Rational method

Rational method was first used in 1889 developed by Emil Kuichling. This Method is widely used to estimate the peak surface runoff rate for design of a variety of drainage structures, such as a length of storm sewer, a storm water inlet, or a storm water detention pond. The Rational Method is most suitable for small urban watersheds that don't have storage such as ponds or swamps. It is best for areas less than 100 acres, but is sometimes used for up to 2 mi² areas.

The Rational Method Equation

The equation that is the centerpiece of the Rational Method is:

q = CiA,

Where q is the peak surface runoff rate in Cubic feet/seconds, from a watershed of area, “A” acres, and runoff coefficient, C, due to a storm of intensity, i in/hr. The units on peak runoff rate, q, are actually acre-in/hr, but the conversion from acre-in/hr to is Cubic feet/seconds very nearly one, so the more common unit, Cubic feet/seconds, is typically used for q. In order to calculate a value for peak runoff rate for a given drainage area, values are needed for the three parameters, A, C, and i.

Following is an example problem that illustrates the application of the Rational method to estimate peak discharges for the construction of Culvert.

Design of Culverts

For the construction or design of culverts we need the maximum rate of runoff at the inlet to a proposed culvert. For this purpose we generally use the 25 to 50 years of return period. After the estimation of the run off rate we construct the cross-section of the culvert according to the runoff rate so that it can convey all the runoff and no water remains there for long time.

As discussed earlier that through Rational method we can find the maximum amount of runoff or discharge that will pass through a point therefore we will use rational equation. For using rational equation the following set of data is collected first.

Site data:

Using a topographic map and a field survey, the area of the drainage basin upstream from the point in question is found. In addition, the following data were measured:

Average overland slope

Length of overland

Length of main basin channel

Slope of channel Roughness coefficient (n)

Type of soil and land cover is determined for the estimation of Runoff coefficient.

Runoff coefficient (C):

The runoff coefficient is the fraction of rainfall striking the drainage area that becomes runoff from that drainage area. It is an empirically determined constant, dependent on the nature of the drainage area surface.

From the topographic survey conducted earlier the type of the cover is determined. The land cover means the area consists of vegetation, forests, or it is concrete/asphalt. Similarly the type of the soil is determined. Then using the table the value of Runoff Coefficient i.e. C is determined.

For areas with a mixture of land uses, a composite runoff coefficient is used. The composite runoff coefficient is weighted based on the area of each respective land use and can be calculated as:

Where:

·         C_W = weighted runoff coefficient

·         C_j = runoff coefficient for area j

·         A_j = area for land cover j (ft²)

·         n = number of distinct land uses

Time of concentration

It is the time required for water to flow from the most remote part of the area to the outlet. When the storm duration equals the time of concentration all parts of the watershed are contributing simultaneously to the discharge at the outlet.

It depends on the watershed factors i.e. Slope, length, type of surface etc

In culvert design the length and slope of the watershed is determined and is used to calculate the Time of Concentration (TOC).

The equation of TOC is                            T_c = 0.0195 L^0.77 S^-0.385

The Rainfall Intensity (I):

Rainfall intensity, i, is the average rate of rainfall in inches per hour. Intensity is selected on the basis of design frequency of occurrence, a statistical parameter established by design criteria, and rainfall duration. For the Rational Method, the critical rainfall intensity is the rainfall having duration equal to the time of concentration of the drainage basin.

Where:

·         P_d = Depth of rainfall (in. or mm) for design storm of duration t_c

·         t_c = drainage area time of concentration (hr.)

After the T_c has been determined, the rainfall intensity should be obtained. For the Rational method, the design rainfall intensity averaging time (i_t) should be that which occurs for the design year storm whose duration equals the time of concentration. The rainfall amounts for various storm frequencies and duration is obtained from the Meteorological Center.

The rainfall intensity is determined by dividing the total rainfall by the duration (time of concentration) in hours.

Area

It is the area (A) of the basin. A map showing the limits of the drainage basin used in design is superimposed on the grading plan showing sub-basins. As mentioned earlier, the configuration of the contributing area with respect to pervious and impervious sub-areas and the flow path should be considered when deciding whether to use all or a portion of the total area.

Peak runoff calculation:

Now the Rational equation is used to calculate the Peak runoff rate.

In terms of British units the equation is:

q= CiA

Where q = Runoff (cubic feet per second)

C = Runoff Coefficient (dimensionless)

A = Area of the water shed (Acre)

Similarly in SI units the equation is:

q = 0.0028CiA

Where q = Runoff (m³ per sec)

A = Area (ha)

After the determination of the peak runoff rate the size of the culvert is determined and then it is checked through other formula i.e. Manning’s equation etc. if the result is same then the design is accurate and is approved.