Water powered machine. A self-contained machine only inputs...

Water powered machine. A self-contained machine only inputs two equal steady streams of hot and cold water at temperatures $T_1$ and $T_2$. Its only output is a single high-speed jet of water. The heat capacity per unit mass of water, $C$, may be assumed to be independent of temperature. The machine is in a steady state and the kinetic energy in the incoming streams is negligible. (a) What is the speed of the jet in terms of $T_1, T_2$ and $T$, where $T$ is the temperature of water in the jet? (b) What is the maximum possible speed of the jet? FIGURE CAN'T COPY.

Water powered machine. A self-contained machine only inputs two equal steady streams of hot and cold water at temperatures $T_1$ and $T_2$. Its only output is a single high-speed jet of water. The heat capacity per unit mass of water, $C$, may be assumed to be independent of temperature. The machine is in a steady state and the kinetic energy in the incoming streams is negligible.
(a) What is the speed of the jet in terms of $T_1, T_2$ and $T$, where $T$ is the temperature of water in the jet?
(b) What is the maximum possible speed of the jet?
FIGURE CAN'T COPY.

Key Concepts

Optimization of Energy Extraction

The maximum theoretical jet speed is achieved when the process extracts the maximum kinetic energy possible from the available temperature difference. This occurs when the final temperature of the output water is minimized, specifically when it equals the lower input temperature. The concept underscores the idea of maximizing energy conversion efficiency by exploiting the greatest possible thermal gradient.

Conversion of Thermal Energy to Kinetic Energy

This concept involves turning thermal energy (related to temperature differences) into mechanical energy (observed as kinetic energy in the jet). In the problem, the difference between the initial thermal energy of the streams and the thermal energy in the mixed jet is converted into the kinetic energy of the jet, a process analogous to using a heat engine where a temperature difference does work on a working fluid.

Mixing of Fluids

Mixing fluids at different temperatures creates a process where the energy from the hotter fluid partially transfers to the colder fluid. When two streams of equal mass flow at different temperatures are combined, the temperature of the resulting mixture is normally the average if no energy is extracted. However, if some energy is diverted—such as the conversion to kinetic energy—the final temperature of the mixture is lower than the simple average.

Thermal Energy and Heat Capacity

Thermal energy is the internal energy contained in a substance, which for water is often calculated using its specific heat capacity (found here as C). This property allows us to relate changes in temperature to changes in energy. The problem uses the heat capacity per unit mass of water to quantify how much energy is stored or released when the temperature of the water changes.

Conservation of Energy

In any physical process, including the mixing of fluids, the total energy is conserved. In this problem, the thermal energy carried by water at different temperatures is partially converted into kinetic energy in the jet. The energy balance equates the sum of the thermal energies of the incoming streams with the thermal energy of the output water plus the kinetic energy carried by the high-speed jet.

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