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23-Chem-A5 Chemical Plant Design and Economics · May 2017

Question 2 of 6: Discounted-Cash-Flow Rate of Return on Two Projects

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

Notes on this paper

Closed-book exam, 3 hours; one aid sheet permitted. Six questions of equal value (20 marks each); five constitute a complete paper — full solutions to all six are given here. Questions 1 and 2 are quantitative (plant material balance and discounted-cash-flow return); Questions 3–6 are design-practice list/essay questions.

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 (process design development Ch. 2, general design considerations: plant location, safety, materials Ch. 3, interest and profitability Ch. 7–10, materials-transfer/pumps Ch. 14); R.H. Perry & D.W. Green, Perry's Chemical Engineers' Handbook (9th ed.) — pump types and selection (Sec. 10), pyrolysis kinetics data; O. Levenspiel, Chemical Reaction Engineering (3rd ed.) — first-order plug-flow space-time behind the reactor sizing in Question 1; supporting Canadian practice from CCOHS and the CSA Z767 / provincial OH&S process-safety-management framework for Question 5.

Question 2: Discounted-Cash-Flow Rate of Return on Two Projects (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. Two mutually exclusive projects, each −$10 M at year 0, five annual cash inflows as tabulated (both total $20 M undiscounted: A back-loaded, B front-loaded), five-year life.

Find. The discounted-cash-flow rate of return (the interest rate that makes net present value zero — the IRR) of each project, and hence which to choose when capital is restricted.

Approach

The DCF rate of return is the discount rate $i$ at which $\text{NPV}(i)=\sum_{t=0}^{5} \dfrac{C_t}{(1+i)^t}=0$. Evaluate NPV at trial rates that bracket zero for each project and interpolate linearly; the higher IRR is preferred because capital is limited.

012345-10 (capital)+1.6+2.8+4.0+5.2+6.4Project A cash flow ($10^6) - back-loadedperiod (year)
Figure 2.1 — Project A cash-flow diagram: −$10 M at year 0 followed by rising ("back-loaded") inflows 1.6→6.4 $106. Deferred returns give the lower IRR (≈ 22 %).
012345-10 (capital)+6.5+5.2+4.0+2.8+1.6Project B cash flow ($10^6) - front-loadedperiod (year)
Figure 2.2 — Project B cash-flow diagram: same −$10 M outlay but falling ("front-loaded") inflows 6.5→1.6 $106. Early recovery of capital gives the higher IRR (≈ 38 %).
  1. Set up the IRR condition. With cash flows $C_t$ ($106), find $i$ such that $$\text{NPV}(i) = -10 + \sum_{t=1}^{5}\frac{C_t}{(1+i)^t} = 0$$ Both projects return $20 M undiscounted on a $10 M outlay, so both are profitable; the timing of the recovery sets the rate.
  2. Bracket Project A. Discounting A's inflows: $$\text{NPV}_A(20\%) = +0.672,\qquad \text{NPV}_A(25\%) = -0.653 \;\;(\$10^6)$$ Linear interpolation between the sign change: $$i_A \approx 20 + 5\times\frac{0.672}{0.672+0.653} = 22.5\%$$ A straight-line interpolation across a 5-point bracket overshoots slightly, because the NPV curve is convex; refining the root until NPV = 0 gives 22.41 %. $\boxed{\text{Project A DCF rate of return} \approx 22.4\%}$
  3. Bracket Project B. The front-loaded inflows discount much more slowly to zero: $$\text{NPV}_B(35\%) = +0.494,\qquad \text{NPV}_B(40\%) = -0.220 \;\;(\$10^6)$$ $$i_B \approx 35 + 5\times\frac{0.494}{0.494+0.220} = 38.5\%$$ Refining that root in the same way gives 38.39 %. $\boxed{\text{Project B DCF rate of return} \approx 38.4\%}$
  4. Decide. Because the two are mutually exclusive and capital is restricted, the project with the higher rate of return on the same $10 M is preferred: $$i_B = 38.4\% \;>\; i_A = 22.4\%$$ $\boxed{\text{Choose Project B}}$ Both recover the same total money, but B returns capital earlier, so each dollar earns a higher effective rate — the decisive factor when funds are scarce and could be redeployed.
QuantityProject AProject B
Total undiscounted return$20.0 M$20.1 M
Cash-flow timingback-loadedfront-loaded
DCF rate of return (IRR)22.4 %38.4 % (preferred)
Check: the roots were confirmed by bisection to NPV = 0 (22.41 % and 38.39 %). If instead the projects had unequal capital, IRR alone can mislead (it ignores project scale) and an incremental-IRR or NPV-at-the-cost-of-capital comparison would be required; here the capital is identical, so ranking by IRR is valid.