272 



not the case it hardly seems possible to apply the mathematical 

 analysis to the problem. 



For the three cases the extrapolated exponent is resp. 1.4, 1.5 a 

 1.6 and 1.65. In the last case, in which we are most certain that 

 the air above the liqnid is in continuous movement, n proves to 

 a^ree quite sufficiently with the theoretical value ^l^-=:^i.&l, which 

 will be deduced hereafter, so that therefore in this case we may be 

 sure, tiiat the air-currents effect the evaporation. In experiments 

 in more quiet air, the values of n approach the value n -=2 J more 

 closely, which value is found by Stefan. 



In the following sections we will give a theoretical treatment ot 

 the diffusion in a flowing gas. As the evaporation from an arbi- 

 trarily formed surface is easily deduced from that of a rectangular 

 one, we firstly choose this last shape, We imagine the space above 

 the plane - = filled with a flowing gas, while the |)lane r = 

 itself is formed by a fixed wall, of which a [)art consists of a surface 

 of the liquid. Let this part have the shape of a rectangle with its 

 sides parallel to the axes of ,v and //, situated at positive y and 

 bounded by the axis of .r. Further we will choose the velocity of 

 the gas to be parallel to the axis of ;// and to be proportional to z, 

 so Vy = az. As namely the gas at the plane c = through external 

 friction must have a velocity equal to zero, we may put: 



Vy =z az -\- a^ z'^ -^ a^ z^ ^ . . . , 



and we may neglect the second and following terms of this series 

 when as will generally be the case, the va{)Our is concentrated in 

 a thin layer above the plane z = 0. 



When we put c for the concentration of the vapour and ]) for 

 the coefficient of diffusion then, as is easily seen, c fulfills the altered 

 equation of diffusion : 



- — X> A c — div {vc) ') (/) 



ot 



Further we suppose that c at the surface of the liquid fulfills the 

 boundary condition : 



1) The last term in the second member may be explained in this way : In the 

 element of volume dx dy dz flows through the element of surface (iy cZs an amount 



Id) 

 of vapour : cvx rfy rfs inward and { cvx + ^— (CTx) } dii dz outward. By computmg 



f ^'^' ' 



these amounts also for the axes of y and z, we get for the total amount that 

 flows outward div {cv) dx dy dz, when r is the velocity, considered as a vector. 



