A cost index is used to scale a known equipment cost from one year to another primarily by correcting for

general inflation in labour and materials over time

the equipment's depreciation schedule

The six-tenths-rule, Cost2 = Cost1*(Capacity2/Capacity1)^0.6, reflects the empirical observation that

equipment cost rises slower than its capacity, since surface area (driving cost) grows slower than volume (driving capacity)

equipment cost rises faster than its capacity as size increases

equipment cost rises slower than its capacity, since surface area (driving cost) grows slower than volume (driving capacity)

cost is entirely independent of capacity

labour cost dominates all equipment cost

Compared to straight-line depreciation, a declining-balance method charges

more depreciation in early years, reducing taxable income earliest

zero depreciation until the asset's final year

depreciation only after the salvage value is reached

Which of the following profitability measures account for the time value of money?

Simple (uncorrected) payback period

Simple accounting rate of return

Internal rate of return (IRR) is defined as the discount rate at which

net present value (NPV) is exactly zero

the payback period is exactly one year

net present value (NPV) is exactly zero

total annualized cost is minimised

depreciation charge equals operating cost

The optimum heat exchanger approach temperature is found by

balancing rising capital cost (smaller approach, more area) against falling operating cost (more heat recovered)

minimising operating cost alone

balancing rising capital cost (smaller approach, more area) against falling operating cost (more heat recovered)

fixing the exchanger area at its maximum manufacturable size

Economic optimum reflux ratio in a distillation column is typically found at

roughly 1.1 to 1.5 times the minimum reflux ratio

exactly the minimum reflux ratio

roughly 1.1 to 1.5 times the minimum reflux ratio

total reflux, for maximum separation

a value entirely unrelated to the minimum reflux ratio

In a single-product batch campaign, the cycle time between successive batches is set by

the slowest processing step (the bottleneck)

the average of all processing steps' times

the slowest processing step (the bottleneck)

the number of products manufactured

A Gantt chart in batch plant scheduling is used chiefly to

visually lay out which equipment is busy with which product's step at every point in time, spotting idle time and clashes

compute the discounted cash flow of the plant

visually lay out which equipment is busy with which product's step at every point in time, spotting idle time and clashes

estimate equipment cost from a cost index

compute the effectiveness factor of a catalyst

Cost Estimation, Profitability, Optimization and Batch Scheduling

Section 9 of the GATE Chemical Engineering (CH) paper — the section that decides whether a technically sound process is also worth building. Start with cost estimation itself: cost indices that move an old cost to today, and depreciation, which spreads a capital cost over the years it earns against. Then the profitability measures used to compare competing projects — rate of return, payback period, discounted cash flow and total annualized cost — each answering a slightly different question about the same numbers. Then optimization in sizing process equipment, where a genuine engineering trade-off (not just cost minimisation) sets the answer. And finally batch plant scheduling, where the same equipment reused across products, not equipment sizing itself, is the actual constraint.

1. Cost estimation, cost indices and depreciation

A cost index (Chemical Engineering Plant Cost Index, Marshall & Swift, and others) lets an old, known equipment or plant cost be scaled to today's cost by a simple ratio, Cost-now = Cost-then * (Index-now / Index-then) — it corrects for general inflation in labour and materials, but says nothing about a specific technology becoming cheaper or more expensive relative to everything else, which a cost index cannot capture. Scaling cost with CAPACITY, not time, uses the six-tenths-rule instead: Cost2 = Cost1 * (Capacity2/Capacity1)^0.6, an empirical observation that most process equipment's cost rises slower than its capacity because surface area (which drives material cost) grows slower than volume (which drives capacity) as size increases.

🧠 Straight-line versus declining-balance depreciation, in one line each

Straight-line depreciation charges the same amount, (initial cost - salvage value)/life, every year — simple, but does not reflect that most equipment actually loses value fastest early on. Declining-balance methods (double-declining-balance, MACRS-style) instead charge a fixed PERCENTAGE of the remaining book value each year, so the depreciation charge is largest in year one and shrinks every year after — this front-loading is often deliberately chosen because it reduces taxable income (and hence tax paid) earliest, when the time value of money makes that saving worth the most.

2. Rate of return, payback period, discounted cash flow and total annualized cost

Four ways to size up the same project
Measure	Accounts for time value of money?	Answers
Simple payback period	No	How many years until cumulative cash flow recovers the investment
Simple (accounting) rate of return	No	Average annual profit as a fraction of investment
Net present value (NPV), via discounted cash flow	Yes	Whether the project creates value once every future cash flow is discounted to today
Internal rate of return (IRR)	Yes	The discount rate at which NPV is exactly zero

Discounted cash flow is the umbrella technique behind both NPV and IRR: every future cash flow is divided by (1+i)^n (i the discount rate, n the year) before being summed, because a rupee received n years from now is worth strictly less than a rupee today — simple payback and simple rate of return ignore this entirely, which is exactly why they can rank two projects differently than NPV or IRR would, and why DCF measures are preferred whenever cash flows are spread unevenly over a long project life. Total annualized cost converts a mix of one-time capital cost and ongoing operating cost into a single equivalent annual number (using an annuity/capital-recovery factor for the capital portion), which is the standard way to compare two designs with different capital-versus-operating cost trade-offs on one number rather than two.

3. Optimization in process design and equipment sizing

Optimum equipment sizing is almost always found by balancing two costs that move in opposite directions as one design variable changes, not by minimising a single cost — the classic example is a heat exchanger's optimum approach temperature, where a SMALLER approach (closer temperatures) needs MORE heat transfer area (higher capital cost) but recovers more heat and burns less utility (lower operating cost), and vice versa; the total-cost curve (capital plus operating, both annualized to the same basis) is minimised somewhere in between, not at either extreme.

🎯 Multistage-contactor optimum trades stages against reflux or solvent

The same logic sizes a multistage contactor — a distillation column or an extraction cascade: more theoretical stages (more capital cost, a taller column) let the process run closer to minimum reflux or minimum solvent-to-feed ratio (lower operating cost), while fewer stages force operation further from that minimum (less capital, more operating cost). Economic optimum reflux for distillation is typically found around 1.1 to 1.5 times the minimum reflux ratio, a compromise that is itself a direct application of the same capital-versus-operating trade-off principle.

4. Batch plant scheduling

A batch plant reuses the same set of vessels across a sequence of processing steps and often across several different products, so its actual constraint is almost never equipment SIZE but equipment AVAILABILITY over time — the same reactor cannot run two batches at once, so scheduling exists to sequence batches through shared equipment without a downstream unit sitting idle waiting for an upstream one, or an upstream unit blocked because a downstream one is still occupied.

In single-product campaigns, the cycle time (time between successive batches leaving the plant) is set by the SLOWEST processing step (the bottleneck), because every other step must wait for it before starting its own next batch — this is exactly the batch-plant analogue of a bottleneck workstation in any production line.
A Gantt chart is the standard tool for laying out a multi-product batch schedule visually — which equipment is busy with which product's which step, at every point in time — and is what a scheduler actually uses to spot idle time and clashes before committing a production sequence.
Overlapping schedules (starting a new batch's early steps in one train while the previous batch is still finishing its later steps in another) shrink the effective cycle time below what a strictly sequential schedule would give, provided the equipment used at each stage does not clash — the whole reason batch scheduling is a genuine combinatorial problem rather than simple arithmetic.

Key takeaways

A cost index scales cost over TIME (general inflation only); the six-tenths-rule scales cost over CAPACITY, because surface area (cost) grows slower than volume (capacity) as equipment scales up.
Declining-balance depreciation front-loads the depreciation charge, reducing taxable income earliest, when the time value of money makes that saving worth the most; straight-line spreads it evenly instead.
Simple payback and simple rate of return ignore the time value of money; NPV and IRR, both built on discounted cash flow, account for it and can rank two projects differently as a result.
Optimum equipment sizing balances two opposing costs (usually capital versus operating) rather than minimising either alone — a heat exchanger's optimum approach temperature and distillation's economic reflux ratio are both examples of the same trade-off.
A batch plant's real constraint is usually equipment availability over time, not equipment size; its cycle time in a single-product campaign is set by the slowest (bottleneck) step.

Practice questions (10)

Attempt each one before opening the answer. Every explanation names the tempting wrong option as well as the right one, because that is where marks are lost.

A cost index is used to scale a known equipment cost from one year to another primarily by correcting for
1. changes in equipment capacity
2. general inflation in labour and materials over time
3. changes in the discount rate
4. the equipment's depreciation schedule
Show answer
Answer: B — general inflation in labour and materials over time
A cost index (CEPCI, Marshall & Swift and similar) tracks general cost inflation in labour, materials and engineering over time, letting an old cost be rescaled by a simple ratio of index values — it says nothing about capacity, discount rate or depreciation, which are handled by entirely different tools (the six-tenths-rule, DCF, and a depreciation schedule respectively).
The six-tenths-rule, Cost2 = Cost1*(Capacity2/Capacity1)^0.6, reflects the empirical observation that
1. equipment cost rises faster than its capacity as size increases
2. equipment cost rises slower than its capacity, since surface area (driving cost) grows slower than volume (driving capacity)
3. cost is entirely independent of capacity
4. labour cost dominates all equipment cost
Show answer
Answer: B — equipment cost rises slower than its capacity, since surface area (driving cost) grows slower than volume (driving capacity)
The exponent 0.6, less than 1, means cost grows slower than capacity — this reflects that material cost tracks surface area while capacity tracks volume, and surface area grows as the square of a linear dimension while volume grows as the cube, so larger equipment is cheaper per unit of capacity.
Compared to straight-line depreciation, a declining-balance method charges
1. the same amount every year
2. more depreciation in early years, reducing taxable income earliest
3. zero depreciation until the asset's final year
4. depreciation only after the salvage value is reached
Show answer
Answer: B — more depreciation in early years, reducing taxable income earliest
A declining-balance method charges a fixed percentage of the remaining book value each year, which is largest when book value is highest — in the earliest years — front-loading the depreciation charge and reducing taxable income (and tax paid) soonest, which is more valuable given the time value of money.
Which of the following profitability measures account for the time value of money?
1. Net present value (NPV)
2. Internal rate of return (IRR)
3. Simple (uncorrected) payback period
4. Simple accounting rate of return
Show answer
Answer: A — Net present value (NPV); B — Internal rate of return (IRR)
NPV and IRR are both built on discounted cash flow, dividing each future cash flow by (1+i)^n before comparing them — simple payback and simple rate of return use raw, undiscounted cash flows and so ignore the time value of money entirely, which is exactly why they can rank projects differently than NPV or IRR would.
Internal rate of return (IRR) is defined as the discount rate at which
1. the payback period is exactly one year
2. net present value (NPV) is exactly zero
3. total annualized cost is minimised
4. depreciation charge equals operating cost
Show answer
Answer: B — net present value (NPV) is exactly zero
IRR is, by definition, the discount rate that makes the sum of all discounted cash flows (the NPV) exactly zero — a project's IRR exceeding the company's minimum acceptable rate of return (hurdle rate) is the standard accept/reject criterion built on this same discounted-cash-flow machinery.
The optimum heat exchanger approach temperature is found by
1. minimising capital cost alone
2. minimising operating cost alone
3. balancing rising capital cost (smaller approach, more area) against falling operating cost (more heat recovered)
4. fixing the exchanger area at its maximum manufacturable size
Show answer
Answer: C — balancing rising capital cost (smaller approach, more area) against falling operating cost (more heat recovered)
A smaller approach temperature needs more heat transfer area (higher capital cost) but recovers more heat and so burns less utility (lower operating cost); the two annualized cost curves are added and the total is minimised at an intermediate approach temperature, not at either cost's own extreme.
Economic optimum reflux ratio in a distillation column is typically found at
1. exactly the minimum reflux ratio
2. roughly 1.1 to 1.5 times the minimum reflux ratio
3. total reflux, for maximum separation
4. a value entirely unrelated to the minimum reflux ratio
Show answer
Answer: B — roughly 1.1 to 1.5 times the minimum reflux ratio
Operating exactly at minimum reflux needs infinite stages (infinite capital); operating at total reflux needs the fewest stages but produces no net product. The economic optimum, balancing capital cost (stages) against operating cost (reboiler/condenser duty), typically falls around 1.1 to 1.5 times the minimum reflux ratio.
In a single-product batch campaign, the cycle time between successive batches is set by
1. the fastest processing step
2. the average of all processing steps' times
3. the slowest processing step (the bottleneck)
4. the number of products manufactured
Show answer
Answer: C — the slowest processing step (the bottleneck)
Every step downstream of the bottleneck must wait for it before it can start its own next batch, so the whole plant's throughput — and hence its cycle time — is paced by the slowest step, exactly the way a bottleneck workstation paces any production line.
A Gantt chart in batch plant scheduling is used chiefly to
1. compute the discounted cash flow of the plant
2. visually lay out which equipment is busy with which product's step at every point in time, spotting idle time and clashes
3. estimate equipment cost from a cost index
4. compute the effectiveness factor of a catalyst
Show answer
Answer: B — visually lay out which equipment is busy with which product's step at every point in time, spotting idle time and clashes
A Gantt chart plots time on one axis and equipment (or product) on the other, showing exactly which unit is occupied by which product's step at every moment — the standard tool a scheduler uses to find idle equipment time and resolve scheduling clashes before committing a production sequence.
An equipment item costs Rs 10 lakh at a capacity of 100 units/hour. Using the six-tenths-rule (Cost2 = Cost1*(Capacity2/Capacity1)^0.6), what is its approximate cost, in lakh rupees, at a capacity of 800 units/hour, given (800/100)^0.6 is approximately 3.48? (Answer to the nearest whole number.)
Numerical answer — type the value.
Show answer
Answer: 35
Cost2 = 10 lakh * (800/100)^0.6 = 10 lakh * 3.48 = 34.8 lakh, approximately 35 lakh rupees. Capacity rose 8-fold but cost rose only about 3.5-fold, exactly the sub-linear scaling the six-tenths exponent is meant to capture.