The Peclet number is a dimensionless number used in mass transfer as the ratio of the movement of a particle due to transport in convection $($forced movement$)$ to diffusion $($random movement$).$ In this example, we will look at the Peclet number of drug particle in an artery.

The Peclet number is defined as: $Pe=\frac{Lu}{D}=\frac{\text{c}\text{o}\text{n}\text{v}\text{e}\text{c}\text{t}\text{i}\text{o}\text{n}}{\text{d}\text{i}\text{f}\text{f}\text{u}\text{s}\text{i}\text{o}\text{n}}=$

$,L$ is the characteristic length $($is equal to the diameter for a sphere$).$

$,u$ is the flow velocity

$D=$ diffusion coefficient $=\frac{kT}{6a}$

$k=$ Boltzmann constant $=1\mathrm{.}38{10}^{-23}{m}^{2}k\frac{g}{K}{s}^2$

$T=$ temperature in $K$

$=$ dynamic viscosity

$a=$ diameter of spherical particle

a$.$ Re$-$write the equation for the Peclet number by substituting in the diffusion coefficient equation.

i$.$ What happens to the Peclet number as the diameter of the particle decreases? What does this mean about how diffusion varies with particle diameter?

ii$.$ What happens to the Peclet number as the temperature increases? What does this mean about the how the diffusion varies with temperature? Which force dominates as temperature increases?

b$.$ A certain pharmaceutical $($drug$)$ is spherical with a diameter of $200nm$ and is in a blood artery with a velocity of $10c\frac{m}{sec}.$ Assume that the body's temperature is $98\mathrm{.}6F.$ Assume that the viscosity of blood at this temperature is $3{10}^{-3}Pa-s.$

i$.$ Calculate the diffusion coefficient of this particle. Be sure to include the units on the diffusion coefficient.

ii$.$ Calculate the Peclet number of the drug.

c$.$ Plot the Peclet number as a function of velocity from $0\mathrm{.}1c\frac{m}{m}in$ to $10c\frac{m}{m}in.$ Please label each axis and put a title on the graph. This graph should be a line and not points since you are plotting a function.

i$.$ What happens to the Peclet number as the velocity increases?

ii As velocity increases, which force dominates more? $($fluid flow or diffusion$)$