Abstract
The discharge of circular sharp-crested orifices is commonly obtained by using experimental data of the discharge coefficient. This procedure is subjected to errors in graph reading. Further, such a procedure also cannot be used for analytical purposes. Presented herein is a high-accuracy explicit equation of the orifice discharge. The equation unifies viscous and potential flows.
1 Introduction
Orifices are used as an emptying device for tanks. The classical discharge equation for circular orifice flow from a side of a large tank is
Lea Citation(1938) explored the relation of C d versus [d(gh)1/2/ν] by utilizing various classical test data. These tests were conducted using mixtures of water and glycerin and a number of oils, resulting in Lea's Citation(1938) curve for C d (). This curve was considered so important that it was redrawn by Rouse Citation(1946) on a different abscissa (Vd/ν) with V = Q/(πd 2/4) being the average orifice flow velocity. For large [d(gh)1/2/ν] the limit value of C d = 0.592 is reached, whereas for large (Vd/ν) or [d(gh)1/2/ν] the plot of Rouse Citation(1946) reaches C d = 0.611 for potential flow. Thus, Rouse rejected data with C d < 0.611 for large [d(gh)1/2/ν]. follows Rouse by using the asymptotic value of C d = 0.611.
Figure 1 Lea's Citation(1938) averaged curve for C d versus [d(gh)1/2/ν]
![Figure 1 Lea's Citation(1938) averaged curve for C d versus [d(gh)1/2/ν]](/cms/asset/e94e7737-6738-434c-ab2b-d135e5fd3ccd/tjhr_a_457348_o_f0001g.gif)
2 Analytical considerations
For small [d(gh)1/2/ν], the relationship C d versus[d(gh)1/2/ν] is linear on a double logarithmic plot (), namely
For very large [d(gh)1/2/ν], the asymptotic discharge coefficient may be fitted to
Following Swamee Citation(1988) by combining EquationEqs (2) and Equation(3)
and fitting the test data of Lea Citation(1938) results in
Equation Equation(4) is shown in , which is fairly accurate to predict C
d
as the errors involved are well within ±1.0%. Combining EquationEqs (1)
and Equation(4)
, the orifice discharge is finally
3 Discussion
Putting ν = 0 for potential flow, EquationEq. (5) reduces to
A perusal of EquationEq. (6) reveals that for very high Reynolds numbers, the orifice discharge varies directly with the opening area, and as a square root of the operating head. In turn, by putting ν → ∞ for highly viscous flow, EquationEq. (5)
reads
4 Conclusions
A unified equation for the discharge coefficient of a sharp-crested orifice flow was established providing a smooth transition between viscous and potential flows.
Notation
C d | = |
discharge coefficient |
d | = |
orifice diameter |
g | = |
gravitational acceleration |
h | = |
hydraulic head |
Q | = |
discharge |
V | = |
average flow orifice velocity |
ν | = |
kinematic fluid viscosity |
References
- Lea , F. C. 1938 . Hydraulics for engineers and engineering students , 6 , London : Arnold .
- Rouse , H. 1946 . Elementary mechanics of fluids , New York : Wiley & Sons .
- Swamee , P. K. 1988 . Generalized rectangular weir equations . J. Hydraul. Eng. , 114 ( 8 ) : 945 – 949 .