We have a catchment with a total area of 9.0 km2, where the runtime principle applies. The area is subdivided in 5 regions (A1 to A5) in such a way that every individual area has a runtime of 30 minutes. The areas drain sequentially. A5 is at the top of the catchment and A1 at the bottom of the catchment near the outlet.
A1 A2 A3 A4 A5
Area (km2) 1.5 2.4 2.4 1.5 1.2
We assume that the rain falls uniformly over the catchment, and that the discharge starts immediately after the start of the rain. The discharge only occurs as a result of the net (i.e. effective) rain.
QUESTION 1 (1 point possible)
Assume rain with a net intensity of 4.5 mm/hr, that lasts (in theory) infinitely long. What will eventually be the discharge in [m³/s] at the outlet of the catchment? If applicable, round your answer to two decimals.
Enter the the discharge in [m³/s] :
Q5 = P1A1+P2A2+P3A3+P4A4+P5A5
Q5 = 11.25 m3/s
0.040*10^6 m^3
To calculate the discharge at the outlet of the catchment, we can use the runoff coefficient which is the ratio of runoff to rainfall. The formula to calculate runoff is:
Runoff (m³/s) = Rainfall (mm/hr) * Catchment Area (km²) * Runoff Coefficient
Given that the net intensity of rain is 4.5 mm/hr and the catchment area is 9.0 km², we need to calculate the runoff coefficient.
The runtime principle states that each individual area drains in 30 minutes. So, we need to convert rainfall intensity from mm/hr to mm/30 min by dividing it by 2. Since the duration of the rainfall is theoretically infinite, it will eventually result in a steady-state condition with constant runoff from each area.
Therefore, the runoff coefficient for each area can be calculated by dividing the individual area by the total catchment area:
Runoff Coefficient (A1) = Area (A1) / Total Catchment Area
Runoff Coefficient (A2) = Area (A2) / Total Catchment Area
Runoff Coefficient (A3) = Area (A3) / Total Catchment Area
Runoff Coefficient (A4) = Area (A4) / Total Catchment Area
Runoff Coefficient (A5) = Area (A5) / Total Catchment Area
Using the given values:
Runoff Coefficient (A1) = 1.5 km² / 9.0 km²
Runoff Coefficient (A2) = 2.4 km² / 9.0 km²
Runoff Coefficient (A3) = 2.4 km² / 9.0 km²
Runoff Coefficient (A4) = 1.5 km² / 9.0 km²
Runoff Coefficient (A5) = 1.2 km² / 9.0 km²
Now, we can calculate the discharge at the outlet using the runoff coefficient and rainfall intensity:
Discharge at Outlet = Rainfall Intensity (mm/30 min) * Total Catchment Area * (Runoff Coefficient (A1) + Runoff Coefficient (A2) + Runoff Coefficient (A3) + Runoff Coefficient (A4) + Runoff Coefficient (A5))
Calculating the values and rounding to two decimal places:
Discharge at Outlet = 4.5 mm/30 min * 9.0 km² * (1.5 km² / 9.0 km² + 2.4 km² / 9.0 km² + 2.4 km² / 9.0 km² + 1.5 km² / 9.0 km² + 1.2 km² / 9.0 km²)
Discharge at Outlet = 0.225 m/30 min * 9.0 km² * (0.167 + 0.267 + 0.267 + 0.167 + 0.133)
Discharge at Outlet = 0.225 m/30 min * 9.0 km² * 0.999
Discharge at Outlet = 0.2025 m/30 min * 9.0 km²
Discharge at Outlet = 1.8225 m³/min
Converting minutes to seconds:
Discharge at Outlet = 1.8225 m³/min * 60 s/min
Discharge at Outlet = 109.35 m³/s
Therefore, the discharge at the outlet of the catchment is approximately 109.35 m³/s.
To find the discharge at the outlet of the catchment, we need to calculate the total effective rainfall and apply the runtime principle.
First, we need to convert the net intensity of rainfall from mm/hr to m/s:
1 mm/hr = 1/1000 m/s = 0.001 m/s
So, the net intensity of rainfall is 4.5 * 0.001 = 0.0045 m/s.
Next, we can calculate the total effective rainfall for each region by multiplying the net intensity of rainfall by the area of the region:
Effective rainfall for A1 = 0.0045 m/s * 1.5 km²
Effective rainfall for A2 = 0.0045 m/s * 2.4 km²
Effective rainfall for A3 = 0.0045 m/s * 2.4 km²
Effective rainfall for A4 = 0.0045 m/s * 1.5 km²
Effective rainfall for A5 = 0.0045 m/s * 1.2 km²
Next, we need to calculate the time it takes for the rainfall to reach the outlet from each region using the runtime principle. Since each region has a runtime of 30 minutes, the time for each region can be calculated as follows:
Time for A1 = 30 minutes
Time for A2 = 2 * 30 minutes = 60 minutes
Time for A3 = 3 * 30 minutes = 90 minutes
Time for A4 = 4 * 30 minutes = 120 minutes
Time for A5 = 5 * 30 minutes = 150 minutes
Now, we can calculate the discharge for each region using the formula: Discharge = Effective rainfall / Time
Discharge for A1 = Effective rainfall for A1 / Time for A1
Discharge for A2 = Effective rainfall for A2 / Time for A2
Discharge for A3 = Effective rainfall for A3 / Time for A3
Discharge for A4 = Effective rainfall for A4 / Time for A4
Discharge for A5 = Effective rainfall for A5 / Time for A5
Finally, to find the total discharge at the outlet, we sum up the discharges from each region:
Total discharge = Discharge for A1 + Discharge for A2 + Discharge for A3 + Discharge for A4 + Discharge for A5
Calculating the above values will give us the discharge in [m³/s] at the outlet of the catchment.