BIOM9311 Mass Transfer in Medicine

Week 1

C

1C

2

x x1 2

dcJ D

dx

C1

C4

x1 x2 x3 x4

membrane

fluid 1 fluid 2

3/03/2015 Week 1 2

What is life?

Life is natures way of keeping meat fresh.

(Dr. Who?, 23/7/05)

Life is a set of ongoing controlled chemical reactions

occurring in prescribed locations.

Reactions need reactants

Delivery of reactants to the reaction site is mass

transfer (in part)

Mass transfer applications

3/03/2015 Week 1 3

Blood processing

Haemodialysis

Oxygenator

Peritoneal dialysis

Haemofiltration

Plasmapheresis

WithIn the body

Lungs

Kidneys

Capillaries

Tissue engineering

Biosensors

Drug delivery

Implants

Skin patches

3/03/2015 Week 1 4

Background we need

We are dealing with solutes in different phases

Liquids (water, blood, lipid bilayer membrane)

Gases (air)

Membranes

We have to know

how species move within phases

what governs the distribution of species between

phases

We need a review of physical chemistry!

3/03/2015 Week 1 5

Background we need (II)

We are interested in mass transfer devices or

physiological systems, often with fluid(s) flowing in

channel(s), always with a diffusion barrier

Haemodialyzer, oxygenator

Blood capillary in tissue, lung, skin

We have to know

The properties of the barrier

The effect of flow rates on the overall rate of transfer

The effect of flow channel dimensions

3/03/2015 Week 1 6

Todays topics

Physical chemistry review

Material balances

Diffusion at a point

The diffusion coefficient

Diffusion across a finite distance

Diffusion across a membrane

Diffusion across a series of diffusion

resistances

Physical Chemistry Review

3/03/2015 Week 1 7

Ideal gas

3/03/2015 Week 1 8

PV nRTT0 = 273.16K (0C) P0 = 1 atm = 760 mmHg

= 101.3 kPa

= 1.013 106 dyne/cm2

V0 = 22.4 L for 1 gram-mole of gas

R = Universal Gas Constant

= 0.0821 atm L/K mol

= 8.31 44 Pa m3/K mol

1.99 cal/K mol

= P0V0/T0 *** (I use this often)

Real gases at atmospheric pressure are approximately

ideal.

Advice: Implement this

in Excel (say). Store

conversion factors for

P and values of R.

Gas mixtures - partial pressure

3/03/2015 Week 1 9

Pn RT

VRT

n

Vi

i i

P P nRT

Vn

RT

Vi i T

Pn

nP x Pi

i

Ti

The partial pressure of a gas is a measure of its concentration

ii

T

nx

n Mole fraction

Thermodynamics - a brief encounter

3/03/2015 Week 1 10

V volume

S entropy

i chemical potential of species i

Gibbs free energy per mole

the driving force for mass transfer by diffusion

(among other processes)

dGG

TdT

G

PdP

G

ndn

P all n T all n i P T all n

i

ii i j

, , , ,

dG SdT VdP dni ii

Gibbs free energy - f (T,P,composition)

Phase equilibrium

3/03/2015 Week 1 11

Given a closed system at equilibrium

G is minimum

dG = 0

Two phases , in equilibrium:

T T

P P

i i

The chemical potential of a

species is the same in both

phases at equilibrium

Phase: 4. Chem. A physically distinct and homogeneous form of matter characterized by

its state (as gas, liquid, or solid) and composition and separated by a bounding

surface from other forms. (OED)

Ideal solutions

3/03/2015 Week 1 12

A binary solution of A and B and its equilibrium vapour phase:

soln vapour

A A

soln vapour

B B

XA XB

PA PB

Therefore

lnvapour oA A ART P

lnsoln oA A ART P Or better:

Asoln

Ao P

PRT A

Ao ln

Ideal solution (rare):

P x PA A Ao A A

oART x

soln ln

But Solution

Vapour

The chemical potential of A

molecules in solution is less

than the chemical potential of

pure A

Osmotic pressure

3/03/2015 Week 1 13

When you dissolve a solute in a solvent,

you lower the chemical potential of the

solvent.

If the solution is separated from pure

solvent by a semi-permeable membrane

(permeable to the solvent but not the

solute), solvent will flow from the pure

side to the solution side.

If sufficient pressure is applied to the

solution, the osmotic flow can be

overcome.

This pressure is the osmotic pressure. It

is the hydrostatic pressure required to

overcome the osmotic flow.

A A+B

A A

Osmotic pressure

3/03/2015 Week 1 14

At equilibrium we have

A A

lno AAA oA

PRT

P

A Ao

P P

but

whereas

So we increase P until there is

no flow

is the osmotic pressure. It is the hydrostatic pressure required to

overcome the osmotic flow. This is the definition of osmotic

pressure.

P

P = P+

A A+B

A A

Osmotic pressure

3/03/2015 Week 1 15

Vant Hoffs Law (ideal solutions)

cRT

c RTii

Where c is the molar concentration of B. If more than one

solute (e.g., an electrolyte that dissociates into 2 or more

ions)

Most solution are not ideal. We refer to the osmolarity, the

concentration of the ideal solution that has the same

osmolarityRT

( ) 19.3 ( / )mmHg c mOsmol L At 37C

Van't Hoff

http://en.wikipedia.org/wiki/Van't_Hoff

Concentrations

3/03/2015 Week 1 16

A concentration is a ratio: amount of X

amount of Y

We will encounter a wide variety ways of expressing a

concentration. For example

moles of i (mol/L)

litre of solutionic

2

2ml O ( )

ml bloodO

STPc

Molar:

Blood gases:

Why specify STP?

Mole fraction: moles A

total molesAx

Vapor/liquid equilibrium: Henry's law

3/03/2015 Week 1 17

Some solubility coefficients in water at 37C (ex Guyton)

(ml (STP)/ml solution/atm)

i ic kP

gas k

Oxygen 0.024

Carbon dioxide 0.57

Carbon monoxide 0.018

Nitrogen 0.012

Helium 0.008

Empirically, there is a linear relationship between the partial

pressure of a gas, Pi, and its equilibrium concentration in

solution, ci.

High solubility of

CO2 in water

has implications

Liquid/liquid equilibrium:

Partition coefficients

3/03/2015 Week 1 18

For species i, at equilibrium

I II

iII

iI

II I

i i ic c

solute phase I Phase II

glycerol water olive oil 0.000068

ethanol " " 0.022

N-propanol " " 0.1

Define a partition coefficient i

such that

Does this

series make

sense?

Write the

formulas of

these three

alcohols.

The

circulation

(for reference)

3/03/2015 Week 1 19

5%

right

heart left

heart

lung

heart

liver

brain

gut

muscle

kidneys

skin,bone

15

%

35%

15%

20%

10%

The mass (material balance)

3/03/2015 Week 1 20

The solutions of many mass transfer problems begin with a

mass (or material) balance on one or more chemical species.

A material balance is expressed in terms of an extensive

property about which conservation statements can be made.

Intensive properties

Concentration in moles/L

Temperature

Density

Extensive properties

Mass in kg or moles of a chemical compound

Number of moles of an element

The volume in L of a fluid

The mass (material balance)

3/03/2015 Week 1 21

Define a species (X) and a control volume.

Identify every process that adds that species to or

removes it from the control volume.

Derive an algebraic equation (if steady state).

Derive a differential equation (if not steady state).

rate of

accumulation

rate of

input

rate of

output

rate of

synthesis

rate of

destruction

input output accumulation

3/03/2015 Week 1 22

Example

Rule: Volume flow rate in = Volume flow rate out

210 ml/min

220 ml/min

175 ml/min

? ml/min

Under what

Assumptions?

Liquid flowing into and out of a device, steady state

3/03/2015 Week 1 23

Example

Assumptions?

210 ml/min

220 ml/min

175 ml/min

185 ml/min

c = 0.9M

c = 0.4M

c = 0.5M

c = ? M

Rule: Volume flow rate in = Volume flow rate out

Rule: Solute X mass flow rate in = Solute X mass flow rate out

Solutions flowing into and out of a device, steady state

3/03/2015 Week 1 24

Example What is rate of solute transfer from white fluid to red fluid?

210 ml/min

220 ml/min

175 ml/min

185 ml/min c = 0.9 M

c = 0.4 M

c = 0.5 M

c = 1.09 M

?N

3/03/2015 Week 1 25

Example 300 ml/min

200 ml/min

300 ml/min

200 ml/min

c = 0 M

Minimum possible c = ? M

Maximum possible c = ? M

c = 2 M Minimum possible c = ? M

Maximum possible c = ? M

3/03/2015 Week 1 26

Example

300 ml/min

200 ml/min

c = 0

c = 2 M Minimum possible c = ?

200 ml/min

300 ml/min

Maximum possible c = ?

3/03/2015 Week 1 27

Example

300 ml/min

200 ml/min

c = 0

c = 2 M

Minimum

possible c = ?

200 ml/min

300 ml/min

Maximum

possible c = ?

Diffusion - microscopic theory (ex Berg: Random walks in biology)

3/03/2015 Week 1 28

The mean kinetic energy (k.e.) of a molecule due to thermal

motion is 3/2 kT split 3 ways among the x, y, z directions. (k is

Boltzmanns constant = R/Navog.)

For the x-direction 21 1

2 2. . xk e m v kT

v kT mx2

12 /

The r.m.s x velocity is therefore

How fast is that?

example: Lysozyme (MW = 14,000) at 310 K

3/03/2015 Week 1 29

7 -1 -11 22 1-1

8.31 10 erg K mol 310K1356cm s

14000 g molx

kT RTv

m M

Of course our lysozyme molecule wont travel very far in a

straight line. It will bump into a water molecule and be sent

off in a random direction.

Hence the random walk

Large molecules move more slowly

The 1D random walk

3/03/2015 Week 1 30

Rules

1. A particle starts at x = 0 and moves L or R with a velocity vx

once every seconds. The distance travelled each step is vx

= .

2. Moving left and right are equally likely and each step is

independent of all previous steps: P(L) = P(R) = 1/2.

3. Each particle is independent of all other particles (dilute).

diffdemo1a.m

3/03/2015 Week 1 31

From rule one, the position of particle i after step n is

Averaging over all particles

We are not making

any progress!

No net movement

x n x ni i( ) ( ) 1

1

1

1

1( ) ( )

1( 1)

1( 1)

( ) ( 1)

N

i

i

N

i

i

N

i

i

x n x nN

x nN

x nN

x n x n

The +s and s will

tend to cancel out,

so if N is large

3/03/2015 Week 1 32

The particles do become spread out, however, some to the

left and some to the right.

A measure of this is the r.m.s displacement

Again, starting from Rule one

Squaring,

taking the average, again using Rule two,

x n21

2( )

x n x ni i( ) ( ) 1

x n x n x ni i i2 2 21 2 1( ) ( ) ( )

x n x n2 2 21( ) ( )

x n n2 2( )

or

3/03/2015 Week 1 33

If we define

we get

or

The spread is proportional to the square root of t.

D

2

2

x t Dt2 2( )

x t Dt21

22( )

Is diffusion fast or slow?

Diffdemo1a.m, Diffdemo2.m, Diffdemo3.m

The characteristic time for a diffusive process is

(forget about the 2)

2

difft L D

Diffusion: Macroscopic theory

3/03/2015 Week 1 34

Flux

3/03/2015 Week 1 35

The flux of X is the flow of X, per unit

area, with respect to a coordinate system.

If the mean relative velocity of X is v and

its concentration is ci, the flux is v

-2 -1 ML TJ cvkg m-2 s-1

Convection

3/03/2015 Week 1 36

The flow of matter (or heat) due to the

bulk motion of a fluid.

If the fluid is moving with velocity v, the

convection flux of species i is

v ci

i iJ c v

If a fluid is flowing in a conduit with cross-

section area A, the total flow of species i is

i i i iN J A c v A Qc Qci

Diffusion

3/03/2015 Week 1 37

There has been net movement

red molecules to the right

yellow molecules to the left

After mixing, no more net movement

before after

Random motion + concentration gradient diffusion

diffdemo2a.m

http://en.wikipedia.org/wiki/Molecular_diffusion

http://en.wikipedia.org/wiki/Diffusion

These 2 wiki pages are of interest primarily for the graphics

http://en.wikipedia.org/wiki/Molecular_diffusionhttp://en.wikipedia.org/wiki/Molecular_diffusion

Ficks (first) Law

3/03/2015 Week 1 38

Postulate: there is a linear relationship between the diffusion

flux of species i and the concentration gradient.

ixi i

dcJ D

dx

In one dimension (x)

Units: [J] = ML-2T-1

[D] = L2T-1

Warning: The length units on J come from c, x, and D. Serious

potential for error!

Diffusion coefficient D

3/03/2015 Week 1 39

Depends on - the diffusing species (size of the molecule)

- the medium in which it diffusing

Much faster in gases than in liquids

An approximate theory follows

Nernst-Einstein equation

3/03/2015 Week 1 40

So

D = thermal energy / friction factor

k Boltzmanns constant =R/Navog

FA friction force on an A molecule moving with velocity vA

fA is a friction factor = FA/vA

DkTv

F

kT

fAB

A

A A

Stokes law

3/03/2015 Week 1 41

The friction factor for a spherical molecule of radius rA in a

medium with viscosity B is

f rA B A 6

DkT

rAB

B A

6

Stokes-Einstein

For a spherical molecule, molecular weight MA, density A , the

radius is

rM

NA

A

A Avog

3

4

13

What is viscosity?

(resistance to flow)

Example

3/03/2015 Week 1 42

For aqueous solutions at 37C, assume solute molecular density ~1

For M = 1000 Da, we find

r = 7.35x10-10 m

D = 4.48x10-10 m2 s-1

1 Da = 1 g mol-1

Advice:

Implement this in Excel (say).

Make a table of MW vs. D

R = 8.314107 erg mol

-1 K

-18.314 J mol

-1 K

-1

T = 310 K

N Avog = 6.02 x 1023 mol-1

A = 1 g cm-3 1000 kg m-3

B = 0.0069 dyne s cm-2 0.00069 Pa s

Some diffusion coefficients at 37C (Keller)

3/03/2015 Week 1 43

Substance Medium D (cm2/sec)

Oxygen Water 3.1 10-5

Carbon dioxide " 2.6 10-5

Urea " 1.9 10-5

Albumin " 1 10-6

Oxygen air 0.15

A series from Colton

3/03/2015 Week 1 44

The one-dimension diffusion equation (Ficks 2nd law)

3/03/2015 Week 1 45

Assume concentration depends on x only and that the

velocity is zero. (Diffusion only.)

x x+dx

diffusion

in at x diffusion

out at x+dx

Material balance: accumulation = input - output

( , )( , ) ( , )x x

c x tdxdydz J x t dydz J x dx t dydz

t

3/03/2015 Week 1 46

Dividing by dxdydz we get

( , ) ( , ) ( , )( , ) x x xJ x t J x dx t J x tc x t

t dx x

xJc

t x

or simply

x

cJ D

x

2

2

c cD

t x

Substitute Ficks first law

to give

Ficks 2nd law in 1D

Steady diffusion through a stationary film

3/03/2015 Week 1 47

Given:

= 0 no flow

c = c(x) one dimension, steady state

Boundary conditions

c(x1) = c1 c(x2) = c2

0c J

t x

Apply Ficks 2nd law

Flux is independent of position if SS. Therefore the

concentration profile is linear.

(steady state)

C

1C

2

x x1 2

dcJ D

dx

3/03/2015 Week 1 48

Substitute Ficks 1st law. We have several ways to express the

flux:

1 2 1 2 1 21 2

1 2 1 2

constantdc

J Ddx

c c DD c c k c c

x x

c c c c

D R

The quantity D/ has the units L T-1.

and is known as the mass transfer coefficient, k.

The inverse, /D is the mass transfer resistance.

Sometimes it is easier to work in terms of k, other times it is more

convenient to work in terms of resistances.

1 2x x

R D

Diffusion through a membrane

3/03/2015 Week 1 49

Assume that

the membrane is homogeneous

the diffusion coefficient of the solute

in the membrane is Dm = 0

Steady State

Again, flux J is independent of position

and is given by

1 2m

m m

DJ c c

C1

C2

C1m

C2m

c1m is the concentration

within the membrane at

surface 1.

3/03/2015 Week 1 50

To express the flux in terms of the concentrations in the fluid

phases, we use partition coefficients

1 1m mc c

1 2 1 2m m

m

DJ c c P c c

We now write

PD

mm m

The permeability of a membrane to a solute is determined by

three factors:

1.The membrane thickness .

2.The chemical interaction between the solute and the

membrane m (in effect, solubility)

3.The frictional interaction of solute and membrane,

through Dm.

Resistances in series

3/03/2015 Week 1 51

Usually we must consider 2 or more

resistances in series, as in this 2-

chamber diffusion apparatus. We want to

relate the flux to the difference between

the measured concentrations in samples

taken from the chambers.

A conceptual model is at the right.

Well mixed x < x1 and x > x4

Unstirred fluid x1< x < x2, x3< x < x4

Membrane x2< x < x3

C1

C4

x1 x2 x3 x4

membrane

fluid 1 fluid 2

3/03/2015 Week 1 52

Assume

= 0

1 = 2 = 3 = 1

SS

For any x1 < x < x4 the flux is independent of x. (Why?)

For the 3 phases we have

constant 1,2,3ndc

J D ndx

3 2 4 32 11 2 3

2 1 3 2 4 3

c c c cc cJ D D D

x x x x x x

3/03/2015 Week 1 53

Solving for c1 - c2 etc.,

The sum

2 11 2 1

1 1

( )J x x Jc c JR

D P

3 22 3 2

2 2

( )J x x Jc c JR

D P

4 33 4 3

3 3

( )J x x Jc c JR

D P

1 4 1 2 31 2 3

J J Jc c J R R R

P P P

3/03/2015 Week 1 54

Defining the overall permeability

we find

or in terms of resistances

Easier to work with resistances when we have multiple layers!

1 4o

JP

c c

1 1 1 1

1 2 3P P P Po

R R R Ro 1 2 3

References

3/03/2015 Week 1 55

Berg HC: Random walks in Biology. Princeton, Princeton

University Press (1993).

Cussler EL: Diffusion: Mass transfer in fluid systems.

Cambridge, Cambridge University Press (1984).

Keller KH: Fluid mechanics and mass transfer in artificial

organs. (1973).

Fournier RL: Basic transport phenomena in biomedical

engineering. Taylor and Francis (2012)

Trusky GA, Yuan Fan, Katz DF: Transport Phenomena in

Biological Systems, Pearson Prentice Hall (2004). GSBME

library.