Develop a simplified form of the general steady-state energy balance appropriate for each operation. State carefully, and justify, any assumptions you make.

Question

A list of common unit operations follows:
(a) Single-pipe heat exchanger
(b) Double-pipe heat exchanger
(c) Pump
(d) Gas compressor
(e) Gas turbine
(f) Throttle valve
(g) Nozzle

Answer 1

The general equation applicable here is



a. We write this for the single stream flowing within the pipe, neglect potential and kinetic energy changes, and set the work term equal to zero. This yields:



In the second and third versions of the LHS, we have used the fact that there is a single inlet and outlet, and that mass conservation requires that the mass flow rate at the inlet and outlet are equal.

b. The equation is here written for the two streams (1 and 2) flowing in the two pipes, again neglecting any potential- and kinetic-energy changes. There is no work, and the heat transfer is internal to the system, between the two streams, making Q?= 0. Thus,



c. For a pump operating on a single liquid stream, the assumption of negligible potential- and kinetic energy changes is reasonable, as is the assumption of negligible heat transfer to the surroundings. Whence,



d. For a properly designed gas compressor the result is the same as in Part (c).



e. For a properly designed turbine the result is the same as in Part (c).



f. The purpose of a throttle is to reduce the pressure on a flowing stream. One usually assumes adiabatic operation with negligible potential- and kinetic-energy changes. Since there is no work, the equation is:



g. The whole purpose of a nozzle is to produce a stream of high velocity. The kinetic-energy change must therefore be taken into account. However, one usually assumes negligible potential-energy change. Then, for a single stream, adiabatic operation, and no work:



The usual case is for a negligible inlet velocity. The equation then reduces to:



where u2 is the exit velocity.