Even when shut down after a period of normal use, a large...

Even when shut down after a period of normal use, a large commercial nuclear reactor transfers thermal energy at the rate of $150 \mathrm{MW}$ by the radioactive decay of fission products. This heat transfer causes a rapid increase in temperature if the cooling system fails (1 watt $=1$ joule/second or $1 \mathrm{W}=1 \mathrm{J} / \mathrm{s}$ and $1 \mathrm{MW}=1 \text { megawatt }) . \quad$ (a) Calculate the rate of temperature increase in degrees Celsius per second ( $^{\circ} \mathrm{C} / \mathrm{s}$ ) if the mass of the reactor core is $1.60 \times 10^{5} \mathrm{kg}$ and it has an average specific heat of $0.3349 \mathrm{kJ} / \mathrm{kg} \cdot^{\circ} \mathrm{C}$. (b) How long would it take to obtain a temperature increase of $2000^{\circ} \mathrm{C},$ which could cause some metals holding the radioactive materials to melt? (The initial rate of temperature increase would be greater than that calculated here because the heat transfer is concentrated in a smaller mass. Later, however, the temperature increase would slow down because the $500,000-\mathrm{kg}$ steel containment vessel would also begin to heat up.)

Even when shut down after a period of normal use, a large commercial nuclear reactor transfers thermal energy at the rate of $150 \mathrm{MW}$ by the radioactive decay of fission products. This heat transfer causes a rapid increase in temperature if the cooling system fails (1 watt $=1$ joule/second or $1 \mathrm{W}=1 \mathrm{J} / \mathrm{s}$ and $1 \mathrm{MW}=1 \text { megawatt }) . \quad$ (a) Calculate the rate of temperature increase in degrees Celsius per second ( $^{\circ} \mathrm{C} / \mathrm{s}$ ) if the mass of the reactor core is $1.60 \times 10^{5} \mathrm{kg}$ and it
has an average specific heat of $0.3349 \mathrm{kJ} / \mathrm{kg} \cdot^{\circ} \mathrm{C}$. (b) How long would it take to obtain a temperature increase of $2000^{\circ} \mathrm{C},$ which could cause some metals holding the radioactive materials to melt? (The initial rate of temperature increase would be greater than that calculated here because the heat transfer is concentrated in a smaller mass. Later, however, the temperature increase would slow down because the $500,000-\mathrm{kg}$ steel containment vessel would also begin to heat up.)

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Key Concepts

Rate of Temperature Increase

The rate of temperature increase in a system can be calculated by dividing the power input by the product of the mass and specific heat capacity of the material. This concept links the energy transfer rate to the temporal evolution of temperature, allowing for predictions of system behavior under heating conditions.

Power and Energy Relationship

Power is defined as the rate of energy transfer measured in watts, where one watt equals one joule per second. Understanding the relationship between power and energy is crucial when determining how quickly temperature changes occur in a system based on the rate of heat input.

Specific Heat Capacity

Specific heat capacity quantifies the amount of heat energy required to raise the temperature of a unit mass of a substance by one degree Celsius. It is a fundamental property used to relate the energy added or removed from a material to its temperature change, as seen in the equation ?T = Q/(mc).

Conservation of Energy in Thermal Systems

The conservation of energy principle in thermal systems states that any energy added to a system results in a corresponding increase in its internal energy, which in the context of heating can be observed as a rise in temperature. This concept underpins the calculation of temperature changes due to externally applied heat power.