I have a hot and cold tap which produce water at fixed temperatures $T_C$ and $T_H$ respectively. I'm want to mix the water from the taps to create a volume $V$ of water with a desired temperature $T_D$.
How do I calculate the ratio of hot to cold water needed? What's the theory behind it?
Best Answer
Energy conservation is the thing. All starting from 1st law of thermodynamics.
No external heat $Q$ or work $W$: $$\Delta U=Q-W=0$$ causes internal energy $U$ to remain unchanged: $$\begin{align} U_{after}&=U_{before}\\ U&=U_{c}+U_{h}\\ mc T&=m_{c}c T_{c}+m_{h}cT_{h}\qquad \leftarrow U=mcT\\ mT&=^*m_{c}T_{c}+m_{h}T_{h}\\ T&=\frac{m_{c}T_{c}+m_{h}T_{h}}{m}\\ T&=\frac{V_c\rho T_{c}+V_h\rho T_{h}}{V\rho}\qquad\leftarrow m=\rho V\\ T&=^{**}\frac{V_c}VT_{c}+\frac{V_h}VT_{h}\\ T&=\frac{V-V_h}VT_{c}+\frac{V_h}VT_{h}\qquad\leftarrow V=^{***}V_c+V_h\\ T&=\left(1-\frac{V_h}V\right)T_{c}+\frac{V_h}VT_{h}\\ \end{align}$$
where $_c$ is cold and $_h$ hot water.
For water at normal pressures and house-hold temperatures and small temperature differences all these assumption are good and differ only negligibly.
The final expression gives you $V_h$, which you can put into $V=V_c+V_h$ to find the other one.