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Steam pipe
Steam is passing through the insulated steel pipe. Minimum insulation layer thickness which is required for the allowable heat losses should be found.
Problem type:
Planeparallel problem of heat transfer.
Geometry:
r_{1} = 60 mm, r_{2} = 80 mm, r_{3} = ?
Given:
Thermal conductivity of steel λ_{steel} = 42 W/(m·K);
Thermal conductivity of asbestos λ_{asbestos} = 0.8 W/(m·K);
Temperature of steam T_{steam} = 150°C,
Temperature of air T_{air} = 20°C,
Convection coefficient on the steam side of the pipe α_{steam} = 100 W/K·m^{2},
Convection coefficient on the air side of the pipe α_{air} = 30 W/K·m^{2},
Allowable power of heat losses F = 2.1 kW/m^{2}.
Task:
Determine the minimum thickness of the insulation (r_{3}r_{2}) required for allowable heat losses.
Solution:
Heat flux passing through any crosssection of the pipe is the same and could be measured at the pipe's surface. To solve the problem it is enough to take a piece of the pipe with arbitrary length, since the flux per meter of the length is the same. For convenience we choose such a length so that square of the steel surface equals to 1 m^{2}, i.e. L_{z} = 1/(2π·r_{2}) = 2.122 m. In this case the heat flux density is numerically equal to the heat flux.
Search of the optimal solution is automated via utility LabelMover, which creates and solves series of the problems with different values of r_{3}, converging to the heat flux value of 2.1 kW.
Results:

r_{3},mm 
F, W/m^{2} 
QuickField 
104.3 
2102 
Theory^{*} 
105 
2100 
Error 
0.7% 
0.1% 
^{*}Rajput R.K.(2010). Engineering Thermodynamics, Third Edition. Sudbury, MA:Jones and Bartlett Publishers, p. 813. Example 15.11.
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