1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Chemical-plant equipment costs vary to size

  1. Jan 10, 2016 #1
    1. The problem statement, all variables and given/known data
    Chemical-plant equipment costs rarely vary in proportion to (i.e., linearly with) size. In the

    simplest case, cost varies with size according to the allometric equation, C = aSb

    The exponent is typically between 0 and 1. For a wide variety of equipment types, it is approximately 0.6.

    1. (a) For 0 < b < 1, show that cost per unit size decreases with increasing size, resulting in an economy of scale.

    2. (b) Consider the case of a spherical storage tank. The size is commonly measured by internal volume Vt. Show that b = 2/3. On what parameters or properties would you expect the quantity a to depend?

    2. Relevant equations
    formulas for derivative

    3. The attempt at a solution
    For part a, I calculated the derivative of C/S with respect to the size, which equals to (b-1)*a*Sb-2. As the value of b-1 is negative, the value of this derivative is always negative, and so the increase in size would result in a decrease in the cost per unit size.
    I don't understand what part b is asking about though. Compared with part a, it doesn't give anything apart from telling me that the tank is spherical. I have completely no clue about how to obtain a specific value for b without any additional information.
    Any help would be much appreciated!
     
  2. jcsd
  3. Jan 10, 2016 #2
    @Ortanul, Hi

    What is the volume of a sphere of radius r?
    Then, what other equation of a sphere could one apply that might indicate its cost with regard to its cost of material. How would you find that?
     
  4. Jan 10, 2016 #3
    @265bits
    Thanks for your reply!
    The equation for calculating the volume is 4/3*pi*r^3
    While for the other equation... Could you please give me a hint?
     
  5. Jan 10, 2016 #4
    Thanks for your reply!
    The equation for calculating the volume is 4/3*pi*r^3
    While for the other equation... Could you please give me a hint?
     
  6. Jan 10, 2016 #5
    What is the volume of the shell. What do you get if you multiply this by the density of the shell metal?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Chemical-plant equipment costs vary to size
  1. Power plants (Replies: 0)

  2. Cogeneration plant (Replies: 6)

Loading...