﻿Long 
  Air-Bubbles 
  in 
  a 
  Vertical 
  Tube, 
  961 
  

  

  bubble 
  at 
  its 
  maximum 
  section, 
  i. 
  e. 
  at 
  its 
  lower 
  end 
  where 
  

   its 
  velocity 
  is 
  V, 
  on 
  writing 
  V=— 
  * 
  — 
  r-^, 
  and 
  simplifying, 
  

  

  this 
  gives 
  very 
  approximately, 
  when 
  ^ 
  is 
  small 
  (say 
  of 
  the 
  

   order 
  of 
  0*1), 
  

  

  dp= 
  — 
  -r 
  ^.dhcm 
  (10) 
  

  

  r 
  «r(A 
  — 
  a) 
  2 
  

  

  With 
  a 
  cylindrical 
  bubble 
  of 
  this 
  area 
  throughout, 
  the 
  loss 
  

   of 
  head 
  in 
  overcoming 
  viscous 
  flow 
  over 
  its 
  whole 
  length 
  

   would 
  be 
  given 
  by 
  

  

  1 
  H 
  2 
  =^-' 
  7 
  i^.Lcm 
  (11) 
  

  

  -as- 
  (A 
  — 
  a)- 
  

  

  Actually, 
  however, 
  the 
  loss 
  from 
  the 
  apex 
  to 
  the 
  point 
  at 
  

   which 
  the 
  bubble 
  becomes 
  sensibly 
  cylindrical, 
  and 
  at 
  which 
  

   its 
  area 
  is 
  a% 
  must 
  be 
  negligibly 
  small, 
  while 
  if 
  L' 
  be 
  the 
  

   length 
  from 
  this 
  point 
  to 
  the 
  base, 
  the 
  true 
  loss 
  is 
  given 
  by 
  

  

  _ 
  v«' 
  (' 
  L 
  ' 
  dv 
  

  

  iH,= 
  — 
  .18^ 
  - 
  (X 
  ^y 
  2 
  . 
  

  

  Writing, 
  as 
  appears 
  to 
  be 
  justified 
  by 
  observation, 
  a 
  = 
  a' 
  — 
  kl, 
  

   this 
  becomes, 
  on 
  integrating 
  : 
  

  

  -v„f**EB 
  

  

  {H 
  2 
  = 
  Ya 
  

  

  L' 
  

  

  ^ 
  '(A-a'XA-a 
  b 
  )' 
  ' 
  

  

  (12) 
  

  

  In 
  order 
  to 
  simplify 
  the 
  resultant 
  calculations 
  it 
  becomes 
  

   desirable 
  to 
  express 
  xH 
  2 
  in 
  terms 
  of 
  A 
  — 
  a 
  and 
  of 
  L, 
  and 
  an 
  

   examination 
  of 
  the 
  enlarged 
  photographs 
  of 
  a 
  number 
  of 
  

  

  bubbles 
  shows 
  that 
  the 
  ratio 
  - 
  — 
  - 
  is 
  very 
  approximately 
  

  

  equal, 
  in 
  every 
  case, 
  to 
  

  

  1L 
  

  

  A 
  — 
  a 
  b 
  

  

  5(A-a'j 
  

   Making 
  this 
  substitution, 
  the 
  viscous 
  loss 
  is 
  given 
  by 
  

  

  Va' 
  Ufju 
  T 
  

  

  iH 
  s 
  

  

  • 
  (A 
  -a') 
  2 
  v 
  

   = 
  KVL 
  

  

  where 
  

  

  (A-aJ 
  2 
  

  

  cm. 
  

  

  (13) 
  

  

  The 
  error 
  involved 
  in 
  this 
  assumption, 
  in 
  bubbles 
  of 
  length 
  

   exceeding 
  two 
  diameters, 
  is 
  very 
  small, 
  and 
  since 
  the 
  viscous 
  

  

  