Discharge equation of a circular sharp-crested orifice

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.

Keywords:

• Discharge coefficient
• discharge measurement
• explicit equation
• flow rate
• orifice
• viscous effect

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

where Q is the discharge, C _d the discharge coefficient, d the orifice diameter, g the gravitational acceleration and h the depth of orifice centre below free surface. The discharge coefficient varies with d(gh)^1/2/ν, where ν is the kinematic fluid viscosity.

Lea (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 (1938) curve for C _d (Fig. 1). This curve was considered so important that it was redrawn by Rouse (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 (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]. Figure 1 follows Rouse by using the asymptotic value of C _d  = 0.611.

Figure 1 Lea's (1938) averaged curve for C _d versus [d(gh)^1/2/ν]

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 (Fig. 1), namely

For very large [d(gh)^1/2/ν], the asymptotic discharge coefficient may be fitted to

Following Swamee (1988) by combining Eqs (2) and (3) and fitting the test data of Lea (1938) results in

Equation (4) is shown in Fig. 1, which is fairly accurate to predict C _d as the errors involved are well within ±1.0%. Combining Eqs (1) and (4), the orifice discharge is finally

3 Discussion

Putting ν = 0 for potential flow, Eq. (5) reduces to

A perusal of Eq. (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, Eq. (5) reads

indicating that for highly viscous flow the orifice discharge is proportional to the operating head h, and proportional to 1.5 power of the opening area. Thus, whereas by doubling the area the discharge is the double of inviscid flow, it is almost three times for viscous flow.

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