A steam pipe is 75 mm external diameter and is 80 m long. It conveys steam at a
rate of 1000 kg/h at a pressure of 2 Mpa. The steam enters the pipe with a
dryness fraction of 0.98 and is to leave the pipe with a dryness of not less than 0.96. The pipe must be insulated – the material to be used has a thermal coefficient of conductivity of 0.08 W/mK.
If the temperature drop across the pipe is negligible, find the minimum thickness of insulation required to meet the conditions. The temperature at the outer surface of the insulation is 270C.
heat transfer
To find the minimum thickness of insulation required, we need to consider the heat transfer through the pipe and insulation material.
1. Calculate the heat transfer due to steam flow:
- Given steam flow rate: 1000 kg/h
- Calculate the specific enthalpy of steam at the given conditions using steam tables or steam table software.
- Subtract the specific enthalpy of liquid water at the same pressure from the specific enthalpy of the incoming steam to get the specific enthalpy of the steam mixture.
- Calculate the heat transfer rate using the specific enthalpy of the steam mixture and the steam flow rate.
2. Calculate the heat transfer through the steam pipe:
- Use the heat transfer rate calculated in step 1 to find the heat transfer per unit length of the pipe.
- Calculate the outer surface area of the pipe using the external diameter and length of the pipe.
- Use the thermal conductivity of the pipe material and the temperature difference between the steam and outer surface of the pipe to calculate the heat transfer per unit length of the pipe.
3. Calculate the heat transfer through the insulation:
- Use the heat transfer rate calculated in step 1 to find the heat transfer per unit length of the insulation.
- Calculate the outer surface area of the insulation using the external diameter and length of the pipe plus insulation.
- Use the thermal conductivity of the insulation material and the temperature difference between the outer surface of the pipe and the outer surface of the insulation to calculate the heat transfer per unit length of the insulation.
4. Set up an energy balance equation:
- The heat transfer rate through the pipe is equal to the heat transfer rate through the insulation:
Heat transfer through the pipe = Heat transfer through the insulation
5. Use the energy balance equation to solve for the minimum thickness of insulation:
- Rearrange the equation to solve for the insulation thickness.
- Substitute the known values and solve for the insulation thickness.
Note: The calculations involve specific enthalpy values, which will require using steam tables or steam table software.
To find the minimum thickness of insulation required, we need to consider the heat transfer that occurs through the pipe and insulation.
First, let's calculate the mass flow rate of the steam. Given that it conveys steam at a rate of 1000 kg/h, we need to convert this to kg/s:
Mass flow rate = 1000 kg/h = (1000/3600) kg/s = 0.2778 kg/s
Next, let's determine the heat transfer through the pipe. The heat transfer equation for a steady-state situation is:
Q = U * A * ΔT
Where:
Q is the heat transfer rate (in Watts),
U is the overall heat transfer coefficient (in W/m^2K),
A is the surface area of the pipe (in m^2), and
ΔT is the temperature difference across the pipe (in K).
Since the temperature drop across the pipe is negligible, ΔT is effectively zero in this case. Therefore, the heat transfer rate through the pipe is also zero.
Now, let's consider the heat transfer through the insulation. The heat transfer equation for a steady-state situation is:
Q = U * A * ΔT
Where:
Q is the heat transfer rate (in Watts),
U is the overall heat transfer coefficient (in W/m^2K),
A is the surface area of the insulation (in m^2), and
ΔT is the temperature difference across the insulation (in K).
To find the overall heat transfer coefficient (U) for the insulation, we can use the following equation:
U = (1/h1 + x/k + 1/h2)
Where:
h1 is the convective heat transfer coefficient for the inner surface (in W/m^2K),
x is the insulation thickness (in m),
k is the thermal coefficient of conductivity for the insulation material (in W/mK), and
h2 is the convective heat transfer coefficient for the outer surface (in W/m^2K).
In this case, the steam enters the pipe with a dryness fraction of 0.98, which means it is almost completely saturated. Therefore, we can assume that the convective heat transfer coefficient for the inner surface (h1) is the same as that for saturated steam at the given pressure.
Next, we need to calculate the convective heat transfer coefficient for the outer surface (h2). This can be determined using empirical correlations or by considering the heat transfer mechanism (e.g., natural convection, forced convection). Without this information, it is not possible to calculate h2 accurately. Thus, h2 is assumed to be known or given.
With the given information, we can now calculate the minimum thickness of insulation required (x):
U = (1/h1 + x/k + 1/h2)
Since we already know the values of U, h1, h2, and k, we can rearrange the equation to solve for x:
x = (U - 1/h1 - 1/h2) * k
Substituting the given values into the equation, we can find the minimum thickness of insulation required.