design excluding the cost of heat losses. And, since we have found the optimum

design (lowest cost) neglecting heat losses, we now know that no design can achieve

a lower cost when heat losses are included. This simple result can be very useful. It

may be possible to find a design, not necessarily known to be optimal, whose cost

including heat losses is acceptably close to that of the optimal design for the lower

bounding problem.

The solution to the complete problem including heat losses is slightly more

complicated, but is easily obtained. To find the extremum of the total variable cost

′

function Ct , we simply take its partial derivative with respect to *d *and set the result

to zero. Before proceeding to do so, however, we must take note of the value of *A*10

in the heat loss term being a function of the pipe diameter. This results from the outer

diameter of the pipe being a function of pipe diameter and insulation thickness. The

appropriate insulation thickness is determined by a separate optimization proce-

dure that would consider the insulation and jacket material costs and the cost of heat

loss. As a result of this separate "sub-optimization," the insulation thickness

becomes a function of the pipe diameter.

For a given set of operating conditions and economic data, the optimal insulation

thickness can be found as a function of the pipe diameter. Here, for the sake of

simplicity, we will assume that the insulation thickness is fixed. We then find an

(2-21)

one of the following form

(2-22)

For a typical set of parameter values

∆*x*i = 0.050 m.

This approximation is within 2% for values of *d *from 0.025 to 1.0 m. Using this

′

approximation for *A*10, we obtain the following equation for Ct

[

]

′

(2-23)

′

If we take the partial derivative of Ct and set the result to zero, we have

{

]2}{1((1γ)/(1+(2∆*x*i)1γ d γ1))}

[

0 = *I*1/*d *ln((4*H*p)γ(*d*γ+ (2∆*x*i)1γ d1))

(5 + *b + c*)*I*3d(6+*b+c*) + *A*9.

(2-24)

This equation can not be solved explicitly for *d*. A solution can be obtained by

using a root-finder technique. An initial estimate needed for the solution can be

found by using the value of *d *obtained from the solution to the lower bounding

problem, which neglects heat losses. The cost associated with this optimal design

that neglects heat losses also provides us with a global lower bound on the actual

cost. In the next section, we consider the solution to a representative case.

11

Integrated Publishing, Inc. |