The exhaust-gas temperature leaving a continuously operating furnace is 260°C, and a proposal is being considered to install a heat exchanger in the exhaust-gas stream to generate low-pressure steam at 105°C. the question to be investigate is whether it is economical to install such a heat exchanger, and if so, to find its optimum size. The following data apply:
Table
year factor year factor year factor
1 0 8 3.008 15 5.275
2 0.476 9 3.376 16 5.552
3 0.937 10 3.730 17 5.801
4 1.379 11 4.060 18 6.058
5 1.810 12 4.384 19 6.295
6 2.224 13 4.696 20 6.5
7 2.622 14 5.002 21 6.703
Flow of exhaust gas-----7.5kg/s
Specific heat of exhaust gas---1.05kJ/(kg*k)
Value of the heat in the form of steam---$1.5 per gigajoule
U value of heat exchanger based on gas-side area---23W/(m^2*K)
Cost of the heat exchanger including installation based on gas-side area----$90 per square meter
Interest rate----8%
Life of installation----5years
Saturated liquid water enters heat exchanger at 105°C and leaves as saturated vapor
1) Develop the equation for the saving as a function of the area, expressed as a uniform annual amount.
To develop the equation for the saving as a function of the area, expressed as a uniform annual amount, we need to consider the cost of the heat exchanger, the value of the heat in the form of steam, and the operating costs.
1) Calculate the heat transfer rate:
The heat transfer rate can be calculated using the formula:
Q = m * Cp * (T2 - T1)
Where:
Q = Heat transfer rate (in Watts)
m = Mass flow rate of the exhaust gas (in kg/s)
Cp = Specific heat of the exhaust gas (in kJ/(kg*K))
T2 = Temperature of the exhaust gas leaving the furnace (in °C)
T1 = Temperature of the exhaust gas entering the heat exchanger (in °C)
Given:
Mass flow rate of the exhaust gas (m) = 7.5 kg/s
Specific heat of the exhaust gas (Cp) = 1.05 kJ/(kg*K)
Temperature of the exhaust gas leaving the furnace (T2) = 260 °C
Assuming the exhaust gas entering the heat exchanger is at the same temperature (T1 = 260 °C)
Q = 7.5 * 1.05 * (260 - 260) = 0 kJ/s
The heat transfer rate is zero since the temperatures at both ends of the heat exchanger are the same. Therefore, there is no heat recovery in this case.
2) Calculate the cost of the heat exchanger:
The cost of the heat exchanger, including installation, is given as $90 per square meter of the gas-side area. However, since there is no heat recovery, the gas-side area of the heat exchanger is zero. Therefore, the cost of the heat exchanger is also zero.
3) Calculate the value of the heat in the form of steam:
The value of the heat in the form of steam is given as $1.5 per gigajoule. To calculate the value of the heat in the form of steam, we need to determine the energy content of the steam generated.
The energy content of the steam can be calculated using the formula:
Energy = m * H_v
Where:
Energy = Energy content of the steam (in Joules)
m = Mass flow rate of the steam (in kg/s)
H_v = Latent heat of vaporization of water at the given pressure (in J/kg)
Since the steam enters and leaves the heat exchanger at the same saturation temperature, the mass flow rate of the steam can be calculated. We can assume that the exhaust gas transfers all of its heat to the steam.
m = Q / (H_v)
Where:
m = Mass flow rate of the steam (in kg/s)
Q = Heat transfer rate (in Watts)
Given:
Q = 0 kJ/s
Therefore, mass flow rate of steam (m) = 0 kg/s
The mass flow rate of steam is zero since there is no heat recovery. Hence, the value of the heat in the form of steam is also zero.
4) Calculate the operating costs:
Operating costs can be calculated by multiplying the cost of the heat in the form of steam by the energy content of the steam.
Operating costs = Value of the heat in the form of steam * Energy content of the steam
Given:
Value of the heat in the form of steam = $1.5 per gigajoule
Energy content of the steam = 0 Joules (since mass flow rate of steam is zero)
Operating costs = $1.5 / gigajoule * 0 J = $0
Since the heat transfer rate and the value of the heat in the form of steam are both zero, the operating costs are also zero.
Therefore, the saving as a function of the area, expressed as a uniform annual amount, is zero.
To develop the equation for the savings as a function of the area, expressed as a uniform annual amount, we need to calculate the amount of heat recovered by the heat exchanger and determine the cost savings resulting from this heat recovery.
1) Calculate the heat recovered by the heat exchanger:
The heat recovered by the heat exchanger is given by the equation:
Heat Recovered (Q) = Mass flow rate * Specific heat * Temperature difference
Here's how to calculate it:
Mass Flow Rate of exhaust gas = 7.5 kg/s
Specific Heat of exhaust gas = 1.05 kJ/(kg * K)
Temperature Difference = (Temperature of exhaust gas) - (Temperature of low-pressure steam)
Temperature of exhaust gas = 260°C
Temperature of low-pressure steam = 105°C
Substituting the values:
Temperature Difference = (260°C - 105°C) = 155°C
Heat Recovered (Q) = 7.5 kg/s * 1.05 kJ/(kg * K) * 155°C
2) Calculate the cost savings resulting from heat recovery:
The cost savings can be calculated by multiplying the heat recovered by the value of the heat in the form of steam, which is given as $1.5 per gigajoule.
To convert the heat recovered from kJ to gigajoules:
Heat Recovered (in gigajoules) = (Heat Recovered (in kJ)) / (10^6)
Cost Savings = Heat Recovered (in gigajoules) * $1.5
3) Express the savings as a uniform annual amount:
To express the savings as a uniform annual amount, we need to calculate the present worth factor (PW) based on the given interest rate, life of installation, and the table provided.
Using the given table, find the factor for the life of installation (5 years) under the "factor" column. The factor for 5 years is 1.810.
Present Worth Factor (PW) = Factor for 5 years = 1.810
Uniform Annual Savings = Cost Savings / Present Worth Factor (PW)
So, the equation for the saving as a function of the area, expressed as a uniform annual amount, is:
Savings (Uniform Annual) = Cost Savings / Present Worth Factor (PW)