Tomato Juice

This juice is easily damaged by high temperature. Consequently its concentration is accomplished with modest heating by hot water in a heat exchanger followed by extraction of water from the juice by action of a vane pump that maintains a low pressure over the juice and ejects water vapor from it in a separator.

The process has two flow streams. The hot water simply enters (is cooled as the juice is heated) and exits. The other stream, juice, enters, is warmed then exits partly as concentrate and partly as vapor exhausted by the vane pump. Use the information of the drawing.

Calculate the mass rate of water that exits through the vane pump as water vapor.

Basic Thermodynamics ~ J. Pohl ©

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Tomato Juice

This juice is easily damaged by high temperature. Consequently its concentration is accomplished with modest heating by hot water in a heat exchanger followed by extraction of water from the juice by action of a vane pump that maintains a low pressure over the juice and ejects water vapor from it in a separator.

The process has two flow streams. The hot water simply enters (is cooled as the juice is heated) and exits. The other stream, juice, enters, is warmed then exits partly as concentrate and partly as vapor exhausted by the vane pump. Use the information of the drawing.

Calculate the mass rate of water that exits through the vane pump as water vapor.
♦  By the diagram, we observe the feed of juice is 250 kilogram per hour while the concentrate produced is 180 kilograms per hour. The mass equation for steady production of concentrate is:

(1) 1

Calculate the mass rate of water for this system. Take the heat exchanger as the system and assume it is frictionless and adiabatic. The event is steady and there is no work. The energy equation is:

(2)2

We approximate the juice as water. They are liquids as they interact in the heat exchanger. We use specific heats to express the enthalpy changes.

(3) 3

(4) 4

Calculate the least horsepower of the vane pump.
♦  We write the energy equation for steady operation of the vane pump. The kinetic and potential energy changes of the flow of vapor through the pump are negligible. The vapor passes through the pump quickly with too little time for heat.

(5) 5

The mass rate of vapor was determined above. The entering and leaving pressures are provided and the entering temperature is 80°C. Being asked for the LEAST work, we set the friction to zero. We enter these numbers into the energy equation:

(6) 6

Our reduced equation applies to a frictionless compression. Still, this equation contains two unknown quantities which are the exiting temperature and the work rate of compression. To solve for the work we require another condition or relation for the process. That condition is that the vapor proceeds along an adiabatic-frictionless path. Water vapor at this low pressure can be approximated as an ideal gas with specific heats and a specific heat ratio of:

(7) 7

We return to the energy equation with this information.

(8) 8

A style of this author is to avoid algebra. The positions of the terms of an equation are part of the meaning of the term. Nothing is gained by rearranging a thermodynamic equation.