Analytic Solution for the Cylinder

Although it may not be practical to solve most vessel drainage problems analytically, we can learn a lot about the behavior of such systems in general by examining the solution of a simple case. By using a computer algebra software package, such as MAPLE or MATHEMATICA.

Rearranging slightly Equation (17) from above, we have our mathematical model for the draining cylinder in a form suitable for submitting to MAPLE for solution.

where we have defined to be the radius of the cylinder whose volume will be exactly the same as that of the cone when the initial liquid level is when measured from the drain orifice.

Exercise: Verify that under these conditions the required cylinder radius is

We assume that you are familiar with the basic operations in MAPLE. You will find that it is frequently a good idea to initialize your variables and parameters in MAPLE, since once numerical values are stored in these symbols, they remain there. That means, that if you want to start your calculations over, you need to re-initialize them. You'll see what we mean as you gain experience.

Below is a typical statement for initializing the MAPLE variables we will use: The MAPLE commands are:

If you like, you can simply use your mouse to `swipe' the commands in the browser window, then paste them in the MAPLE window. Then press the enter key.

Now to submit the equation we wish to solve to MAPLE, we define, for convenience, a variable called de1 in which to store the equation. The necessary MAPLE command is

Next we turn this ordinary differential equation over to the differential equation solver in MAPLE, called dsolve. We will will define a variable y1(t) to be the solution of de1.

The solution is not very elegant looking. In fact, it's virtually impossible to tell anything about the behavior of this physical system by looking at the solution to the mathematical model. If a picture is worth a thousand words, then a plot of the solution is worth about the same. So let's choose some numerical values and plot the results to see what the solution looks like.

First, let's tell MAPLE what is in terms of variables we will use later in the calculations involving the cones.

Then let us choose some reasonable physical values in the SI system of units, with all lengths in meters and g in meters/sec² .

Note carefully that we have made certain that the radius of the drain is smaller than the minor radius of the cone r. Now to produce a plot let's call on MAPLE again. Since we have no prior knowledge of how long the cylinder might take to drain (in seconds), we might try solving algebraically for t in our solution from above. But we see right away that y(t)=y1(t) is quadratic in t in a very messy way. However, if we examine our solution in light of the fact that so that , then we will find that the time for the cylinder to empty completely will be about 30 minutes, or 1760 seconds.

Exercise: Go back to Equation (17) and eliminate the radical in the denominator,

and obtain the solution to this equation using MAPLE. (This model neglects the velocity of the free surface of the liquid in computing the exit velocity. That is, it assumes .) Now use the resulting solution to estimate the drainage time by solving for t when y(t)=0. You can do this using the solve and evalf commands in MAPLE.

To produce the plot of vs t, the appropriate MAPLE statement is: