Chapter 8, Problem 8.32P

Batch processes are often used in chemical and pharmaceutical operations to achieve a desired chemicalcomposition for the final product. Related heat transferprocesses are typically transient, involving a liquid a fixed volume that may be heated from room temperature to a desired process temperature, or cooled fromthe process temperature to room temperature. Considera batch process for which a pharmaceutical (the coldfluid. c) ¡s poured into an insulated, highly agitated vessel (a stirred reactor) and heated by passing a hot fluid(h) through a submerged heat exchanger coil of thin-walled tubing and surface area $A_S$ . The flow rate, $m_h$ ,mean inlet temperature, $T_{hi}$ c. and specific heat, $c_{ph}$ , ofthe hot fluid are known, as are the initial temperature, $T_{c,i}<T_{h,i}$ , the volume, $V_c$ , mass density, ${\rho }_c$ , and specificheat, $c_{v,c}$ , of the pharmaceutical. Heat transfer from thehot fluid to the pharmaceutical is governed by an overall heat transfer coefficient U.

(a) Starting from basic principles, derive expressions thatcan be used to determine the variation of $T_c$ and $T_{\text{h},0}\left(t\right)$ with time during the heating process. Hint: Two equations may be written for the rate of heat transfer, $q\left(t\right)$ ,to the pharmaceutical. one based on the log-mean temperature difference and the other on an energy balance for flow of the hot fluid through the tube. Equate these expressions to determine $T_{\text{h},0}\left(t\right)$ as a function of $T_c\left(t\right)$ and prescribed parameters. Use the expression for $T_{\text{h},0}\left(t\right)$ and the energy balance for flow through the tube with an energy balance tor a control volume containing the pharmaceutical to obtain an expression for $T_c\left(t\right)$ .

(b) Consider a pharmaceutical of volume $V_c=1{\text{m}}^3$ , density ${\rho }_c=1100{\text{kg/m}}^3$ , specific heat $c_{v,c}=2000\text{J/Kg}\cdot \text{K}$ , and an initial temperature of $T_{c,i}=25°C$ . A coiledtube of length $L=40m$ , diameter $\text{D = 50 mm}$ , and coil diameter $C=500mm$ is submerged in the vessel, and hot fluid enters the tubing at $T_{h,i}=200°\text{C}$ and $m_h=2.4\text{kg/s}$ . The convection coefficient at the outer surface of the tubing may be approximated as $h_0=1000{\text{W/m}}^2\cdot \text{K}$ , and the fluid properties are $c_{p,h}=2500\text{J/kg}\cdot \text{K,}{\mu }_{\text{h}}\text{=0}\text{.02N}\cdot {\text{s/m}}^{\text{2}},k_h=0.260\text{W/m}\cdot \text{K}$ , and ${\mathrm{Pr}}_h=20$ . For the foregoing conditions, compute and plot the pharmaceutical temperature T and the outlet temperature $T_{h,0}$ as a function of time over the range $0\le t\le 3600\text{s}$ . How long does it take to reach a batch temperature of $T_c=160°\text{C}$ ? The process operator may control the heating time by varying $m_h$ . For $1\le m_h\le 5\text{kg/s}$ , explore the effect of the flow rate on the time $t_c$ , required to reach a value of $T_c=160°\text{C}$ .