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 savings as a function of the area, we need to consider the energy transferred by the heat exchanger and the cost of the steam generated.
First, let's define some variables:
A = Area of the heat exchanger (in square meters)
Qdot = Energy transferred by the heat exchanger (in watts)
Qdot_steam = Energy required to generate steam (in watts)
Cost_steam = Cost of steam generated per unit energy (in dollars per joule)
Cost_annual = Saving as a uniform annual amount (in dollars per year)
The energy transferred by the heat exchanger can be calculated using the formula:
Qdot = U * A * (T_g - T_sw)
Where:
U = U value of the heat exchanger based on gas-side area (in watts per square meter per Kelvin)
T_g = Temperature of the exhaust gas (in Kelvin)
T_sw = Temperature of the saturated water entering the heat exchanger (in Kelvin)
The energy required to generate steam can be calculated using the formula:
Qdot_steam = m_dot * (h_fg + c_p_water * (T_steam - T_sw))
Where:
m_dot = Flow rate of exhaust gas (in kilograms per second)
h_fg = Enthalpy of vaporization of water at saturated conditions (in joules per kilogram)
c_p_water = Specific heat of water (in joules per kilogram per Kelvin)
T_steam = Temperature of the steam generated (in Kelvin)
The saving as a uniform annual amount can be calculated using the formula:
Cost_annual = (Qdot - Qdot_steam) * Cost_steam
You can substitute the given values into these formulas to find the equation for savings as a function of the area.
To develop the equation for the saving as a function of the area, expressed as a uniform annual amount, we need to consider the following factors:
1. Heat Transfer Calculation:
The heat transfer rate in the heat exchanger can be calculated using the formula:
Q = m * cp * ΔT
Where:
Q = Heat transfer rate (in Watts)
m = Mass flow rate of exhaust gas (in kg/s)
cp = Specific heat of exhaust gas (in kJ/(kg*K))
ΔT = Temperature difference between exhaust gas and steam (in K)
2. Savings Calculation:
The amount of heat transferred to the steam can be converted to energy in gigajoules (GJ) using the formula:
Energy in GJ = Q * (1 / 10^9)
The cost of the heat in the form of steam can be calculated using the formula:
Cost of heat in $ = Energy in GJ * Cost per GJ
Since the heat exchanger operates continuously, the cost is considered per year. Therefore, the savings in each year will be:
Savings = Cost of heat in $ * (1 - (1 + r)^-n) / r
Where:
r = Interest rate (in decimal form)
n = Life of installation (in years)
3. Cost of Heat Exchanger:
The cost of the heat exchanger, including installation, is given as a function of the gas-side area. Therefore, the cost of the heat exchanger for a given area A can be calculated as:
Cost of heat exchanger = Area * Cost per square meter
Now, let's put all these factors together to develop the equation for the saving as a function of the area, expressed as a uniform annual amount.