代做3900-Numerical、代写R编程、代做R语言、代写Eq留学生

日期：2018-11-19 09:30

Final Projects,Math 3900-Numerical Analysis I

Name:

Project1:

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 3

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).