Go The recovery time of a hot water heater is the time...

Go The recovery time of a hot water heater is the time required to heat all the water in the unit to the desired temperature. Suppose that a 52 -gal $\left(1.00 \mathrm{gal}=3.79 \times 10^{-3} \mathrm{~m}^{3}\right)$ unit starts with cold water at $11^{\circ} \mathrm{C}$ and delivers hot water at $53^{\circ} \mathrm{C}$. The unit is electric and utilizes a resistance heater $(120 \mathrm{~V} \mathrm{ac}, 3.0 \Omega)$ to heat the water. Assuming that no heat is lost to the environment, determine the recovery time (in hours) of the unit.

Key Concepts

Energy Required for Heating

This concept involves calculating the amount of energy needed to raise the temperature of a substance using the equation Q = m × c × ?T, where Q is the energy, m is the mass of the substance, c is its specific heat capacity, and ?T is the change in temperature. It is essential for determining how much energy must be supplied to achieve the desired temperature change.

Specific Heat Capacity

Specific heat capacity is a property of a material that indicates the amount of energy required to raise the temperature of one unit mass of the substance by one degree. This concept is crucial for calculating energy requirements for heating processes as it influences the total energy needed.

Unit Conversions in Energy Calculations

This involves converting between different units of measurement (such as converting gallons to cubic meters and then to mass using density, or converting energy units and time units) to correctly apply formulas in a consistent system. Accurate unit conversion is fundamental in ensuring that calculations yield meaningful and correct results.

Resistance Heating and Power Calculation

Resistance heating involves converting electrical energy into heat through the resistance offered by a material. The power delivered by an electrical heating element can be calculated using the formula P = V² / R, where V is the voltage and R is the resistance. Understanding this concept allows one to determine the rate at which energy is supplied to the system.

Relationship between Energy, Power, and Time

This key concept states that energy is the product of power and time (Energy = Power × Time). This relationship is used to determine the duration required for a process when the energy requirement and the rate of energy input (power) are known. It is fundamental in calculating recovery or heating times in various applications.

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