In the past, calculating the flow in a submerged Parshall Flume has been difficult.  The solutions available were graphs (Robinson, Bureau of Reclamation), limited (size) equations (Wahl), or long, complex equations (ISO 9826:1992).

A better solution was needed – one that was non-graphical, applicable to all Parshall flume sizes, and that was simpler in complexity.

A different approach to submerged flow was taken buy Hyatt (1965) for trapezoidal flumes.  Skoegerboe, Walker, and Robinson (1965) verified the approach for rectangular measuring flumes.  Later that same year Skogerboe, Hyatt, Johnson, and England applied the approach to submerged Parshall flumes.

The resulting equation shown below accomplishes the goals of:

• Non-graphical in nature
• Applicable to all Parshall flume sizes
• Simpler in complexity

Universal Equation for Submerged Flow in Parshall Flumes

The equation below is application in both Imperial and SI units.  Constants and exponents are presented in both systems of measurement in the table below.  

Where the constants and exponents (in both Imperial and SI units) are listed in the table below:

W	Imperial		SI		nf	ns	St
	Cf	Cs	Cf	Cs
1-Inch	4.06	3.59	2.38	2.10	1.55	1.000	0.56
2-Inch	4.06	3.67	2.38	2.15	1.55	1.000	0.61
3-Inch	3.97	3.66	2.32	2.14	1.55	1.000	0.64
6-Inch	4.12	3.32	2.50	2.02	1.58	1.080	0.55
9-Inch	4.09	3.35	2.34	1.91	1.53	1.060	0.63
12-Inch	4.00	3.11	2.26	1.76	1.52	1.080	0.62
18-Inch	4.00	2.95	2.32	1.71	1.54	1.115	0.64
24-Inch	4.00	2.97	2.34	1.74	1.55	1.140	0.66
36-Inch	4.00	2.87	2.37	1.70	1.56	1.160	0.68
48-Inch	4.00	2.78	2.40	1.66	1.57	1.185	0.70
60-Inch	4.00	2.71	2.43	1.65	1.58	1.205	0.72
72-Inch	4.00	2.64	2.46	1.62	1.59	1.230	0.74
84-Inch	4.00	2.59	2.49	1.61	1.60	1.250	0.76
96-Inch	4.00	2.55	2.49	1.59	1.60	1.260	0.78
120-Inch	4.01	2.48	2.47	1.52	1.59	1.275	0.80
144-Inch	3.96	2.45	2.43	1.50	1.59	1.275	0.80

 

S_t is the transition submergence - where flow in the flume transitions from free-flow to submerged flow (and the above equation is applied).

For high level of submergence, S > 0.95, little faith can be placed in the calculation as very minor errors in determining the Ha or Hb levels become critical.  Practically speaking, the universal submergence equation should not be used on Parshall flumes experiencing submergence ratios above 90%.

 

Sources:  Skogerboe, G., Hyatt, L., Johnson, R., England, D., Submerged Parshall Flumes of Small Size, 1965.  Merkley, G. Irrigation Conveyance & Control:  Flow Measurement & Structure Design, Utah State University, 2004.