Models of Diffusion and Heat Transfer
This chapter describes the theory and techniques for modelling the transport processes of diffusion and heat in COMSOL. It begins with a summary of Fick’s Laws of diffusion and the physics of convective transport, before moving on to heat transfer and the
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Models of Diffusion and Heat Transfer
7.1 Diffusion Diffusion refers to the net transport of molecules from regions of high concentration to low concentration as a result of random molecular motion. Diffusion processes play an important role in a range of biological systems and bioengineering applications. Typical examples include: • • • •
exchange of metabolites between a cell and its environment diffusion of water through a semi-permeable membrane (osmosis) exchange of oxygen and carbon dioxide in the alveoli of mammalian lungs drug delivery medical devices
7.1.1 Fick’s Laws of Diffusion In 1855, Adolf Fick1 formulated two important laws governing diffusion processes based on his experimental observations using salt solutions. The first law states that the amount of substance transported by diffusion per unit time is proportional to the gradient in substance concentration. The rate of substance transported per unit area is also known as the material flux, and is a vector quantity having both magnitude and direction. Fick’s first law can be stated mathematically in terms of the material flux as Γ = −D∇c (7.1) where Γ is the material flux in SI units of mol m-2 s-1 , c is the concentration with SI units of mol m-3 , and D is the diffusion coefficient of the substance in a given medium in SI units of m2 s-1 . Note that the negative sign in Eq. 7.1 indicates that diffusion occurs down the concentration gradient, from regions of high to low concentration. 1 German
physiologist and physician, 1829–1901.
© Springer-Verlag Berlin Heidelberg 2017 S. Dokos, Modelling Organs, Tissues, Cells and Devices, Lecture Notes in Bioengineering, DOI 10.1007/978-3-642-54801-7_7
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7 Models of Diffusion and Heat Transfer
Fick’s second law describes the transient changes of concentration with time, and is written mathematically as a PDE in the concentration variable c. It can be derived from the divergence theorem (Eq. 4.5) for any vector field F:
(∇ · F) dV = V
F · dS S
where V denotes a closed region with boundary S. Substituting the material flux for F, we obtain: ∂ (∇ · Γ ) dV = Γ · dS = − c dV ∂t V V S ∂c ∇ · (−D∇c) dV = − dV V V ∂t Swapping the left and right-hand sides, and cancelling the negative sign from both, we obtain: ∂c dV = ∇ · (D∇c) dV V ∂t V Since this holds for any arbitrary volume V , it follows that the integrands must be identically equivalent. Namely, ∂c = ∇ · (D∇c) ∂t
(7.2)
Typical values for the diffusion coefficient D are 0.6 × 10−9 – 2 × 10−9 m2 s−1 for individual ions and 10−11 – 10−10 m2 s−1 for biological molecules. D itself is proportional to the velocity squared of the diffusing molecules and inversely proportional to their size. Equation 7.2 is automatically implemented in COMSOL using the Chemical Species Transport| Transport of Diluted Species physics interface,2 but can also be readily implemented using the Mathematics PDE interfaces in the base COMSOL package.
7.1.2 Example: Diffusion and Uptake into a Spherical Cell A spherical cell of radius 50 µm is taking in
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