What is the complete derivation of the nonhomogeneous fluid flow rate equation?

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Discussion Overview

The discussion revolves around the derivation of the general solution for the nonhomogeneous fluid flow rate equation, as referenced from a specific textbook. Participants are seeking clarity on the mathematical steps involved in the derivation process.

Discussion Character

  • Technical explanation

Main Points Raised

  • One participant requests the complete derivation of the nonhomogeneous equation from a specific textbook.
  • Another participant provides a partial derivation using calculus, presenting equations related to the change in concentration over time.
  • Subsequent posts highlight a missing step in the derivation, specifically the completion of the equation involving the natural logarithm of the concentration difference.
  • There is a reiteration of the need for clarity on the equation involving the logarithmic expression, with a specific reference to its equivalence to another expression presented earlier.

Areas of Agreement / Disagreement

Participants do not appear to reach a consensus, as there are requests for further clarification and completion of the derivation steps, indicating ongoing uncertainty and debate regarding the mathematical process.

Contextual Notes

The discussion highlights missing steps in the derivation and dependencies on specific mathematical assumptions, particularly concerning the logarithmic transformation and its implications for the solution.

fahadismath
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does anyone know the derivation of the general solution of the nonhomogeneous equation shown in the image (book name: Devendra K. Chaturvedi - Modeling and Simulation of Systems Using MATLAB and Simulink -CRC Press (2010))
Capture.PNG
 
Last edited by a moderator:
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This is basic calculus.
\begin{split}<br /> \frac{dC}{dt} &amp;= \frac{F}{V}(C_0 - C) \\<br /> \int \frac{1}{C_0 - C}\frac{dC}{dt} \,dt &amp;= \int \frac{F}{V}\,dt \\<br /> \int \frac1{C-C_0}\,dC &amp;= \ln |k| - \frac{F}{V}t \\<br /> \ln |C - C_0| &amp;= \\<br /> C(t) &amp;= C_0 + ke^{-Ft/V}.\end{split} (We can drop the absolute value signs since C - C_0 and k must have the same sign.)
 
in your derivation you didn't complete this step ln|C-Co|= ? KINDLY write the complete equation
 
fahadismath said:
in your derivation you didn't complete this step ln|C-Co|= ? KINDLY write the complete equation
It equals the same as the right side of the line above it: ##\ln |k| - \frac{F}{V}t##.
 

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