Chapter 5, Problem 5.19P

Batch processes are often used in chemical and pharmaceutical operations to achieve a desired chemical composition for the final product and typically involve a transient heating operation to take the product from room temperature to the desired process temperature. Consider a situation for which a chemical of density $\rho =1200{\text{kg/m}}^{\text{3}}$ and specific heat $c=2200\text{J/kg}\cdot \text{K}$ occupies a volume of $V=2.25{\text{m}}^{\text{3}}$ in an insulated vessel. The chemical is to be heated from room temperature, $T_i=300\text{K,}$ to a process temperature of $T=450\text{K}$ by passing saturated steam at $T_h=500\text{K}$ through a coiled, thin-walled, 20-mm-diameter tube in the vessel. Steam condensation within the tube maintains an interior convection coefficient of $h_i=10,000{\text{W/m}}^{\text{2}}\cdot \text{K,}$ while the highly agitated liquid in the stirred vessel maintains an outside convection coefficient of $h_o=2000{\text{W/m}}^{\text{2}}\cdot \text{K}\text{.}$

If the chemical is to be heated from 300 to 450 K in 60 min, what is the required length L of the submerged tubing?