1.3 Engineering Economics, Life Cycle Cost Analysis & Equipment Financial Evaluation

1.3 Engineering Economics, Life Cycle Cost Analysis & Equipment Financial Evaluation

HVAC design engineers are routinely tasked with evaluating competing mechanical system designs, sizing energy-efficiency retrofits, and justifying higher initial capital expenditures through lifetime operating energy savings. The NCEES PE Mechanical Reference Handbook provides discrete compounding interest tables and economic formulas to perform Life Cycle Cost Analysis (LCCA), Net Present Value (NPV) evaluations, and Equivalent Uniform Annual Cost (EUAC) determinations.

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1. Time Value of Money & Cash Flow Notation

All engineering economic evaluations rest upon the principle that a dollar today is worth more than a dollar in the future due to its earning capacity (interest rate, $i$, or discount rate / Minimum Attractive Rate of Return, $\text{MARR}$). Cash flows are modeled as discrete lump sums or annual series occurring at the end of discrete compounding periods ($n$, in years).

  P (Present Worth at Year 0)
  |
  v
--+-------+-------+-------+-------+-------+--> Time (Years)
  0       1       2       3      ...      n
          ^       ^       ^               ^
          |       |       |               | + S (Salvage Value)
          A       A       A               A (Uniform Annual Series: Energy + Maint)

Primary Economic Variables

• $P$: Present worth or initial capital investment (at time $t = 0$)
• $F$: Future worth or single lump-sum value (at end of period $n$)
• $A$: Equivalent Uniform Annual Worth / Cost occurring at the end of each period for $n$ periods
• $G$: Uniform gradient series (annual cash flow increases/decreases by a constant amount $G$ each period)
• $S$: Salvage or residual value of equipment at the end of its useful life ($t = n$)
• $i$: Effective interest rate or discount rate per period (decimal or percentage)
• $n$: Number of interest periods (equipment service life in years)

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2. Compound Interest Factors & Mathematical Formulations

The standard NCEES compound interest factors allow direct conversion between cash flow formats.

Complete Summary of NCEES Interest Factors

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3. Life Cycle Cost Analysis (LCCA) Framework

Life Cycle Cost Analysis evaluates the total cost of owning, operating, and maintaining an HVAC system over its designated service life. For a given mechanical system alternative, the total Present Worth Life Cycle Cost ($\text{LCC}_P$) is defined as:

$$\text{LCC}_P = C_{\text{cap}} + C_{\text{inst}} + \text{PW}(C_{\text{energy}}) + \text{PW}(C_{\text{maint}}) + \text{PW}(C_{\text{replace}}) - \text{PW}(S)$$

Where:

• $C_{\text{cap}} + C_{\text{inst}}$: Total initial turnkey installed capital cost ($t = 0$)
• $\text{PW}(C_{\text{energy}}) = C_{\text{annual energy}} \times (P/A, i, n)$
• $\text{PW}(C_{\text{maint}}) = C_{\text{annual maint}} \times (P/A, i, n)$ (or includes $(P/G, i, n)$ for escalating maintenance)
• $\text{PW}(C_{\text{replace}}) = C_{\text{overhaul}} \times (P/F, i, n_{\text{replace}})$ (e.g., compressor rebuild at year 10)
• $\text{PW}(S) = S \times (P/F, i, n)$ (salvage value deducted)

Equivalent Uniform Annual Cost (EUAC)

To compare equipment alternatives with different service lives, convert total present worth into Equivalent Uniform Annual Cost:

$$\text{EUAC} = \text{LCC}_P \times (A/P, i, n)$$

The alternative with the lowest EUAC is the most cost-effective engineering choice.

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4. Economic Evaluation Metrics: Payback, NPV & IRR

+-------------------------------------------------------------------------------------+
| COMPARISON OF FINANCIAL DECISION METRICS                                            |
+-------------------------------------------------------------------------------------+
| Metric             | Time Value of Money? | Reinvestment Rate? | Exam Decision Rule  |
| ------------------ | -------------------- | ------------------ | ------------------- |
| Simple Payback     | NO                   | N/A                | Lower is better     |
| Discounted Payback | YES                  | Discount rate (i)  | Lower is better     |
| Net Present Value  | YES                  | MARR               | Select Max NPV > 0  |
| Internal Rate (IRR)| YES                  | Calculated IRR     | Select if IRR > MARR|
+-------------------------------------------------------------------------------------+

1. Simple Payback Period (SPP)

$$\text{SPP} = \frac{\Delta \text{Initial Investment}}{\Delta \text{Annual Operating Cost Savings}}$$

Limitation: SPP completely ignores cash flows beyond the payback cutoff and disregards the time value of money.

2. Discounted Payback Period (DPP)

The smallest integer number of years $k$ such that cumulative discounted annual savings equal or exceed the initial incremental investment:

$$\sum_{t=1}^{k} \frac{\Delta A_t}{(1 + i)^t} \ge \Delta P$$

3. Net Present Value (NPV) & Incremental Analysis

For independent projects, select any project with $\text{NPV} > 0$. For mutually exclusive alternatives (e.g., choosing between Standard Chiller A and High-Efficiency Chiller B):

1. Rank alternatives in order of increasing initial capital cost ($P_A < P_B$).
2. Compute incremental initial cost: $\Delta P = P_B - P_A$.
3. Compute incremental annual savings: $\Delta A = A_A - A_B$.
4. Compute incremental Net Present Value:

$$\Delta \text{NPV} = -\Delta P + \Delta A \cdot (P/A, i, n) + \Delta S \cdot (P/F, i, n)$$

1. If $\Delta \text{NPV} > 0$ (or $\Delta \text{IRR} > \text{MARR}$), the extra investment in Alternative B is economically justified.

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5. Comprehensive Worked Example: Chiller Financial Evaluation

Problem Statement

A hospital facility is evaluating two competing 400-ton water-cooled centrifugal chillers for a central plant replacement. The facility operates 3,500 equivalent full-load hours (EFLH) annually, the cost of electricity is $0.12 / kWh, the discount rate ($\text{MARR}$) is 8.0%, and the project analysis horizon is 15 years.

• Option A (Standard Efficiency Chiller):
  • Installed Cost: $180,000
  • Full-Load Efficiency: $0.62\text{ kW/ton}$
  • Annual Maintenance: $6,000 / year
  • Salvage Value at Year 15: $15,000
• Option B (Premium Magnetic-Bearing VFD Chiller):
  • Installed Cost: $260,000
  • Full-Load Efficiency: $0.48\text{ kW/ton}$
  • Annual Maintenance: $4,500 / year
  • Salvage Value at Year 15: $25,000

Step 1: Calculate Annual Electrical Energy Consumption & Costs

$$\text{Annual Energy (kWh)} = \text{Tonnage} \times \text{Efficiency (kW/ton)} \times \text{EFLH (hrs)}$$

• Option A:

  • $\text{Power Demand} = 400\text{ tons} \times 0.62\text{ kW/ton} = 248\text{ kW}$
  • $\text{Annual kWh} = 248\text{ kW} \times 3,500\text{ hrs} = 868,000\text{ kWh}$
  • $\text{Annual Energy Cost} = 868,000\text{ kWh} \times \text{\textdollar}0.12/\text{kWh} = \text{\textdollar}104,160 / \text{year}$
  • $\text{Total Annual Operating Cost}_A = \text{\textdollar}104,160 + \text{\textdollar}6,000 = \text{\textdollar}110,160 / \text{year}$

• Option B:

  • $\text{Power Demand} = 400\text{ tons} \times 0.48\text{ kW/ton} = 192\text{ kW}$
  • $\text{Annual kWh} = 192\text{ kW} \times 3,500\text{ hrs} = 672,000\text{ kWh}$
  • $\text{Annual Energy Cost} = 672,000\text{ kWh} \times \text{\textdollar}0.12/\text{kWh} = \text{\textdollar}80,640 / \text{year}$
  • $\text{Total Annual Operating Cost}_B = \text{\textdollar}80,640 + \text{\textdollar}4,500 = \text{\textdollar}85,140 / \text{year}$

Step 2: Determine Compound Interest Factors ($i = 8\%$, $n = 15$)

From NCEES interest tables (or formulas):

$$(P/A, 8\%, 15) = \frac{(1 + 0.08)^{15} - 1}{0.08(1 + 0.08)^{15}} = \frac{3.17217 - 1}{0.08 \times 3.17217} = 8.5595$$ $$(P/F, 8\%, 15) = (1 + 0.08)^{-15} = \frac{1}{3.17217} = 0.31524$$

Step 3: Compute Life Cycle Cost (Present Worth) for Both Options

$$\text{LCC}_{P,A} = \$180,000 + (\$110,160 \times 8.5595) - (\$15,000 \times 0.31524) = \$180,000 + \$942,915 - \$4,729 = \$1,118,186$$ $$\text{LCC}_{P,B} = \$260,000 + (\$85,140 \times 8.5595) - (\$25,000 \times 0.31524) = \$260,000 + \$728,756 - \$7,881 = \$980,875$$

Step 4: Incremental Analysis & Decision

$$\Delta \text{NPV}_{B-A} = \text{LCC}_{P,A} - \text{LCC}_{P,B} = \$1,118,186 - \$980,875 = +\$137,311$$ $$\Delta \text{Annual Operating Savings} = \$110,160 - \$85,140 = \$25,020 / \text{year}$$ $$\text{Simple Payback Period} = \frac{\Delta \text{Capital Cost}}{\Delta \text{Annual Savings}} = \frac{\$260,000 - \$180,000}{\$25,020} = \frac{\$80,000}{\$25,020} = 3.20\text{ years}$$

Conclusion: Option B results in a net present value savings of $137,311 over 15 years with a simple payback of 3.2 years. Option B is strongly recommended.