where: 



/i/=heat of fusion of water 



therefore Qf=^79.7 gm cals/gmsX2.7X10'' gms = 

 2 1 5.0 X 10'- gm cals. This means that 230.9 X 10'^ 

 gm cals of heat are needed to melt this model berg 

 in the 20-day period. However, only 0.505X10'^ 

 gm cals have been accounted for. Therefore the 

 remaining heat must come from sensible heat 

 transfer from the environment by conduction and 

 convection. Because the heat transferred by 

 conduction and convection amounts to approx- 

 imately 500 times the amount of heat added by 

 radiation, the terms (Qs—Q,), Qu, and Qau^, can 

 be dropped from the heat budget equation as 

 negligible and the most attention must be given 

 to conduction and convection. This is somewhat 

 contrary to what is found in the literature con- 

 cerning studies of glaciers. Heat budgets pre- 

 sented by Hoinkes (1964) for various glaciers in 

 the northern hemisphere show that short-wave 

 radiation accounts for up to 89 percent of the 

 source of incoming heat after consideration of 

 albedo. This means only 11 percent of the in- 

 coming heat is suppUed by environmental tem- 

 peratures. These figures are not apphcable to 

 an iceberg due to the percent of surface area 

 exposed to solar radiation and because of the 

 berg's warmer environmental subarctic tempera- 

 tures. The submergence of a berg in a compara- 

 tively warm liquid, with a high specific heat, 

 cannot be compared with the exposure of a glacier's 

 surface to arctic winds. 



The problem of examining the conductive- 

 convective terms, Qa and Q^, is complicated by 

 the turbulence of the environmental fluids and the 

 random surface configurations of the visible and 

 submerged portions of the berg. For these 

 reasons, the terms Qa and Q^,, the conductive- 

 convective heat flow from the fluid environment 

 to the berg must be examined from first principles 

 of thermodynamics in order to gain a more 

 thorough undertanding of the problem. Prandtl 

 (1952) analyzed the heat flow, from a moving 

 fluid, through a boundary. The following discus- 

 sion is based on his treatment. In the case of 

 laminar flow of fluid past a boundary, i.e., water or 

 air flow past a berg's surface, it is a fundamental 

 concept that a velocity gradient is established as 

 the boundary is approached. This gradient 

 ranges between the velocities of at the boundary 

 to maximum flow at a given distance away from 

 the boundary. The gradient is the result of fluid 



friction. The thickness of the velocity gradient, 

 or boundary layer, is a function of the viscosity of 

 the fluid and the velocity of flow. In the case of 

 laminar flow, all movement is parallel to the 

 direction of flow. 



Heat transfer in laminar flow is given by the 

 equation of continuity as follows: 



bx dy dz 







Qi is the heat transfer along the direction of flow 

 given by Qi = CppuT, where Cp is t!ie constant 

 pressure specific heat, p is the density, T is the 

 temperature and u is the rate of flow. Qo is the 

 heat lost in the direction of maximum temperature 

 gradient, in this case to the boundary due to con- 



duction, and is given by ^2= — ^ ^ where k is the 



molecular thermal conductivity coefficient. Since 

 the thermal conductivity of fluids is small, there 

 is an abrupt fall in temperature as the boundary 

 is approached. The thickness of the thermal 

 boundary layer in laminar flow is a function of the 

 velocity distribution near the boundary and the 

 thermal conductivity of the fluid. A low thermal 

 conductivity and high viscosity will form a thick 

 heat barrier giving a low heat transfer compared 

 to higher conductivity and low viscosity for the 

 same fluid velocities. Q3 is the same as Q2 in tne 

 h direction. No thermal gradient is assumed in 

 the direction of flow due to flow velocity («), 

 hence no molecular transfer. In nature, flow is 

 seldom laminar. In the case of fluid movement 

 around icebergs, configuration and surface rough- 

 ness would rule out laminar flow entirely. The 

 above relationship must be modified to include 

 turbulence. 



In turbulent flow the boundary layer concept is 

 retained, however, great reduction in its thickness 

 occurs. Turbulence provides still another mech- 

 anism for heat transfer; that of apparent heat 

 conduction. The temperature distribution away 

 from the boundary, in the turbulent case, is 

 uniform due to mixing. As the boundary is 

 approached tliere is an abrupt fall in temperature. 

 This provides for a heat flow in the direction 

 of the maximum gradient. This is given by 



Q=—c„A„-r- where A, is the eddy exchange 

 t '^ ^ on 



coefficient. 'I'his type of heat flow disappears as 



the laminar subboundary layer is readied wliere 



molecular transfer must complete the flow. The 



45 



