Louisiana Tech University Ruston, LA Slide 1 Mass Transport Steven A. Jones BIEN 501 Friday, April 13, 2007

Louisiana Tech University Ruston, LA Slide 2 Mass Transport Major Learning Objectives: 1.Obtain the differential equations for mass transfer. 2.Compare and contrast these equations with heat transfer equations. 3.Apply the equations to nitric oxide transport in the body.

Louisiana Tech University Ruston, LA Slide 3 Mass Transport Minor Learning Objectives: 1.Review Continuum. 2.Describe various definitions of concentration. 3.Write down the equation for conservation of mass for a single species and a multi-component system. 4.Obtain the solution for 1-dimensional diffusion of a substance with a linear reaction. 5.Linearize a nonlinear partial differential equation. 6.Apply the Laplace transform to a partial differential equation.

Louisiana Tech University Ruston, LA Slide 4 Continuum Concept Properties are averaged over small regions of space. The number of blue circles moving left is larger than the number of blue circles moving right. We think of concentration as being a point value, but it is averaged over space.

Louisiana Tech University Ruston, LA Slide 5 Time Derivative The Eulerian time derivative still applies. 1.For mass transport, B is usually a concentration. 2.There are various ways to refer to concentration.

Louisiana Tech University Ruston, LA Slide 6 Expressions for Concentration Mass per Unit Volume. g/cm 3 Moles per Unit Volume. moles/cm 3 Example: NO has molecular weight of 30 (one nitrogen, molecular weight 14 g/mole, one oxygen, molecular weight 16 g/mole ). A 1  g/liter solution of NO has a molar concentration of 1  g/liter / 30  g/  mole, or 33 nmole/liter. Mass fraction. is the mass of species A divided by the total mass of all species. Molar fraction x (A) is the number of moles of species A divided by the total number of moles of all species.

Louisiana Tech University Ruston, LA Slide 7 Expressions for Concentration To convert mass concentration to molar concentration, it is necessary to divide by the molar mass of the species. Two species may have the same mass concentration, but vastly different molar concentrations (e.g. NO vs. a large protein. If they had the same mass concentration, the molar concentration of the protein would be much smaller than the molar concentration of the NO). If we have a pure solution of something, then the mass fraction is 1. Similarly, molar fraction may be vastly different from the mass fraction.

Louisiana Tech University Ruston, LA Slide 8 Properties of a Multicomponent Mixture Density: Mass-Averaged Velocity: Rate of reaction (rate of production): Molar-Averaged Velocity: g/(cm 3 -s)

Louisiana Tech University Ruston, LA Slide 9 Density of the System To convince yourself that Consider the following thought experiment. Add 1 liter of water to 1 liter of another liquid with density 1.2 g/cm 3. Although the density of water is 1 g/cm 3, the density of water in the mixture is (1 liter x 1 g/cm 3 )/(2 liters). I.e., in the mixture, there is 0.5 g/cm 3 of water. Similarly, there is 0.6 g/cm 3 of the other liquid. The density of the mixture is then 0.5 g/cm g/cm 3, or 1.1 g/cm 3, as expected. It is important to remember that  (A) is the density in the mixture (0.5 g/cm 3 for water), not the density of the substance itself (1 g/cm 3 ).

Louisiana Tech University Ruston, LA Slide 10 Density + 1 Liter Mass 1 Kg  =1 Kg/L 1 Liter Mass 1.2 Kg  =1.2 Kg/L = 2 Liter Mass 2.2 Kg  =1.1 Kg/L  a  =1 Kg/(2 L) = 0.5 Kg/L  b =1.2 Kg/(2 L) = 0.6 Kg/L  =  a +  b

Louisiana Tech University Ruston, LA Slide 11 Conservation of Mass In terms of mass-averaged velocity, for a single species: For a the entire system: The last equality is a result of conservation of mass. It would not hold for molar concentration. Consequence of the individual mass balances, not an independent equation.

Louisiana Tech University Ruston, LA Slide 12 Species Velocity The difference between species velocity and mass averaged velocity adds an increased level of complexity. Consider two species, where one is dominant. In I.e. velocity of both species will be about the same.

Louisiana Tech University Ruston, LA Slide 13 Mass Flux With respect to a fixed coordinate system. With respect to a mass-averaged velocity. Which leads to Tells us how rapidly substance A is moving with respect to the other substances. Compare to slide 11:

Louisiana Tech University Ruston, LA Slide 14 Subtle Point If I have the following: z =0 Will there ever be any NO upstream of z =0?

Louisiana Tech University Ruston, LA Slide 15 Fick’s Law of Diffusion In Use Fick’s Law To get

Louisiana Tech University Ruston, LA Slide 16 Fick’s Law of Diffusion If density and diffusion coefficient are constant: Becomes Which is like the equation for heat transfer. In many cases, the mass-averaged velocity will be uncoupled from the mass transport, so the equations can be solved independently of one another.

Louisiana Tech University Ruston, LA Slide 17 Sources of Nonlinearity 1.Reaction rates may be nonlinear. I.e., reaction may depend on higher powers of concentration. 2.When concentrations are high, changes in mass fractions may affect the overall density. E.g. for a 2-component system:

Louisiana Tech University Ruston, LA Slide 18 Fick’s Law of Diffusion Then:

Louisiana Tech University Ruston, LA Slide 19 Example – Transport of NO Consider the diffusion of Nitric Oxide from a monolayer of platelets: NO 2 nd order reaction depends on concentration. (where J 0 is constant)

Louisiana Tech University Ruston, LA Slide 20 Example – Transport of NO The reaction between NO and O 2 - is nonlinear in that it is the product of the NO concentration and the O 2 - concentration, and as NO is consumed, so is O 2 -. However, if we assume that the O 2 - concentration is constant, then we can “linearize” the equation such that:

Louisiana Tech University Ruston, LA Slide 21 Transport of NO Modeling Assumptions 1.Platelets adhere in a monolayer along a wall placed at z=0 and simultaneously begin to produce NO with constant flux in the positive z – direction. 2.Diffusion obeys Fick’s law. 3.All densities and diffusion coefficients are constant. 4.The wall is infinite in width and height so that diffusion occurs in the z – direction only. 5.No convection. 6.Consumption of NO by O2- follows a first order reaction (not really). 7.Constant flux of NO from the monolayer. 8.Initial NO concentration is zero. 9.Concentration of NO at infinity is zero.

Louisiana Tech University Ruston, LA Slide 22 Fick’s Law of Diffusion One-dimensional Diffusion: General Law:

Louisiana Tech University Ruston, LA Slide 23 Fick’s Law of Diffusion Rate of increase Conservation of Mass: Diffusive transport

Louisiana Tech University Ruston, LA Slide 24 Initial and Boundary Conditions 1. Initial concentration of NO throughout the medium is zero: 2. The concentration must go to zero for large values of z. 3.The flux of NO through the surface at z=0 is constant and equal to J 0 for all t > 0, i.e.: Where u(t) is the unit step function.

Louisiana Tech University Ruston, LA Slide 25 Laplace Transform Property Recall that: We will use L(s,z) to represent the Laplace transform of C A (t,z).

Louisiana Tech University Ruston, LA Slide 26 Solution to the Governing Equation Use the Laplace transform to transform the time variable in the governing equations and boundary conditions. The governing PDE transforms from: To: Or:

Louisiana Tech University Ruston, LA Slide 27 Transformed BC’s 1. Initial condition was already used when we transformed the equation. 2.The concentration must go to zero for large values of z. 3.The flux of NO through the surface at z=0 is constant and equal to J 0 for all t > 0, i.e.:

Louisiana Tech University Ruston, LA Slide 28 Match Boundary Conditions With: The 2 nd term will go to infinity for infinite z, so B=0. To apply constant flux, we must differentiate: Thus:

Louisiana Tech University Ruston, LA Slide 29 Invert the Laplace Transform The solution for the Laplace transform is: So the solution for the concentration is the inverse Laplace transform:

Louisiana Tech University Ruston, LA Slide 30 Invert the Laplace Transform In the Laplace transform, division by powers of s is the same as integration in the time domain from 0 to t. The result of this application is:

Louisiana Tech University Ruston, LA Slide 31 Invert the Laplace Transform Now we can use the time shifting property of Laplace transforms:

Louisiana Tech University Ruston, LA Slide 32 Invert the Laplace Transform Compare the form of the inverse Laplace transform in: To the following form from a table of Laplace transforms:

Louisiana Tech University Ruston, LA Slide 33 Invert the Laplace Transform Then: So:

Louisiana Tech University Ruston, LA Slide 34 Example Concentration Profiles Increasing time 5 s