Flicks diffusion equation - metabolism rate

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SUMMARY

The discussion focuses on solving the Flicks diffusion equation related to the metabolism rate of an organic chemical in a bio-film reactor. The concentration parameters include an inlet concentration (Cin) of 2 mg/L and a flow rate (f-in) of 20 m³/hr. The equation presented is e*dc/dt = d*(d²C/Dz²) - R, where C3 represents the concentration in the bio-film, and constants e (porosity), d (diffusion coefficient), and R (metabolism rate) are defined. The user seeks assistance in applying separation of variables to find a particular solution to the equation.

PREREQUISITES
  • Understanding of Flicks law of diffusion
  • Knowledge of bio-film reactor dynamics
  • Familiarity with separation of variables in differential equations
  • Basic concepts of metabolic rates in biochemical processes
NEXT STEPS
  • Study the application of separation of variables in solving partial differential equations
  • Research the specific characteristics of bio-film reactors and their impact on diffusion
  • Explore the derivation and application of the heat equation in biochemical contexts
  • Investigate methods for approximating solutions to nonlinear differential equations
USEFUL FOR

Researchers, chemical engineers, and biochemists involved in bio-film reactor design and optimization, as well as students studying diffusion processes in biochemical systems.

gladerider
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So i have an organic chemical in a bio-film reactor being diffused into a bio-film and also being metabolized at constant rate R by bacteria.

the concentration into the reactor Cin = 2mg/L with f-in at 20 m^3/hr
the concentration in the reactor is C1
C3 is the concentration in the bio-film

flicks law of diffusion:

e*dc/dt = d*(d^2C/Dz^2) - R where c = C3

where e, d, and R are constants.
e-porosity
d - diffusion coefficient

with bc's ;
dC3/dz = 0 @ z = 2micrometers
C3 = C1(.005mg/l) @ z = 0


I've was trying to solve with separation of variables but I'm rusty and running into some problems... Please help!
 
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The general solution will be a particular solution + the general solution of the heat equation. I hope we can find a particular solution (even approximate) in polynomials is z,t .
 

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