Parameters
Illustrative diffusivities, not certified material data
Diffusion coefficient D [m²/s]
10⁻¹⁴ ~ 10⁻⁸ m²/s(Log Scale)
Observation length / slab thickness L [mm]
mm
Boundary model (surface C/Cs=1)
Initial concentration ratio C0/Cs
Number of displayed times
Maximum time t_max [h]
h
Initial concentration profile
Click on the chart to set initial concentration distribution points. A concentration point is added at the clicked position with linear interpolation.
Playback Controls
Concentration Distribution Snapshot
Results
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Diffusion length sqrt(Dt) [um]
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10% concentration-increment depth [μm]
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Average concentration Cbar/Cs
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Normalized surface flux J₀/Cs [m/s]
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Time to 50% mean concentration increase [h]
Diff
CAE Applications
For comparison with analysis software, match units, the location of x=0, initial profile, observation interval, fixed surface concentration and zero back-face flux. Numerical agreement is not material safety certification.
Theory & Key Formulas
Diffusion equation for constant D
$$\frac{\partial c}{\partial t}=D\frac{\partial^2 c}{\partial x^2}$$
Semi-infinite solution for uniform q=C₀/Cs (t>0)
$$c(\xi,t)=q+(1-q)\,\mathrm{erfc}\!\left(\frac{\xi}{2\sqrt{Fo}}\right)$$
Finite slab: λₙ=(n+1/2)π and Fo=Dt/L² (t>0)
$$c(\xi,t)=q+(1-q)\left[1-\sum_{n=0}^{\infty}\frac{2\sin(\lambda_n\xi)}{\lambda_n}e^{-\lambda_n^2Fo}\right]$$
Finite-slab mean concentration
$$\bar c=q+(1-q)\left[1-\sum_{n=0}^{\infty}\frac{2}{\lambda_n^2}e^{-\lambda_n^2Fo}\right]$$
Normalized surface flux for a uniform initial profile
$$\left.\frac{J_0}{C_s}\right|_{\rm semi}=(1-q)\sqrt{\frac{D}{\pi t}},\qquad \left.\frac{J_0}{C_s}\right|_{\rm finite}=\frac{2D(1-q)}{L}\sum_{n=0}^{\infty}e^{-\lambda_n^2Fo}$$
These closed forms apply to uniform initial concentration. A drawn profile is evolved using Gaussian kernels / eigenfunctions for the same boundaries; the curve, mean and flux use that same initial profile. Animation frames are not numerical time steps controlling solution stability.