A submarine is to be designed to provide a comfortable temperature of no less than 21°C for the crew. The submarine is idealized by a cylinder 9 m in diameter and 61 m in length, as shown. The combined heat transfer coefficient on the interior is about 14 W/(m2 K), while on the outside the heat transfer coefficient is estimated to vary from about 57 W/(m2 K) (not moving) to 847 W/(m2 K) (top speed). For the following wall constructions, determine the minimum size in kilowatts of the heating unit required if the sea water temperatures vary from 1.1 to 12.8°C during operation. The walls of the submarine are (a) 2 cm aluminum (b) 1.8 cm stainless steel with a 2.5 cm thick layer fiberglass insulation on the inside and (c) of sandwich construction with a 1.8 cm thick layer of stainless
steel, a 2.5 cm thick layer of fiberglass insulation, and a 0.6 cm thick layer of aluminum on the inside. What conclusions can you draw?
GIVEN
• Submarine
? Inside temperature (Ti) > 21°C
• Can be idealized as a cylinder
? Diameter (D) = 9 m length (L) = 61 m
• Combined heat transfer coefficients
? Inside ( ci h ) = 14 W/(m2 K)
? Outside ( co h ): not moving = 57 W/(m2 K)
? top speed: 847 W/(m2 K)
• Sea water temperature (To) varies: 1.1°C < To < 12.8°C FIND
Minimum size of the heating unit (q) in kW for (a) 2 cm thick aluminum walls (b) 1.8 cm thick stainless steel with 2.5 cm thick layer of fiberglass insulation (c) Sandwich of 1.8 cm thick stainless steel, 2.5 cm thick layer of fiberglass insulation, and 0.6 cm
layer of aluminum
ASSUMPTIONS
• Steady state prevails
• Heat transfer can be approximated as heat transfer through a flat plate with the surface area of the cylinder
• Constant thermal conductivities
• Contact resistance between the difference materials is negligible
SKETCH
PROPERTIES AND CONSTANTS
The thermal conductivities are
Aluminum (ka) = 236 W/(m K) at 0°C
Stainless steel (ks) = 14.36 W/(m K) at 20°C
Fiberglass insulation (kfg) = 0.035 W/(m K) at 20°C
The thermal circuits for the three cases are shown below
The total surface area of the idealized submarine (A) is
(a) For case (a) the total resistance is
The heat transfer through the wall is
By examination of the above equation, the heater requirement will be the largest when To is at its
minimum value and hco is at its maximum value
(b) Similarly, for case (b), the total resistance is
The size of heater needed is
(c) The total resistance for case (c) is
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