a. Steam at 230oC flows through a steel pipe (� = 15��/�.
) whose inner and outer
diameters are 9 cm and 10 cm, respectively in an environment at 15oC. The pipe is
insulated with 5 cm thick fiberglass (� = 0.035��/�.
). The given heat transfer
coefficients on the inside and the outside of the pipe are 170��/��.
and 38��/
��.
respectively.
Calculate the heat transfer rate across the steel pipe.
To calculate the heat transfer rate across the steel pipe, we need to use the formula for calculating heat transfer through a cylindrical conductor:
Q = 2πkL(T1 - T2) / ln(r2 / r1)
where:
Q is the heat transfer rate
k is the thermal conductivity of the material (in this case, steel)
L is the length of the pipe
T1 and T2 are the temperatures on the inner and outer surfaces of the pipe, respectively
r1 and r2 are the inner and outer radii of the pipe, respectively
Given data:
- Steam temperature (T1) = 230oC
- Environment temperature (T2) = 15oC
- Inner diameter of the pipe (2r1) = 9 cm
- Outer diameter of the pipe (2r2) = 10 cm
- Insulation thickness (L) = 5 cm
- Thermal conductivity of steel (k) = 15 W/(m·K)
First, we need to convert the temperatures from Celsius to Kelvin:
T1 = 230 + 273 = 503 K
T2 = 15 + 273 = 288 K
Next, let's convert the dimensions from centimeters to meters:
r1 = 9 cm / 100 = 0.09 m
r2 = 10 cm / 100 = 0.1 m
L = 5 cm / 100 = 0.05 m
Now, we can substitute the values into the formula and calculate the heat transfer rate:
Q = (2π * 15 * 0.05 * (503 - 288)) / ln(0.1 / 0.09)
Using a calculator, we can solve this equation step by step:
Step 1: Calculate the temperature difference:
ΔT = T1 - T2 = 503 - 288 = 215 K
Step 2: Calculate the natural logarithm of the radii ratio:
ln(r2 / r1) = ln(0.1 / 0.09) ≈ 0.1054
Step 3: Calculate the heat transfer rate:
Q = (2π * 15 * 0.05 * 215) / 0.1054 ≈ 1025.6 W
So, the heat transfer rate across the steel pipe is approximately 1025.6 watts.