The rational method, also called the Lloyd-Davies method in the United Kingdom, was developed in 1851 by Mulvaney. It is an equilibrium-based approach to peak flow
estimation that uses rainfall intensity data and watershed characteristics to predict peak flows for a rainfall event. This method was originally presented in American hydrologic literature by Kuichling (1889) and has been a staple of American hydro-logic practice since that time. The rational method is especially popular in storm sewer design because of its simplicity, and because storm sewer design typically requires only peak discharge data.
At the most fundamental level, the rational method assumes that an equilibrium (that is, a steady state) is attained such that the effective rainfall inflow rate of water onto a drainage basin is equal to the outflow rate of water from the basin. If one expresses the volumetric effective inflow rate as the product of the basin area A and the effective rainfall intensity ie, then the outflow rate Q is obtained as Q = ieA. Further, if one accounts for abstractions using a runoff coefficient, then the effective intensity is a product of the actual rainfall intensity and the runoff coefficient, resulting in
Q = CiA (5.21)
where Q = runoff rate (ac-in/hr, ha-mm/hr) C = runoff coefficient (see Table 5.2)
i = rainfall intensity (in/hr, mm/hr) A = drainage area (ac, ha)
Because 1 ac-in/hr = 1.008 cfs | 1 cfs, engineers performing calculations by hand in U.S. customary units typically ignore the conversion factor and simply assume that the discharge Q is in units of cfs. This conversion factor is not ignored in computer applications.
Table 5.11 lists recommended runoff coefficients corresponding to various land uses.
It should be noted, however, that some locales have developed runoff coefficient tables that also consider soil type and/or storm return period. Coefficients should be selected carefully for proper application to a particular locale.
When a drainage basin consists of a mixture of land uses, a composite runoff coeffi-cient may be computed for the basin by weighting individual runoff coefficoeffi-cients for each land use by their respective areas, as demonstrated in Example 5.8.
The time of concentration used to find ie is the smallest time for which the entire basin is contributing runoff to the basin outlet; therefore, the storm duration must be at least as long as the time of concentration if a steady-state condition is to be achieved. Also, steady-state conditions dictate that the storm intensity be spatially and temporally uni-form. It is not reasonable to expect that rainfall will be spatially uniform over a large drainage basin, or that it will be temporally uniform over a duration at least as long as the time of concentration when tc (and hence A) is large. Therefore, these conditions limit the applicability of the rational method to small drainage basins. An upper limit of 200 acres (80.9 ha) has been suggested by some, but the limit should really depend on the storm characteristics of the particular locale. These local characteristics may
limit the applicability of the rational method to basins smaller than 10 acres (4 ha) in some cases.
Example 5.8 – Determining the Weighted Runoff Coefficient. Estimate the runoff coefficient for a drainage basin that is made up of 6 ha of park and 12 ha of medium-density, single-family housing.
Solution: From Table 5.11, the runoff coefficients for the park and residential areas are estimated to be 0.20 and 0.40, respectively. The composite runoff coefficient for the entire drainage basin of 18 ha is therefore
C = [6(0.20) + 12(0.40)]/18 = 0.33
Additional assumptions associated with the rational method are that the runoff coeffi-cient is a constant and does not change during the duration of the storm, and that the recurrence intervals of the rainfall and corresponding runoff are equal (see, for exam-ple, Schaake, Geyer, and Knapp, 1967).
Table 5.11 Runoff coefficients for use in the rational method
(Schaake, Geyer, and Knapp, 1967)
Type of Area or Development C
Types of Development
When several drainage basins (or subbasins) discharge to a common facility (such as a storm sewer or culvert), the time of concentration should be taken as the longest of all the individual times of concentration. Further, the total drainage area served (the sum of the individual basin areas) should be no larger than the 200-acre limit (or smaller where applicable) of the rational method.
The basic steps for applying the rational method are as follows:
Step 1: Estimate tc and apply I-D-F data. Develop or obtain a set of intensity-duration-frequency (IDF) curves for the locale in which the drainage basin resides (see Chapter 4). Estimate the time of concentration of the basin using one of the techniques presented in Section 5.3 or other applicable method. Assume that the storm duration is equal to the time of concentration and use the IDF curves to deter-mine the precipitation intensity for the recurrence interval of interest. Note that the assumption that the storm duration and time of concentration are equal is conservative in that it represents the highest intensity for which the entire drainage area can con-tribute.
The rational method is frequently used in the design of storm sewer systems, and an important part of the hydraulic analysis is determining the time of concentration for each component in the system so that design flows can be computed correctly. The time of concentration for an inlet’s contributing drainage area is used to compute the inlet design flow. However, finding the appropriate time of concentration for use in computing the design flow for a pipe additionally involves the consideration of travel time through any upstream piping. The longest total travel time to the pipe is the con-trolling time of concentration used in design flow calculation. An illustration of this approach is provided later in the text in Example 11.1 (page 424).
Step 2: Compute watershed area. The basin area A can be estimated using topographic maps, computer tools such as CAD or GIS software, or by field recon-naissance.
Step 3: Choose C coefficient. The runoff coefficient C can be estimated using Table 5.11 if the land use is homogeneous in the basin, or a composite C value can be estimated if the land use is heterogeneous (see Example 5.5).
Step 4: Solve for peak flow. Finally, the peak runoff rate from the basin can be computed using Equation 5.21.
The following example illustrates the use of the rational method for several subbasins draining into a common storm sewer system.
Example 5.9 – Computing Peak Flow with the Rational Method. A proposed roadway culvert near Memphis, Tennessee is to drain a 5-ha area with a runoff coefficient of 0.65.
The time of concentration for the drainage area is 15 min. Use the rational method to compute the peak discharge (in m3/s) for a 25-year design storm. From Example 4.2 (page 90), the 15-min inten-sity for a 25-year storm in Memphis is 5.96 in/hr (151.4 mm/hr).
Solution: From Equation 5.21, the peak flow is computed as Q = CiA = (0.65)(151.4 mm/hr)(5 ha) = 492.05 ha-mm/hr Converting the units, Q is 1.37 m3/s.
Additional Duration and Area Considerations. Differences in land sur-face characteristics within a drainage basin can give rise to peak discharges larger than those computed assuming basin homogeneity. For example, the impervious sur-faces within a basin can be subdivided into two parts, one consisting of directly con-nected impervious areas and the other consisting of indirectly concon-nected or unconnected impervious areas. In a residential subdivision, the directly connected impervious areas consist of roadway and driveway surfaces from which runoff flows directly to stormwater inlets. Runoff from indirectly connected impervious areas passes over pervious areas prior to reaching channels, gutters, or inlets; thus, an infil-tration opportunity exists for runoff from these areas. Rooftop areas are a common example of unconnected impervious areas in a residential subdivision.
One problem with using the rational method as previously described is that it may under-predict peak flow for heterogeneous drainage basins. This under-prediction occurs because the time of concentration for the basin as a whole is longer than that for the directly connected impervious areas alone. Thus, the intensity for the directly connected impervious area considered alone will be higher than that of the basin as a whole. In fact, it may be so much higher that the peak flow computed for the directly connected impervious area only is greater than that computed for the whole basin, despite the fact that its area is smaller. The higher peak flow estimate should govern in such cases. Example 11.2 (page 427) illustrates this procedure.