Time of concentration and time-area graph

Time of concentration

The time of concentration ($t_c$) is the time taken by rain falling on the most remote point of the catchment to reach the point of the drain or sewer under design.

$$t_c = t_e + t_f$$

• $t_e$ = time of entry (inlet time): the time for water to flow overland and through gutters to the first inlet (5-20 min).
• $t_f$ = time of flow: the travel time in the sewer, $t_f=L/v$.

Why it is important

1. The peak discharge occurs when the whole catchment contributes, i.e. when the rain has lasted at least $t_c$. So the design storm duration is taken equal to $t_c$.
2. The rainfall intensity $i$ depends on duration: short storms are more intense. A wrong $t_c$ gives a wrong $i$, and so a wrong $Q = CiA/360$.
3. If $t_c$ is taken too short, $i$ is too high and the sewer is oversized and costly; if it is taken too long, the sewer is undersized and floods.
4. It is needed to fix the size of every downstream section, since $t_c$ increases down the sewer.
5. It decides how large an area a storm drain of a given size can serve.

Time-area graph

It is a plot of the contributing area against time, used to find how the runoff builds up.

Area
  A |                    ________ whole area
    |                 .'
    |              .'
    |           .'
    |        .'
    |     .'
    +----+-----+-----+-----> time
    0   t1    t2    tc

• The catchment is divided by lines of equal travel time to the outlet (isochrones): 0, t1, t2 ... tc.
• The area between two successive isochrones is measured, and the cumulative area is plotted against time.
• The graph rises with time and becomes constant at $t_c$, when the entire area A contributes.
• For a rain of duration less than $t_c$, only the area read from the graph at that time contributes, so the runoff is smaller.
• For a uniform rainfall, the runoff hydrograph is obtained as $Q = C\,i\,A_t/360$, where $A_t$ is the area (ha) read from the graph at time $t$. Its shape shows that the peak occurs at $t_c$ (rational method).

For a nearly rectangular catchment the graph is a straight line; for a fan-shaped one it is curved.