OCR Text |
Show - 4 - The above considerations make it possible to write dl - = ^ P Jj_ ( 5) and < PP ^ ? L£- d- a" r +_^ L-/ Q£ f ( 6) Hz? dt cL2- x dT^ \ d. 7. J Employing equation ( 3) for vapor pressure, equations ( 5) and ( 6) become dp ( 7) d. 7. - . T - r oLT oLT.. '& + P ^ . X, Vett and , x^ ( 8) now where x = L/ RT. The rate of accumulation of solid, r, becomes r, &*-££+•£. (**-**) C£ f ( 9) The first term on the right hand side ( r. h. s.) is proportional to the second derivative of temperature, and is thus equivalent to an expression obtained by Yosida ( 5). The second term on the r. h. s, is proportional to the temperature gradient squared. This term is especially important in the presence of shallow s in cold climates. The necessity for this additional term can be established by considering the hypothetical case of a uniform temperature gradient, dT/ dz = const, A plot of p ( ordinate) against T or z ( abcissa) is, like other vapor pressure curves, concave up, and the term ^ P / 3 7 -^ Y~ ' s finite and positive. Thus while the first term on the r. h. s, is zero, there is a finite accumulation rate r which must be assigned to the second term on the r. h. s. The latter quantity will always contribute to some extent whether the temperature gradient is constant or not. |