Time of Concentration

What is the maximum flow that will flow off a 100 hectare construction site in a 20 year storm?

The answer can be found with Q = CiA, but we need to know the time of concentration to determine the intensity.

A general discussion of the Rational Method (Q=CiA) can be seen by clicking the box with Q=CiA on the homepage.

The time of concentration is the name given to the time it would take for a drop of rainwater in a reasonably intense storm to get from the furthest point on the catchment to the bottom of the catchment.

The Simple Way

The simplest way to estimate time of concentration (tc ) is to:

1) estimate the distance from the top of the catchment to the bottom,

2) estimate the speed of the flow (0.5 metre per second usually serves pretty well) and

3) calculate the time it takes to travel that distance.

A 100 hectare catchment is (100 ha x 10,000 m2/ha) 1,000,000 m2. Assuming a square 1000 metres on each side and assuming that the diagonal down the middle is the longest distance from the top of the catchment to the bottom, we get a 1400m distance from the top to bottom. Calculating the velocity depends on the slope of the catchment and the roughness of the flow path just as shown in the discussion of Manning’s equation. With a relatively flat area (1 to 2%) and a smooth soil flow path, as expected on a construction site, the velocity can be assumed to be about 0.5 metres per second. Time of concentration can be calculated as follows:

If the catchment was steeper the velocity would be quicker and the time of concentration would be shorter. If the flow path was rougher and partly forested the velocity would be slower and the time of concentration longer.

For complicated catchments with some steep parts, some forested parts, some concrete channels and other mixes of factors that would influence the velocity; it is best to divide up the flow path, so that an average velocity can be determined.

The Australian Rainfall and Runoff Way

A formula for time of concentration based on typical conditions in eastern New South Wales has been developed in Australian Rainfall and Runoff (1987)

time of concentration = tc = 0.76A0.38 , with tc in hours and A = area, in square kilometres

100 hectares = 1 square kilometre, so the calculation is very simple for us.

tc = 0.76 x (1)0.38 = 0.76 hours = 45.6 minutes

Other Ways

ARR lists several other methods of calculating time of concentration for different parts of Australia. Richard McCuen lists 12 different formulas for calculating tc in Hydrologic Analysis and Design (1989). Most of the equations include variables for length of the flow path and slope. Many of the equations use other factors such as Manning’s “n,” channel factors, runoff coefficient, rainfall intensity and hydraulic radius.

The Federal Aviation Administration equation gives a good example:

tc = 0.03 (1.1 – C) L0.5 / S0.333

C = Runoff Coefficient = 0.55 for this example

L = length of the flow path, 1400 metres must be converted to 4600 feet because the Americans can’t make themselves learn the metric system

S = slope = 2%

tc = 0.03 (1.1 – 0.55) x (4600)0.5 / (2)0.333

tc = 0.03 x 0.55 x 67.8 / 1.26 = 0.89 hours = 53.3 minutes

This generally agrees with the other two calculations.

Calculating the necessary size of the channel to carry the maximum 20 year storm

How big would a channel have to be to carry the water on a 2% slope from the maximum flow off a 100 hectare construction site in a 20 year storm?

If we assume that the time of concentration is 45 minutes for our 100 hectare construction site near the Homebush Olympic site, then we can go to the table below and look up the 20 year, 45 minute storm and get the rainfall intensity. The 45 minute intensity is greater than the 45.6, 46.7 or 53.3 minute intensity because the ARI storm gets more intense a the period gets shorter. Therefore the 45 minute storm is a conservative choice for the design of the channel.

The table shows the rainfall intensity to be 82.8 mm/ hour.

We can assume C = 0.55 as the worst case from the reporting of Runoff Coefficients from the CALM manual Urban Erosion and Sediment Control, 1992 as discussed in the Q=CiA page.

Q=CiA = 0.55 x 82.8 mm/hr x 100 hectares x 10,000 m2 / ha x (1 m / 1000mm) x (1 hour / 3600 sec)

Q =  (0.55 x 82.8 x 100 x 10,000) / (1000 x 3600) = 12.6 m3/sec

As described in the Manning’s equation page, the complexity of even this simplest of channel types, means that the solution is usually best found using trial and error with a spreadsheet.

Q = bd (bd / b + 2d )2/3 s1/2 / n

Assuming a slope of 2% and a Manning’s “n” of 0.013 for a concrete channel, a bottom width of two metres and a depth of one metre the flowrate is 13.7 m3 / sec. More trial and error is shown in the table below.

 2 m wide rectangular channel, 12.6 m3/sec depth flowrate (m3/sec) velocity m/sec 1.00 13.7 6.8 0.90 11.9 6.6 0.94 12.6 6.7 1.20 17.4 7.3

A 2 metre wide, 1.2 metre deep channel would handle the flow and leave some freeboard in case there was a blockage or some other factor that might increase the flow. The extra 26 cm of depth would carry an additional 40 % of flow.

The velocities might be a little too high or the depth too great for the conditions. If land is available it might be better to try a four metre wide channel. The table below shows the trial and error for a 4 m wide channel carrying 12.6 m3/sec

 4 m wide rectangular channel, 12.6 m3/sec depth flowrate (m3/sec) velocity m/sec 0.50 11.8 5.9 0.54 13.3 6.1 0.52 12.5 6.0 0.65 17.6 6.8

The velocity has not changed significantly but a shallower channel can be built. With an additional 13 cm an extra 40% of flow can be carried.

Rational Method C values for disturbed sites

Bare packed soil, smooth = 0.25 to 0.55

Bare packed soil, rough = 0.15 to 0.45

From Urban Erosion and Sediment Control, 1992, Department of Conservation and Land Management, page 29

Storm Intensity in millimetres per hour

at Homebush Olympic Site, Sydney, New South Wales, Australia

 Duration Average Storm Recurrence Interval (years) 1 2 5 10 20 50 100 5 min 98.6 126.5 161.9 180.8 207.2 241.4 267.4 10 min 75.7 97.5 126.1 141.5 162.8 190.7 211.8 15 min 63.3 81.8 106.4 119.9 138.4 162.6 181.0 30 min 45.0 58.3 76.9 87.2 101.2 119.6 133.7 45 min 36.1 46.9 62.4 71.1 82.8 98.3 110.2 60 min 30.7 40.0 53.5 61.2 71.4 85.0 95.5 3 hr 15.2 19.8 26.4 30.1 35.1 41.7 46.8 6 hr 9.7 12.6 16.7 19.0 22.2 26.4 29.6 12 hr 6.2 8.0 10.6 12.1 14.1 16.7 18.7 24 hr 4.02 5.23 6.94 7.90 9.19 10.90 12.21 72 hr 1.92 2.50 3.31 3.77 4.38 5.19 5.81

Calculated using the algebraic procedures in Chapter 2 of Australian Rainfall and Runoff (1987).