While the design is for the deep ocean, say 20,000 feet, the 

 immediate results must be related to 1 atmosphere; comparable values of 

 the important phvsical properties and resulting dimensionless numbers 

 are tabulated in Table 1. Because of the small change in physical prop- 

 erties in the direction of improved heat transfer with depth, results 

 from the 1 -atmosphere tests are slightly conservative--aetal surface 

 temperatures will be slightly lower in the actual deep-ocean applications. 

 The f(Pr) is shown as a function of Pr, Figure 3, and the values for the 

 shallow and deep water indicated in the curve. For the remainder of 

 the report, shallow water values only will he used. 



OPTIMIZATION OF EXTENDED SURFACES --ITERATIVE APPROACH 



Referring to Equation 1 and Figure 1, it is obvious that the only 

 physical aspect of the heat rejection surface related to h is the height, 

 L, which is fixed by the dimensions of the thermoelectric elements to be 

 used. The other important parameters are the physical properties of the 

 fluid, the temperature difference (At), and the exponent --which is assumed 

 to be 1/4, as the Crashoff's number is about 10 , the approximate upper 

 limit for laminar flew (Braun, 1965). Table 2 gives some estimates of 

 the effect o* varying the surface temperature on the convection film 

 coefficient, h, the resulting heat rejection of a bare cylinder, and 

 the increase in area per each of the 12 modules necessary to reject the 

 design value of 8,850 Btu per hour per module. 



Because all of the heat transferred from the base area through the 

 extended fin surface must be moved through the fin rjtal with a loss in 

 temperature proportional to the fin conductivity, there is an obvious 

 tradeoff between long, slender fins (dimension H, Figure 2) to obtain a 

 maximum usable area and short, stubby fins to produce a high surface 

 temperature (minimize temperature drop) and a resulting high convection 

 coefficient, h. Because the heat transferred is proportional to the area 

 and to h, and because h varies as the l/4th power of the temperature 

 difference, there is no simple linear relationship alloving a direct 

 approach to establishing proportions. Using criteria listed earlier, a 

 first design of a possible shape is snown, Figure 4. This stubby fin has 

 approximately 2.5 times the area of the base cylinder and from Figures 

 5, 6, and 7, this fin should have a surface temperature of about 130°F 

 when transferring the necessary heat to 40°F seawater at low pressures, 

 if made of copper. 



A similar, much longer c in made of aluminum (K = 1,500, Table 3) 

 would have temperatures approximately as shown in Figure 8. This is the 

 first result of the use of an analog computer which utilizes a high 

 resistance paper to simulate heat conduction and a millivolt meter to 

 read voltages, which simulate temperature. In Figure 2, the surface 

 was divided into ten segments and the surface temperature estimated 

 for each. The method is very rapid and powerful, in that simple shapes 

 can be rapidly produced, and substitution of various fin materials can 



