k x 



s 



.0053 



c l 



E 



-20° C. 



OC T 



S 



.0118 



k 2 



B 



.00143 



<*2 



s 



-1°C. 



oc 2 



= 



.00143 



L 



s 



80.0 



>°i = 0.92. 



Using the value of b found from Figure 1, the equation relating the 

 thickness of ice to the time, for an ice surface temperature of -20°C, 

 and water temperature of -1°C., is 



£ = 0.053l,^/t , (20) 



where f = ice thickness in cm. and t = time in seconds. An evaluation 

 of (20) for 14 hours gives an ice thickness of 12 cm. as compared to 

 8 cm. from the practical forecast curves of Figure 12. The differences 

 may be due to the different values of constants used, i.e. V = 20, N - 0. 



Stefan In a similar fashion derived an expression for the ice thick- 

 ness as a function of time. He made a further simplification by assuming' 

 the water temperature to be 0°C. This is merely a special eass of 

 Neumann's solution and can be obtained from it by making C 2 * in equation 

 (19). To a first approximation Stefan's equation is 



£ 2 = - 2C ' c ' oc ' ! ' (21) 



f = ice thickness, 

 Cl ■ temperature of the ice surface, 



L - latent heat of fusion, 

 c^ ■ specific heat of ice, 

 oc i = thermal diffusivity of ice, and 



t = tiTie. 



(o= 1 is available from tables but is equal to the thermal conductivity 

 divided by the product of specific heat times density—all available 

 from tables). 



Equations (19) and (21) are not suited for practical application, 

 since the effects of initial thickness of ice, thickness and density of 

 snow cover, wind velocity, cloud cover, humidity of the air, and salinity 

 of sea water are neglected, and in addition such a long and tedious com- 

 putation is not suited for practical use. To take the meteorological 

 conditions into account, it is necessary to form heat budget equations. 



