not done; and such a bellmouth may also have the form of the contracted jet. Fig. 19 shows the proportions of such a bellmouth or bell-mouthed orifice, which approximates to the form of the contracted jet sufficiently for any practical purpose.
For such an orifice L. J. Weisbach found the following values of the coefficients with different heads.
| Head over orifice, in ft. = h | ·66 | 1·64 | 11·48 | 55·77 | 337·93 |
| Coefficient of velocity = cv | ·959 | ·967 | ·975 | ·994 | ·994 |
| Coefficient of resistance = cr | ·087 | ·069 | ·052 | ·012 | ·012 |
As there is no contraction after the jet issues from the orifice, cc = 1, c = cv; and therefore
§ 20. Coefficients for Sharp-edged or virtually Sharp-edged Orifices.—There are a very large number of measurements of discharge from sharp-edged orifices under different conditions of head. An account of these and a very careful tabulation of the average values of the coefficients will be found in the Hydraulics of the late Hamilton Smith (Wiley & Sons, New York, 1886). The following short table abstracted from a larger one will give a fair notion of how the coefficient varies according to the most trustworthy of the experiments.
Coefficient of Discharge for Vertical Circular Orifices, Sharp-edged,
with free Discharge into the Air. Q = cω √ (2gh).
| Head measured to Centre of Orifice. | Diameters of Orifice. | ||||||
| ·02 | ·04 | ·10 | ·20 | ·40 | ·60 | 1·0 | |
| Values of C. | |||||||
| 0·3 | .. | .. | ·621 | .. | .. | .. | .. |
| 0·4 | .. | ·637 | ·618 | .. | .. | .. | .. |
| 0·6 | ·655 | ·630 | ·613 | ·601 | ·596 | ·588 | .. |
| 0·8 | ·648 | ·626 | ·610 | ·601 | ·597 | ·594 | ·583 |
| 1·0 | ·644 | ·623 | ·608 | ·600 | ·598 | ·595 | ·591 |
| 2·0 | ·632 | ·614 | ·604 | ·599 | ·599 | ·597 | ·595 |
| 4·0 | ·623 | ·609 | ·602 | ·599 | ·598 | ·597 | ·596 |
| 8·0 | ·614 | ·605 | ·600 | ·598 | ·597 | ·596 | ·596 |
| 20·0 | ·601 | ·599 | ·596 | ·596 | ·596 | ·596 | ·594 |
At the same time it must be observed that differences of sharpness in the edge of the orifice and some other circumstances affect the results, so that the values found by different careful experimenters are not a little discrepant. When exact measurement of flow has to be made by a sharp-edged orifice it is desirable that the coefficient for the particular orifice should be directly determined.
The following results were obtained by Dr H. T. Bovey in the laboratory of McGill University.
Coefficient of Discharge for Sharp-edged Orifices.
| Head in ft. | Form of Orifice. | |||||||
| Circular. | Square. | Rectangular Ratio of Sides 4:1 | Rectangular Ratio of Sides 16:1 | Tri- angular. | ||||
| Sides Vertical. | Diagonal vertical. | Long Sides vertical. | Long Sides hori- zontal. | Long Sides vertical. | Long Sides hori- zontal. | |||
| 1 | ·620 | ·627 | ·628 | ·642 | ·643 | ·663 | ·664 | ·636 |
| 2 | ·613 | ·620 | ·628 | ·634 | ·636 | ·650 | ·651 | ·628 |
| 4 | ·608 | ·616 | ·618 | ·628 | ·629 | ·641 | ·642 | ·623 |
| 6 | ·607 | ·614 | ·616 | ·626 | ·627 | ·637 | ·637 | ·620 |
| 8 | ·606 | ·613 | ·614 | ·623 | ·625 | ·634 | ·635 | ·619 |
| 10 | ·605 | ·612 | ·613 | ·622 | ·624 | ·632 | ·633 | ·618 |
| 12 | ·604 | ·611 | ·612 | ·622 | ·623 | ·631 | ·631 | ·618 |
| 14 | ·604 | ·610 | ·612 | ·621 | ·622 | ·630 | ·630 | ·618 |
| 16 | ·603 | ·610 | ·611 | ·620 | ·622 | ·630 | ·630 | ·617 |
| 18 | ·603 | ·610 | ·611 | ·620 | ·621 | ·630 | ·629 | ·616 |
| 20 | ·603 | ·609 | ·611 | ·620 | ·621 | ·629 | ·628 | ·616 |
The orifice was 0·196 sq. in. area and the reductions were made with g = 32·176 the value for Montreal. The value of the coefficient appears to increase as (perimeter) / (area) increases. It decreases as the head increases. It decreases a little as the size of the orifice is greater.
Very careful experiments by J. G. Mair (Proc. Inst. Civ. Eng. lxxxiv.) on the discharge from circular orifices gave the results shown on top of next column.
The edges of the orifices were got up with scrapers to a sharp square edge. The coefficients generally fall as the head increases and as the diameter increases. Professor W. C. Unwin found that the results agree with the formula
where h is in feet and d in inches.
Coefficients of Discharge from Circular Orifices. Temperature 51° to 55°.
| Head in feet h. | Diameters of Orifices in Inches (d). | ||||||||
| 1 | 114 | 112 | 134 | 2 | 214 | 212 | 234 | 3 | |
| Coefficients (c). | |||||||||
| ·75 | ·616 | ·614 | ·616 | ·610 | ·616 | ·612 | ·607 | ·607 | ·609 |
| 1·0 | ·613 | ·612 | ·612 | ·611 | ·612 | ·611 | ·604 | ·608 | ·609 |
| 1·25 | ·613 | ·614 | ·610 | ·608 | ·612 | ·608 | ·605 | ·605 | ·606 |
| 1·50 | ·610 | ·612 | ·611 | ·606 | ·610 | ·607 | ·603 | ·607 | ·605 |
| 1·75 | ·612 | ·611 | ·611 | ·605 | ·611 | ·605 | ·604 | ·607 | ·605 |
| 2·00 | ·609 | ·613 | ·609 | ·606 | ·609 | ·606 | ·604 | ·604 | ·605 |
The following table, compiled by J. T. Fanning (Treatise on Water Supply Engineering), gives values for rectangular orifices in vertical plane surfaces, the head being measured, not immediately over the orifice, where the surface is depressed, but to the still-water surface at some distance from the orifice. The values were obtained by graphic interpolation, all the most reliable experiments being plotted and curves drawn so as to average the discrepancies.
Coefficients of Discharge for Rectangular Orifices, Sharp-edged, in Vertical Plane Surfaces.
| Head to Centre of Orifice. | Ratio of Height to Width. | |||||||
| 4 | 2 | 112 | 1 | 34 | 12 | 14 | 18 | |
| Feet. | 4 ft. high. 1 ft. wide. | 2 ft. high. 1 ft. wide. | 112 ft. high. 1 ft. wide. | 1 ft. high. 1 ft. wide. | 0·75 ft. high. 1 ft. wide. |
0·50 ft. high. 1 ft. wide. | 0·25 ft. high. 1 ft. wide. | 0·125 ft. high. 1 ft. wide. |
| 0·2 | .. | .. | .. | .. | .. | .. | .. | ·6333 |
| ·3 | .. | .. | .. | .. | .. | .. | ·6293 | ·6334 |
| ·4 | .. | .. | .. | .. | .. | ·6140 | ·6306 | ·6334 |
| ·5 | .. | .. | .. | .. | ·6050 | ·6150 | ·6313 | ·6333 |
| ·6 | .. | .. | .. | ·5984 | ·6063 | ·6156 | ·6317 | ·6332 |
| ·7 | .. | .. | .. | ·5994 | ·6074 | ·6162 | ·6319 | ·6328 |
| ·8 | .. | .. | ·6130 | ·6000 | ·6082 | ·6165 | ·6322 | ·6326 |
| ·9 | .. | .. | ·6134 | ·6006 | ·6086 | ·6168 | ·6323 | ·6324 |
| 1·0 | .. | .. | ·6135 | ·6010 | ·6090 | ·6172 | ·6320 | ·6320 |
| 1·25 | .. | ·6188 | ·6140 | ·6018 | ·6095 | ·6173 | ·6317 | ·6312 |
| 1·50 | .. | ·6187 | ·6144 | ·6026 | ·6100 | ·6172 | ·6313 | ·6303 |
| 1·75 | .. | ·6186 | ·6145 | ·6033 | ·6103 | ·6168 | ·6307 | ·6296 |
| 2 | .. | ·6183 | ·6144 | ·6036 | ·6104 | ·6166 | ·6302 | ·6291 |
| 2·25 | .. | ·6180 | ·6143 | ·6029 | ·6103 | ·6163 | ·6293 | ·6286 |
| 2·50 | ·6290 | ·6176 | ·6139 | ·6043 | ·6102 | ·6157 | ·6282 | ·6278 |
| 2·75 | ·6280 | ·6173 | ·6136 | ·6046 | ·6101 | ·6155 | ·6274 | ·6273 |
| 3 | ·6273 | ·6170 | ·6132 | ·6048 | ·6100 | ·6153 | ·6267 | ·6267 |
| 3·5 | ·6250 | ·6160 | ·6123 | ·6050 | ·6094 | ·6146 | ·6254 | ·6254 |
| 4 | ·6245 | ·6150 | ·6110 | ·6047 | ·6085 | ·6136 | ·6236 | ·6236 |
| 4·5 | ·6226 | ·6138 | ·6100 | ·6044 | ·6074 | ·6125 | ·6222 | ·6222 |
| 5 | ·6208 | ·6124 | ·6088 | ·6038 | ·6063 | ·6114 | ·6202 | ·6202 |
| 6 | ·6158 | ·6094 | ·6063 | ·6020 | ·6044 | ·6087 | ·6154 | ·6154 |
| 7 | ·6124 | ·6064 | ·6038 | ·6011 | ·6032 | ·6058 | ·6110 | ·6114 |
| 8 | ·6090 | ·6036 | ·6022 | ·6010 | ·6022 | ·6033 | ·6073 | ·6087 |
| 9 | ·6060 | ·6020 | ·6014 | ·6010 | ·6015 | ·6020 | ·6045 | ·6070 |
| 10 | ·6035 | ·6015 | ·6010 | ·6010 | ·6010 | ·6010 | ·6030 | ·6060 |
| 15 | ·6040 | ·6018 | ·6010 | ·6011 | ·6012 | ·6013 | ·6033 | ·6066 |
| 20 | ·6045 | ·6024 | ·6012 | ·6012 | ·6014 | ·6018 | ·6036 | ·6074 |
| 25 | ·6048 | ·6028 | ·6014 | ·6012 | ·6016 | ·6022 | ·6040 | ·6083 |
| 30 | ·6054 | ·6034 | ·6017 | ·6013 | ·6018 | ·6027 | ·6044 | ·6092 |
| 35 | ·6060 | ·6039 | ·6021 | ·6014 | ·6022 | ·6032 | ·6049 | ·6103 |
| 40 | ·6066 | ·6045 | ·6025 | ·6015 | ·6026 | ·6037 | ·6055 | ·6114 |
| 45 | ·6054 | ·6052 | ·6029 | ·6016 | ·6030 | ·6043 | ·6062 | ·6125 |
| 50 | ·6086 | ·6060 | ·6034 | ·6018 | ·6035 | ·6050 | ·6070 | ·6140 |
§ 21. Orifices with Edges of Sensible Thickness.—When the edges of the orifice are not bevelled outwards, but have a sensible thickness, the coefficient of discharge is somewhat altered. The following table gives values of the coefficient of discharge for the arrangements of the orifice shown in vertical section at P, Q, R (fig. 20). The plan of all the orifices is shown at S. The planks forming the orifice and sluice were each 2 in. thick, and the orifices were all 24 in. wide. The heads were measured immediately over the orifice. In this case,
§ 22. Partially Suppressed Contraction.—Since the contraction of the jet is due to the convergence towards the orifice of the issuing streams, it will be diminished if for any portion of the edge of the orifice the convergence is prevented. Thus, if an internal rim or border is applied to part of the edge of the orifice (fig. 21), the convergence for so much of the edge is suppressed. For such cases G. Bidone found the following empirical formulae applicable:—
