A stream of warm water is produced in a steady-flow mixing...

A stream of warm water is produced in a steady-flow mixing process by combining $1.0 \mathrm{kg} \mathrm{s}^{-1}$ of cool water at $298.15 \mathrm{K}\left(25^{\circ} \mathrm{C}\right)$ with $0.8 \mathrm{kg} \mathrm{s}^{-1}$ of hot water at $348.15 \mathrm{K}$ $\left(75^{\circ} \mathrm{C}\right) .$ During mixing, heat is lost to the surroundings at the rate of $30 \mathrm{kW}$. What is the temperature of the warm-water stream? Assume the specific heat of water constant at $4.18 \mathrm{kJ} \mathrm{kg}^{\prime} \mathrm{K}^{-1}$.

A stream of warm water is produced in a steady-flow mixing process by combining $1.0 \mathrm{kg} \mathrm{s}^{-1}$ of cool water at $298.15 \mathrm{K}\left(25^{\circ} \mathrm{C}\right)$ with $0.8 \mathrm{kg} \mathrm{s}^{-1}$ of hot water at $348.15 \mathrm{K}$
$\left(75^{\circ} \mathrm{C}\right) .$ During mixing, heat is lost to the surroundings at the rate of $30 \mathrm{kW}$. What is the temperature of the warm-water stream? Assume the specific heat of water constant at $4.18 \mathrm{kJ} \mathrm{kg}^{\prime} \mathrm{K}^{-1}$.

Key Concepts

Mixing Process

The mixing process involves combining two or more streams with different thermal or physical properties to produce a single, homogenized output. Understanding this concept requires a careful consideration of both energy and mass conservation principles, as well as any additional losses (such as heat dissipation) that occur during the mixing process.

Heat Loss

Heat loss refers to energy that escapes from a system to its surroundings, deviating from an ideal, closed energy balance. Accurately accounting for heat losses is essential in real-world applications as it affects the net energy available for processes like mixing, thereby influencing final temperatures and system performance.

Specific Heat Capacity

Specific heat capacity is a material property that indicates the amount of energy required to change the temperature of a unit mass of a substance by one degree. It plays a critical role in energy transfer calculations, particularly in thermal mixing and heating/cooling processes, as it relates the energy exchanged to the resulting temperature change.

Steady-Flow Energy Balance

This concept involves applying the first law of thermodynamics to systems where mass and energy continuously cross the boundaries, such as mixing streams. In these situations, the total energy entering the system—including enthalpy flows and any work or heat interactions—must equal the energy leaving the system, leading to an energy balance equation. This principle is fundamental when calculating outlet conditions in many engineering applications involving steady flows.

Mass Flow Rate

Mass flow rate is the measure of the mass of a substance flowing through a given cross-section per unit time. In mixing processes, different streams may have different mass flow rates, and these rates are key factors in determining the weighted contributions of the streams to the overall energy and mass balances of the system.

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