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23-Chem-A5 Chemical Plant Design and Economics · December 2018

Question 2 of 6: Minimum Selling Price by DCFROR

Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)

Notes on this paper

National Exams — December 2018 — 16-Chem-A5 Chemical Plant Design and Economics. Three-hour, closed-book exam; one two-sided aid sheet and an approved Sharp/Casio calculator permitted. Six equally weighted (20-mark) questions are posed and the candidate answers any five; only the first five are marked. All six are answered below for completeness. Question 1 is a process-synthesis flowsheet (catalytic propane dehydrogenation), Question 2 is a numerical discounted-cash-flow (DCFROR) minimum-selling-price calculation, and Questions 3–6 are qualitative design/economics essays (pilot-plant investigation, technical design factors, economic design factors, and a solvent-emission abatement scheme).

Reference texts: M.S. Peters, K.D. Timmerhaus & R.E. West, Plant Design and Economics for Chemical Engineers (5th ed., McGraw-Hill) — the exam's named primary text (pilot plants Ch. 3, general design considerations and plant-location factors Ch. 2–3, interest/depreciation/profitability Ch. 7–10); R.K. Sinnott & G. Towler, Chemical Engineering Design (Coulson & Richardson vol. 6) — economic analysis, cash-flow/DCFROR and flowsheeting; R. Smith, Chemical Process Design and Integration (2nd ed., Wiley) — reaction–separation–recycle structure; supporting Canadian practice from CCOHS, provincial OH&S regulation and Environment and Climate Change Canada air-emission guidance.

Question 2: Minimum Selling Price by DCFROR (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given.

SymbolQuantityValue
$P$Production rate50,000 tons/yr
$I$Investment (year 0)$10\times10^{6}$
$n$Project life5 yr
$FC$Fixed operating cost$7.5\times10^{5}$/yr
$VC$Variable cost (excl. raw materials)$40/ton
$RM_0$Stoichiometric raw-material cost$80/ton
$Y$Yield (product per raw material)80%
$t$Tax rate35%
$i$Minimum acceptable DCFROR15%/yr

Find. The selling price $S$ ($/ton) that makes the net present value of the project exactly zero at a 15% discount rate — i.e. the price at which the project just earns the minimum acceptable DCFROR.

Approach

Set the net present value to zero at 15%: the uniform after-tax cash flow must repay the $\$10\text{ M}$ investment over 5 years, so the required annual cash flow is $I \div (P/A)_{15\%,5}$. Express the after-tax cash flow as a function of the selling price—remembering the raw-material cost must be grossed up by the yield and that straight-line depreciation is a non-cash tax shield—then solve for $S$.

012345-$10M (TCI)+$2.98M/yr+$2.98M/yrQ2 minimum-price cash flow (M$, after-tax, i=15%)period (year)
Figure 2.1 — After-tax cash-flow diagram for the minimum-price case: a $10 M capital outlay at year 0 followed by the uniform $2.98 M/yr after-tax cash flow that just recovers the investment at a 15% DCFROR over 5 years.
  1. Gross up the raw-material cost by the yield. Because only 80% of the raw material ends up as product, each ton of product actually consumes $1/Y$ tons of raw material: $$RM = \frac{RM_0}{Y} = \frac{80}{0.80} = \$100/\text{ton}$$ so the total variable cost is $VC + RM = 40 + 100 = \$140/\text{ton}$.
  2. Annual cash operating cost (excluding depreciation). $$C = FC + P\,(VC + RM) = 750{,}000 + 50{,}000(140) = \$7{,}750{,}000/\text{yr}$$
  3. Straight-line depreciation. The $10 M investment written off over 5 years with no salvage gives a non-cash charge that shields tax: $$D = \frac{I}{n} = \frac{10{,}000{,}000}{5} = \$2{,}000{,}000/\text{yr}$$
  4. Present-worth annuity factor and required cash flow. At $i=15\%$ over $n=5$ yr, $$\left(\frac{P}{A}\right)_{15\%,5} = \frac{1-(1.15)^{-5}}{0.15} = 3.35216$$ For NPV = 0 the uniform after-tax cash flow must recover the investment: $$CF_{req} = \frac{I}{(P/A)} = \frac{10{,}000{,}000}{3.35216} = \$2{,}983{,}156/\text{yr}$$
  5. After-tax cash flow as a function of price. With revenue $R = P\,S$, taxable income $=R-C-D$, and depreciation added back: $$CF = (R - C - D)(1-t) + D$$ Set $CF = CF_{req}$ and solve for $R$: $$R - C - D = \frac{CF_{req} - D}{1-t} = \frac{2{,}983{,}156 - 2{,}000{,}000}{0.65} = \$1{,}512{,}548$$ $$R = 1{,}512{,}548 + C + D = 1{,}512{,}548 + 7{,}750{,}000 + 2{,}000{,}000 = \$11{,}262{,}548/\text{yr}$$
  6. Selling price. $$S = \frac{R}{P} = \frac{11{,}262{,}548}{50{,}000} = \$225.25/\text{ton}$$ ==**Minimum selling price ≈ $225/ton ($225.25/ton) to earn a 15% DCFROR.**==
QuantityValue
Effective raw-material cost$100/ton
Total variable cost$140/ton
Annual cash operating cost $C$$7,750,000/yr
Straight-line depreciation $D$$2,000,000/yr
$(P/A)_{15\%,5}$3.35216
Required annual cash flow$2,983,156/yr
Minimum selling price $S$$225.25/ton (≈$225)
Check: substituting $S=\$225.25$/ton back gives an after-tax cash flow of $\$2.983\text{ M/yr}$, whose present worth at 15% over 5 yr is $2.983\text{M}\times3.35216 = \$10.0$ M, exactly recovering the investment (NPV = 0). Two traps sink the naive answer: (i) forgetting to gross up the raw material by the 80% yield ($\$80$→$\$100\text{/ton}$) understates cost, and (ii) omitting the depreciation tax shield—treating the whole thing on a pre-tax basis or ignoring the $D$ add-back—over-prices the product to about $\$247\text{/ton}$.