Question $5\mathrm{.}6)$ Consider an ordinary shower where hot water at $\backslash (140^\{\backslash$circ$\}\backslash$mathrm$\{$F$\}\backslash )$ is mixed with cold water at $\backslash (50^\{\backslash$circ$\}\backslash$mathrm$\{$F$\}\backslash ).$ If it is desired that a steady stream of warm water at $\backslash (110^\{\backslash$circ$\}\backslash$mathrm$\{$F$\}\backslash )$ be supplied, determine the ratio of the mass flow rates of the hot to cold water. Assume the heat losses from the mixing chamber to be negligible and the mixing to take place at a pressure of $20$ psia $.$

Answer to Q $5\mathrm{.}6)$

In a shower, cold water is mixed with hot water at a specified temperature. For a specified mixture temperature, the ratio of the mass flow rates of the hot to cold water is to be determined.

Assumptions

$1)$ This is a steady$-$flow process since there is no change with time at any point and thus $\backslash (\backslash$Delta m$\_\{$c v$\}=0\backslash )$ and $\backslash (\backslash$Delta E$\_\{\backslash$mathrm$\{$cv$\}\}=0\backslash ).$

$2)$ The kinetic and potential energies are negligible, $\backslash (\backslash$Delta k e$=0\backslash )$ and $\backslash (\backslash$Delta p e$=0\backslash ).$

$3)$ Heat losses from the system are negligible and thus $\backslash (\backslash$dot$\{$Q$\}=0\backslash ).$

$4)$ There is no work interaction involved.

Analysis We take the mixing chamber as the system $($figure above$).$ This is a control volume since mass crosses the system boundary during the process. We observe that there are two inlets and one exit. Under the stated assumptions and observations, the mass and energy balances for this steady$-$flow system can be expressed in the rate form as follows:

$[$Left as an exercise$]$

Final answer is $0\mathrm{.}2$

Discussion Note that the mass flow rate of the hot water must be twice the mass flow rate of the cold water for the mixture to leave at $\backslash (110^\{\backslash$circ$\}\backslash$mathrm$\{$F$\}\backslash ).$