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How to Estimate Time of Concentration (tc)

This article provides the fundamentals required for designing the stormwater management facility, which includes the calculation of Time of Concentration, tc, Time of Overland Flow Time, to and Time of Drain, td.

Project file example 

Time of Concentration, tc #

According to MSMA, tc is the travel time of runoff flows from the most hydraulically remote point upstream in the contributing catchment area to the point under consideration downstream. There are two methods of calculating the tc which are

  1. total of to and td,
  2. using Bransby Williams Equation. 

First Method (total of to and td) #

Time of Overland Flow Time, to #

To is estimated based on the land surface condition which determines the catchment roughness and also is affected by the distance and slope of the water flow in the catchment. The Horton’s Roughness for various land surfaces are given in Table 2.2. 

Formula to calculate the to:

$$ t_0=\frac{(107.n.L^{1/3})}{s^{1/5}}\ $$

to = Overland sheet flow travel time (minutes)

Lo = Distance of flow path travel from the catchment area towards the drain (m)

n = Horton’s roughness value (Table 2.2)

S = Slope of overland surface (%)

Example of calculation of to based on Catchment 1:

How to calculate the Lo #

Based on the project file attached, see Catchment 1. The distance (L) is calculated based on the estimation of the flow path in the catchment towards the drain. The unit used is in meters.

The L is also auto calculated in the MiTS. You can refer the L in the catchment spread input.

Lo 133.692 m

How to calculate the S #

The S is the slope of the overland flow in the catchment and it is calculated in percentage (%). It is calculated from a point from the upstream to another point of the downstream which is at the drain. The S is calculated by using the same line used to calculate the Lo. For example, 
Upstream elevation : 31.921 m
Downstream elevation
: 30.946 m

Basic equation of gradient is applied to calculate the slope.

$$ Slope,\;S_o=\;\frac{Vertical\;Dis\tan ce}{Horizontal\;Dis\tan ce}\times100\ $$

$$ \;\;\;\;\;\;\;\;\;\;\;\;\;\;=\frac{31.921-30.946}{133.692}\times100\ $$
$$ \;\;\;\;\;\;\;\;\;\;\;\;\;\;=\;0.729\;\%\ $$

The slope is also auto calculated in MiTS. See catchment spread input. 

Slope 0.729 %

How to determine the Horton’s Roughness, n #

Typical values of Horton’s Roughness are given in Table 2.2 in MSMA 2.

Select appropriate n for the surface condition. In this example, we will be choosing n for Average Grassed 0.045. 

Therefore, to of Catchment 1:
$$ t_0=\frac{(107.n.L^{1/3})}{s^{1/5}}\ $$

$$ t_o=\frac{(107\;.\;0.045\;.\;133.692^{1/3})}{0.729^{1/5}}\ $$

$$ \;\;\;=\;26.225\;min\ $$

To is auto calculated in MiTS.

Time of Drain, td

The td can be calculated by using Constant Average Velocity with the assumption of Velocity, V 1 m/s, or by using Manning Equation. You can change the settings in the “Options > Project Parameters > Drainage > Design > Drain Flow Time”. 

For this example, we will be using Constant Average Velocity for the td calculation. The equation used is:

Td = L/V

Ld = Length of Drain (m)

V = Velocity (m/s)

How to calculate the Ld #

The L is the length of the drain. For example, the length of Drain 1 is the Ld for that drain.

It is auto measured in MiTS. See Drain Spread Input.

Therefore, td for Drain 1:

Td = L/V

Td = 100/60

= 1.667 min

Td is auto calculated in MiTS.

Td 1.667 min

Therefore, tc of Drain 1 is the total sum of to and td. Thus,

Tc = to + td

= 26.225 + 1.667

= 27.892 min

Tc is auto calculated in MiTS.

How about the consecutive drains? How to determine the tc?

The next drain tc calculation will consider the tc from the previous drain as well. See Drain 2 as an example. 

1. Calculate the to for Drain 2 which is from Catchment 2 and results in 23.537 min.

2. Calculate Drain 2 td. The td is 1.667 min

3. Take the longer tc between these two equations:

Equation a : tcD1 + tdD2

= 27.891 + 1.667

= 29.558 min

Equation b = toD2 + tdD2

= 23.537 + 1.667

= 25.204 min

Equation a results to higher value, therefore 29.558 min is adopted for tc of Drain 2.

MiTS will detect whichever higher to be the tc

4. Repeat the steps towards the end of the network. 

How to adapt the first method to tc calculation in Pond Design? #

If you have multiple branches of networks, there would be multiple values of tc which carries the tc of each of the networks.

Based on MSMA 2 Chapter 7 Detention Pond, the tc is chosen based on the longest tc among the three different networks. Since drainage module auto calculates the tc, you can simply refer to the table and find the longest tc among those three networks. 

Based on the output, the longest tc lies at the network 2 with the longest tc 53.44 min. Therefore, override the tc 53.44 min in the pond module. 

If you still need to plug in the values of to and td in the pond design, therefore you only need to  fill the input based on the network 2. 

Second Method (Bransby Williams) #

Another method of calculating tc for Pre Development is by using Bransby Williams Equation. This method is better suited for catchments which consist of natural streams or catchments without well-defined drainage channels. 

You can change the method of Pre Development tc in Pond Module by changing the settings in “Options > Project Parameters > Det. Facility > Pond > Design > Pre Development > Tc Calculation Method”

The equation of Bransby Williams:

$$ t_c=\;\frac{F_c\;.\;L}{A^{1/10}\;.\;S^{1/5}}\ $$

Tc = time of concentration (min)
Fc = conversion factor, 92.5 when area is in ha (MiTS is using ha)
L = Length of flow path from the summit of catchment to outlet (km)
*input the L in MiTS in meter, it will be converted automatically to km
A = Catchment area (ha)
S = Slope of stream flow path from the summit of catchment to outlet(m/km)

Summit of the catchment and the area of consideration downstream

How to calculate L #

The L is calculated from the peak of catchment towards the downstream. The unit used in the equation is in km. However, you only need to key in the value in MiTS in meters, which software will automatically change the unit to km to suit the equation. 

The L is measured as example below:

Length of flow from the peak area to downstream 530.82 m

How to determine the A #

Measure the catchment area in ha. Then insert the value in A box. For this example, the catchment area is 18.755 ha.

How to calculate the S

S is the slope of the flow path which is calculated in m/km. It is calculated in m/km from the upstream point towards the downstream point. The S is calculated by using the same equation of slope. 

Upstream elevation: 34.729 m

Downstream elevation: 23.389 m

$$ S=\frac{Vertical\;Distance}{Horizontal\;Distance\;in\;km}\ $$

 $$ \;\;=\frac{34.729-23.389}{530.82/1000}\ $$

 $$ \;=21.36\;m/km\ $$

Time of Concentration, tc #

Therefore, the tc by using Bransby Williams is:
$$ t_c=\;\frac{F_c\;.\;L}{A^{1/10}\;.\;S^{1/5}}\ $$

$$ t_c\;=\;\frac{92.5\;\times\;0.53082}{18.755^{1/10}\;\times\;S^{1/5}}\ $$

 $$ \;\;\;\;\;=\;19.85\;min\ $$

MiTS result:

You may also be interested in :
How to determine the ‘S (%)’ and ‘L (m)’ value for catchment
Modelling Detention Drain in MiTS 2

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