Scale, capacity and feedstock: which chemical plants get built: Economies of scale: why big plants make cheap tonnes
Understanding Chemicals / Module 7: Scale, capacity and feedstock: which chemical plants get built Part 1 of 5Economies of scale: why big plants make cheap tonnes
A chemical plant twice the size does not cost twice as much to build or staff, but it needs twice the raw material. That asymmetry explains why commodity chemicals are made in very large plants and why small plants usually lose. This part shows where the rule comes from, puts numbers on it with a sulphuric acid plant, and shows which cost lines it touches.
1.1 Capacity grows with volume while cost grows with surface, so bigger units are cheaper per tonne
Most process equipment is a container: a tank, a column, a reactor shell. What it holds rises with its volume, which goes with the cube of its length. What it costs is mostly the steel that wraps that volume, which goes with the square of its length. So cost rises with capacity to the power of about two-thirds. Wittcoff calls this the square-cube law. It fails for some items: a heat exchanger's duty depends on its area, so its cost rises almost in step with capacity, while a control room hardly changes at all. Across a modern petrochemical plant, cost tends to rise with capacity to the power 0.6 (pdf p. 58).
Brennan writes this as I = k x Q^b, where I is fixed capital, Q is capacity and b is the scale exponent. The value of b depends on the flowsheet. Where each step is one large item (a single-stream or 'single-train' plant), b is about 0.5 to 0.6. Where capacity comes from many identical items in parallel, b approaches 1; nominally parallel plants sit at 0.8 to 0.9 because some costs are shared. Mixed plants can be weighted: an ethylene plant is roughly 30% cracking furnaces (b about 0.9) and 70% single-stream purification (b about 0.6), giving 0.3 x 0.9 + 0.7 x 0.6 = 0.69 (pdf pp. 66-67).
Labour behaves the same way: doubling a unit may raise its labour cost only 10 to 20% because of automation (Wittcoff, pdf p. 59). What scale does not change is the raw material and, usually, the energy needed per tonne, which are set by chemistry and yield. Scale shrinks only the capital-related and staffing costs per tonne (Brennan, pdf p. 188).
| Plant or equipment | Exponent b | Basis |
|---|---|---|
| Single-stream plant | 0.5 to 0.6 | Accumulated empirical experience |
| Parallel-stream plant | 0.8 to 0.9 | Shared costs stop b reaching 1 |
| Ethylene plant (30% furnaces, 70% purification) | 0.69 | Weighted estimate |
| Chlor-alkali plant (50% cells, 50% brine and purification) | 0.75 | Weighted estimate |
| Modern petrochemical plant overall | about 0.6 | Wittcoff |
| Equipment items, 662 cases | average 0.6 | 71% between 0.4 and 0.8 |
| Plants growing by expansion (cumulative capital) | average 0.64 | Australian plant studies |
- Wittcoff (2004), Ch 1 The Chemical Industry · pdf pp. 58-59
- Brennan (2020), Ch 3 Capital cost estimation; Ch 7 Process technology evolution and adoption · pdf pp. 66-67, 75, 188
1.2 A worked example: one sulphuric acid design at three sizes
Brennan's Example 6.8 is a sulphur-burning Sulphuric acid plant selling to neighbouring fertiliser and chemical plants. The base design makes 150,000 t/year of 98% acid. Fixed capital is $38.2 million for the process units plus $9.5 million for storage, utilities and other offsites: $47.7 million in all (pdf p. 174).
The variable cost is small. Each tonne of acid uses 0.32 t of sulphur at $80/t ($25.60) and 70 m3 of cooling water at 12 cents ($8.40). Burning sulphur releases so much heat that the plant exports 0.2 MWh of power at $100 ($20 credit) and 0.6 t of steam at $12 ($7.20 credit). Net variable cost is 25.6 + 8.4 - 20 - 7.2 = $6.80/t. Fixed cash costs (labour and overheads $2.0 million, maintenance, insurance and tax $2.8 million, administration $0.4 million) total $5.2 million a year, or $34.70/t. Cash cost is about $41.50/t against a selling price of $100/t (pdf pp. 174-176).
Brennan rescales the plant with b = 0.6, since sulphur-burning plants are single stream. A 100,000 t/year plant costs 47.7 x (100/150)^0.6 = 47.7 x 0.78 = $37.4 million; a 600,000 t/year plant costs 47.7 x 4^0.6 = 47.7 x 2.30 = $109.6 million. Labour barely moves and maintenance scales with capital (pdf pp. 176-177).
Quadrupling capacity from 150,000 to 600,000 t/year cuts cash cost from $41.50 to $22.10/t, and cash cost plus 15-year depreciation from $62.70 to $34.30/t, with no change in the variable line. Yet Brennan finds the 150,000 t/year plant only marginally profitable: allowing for its sales ramp-up, NPV at 10% is just $2.9 million over 15 years (pdf p. 176). The engineering argues for the big plant. Whether 600,000 t/year can be sold nearby is a market question.
| Item | 100 kt/yr | 150 kt/yr (base) | 600 kt/yr |
|---|---|---|---|
| Fixed capital ($ million) | 37.4 | 47.7 | 109.6 |
| Capital per annual tonne ($) | 374 | 318 | 183 |
| Variable cost net of steam and power credits ($/t) | 6.8 | 6.8 | 6.8 |
| Fixed cash costs ($ million/yr) | 4.6 | 5.2 | 9.2 |
| Fixed cash cost ($/t) | 46.0 | 34.7 | 15.3 |
| Total cash cost ($/t) | 52.8 | 41.5 | 22.1 |
| Depreciation over 15 years ($/t) | 24.9 | 21.2 | 12.2 |
| Cash cost plus depreciation ($/t) | 77.7 | 62.7 | 34.3 |
Selling price $100/t. Recomputed from Brennan's cost components; his printed totals for the base and 600 kt cases contain arithmetic slips.
Source: Brennan (2020), Example 6.8 Sulphuric acid manufacture, pdf pp. 174-177- Brennan (2020), Ch 6 Some examples in process economics, Example 6.8 · pdf pp. 174-177
- Kent (2012), Ch 22 Synthetic Nitrogen Products · pdf p. 907
1.3 Scale matters most where fixed costs dominate the cost sheet
The acid plant shows why the prize from scale depends on the cost structure. Sulphur burns in dried air to sulphur dioxide. The gas is cooled in waste-heat boilers to 420-440 C and passed over vanadium pentoxide catalyst, where it combines with more oxygen to form sulphur trioxide, which is absorbed in acid held at 98.5-99% (Kent, pdf pp. 1013-1016). Every step releases heat, hence the credits for steam and power. With variable cost netted down to $6.80/t, nearly everything left is fixed cost, which is exactly what scale shrinks.
Brennan's scale curves make the general point. He plots production cost relative to the largest available plant against size for ethylene, chlor-alkali and aluminium smelting. The aluminium curve is the flattest, even though smelters are more parallel-streamed, because electricity, a variable cost, is such a large share of the total (pdf pp. 125-126). The penalty for being small depends on how far below the largest plant you are, how the flowsheet is built, and the split between fixed and variable cost (pdf p. 188).
Penalties also add up along a chain. PVC pipe needs ethylene and chlorine plants, a vinyl chloride monomer plant, a PVC plant and a pipe plant. If each is small, each passes a higher cost to the next (pdf p. 189).
Ammonia is the opposite case. Natural gas is 70% to over 85% of its cash cost (Kent, pdf p. 907), so the gas price at the site matters more than the size of the plant. Part 4 returns to this.
S + O2 → SO2; combustion heat raises high-pressure steam
2 Gas coolingSecond waste-heat boiler cools gas to 420-440 C for the converter
3 Converter, V2O5 catalyst beds2SO2 + O2 → 2SO3; 150-200 L catalyst per t/day of capacity in a double-absorption plant
4 Intermediate and final absorbersSO3 absorbed in 98.5-99% acid; absorbing between beds raises overall conversion
5 Acid cooling and dilutionHeat removed; product adjusted to 98% H2SO4
Outputs Sulphuric acid 98%1 t Exported power0.2 MWh per t Exported steam0.6 t per tKent puts the heat released in a 1,000 t/day plant at about 63 MW, which is why these plants are net energy exporters.
Source: Brennan (2020) Example 6.8; Kent (2012) Ch 25, Brennan pdf p. 174; Kent pdf pp. 1013, 1016-1017- Brennan (2020), Ch 4 Operating cost estimation; Ch 6 Example 6.1; Ch 7 · pdf pp. 125-126, 160, 188-189
- Kent (2012), Ch 25 Sulfur and Sulfuric Acid; Ch 22 · pdf pp. 907, 1013-1017
See it in the table
H₂O WaterGas Natural gasAl AluminiumSt SteelV VanadiumH2SO4 Sulphuric acidCl2 ChlorineO2 OxygenNH3 AmmoniaC2H4 EthyleneVCM Vinyl chloride monomerPVC PVCCheck your understanding
Using the six-tenths rule, roughly how much more does a plant four times larger cost to build?
4^0.6 = 2.30. Brennan's 600,000 t/year acid plant costs 47.7 x 2.30 = $109.6 million against $47.7 million for 150,000 t/year.Which cost per tonne stays the same when the sulphuric acid plant is quadrupled in size?
Raw material use and energy per tonne are set by chemistry and yield, not size. The net variable cost stays at $6.80/t at every capacity; only fixed and capital-related costs per tonne fall.Why is the scale curve for aluminium smelting flatter than for ethylene or chlor-alkali?
Brennan notes that the high share of variable cost from electricity lowers the aluminium scale curve, even though smelters are more parallel-streamed.