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Thread: Please! Help me out with an analytical solution for the drying problem

  1. #1
    Feb 2009

    Please! Help me out with an analytical solution for the drying problem

    Dear all,

    Now, mathematics really got me struck during my graduation project on drying of milk particles in a fluidized bed. For this project I am trying to calculate the profile for surface evaporation according to Crank (1955). The analytical solution is quite diffucult and displays lots of numerical errors in Excel.

    Being practically desperate I hope someone could help me out with this one.

    I will first explain the total problem and thereafter the practical issue.

    We consider a moist particle of 200 mu within a an air stream. The general formula is for the concentration on time = t and position = r:


    \frac{C_(r,t)-C_0}{C_1-C_0} = \langle\frac{2LR}{r}\rangle\Sigma_{n=1}^{\infty}\l brace\frac{e^{-D\beta_{n}^{2}}\frac{t}{R^{2}}}{\left(\beta_{n}^{2 }+L\left( L-1\right) \right) }\langle\frac{\sin\left(\beta_{n}\frac{r}{R} \right)}{\sin\left(\beta_{n}\right)} \rangle\rbrace$

    The $\displaystyle \beta $ 's are the roots of the following equation:
    $\displaystyle \beta_{n}cot(\beta_{n})+L-1 $

    $\displaystyle L= \frac{Rh}{D} $

    In the formula the letters have the following meaning:
    $\displaystyle C_(r,t) =$ concentration at time = t and place r in particle
    $\displaystyle C_0 =$ initial concentration within the particle at t = 0
    $\displaystyle C_1 =$ concentration around the particle at
    $\displaystyle R =$ 1/2 particle diameter
    $\displaystyle D =$ Diffusion coefficient
    $\displaystyle h =$ mass transition coefficient between particle and air
    $\displaystyle e =$ Euler number

    Now the real problem. The formula will be used to describe the profile of C over r and t. Resturcturing the formula clearly shows that there are several limits.
    C_(r,t) =C_0 + \langle C_1-C_0 \rangle \langle2LR\rangle\Sigma_{n=1}^{\infty}\lbrace \langle\frac{1}{r}\rangle \langle \frac{e^{-D\beta_{n}^{2}}\frac{t}{R^{2}}}{\left(\beta_{n}^{2 }+L\left( L-1\right)\right) }\rangle\langle\frac{\sin\left(\beta_{n}\frac{r}{R } \right)}{\sin\left(\beta_{n}\right)} \rangle\rbrace

    Some of these limits are not a big problem, like: $\displaystyle r=0$ or $\displaystyle \beta_{n}^{2} = -L^{2}+L$ YET...

    Since the diffusion coefficient is quite low $\displaystyle 10^{-12}$ to $\displaystyle 10^{-18}$, the $\displaystyle \beta$'s go to practically $\displaystyle \pi$.(not really pi) And this makes the $\displaystyle \sin\left( \beta_{n}\right) $ go to zero which means that $\displaystyle \frac{\sin\left(\beta_{n}\frac{r}{R} \right)}{\sin\left(\beta_{n}\right)}$ goes to infinity.

    The question to you now is whether you could help me out solving the following limit analytically:

    $\displaystyle \lim_{\beta_{n}\rightarrow N*\pi}\langle \frac{\sin\left(\beta_{n}\frac{r}{R} \right)}{\sin\left(\beta_{n}\right)}\rangle


    Please not that the l'hopital rule can probably not be applied since only the denominator is 0. And note that $\displaystyle \beta$ is not really $\displaystyle \pi$! Has someone got another option?

    Or even better to solve analytically:

    $\displaystyle \lim_{\beta_{n}\rightarrow N*\pi}Sigma_{n=1}^{\infty}\lbrace \langle\frac{1}{r}\rangle \langle \frac{e^{-D\beta_{n}^{2}}\frac{t}{R^{2}}}{\left(\beta_{n}^{2 }+L\left( L-1\right)\right) }\rangle\langle\frac{\sin\left(\beta_{n}\frac{r}{R } \right)}{\sin\left(\beta_{n}\right)} \rangle\rbrace


    with the $\displaystyle \beta_{n}cot(\beta_{n})+L-1 $ as input for the $\displaystyle \beta$'s

    Could please someone help me out? I'm struggelling for weeks now. Excel turns mad when I try to solve this equation, since Excel uses taylor approximations for the sinus. Even Maple 11 could not help me with the rootfinding?

    Wish you all the best

    Last edited by Mutz; Feb 24th 2009 at 07:20 AM.
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