代做3900-Numerical作业、代写R编程作业、代做R语言作业、代写Eq留学生作业
Final Projects, Math 3900-Numerical Analysis I
Name:
Project 1:
The ideal gas law is given by
=(1)
where P is the absolute pressure, = !
!
is the molar density, V is the volume, n is the
number of moles, R is the universal gas constant, and T is the absolute temperature.
Although this equation is widely used by engineers and scientists, it is accurate over only
a limited range of pressure and temperature. Furthermore, Eq. 1 is more appropriate for
some gases than for others.
An alternative equation of state for gases is given by
= !"#
!!!" ! (2)
known as the van der Waals equation, where a and b are empirical constants that depend
on the particular gas.
A chemical engineering design project requires that you accurately estimate the
molar density = !
!
of both carbon dioxide and oxygen for a number of different
temperature and pressure combinations so that appropriate containment vessels can be
selected. It is also of interest to examine how well each gas conforms to the ideal gas law
by comparing the densities as calculated by Eqs. (1) and (2). The following data are
provided:
R = 0. 082054 L atm/( mol K)
Carbon dioxide empirical constants
a = 3. 592
b = 0. 04267
Oxygen empirical constants
a = 1. 360
b = 0. 03183
The design pressures of interest are 1, 10, and 100 atm for temperature combinations of
300, 500, and 700 K. For each of these combinations find the molar density using Eqs.
(1) and (2) and compare the results.
Project 2:
We can use Lagrange Interpolation to study a trend analysis problem such as a
falling parachutist. Assume that we have developed instrumentation to measure the
velocity of the parachutist. The measured data obtained for a particular test case is
Time (s) Measured Velocity v (cm/s)
1 800
3 2310
5 3090
7 3940
13 4755
Our problem is to estimate the velocity of the parachutist at t =10 s to fill in the large gap
in the measurements between t =7 and t =13 s. We are aware that the behavior of
interpolating polynomials can be unexpected. Therefore, you will construct polynomials
of orders 4, 3, 2, and 1 and compare the results. Make plots of the constructed
polynomials between t=7 and t=13 and tell me which order polynomial or polynomials
best fits the data between these two values. Also explain why this or these particular
polynomials fit the data best.
Project 3:
Heat calculations are employed routinely in chemical and bioengineering as well
as in many other fields of engineering. One problem that is often encountered is the
determination of the quantity of heat required to raise the temperature of a material. The
characteristic that is needed to carry out this computation is the heat capacity c. This
parameter represents the quantity of heat required to raise a unit mass by a unit
temperature. If c is constant over the range of temperatures being examined, the required
heat (in calories) can be calculated by
=(1)
where the heat capacity of water is approximately 1 cal/(g . 0
C). Such a computation is
adequate when is small. However, for large ranges of temperature, the heat capacity is
not constant and, in fact, varies as a function of temperature. For example, the heat
capacity of a material could increase with temperature according to a relationship such as
= 0.132 + 1.56 10!!+ 2.64 10!!! (2)
In this instance you are asked to compute the heat required to raise 1000 g of this material
from ?100 to 200 0
C. Therefore, we can calculate the average value of c(T) by the
following
= ! ! !" !!
!!
!!!!!
(3)
which can be substituted into Eq. 1 to get
= !!
!! (4)
where = !!. Now because, for the present case, c(T ) is a simple quadratic,
can be determined analytically. Eq. 2 is substituted into Eq. 4 and the result integrated to
yield an exact value of = 42,732 . Using the following table of values of c for
various values of T
T, 0
C c, cal/(g 0
C)
-100 0.11904
-50 0.12486
0 0.13200
50 0.14046
100 0.15024
150 0.16134
200 0.17376
Use the composite Simpson’s rule to compute an integral estimate of = 42,732 ???.
How does your integrated estimate value agree with this result? Why does this agree or
not agree?
Project 4:
Fick’s first diffusion law states that
= !"
!" (1)
where mass flux = the quantity of mass that passes across a unit area per unit time
(g/cm2
/s), D = a diffusion coefficient (cm2
/s), c = concentration, and x = distance (cm).
An environmental engineer measures the following concentration of a pollutant in the
sediments underlying a lake (x = 0 at the sediment-water interface and increases
downward):
x, cm 0 1 3http://www.6daixie.com/contents/18/2183.html
c, 10-6 g/cm3 0.06 0.32 0.60
Use the best numerical differentiation technique available to estimate the derivative at x =
0. Employ this estimate in conjunction with Eq. 1 to compute the mass flux of pollutant
out of the sediments and into the overlying waters (D = 1.52*10 6 cm
2
/s).
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