The assumption is introduced that the ratio q will 



r 

 remain constant across the width of the boundary 

 layer. This is reasonable since the momentum 

 lost in the turbulent layer is that transferred to 

 the laminar layer in the absence of friction, and 

 the heat lost from the turbulent layer is that 

 transmitted through the laminar layer. The 

 reader is referred to Schlicting (1960) for a com- 

 plete treatment of this development. The end 

 result of the integration and combination of the 

 above equatic^ns, (2) and (3), is an expression for 

 the local coefficient of heat transfer across a 

 boundary : 



(4) 



[■+*«-0] 



where : 



T=stress in the laminar subboundary layer 

 [7m = average fluid velocity at a great distance 

 from the boundary surface 

 u, = average fluid velocity at the top of the 



laminar subboundary layer 

 P=ratio of molecular momentum and 



thermal transfer coefficients 

 P, = ratio of turbulent momentum and 

 thermal transfer coefficients 

 </ = gravity 

 Cp = specific heat 



A basic problem of measurement of the boundary 

 layer stress (t) remains in equation (4). Simpli- 

 fication of this equation is required before the 

 calculation of Qa and Qw can be attempted and 

 used in practical deterioration problems. 



Statistical Approach 



Certain simplifications of equation (4) will now 

 be made. P, can be set equal to 1, which implies 

 that the turbulent mechanism for the exchange of 

 momentum is the same as that for heat. Ac- 

 cording to Schlictuig ( 1 960) , because the boimdary 

 velocity and temperature profiles are quite similar, 

 little error will result from this assumption. The 

 Prandtl number P will equal 1 when dealing with 

 gases, but will differ considerably in liquids. 

 This means that the heat transfer between the 

 air-ice boundary and the water-ice boundary 

 should receive sliglitly different treatment. 



Tlie stress term in the above equation can be 



further broken down into r= — ;— '• With w, =0 at 



dy 



the wall, and linearizing the gradient, the equation 

 (4) will now take the form: 



[C'„+17,(^-.)]: 



Now setting P=\ and P,= \ as stipulated above 

 for the air exposed portion of the berg, the equa- 

 tion evolves down to: 





(5) 



with F being equal to a number made up by 

 gravity, specific heat, viscosity, and the term 



y 



- representing the laminar subboundary layer 



velocity gradient. This gradient is impractical to 

 determine in the field at this time and requires 

 that an empirical approach be attempted. A 

 general expression for heat transfer is: 



q = a{T„ — T^) 



(6) 



where: 



T'„ = average air temperature at a great 



distance from the berg 

 7"^ = surface temperature of the iceberg, 



assumed to be 0° C 

 a = coefficient of heat transfer 



Thus combming equations (5) and (6) the rate of 

 heat transfer per unit area from the air environ- 

 ment to the berg becomes: 



5=a7'„==^ /''gmcals/cm2/sec (7) 



For a water environment the relationship is not 

 as concise because P^^l, therefore 



gCpUin 



{U^+uAP-\)] 



y 



(8) 



The measurement of [/„ for water would be very 

 difficult if not impossible. Water movement rela- 

 tive to the berg will be very small in the horizontal 

 and totally negligible when compared to the 

 turbulence caused by the rolling of a berg. Water 

 velocities of several knots are possible as a berg 

 oscillates or rolls in a seaway. This is a variable 

 and depends upon the bergs size stability and the 



47 



