Answer :
The rate of heat loss from the steam line is approximately 9,817.5 Watts and the annual cost is approximately $0.8572.
To calculate the rate of heat loss from the steam line, we can use the formula for heat transfer by convection:
Q_conv = h × A × ΔT
Where:
Q_conv is the rate of heat transfer by convection,
h is the convective heat transfer coefficient,
A is the surface area of the pipe,
ΔT is the temperature difference between the surface of the pipe and the surroundings.
(a) Let's calculate the rate of heat loss from the steam line:
Given data:
Pipe length (L) = 25 m
Pipe diameter (D) = 100 mm = 0.1 m
Pipe surface temperature (T_s) = 150ºC
Surrounding temperature (T_surroundings) = 25ºC
Convective heat transfer coefficient (h) = 10 W/m²K
First, we need to calculate the surface area of the pipe using its diameter:
A = π × D × L
A = 3.14159 × 0.1 m × 25 m
A ≈ 7.854 m²
Next, we can calculate the temperature difference:
ΔT = T_s - T_surroundings
ΔT = 150ºC - 25ºC
ΔT = 125ºC
Now, we can calculate the rate of heat loss:
Q_conv = h × A × ΔT
Q_conv = 10 W/m²K × 7.854 m² × 125ºC
Q_conv ≈ 9,817.5 W
Therefore, the rate of heat loss from the steam line is approximately 9,817.5 Watts.
(b) To calculate the annual cost of heat loss, we need to consider the energy consumed by the gas-fired boiler to generate the steam.
Given data:
Efficiency of the gas-fired boiler (η_f) = 0.90
Price of natural gas (C_g) = $0.01 per MJ
The energy consumed by the gas-fired boiler can be calculated using the formula:
Energy_consumed = Energy_required / η_f
To find the energy required, we can use the rate of heat loss:
Energy_required = Q_conv × time
Assuming the steam line operates continuously throughout the year (24 hours/day, 365 days/year):
Energy_required = Q_conv × (24 hours/day) × (365 days/year)
Energy_required = 9,817.5 W × 24 × 365 J
Energy_required ≈ 85,723,050 J
Converting the energy to megajoules (MJ):
Energy_required_MJ = Energy_required / 1,000,000
Energy_required_MJ ≈ 85.72 MJ
Now, we can calculate the annual cost of heat loss:
Cost = Energy_consumed × C_g
Cost = Energy_required_MJ × $0.01/MJ
Cost ≈ 85.72 MJ × $0.01/MJ
Cost ≈ $0.8572
Therefore, the annual cost of heat loss from the steam line is approximately $0.8572.
Learn more about Heat loss.
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