﻿170 
  JOURNAL 
  OP 
  THS 
  WASHINGTON 
  ACADEMY 
  OF 
  SCIKNCES 
  VOL. 
  12, 
  NO. 
  7 
  

  

  medium 
  is 
  denoted 
  by 
  5 
  and 
  its 
  rate 
  of 
  flow 
  in 
  mass 
  units 
  per 
  unit 
  of 
  

   time 
  by 
  M, 
  then 
  the 
  temperature 
  elevation 
  will 
  be 
  given 
  by 
  an 
  equa- 
  

   tion 
  of 
  the 
  type 
  

  

  A 
  = 
  F{\,H,D,M,S) 
  (9) 
  

  

  in 
  which 
  D 
  denotes 
  some 
  linear 
  dimension 
  of 
  the 
  sample. 
  (This 
  

   equation 
  is 
  only 
  approximately 
  complete; 
  while 
  serving 
  well 
  enough 
  

   as 
  it 
  stands 
  for 
  the 
  purpose 
  of 
  illustration, 
  it 
  can 
  in 
  practice 
  be 
  made 
  

   more 
  exact 
  by 
  introducing 
  the 
  additional 
  variables 
  p, 
  fx 
  and 
  X' 
  to 
  

   denote, 
  respectively, 
  the 
  density, 
  viscosity, 
  and 
  thermal 
  conductivity 
  

   of 
  the 
  cooling 
  agent, 
  which 
  will 
  have 
  some 
  influence 
  on 
  the 
  rate 
  of 
  

   heat 
  transfer, 
  though 
  not 
  so 
  much 
  as 
  the 
  quantities 
  M 
  and 
  5.) 
  Equa- 
  

   tion 
  9 
  can 
  be 
  further 
  developed 
  by 
  dimensional 
  theory, 
  and 
  then 
  solved 
  

   for 
  the 
  conductivity 
  X, 
  whereupon 
  it 
  goes 
  over 
  into 
  the 
  form 
  

  

  In 
  the 
  second 
  part 
  of 
  this 
  equation 
  x 
  has 
  been 
  written 
  for 
  H/D 
  A 
  and 
  

   y 
  for 
  MS 
  A/H. 
  Plot 
  observed 
  values 
  of 
  y/yo 
  as 
  ordinate 
  against 
  x/xo 
  

   as 
  abscissa, 
  and 
  denote 
  by 
  Xi/xo 
  the 
  abscissa 
  of 
  the 
  point 
  where 
  simi- 
  

   larity 
  occurs 
  ; 
  that 
  is, 
  the 
  point 
  where 
  the 
  empirical 
  curve 
  crosses 
  

   the 
  line 
  y/yo 
  = 
  l. 
  Referring 
  therefore 
  to 
  Equation 
  10, 
  the 
  relative 
  

   conductivity 
  will 
  evidently 
  be 
  given 
  by 
  the 
  formula 
  

  

  X 
  .^1 
  , 
  , 
  

  

  - 
  =- 
  (11) 
  

  

  Ao 
  Xq 
  

  

  In 
  the 
  more 
  exact 
  solution 
  suggested 
  above, 
  the 
  consideration 
  of 
  

   p 
  will 
  introduce 
  an 
  additional 
  argument 
  p^-HD^M^ 
  into 
  Equation 
  

   10, 
  while 
  the 
  recognition 
  of 
  ^ 
  will 
  add 
  an 
  argument 
  of 
  the 
  form 
  ixD/M, 
  

   and 
  so 
  on 
  if 
  additional 
  correction 
  terms 
  are 
  included. 
  In 
  order 
  to 
  

   apply 
  the 
  routine 
  reasoning 
  above, 
  which 
  was 
  based 
  on 
  Equation 
  

   10, 
  these 
  new 
  arguments 
  must 
  now 
  be 
  held 
  constant, 
  which 
  may 
  or 
  

   may 
  not 
  be 
  experimentally 
  practicable, 
  although 
  possible 
  in 
  principle 
  

   if 
  suitable 
  facihties 
  are 
  provided. 
  For 
  example, 
  to 
  keep 
  the 
  argument 
  

   p'^HD'^/M^ 
  constant, 
  it 
  is 
  sufficient 
  to 
  increase 
  the 
  mass 
  flow 
  in 
  pro- 
  

   portion 
  to 
  the 
  cube 
  root 
  of 
  the 
  heat 
  input, 
  whenever 
  the 
  latter 
  is 
  

   changed. 
  

  

  General 
  formulation. 
  — 
  The 
  procedure 
  illustrated 
  above 
  may 
  be 
  

   outlined 
  in 
  more 
  general 
  terms 
  as 
  follows: 
  

  

  1. 
  Develop 
  the 
  appropriate 
  dimensionless 
  equation 
  for 
  some 
  chosen 
  

  

  