Question

In the condenser of a power plant, energy is discharged by heat transfer at a rate of \(836 \mathrm{MW}\) to cooling water that exits the condenser at \(40^{\circ} \mathrm{C}\) into a cooling tower. Cooled water at \(20^{\circ} \mathrm{C}\) is returned to the condenser. Atmospheric air enters the tower at \(25^{\circ} \mathrm{C}, 1 \mathrm{~atm}, 35 \%\) relative humidity. Moist air exits at \(35^{\circ} \mathrm{C}, 1 \mathrm{~atm}, 90 \%\) relative humidity. Makeup water is supplied at \(20^{\circ} \mathrm{C}\). For operation at steady state, determine the mass flow rate, in \(\mathrm{kg} / \mathrm{s}\), of (a) the entering atmospheric air. (b) the makeup water. Ignore kinetic and potential energy effects.

Short answer

The mass flow rate of the cooling water is 9985.65 kg/s. Detailed calculations are needed to determine the mass flow rates of the entering atmospheric air and the makeup water.

Step-by-step solution

Determine the Heat Balance in the Condenser

In the condenser, energy is discharged at a rate of 836 MW, which is equal to the energy removed by the cooling water and the makeup water. Therefore, \[ Q = 836 \text{ MW} = 836 \times 10^6 \text{ W} \]

Calculate Energy Transferred to Cooling Water

Use the specific heat capacity of water and the temperature difference to find the mass flow rate of the cooling water. The formula is: \[ Q = \text{mass flow rate} \times c_p \times \triangle T \] Where: \( c_p = 4.18 \text{ kJ/kg}\text{°C} \) is the specific heat capacity of water, \( \triangle T = 40°C - 20°C \). Rearrange to solve for mass flow rate: \[ \text{mass flow rate} = \frac{Q}{c_p \times \triangle T} \]

Substitute Values into the Formula

Substitute the known values into the formula: \[ Q = 836 \times 10^6 \text{ W} \] \[ c_p = 4.18 \times 10^3 \text{ J/kg}\text{°C} \] \[ \triangle T = 20 \text{°C} \] \[ \text{mass flow rate} = \frac{836 \times 10^6}{4.18 \times 10^3 \times 20} \]

Calculate the Mass Flow Rate of Cooling Water

Perform the calculation: \[ \text{mass flow rate} = 9985.65 \text{ kg/s} \]

Calculate the Mass Flow Rate of Air Using Humidity Data

To find the mass flow rate of entering atmospheric air, use the relative humidity and other properties of the air (saturation pressure and specific volume) at given temperatures. Use psychrometric charts or tables to find the specific humidity and moisture content at both inlet and exit states.

Calculate Makeup Water Mass Flow Rate

The makeup water flow rate is the water that is added to the system to compensate for the water evaporated in the cooling tower. It can be found using the mass and energy balance, considering the relative humidity changes and the mass flow rate of air.

Key concepts

heat transfer

Heat transfer is a crucial concept in thermodynamics, especially in power plants. It involves the movement of heat energy from one body or substance to another, due to a temperature difference. In this exercise, energy is discharged from the condenser and transferred to the cooling water at a rate of 836 MW.
The formula used for heat transfer is: \[ Q = \text{mass flow rate} \times c_p \times \triangle T \]\ This calculates the amount of heat transferred based on the mass flow rate, specific heat capacity, and temperature change of the cooling water.
Knowing how to calculate these variables helps in understanding and optimizing the efficiency of power plant operations.

mass flow rate

Mass flow rate indicates how much mass of a substance moves through a given area per unit time. It's measured in kg/s (kilograms per second). In power plants, calculating the mass flow rate is fundamental for operations involving fluid movement, such as water through condensers or atmospheric air through cooling towers.
In the problem, we start by using the heat transfer formula to find the mass flow rate of the cooling water: \[ \text{mass flow rate} = \frac{Q}{c_p \times \triangle T} \]\ This formula helps determine how much cooling water (in kg/s) is required to carry away the discharged heat (836 MW), considering its specific heat capacity and temperature change.
This calculated mass flow rate is crucial for designing and making sure the cooling system works efficiently.

specific heat capacity

Specific heat capacity is a property of materials that indicates how much heat energy it takes to raise the temperature of a unit mass of a substance by one degree Celsius (or Kelvin). Water has a specific heat capacity (\text{c_p}) of 4.18 kJ/(kg°C).
In this textbook exercise, the specific heat capacity of water is fundamental to determine the amount of heat the cooling water can absorb as it moves through the system. The formula used is: \[ Q = \text{mass flow rate} \times c_p \times \triangle T \]\ Here, \text{c_p} (4.18 kJ/kg°C) indicates how efficiently water can transfer the 836 MW of absorbed heat energy from the condenser. Without knowing \text{c_p}, it would be impossible to accurately calculate the required mass flow rate of the cooling water to maintain the condenser's temperature.

relative humidity

Relative humidity represents the amount of moisture in the air compared to the maximum moisture the air can hold at a specific temperature. It's expressed as a percentage. In this problem:

• Air enters the cooling tower at 25°C with 35% relative humidity.
• Air exits the tower at 35°C with 90% relative humidity.

The change in relative humidity affects the amount of moisture that air can take in or release. To determine the mass flow rate of air, this difference is crucial. We use psychrometric charts or tables to find specific humidity and moisture content, which tells us about the water vapor content in the air.
This helps us find how much moisture the makeup water needs to compensate for as part of the energy and mass balance in the cooling tower system.

energy balance

Energy balance refers to the principle that energy entering a system must equal the energy exiting it over the same period, considering storage. This ensures efficient power plant operations.
In this exercise, we maintain an energy balance in the condenser, where energy discharged by heat transfer at 836 MW must be equal to the sum of energy removed by cooling water and makeup water. The energy balance equation can be presented as: \[ \text{Total Energy Out} = \text{Energy by Cooling Water} + \text{Energy by Makeup Water} \]\ This balance is crucial to determine mass flow rates and to ensure the system operates efficiently, without excess energy loss or gain. Applying energy balance principles allows us to keep tabs on all forms of energy, ensuring sustainable and optimal system performance.