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This solution is admissible in that the velocity in nondecreasing in going from the B C to the I C . However, it in not unique . Several values of S 2 will give admissible solutions. Suppose that the value shown here is a solution. Also suppose that dispersion across the shock causes the presence of other values of S between S 1 and S 2 . 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 S 2 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.

d x d t d S = 0 = d f d S S

With uniform initial and boundary conditions, the origin of all changes in saturation is at x = 0 and t = 0 . If f ( S ) 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.

x ( S ) = d x d t d S = 0 t = d f d S S t x Δ S = d x d t Δ S t = Δ f Δ S t

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 .

t Δ S = x Δ f Δ S , x = 1 t S = x d f d S S , x = 1

The breakthrough time , t B T , 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).

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Transport phenomena. OpenStax CNX. May 24, 2010 Download for free at http://cnx.org/content/col11205/1.1
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