Production Cost per Kilogram of Total Product: 23-Chem-A5 Chemical Plant Design and Economics, Undated paper, Q3

Question 3: Production Cost per Kilogram of Total Product (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. The cost-data block, printed at the top of page 4 as the continuation of this question. The reaction-product slate (EDA 74%, PIP 4%, HEP 4%, DETA 8%, AEP 10%, summing to 100%) identifies the chemistry and confirms that the 10,000 t/yr is total product, which is the basis every cost below is quoted on; no individual product yield is needed for this question.

Item	Value
Fixed capital cost, FCI	$10.3 million
Interest / return on capital	10%/yr
Electricity	$3.56\times10^{6}\,\text{J/kg}$ @ $4.17\times10^{-7}\,\text{cents/J}$
Steam (1 atm)	18.1 kg/kg @ 0.0113 cents/kg
Cooling water	$0.1528\,\text{m}^3/\text{kg}$ @ $3.17\,\text{cents/m}^3$
Operating labour	$8.61/hr
Land + land development	1.5% + 2.1% of depreciable capital
Raw materials	80 cents/kg
Production rate / hours	10,000 t/yr; 8000 h/yr (333 days)

Find. The production cost per kilogram of total product.

Check — stated assumptions

The recovered data fix the raw-material, utility and capital charges but omit the number of operators, maintenance, plant overhead, insurance and general expenses. To close the problem we assume (i) four operators are on the payroll continuously; (ii) straight-line depreciation over a 10-year life on the FCI; (iii) the 10% charge is a return on total capital investment (FCI + land + land development). The result is therefore a direct-plus-capital estimate; a full Peters–Timmerhaus total product cost (adding maintenance, supplies, overhead, insurance and general expenses) would be higher. The utility and labour rates are used exactly as printed and are mutually consistent for a paper of this vintage: $4.17\times10^{-7}$ cents/J is $1.50$ cents/kWh, alongside steam at $0.0113$ cents/kg, water at $3.17\ \text{cents/m}^3$ and labour at $8.61/hr. both were checked against the reassembled page and are wrong.)

Approach. Build the cost per kilogram additively — raw materials + utilities + operating labour + depreciation + capital charge — converting each annual figure to a per-kg basis with the 10,000 t/yr (= $10^{7}\,\text{kg/yr}$) output.

Basis. Annual output $=10{,}000\,\text{t/yr}\times1000=10^{7}\,\text{kg/yr}$; operating time 8000 h/yr (the paper’s “333 days” of uninterrupted operation).
Raw materials. Given directly: $c_{\text{RM}}=80\ \text{cents/kg}=\boxed{\$0.800/\text{kg}}$.
Utilities (per kg of product). Electricity $=3.56\times10^{6}\times4.17\times10^{-7}=1.485\ \text{cents}$; steam $=18.1\times0.0113=0.205\ \text{cents}$; water $=0.1528\times3.17=0.484\ \text{cents}$. Total $c_{\text{util}}=2.173\ \text{cents/kg}=\boxed{\$0.02173/\text{kg}}$ — electricity is the largest of the three utilities, but all of them together are small beside the feedstock.
Operating labour. With four operators on the payroll, $C_{\text{lab}}=4\times\$8.61/\text{h}\times8000\,\text{h}=\$275{,}520/\text{yr}$, so per kg $c_{\text{lab}}=275{,}520/10^{7}=\boxed{\$0.02755/\text{kg}}$.
Depreciation (straight line, 10 yr). $D=\dfrac{\$10.3\times10^{6}}{10}=\$1.03\times10^{6}/\text{yr}$, so $c_{D}=1.03\times10^{6}/10^{7}=\boxed{\$0.1030/\text{kg}}$.
Capital charge (10% on TCI). Land $=0.015(10.3\text{M})=\$0.1545\text{M}$ and land development $=0.021(10.3\text{M})=\$0.2163\text{M}$, so $\text{TCI}=10.3+0.1545+0.2163=\$10.671\text{M}$. The annual return $0.10\times10.671\text{M}=\$1.067\text{M/yr}$ gives $c_{i}=1.067\times10^{6}/10^{7}=\boxed{\$0.1067/\text{kg}}$.
Total production cost. Summing, $$c=0.800+0.02173+0.02755+0.1030+0.10671=\$1.0590/\text{kg}.$$ Rounding, the production cost is $\boxed{\approx\$1.06\ \text{per kg of total product}}$. The answer is dominated by the 80 cents/kg of raw material; note also that it is insensitive to the operator count — going from four operators to eight would add only 2.8 cents/kg.

Cost component	$/kg	Share
Raw materials	0.800	75.5%
Utilities (electricity, steam, water)	0.02173	2.1%
Operating labour (4 operators)	0.02755	2.6%
Depreciation (SL, 10 yr)	0.1030	9.7%
Capital charge (10% of TCI)	0.10671	10.1%
Production cost	$1.06	100%

Concept & Theory

Topic. Estimating the cost of production per unit of product from a limited data set (the Peters–Timmerhaus total-product-cost anatomy).

Key relations. Total product cost = manufacturing cost + general expenses; manufacturing cost = direct production costs (raw materials, labour, utilities, maintenance, supplies) + fixed charges (depreciation, taxes, insurance, capital return) + plant overhead. Here only the materials, utilities, labour, depreciation and capital-return terms are supplied.

Why this works. Each line item is naturally an annual cost; dividing by the annual output puts every term on the common $/kg basis so they can be summed. Raw materials dominate (~76%), which is typical of a bulk chemical — a reminder that yield and raw-material selection drive the economics more than any single utility.

Common pitfall. Mixing bases (adding an annual dollar figure to a per-kg figure), or reporting this as the full production cost when maintenance, overhead, insurance and general expenses were never given — state that the estimate is direct-plus-capital only.

Source. Peters, Timmerhaus & West, Plant Design and Economics for Chemical Engineers, Ch. 6 (estimation of total product cost).

Question 3 of 6: Production Cost per Kilogram of Total Product

Question 3: Production Cost per Kilogram of Total Product (20 marks)