(Solved): please solve this problem, i need clear steps... and please dont use whst is alrwady on chegg i ...

The overhead vapor from a distillation column is totally condensed in a water-cooled condenser at \( 120^{\circ} \mathrm{F} \) and \( 220 \mathrm{psig} \), and the condensed stream is sent to a reflux drum at the same temperature and pressure. The vapor design flow rate is \( 10,000 \mathrm{lb} \mathrm{mr} \), and the average latent heat of vaporization is \( 125 \mathrm{Btu} / \mathrm{b}_{\mathrm{m}} \). The cooling water inlet and outlet temperatures are \( 80^{\circ} \mathrm{F} \) and \( 110^{\circ} \mathrm{F} \), respectively. (a) Calculate the cooling water flow rate in gpm at the design conditions. Assume the density of water is \( 62.4 \mathrm{lb} w / \mathrm{ft}^{3} \) and its specific heat is \( 1.0 \mathrm{Btu} / \mathrm{b}_{\mathrm{m}}{ }^{\circ} \mathrm{F} \). Also, \( 7.48 \mathrm{gal} \) of liquid \( =1 \mathrm{ft}^{3} \) of liquid. (Note - If you cannot calculate a cooling water flow rate in part (a), assume the cooling water flow rate is 100 gpm at design conditions The cooling water pressure drop through the condenser is \( 5 \mathrm{psi} \) at the design conditions and it is proportional to the volumetric flow rate squared. A control valve is installed in the cooling water line, and the total pressure drop over the condenser and control valve is constant at \( 35 \mathrm{psi} \). The pressure in the reflux drum is measured by a pressure sensoritransmitter that has a range of \( 150-250 \mathrm{psig} \) and an output signal range of \( 4-20 \mathrm{~mA} \). A proportional controller with a gain of \( \mathrm{K}_{\mathrm{c}}=4 \) is used to control pressure in the reflux drum by manipulating the cooling water flow rate. The current signal \( (\mathrm{mA}) \) from the controller is converted to an air pressure signal (psig) in the I/P transducer. The cooling water pressure drop through the condenser is \( 5 \mathrm{psi} \) at the design conditions and it is proportional to the volumetric flow rate squared. A control valve is installed in the cooling water line, and the total pressure drop over the condenser and control valve is constant at \( 35 \mathrm{psi} \). The pressure in the reflux drum is measured by a pressure sensor/transmitter that has a range of \( 150-250 \mathrm{psig} \) and an output signal range of \( 4-20 \mathrm{~mA} \). A proportional controller with a gain of \( \mathrm{K}_{\mathrm{c}}=4 \) is used to control pressure in the reflux drum by manipulating the cooling water flow rate. The current signal \( (\mathrm{mA}) \) from the controller is converted to an air pressure signal (psig) in the I/P transducer. (b) Determine the valve size coefficient \( C_{v} \) if the flow rate through a wide-open valve must be twice the design flow rate. (c) Specify the action of the control valve and controller. (d) Determine the valve opening \( \mathrm{f}(\mathrm{l}) \) at the design flow rate. (e) Determine the current signal from the pressure transmitter at the design flow rate.