In a serious accident, nuclear power plants reactor vessel cracks and all of the cooling water drains out. Although nuclear fission stops, radioactive decay continues to heat the reactor's 2.5× 10 to the 5 power kg uranium core at rate of 120 Mega-Watts. Once the core reaches its melting point, how much energy will it takes to melt the core? How long will the melting take?
Uranium (melting point,K 1406)
Lf,Kj/kg(82.8)
Boiling point,K(4091)
Kj/kg(1875)
hEAT TO MELT=HEATFUSION*MASS
How long? time=heattomelt/120E6 seconds
To calculate the energy required to melt the core, we need to determine the heat required to raise the temperature from room temperature to the melting point, and then add the heat of fusion to convert it from a solid to a liquid.
First, let's calculate the heat required to raise the temperature from the current temperature to the melting point.
Heat required = mass × specific heat × change in temperature
Here, the mass of the uranium core is given as 2.5 × 10^5 kg (kilograms), and the specific heat capacity of uranium is not mentioned. However, we can estimate it as an average value around 120 J/(kg·K) (joules per kilogram per Kelvin).
The change in temperature is calculated as the difference between the melting point and the current temperature, which is assumed to be room temperature (around 298 K).
So, the heat required to raise the temperature is:
Heat required = 2.5 × 10^5 kg × 120 J/(kg·K) × (1406 K - 298 K)
Next, we need to add the heat of fusion to convert the uranium from a solid to a liquid. The heat of fusion is given as 82.8 kJ/kg (kilojoules per kilogram).
Heat of fusion = mass × heat of fusion
Heat of fusion = 2.5 × 10^5 kg × 82.8 kJ/kg
Finally, we can calculate the total energy required:
Total energy required = heat required + heat of fusion
Now let's calculate the time it will take for the core to melt.
The rate at which the core is being heated is given as 120 Mega-Watts or 120 × 10^6 W (watts). We can convert this to Joules per second (J/s).
Rate of heating = 120 × 10^6 W = 120 × 10^6 J/s
To find the time it takes to melt the core, we can divide the total energy required by the rate of heating:
Melting time = Total energy required / Rate of heating
Keep in mind that these calculations are simplified estimates and do not take into account various factors like heat losses or variations in the heating rate.
By plugging in the values mentioned above into the respective formulas, you can find the exact amount of energy required to melt the uranium core and the time it will take based on the heating rate.