Diffusivity

Diffusivity is the measure of the ability of a substance to allow something to pass by diffusion. Fick’s Law states the relationship between diffusion flux, J, and the concentration gradient as the change in concentration to the change in position.

 

J = -D*dϕ/dx

 

This means that diffusion flux increases as the concentration rate decreases (diffusion rate increases), where D is the diffusion coefficient of the substance, and dϕ/dx is the concentration gradient. 

Applications

In the Anemoi UV Air Purifier, there is an internal fan that takes in the air and Micrococcus luteus bacteria solution around the purifier and transports the solution through the purifying procedure. However, there should be an ideal angular velocity of the fan such that most of the bacteria are concentrated enough within the enclosed area of the circulation to be transported through the purifier, and the bacteria are able to enter the purifier through the fan. High angular velocities may greatly diffuse the bacteria in the enclosed area, though low angular velocities would not provide enough suction to draw the bacteria in. A relationship between diffusion flux and circulation can be made to identify the most efficient angular velocity of the fan. It is evident that the diffusion rate is directly proportional to the circulation of the fluid:

 

J = k𝛤 + h

 

If the contents of fluid are mixed by a rotational force, then the diffusion flux through the fan and in the enclosed area would increase. As a result, higher circulation would increase diffusion flux and since circulation is a scalar quantity, the lowest measure of 0 circulations (no spin) still allows the substance to be diffused. Therefore, the diffusion term k𝛤 can be added to the original Fick’s Law equation to form a theoretical relation between the components of diffusion flux and circulation.

 

J = k𝛤 - D*dϕ/dx

 

The k in the equation is the proportional circulation coefficient that will affect the quality of the circulation of the fluid. Since the unit of diffusion flux is kg/m s, and the unit of circulation is m /s, the unit of k should be kg/m . The angular momentum of the fan providing the force to the fluid and the time elapsed since the circulation would be directly proportional to the diffusion flux of the bacteria. Since angular momentum, L, is the product of the sum of all of the moments of inertia I of the fan and the angular velocity of the fan (L = IΩ), increased angular velocity would increase the angular momentum of the fan, and therefore the diffusion flux of the bacteria. The cylindrical volume with a base of the enclosed area and a diminishing height will instead decrease the diffusion flux as there is less substance per region and is inversely proportional to diffusion flux. Therefore, the revised relation between circulation and diffusion flux is

 

J = Lt𝛤/V  - D*dϕ/dx

 

The desired diffusion flux of the Micrococcus luteus bacteria is 10  kg/m s and the concentration gradient is 10    kg/m . The diffusion coefficient of the Micrococcus luteus bacteria is 500 µm /s = 5 * 10    m /s. The fan has a radius of 21 cm (0.21 m) or 8.27 in and the depth of the volume of the fluid region is approximately 91 cm (0.91 m) or 35.83 in, so the volume of the fluid region is 0.040131*π m . The time that the fan takes to allow 10  kg/m s of the bacteria to flow would be decided to be 12 hours or 432000 s. Circulation is 2 times the angular velocity times the enclosed area, so 𝛤 = 2πΩr  = 0.0882*π*Ω m . To find the total moments of inertia of the fan, the individual moments of inertia of the components of the fan must be computed and summed. The fan consists of nine sectors of a ring and a tube. The sum of the moments of inertia of the nine ring sectors along the x-axis equals the moment of inertia of the entire ring along the x-axis (Σ I   = I   = m (r  + r  )/2), and the moment of inertia of the tube is I    = m (r  + r  )/2. The radii (r , r , r ) of the ring and tube are respectively 0.21 m, 0.03 m, and 0.02 m. The material used for the fan is High Impact Polystyrene (HIPS) and its density ρ is 1.045 g/cm  or 1045 kg/m . The volume of all of the ring sectors combined is πh (r   - r  ) and the volume of the tube is πh (r   - r  ). The height h of the sectors is 0.001 m and the height h  of the tube is 0.05 m. The total volume of the ring sectors is 0.0000432*π m  and the volume of the tube is 0.000025*π m . Since mass m is the product of the density of the material and its volume (m = ρV), the mass of the ring sectors is m  = 0.045144*π kg and m  = 0.026125*π kg. Thus, the moment of inertia of the full ring is 0.00101574*π kg m  and the moment of inertia of the tube is 0.00001698125*π kg m ; the total moment of inertia of the fan is 0.00103272125*π kg m  and the angular momentum of the fan is 0.00103272125*π*Ω kg m . Plugging all of the individual values into the revised diffusion flux equation, the statement changes from

 

J = Lt𝛤/V  - D*dϕ/dx

 

to the following statement

 

10  kg/m s = 2 * 0.00103272125 kg m  * 432000 s * 0.0882 m  * π  * Ω  / (0.040131*π m )  - 5 * 10    m /s * 10    kg/m 

 

10  kg/m s = 4496.201360544218 kg s/m  * Ω  - 5 * 10    kg/m s

 

(10  + 5*10   ) kg/m s = 4496.201360544218 kg s/m  * Ω 

 

Ω  = 222.40996783982123 1/s 

 

Ω = ± 14.9134 1/s

 

Ω = ± 15 1/s

 

As a result of the computations, the most efficient angular velocity of the fan, in this case, is 15 1/s. The fan can spin clockwise or counterclockwise, though it must spin at a magnitude of 15 revolutions per second. Since the fan blades are tilted at an angle and are concave, the direction of the angular velocity is crucial; therefore, the fan would spin with the concave side of the fan blades moving first.