Math Conversion

Question

The drainage basin of the Delaware River encompasses about 35,500 km^2 (= 35 x 109 m^2). The average annual precipitation for the drainage area is about 44 inches/year (= ~1.1 m/yr) and the evaporation reaches 23 inches/year (= ~0.6 m/yr). What is the flow rate of total flow at the mouth of the Delaware Bay? your units must be in m^3/seconds

Answer 1

In order to get a flow rate out of this, the useful idea is that whatever rain falls on the basin and does not evaporate has to leave through the river. So the first thing to find is the NET amount of water, not the total rainfall.

Precipitation is about 1.1 m/yr and evaporation is about 0.6 m/yr, so:

1.1 - 0.6 = 0.5 m/yr

That 0.5 metres per year is a depth. Picture half a metre of water sitting across the entire basin every year. To turn a depth into a volume, we multiply by the area it is spread over:

0.5 m/yr x 35 x 10^9 m^2 = 17.5 x 10^9 m^3/yr

If you notice, the units do the bookkeeping for us. Metres times metres squared gives metres cubed, which is exactly the volume unit we want.

Remember, the question asks for cubic metres per SECOND, so the last step is converting years into seconds. There are 365 days in a year, 24 hours in a day, 60 minutes in an hour and 60 seconds in a minute:

365 x 24 x 60 x 60 = 31,536,000 seconds in a year

Now divide the yearly volume by the number of seconds in a year:

17.5 x 10^9 / 31,536,000 = 555

So, our final answer is a flow rate of about 555 m^3/s.

As a sanity check, that number is in the right neighborhood for a river the size of the Delaware, which is a good sign we did not lose a factor of a thousand somewhere. Losing track of a power of ten is by far the most common mistake on conversion problems like this, so it is always worth asking at the end whether the answer is physically believable.

Answer 2

So we are given the area: 35.5 x 10⁹ m²

We want to find: Some flow rate in m³s^-1

SOLUTION:

1. Find avg. precipitation in terms of volume per year
2. Find avg. evaporation in terms of volume per year
3. Combine to find flow rate per year
4. Use dimensional analysis to convert to per second. Basically, we multiplying by 1/1, so the units change, but the actual value doesn't
5. Years > Days > Hours > Minutes > Seconds
6. 1 year * (365 days/1 year) * (24 hrs/1 day) * (60 minutes/ 1 hour) * (60 seconds/1 minute) = 3.1536x10⁷ seconds in 1 year
7. Answer should be in m³s^-1