﻿OF 
  SINGLE-ACTING 
  EXPANSIVE 
  STEAM-ENGINES. 
  201 
  

  

  Economy 
  of 
  Single- 
  Acting 
  Expansive 
  Engines. 
  

  

  (37.) 
  By 
  increasing 
  the 
  ratio 
  of 
  expansion 
  in 
  a 
  Cornish 
  engine, 
  the 
  quantity 
  of 
  

   steam 
  required 
  to 
  perform 
  a 
  given 
  duty 
  is 
  diminished 
  ; 
  and 
  the 
  cost 
  of 
  fuel, 
  and 
  

   of 
  the 
  boilers, 
  is 
  lowered. 
  But 
  at 
  the 
  same 
  time, 
  as 
  the 
  cylinders 
  and 
  every 
  part 
  

   of 
  the 
  engine 
  must 
  be 
  made 
  larger, 
  to 
  admit 
  of 
  a 
  greater 
  expansion, 
  the 
  cost 
  of 
  

   the 
  engine 
  is 
  increased. 
  It 
  thus 
  becomes 
  a 
  problem 
  of 
  maxima 
  and 
  minima 
  to 
  

   determine 
  what 
  ratio 
  of 
  expansion 
  ought 
  to 
  be 
  adopted 
  under 
  given 
  circum- 
  

   stances, 
  in 
  order 
  that 
  the 
  sum 
  of 
  the 
  annual 
  cost 
  of 
  fuel, 
  and 
  the 
  interest 
  of 
  the 
  

   capital 
  employed 
  in 
  construction, 
  may 
  be 
  the 
  least 
  possible, 
  as 
  compared 
  with 
  the 
  

   work 
  done. 
  

  

  That 
  this 
  problem 
  may 
  admit 
  of 
  a 
  definite 
  solution, 
  the 
  following 
  five 
  quanti- 
  

   ties 
  must 
  be 
  given 
  : 
  — 
  

  

  P 
  l5 
  the 
  initial 
  pressure 
  in 
  the 
  cylinder. 
  

  

  F, 
  the 
  resistance 
  not 
  depending 
  on 
  the 
  useful 
  load. 
  

  

  I 
  n, 
  the 
  amount 
  of 
  the 
  length 
  of 
  the 
  effective 
  strokes 
  made 
  in 
  unity 
  of 
  time. 
  

  

  h, 
  the 
  annual 
  cost 
  of 
  producing 
  unity 
  of 
  weight 
  of 
  steam 
  in 
  unity 
  of 
  time, 
  

   which 
  consists 
  of 
  two 
  parts 
  ; 
  the 
  price 
  of 
  fuel, 
  and 
  the 
  interest 
  of 
  the 
  cost 
  of 
  the 
  

   boilers. 
  

  

  k, 
  the 
  interest 
  of 
  the 
  cost 
  of 
  the 
  engine, 
  per 
  unit 
  of 
  area 
  of 
  piston. 
  

  

  Hence 
  the 
  annual 
  expenditure 
  to 
  be 
  taken 
  into 
  consideration, 
  reduced 
  to 
  unity 
  

   of 
  weight 
  of 
  steam, 
  is 
  

  

  h 
  + 
  k 
  ™ 
  =h 
  + 
  k-j^ 
  

   W 
  In 
  

  

  And 
  the 
  useful 
  effect 
  of 
  unity 
  of 
  weight 
  of 
  steam 
  being 
  

  

  V^Z-Fs) 
  

  

  The 
  problem 
  is 
  to 
  determine 
  the 
  ratio 
  of 
  expansion 
  s, 
  so 
  that 
  

  

  h 
  + 
  k^- 
  

   In 
  

  

  shall 
  be 
  a 
  maximum. 
  

  

  Dividing 
  the 
  numerator 
  of 
  this 
  fraction 
  by 
  V 
  1 
  P 
  x 
  , 
  and 
  the 
  denominator 
  by 
  

  

  k 
  V 
  

  

  -— 
  -, 
  both 
  of 
  which 
  are 
  constants 
  in 
  this 
  problem, 
  we 
  find 
  that 
  it 
  will 
  be 
  solved 
  

  

  by 
  making 
  the 
  ratio 
  

  

  F 
  

  

  . 
  . 
  (62.) 
  

  

  h 
  In 
  

  

  kY 
  t 
  

   a 
  maximum. 
  

  

  The 
  algebraical 
  solution 
  would 
  be 
  extremely 
  complicated 
  and 
  tedious. 
  The 
  

  

  