Question

In cost estimation, the six-tenths rule relates cost to equipment capacity by:

• A. Cost proportional to \(\text{Capacity}^{0.6}\)
• B. Cost proportional to \(\text{Capacity}^{1.0}\)
• C. Cost proportional to \(\text{Capacity}^{0.4}\)
• D. Cost proportional to \(\text{Capacity}^{1.6}\)

Hint:

The exponent \(0.6\) represents economies of scale.
If the exponent were \(1.0\), the cost would scale linearly with size (no economies of scale).
If the exponent is less than \(1.0\), scaling up is economically advantageous because the cost per unit capacity decreases as size increases.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The question asks for the mathematical relationship between equipment cost and equipment capacity as defined by the "six-tenths rule" used in chemical plant cost estimation.

Step 2: Key Formula or Approach:
The six-tenths rule (also known as Williams' rule) is an empirical scaling relationship used by cost engineers to estimate the purchased cost of a piece of equipment when the cost of a similar equipment of a different capacity is known.
The mathematical formula is:
\[ \frac{C_2}{C_1} = \left(\frac{q_2}{q_1}\right)^{a} \] where:
\(C_1\) is the cost of equipment with capacity \(q_1\).
\(C_2\) is the cost of equipment with capacity \(q_2\).
\(a\) is the scaling exponent, which is empirically taken as \(0.6\) for many standard chemical process units.

Step 3: Detailed Explanation:
Let us analyze the relationship when the scaling exponent \(a = 0.6\):
From the scaling equation, we can rewrite the cost of the new equipment as:
\[ C_2 = C_1 \cdot \left(\frac{1}{q_1}\right)^{0.6} \cdot q_2^{0.6} \] Since \(C_1\) and \(q_1\) are known constant references for the existing equipment, the term \(C_1 \cdot (1/q_1)^{0.6}\) is a constant, which we can call \(k\):
\[ C_2 = k \cdot q_2^{0.6} \] This shows that the cost of the equipment is directly proportional to its capacity raised to the power of \(0.6\):
\[ \text{Cost} \propto \text{Capacity}^{0.6} \] This exponent of \(0.6\) reflects the economies of scale in equipment manufacturing.
For instance, the cost of a vessel or tank is related to its surface area (which scales with volume to the \(2/3\) or \(0.67\) power), while its capacity is related to its volume.
Thus, doubling the volume capacity does not double the material and fabrication cost, resulting in a scaling exponent less than 1.0.

Step 4: Final Answer
The six-tenths rule states that cost is proportional to \(\text{Capacity}^{0.6}\), which corresponds to option (A).