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

1. ## 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:

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

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.
$\displaystyle C_(r,t) =C_0 + \langle C_1-C_0 \rangle \langle2LR\rangle\Sigma_{n=1}^{\infty}\lbrace \langle\frac{1}{r}\rangle \langle \frac{e^{-D\beta_{n}^{2}}\frac{t}{R^{2}}}{\left(\beta_{n}^{2 }+L\left( L-1\right)\right) }\rangle\langle\frac{\sin\left(\beta_{n}\frac{r}{R } \right)}{\sin\left(\beta_{n}\right)} \rangle\rbrace$

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

jurjen