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23-Chem-A5 Chemical Plant Design and Economics · May 2014

Question 3 of 6: Depreciation

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

Notes on this paper

National Exams — May 2014 — 04-Chem-A5 Chemical Plant Design and Economics. Three-hour, closed-book exam; any non-communicating 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. Questions 1, 5 and 6 are conceptual design / management / safety questions answered as organised prose; questions 2, 3(i) and 4 contain the numerical work (cost–capacity scaling of a heat exchanger, sinking-fund depreciation, and simple/compound loan interest), and every boxed figure.

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 (cost estimation Ch. 6, interest and investment Ch. 7, depreciation Ch. 9, profitability Ch. 10, optimum design Ch. 11, plant safety and loss prevention); R. Turton et al., Analysis, Synthesis, and Design of Chemical Processes (4th ed., Prentice Hall) — flowsheet synthesis and process development; T.M. Duncan & J.A. Reimer, Chemical Engineering Design and Analysis (Cambridge, 1998) — the source of the boiling-point data used in Question 1; supporting Canadian tax practice from the Canada Revenue Agency Capital Cost Allowance classes and the half-year rule.

Question 3: Depreciation (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.

(i) Asset value at the end of year 10

Given. Original value $V_0 = \$30{,}000$; service life $n = 15$ yr; salvage (scrap) value $V_s = \$4000$; depreciation-fund interest rate $i = 6\% = 0.06$; evaluate at age $a = 10$ yr.

Find. The book (asset) value $V_{10}$ at the end of the 10th year.

Approach. The phrase "interest rate for the depreciation fund" signals the sinking-fund method: a uniform annual deposit $R$ earns interest at $i$ so that the accumulated fund exactly equals the depreciable value $(V_0-V_s)$ after $n$ years; the asset value at any age is $V_0$ less the fund accumulated to that age.

  1. Size the uniform sinking-fund deposit. The deposit that grows to the total depreciation $(V_0-V_s)$ over $n$ years is $$R = (V_0 - V_s)\,\frac{i}{(1+i)^{n}-1} = (\$26{,}000)\frac{0.06}{(1.06)^{15}-1} = \frac{\$1560}{1.39656} = \boxed{\$1117\text{/yr}}$$
  2. Accumulate the fund to the end of year 10. The depreciation charged (fund value) after $a=10$ years is the future worth of the deposits: $$D_{10} = R\,\frac{(1+i)^{a}-1}{i} = \$1117\,\frac{(1.06)^{10}-1}{0.06} = \$1117(13.181) = \$14{,}723$$
  3. Book value = original value − accumulated depreciation. Substituting, $$V_{10} = V_0 - D_{10} = \$30{,}000 - \$14{,}723 = \boxed{\$15{,}277}$$ Equivalently, in one line, $V_{10}=V_0-(V_0-V_s)\dfrac{(1.06)^{10}-1}{(1.06)^{15}-1}=\$15{,}277$.
QuantityValue
Annual sinking-fund deposit $R$$\$1117$/yr
Accumulated depreciation to yr 10$\$14{,}723$
Asset (book) value at end of yr 10$\$15{,}277$
Check: assumptions stated as the question requests — (1) the specified fund interest rate makes this a sinking-fund calculation, not straight-line; (2) deposits are made at year-end (ordinary annuity); (3) salvage is realised only at end of life, so it does not enter the year-10 book value except through $R$. Under plain straight-line the year-10 value would instead be $\$30{,}000-10(\$26{,}000/15)=\$12{,}667$; the sinking-fund value is higher because it back-loads the write-off.

(ii) Two methods of computing depreciation — pros and cons

Straight-line (SL) method. The depreciable value is written off in equal annual amounts, $d = (V_0-V_s)/n$, so the book value falls linearly. Advantages: simple, transparent, easy to audit, and the constant charge smooths reported earnings. Disadvantages: it ignores the time value of money, and it poorly matches reality for assets that lose most of their worth early or that become obsolete before their nominal life ends.

Declining-balance (DB) method. A fixed fraction $f$ of the remaining book value is charged each year, $d_k = f\,V_{k-1}$, giving large early charges that taper off — an accelerated method. Advantages: better matches the true early loss of value of most process equipment, and, by front-loading the deduction, defers income tax and improves early cash flow — the basis of the Canadian Capital Cost Allowance (CCA) system, where each asset class has a prescribed rate and the "half-year rule" allows only half the normal rate in the year of acquisition. Disadvantages: more complex, never reaches zero on its own (a salvage floor or a switch to straight-line must be imposed), and the heavier early write-off depresses early book profit.

(Other recognised methods include sum-of-the-years-digits, another accelerated scheme, and the sinking-fund method used in part (i), which is the slowest write-off because the fund earns interest.)