A standard 4 10 cm steel pipe (ID = 10.066 cm., OD = 11.25 cm) carries superheated steam at 650°C in an enclosed space where a fire hazard exists, limiting the outer surface temperature to 38°C. To minimize the insulation cost, two materials are to be used; first a high temperature (relatively expensive) insulation is to be applied to the pipe and then magnesia (a less expensive material) on the outside. The maximum emperature of the magnesia is to be 315°C. The following constants are known.

Question

(a) Specify the thickness for each insulating material.

(b) Calculate the overall heat transfer coefficient based on the pipe OD.

(c) What fraction of the total resistance is due to (1) steam-side resistance, (2) steel pipe

resistance, (3) insulation (combination of the two), and (4) outside resistance?

(d) How much heat is transferred per hour, per foot length of pipe?

GIVEN

• Steam filled steel pipe with two layers of insulation

• Pipe inside diameter (Di) = 10.066 cm=0.10066 m

• Pipe outside diameter (Do) = 11.25 cm=0.1125 m

• Superheated steam temperature (Ts) = 650°C

• Maximum outer surface temperature (Tso) = 38°C

• Maximum temperature of the Magnesia (Tm) = 315°C

• Thermal conductivities

? High-temperature insulation (kh) = 0.1 W/(m K)

? Magnesia (km) = 0.076 W/(m K)

? Steel (ks) = 43 W/(m K)

• Heat transfer coefficients

? Steam side ( ci h ) = 500 W/(m2 K)

? Outside ( co h ) = 11 W/(m2 K)

• Ambient temperature (Ta) = 21°C FIND

(a) Thickness for each insulation material

(b) Overall heat transfer coefficient based on the pipe OD (c) Fraction of the total resistance due to

? Steam-side resistance

? Steel pipe resistance

? Insulation

? Outside resistance (d) The rate of heat transfer per unit length of pipe (q/L) ASSUMPTIONS

• The system is in steady state

• Constant thermal conductivities

• Contact resistance is negligible

SKETCH

Answer 1

The thermal circuit for the insulated pipe is shown below



The values of the individual resistances can be evaluated with



The variables in the above equations are





The temperatures for this problem are



There are five unknowns in this problem: q/L, T1, T2, r3, and r4. These can be solved for by writing the

equation for the heat transfer through each of the five resistances and solving them simultaneously.

1. Steam side convective heat transfer



2. Conduction through the pipe wall



3. Conduction through the high temperature insulation



4. Conduction through the magnesia insulation



5. Air side convective heat transfer



To maintain steady state, the heat transfer rate through each resistance must be equal. Equations [1] through [5] are a set of five equations with five unknowns, they may be solved through numerical iterations using a simple program or may be combined algebraically as follows

Substituting Equation [1] into Equation [2] yields





Substituting this into Equation [3] and combining the result with Equation [1]



Substituting this into Equation [4] and combining the result with Equation [1]



Finally, substituting this into Equation [5] and combining the result with Equation [1]