Question

The compressed air needs of a factory are met by a 150 hp compressor located in a room that is maintained at 20 ° C. In order to minimize the work of the compressor, its intake port is connected to the outside by means of a 11m long and 20 cm diameter duct made of thin aluminum foil. The compressor admits air at a rate of 0.27 m3 / s at the outdoor conditions of 10 ° C and 95 kPa. If you discount the thermal resistance of the duct and take the coefficient of heat transfer on the outside surface of the duct as 10 W / m2 ° C, determine

a) The power used by the compressor to overcome the pressure drop in this pipeline.
b) The rate of heat transfer to the colder incoming air.
c) The rise in temperature of the air as it flows through the duct.

Answer 1

To determine the rate of heat transfer to the colder incoming air and the rise in temperature of the air as it flows through the duct, we need to consider the energy balance of the system.

Step 1: Calculate the heat transfer rate:

The heat transfer rate can be calculated using the equation:

Q = m_dot * C_p * ΔT

where Q is the heat transfer rate, m_dot is the mass flow rate, C_p is the specific heat capacity, and ΔT is the temperature difference.

First, we need to calculate the mass flow rate using the given information:

m_dot = ρ * A * V

where m_dot is the mass flow rate, ρ is the density of air, A is the cross-sectional area of the duct, and V is the velocity of the air.

The density of air can be calculated using the ideal gas law:

ρ = P / (R * T)

where P is the pressure, R is the gas constant, and T is the temperature.

Using the given values:
P = 95 kPa = 95000 Pa
T = 10 °C = 10 + 273 = 283 K
R = 8.314 J/(mol·K)

ρ = 95000 / (8.314 * 283) = 42.58 kg/m^3

The cross-sectional area of the duct can be calculated using the diameter:

A = π * (d/2)^2

where A is the cross-sectional area and d is the diameter.

Using the given values:
d = 20 cm = 0.2 m

A = π * (0.2/2)^2 = 0.0314 m^2

Next, we can calculate the velocity of the air using the given volumetric flow rate:

V = m_dot / ρ * A

Using the given value:
V = 0.27 / (42.58 * 0.0314) = 0.201 m/s

Now, we can calculate the heat transfer rate:

Q = m_dot * C_p * ΔT

The specific heat capacity of air at constant pressure (C_p) is approximately 1005 J/(kg·K).

Using the given values:
ΔT = 20 °C - 10 °C = 10 K

Q = 0.27 * 1005 * 10 = 2709.45 J/s

Therefore, the rate of heat transfer to the colder incoming air is 2709.45 J/s.

Step 2: Calculate the rise in temperature:

To determine the rise in temperature of the air as it flows through the duct, we need to consider the thermal resistance of the duct.

The thermal resistance can be calculated using the equation:

R = L / (k * A)

where R is the thermal resistance, L is the length of the duct, k is the thermal conductivity of aluminum, and A is the cross-sectional area of the duct.

Using the given values:
L = 11 m
k = 200 W/(m·°C)

R = 11 / (200 * 0.0314) = 17.51 °C/W

Now, we can calculate the rise in temperature:

ΔT = Q * R

Using the calculated value for Q:
ΔT = 2709.45 * 17.51 = 47429.65 °C

Therefore, the rise in temperature of the air as it flows through the duct is 47429.65 °C.

In summary, the rate of heat transfer to the colder incoming air is 2709.45 J/s, and the rise in temperature of the air as it flows through the duct is 47429.65 °C.