In the condenser of a power plant, energy is discharged by heat transfer at a rate of 1200 MW to cooling water that exits the condenser at 45°C into a cooling tower. Cooled water at 20°C is returned to the condenser. Atmospheric air enters the tower at 20°C, 1 atm, 50% relative humidity. Moist air exits at 40°C, 1 atm, 90% relative humidity. Makeup water is supplied at 15°C. Ignore kinetic and potential energy effects. For operation at steady state, determine: (a) the volumetric flow rate of atmospheric air entering, in m3/sec. (b) the mass flow rate of makeup water, in kg/sec. (c) the dew point of air exiting the cooling tower.

Answers

Answer 1

a) the volumetric flow rate of atmospheric air entering the cooling tower is 63.29 m³/sec.

b) the mass flow rate of makeup water is 47937.5 kg/sec.

c) the dew point of air exiting the cooling tower is 25.9°C.

(a) The volumetric flow rate of atmospheric air entering, V1 = ?

Density of atmospheric air at the entering condition, ρ3 = 1.184 kg/m³

The volumetric flow rate of atmospheric air entering the cooling tower can be determined using the formula:

V₁ = Q / (ρ₃ × Cp × (t₄ - t₃))

Where, Cp is the specific heat of air at constant pressure = 1005 J/kg K

Substituting the values in the above equation:

V₁ = 1200 × 10^6 / (1.184 × 1005 × (40 - 20)) = 63.29 m³/sec

Thus, the volumetric flow rate of atmospheric air entering the cooling tower is 63.29 m³/sec.

(b) The mass flow rate of makeup water, m5 = ?The cooling water returning to the condenser is at a temperature of 20°C.

Assuming no heat losses, the energy transferred from the condenser to the cooling water is equal to the energy taken up by the air-water mixture entering the tower.

Therefore, the energy balance equation becomes:

Q = mₐ × Cpₐ × (t₄ - t₃) + mₕ × Cpₕ × (t₅ - t₄)

Where, mₐ is the mass flow rate of atmospheric air, mₕ is the mass flow rate of water, and Cpₐ and Cpₕ are the specific heats of air and water, respectively.

Rearranging the equation we get,

mₕ = (Q - mₐ × Cpₐ × (t₄ - t₃)) / (Cpₕ × (t₅ - t₄))

Substituting the values in the above equation:

mₕ = (1200 × 10^6 - 63.29 × 1.184 × 1005 × (40 - 20)) / (4.18 × (20 - 15)) = 47937.5 kg/sec

Thus, the mass flow rate of makeup water is 47937.5 kg/sec.

(c) The dew point of air exiting the cooling tower, tdp = ?

To calculate the dew point temperature, we need to calculate the partial pressure of water vapor in the air mixture at the exit condition.

Using the formula for the saturation pressure of water vapor:

pₛ = exp(77.3450 + 0.0057T - 7235 / T)

where T is the temperature in K (Kelvin)

At the inlet condition:

ps3 = exp(77.3450 + 0.0057(20 + 273) - 7235 / (20 + 273))= 2.339 kPa

At the exit condition:

ps4 = φ4 × ps4 = 0.9 × 5.85 = 5.265 kPa

Therefore, the partial pressure of water vapor in the air-water mixture is 5.265 kPa.

At the dew point condition:

p = ps4 = 5.265 kPa

Using the steam tables, at a pressure of 5.265 kPa, the dew point temperature is 25.9°C

Therefore, the dew point of air exiting the cooling tower is 25.9°C.

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Answer 2

(a)The volumetric flow rate of atmospheric air entering the cooling tower in m3/sec is 5.197 m3/sec

(b)The mass flow rate of makeup water supplied in kg/sec is 78.89 kg/sec

(c)The dew point of air exiting the cooling tower is 35.23°C.

A condenser is a machine that converts a gaseous substance to a liquid one. In a power plant, the condenser is used to transform steam from the turbine into water so that it may be used again. The steam produced from the power generation process is first driven through a turbine. When it exits the turbine, the steam is led to a condenser where it is cooled by water. The volumetric flow rate is the measure of how much space is occupied by a fluid or gas that passes through a given area. It is calculated by dividing the mass flow rate by the density of the fluid or gas.

Here, the volumetric flow rate of atmospheric air entering the cooling tower can be determined using the formula below;{eq}\text{Volumetric flow rate, Q} = \frac{\text{Mass flow rate}}{\text{Density}} {/eq}Density of atmospheric air at 20°C, 1 atm, and 50% relative humidity can be found using steam tables. For these conditions, the density of air is 1.1769 kg/m3.

The rate at which makeup water is supplied is equal to the rate at which cooling water leaves the tower. This is because the mass of water leaving is equal to the mass of makeup water supplied. Thus, the mass flow rate of makeup water supplied is 7.198 × 106 kg/sec.(c)To determine the dew point temperature of the moist air, we need to use a psychrometric chart.

First, we need to determine the humidity ratio of the atmospheric air. The chart shows that the relative humidity is 50% and the temperature is 20°C. At these conditions, the humidity ratio is 0.00845 kg H2O/kg dry air. Now, the humid air exits at 40°C and 90% relative humidity. On the chart, we go to 40°C and follow the 90% relative humidity line to the left until it intersects the line of 0.00845 kg H2O/kg dry air. We then read the corresponding dew point temperature from the chart, which is found to be 35.23°C.

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