﻿Some 
  Problems 
  of 
  Evaporation. 
  271 
  

  

  When 
  the 
  air 
  is 
  in 
  motion, 
  k 
  at 
  a 
  distance 
  from 
  a 
  boundary 
  

   is 
  almost 
  wholly 
  due 
  to 
  turbulence, 
  and 
  is 
  practically 
  

   independent 
  of 
  position. 
  The 
  velocity 
  near 
  a 
  boundary 
  

   rapidly 
  increases 
  from 
  zero 
  to 
  about 
  half 
  its 
  amount 
  at 
  a 
  

   considerable 
  distance 
  ; 
  this 
  transition 
  is 
  accomplished 
  in 
  a 
  

   thin 
  layer 
  of 
  shearing, 
  whose 
  thickness 
  in 
  centimetres 
  is 
  

   estimated 
  at 
  40/U, 
  where 
  U 
  is 
  the 
  velocity 
  in 
  centimetres 
  

   per 
  second 
  at 
  a 
  considerable 
  distance 
  *. 
  In 
  outdoor 
  problems 
  

   it 
  is 
  therefore 
  usually 
  of 
  the 
  order 
  of 
  a 
  millimetre 
  or 
  smaller. 
  

   The 
  concentration 
  and 
  temperature 
  also 
  change 
  rapidly 
  

   within 
  this 
  layer, 
  and 
  in 
  most 
  cases 
  it 
  will 
  be 
  justifiable 
  to 
  

   assume 
  that 
  at 
  the 
  outer 
  boundary 
  of 
  it 
  each 
  is 
  constant 
  and 
  

   midway 
  between 
  the 
  values 
  at 
  the 
  surface 
  and 
  at 
  a 
  great 
  

   distance. 
  Outside 
  this 
  layer 
  transference 
  of 
  heat 
  and 
  vapour 
  

   will 
  take 
  place 
  according 
  to 
  equation 
  (3), 
  where 
  k 
  is 
  now 
  

   put 
  equal 
  to 
  the 
  eddy 
  viscosity. 
  

  

  If 
  the 
  dimensions 
  of 
  the 
  liquid 
  surface 
  are 
  of 
  order 
  I, 
  and 
  

   the 
  time 
  needed 
  for 
  any 
  considerable 
  change 
  of 
  condition 
  

   over 
  it 
  is 
  of 
  order 
  t, 
  we 
  see 
  that 
  the 
  terms 
  like 
  dV/d£, 
  

   u^V/'dx, 
  and 
  #B 
  2 
  V/B«# 
  2 
  are 
  relatively 
  of 
  orders 
  

  

  1/t, 
  U/Z, 
  and 
  k/l 
  2 
  . 
  

  

  Taking 
  k 
  to 
  be 
  of 
  the 
  order 
  of 
  10 
  3 
  cm. 
  2 
  /sec, 
  which 
  is 
  a 
  

   somewhat 
  small 
  estimate 
  for 
  outdoor 
  problems, 
  t 
  = 
  1 
  second, 
  

   and 
  U 
  = 
  400 
  cm./sec, 
  we 
  see 
  that 
  the 
  first 
  term 
  is 
  small 
  

   compared 
  with 
  the 
  last 
  provided 
  I 
  is 
  less 
  than 
  10 
  cm., 
  and 
  for 
  

   slower 
  variations 
  this 
  term 
  will 
  be 
  small 
  for 
  still 
  larger 
  values 
  

   of 
  I. 
  Again, 
  we 
  see 
  that 
  the 
  second 
  term 
  is 
  small 
  compared 
  

   with 
  the 
  last, 
  provided 
  I 
  is 
  less 
  than 
  2 
  cm. 
  ; 
  otherwise 
  it 
  must 
  

   be 
  taken 
  into 
  account. 
  In 
  indoor 
  problems 
  U 
  is 
  probably 
  not 
  

   far 
  from 
  zero, 
  & 
  = 
  0*24 
  cm. 
  2 
  /sec, 
  and 
  the 
  whole 
  of 
  d/dt 
  can 
  

   therefore 
  be 
  neglected 
  for 
  a 
  surface 
  1 
  cm. 
  across 
  provided 
  

   the 
  saturation 
  near 
  it 
  does 
  not 
  change 
  considerably 
  in 
  

   8 
  seconds. 
  

  

  When 
  I 
  is 
  sufficiently 
  small 
  to 
  satisfy 
  both 
  these 
  conditions 
  

   the 
  equation 
  of 
  transference 
  reduces 
  to 
  V 
  2 
  V" 
  = 
  0, 
  subject 
  to 
  

   the 
  same 
  boundary 
  conditions 
  as 
  before. 
  Without 
  loss 
  of 
  

   generality 
  we 
  can 
  take 
  the 
  concentration 
  at 
  a 
  great 
  distance 
  

   to 
  be 
  zero. 
  For 
  if 
  it 
  is 
  actually 
  Yd, 
  Y— 
  Y 
  d 
  will 
  satisfy 
  the 
  

   same 
  differential 
  equation 
  and 
  will 
  be 
  constant 
  over 
  all 
  the 
  

   wetted 
  surfaces, 
  and 
  hence 
  if 
  a 
  problem 
  is 
  solved 
  for 
  Vd=0 
  

   the 
  solution 
  for 
  any 
  other 
  value 
  of 
  Y 
  d 
  can 
  at 
  once 
  be 
  found 
  

   by 
  merely 
  writing 
  Y 
  — 
  Yd 
  for 
  V 
  throughout. 
  

  

  Now, 
  V 
  2 
  V 
  = 
  is 
  the 
  equation 
  of 
  steady 
  diffusion 
  and 
  

  

  * 
  Private 
  communication 
  from 
  Major 
  G. 
  I. 
  Taylor. 
  

   U2 
  

  

  