Problem set 6 Due: Thursday, Dec 17, 1998
You should e-mail an Xcel workbook and a Maple workbook as attachments.
1. Excel (Problem suggested by Eric)
Eric's research is explosive stuff. He pours liquid lead into water.
You better not be too close when he does this.
If a lead drop of diameter D hits the water at speed V ,
what happens depends largely on the value of a nondimensional parameter
called the Weber number
We = rho D V2 / sigma
where rho is the density of the water and sigma is the surface
tension at the lead-water interface. Nondimensional parameters are key in
physical processes. Doing the experiments in France, where they use Meters and
Kilograms, will produce exactly the same effect as here in Wisconsin, where
we prefer Inches and Pounds, as long as the Weber number is the same for
both experiments. A small drop with high velocity will have the same relative
effects as a drop 4 times as big with half as small velocity (remember the concept
of relative accuracy, that too was a nondimensional number, and when we scaled out
R in all those problems before, we used nondimensional numbers too).
Both roh and sigma are
functions of temperature. Your boss needs you to write an Xcel spreadsheet
that will provide the Weber number for making lead shot
for a lead temperature range of 340 to 1000C and a water quench temperature
range from 20 to 100C. The We is important
in determining the smoothness of the shot. The process variables (inputs)
the technicians control are: lead melt temperature, fall height above the
water, water temperature, and drop diameter. Provide the following:
- 1) A spreadsheet that will calculate the We number where the technicians can
change diameter, melt and water temperature and fall distance;
- 2) Use the
density and surface data
(provided as functions of temperature)
as look-up tables. Have the data as a separate worksheet;
- 3) Produce a double y-axis graph plotting the Weber number and the lead
surface tension as a function of temperature (0 to 1000C). Hold D constant
at 1/4", water temperature at 50C and drop velocity constant 2cm/sec;
- 4) Produce a 3-D plot (a real one, not a Bill Gates look-alike 3-D plot
where spacing is even between data points: be careful)
of the Weber number as a function of water
temperature (from 20 to 100C) and Velocity (0 to 20cm/sec). Hold drop
temperature (400C) and diameter (1/4") constant .
Your Workbook should have at least 5 Worksheets: 1) User interface for your
boss (who is Dilbert and not very smart) so make it user friendly and check
spelling, 2) sheet for the density data, 3) sheet for the surface tension
data, 4) double y graph, 5) 3D plot, 6) more if you need them.
2. Maple
- Expand J0(x) , our favorite Bessel function of
the first kind of order 0, in powers of x around both x=0
and x=1 up to eighth order (i.e. up to and including 8th powers of x).
- Calculate the limit for x going to zero of (sin (a x))/ x .
- Calculate the limit for x going to zero of (ea x^2 - cos(b x))
/ x2.
- Calculate the indefinite integral of sqrt(1 - cos x) . Check
that the answer is correct. Evaluate the result at x=Pi/2 and 0 .
Fabian Waleffe
Wed Dec 09 20:12 CDT 1998