equation of continuity shown by Prandlt (1952) 

 now takes the form of: 



P<-pU-^-^. ^(Cp^,+ ^) ^j- 



dx dy 



-i{(^^.-f^)f}-o 



Assume now an element of fluid on the boundary 

 surface with one side acting as the heat flow 

 boundary and with no heat flow assumed in the 

 direction as shown below. 



It can be assumed that the heat (Q) lost through 

 the boundary surface must pass through the 

 boundary layer and must equal the heat lost 

 between the sections y^ z, and ?/2 22- This is true 

 even in the turbulent case since A, is zero in the 

 laminar subboundary layer. The continuity ex- 

 pression shown above now reduces to 



pCpU 



dT 

 dx' 



by \ dy J 



or over the area of the surfaces 



c,pjju{T,-T,)dydz= fjk (^yxdz 



From this expression it is apparent that the heat 

 transfer through the boundary surface is controlled 

 entirely by the thickness of the lammar sub- 

 boundary layer and the thermal conductivity. 

 This means that if u is low enough, assuming 

 good mixing within the elemental volume, the 

 heat flow through the boundary will be limited 

 by the amount of heat coming in. In this case 

 the temperature T.2 would be equal to the temper- 

 ature at the boundary. As u increases, T2 will 

 become greater than the boundary temperature 

 and approach Tj in value. At the same tune 



more heat will be provided for boundary transfer. 

 When this occurs, the heat flow across the bound- 

 ary tends toward an equilibrium condition where 



Ii—To^O, and ;r >^ — 



dy dy 



tion is never reached as long as (u) increases. 



The equilibrium condi- 



dT 



This is because the gradient ~ will increase with 



(u) as the thickness of the laminar subboundary 



layer is reduced. 



The difficulty in evaluating the conductive- 



convective heat transfer relationship shown above 



is lack of knowledge of the temperature gradient 



dT 



-5— near the surface of the berg. The gradient is 



impossible to measure due to its minute thickness 



when turbulent flow exists. It does not lend itself 



to modeling due to the variable effects of large and 



small eddies created by the erratic surface shapes 



found on bergs. Schlicting (1960) shows the 



Reynolds analogy which allows the derivation of 



relations for heat transfer from relations for 



turbulent flow. This is convenient since fluid 



velocity measurements near boundaries have been 



made and eventually may be attempted in the 



vicinity of an iceberg. In the Reynolds analogy, 



A u 



Prandtl numbers given by P i = -r andP=^Cp t 



A^ K 



are defined showing the relationship of the eddy 

 and molecular transfer coefficients for convenience 

 of notation. The basic equation showing momen- 

 tum transfer and heat transfer, within both the 

 turbulent and laminar subboundary layers is given 

 by: 



= (M+^r) -T- (stress) 



(2) 



q=-Cj,g\ 



\dT 



(heat) 



- = velocity gradient in a particular layer 



(3) 



where : 

 du 

 df 



(/r_ temperature gradient in a particular 

 dy layer 



/i = molecular coefficient of viscosity 

 .4r = eddy coefficient of momentum transfer 

 y4j = eddy coefficient of thermal conductivity 



17 = gravity 

 fp = specific heat 



/: = molecular thermal conductivity 



46 



