A simple model of accumulation and flow.

waterbucketNoDrain.gif (6015 bytes) 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 easy to calculate from

  C = S t

Complete the table below for S=2.0 gal/min (gallons per minute)

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.

   Bucket content in 3 minutes will be gallons

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Pretty easy so far. Let's explore a more realistic model.

A bucket with a leak.  A more realistic model of global atmospheric pollution.

click to enlarge 

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.

Letís 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.

Hereís 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

      =-(1gal/min) Dt

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 letís assume that it is.   After 1 minute the new removal rate would then be Ė0.9 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. Copy and paste answers into a text editor for later e-mail.

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.

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   After 10 minutes the approximate (from your table) water content is gallons