5 litres Cd = 0.6905 percentage error = 1.36 For an error of +0.5 seconds Cd = 0.6755 Percentage error = 0.84 For an error of +0.5 mm Cd = 0.6599 Percentage error = 3.22 Conclusion It was also concluded that for the sharp crested weirs that had openings that contained water heads with heights below 2cm were considered or regarded as the non-aerated weirs and that the that particular water head was considered to the minimum below which the head was regarded as negligible. It was established that experiments involving water profile surfaces were carried out for purposes of achieving the determination of the appropriate measurement of the point gauge location and this then lead to the conclusion that the lower drawn effect of weirs that are sharp crested are to be neglected. Another conclusion that was obtained was that for sharp crested weirs that are rectangular in nature, were to taken in to consideration as two distinct parts. The two parts included the contracted weir and the silt weir. From the experimental collected data was determined that that the discharge expression should also be able to accommodate fluids with large discharge values. Otherwise the experiment was successfully performed and the attainment of the of the experimental objectives also realized which involved the determination as well as the comparison of between the characteristics and features of operation of both the rectangular and the v notch types of sharp crested weirs. Experiment 3-Flow through an orifice Aim The aim of the experiment was to determine the coefficient of contraction, the discharge through the orifice as well as the velocity with which the fluids pass the orifice for a particular given fluid whereby in this case the fluid under consideration was water. Theory Flow meter finds several uses in industrial applications to assist in the measurement of volumetric flow rate on several fluids in the industries (Alexander, 2000). Differences in the pressure types have led to the introduction of head flow meters that are purposely for the measurement of the rate of flow by way of introduction of a constriction in the course of the flow. The difference in pressure that results from the constriction is related the flow rate of the fluid passing through it and is described by the use of the Bernoulli’s theorem. If a pipe that is carrying a fluid flow stream is subjected to constriction at any one point along its length, an increase in velocity is observed and therefore the kinetic energy also increases at that point where constriction is made to happen. In accordance to the Bernoulli’s expression with relation to energy balance the has to be a corresponding pressure reduction whenever the pipe is subjected to constriction at any particular point and therefore is possible to calculated thee discharge rate from the point of construction when the cress sectional area for the constriction is and the pressure at that point are known. Other parameters that also need to be known are the discharge coefficient as well as the density of the fluid. The coefficient of discharge for a fluid is referred to as the ratio of the actual fluid flow to the theoretical value of the fluid flow. The coefficient of discharge creates allowance for the effects of friction as well as contraction of the stream. The head flow meters that are in wide usage include the Pitot tube the orifice meter as well as the venture meter. However the orifice meter and the venture meters are have got frequent applications due to the fact that they more convenient for use. The venture meter finds a wide application of usage in particular in fluids involving large volumes flow such as gasses and liquids since it experiences low pressure losses. The discharge determination, and the velocity determination as well as coefficient of contraction determination employ the use of Bernoulli expression where the determination of the flow through an orifice can be obtained as follows; ……………………… ………………… (I) Reynolds number which has got no dimensions is then obtained for use in the determination of the mechanic features as well as behaviors of the fluid as shown; …………………………………………………………………. (II) Where, is the fluid density (Kg/m3) is the fluid velocity (m/s) is the cross sectional diameter (m) = is the dynamic viscosity of the fluid Figure 1: sketch of the orifice installation Apparatus Stop watch Measuring cylinder Thermometer Bench of hydraulics Water tank having an orifice Marking pen (non-permanent), graph paper and a clean transparent sheet Method During the commencement of the experiment, a setup consisting of a hydraulic bench was properly installed. The installation of this set up was performed with a lot of care to ensure that instrumental errors were avoided as much as possible. The careful installation was achieved by way of directing all the overflowing fluid to a pump and then making proper use of the well calibrated and visible instruments of measurement. The observation, taking of data values and recording of the temperature of water and the measurement of the orifice diameter were performed. The path of the fluid jet was carefully observed and its measurements carefully recorded by way of placing a graph paper at the underside of the board which helped in determining indicating the origin of the distances of the trajectories. The trajectory distances were established as paths that were then recorded in terms of their lengths. The measurement and recording processes were then brought to stoppage before the above procedure was entirely done once again and another set of values of data observed, measured and recorded. This was followed by the process of sampling of the recoded values where the sampled optimally correct values were presented as the results of the experiment. Results The experimental procedural steps were completed and a set of data values obtained and presented in a tabulated format as shown; Temperature of water: 21 degrees Celsius Orifice diameter: 6 mm Range of the head values obtained: between 270 mm and 240 mm Head H (mm) Volume Water (L) Time T (sec) Q (L/s) 350 0.420 30 0.014 310 0.390 30 0.013 270 0.360 30 0.012 Trajectories Head H (mm) X1 (mm) Y1 (mm) X2 (mm) Y2 (mm) X3 (mm) Y3 (mm) X4 (mm) Y4 (mm) 350 50 9 100 15 150 26 200 40 310 50 10 100 17 150 27 200 42 270 50 10 100 20 150 31 200 48 Head H (mm) X5 (mm) Y5 (mm) X6 (mm) Y6 (mm) X7 (mm) Y7 (mm) X8 (mm) Y8 (mm) 350 250 58 300 79 350 105 400 135 310 250 61 300 85 350 115 400 149 270 250 70 300 100 350 130 400 170 Head (mm) Cv Cd Cc Actual Velocity (m/s) Reynolds Number 350 0.776859 0.698766 0.683429 2.123772 9168.88 310 0.780047 0.716111 0.695007 1.953547 8414.26 270 0.797041 0.709887 0.7309 1.799265 7584.51 Discussion It was determined from the analysis of the results obtained that the relationship between the aspects as well as factors that assisted in the establishment of the properties and also the behavior of the of the fluid which in this case was taken to be water largely depended of the surface of passage of that particular fluid. It was also found that a relationship existed between the fluid properties on one side and the velocity as well as the coefficient of discharge exhibited contraction or constriction through an orifice on the other side. This relationship was further described through the use of the Reynolds number whereby it was observed and noted that there was an increasing trend in the amount of discharge as he Re number was kept at constant. During the experimental evaluation and analysis of the results and observations noted it was established that all theoretical as well as actual velocity values were related to each other the relation of the Re number. The Re number was computed as; Reynold Number = ………………………………………………….… (III) Re_380=9168.88 Re_320 = 8414.26 Re_260 = 7584.5 The following expressions were applied in the determination of both theoretical as well as the actual values of velocities. He computations were done using the values obtained from the results and the results tabulated as shown below; V. actual=……………………………………………………….….. (IV) V. theoretical= ……………………………………………………….. (V) Analysis results Head X1 Y1 X2 Y2 X3 Y3 X4 Y4 350 50 11 100 19 150 28.5 200 30 Velocity 1.055 1.605 1.96683 2.566 Head X5 Y5 X6 Y6 X7 Y7 X8 Y8 350 250 59 300 71 350 100 400 123 Velocity 2.278 2.4922 2.45 2.52467 Average 2.1172 310 50 12 100 19 150 30 200 42 Velocity 1.0103629 1.6059 1.917028 2.16024 310 250 62 300 88 350 118 400 149 Velocity 2.2225 2.2386 1.78684 2.29384 V. average 1.9044             V. theoretical 2.5044 270 50 14 100 22 150 36 200 54 Velocity 0.93541 1.4924 1.75 1.9051 270 250 75` 300 102 350 140 400 170 Velocity 2.02007 2.0793 2.0706 2.1475 Average 1.8001               Theoretical velocity 2.25743 The properties of water at 13 degrees Celsius = 0.8903E-3 Pa*s = 999.1 kg m-3 Area of orifice , where r = 0.003m Conclusion AS was learnt from the analysis and evaluation of the experimental results, the experiment was a successful one since the objectives of the experiment were to a larger extent realized. However, this was not without experimental challenges which ranged unfavorable experimental conditions to errors that were encountered especially in the process of observation and measurements of various quantities. References Alexander, JS 2000, A Physical Introduction to Fluid Mechanics, John Wiley, New York. John, JB 2000, Practical Fluid Mechanics for Engineering Applications, Marcel Dekker Incorporated, New Jersey. Nicholas, PC 1990, Practical Fluid Mechanics for Engineers and Scientists, Technomic Pub. Co., London. Read More