Steady State Investigation Notes

Steady State Investigation Overview

  • Focus on steady state problems (no time derivative, diffusion only).

  • Concentration profiles determined from body and boundary conditions.

Diffusion Fundamentals

  • Consider diffusion in one-dimensional continuum systems (e.g., solid tissue).

  • Analyze diffusion across different geometries (e.g., cylindrical, spherical).

  • Understanding resistance to diffusion and relevant coefficients (e.g., absorption coefficient).

Reaction in Steady State Problems

  • Introduction of reactions leads to consumption/production of substances (e.g., drug metabolism).

  • Concentration profiles affected by metabolic processes (higher consumption equals lower concentration).

  • Key examples: Drug transport/metabolism and photosynthesis CO₂ diffusion.

Mathematical Representation

  • Reaction term as a sink; impacts diffusion model.

  • General form involving first-order reaction term (proportional to concentration).

  • Steady state assumption: no convection and simplified governing equations using differential equations.

  • Homogeneous differential equation form affording characteristic equation solutions.

Concentration Profile and Flux

  • Without a reaction: linear concentration profile.

  • With reaction: concentration profile displays exponential decay.

  • Flux analysis shows increased flux at surface due to reaction increases.

  • Mathematical demonstration involves comparing specific expression terms and using hyperbolic cotangent results.

Problem-Solving Steps

  1. Visualize problem situation (sketch).

  2. Identify governing equations.

  3. Determine boundary conditions and initial conditions.

  4. Assess reaction type (zero, first, second order).

  5. Document all assumptions during analysis.

  6. Solve equations using boundary conditions to find constants.

  7. Validate the solution against problem norms.

Application: Drug Delivery Patch Example

  • Consider transdermal diffusion of drug (e.g., scopolamine).

  • Positive boundary conditions: flux at patch-skin interface equals flux into systemic circulation.

  • Instantaneous removal at blood vessel boundary results in zero concentration at interface.