Chemical-plant equipment costs rarely vary in proportion to size. In the simplest case, cost C varies with size S according to the allometric equation

Question

(a) For 0 < ? < 1, show that cost per unit size decreases with increasing size. (“Economy of scale.”)

(b) Consider the case of a spherical storage tank. The size is commonly measured by internal volume Show why

one might expect that ? = 2/3. On what parameters or properties would you expect quantity ? to depend?

Answer 1

a. The cost per unit size is If ? is less than 1, then the exponent (? – 1) is negative, and the cost per size decreases with increasing size (S).

b. To a good first approximation, the cost of the tank should be proportional to the amount (mass) of material required to construct the tank. If the wall thickness, t, is independent of the tank volume, and is small compared to the tank diameter, then the mass of material required to make the tank will be where d is the diameter of the spherical tank and ? is the density of the material from which it is constructed.

The tank volume is

Or, writing the diameter in terms of the volume,

Substituting this into the expression for the mass of material gives

This shows that the exponent should be approximately ? = 2/3. It also shows that the cost should depend on the thickness of the tank walls, which will, in turn, depend on the maximum pressure that the tank must be able to withstand. Other factors that will influence the quantity ? should include the material of construction and the fabrication techniques used. The material of construction will depend on the need for corrosion resistance and other such factors.