Calculator Inputs
Example Data Table
| Case | L (m) | D (m²/s) | k (1/s) | S (mol/m³·s) | Left C | Right C | Use Case |
|---|---|---|---|---|---|---|---|
| Liquid film | 0.010 | 1.2E-9 | 0.035 | 0 | 2.0 | 0.3 | Absorption with reaction |
| Membrane layer | 0.002 | 6.5E-10 | 0.012 | 0.001 | 1.5 | 0.1 | Drug diffusion |
| Catalyst washcoat | 0.0008 | 2.1E-10 | 0.18 | 0 | 0.9 | 0.2 | Reactant depletion |
Formula Used
This calculator models a one dimensional steady diffusion-reaction boundary value problem.
Finite difference form for each interior node:
Flux is estimated from Fick law:
Important dimensionless checks:
Thiele style value = L√(k / D)
Diffusion time = L² / D
How to Use This Calculator
- Enter the physical length of the film, membrane, or catalyst layer.
- Enter the molecular diffusivity of the species.
- Add the first order reaction rate constant.
- Use a source term when material is generated inside the domain.
- Select boundary types for the left and right sides.
- Enter fixed concentrations or fixed fluxes as needed.
- Increase grid nodes for steep reaction zones.
- Submit the form and review the result section above the form.
- Use CSV for detailed profile data.
- Use PDF for a compact report.
Chemistry Boundary Value Problem Guide
Why Boundary Values Matter
Boundary value problems appear often in chemistry. They describe a field between two known edges. The field may be concentration, temperature, potential, or pH. In this calculator, the main field is concentration inside a one dimensional film, membrane, catalyst pellet, or stagnant liquid layer.
Diffusion and Reaction
The model is useful when diffusion and reaction act together. Reactants enter at a boundary. They move through the domain by molecular diffusion. They may also disappear by first order reaction. A uniform source term can represent generation, dosing, or internal production.
Numerical Method
This page uses a finite difference grid. The domain is divided into equal nodes. Each interior node receives a balance equation. Neighboring nodes estimate the second derivative. Boundary rows enforce either concentration or flux values. The resulting linear system is solved directly. This makes the calculator flexible for mixed chemistry cases.
Interpreting Results
The output helps compare transport and reaction strength. The Damköhler number shows whether reaction competes strongly with diffusion. A small value means the profile stays almost linear. A large value means concentration can fall sharply inside the film. The Thiele style value gives another quick reaction-diffusion signal.
Flux and Balance Checks
Flux values also matter. They estimate material moving across each boundary. The left and right fluxes show net transport direction. The integrated reaction and source rates give an area based balance. A small residual suggests the numerical grid is consistent.
Unit Selection
Use realistic units. Length should be in meters. Diffusivity should be in square meters per second. Rate constants should be per second. Concentration should use moles per cubic meter. The source term should use moles per cubic meter per second.
Accuracy Tips
Increase the node count for steep profiles. A larger grid usually improves accuracy. It also takes more computation. Check the mesh warning before trusting a sharp curve. Export the CSV when you need profile data. Export the PDF when you need a quick report for class, lab notes, or process review.
Advanced Study
For advanced studies, compare several runs. Change one value at a time. Watch how the curve, flux, and residual change. This habit separates physical effects from input mistakes. It also makes reports easier to explain to students, engineers, and reviewers. It supports design review meetings.
FAQs
1. What does this calculator solve?
It solves a steady one dimensional chemistry boundary value problem. The model includes diffusion, first order reaction, source generation, concentration boundaries, and flux boundaries.
2. Which equation is used?
It uses D times the second concentration derivative, minus kC, plus S equals zero. This is a common steady diffusion-reaction balance.
3. What units should I enter?
Use meters for length, square meters per second for diffusivity, per second for reaction rate, and moles per cubic meter for concentration.
4. What does positive flux mean?
Positive flux follows the positive x direction. The calculator uses Fick law, so flux depends on the negative concentration gradient.
5. What is the Damköhler number?
It compares reaction strength with diffusion strength. Higher values suggest reaction has a stronger effect on the concentration profile.
6. Why should I increase grid nodes?
More nodes make the finite difference grid finer. This helps when concentration changes sharply near a boundary or reaction zone.
7. Can I use flux boundaries?
Yes. Select fixed flux for either boundary. Then enter the desired boundary flux in moles per square meter per second.
8. Is the result exact?
It is a numerical approximation. Accuracy depends on the model assumptions, input units, grid size, and physical realism of the values.