Fick's First Law Diffusive Flux, Peclet (Pe) & Schmidt (Sc) Numbers

Declare direction, concentration units and the transport model

Let position x increase from C1 at x=0 to C2 at x=dx. With constant diffusivity D and a linear concentration profile, J=-D(C2-C1)/dx gives the diffusive molar flux in that positive direction. Concentrations are mol/m³, D is m²/s, dx is metres, and J is mol/(m² s).

A positive J means diffusion from endpoint 1 toward endpoint 2; reversing the concentration difference reverses the sign. Fick’s local gradient law also applies in time-dependent models. The finite difference used here assumes a linear profile, as in a steady one-dimensional, constant-D, no-source layer.

This is diffusive flux relative to the bulk motion, not total advective-plus-diffusive flux. A membrane may require partitioning, effective diffusivity, reactions or other transport terms. Equal endpoint concentrations yield zero in this linear model; they do not prove system-wide thermodynamic equilibrium.

Mass Peclet number is |v|L/D and Schmidt number is nu/D. L is the chosen characteristic length, not necessarily the layer thickness dx. A large Pe compares transport scales along that length; it does not eliminate diffusion in other directions. A value of Sc alone does not identify a gas or liquid phase.

Signed molar diffusive flux in D2

Reverse C1 and C2 to reverse the flux. Keep the unrounded value for subsequent calculations.

=IF(COUNT(B2:B5)<>4,"Enter numeric diffusion inputs",IF(OR(B2<=0,B3<0,B4<0,B5<=0),"Require D>0, concentrations>=0 and thickness>0",-B2*(B4-B3)/B5))

Mass Peclet magnitude in D3

The velocity sign indicates direction, while this dimensionless magnitude uses ABS(v). Zero velocity gives Pe=0.

=IF(COUNT(B2,B6:B7)<>3,"Enter numeric Peclet inputs",IF(OR(B2<=0,B7<=0),"Require D>0 and length>0",ABS(B6)*B7/B2))

Schmidt number in D4

The two diffusivities must use the same area/time units.

=IF(COUNT(B2,B8)<>2,"Enter numeric Schmidt inputs",IF(OR(B2<=0,B8<=0),"Require D>0 and viscosity>0",B8/B2))

Reynolds number for the same velocity and length in D5

Using the same v,L,nu and D gives the algebraic identity Pe=Re×Sc. This is a useful units check, not a separate physical validation.

=IF(COUNT(B6:B8)<>3,"Enter numeric Reynolds inputs",IF(OR(B7<=0,B8<=0),"Require length>0 and viscosity>0",ABS(B6)*B7/B8))

Further reading

• Microsoft Support — ROUND function
• COMSOL — Convection–diffusion equation and Peclet number