$4\mathrm{.}13.$ Our highly efficient plant has a nitrogen stream at $2\mathrm{.}53$ bar and $305\mathrm{.}5K$ which is presently

vented to the atmosphere. Management would like to use this stream to satisfy some of

the heating or cooling requirements of other processes. Our old friend, Rocky Jones, has

devised a black box that will produce equal amounts of a hot stream at $500K$ and a cold

stream at $111K$ and thus satisfy simultaneously some heating and cooling requirements

$($see Figure P$4\mathrm{.}13).$ Furthermore, Rocky claims that his device will be self$-$sustaining

because no additional heat or work need to be supplied to the device.

$$

$|$ I K

Figure P$4\mathrm{.}13$

One of our new engineers, Barry Goldfinder, claims that he has a black$-$box device that

will produce equal amounts of hot stream of $533K$ and a cold stream of $77\mathrm{.}7K.$ His device

will also be self$-$sustaining.

$($a$)$ Are either $($or both$)$ of these devices possible? Explain.

$($b$)$ Describe a process that will satisfy the requirements of Rocky's black box.

Calculate all heat and work interactions for each device used in this process and

indicate now the devices should be arranged so that no additional heat or iwork

need be supplied by the environment.

Assume that the heat capacity of nitrogen is $29\mathrm{.}3\frac{J}{m}$olK, independent of temperature or

pressure.