High School

An overhead 25-meter-long, uninsulated industrial steam pipe with a diameter of 100 mm is routed through a building where the walls and air are at 25ºC. The pressurized steam maintains a pipe surface temperature of 150ºC. The coefficient associated with natural convection is [tex]h = 10 \, \text{W/m}^2\text{K}[/tex], and the surface emissivity is [tex]\varepsilon = 0.8[/tex].

(a) What is the rate of heat loss from the steam line?

(b) If the steam is generated in a gas-fired boiler operating at an efficiency of [tex]\eta_f = 0.90[/tex] and natural gas is priced at [tex]C_g = \$0.01[/tex] per MJ, what is the annual cost of heat loss from the line?

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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