0.6 Solution of the partial differential equations  (Page 13/13)

 Page 13 / 13

This solution is admissible in that the velocity in nondecreasing in going from the $BC$ to the $IC$ . However, it in not unique . Several values of $S2$ will give admissible solutions. Suppose that the value shown here is $\mathbf{a}$ solution. Also suppose that dispersion across the shock causes the presence of other values of $S$ between $S1$ and $S2$ . There are some values of $S$ that will have a velocity (slope) greater than that of the shock shown here. These values of $S$ will overtake $S2$ and the shock will go the these values of $S$ . This will continue until there is no value of S that has a velocity greater than that of the shock to that point. At this point the velocity of the saturation value and that of the shock are equal. On the graphic construction , the cord will be tangent to the curve at this point. This is the unique solution in the presence of a small amount of dispersion.

Composition (Saturation) Profile The composition (saturation) profile is the composition distribution existing in the system at a given time.

Composition or Flux History : The composition or flux appearing at a given point in the system, e.g., $x=1$ .

Summary of equations

The dimensionless velocity that a saturation value propagates is given by the following equation.

${\left(\frac{dx}{dt}\right)}_{dS=0}=\frac{df}{dS}\left(S\right)$

With uniform initial and boundary conditions, the origin of all changes in saturation is at $x=0$ and $t=0$ . If $f\left(S\right)$ depends only on $S$ and not on $x$ or $t$ , then the trajectories of constant saturation are straight line determined by integration of the above equation from the origin.

$\begin{array}{ccc}\hfill x\left(S\right)& =& {\left(\frac{dx}{dt}\right)}_{dS=0}t=\frac{df}{dS}\left(S\right)\phantom{\rule{0.277778em}{0ex}}t\hfill \\ \hfill x\left(\Delta S\right)& =& {\left(\frac{dx}{dt}\right)}_{\Delta S}t\phantom{\rule{1.em}{0ex}}=\frac{\Delta f}{\Delta S}\phantom{\rule{0.277778em}{0ex}}t\hfill \end{array}$

These equations give the trajectory for a given value of $S$ or for the shock. By evaluating these equations for a given value of time these equations give the saturation profile .

The saturation history can be determined by solving the equations for $t$ with a specified value of $x$ , e.g. $x=1$ .

$\begin{array}{c}t\left(\Delta S\right)=\frac{x}{\frac{\Delta f}{\Delta S}},\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}x=1\\ t\left(S\right)=\frac{x}{\frac{df}{dS}\left(S\right)},\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}x=1\phantom{\rule{0.277778em}{0ex}}\end{array}$

The breakthrough time , ${t}_{BT}$ , is the time at which the fastest wave reaches $x=1.0$ . The flux history (fractional flow history) can be determined by calculating the fractional flow that corresponds to the saturation history.

Summary of diagrams

The relationship between the diagrams can be illustrated in a diagram for the trajectories. The profile is a plot of the saturation at $t={t}_{o}$ . The history at $x=1.0$ is the saturation or fractional flow at $x=1$ . In this illustration, the shock wave is the fastest wave. Ahead of the shock is a region of constant state that is the same as the initial conditions.

New references

Carslaw, H. S. and Jaeger, J. C., Conduction of Heat in Solids, Oxford, (1959).
Churchill, R. V., Operational Mathematics, McGraw-Hill, (1958).
Courant, R. and Hilbert, D., Methods of Mathematical Physics, Volume II Partial Differential Equations, Interscience Publishers, (1962).
Hellums, J. D. and Churchill, S. W., "Mathematical Simplification of Boundary Value Problems," AIChE. J. 10, (1964) 110.
Jeffrey, A., Quasilinear Hyperbolic Systems and Waves, Pitman, (1976)
Lax, P. D., Hyperbolic System of Conservation Laws and the Mathematical Theory of Shock Waves, SIAM, (1973).
LeVeque, R. J., Numerical Methods for Conservation Laws, Birkhauser, (1992).
Milne-Thompson, L. M., Theoretical Hydrodynamics, 5th ed. Macmillan (1967).
Morse, P. M. and Feshbach, H., Methods of Theoretical Physics, (1953)
Rhee, H.-K., Aris, R., and Amundson, N. R., First-Order Partial Differential Equations: Volume I&II, Prentice-Hall (1986, 1989).

what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!