Question

In a food processing plant, a steel pipe (thermal conductivity 17 W/m

∘
C, internal diameter 5 cm; thickness 3 mm ) is being used to transport a liquid food. The inside surface temperature of the pipe is at 95
∘
C. A 4−cm thick insulation (thermal conductivity 0.03 W/m
∘
C ) is wrapped around the pipe. The outside surface temperature of the insulation is 30
∘
C. Show the problem givens on a figure and calculate the rate of heat transfer per unit length of the pipe.

Answer 1

To calculate the rate of heat transfer per unit length of the pipe, use the formula for heat conduction.

Step-by-step explanation:

To calculate the rate of heat transfer per unit length of the pipe, we can use the formula for heat conduction:

Q/t = (k * A * ΔT) / d

Where:

• Q/t is the rate of heat transfer per unit time
• k is the thermal conductivity of the pipe
• A is the surface area of the pipe
• ΔT is the temperature difference across the pipe
• d is the thickness of the pipe

Using the given values:

• k of the steel pipe is 17 W/m°C
• A = π * (r² - (r - d)²) where r is the internal radius of the pipe
• ΔT = 95°C - 30°C
• d is given as 3 mm

With these values, you can calculate the rate of heat transfer per unit length of the pipe.

Answer 2

The rate of heat transfer per unit length of the steel pipe with an inner surface temperature of 95°C and covered in 4-cm thick insulation with an outer surface temperature at 30°C is approximately 2.06 W/m.

Step-by-step explanation:

To calculate the rate of heat transfer per unit length of the pipe, we can use the formula for heat conduction through a cylindrical wall, which is derived from Fourier’s law of thermal conduction:

Q/t = 2πL(T1 - T2)/[ln(r2/r1)/k1 + ln(r3/r2)/k2]

Where:

• Q/t is the rate of heat transfer (Watts/m)
• L is the length of the pipe (which will cancel out, as we are looking for heat transfer per unit length)
• T1 and T2 are the temperatures at the inner and outer surfaces, respectively
• r1 is the inner radius of the steel pipe
• r2 is the outer radius of the steel pipe
• r3 is the outer radius of the insulation
• k1 is the thermal conductivity of the steel
• k2 is the thermal conductivity of the insulation

The given values are:

• Internal diameter of the steel pipe = 5 cm, so r1 = 2.5 cm = 0.025 m
• Thickness of the steel pipe = 3 mm, so r2 = r1 + 0.003 m = 0.028 m
• Thickness of the insulation = 4 cm, so r3 = r2 + 0.04 m = 0.068 m
• Inner surface temperature (T1) = 95°C
• Outer surface temperature (T2) = 30°C
• Thermal conductivity of steel (k1) = 17 W/m°C
• Thermal conductivity of insulation (k2) = 0.03 W/m°C

Plugging in the values:

Q/t = 2π(95 - 30)/[ln(0.028/0.025)/17 + ln(0.068/0.028)/0.03]

Q/t = 130π/[0.22239/17 + 1.8718/0.03]

Q/t ≈ 130π/[0.01308 + 62.393]

Q/t ≈ 130π/62.406

Q/t ≈ 2.06 Watts per meter (rounded to two decimal places)

Therefore, the rate of heat transfer per unit length of the pipe covered with the given insulation is approximately 2.06 W/m.