Steady State Diffusion Ficks 2nd Law

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Steady State DiffusionFicks 2nd Law
27-216 Spring 2004 A. D. Rollett
Slides from Diffusion Modules by Glicksman,
except where marked
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Fluxes

• Atoms/ions/molecules jump a lot (rn) but only a
  short NET distance rvn.
• All this jumping to a flux of material,
• when a concentration gradient exists

Examinable
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Objectives

• Steady State what does this mean?
• Now that we know what diffusion is and the
  physics behind it, how do we use it in
  engineering problems?
• Our 1st diffusion problem Case study

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What does steady state mean?

• The meaning of steady state is very simple
• The concentration profile does not change
• The amount of solute in any given volume does not
  change
• We do not have to worry about changes in
  concentration with time, so t can be omitted
  from the equation
• Example (from fluid flow!) fish ladders

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http//www.nwd.usace.army.mil/ps/colrvbsn.htm
Examinable
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What do we need to know?
Examinable
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Our 1st diffusion problem A special case study
Steady State
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Design of Hydrogen Reactor Vessel

• A plant for hydrogenation of a hydrocarbon
  (current example of research in this area
  formation of methanol by hydrogenation of
  supercritical CO2) is being designed. The
  reaction vessel is to be made of low-alloy steel
  and the wall thickness depends primarily on how
  large a loss rate of hydrogen can be tolerated
  through the wall.
• The boundary conditions are as follows
• Reactor temperature 450C
• Hydrogen pressure in reactor 75 atmospheres
• Hydrogen pressure outside reactor 1 atmosphere
• Length of vessel 1 m
• Inner diameter, 2r1, of vessel 0.1 m.

PG section 13.1
1 m
NB for this problem, J total flow, jflux
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Steady-State Diffusion in Cylinders Constant D
Ficks law reduces to an ODE (Laplaces
Equation) in a single spatial variable, r .
The boundary conditions are
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Steady-State Diffusion in Cylinders Constant D
Steadystate diffusion solution for cylinders,
rinner 1.
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Design of Hydrogen Reactor Vessel, contd.

• Since we are discussing a cylindrical vessel we
  need a particular solution (to be discussed in a
  later lecture) for steady-state diffusion,
  despite the fact that we are dealing with 1D
  diffusionThe inner and outer radii are
  rinside and routside, respectively, and Cinside
  and Coutside are the concentrations at the inner
  and outer radii.
• Now this is a solution to the differential
  equation,
• At steady state, we also know from Ficks 1st Law
  that
• Also, the mass flow rate, J, at a given radius
  must be equal to the flux multiplied by the area,
  which is the perimeter, 2pr, multiplied by the
  length, l J jarea 2prlj

Examinable
A. Rollett 04
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Design of Hydrogen Reactor Vessel, contd.

• Combining the mass flow rate with Ficks 1st Law
  gives this
• Now lets insert the particular solution for the
  cylinder into this relationship
• Now we need some additional information because
  we are dealing with gases we have to delve into
  permeability.

Examinable
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Permeability

• First, we need to be able to relate gas pressures
  to concentrations based on chemical equilibrium.
  In this case we use Sieverts Law to relate
  solubility (equivalent to concentration), S, to
  (hydrogen) gas pressure, p, where K is the
  (reaction) equilibrium constant S Kvp.
  The corresponding chemical reaction is 1/2
  H2(g) H (in solution).
• Permeability, P, is defined as the product of
  diffusivity and reaction constant for the above
  reaction P P0 exp-Q/RT
• Implicit in this definition is some information
  about diffusion, so lets expand this definition.

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Permeability, contd.

• Permeation of a membrane by a chemical species
  depends on the concentration gradient and the
  diffusivity. Given concentrations defined by gas
  pressures in equilibrium with each surface (e.g.
  gas permeation of a metal foil), we can write
  Ficks 1st Law in this form

From Poirier Geiger note the non-SI units
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Design of Hydrogen Reactor Vessel, contd.

• Now, finally, we can insert values for
  concentrations related to the gas pressures
  defined in the design problem
• From Table 13.1 in PG we can estimate the
  permeability that we need for hydrogen in steel
  as P 8.4 10-6 cm3(STP) s-1 atm-1/2.
• Inserting this we obtain

Examinable
A. Rollett 04
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Design of Hydrogen Reactor Vessel, contd.

• For the design, we choose a rate at which we can
  afford to lose hydrogen from the reactor vessel.
  This then determines the outer radius of the
  vessel.

PG section 13.1
Examinable
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Divergence Operators
A useful shorthand notation in math is divergence
or div, which stands for the sum of the 1st
spatial derivatives of a vector function (e.g.
flux).
Cartesian Coordinates
Cylindrical Coordinates
Spherical Coordinates
contrast with grad, the 1st spatial
derivative of a field,but also written as ?.
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Ficks First Law vector form
One application of div is in writing Ficks 1st
Law.
Vector form of Ficks first law
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Ficks 2nd Law

• Restricting conditions such that there is no
  change in concentration with time is too
  restrictive for real life! Many real problems
  are transient.
• What happens if concentrations can change with
  time? Answer we have to be concerned with
  accumulations and losses of material in a given
  volume of material.
• Ficks 2nd Law time rate of change of
  concentration diffusion coefficient second
  spatial derivative of concentration gradient

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Ficks Second Law(Linear Diffusion Equation)
or
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Ficks Second Law (Cartesian Coordinates)
Laplacian
The Laplacian is a differential operator, i.e.
something that you do to operate on a quantity
or variable to obtain the second spatial
derivative
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What does transient mean?

• The meaning of transient is very simple
• The concentration profile does change
• The amount of solute in any given volume changes
  with time
• We do have to worry about changes in
  concentration with time, so t must be in the
  equation(s)
• Example (from fluid flow!) think about filling
  a fish ladder for the first time
• Where does Ficks 2nd Law come from? Answer we
  have to think about net accumulation ratesin
  control volumes

Examinable
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A. Rollett 04
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Gradient Vector Field
dC/dy
dC/dx
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Mass-flow Control Volume
This is where we derive Ficks 2nd Law
Control volume indicating mass flows in the
y-direction occurring about an arbitrary point P
in the Cartesian coordinate system, x,y,z.
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Mass Conservation
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Mass Conservation
or
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Exercises
1. Given the concentration field Csin(?x)
g/cm3, plot the gradient field, and then
determine the flux field using Ficks 1st law,
assuming that the diffusion coefficient, D1
cm2/s. The use of a mathematical software
package or spreadsheet provides an invaluable aid
in solving diffusion problems such as suggested
by these exercises, and many others introduced in
later chapters of this book. In one dimension,
the gradient is defined as
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Exercises
The flux field may be found using Ficks 1st law.
Gradient Plot
Flux Plot
Note These plots differ by a minus sign!
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Exercises
2. A twodimensional concentration field is given
by C(x,y)-sin(?x)cos(?y). Calculate the
gradient field, and make plots of it and the
given concentration field.
Concentration field
Gradient field
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Summary

• Now that we know what diffusion is and the
  physics behind it, how do we use it in
  engineering problems? Find boundary conditions,
  find diffusion constant data and solve PDE
• Steady state no change with time
• Our 1st diffusion problem Case study of gas
  permeation through a wall
• What about problems in which the concentration
  changes with time? We have to concern ourselves
  with net accumulation rates in a given volume.
  By taking limits (standard calculus) we arrive at
  Ficks 2nd Law.

Examinable
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