Hydraulic Efficiency of Grate and Curb Inlets - Urban Drainage and ...

where:
q 1 = dependent parameter;
q 2 …q n = n-1 independent parameters;
f = function relating dimensional analysis parameters q; and
g = unspecified function different from f.
The Buckingham Pi theorem states that, given a relation among n parameters in the form
of Equation 2-17, the n parameters may be grouped into n-m independent dimensionless ratios
also called Pi (Π) groups (Fox, 2006). In functional form this is expressed as Equation 2-18:
Π = G Π , Π , ..., Π ) or G Π , Π , ..., Π ) 0 Equation 2-18
1 1( 2 3 n−m
(
1 2 n−m
=
where:
Π = Pi parameter; and
G = function relating the dimensionless Pi parameters, related to the function f.
The number m is often, but not always, equal to the number of dimensions required to
specify the dimensions of all the parameters (q i ) of the problem or phenomena in question. The
n-m dimensionless Pi parameters obtained from this procedure are independent of one another.
The Buckingham Pi theorem does not predict the functional form of G, which must be
determined experimentally.
2.6 Significant Parameter Groups for Calculating Inlet Efficiency
A review of available literature has shown that the complex nature of street inlet flow has
precluded the development of purely theoretical equations. Often the approach of developing
empirical equations has been used. Physical variables related to gutter flow and inlet
characteristics are typically identified and combined into meaningful parameter groups using
dimensional analysis. Tests are performed on parameter groups to quantify their relevance.
Although the method of dimensional analysis is universally applicable to development of
parameter groups, there are many forms that these dimensionless groups may take depending
upon what parameters are used. Two of the larger studies conducted on the topic of inlet
efficiency were the FHWA study on bicycle-safe grate inlets described previously (FHWA,
1977) and a study completed at The Johns Hopkins University (Li, 1956). Equations developed
from the FHWA study were incorporated into HEC 22 and were presented previously. The
Johns Hopkins University study took a slightly different approach of regression analysis. For an
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