Answer
$\dfrac{\dot{m}_h}{\dot{m}_c}=2.08$
$\dot{Q}=641\ kWh$
Work Step by Step
$\dot{m}_c+\dot{m}_h=\dot{m}_m$
$\dot{m}_ch_c+\dot{m}_hh_h=\dot{m}_mh_m$
$\dot{m}_ch_c+\dot{m}_hh_h=(\dot{m}_c+\dot{m}_h)h_m$
$\dfrac{\dot{m}_h}{\dot{m}_c}=\dfrac{h_m-h_c}{h_h-h_m}$[
$\dfrac{\dot{m}_h}{\dot{m}_c}=\dfrac{c(T_m-T_c)}{c(T_h-T_m)}$
Given $T_m=42°C,\ T_c=15°C,\ T_h=55°C$
$\dfrac{\dot{m}_h}{\dot{m}_c}=2.08$
$\dot{m}=1\ kg/L\times(13.5-10.5)L/min\times5\ min/person.day\times4\ persons\times365\ days/year$
$\dot{m}=20,440\ kg/year$
$\dot{Q}=\dot{m}c(T_m-T_c),\ c=4.18\ kJ/kg.K$
$\dot{Q}=641\ kWh$
