116 THE ROYAL SOCIETY OF CANADA 



rate of growth of a frozen layer which at any time t has attained a 

 depth X may be calculated as follows. We denote by K the heat 

 conductivity of the frozen medium in C.G.S. units: Ey denotes the 

 emissivity of the frozen medium with respect to air in calories per 

 sq.cm. per second. We also denote by pf the density of the frozen 

 medium and by L the latent heat absorbed or released in the "freezing 

 up" or "thawing out" of unit mass of the porous medium (including 

 the water contained in the pores). If we write hf=KflEf the loss of 

 heat from an element of volume of thickness dx and unit area at a 

 distance x from the surface is: 



(x + hj) 

 This heat is derived from the "freezing up" of the element of volume 

 which contributes an amount pf.L.dx 



We thus obtain the equation for x 



{x^hj) — = -^- 

 d/ Lpf 



of which the solution which makes x = when / = is 



x(hf + ix)= Y^ / odt = -^ et (1) 



where d is the average temperature during the interval to /. 



Similarly, if the "thawing out" takes place when the medium or 

 part of it is exposed to water at a temperature +^' the depth x' 

 thawed out in time t' is given by 



x' (/^; +lx) = i^ e't' (2) 



where Kj„ and p,j, refer to the water-soaked medium and h^ refers 

 to the surface conductivity of the interface with respect to water. 



These equations with appropriate constants apply as they stand 

 to the penetration of frost into wet concrete exposed to changes of air 

 temperature. Equation (1) holds for the rate of growth of ice over 

 water and has been found to give a good representation of existing 

 observations with the following constants (distinguished by the 

 sufifixi) expressed in C.G.S. units: L = 80 calories, pi = 0-9, K,= -0057, 

 E,= -00024 calories per sq. cm. per second /îj = Kj/E, = 24 cm. The 

 value given for h, was obtained from observations by the writer on 

 the rate of growth of ice under open air conditions with practically 

 no wind. 



In the application of equations (1) and (2) to a porous medium 

 it must be kept in mind that surface conductivities vary very greatly 



