# Mixing hot and cold taps to a desired temperature

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?

• The final temperature is a weighted average (weighted in terms of the volumes of hot and cold water used) of the hot and cold temperatures. – Chet Miller Dec 20 '16 at 12:21

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.

• At $^*$ we assumed specific heat capacities $c$ constant.
• At $^{**}$ we assumed densities $\rho$ constant.
• At $^{***}$ we assumed volumes $V$ constant (incompressible liquid).

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.

• Just for completeness the anwser is $\frac{V_h}{V} = \frac{T-T_c}{T_h-T_c}$ – joshlk Dec 22 '16 at 15:02

The amount of heat in cold water is $Q_C = c m_C T_C$, the amount of heat in hot water is $Q_H = c m_H T_H$, where $c$ is the heat capacity. The heat of the mixture would be $Q_H-Q_C = c (m_C+m_H) T_D$. Combining this with previous formulas you get: $T_D = \frac{m_H}{m_H+m_C}T_H + \frac{m_C}{m_H+m_C}T_C$, which is a weighted average as pointed out in comments. By reshuffling this you can get the ratio between the volumes.