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# Diffusion and heat transfer analysis - Fick's and Fourier's equations comparison

## Diffusion and heat transfer analysis - Fick's and Fourier's equations comparison

(OP)
Dear fellow researchers,

I am having problems in a Abaqus standard diffusion analysis - water vapour transfer in wooden elements. I ran different analyses:
- mass diffusion analysis of water vapour transfer in wood (using material with defined diffusion coefficients and solubility and surface concentration flux for moisture "load", Fick's law was used for diffusion)
- heat tranfer analysis, using Fourier equations.

Fourier's equations (25) for heat transfer and Fick's equations (24) for mass diffusion are mathematically analogous.

With the help of research in [1], I set the material characteristics for heat transfer in mass diffusion model, the load was in both cases applied as a factorized relative humidity for 2 years time (cycles of drying and wetting of wood, however the point was just to get the same response with Fourier's and Fick's equations).

The response I got in one point is not the same. Although i got the same range of values (__* 10^-10), the difference factor between both values is around 1.5.
Does anybody got any idea, what else could be wrong? I thank you all for the answers.

With kind regards, Uros

[1] Stefania Fortino, Florian Mirianon, Tomi Toratti: A 3D moisture-stress FEM analysis for time dependent problems in timber structures.

where ρ, cT and T are the density, the specific heat and the temperature of wood, respectively,
while λ represents the second-order thermal conductivity tensor. The analogy between
(23) and (25) is obtained by posing cT = 1 and λ = ρD.

### RE: Diffusion and heat transfer analysis - Fick's and Fourier's equations comparison

Even with forced equivalence of governing differential equations, procedures used in Abaqus for these two analysis step types (mass transfer and heat transfer) may not be the same. Take a look at the article titled „On shape stability of panel paintings exposed to humidity variations - Part 1: Modelling isothermal moisture movement” by S. Reijnen et al. Authors compare the two approaches and highlight the differences.

I would start from a very simple single elements test and proceed to full model only when good agreement between the results is obtained.

### RE: Diffusion and heat transfer analysis - Fick's and Fourier's equations comparison

(OP)
FEA way, thank your for the useful article and information.

There are of course differences between both analysis types, Reijnen wrote it best:
"Heat transfer analysis based on Fourier's law can only use temperature gradient as the driving force behind the diffusive process. Driving forces like pressure, temperature and concentration can control the diffusive process. Consequently, there has to be translated to a temperature gradient."

I believe I found the right parameters, that connect both types of analysis:
In Fourier's equations, both density and specific heat of wood were set to 1 (it is the product that matters), the same values were used for diffusion coefficients and conductivity (heat transfer) coefficients and for surface flux (concentration or heat). In the attachement, results are shown, where the same response was obtained for concentration evolution through time (t). Basically, concentration and temperature field was compared in the analysis.

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