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Some Quicktime movies of a simple bucketA
simple model of accumulation and flow.
click
to enlarge
Some Quicktime movies of a simple bucket
Here are Questions in compact form Assign1(pdf_254k)
Imagine
a 10 gallon bucket that is filled by a faucet.
We refer to the faucet as the source of water flowing into the bucket, S.
If S=2.0 gallon per minute (2.0 gal/min) then the bucket will become
completely filled in 5 minutes. The
amount of water in the bucket at any given time will be referred to as the
bucket content, C.
If
we start with an empty bucket and turn on the faucet at a constant flow (S in
gallons per minute), then the
water content C at any time t is
C = S t
QuesA1:
Check your understanding. If the bucket starts out empty (C0=0) and is filled at a rate of S=3.0 gal/min, how much water will be in the bucket in 3.0 minutes? The symbol C0 stands for the initial bucket content at time t=0.**********************************
Pretty easy so far. Let's explore a more realistic model.
A bucket with a leak. A more realistic model of global atmospheric pollution.
Now
imagine that the bucket has a leak in it.
The rate at which water flows out of the hole depends on how full the
bucket is. This is because the
fuller the bucket, the greater the water pressure forcing water out of the hole.
Lets say that the flow rate out of the bucket is,
C/(lifetime)
Where
the liftime is a
characteristic time related to how fast the bucket empties.
The life-time depends of the size of the hole and the fluid's viscosity.
Let’s clarify what is meant by life-time.
Assume that the bucket is initially full C0 =10 gallons. The life-time is defined such that if it equals 2 minutes then it would take 2 minutes for the bucket to loose 63 % of its water content (drop to 37% it's starting value). The 37 % value comes from the fact that (1/e) is about 0.37. The number (e) is sometimes called the natural number and is a number that pops up whenever the rate of change of something depends on how much of it exists (like water out of the bucket). The bigger the hole in the bucket the shorter the life-time for water in the bucket.
For a life-time
of
10 minutes then the initial removal rate
is
C/10 minutes=0.1C (units
of gal/min)
For
any water content, 10 % of
what's in the bucket is lost each minute.
Here’s
a concrete example. Assume that the
starting water content is 10 gallons, the life-time is t=10 minutes, and the faucet
has been turned off. The removal
rate would start out at 1 gal/min (C/t).
In
a time interval, Dt, the change
in water content would be
The
water level would drop to about 9 gallons (DC=-1
gal) in Dt=1 minute.
If the removal rate stayed at 1.0 gal/min, then the bucket would be
completely empty in
one life-time (10 minutes). This is
not the case since the removal rate gets slower as the bucket empties.
In fact the removal rate is not even 1.0 gal/min for the complete first
minute but for this discussion let’s assume that it is.
After 1 minute the new removal rate would then be –0.9 gal/min, so in
the next 1.0 time interval the water content would drop by about 0.9 gallons to
about 8.1 gallons. Using this
logic, complete the table below.
QuesA2:
To help you check your answers, the graph below shows:
1) How the bucket drains if the drain rate is a constant 1 gal/min (black line). Note that the bucket completely drains under this assumption in one life-time.
2) How the bucket drains if the drain rate at any instant is always equal to 10% of what's in the bucket. (red line) For this exact solution 37% of the water remains in the bucket after one life-time.
3) How the bucket drains if we assume the for each 1-minute time interval the drain rate is constant and equal to 10% of what's in bucket at the start of the 1-minute time interval. (blue diamonds) This is what we did for the above table. The errors introduced by this approximation are relatively small.
