A power plant, having a Carnot efficiency, produces 1000 MW of electrical power from turbines th

A power plant, having a Carnot efficiency, produces 1000 MW...

Key Concepts

Heat Transfer and Specific Heat Capacity

Heat transfer involves the movement of thermal energy from one body or substance to another due to a temperature difference. Specific heat capacity is a material property that indicates the amount of heat required to change the temperature of a unit mass of a substance by one degree. These principles are used to determine how much heat energy is needed to raise the temperature of the river water by a specified amount, linking the waste heat output of the power plant to the resulting temperature increase in the environment.

Mass Flow Rate Calculation

Mass flow rate calculation connects the energy transferred to an amount of substance that undergoes a temperature change. By applying the relation Q = m*c*?T, where Q is the heat energy, m is the mass, c is the specific heat capacity, and ?T is the temperature change, one can determine the required flow rate (mass per unit time) of the river water that absorbs the waste heat. This concept ties together the energy concerns of the power plant’s operation with environmental impact assessments.

Carnot Efficiency

Carnot efficiency represents the maximum theoretical efficiency possible for a heat engine operating between two thermal reservoirs. It is defined by the temperatures of the hot and cold reservoirs (in absolute units) and establishes an ideal limit: no real engine can be more efficient. This concept is central in understanding the energy conversion limits in power plants and the unavoidable waste of heat energy due to the temperature difference between the high-temperature source and the low-temperature sink.

Heat Engine Energy Conversion

Heat engine energy conversion refers to the process through which thermal energy is transformed into mechanical work. The efficiency of this conversion is determined by the fraction of input heat energy that is converted into useful work, with the remainder typically being expelled as waste heat. In the context of a power plant, this concept is crucial for evaluating how much heat must be input to generate a given amount of electrical power.

Transcript

00:01 Hello everyone in this problem it is given power plant having carnot efficiency it produces thousand megawatt electrical power the power output of the plant is given thousand megawatt that is 10 to the power 9 watt turbine of the power plant taking the steam at 500 kelvin so temperature of hot reservoir is 500 kelvin and it rejecting the heat to the water at 300 kelvin so temperature of cold reservoir is 300 kelvin due to rejection of heat temperature of the water gets increased by 6 kelvin we have to find the rate of flow of water in the river that is dm upon dt we have to calculate let us start solve efficiency of carnot engine would be 1 minus temperature of cold reservoir to temperature of hot temperature of cold reservoir is 300 and that is hot is 500 so this efficiency of you can also define by e i have a habit to write with ether so it would be 200 by 500 that is 0 .4.

02:06 Now for carnot cycle we can write temperature of cold reservoir divided by temperature of hot reservoir to be equal to amount of the heat rejected to the cold reservoir to amount of the heat absorbed from the hot reservoir so from here we can write amount of the heat rejected to the cold reservoir is amount of the heat absorbed upon tc upon t s substitute the value so amount of the heat rejected will be q into h that is 300 by 500 so we can write it is 3 by 5 of heat absorbed from the hot reservoir now this value we can calculate by the efficiency efficiency will be mechanical work done upon heat absorbed so it would be w upon e substitute in equation to this value so we will get heat rejected to the water will be 3 by 5th or efficiency is 0 .4 so it becomes 3 by 2 mechanical work done now let us see here this amount of the heat absorbed by the water is mc delta t so mass of water can be written as differentiate with respect to time so rate of flow of water in the river, dm upon dt, 3y2, 1 upon c delta t into dw upon dt, this is the power output of the plant.

04:14 D .y2, c is the specific heat.

04:18 4186, increase in temperature of water is 6 and that is power and that is given 10 to the power 9 watt...

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