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Bioengineering

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    Molecular Diffusion

    Molecular diffusion is the transport of one mass component from a region of higher concentration to a region of lower concentration, making it a passive process.

    It equilibrates differences in concentration, due to random (Brownian) molecular movement. As temperature increases, molecular kinetic energ also increases, speeding up the process.

    It is analogous to diffusive heat transfer (thermal conduction), but can be more complicated if numerous chemical species are involved.

    When concentration gradient is the driving force, the rate of mass transport per unit area (diffusive flux) is related to it via Fick's Law:

    Fick's Law
    $$ j_{A,x} = -D_{AB} \frac{dc_{A}}{dx} $$
    where
    • \( j_{A,x} \) = diffusive flux of A, in x-direction \( \left [ \frac{kg}{m^{2} s}, \frac{kmol}{m^{2} s} \right] \)
    • \( D_{AB} \) = mass diffusivity of A in B (diffusion coefficient) \( \left[\frac{m^{2}}{s}\right] \)
    • \( c_A \) = concentration of A \( \left[\frac{kg}{m^{3}}, \frac{kmol}{m^{3}}\right] \)
    • \( x \) = distance \( [m] \)

    Consider a 1D diffusion process along the \( x \) direction. At any time, molecules have moved varying distances from the origin independently, with equal probability of moving left or right. Mean-square displacement \( \langle x^2 \rangle \) is the average of the square of displacements, and serves as a measure of how much molecules have spread.

    Diffusivity is defined as one half of the mean-square displacement per unit time in 1D:

    Diffusivity (1D)
    $$ D = \frac{\langle x^{2} \rangle}{2t} $$
    where
    • \( D \) = diffusivity \( \left[\frac{m^{2}}{s}\right] \)
    • \( \langle x^{2} \rangle \) = mean-square displacement \( [m^{2}] \)
    • \( t \) = time \( [s] \)

    The average diffusion velocity is then:

    Average Diffusion Velocity
    $$ u_{diff} = \frac{\sqrt{\langle x^{2} \rangle}}{t} = \frac{\sqrt{2Dt}}{t} = \sqrt{2D/t} $$

    where

    • \( u_{diff} \) = average diffusion velocity \( \left[\frac{m}{s}\right] \)

    Porous Media Flow

    Darcy's Law

    Porous mediums can be approximated as a bundle of tubes of varying diameter, embedded within a solid matrix. Movement of liquid in a porous material is described by Darcy's Law:

    Darcy's Law
    $$ n^v = -K\frac{d \mathcal{H} }{ds} $$
    where
    • \( n^v \) = volumetric flux (volume/area · time) \( \left[\frac{m^{3}}{m^{2} s}\right] = \left[\frac{m}{s}\right] \)
    • \( \mathcal{H} \) = hydraulic potential \( [m] \)
    • \( s \) = distance along the flow path \( [m] \)
    • \( K \) = hydraulic conductivity \( \left[\frac{m}{s}\right] \)

    The hydraulic (water) potential \( \mathcal{H} \) drives flow, and is defined as:

    Hydraulic Potential
    $$ \mathcal{H} = h + z $$
    where
    • \( h \) = pressure (hydrostatic or pneumatic) and matrix potential — physical forces which bind water to the porous matrix; attractive force between liquid and solid phases
    • \( z \) = gravitational potential due to difference in depth in the vertical direction (i.e., from surface), in the direction of gravity

    Hydraulic conductivity \( K \) describes the ease with which a fluid can be transported through a porous matrix — it is the ratio of volumetric flux to the hydraulic gradient causing the flow. It depends on properties of both the solid matrix (pore size distribution, shape of pores, porosity, tortuosity) and the fluid (density, viscosity):

    Hydraulic Conductivity
    $$ K = \frac{\rho g}{\mu} \frac{1}{8\tau} \sum_{i} \Delta\beta_i r_{i}^{2} $$
    where
    • \( \rho \) = fluid density
    • \( \mu \) = fluid viscosity
    • \( \tau \) = tortuosity — how twisted or complicated the flow path is (approximately 1–2 for soils)
    • \( \Delta\beta_i \) = volume fraction of pores with radius \( r_i \) (area fraction, 0–1), defined as \( \Delta\beta_i = \frac{\omega_i \pi r_i^2}{A} \) where \( \omega_i \) is the number of pores in the \( i^\text{th} \) pore size class with radius \( r_i \)
    • \( r_i \) = radius of pores in size class \( i \) \( [m] \)

    The sole effect of the matrix properties can be isolated as the permeability (\( k \)):

    Permeability
    $$ k = \frac{1}{8\tau} \sum_{i} \Delta\beta_i r_i^2 $$
    where \( k \) has units of \( [m^2] \). Hydraulic conductivity can then be written as:
    Hydraulic Conductivity (from Permeability)
    $$ K = \frac{k \rho g}{\mu} $$

    separating the effect of the matrix (captured in \( k \)) from the effect of the fluid properties (\( \rho \), \( \mu \)).

    Porosity and Flux

    Retention of water in a porous medium results from attractive forces between the solid and liquid phases.

    Volumetric porosity (\( \phi \)) is the ratio of the volume of void space (pore volume) to the bulk volume of a porous medium. Average volumetric porosity is considered the same as average areal porosity at a cross section of the material:

    Porosity
    $$ \phi = \frac{\text{pore volume}}{\text{total volume}} = \frac{\text{pore area}}{\text{total area}} $$

    Volumetric flux (\( n^v \)) is sometimes called "Darcy velocity", but it is not the true average velocity through the cross-sectional area. The true average velocity through the pores is much higher than that through the medium (solid matrix):

    Average Pore Velocity
    $$ v_{avg} = \frac{n^v}{\phi} $$

    The average velocity is a vector with direction and magnitude.

    Capillary Flow

    Capillary flow describes capillary action (flow of liquid through a narrow space) within a porous solid medium. It is the balance between the relative attraction of the molecules of the fluid for each other (liquid-liquid) and for those of the surface (liquid-solid).

    When liquid-solid forces are stronger, the liquid "climbs" against gravity (and other forces). This phenomena, called capillary rise, equates hydrostatic pressure \( P = \rho g h \) and capillary action pressure \( \frac{2\gamma}{r} \) to yield:

    Capillary Rise
    $$ h = \frac{2\gamma}{\rho g r} $$
    where
    • \( h \) = height of the column or capillary \( [m] \)
    • \( \gamma \) = water surface tension \( \left[\frac{N}{m}\right] \) (look up values)
    • \( \rho \) = fluid density \( \left[\frac{kg}{m^3}\right] \)
    • \( g \) = gravitational acceleration \( \left[\frac{m}{s^2}\right] \)
    • \( r \) = radius of tube \( [m] \)

    Capillary rise increases significantly as tube radius decreases. If the tube diameter is sufficiently narrow, liquid is "propelled" upward by the combination of surface tension and liquid-surface adhesive forces.

    Unsaturated Flow

    Recall that in a porous solid, capillarity causes liquid to be attracted more strongly when there is less of it (i.e., at lower concentration); causing liquid to flow across its concentration gradient. In unsaturated environments, the difference in negative pressures (matric potential) drives flow instead.

    Unsaturated flow is mathematically similarly to molecular diffusion, and can be modeled as:

    Unsaturated Flow
    $$ n = -D_{cap}\frac{dc}{ds} $$

    where \( D_{cap} \) is capillary diffusivity; material-specific, which is derived experimentally.

    Osmotic Flow

    Osmosis is the transport of a solvent from low to high solute concentration through a semi-permeable membrane. The porous structure allows passage of solvent but not the solute/medium, until a hydraulic pressure (\( \Pi \)) develops to oppose the osmotic flow. Osmosis occurs in the direction of reducing the solute concentration gradient (i.e., equalizing concentration) — for example, water flowing across cell membranes.

    The relationship between hydraulic pressure and solute concentration is described by Van't Hoff law:

    Van't Hoff Law
    $$ \Pi = cRT $$
    where
    • \( \Pi \) = hydraulic or osmotic pressure
    • \( c \) = total concentration of the solute
    • \( T \) = absolute temperature \( [K] \)
    • \( R \) = gas constant \( \left[\frac{J}{mol \cdot K}\right] \)
    Gummy bear osmosis experiment: water travels against the solute concentration gradient through the gelatin, swelling the candy. (Martin Leigh, Getty Images)

    Summary Table

    Mode Description Described by
    Molecular diffusion Spontaneous movement of mass (solid, liquid, gas) down a concentration gradient due to random molecular movement Fick's Law: \( j_{A,x} = -D_{AB}\frac{dc_{A}}{dx} \)
    Capillary diffusion Movement of mass due to capillary action in a porous medium (mathematically similar to molecular diffusion)
    Bulk flow through porous media Bulk movement of a fluid in a porous media due to hydraulic forces (pressure, matric forces, gravity) Darcy's Law: \( n^{v} = -K\frac{\partial \mathcal{H}}{\partial s} \)
    Dispersion Spreading of fluid from its bulk flow path (similar effects to molecular diffusion, but different mechanisms; due to turbulence in fluid) \( j_{A,x} = -E_{x}\frac{dc_{A}}{dx} \)
    Convection Addition of bulk flow to diffusion or dispersion \( N_{A_{1-2}} = h_{m}A(c_{1}-c_{2}) \)