For a packed bed, the Ergun equation provides a relationship between the pressure drop per unit length and the properties of the bed,

$\frac{P}{L}=150(\frac{v_0}{D_p^2})\frac{(1-lon{)}^2}{lo{n}^3}+\frac{7}{4}(\frac{v_0^2}{D_p})\frac{1-lon}{lo{n}^3}$

where $$ is the fluid viscosity, $v_0$ is the superficial velocity, $D_p$ is the particle diameter, $$ is the fluid density, and $$ is the void fraction of the bed. Consider a $1\mathrm{.}5m$ long packed bed containing $5cm$ diameter particles. Fluid of density $2\frac{g}{c}{m}^3$ and viscosity $1cP$ flows through this bed at a superficial velocity of $0\mathrm{.}1\frac{m}{s}.$ The pressure drop in the bed is $416Pa.$

Write a program that uses Newton's method to determine the void fraction of the bed and outputs it to screen along with the number of iterations and the value of $|f(x)|.$

$2.$ Write a program that uses one of the other methods, rather than Newton's method, to determine the void fraction in the previous question.

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