Fluid Discharged Distance Coordinates Equations and Calculator

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Fluid Discharged Distance Coordinates Equations and Calculator

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Fluids Flow Design and Engineering

Fluid Stream From a Container Distance and Coordinates Equations and Calculator

Fluid discharged from an orifice in a tank gets its initial velocity from the conversion of potential energy. After discharge, no additional energy conversion occurs, and all subsequent velocity changes are due to external forces. (See Fig. 1.)

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Figure 1 Coordinates of a Fluid Stream

In the absence of air friction (drag), there are no retarding or accelerating forces in the x-direction on the fluid stream. The x-component of velocity is constant. Projectile motion equations can be used to predict the path of the fluid stream.

Eq. 1
Horizontal Discharge at nozzle
v_x = v_o = C_v ( 2 · g · h )^0.5

Eq. 2
Horizontal distance
x = v_o · t = v_o ( 2 · y / g )^0.5 = 2 · C_v ( h · y )^0.5

After discharge, the fluid stream is acted upon by a constant gravitational acceleration. The y-component of velocity is zero at discharge but increases linearly with time.

Eq. 3
Vertical discharge velocity
v_y = g · t

Eq. 4
Vertical distance
y = g · t² / 2 = g · x² / ( 2 v_o² ) = x² / ( 4 · h · C_v² )

Eq. 5
Volumetric flow rate
V = C_d · A_o ( 2 · g · h )^0.5

Where

v_x = Fluid velocity horizontal, ft/sec, m/s
v_y = Fluid velocity vertical, ft/sec, m/s
v_o = Acual fluid velocity, ft/sec, m/s
V = volumetric flow rate, ft³/sec, m³/s
A_o = area of discharge orifice, ft², m²
x = x -coordinate of position, ft, m
y = y -coordinate of position, ft, m
h = height / depth of head, ft, m
t = time, seconds (s)
g = gravitational acceleration, 32.2 ft/s², (9.81 m/s²)
C_v = coefficient of velocity, empirical factor that accounts for the friction and turbulence at the orifice, see Table 1
C_d = Approximate Orifice Coefficients for Turbulent Water, see Table 1

Table 1 Approximate Orifice Coefficients for Turbulent Water

Illustration	Description	C_d	C_v
A	sharp-edged	0.62	0.98
B	round-edged	0.98	0.98
C	short tube* (fluid separates from walls)	0.61	0.61
D	sharp tube (no separation)	0.82	0.82
E	short tube with rounded entrance	0.97	0.98
F	reentrant tube, length less than one-half of pipe diameter	0.54	0.99
G r	eentrant tube, length 2 to 3 pipe diameters	0.72	0.72
H	Borda	0.51	0.98
(none)	smooth, well-tapered nozzle	0.98	0.99

*A short tube has a length less than approximately 3 diameters.

Illustrations for Table 1

Source:

Civil Engineering Reference Manual, Fourteenth Edition
Michael R. Lindeburg, PE