Question

The `six-tenths rule' is used to estimate

• A. Equipment cost from size
• B. Plant cost from capacity
• C. Operating labor from production rate
• D. Maintenance cost from equipment cost

Hint:

- If \(x = 0.6\): Increasing plant capacity by a factor of 2 increases costs by only \(2^{0.6} \approx 1.52\) times (demonstrating clear economies of scale). - If \(x = 1.0\): Costs scale linearly, meaning there are no financial advantages to scaling up.

Answer 1

The Correct Option is B

Concept: In preliminary industrial plant design, engineers use empirical scaling relationships to estimate the capital cost of a new plant based on historical cost data from an existing operational plant of a different capacity. This approach leverages economies of scale.

Step 1: Explaining the cost-capacity power-law relationship.
The capital cost of a plant does not scale linearly with its production capacity. Instead, it follows a non-linear power-law relationship known as the Williams rule or the six-tenths rule: \[ \frac{C_2}{C_1} = \left(\frac{V_2}{V_1}\right)^x \] Where:

• \(C_2\) is the estimated cost of the target plant with capacity \(V_2\).

• \(C_1\) is the known historical cost of a reference plant with capacity \(V_1\).

• \(x\) is the empirical cost-capacity scaling exponent.

Step 2: Justifying the exponent value.
For a broad variety of chemical processing plants, the average value of the exponent \(x\) is approximately 0.6. This non-linear behavior arises because equipment capacity scales with three-dimensional volume (\(\propto r^3\)), whereas equipment manufacturing costs scale with the material surface area (\(\propto r^2\)). This relationship (\(r^2 \propto (r^3)^{2/3}\)) provides a theoretical basis for the empirical exponent value. Thus, the rule is used to estimate plant cost from capacity modifications, matching option (2).