COST/CAPACITY RELATIONSHIPS from first principles: Part 1

{{C2/C1 = (Q2/Q1)E = QE (1)where:C1 is the (known) capital cost of plant 1;C2 is the (unknown) capital cost ofplant 2 ;Q1 is the (known) throughput of plant 1;Q2 is the (known) throughput of plant 2;Q is the ratio Q2/Q1; andE is the exponent.}}

As we have often discussed on this page before, E is usually in the 'two-thirds' range of 0.6 - 0.75.

However, a problem arises when E is not known. In other words, a single cost may be evaluated for a prospective novel process. But further similar estimates are needed to assist in the economic evaluation of the project; the assessment of several capital and corresponding operating costs against predicted product selling prices and volumes are essential to the analysis which leads to optimum plant sizing.

The capital cost aspect of this question is the subject of this and next month's Costimator.

With only one point on the graph of capital cost vs capacity, how do we extrapolate with some degree of certainty the cost of, say, a plant of 50 per cent greater capacity.

Taking the factored budget estimating approach, equation (1) suggests:

{{C2 = QESgmcm1 = Sgmcm1Qnm (2)m m}}

In equation (2) cm1 is the cost of item m in project 1 and gm is a factor which accounts for the installation, engineering and related costs of item m. The quantity gm is assumed to be the same for plants 1 and 2 and nm is the individual exponent for each item m.

Cancelling the quantity gm in equation (2), rearranging and taking logarithms, we obtain equation (3):

{{E = {log(Scm1Qnm) - log(Scm1)}/logQm m}}

Although equation (3) does imply that a power law (the Eth!) is indeed applicable to integrated sets of main plant items (MPIs), it is rather inconvenient during the early stages of development. That is just at the time when a simpler global relationship of this kind is most needed. First, equation (3) demands a knowledge of the separate costs of all the base case items. Second, it is a function of Q, which is a further encumbrance.

A solution to the problem of calculating E was brought to mind in my review last month of Dr Brennan's book Process industry economics. In there it is mentioned (page 51) that 'Allen and Page proposed a relationship for relating the exponent b (our E) to the exponents bi (effectively our nm) for base case individual equipment items.' The equation quoted is:

{{b = Scibi/Sci (4)J J}}

where ci is the purchased cost of MPIs of base capacity.

The relevant reference is:

Allen DH and Page RC, Revised technique for predesign cost estimating, Chemical Engineering 82, 142-145 (1975).

While I would not recommend the short-cut estimation method developed by Dr Page, the formula in equation (4) is certainly valid and useful.

Another version of equation (4), which is sometimes even more useful, is to define a weighting factor, wm, which replaces cm1 such that:

wm = cm/cz (5)

Thus, dividing both sides of equation (4) by cz, and using our terminology, equation (4) may be rewritten:

{{E = Scm1nm/Scm1 (6)m m

which is the same as:

E = Swmnm/Swm (7)m m}}

In equation (7), E is independent of cm1 and nm is known for most major plant items. It does, however, depend upon a knowledge of w for each item. wm, assumed constant for item m, is the ratio of the cost of that item to the cost of a standard item (eg, a specific pump) sized for the same representative throughout.