In Class Transport PPT

8/11/2019 In Class Transport PPT

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CME 490 590 Fall 2014Comfort

Two major modes of mass transport

1) Convectionbulk fluid motion

2) Diffusionmolecular motion

Random thermodynamically driven molecule motion

Molecules like to be uniformly dispersed

xAv x

Density of target

molecule A x velocity in

the x direction

Lets look at a control

volume


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What is a flux (N)?

Flux = the amount of material crossing a unit area normal to thedirection of transport in a given unit of time

Units [=] mass/area-time


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Stresses = Force/Surface Area

Shear stress- force applied tangentially (joints, eyelids on eyeballs)

Normal stress- force applied perpendicular to surface (blood, fluid flow) Viscositymeasure of frictional resistance of fluid to the flow

(higher viscosity = thicker fluid)

Density- property of how closely material is packed or arranged

Kinematic viscosity= (viscosity/density) similar to diffusivity [=] m2/s

Reynolds Numbertells us laminar/turbulent flow = inertial forces/viscous forces


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Dimensionless Analysis

Reynolds Number, Re = Lv/ (Inertial forces)/(viscous forces)

Peclet Number, Pe = vL/Dij (Mass transport by convection)/(mass transport by diffusion) >1, convection will be favored,


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Diffusion / Convection Time Scales

Quantity Length Scale

(m)

Proteins and nucleic acids 10-8

Organelles 10-7

Cells 10-5to 10-6

Capillary spacing 10-4

Organs 10-1

Whole body 100

Relevant Length Scales in

Biological Systems

0.0001

0.001

0.01

0.1

1

10

100

1000

10000

100000

0.00001 0.0001 0.001 0.01 0.1

Time,s

Distance, cm

Effect of distance on diffusion

and convection times

Adapted from: Transport Phenomena in Biological Systems by GA Truskey, F Yuan, and DF Katz


7/21CME 490 590 Fall 2014Comfort

Relative Importance of Diffusion and Convection

Molecule MW (g/mole) Dij(cm2/ s) Diffusion Time,

L2/Dij

Pe = Lv/Dij

Oxygen 32 2 x 10-5 5 0.05

Glucose 180 2 x 10-6 50 0.50

Insulin 6,000 1 x 10-6 100 1.0

Antibody 150,000 6 x 10-7 167 1.67

Particles Diameter Dij(cm2/s) Diffusion Time

(s)

Pe

Virus 0.1 m 5 x 10-8 2,000 20

Bacterium 1 m 5 x 10-9 20,000 200

Cell 10 m 5 x 10-10 200,000 2,000

For L = 100 m, and if v = 1 m/s, the time for convection is always equal to L/v = 100 s for all molecules and particles.

Adapted from: Transport Phenomena in Biological Systems by GA Truskey, F Yuan, and DF Katz



Nomenclature:

Fixed axes mass flux (what were used to):

Mass, nAz = ii[=]kg/m2s

Molar,Naz= CiVi[=] mol/m2s

Moving axes mass flux (diffusion): Mass,jAz

Molar,JAz

A

wn

mass flow rate

Notes:

1) All fluxes have 2 subscripts:

Letter 1: Molecule/component that we are examining flux of

Letter 2: Direction of flux (x, y, z)

2) Lowercase is mass, uppercase is molar



Nomenclature continued:

zAzAAz vvj

**

zAzAAz vvcxJ

zAzAAz vvcxJ

mass average velocity of mixture in z-direction

this form is seldom used

molar average velocity of mixture in z-directionconcentration

mole fraction of A

AzAAz vn

mass fraction of species A

g/cm3*cm/s [=] g/cm2-s

mass average velocity of mixture in z-direction



Solving mass transport problems

Start with a mass balance (Component i):

Typical boundary conditions:

Set flux at boundary

Set concentration at boundary

Relationship between 2 species (i.e. Henrys law)

Known reaction rate

)()(

)()()(

inConsumptioiGeneration

iMolesiMolesiAccum exitingentering



Control Volume Balance

Rate of

Accumulation ofmass of iin

control volume

=

Transport of

mass of iinto

control volume

-

Transport of mass of

iout of control

volume

+

Gain/loss of mass of

idue to chemical

reaction

Physical constraints on systemsBoundary Conditions

Set concentration at boundary Gas-liquid Interface

Relationship between 2 species (i.e. Henrys law)

Impermeable SolidNo flux at surface

Set flux at boundary

Permeable Solid

Chemical ReactionNix|1= Rix|2

Known reaction rate



Diffusion in a thin slab

Diffusive mass transport

Concentration profile in a thin slab

Helium slowlypenetratinga solid slab throughdiffusion

Top surface replacedquickly(e.g., gas is swept awayby a stream of air)

Ais the mass fraction of species A

For this case, the solid sets the average velocity to zero

and thus this is the same as fixed axes.

Silica fused plate, top & bottom in

contact with air

Air below is replaced with

He



Ficks Law

Flux of component A through species B in the z-direction due to a concentration gradient in the z-

direction

dz

dCD

dz

dDj

LD

A

w

AAB

AABAz

AAB

diffusivity

mass fraction of species A

mixture density

mass flow rate

Concentration of species A

Unit check:


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Equations of change - Mass

Mass balance equation of change (multi-componentequation of continuity)

Mass/Molar flux formAppendix B.10 (BSL)Table 7.1 (TPBS)

Mass/Mole fraction form (assuming Ficks law)Appendix B.11 (BSL)Table 7.2 (TPBS)

Definitions/interrelationshipsTables 17.7-1, 17.7-2, 17.8-1, 17.8-2 (BSL)


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Appendix B.10 (BSL)


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Appendix B.11 (BSL)


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Important Interrelationships


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Important Interrelationships

From TPBS


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Solving problems

Governing equation needs to be simplified through use

of continuity equations

Evaluate/determine occurring reactions (RA)

Apply boundary conditions

Solve

zBzAA

AABzA NNxz

xcDN

Typically need to eliminate one of these to solve


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1. Diffusion Through Skin (Cartesian)

Determine the pseudo steady state concentration

profile of a solute as it diffuses through skin. Initial

concentration at the surface is determined to be a

function of the solution concentration in a patch (at

x=0, CM=*Co) and the concentration at thecapillary is expected to be a function of the solute

concentration in blood (at x=L, CM=*CL)


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C f t

Cylindrical System Example (Annulus)

A drug is supplied in the blood. The concentration at

the interior arterial wall in Cb(r=Rb). At the outer

part of the arterial wall, conc=C0(r=R0). Determine

the steady state concentration profile and flux of the

drug into the artery at Rb. Neglect chemicalreactions.

In Class Transport PPT

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