A nuclear power plant has an electrical power output of 1200 MW and operates with an efficiency of 31%. If excess energy is carried away from the plant by a river with a flow rate of 1.0 multiplied by 10^6 kg/s, what is the rise in temperature of the flowing water?
If the plant efficiency is 31%, 69% of the nuclear reaction heat is carried away in cooling water. That would be (.69/.31)*1200 MW or 2670*10^6 Watts.
Transferring that heat to water flowing at 10^6 kg/s will raise the temperature by an amount deltaT, such that
2670*10^6 W = dM/dt*C*(deltaT)
deltaT = (2670*10^6 J/s)/(10^6 kg/s*4184 J/kg*C)= 0.64 C
To find the rise in temperature of the flowing water, we need to calculate the amount of energy transferred from the power plant to the river.
Step 1: Calculate the amount of energy transferred from the power plant
The electrical power output of the power plant is given as 1200 MW (megawatts). We can convert this to watts by multiplying by 10^6.
1200 MW = 1200 x 10^6 watts.
The efficiency of the power plant is given as 31%. This means that only 31% of the energy produced is converted into electrical energy. We can calculate the amount of energy produced by dividing the electrical power output by the efficiency.
Energy produced = (1200 x 10^6) watts / 0.31.
Step 2: Calculate the heat absorbed by the flowing water
The energy transferred from the power plant is equal to the heat absorbed by the flowing water.
Heat absorbed = Energy produced.
Step 3: Calculate the rise in temperature of the flowing water
To calculate the rise in temperature, we need to use the specific heat capacity formula.
Q = mcΔT
Where:
Q = Heat absorbed (energy transferred)
m = Mass of the water flow rate
c = Specific heat capacity of water
ΔT = Rise in temperature
The flow rate of the river is given as 1.0 x 10^6 kg/s. Since the mass flow rate is the same as mass in this case, we can use this value for 'm'.
The specific heat capacity of water is approximately 4.186 J/g°C. We need to convert the mass flow rate from kg to g to use this value.
Specific heat capacity of water = 4.186 J/g°C.
Now, we can rearrange the equation to solve for ΔT.
ΔT= Q / (mc)
Step 4: Substitute the values and calculate the rise in temperature.
Substituting the known values into the equation, we get:
ΔT = Heat absorbed / (m x c)
ΔT = Energy produced / (m x c)
= (1200 x 10^6) watts / (1.0 x 10^6 kg/s x 4.186 J/g°C)
= (1200 x 10^6) / (1.0 x 10^6 x 4.186)
= 0.285 °C
Therefore, the rise in temperature of the flowing water is approximately 0.285 °C.
To calculate the rise in temperature of the flowing water, we can use the principle of energy conservation. The energy output of the power plant is known, and if we assume that all of this energy is transferred to the water, we can calculate the rise in temperature.
Here's how we can do it step by step:
1. Calculate the thermal power output of the nuclear power plant:
- The electrical power output is given as 1200 MW (megawatts).
- The efficiency of the power plant is given as 31%.
- Since power = energy / time, we can convert the electrical power to thermal power using the efficiency.
- Thermal power output = electrical power output / efficiency.
Therefore, the thermal power output of the nuclear power plant is (1200 MW) / (0.31) = 3870.96 MW.
2. Calculate the energy transferred to the water per unit time:
- Energy transferred to water = thermal power output.
- Since power = energy / time, we need to calculate the time.
The time can be calculated by dividing the mass of water flowing per unit time (flow rate) by the mass of water required to increase its temperature by 1 degree Kelvin (specific heat capacity).
3. Calculate the rise in temperature:
- The flow rate of the river is given as 1.0 multiplied by 10^6 kg/s (kilograms per second).
- Assume the specific heat capacity of water is 4.18 kJ/(kg·K) (kilojoules per kilogram per Kelvin).
- The rise in temperature can be calculated using the relationship:
Energy transferred to water = mass of water * specific heat capacity * temperature rise.
Rearranging the equation, we have:
Temperature rise = Energy transferred to water / (mass of water * specific heat capacity).
4. Substitute the values into the equation and calculate the rise in temperature:
- Energy transferred to water = (3870.96 MW) = (3870.96 * 10^6 kJ/s) (since 1 MW = 10^3 kW = 10^6 J/s = 10^6 kJ/s).
- Mass of water = flow rate = (1.0 * 10^6 kg/s).
- Specific heat capacity = 4.18 kJ/(kg·K).
Temperature rise = (3870.96 * 10^6 kJ/s) / [(1.0 * 10^6 kg/s) * (4.18 kJ/(kg·K))].
After performing the calculations, you will obtain the rise in temperature of the flowing water.