Chapter 10, Problem 10.29P

A steam coil is immersed in a stirred tank. Saturated steam at 7.50 bar condenses within the coil, and the condensate emerges at its saturation temperature. A solvent with a heat capacity of 2.30 kJ/(kg? °C) is fed to the tank at a steady rate of 12.0kg/min and a temperature of 25°C, and the heated solvent is discharged at the same flow rate. The tank is initially filled with 760 kg of solvent at 25°C, at which point the flows of both steam and solvent are commenced. The rate at which heat is transferred from the steam coil to the solvent is given by the expression

   $\stackrel{\dot{}}{Q}=UT\left(T_{steam}-T\right)$

where UA (the product of a heat transfer coefficient and the coil surface area through which the heat is transferred ) equals 11.5 kJ/(min?°C). The tank is well stirred, so that the temperature of the contents is spatially uniform and equals the outlet temperature.

1. Prove that an energy balance on the tank contents reduces to the equation given below and supply an initial condition.

   $\frac{dT}{dt}=1.50°C/\min -0.0224T$

1. Without integrating the equation, calculate the steady-state value of T and sketch the expected plot of T versus t, labeling the values of T_bat t = 0 and t ? 8.

Integrate the balance equation to obtain an expression for T(t) and calculate the solvent temperature after 40 minutes.

The tank is shut down for routine maintenance, and a technician notices that a thin mineral scale has formed on the outside of the steam coil. The coil is treated with a mild acid that removes the scale and reinstalled in the tank. The process described above is run again with the same steam conditions, solvent flow' rate, and mass of solvent charged to the tank, and the temperature after 40 minutes is 55°C instead of the value calculated in Part (c). One of the system variables listed in the problem statement must have changed as a result of the change in the stirrer. Which variable would you guess it to be, and by what percentage of its initial value did it change?