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Integration help

  1. Dec 1, 2015 #1
    1. The problem statement, all variables and given/known data

    I am working on a mass transfer problem and have this equation:

    d/dt(Vc1) = Ak[c1(sat) - c1]


    2. Relevant equations

    Initial conditions:

    x=0, c1=0

    3. The attempt at a solution

    I know that the result from integration should be:

    c1/c1(sat) = 1 – e-(kA/V)t

    But I don't understand the integration that got there. If someone could show me the steps, that would be very helpful!
    Thanks!
     
  2. jcsd
  3. Dec 1, 2015 #2

    Mark44

    Staff: Mentor

    What are the symbols here? What's the difference between c1 and c1(sat). Does Vc1 mean V * c1 or is it ##V_{c_1}##
     
  4. Dec 1, 2015 #3
    Thanks for the response, c1 is the concentration c1(sat) is the saturated concentration. It is supposed to be V*c1.
     
  5. Dec 1, 2015 #4

    RUber

    User Avatar
    Homework Helper

    ## \frac{d}{dt}Vc_1 =Ak [ c_1 (sat) - c_1 ]##
    This problem is easier to do as a differential equation.

    Let's call this ##V c'(t) = Ak c_{sat} - Akc(t) ##.
    This gives:
    ##V c'(t) +Akc(t) = Ak c_{sat}\\
    c'(t) + \frac{Ak}{V} c(t) =\frac{Ak}{V} c_{sat} ##
    A general solution to
    ##c'(t) + \frac{Ak}{V} c(t)=0## is ##c(t) = N e^{- \frac{Ak}{V} t} ##.
    Where N is a constant.
    Using this general solution, you can solve for a particular solution that satisfies the differential form you were given and initial conditions.
    Notice that you can just add ##c_{sat}## to ##c(t) ## without affecting the derivative, since it is a constant.
    So a solution to
    ##c'(t) + \frac{Ak}{V} c(t)= \frac{Ak}{V}c_{sat}## is ##c(t) = N e^{- \frac{Ak}{V} t} +c_{sat} ##.
    Now, use your initial condition to solve for N.
    ##c(t) = N e^{- \frac{Ak}{V} t} +c_{sat} ## with ##c(0) = 0##.
    ...
    From there, you should see how the solution came about.
     
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