PE Mechanical: HVAC and Refrigeration formula sheet
135 key equations from 14 exam topics, each with what it is for and where it lives in the NCEES PE Mechanical Reference Handbook. No signup. These are the equations from the free chapters of our PE Mechanical: HVAC and Refrigeration study handbook; the full handbook covers every topic at this depth, with worked examples and the traps that cost points.
135
Equations
14
Exam topics
14
Concepts
Psychrometrics & Air Processes14% of the exam
Moist-air properties, the psychrometric chart, and sensible, latent, mixing, humidification, and dehumidification processes at sea level and altitude.
Moist-Air Properties and the Psychrometric Chart
The six interlocking properties of moist air — dry-bulb, wet-bulb, dew point, humidity ratio, enthalpy, specific volume — and how to fix a state and read processes on the chart.
- Humidity ratio from partial pressures
- $W$ in $\text{lb}_w/\text{lb}_{da}$; $p_w$ = vapor partial pressure, $p$ = total pressure (psia). $7000\,\text{gr}=1\,\text{lb}_w$.
- Saturation humidity ratio
- Maximum $W$ at temperature $t$; $p_{ws}$ = saturation pressure of water at $t$ (psia).
- Relative humidity
- Ratio of actual to saturated vapor partial pressure at the same dry-bulb $t$ and $p$.
- Dew point condition
- $t_d$ = temperature where saturation pressure equals the actual vapor pressure; set by $W$ alone.
- Humidity ratio from wet-bulb
- $t$ = dry-bulb, $t^{*}$ = wet-bulb (°F), $W_s^{*}$ = saturation humidity at $t^{*}$. ASHRAE USCS correlation (not in the NCEES handbook, which gives $W$ only from partial pressures).
- Moist-air specific enthalpy
- $\text{Btu/lb}_{da}$; $t$ in °F. First term dry-air sensible, parenthesis vapor latent + sensible.
- Moist-air specific volume
- $\text{ft}^3/\text{lb}_{da}$; $t$ in °F, $p$ in psia. Per pound of DRY air, not mixture.
- Degree of saturation
- Dimensionless; close to but not equal to relative humidity $\phi$.
Where it lives: NCEES PE Mechanical Reference Handbook — §7 Psychrometrics · NCEES PE Mechanical Reference Handbook — §7.4 Thermodynamic Properties of Moist Air · ASHRAE Handbook—Fundamentals, Chapter 1, Psychrometrics
Read the Psychrometrics & Air Processes chapterHeating & Cooling Loads10% of the exam
Envelope conduction, solar and internal gains, infiltration and ventilation loads, supply-air flow from room sensible heat, and reheat.
Envelope Conduction Loads: U-Factor, R-Value and CLTD
Build the overall U-factor from series and parallel thermal resistances, then drive heating loads with UAΔT and cooling loads with the CLTD method.
- Layer resistance (conduction)
- Unit thermal resistance of a solid layer; $L$ = thickness (ft or in), $k$ = conductivity (Btu·in/(h·ft²·°F) or Btu/(h·ft·°F)). Units of $R$ are h·ft²·°F/Btu.
- Series resistance and U
- Add layer and film resistances in series; $U$ is the reciprocal. $R_i \approx 0.68$ (still indoor air), $R_o \approx 0.17$ winter / $0.25$ summer.
- Parallel-path (area-weighted) U
- For framing and other parallel paths; $f_i$ = area fraction, $U_i$ = transmittance of path $i$. Weight U-values, never R-values.
- Steady-state heating conduction
- Design heating load through a surface, Btu/h. No solar or internal credit; $A$ in ft², $\Delta T$ in °F.
- Heat gain through interior partitions
- Gain from an adjacent unconditioned space at temperature $t_b$ to the conditioned space at $t_i$ (handbook §9.1.9).
- CLTD cooling load (opaque)
- Cooling load through a sunlit wall or roof, Btu/h. CLTD (°F) embeds sol-air temperature and mass time lag.
- CLTD temperature correction
- Adjusts a table value (78 °F indoor, 85 °F mean outdoor basis) to actual design temperatures; $t_{o,\text{mean}} = t_{o,\text{design}} - \tfrac12(\text{daily range})$.
Where it lives: NCEES PE Mechanical Reference Handbook — §9 Heating, Ventilation, and Air Conditioning (§9.1.9 interior surfaces, §9.1.11 thermal resistance, §9.1.13 fenestration U-factors) · ASHRAE Handbook—Fundamentals, Ch. 18 (Nonresidential Cooling and Heating Load Calculations) and Ch. 26 (Material Properties) · ASHRAE Standard 90.1 — Energy Standard for Buildings
Read the Heating & Cooling Loads chapterAir Distribution & Ductwork6% of the exam
Duct sizing and friction, equal-friction and static-regain methods, fan total/static pressure, terminal devices, diffusers, and VAV airflow.
Duct Sizing: Friction, Velocity Pressure, Equal-Friction and Static Regain
How the friction chart, velocity pressure (V/4005)², equivalent diameter, and the equal-friction and static-regain methods turn an airflow schedule into a sized, balanced duct system.
- Continuity (duct velocity)
- $V$ in fpm, $Q$ in cfm, $A$ in $\text{ft}^2$. The first line of nearly every duct calculation.
- Velocity pressure (general)
- $p_v$ in in. wg, $V$ in fpm, $\rho$ in $\text{lbm/ft}^3$. Use this form whenever air is non-standard (altitude, hot air).
- Velocity pressure (standard air)
- Valid only at $\rho = 0.075\ \text{lbm/ft}^3$; $4005 = 1097/\sqrt{0.075}$.
- Total pressure
- Total = static + velocity, all in in. wg. Fans and Bernoulli work in total pressure; friction consumes it.
- Darcy friction loss
- $\Delta p_f$ in in. wg; $L$ in ft, $D_h$ in in., $f$ from Colebrook at $\varepsilon=0.0003\ \text{ft}$. The friction chart evaluates this for you.
- Hydraulic diameter
- For non-circular duct; $A$ = area, $P$ = wetted perimeter (consistent units). Used inside the Darcy relation.
- Circular equivalent diameter
- Round diameter (in.) giving equal airflow, friction, and length to a rectangle of sides $a,b$ (in.). Equal friction, NOT equal area.
- Equal-friction principle
- Hold one friction rate (e.g. $0.10\ \text{in. wg}/100\ \text{ft}$); read each section's size from the chart at its airflow.
- Static-regain recovery
- Static recovered as air slows; recovery factor $R\approx 0.5$–$0.75$. Set equal to downstream friction to keep static constant.
- Local loss coefficient
- Dimensionless ratio of a fitting's total-pressure loss to the reference velocity pressure; ties duct loss to fittings (see the fittings concept).
Where it lives: NCEES PE Mechanical Reference Handbook — §9.3.6 Duct Design · ASHRAE Handbook—Fundamentals, Ch. 21 Duct Design · ASHRAE Standard 90.1 — Energy Standard
Read the Air Distribution & Ductwork chapterVentilation & Indoor Air Quality5% of the exam
Outdoor-air requirements, dilution and contaminant balances, filtration efficiency (MERV), mixing, exhaust, and pressurization.
Outdoor-Air Rates, Dilution and the Ventilation-Rate Procedure
How ASHRAE 62.1 sizes minimum outdoor air from people-plus-area components, why a single critical zone sets system intake, and the efficiency math behind it.
- Breathing-zone outdoor airflow
- Minimum outdoor air for the occupied zone. $R_p$ = per-person rate (cfm/person), $P_z$ = people, $R_a$ = per-area rate (cfm/ft$^2$), $A_z$ = floor area (ft$^2$).
- Zone outdoor airflow
- Supply outdoor air after correcting for distribution. $E_z$ = zone air-distribution effectiveness (1.0 cooling ceiling supply, 0.8 heating ceiling supply, 1.2 displacement).
- Uncorrected outdoor-air intake
- Sum of zone requirements with occupant diversity $D$ applied to the people term only. $D = $ peak system population / sum of zone populations.
- Outdoor-air fraction (system)
- Average uncorrected outdoor fraction of the system primary airflow $V_{ps}$ (cfm).
- Outdoor-air fraction (critical zone)
- Outdoor fraction required in a zone's primary supply $V_{pz}$. The largest $Z_p$ flags the critical zone.
- Zone ventilation efficiency
- Appendix-A single-zone estimate. $<1$ when the zone needs a richer outdoor mix than the system average.
- System ventilation efficiency
- Smallest zone efficiency governs; dimensionless, typically 0.6–1.0.
- Corrected outdoor-air intake
- Actual outdoor air the AHU must draw so the critical zone is satisfied (cfm).
- Single-zone intake
- Collapsed procedure when one zone is served by one unit (100% outdoor or single-zone recirculating).
Where it lives: NCEES PE Mechanical Reference Handbook — §9 Heating, Ventilation, and Air Conditioning (§9.1.2 Human Oxygen Consumption) · ANSI/ASHRAE Standard 62.1 — Ventilation for Acceptable Indoor Air Quality · ASHRAE Handbook — Fundamentals
Read the Ventilation & Indoor Air Quality chapterHydronic, Steam & Piping Systems7% of the exam
Pipe head loss and pump head, hydronic coil capacity, glycol corrections, steam and condensate sizing, and expansion-tank sizing.
Pipe Head Loss, Pump Head and the 500 Water Coefficient
How to size a hydronic loop end to end: friction and fitting head loss, total pump head from the critical circuit, and the q = 500·GPM·ΔT heat-transport rule.
- Reynolds number
- Flow regime for pipe flow (handbook §3.4.1): laminar $Re<2{,}000$, critical/unstable zone $2{,}000$–$4{,}000$, transition zone $4{,}000$–$12{,}000$, fully turbulent $Re>12{,}000$. $D$ in ft, $v$ in fps, $\nu$ kinematic viscosity in ft²/s.
- Water velocity shortcut
- $v$ in fps from $Q$ in gpm and inside diameter $d$ in inches; keep mains at 4–8 fps to limit noise and erosion.
- Darcy–Weisbach head loss
- Universal friction loss (ft) for any fluid; $f$ from Moody/Colebrook, or $64/Re$ if laminar. Use whenever the fluid is not standard water.
- Laminar friction factor
- Exact for $Re < 2000$; independent of roughness.
- Swamee–Jain (turbulent f)
- Explicit fit to the Colebrook equation; $\varepsilon$ is absolute roughness ($\approx 0.00015$ ft for commercial steel).
- Hazen–Williams (water)
- Head loss in ft per 100 ft; $Q$ gpm, $d$ inches. $C\approx$ 150 plastic, 130 copper/cast iron, 120 new welded steel, ~100 old steel (handbook §3.4.2.3). Water only.
- Minor (fitting) loss
- $K$ is the geometry-dependent loss coefficient (handbook §3.4.2.4, Minor Losses in Pipe Fittings, Contractions, and Expansions); sum the $K$'s along the critical circuit and apply at local velocity.
- Equivalent length
- Straight-pipe length giving the same loss as a fitting; alternative to the $K$ method — do not use both on one fitting.
- Total pump head
- Closed loop: no static lift, only friction + fittings + equipment drop along the worst (critical) path.
- Water horsepower
- $Q$ gpm, $h$ ft, $\Delta P$ psi; constants bake in water density (handbook §3.7.5).
- Brake horsepower
- Divide hydraulic power by pump efficiency; multiply by specific gravity for non-water fluids. Sizes the motor.
- Sensible water heat transport (500 rule)
- $q$ Btu/h, $Q$ gpm, $\Delta T$ °F. $500 = 8.33\,\text{lb/gal}\times 60\,\text{min/h}\times 1.0\,\text{Btu/lb·°F}$; valid for water only.
Where it lives: NCEES PE Mechanical Reference Handbook — §3 Hydraulics, Fluids, and Pipe Flow (§3.4 Fluid Flow, §3.7 Fluid Flow Machinery — §3.7.5 Pump Power Equation) · ASHRAE Handbook—Fundamentals, Pipe Sizing (Ch. 22) · ASHRAE Handbook—HVAC Systems and Equipment, Hydronic Heating and Cooling (Ch. 13)
Read the Hydronic, Steam & Piping Systems chapterHVAC Controls & Instrumentation5% of the exam
Valve and damper authority and Cv, sensors and actuators, economizer and reset strategies, and PID feedback response.
Control-Valve Sizing: Cv, Authority and Characteristic
Size a modulating valve from its flow coefficient, protect controllability with valve authority, and match an equal-percentage trim to a nonlinear coil for linear control.
- Flow coefficient (general fluid)
- $Q$ in gpm, $\Delta P$ across the valve in psi, $S_g$ = specific gravity (1.0 for water). Defines the valve size.
- Flow from Cv
- Square-root law: flow through a fixed opening scales with $\sqrt{\Delta P}$. Use to back-check actual drop at a chosen $C_v$.
- Cv for water
- $S_g = 1$ simplification. $C_v$ equals the gpm of $60^\circ\text{F}$ water at $1\,\text{psi}$ drop.
- Equal-percentage characteristic
- $x$ = fractional stroke (0–1), $R$ = rangeability (max/min controllable flow). Flow rises by a constant percentage per equal stroke step.
- Rangeability
- Ratio of maximum to minimum controllable flow at a stated characteristic; larger is better for turndown.
- Valve authority
- Open-valve drop divided by total variable branch drop (valve + coil + balancing + fittings) at full flow. Target $\beta \ge 0.5$.
- Installed characteristic
- Maps inherent flow fraction $f$ to installed flow fraction $\varphi$ given authority $\beta$. Lower $\beta$ distorts toward quick-opening.
- Valve gain
- Slope of the installed flow-vs-stroke curve. Good control wants this roughly constant across the operating range.
Where it lives: NCEES PE Mechanical Reference Handbook — §11 Temperature Controls (§11.3 Control Valves) · ASHRAE Handbook — Fundamentals, Ch. 7 Fundamentals of Control · ASHRAE Handbook — HVAC Systems and Equipment, Valve characteristics and selection
Read the HVAC Controls & Instrumentation chapterRefrigeration Cycles & Systems8% of the exam
Vapor-compression cycles on the p-h diagram, COP and capacity, refrigerant properties and selection, subcooling, superheat, and food/process refrigeration loads.
The Vapor-Compression Cycle: COP, Capacity and the p-h Diagram
The four-process vapor-compression cycle on the pressure-enthalpy diagram, refrigeration effect and compressor work, and how COP, tons, and kW/ton fall out of enthalpy differences.
- Refrigeration effect
- Useful cooling per unit mass (Btu/lb). $h_4 = h_3$ because expansion is isenthalpic.
- Compressor work
- Ideal (isentropic) work per unit mass (Btu/lb); $h_2$ at condenser pressure on the $s = s_1$ line.
- Heat rejected
- Condenser duty per unit mass; energy balance closes the loop.
- Capacity from mass flow
- Evaporator duty (Btu/h) with $\dot m$ in lb/h; divide by 12,000 for tons.
- Ton of refrigeration
- Standard refrigeration capacity unit; use to convert between tons and Btu/h or kW.
- Coefficient of performance (refrigerator)
- Cooling delivered per unit compressor work (dimensionless).
- COP heat pump
- Same hardware billed for heat rejected; always one greater than the cooling COP.
- EER and kW/ton
- Mixed-unit restatements of COP. High EER and LOW kW/ton both mean efficient.
- Carnot (reversed) COP
- Thermodynamic ceiling; $T_L$, $T_H$ ABSOLUTE ($^\circ\text{R}$ or K). Real plant 45–60 % of this.
- Compressor isentropic efficiency
- Ratio of ideal to actual work; solve for actual exit enthalpy $h_{2a}$ to size real power.
- Compressor power
- Btu/h; divide by 2,544 for hp or by 3,412 for kW.
- Volumetric flow / displacement
- Suction volume flow (ft³/min); $v_1$ = specific volume of suction vapor. Sets compressor displacement.
Where it lives: NCEES PE Mechanical Reference Handbook — §8 Refrigeration · NCEES PE Mechanical Reference Handbook — §4.5 Thermodynamic Cycles and Compressors · ASHRAE Handbook—Fundamentals, Ch. 2 Thermodynamics and Refrigeration Cycles
Read the Refrigeration Cycles & Systems chapterCooling Towers & Heat Rejection5% of the exam
Tower range, approach, and effectiveness, evaporation and makeup water, cycles of concentration, and condenser-water heat balance.
Cooling-Tower Range, Approach, Effectiveness and Water Balance
Range, approach, and effectiveness define what a cooling tower delivers; the wet-bulb sets the thermodynamic floor and the 500-coefficient ties heat rejection to water flow.
- Range
- Tower range (°F): entering hot minus leaving cold water. Equals the condenser-water rise across the chiller in a closed loop; set by load and flow.
- Approach
- Approach (°F): leaving cold water minus entering-air WET-BULB. Fixed by tower size and design wet-bulb; never zero.
- Cold-water temperature
- Leaving water at a given approach tracks the wet-bulb one-for-one; the quick way to re-estimate at a new design wet-bulb.
- Tower effectiveness
- Dimensionless thermal effectiveness; cold-side limit is the wet-bulb. Real towers ≈ 0.5–0.8.
- Water-side heat rejection
- Sensible heat carried by the circulating water (Btu/hr). The 500 = 8.34 lb/gal × 60 min/hr × 1 Btu/lb·°F assumes STANDARD water.
- Required water flow
- Inverts the rejection equation; with a 10°F range and 15,000 Btu/hr per cooling-tower ton this gives the 3 gpm/ton rule.
- Cooling-tower ton
- Industry rating adding compressor heat to the 12,000 Btu/hr refrigeration ton at ≈0.88 kW/ton. Do not confuse with the evaporator ton.
- Heat-rejection from refrigeration tons
- Adds heat of compression to the chiller load; gives Btu/hr to reject for an electric chiller (3412 Btu/hr per kW).
Where it lives: NCEES PE Mechanical Reference Handbook — §9.3.9 Cooling Towers and Fluid Coolers · NCEES PE Mechanical Reference Handbook — §8.3 Condensers · ASHRAE Handbook — HVAC Systems and Equipment, Cooling Towers · Cooling Technology Institute (CTI) — Cooling Tower Performance terminology
Read the Cooling Towers & Heat Rejection chapterBoilers, Furnaces & Combustion6% of the exam
Fuel heating values, air-fuel ratio and excess air, combustion and thermal efficiency, boiler/furnace fuel rate, and flue-gas losses.
Heating Values, Combustion Efficiency and Fuel Rate
Higher vs lower heating value, combustion vs thermal efficiency, and how to back the fuel-input rate (and boiler horsepower) out of a required output.
- Higher vs lower heating value
- $m_w$ = mass of water formed per unit fuel (lb water / lb fuel), $h_{fg}\approx 1{,}060\,\text{Btu/lb}$ at the heating-value reference temperature (~60 °F, where the product water is condensed to liquid). HHV counts the latent heat of the product water; LHV does not.
- Fuel-to-load (overall) efficiency
- Useful delivered heat divided by fuel energy input; debits stack, radiation, and blowdown losses. $\dot{Q}_{out}$ in Btu/h, $\dot m_f$ in lb/h (or use $HHV_{vol}$ with volumetric flow). Use HHV with HHV-basis fuel data. Note: the handbook's §10.3 "thermal efficiency" is the stack-only value below, not this one.
- Combustion efficiency (= handbook §10.3 thermal efficiency)
- What a flue-gas analyzer reports; equals 100% minus the dry-gas and moisture stack losses. The NCEES handbook §10.3 prints this exact stack-loss-based form and calls it "thermal efficiency." Always higher than the fuel-to-load efficiency, which also subtracts radiation and blowdown.
- Fuel-input rate from output
- Required fuel energy input (Btu/h). Use the efficiency on the same heating-value basis as the fuel data.
- Fuel quantity (mass or volume)
- Mass rate (lb/h) for oil/coal using mass HHV (Btu/lb); volumetric rate (ft³/h or gal/h) for gas/oil using volumetric HHV (Btu/ft³ or Btu/gal).
- Gas input rate with meter correction
- Handbook §10.6. $T_s=520\,^\circ\text{R}$, $P_s=14.735\,\text{psia}$; $T$,$P$ are absolute meter conditions; $VFR$ is metered volumetric flow (ft³/h).
- Efficiency basis conversion
- Convert a lower-heating-value efficiency to the higher-heating-value basis (and inverse). For methane $LHV/HHV = 0.900$.
- Boiler horsepower
- Output rating. $\dot Q_{out}$ in Btu/h; $\dot m_s$ in lb/h of steam from and at $212\,^\circ\text{F}$. 1 BHP also $\approx 9.81\,\text{kW}$.
- Boiler steam output (real feedwater)
- $h_s$ = enthalpy of generated steam, $h_{fw}$ = feedwater enthalpy (Btu/lb). Reduces to the latent heat only when feedwater is saturated liquid at 212 °F.
- Hot-water output (standard coefficient)
- Sensible heat of water; $500 = 8.33\,\tfrac{\text{lb}}{\text{gal}}\times 60\,\tfrac{\text{min}}{\text{h}}\times 1\,\tfrac{\text{Btu}}{\text{lb}\,^\circ\text{F}}$. $\Delta T$ in °F; valid for water near room temperature.
Where it lives: NCEES PE Mechanical Reference Handbook — §10 Combustion and Fuels · NCEES PE Mechanical Reference Handbook — §6 Steam (boiler plant basics) · ASHRAE Handbook—Fundamentals, Ch. 28 Combustion and Fuels · ASHRAE Handbook—HVAC Systems and Equipment, Boilers chapter
Read the Boilers, Furnaces & Combustion chapterHeat Exchangers & Coils7% of the exam
LMTD and effectiveness-NTU methods, shell-and-tube and plate-and-frame exchangers, correction factors, and cooling/heating coil duty.
The LMTD Method and the F Correction Factor
Counterflow vs parallel flow, the log-mean temperature difference, and the F factor that lets Q = U·A·LMTD·F size shell-and-tube and cross-flow exchangers.
- Heat-exchanger rate equation
- Duty $Q$ (Btu/h); $U$ overall coefficient (Btu/h·ft²·°F) on area $A$ (ft²); $\Delta T_{lm}$ counterflow LMTD; $F$ correction factor ($=1$ for pure counter/parallel flow and for phase-change streams).
- Log-mean temperature difference
- $\Delta T_1,\Delta T_2$ are the terminal temperature differences (°F or °R — same value). Reduces to $\Delta T_1$ when the two ends are equal.
- Counterflow terminal differences
- Hot inlet pairs with cold outlet, hot outlet with cold inlet. Use for the LMTD baseline even on multipass units.
- Parallel-flow terminal differences
- Both streams enter the same end. Always gives a smaller LMTD than counterflow for identical temperatures.
- First-law duty
- Energy balance on one stream; for water $\dot m c_p \approx 500\,(\text{gpm})$ Btu/h·°F. Both streams carry the same $Q$ (adiabatic shell).
- F-factor parameter P (effectiveness)
- Tube-side temperature rise over the maximum available difference; $t$ = tube (usually cold) stream, $T$ = shell stream. $0\le P\le 1$.
- F-factor parameter R (capacity ratio)
- Ratio of the two stream temperature changes; equals $\dot m c_p$ of tube side over that of shell side. $R$ may be above or below 1.
- F factor, 1 shell pass / 2+ tube passes
- Closed form for the most common shell-and-tube geometry (valid $R\ne1$). The exam usually supplies $F$ from a chart; know how $P$ and $R$ feed it.
- Equal-end LMTD limit
- Avoids the 0/0 logarithm; common when one side is condensing or evaporating at constant temperature.
- Required area
- Sizing form. $A$ is on the same reference surface as $U$ (inside or outside).
Where it lives: NCEES PE Mechanical Reference Handbook — §5.5 Heat Exchangers · ASHRAE Handbook—Fundamentals, Heat Transfer chapter · Incropera & DeWitt, Fundamentals of Heat and Mass Transfer
Read the Heat Exchangers & Coils chapterChillers, Heat Pumps & Thermal Storage6% of the exam
Chiller kW/ton and IPLV, heat-pump heating COP and balance point, condenser/evaporator duty, and chilled-water and ice thermal storage.
Chiller Performance: kW/ton, COP, EER and IPLV
Translate freely among kW/ton, COP and EER, weight the four part-load points into IPLV, and connect water- vs air-cooled chillers to the condenser and evaporator approach.
- Cooling COP
- Dimensionless; cooling effect $\dot Q_L$ over compressor input $\dot W$, both in the same units. Convert Btu/h to W with $3.412\ \text{Btu/h}=1\ \text{W}$.
- Ton of refrigeration
- The conversion that links every chiller rating; from §4 Thermodynamics ($3{,}516\ \text{W}$ per ton).
- kW/ton to COP
- Inverse efficiency — lower is better. $3.516$ is kW of heat per ton. Typical water-cooled $0.5\text{–}0.6$, air-cooled $1.0\text{–}1.2$.
- EER definition
- Cooling in Btu/h per watt of input; quoted as a bare number but carries units $\text{Btu/(W}\cdot\text{h)}$.
- EER from kW/ton and COP
- The 12 is $12{,}000\ \text{Btu/h}$ per ton divided by 1000 W; the $3.412$ is Btu/h per watt.
- IPLV in kW/ton (linear approximation)
- Linear weighted sum of kW/ton at 100/75/50/25% load ($A,B,C,D$); a close approximation to the AHRI 550/590 harmonic form. Weights are operating-hour fractions.
- IPLV — AHRI 550/590 form
- The reciprocal (harmonic) weighting AHRI 550/590 actually defines; required when $A,B,C,D$ are efficiencies (COP or EER, higher better) and exact for kW/ton too.
- Condenser approach
- Saturated condensing temperature minus leaving condenser-water temperature (°F). Widens with fouling.
- Evaporator approach
- Leaving chilled-water temperature minus saturated suction temperature (°F).
- Condenser heat rejection
- First-law balance: heat the condenser must reject equals evaporator cooling plus compressor work (Btu/h).
- Condenser water flow
- $\dot Q_{\text{rej}}$ in Btu/h, $\Delta T$ = condenser-water range (°F). The 500 bakes in water density and $c_p$ ($8.34\times60$).
Where it lives: NCEES PE Mechanical Reference Handbook — §8 Refrigeration · NCEES PE Mechanical Reference Handbook — §4 Thermodynamics (COP, ton definition) · AHRI Standard 550/590 — Performance Rating of Water-Chilling Packages · ASHRAE Handbook — HVAC Systems and Equipment, Liquid-Chilling Systems
Read the Chillers, Heat Pumps & Thermal Storage chapterPumps, Compressors & Fans7% of the exam
Affinity laws, pump and fan power and efficiency, NPSH and cavitation, operating point, parallel/series operation, and compressor work.
Pump and Fan Affinity Laws and the Operating Point
The affinity laws (flow ∝ N, head ∝ N², power ∝ N³), riding the pump/fan curve, and why a VFD beats a throttling valve on energy.
- Affinity laws — speed change
- Fixed impeller, variable speed $N$ (rpm). Flow linear, head/pressure square, power cube. Valid at constant efficiency.
- Affinity laws — impeller diameter
- Fixed speed, trimmed impeller diameter $D$. Same 1-2-3 exponents as speed change.
- Specific-gravity scaling
- Head in feet of fluid is density-independent; pressure (psi) and brake horsepower scale linearly with $SG$.
- Fan law form (pressure)
- $P$ = fan total, static, or velocity pressure (in. wg). Fan pressure also scales with air density $\rho$ — important at altitude or elevated temperature.
- Fan law form (power)
- Fan shaft power scales with speed cubed and linearly with air density. Per NCEES §9.3.8.3 (dry standard air, 14.696 psia, 70°F unless stated).
- Friction-system power reduction
- VFD savings estimate on a friction-dominated system ($h\propto Q^2$ through the origin). Overstates savings if static head is present.
- Head as a parabola through the origin
- Combines the flow and head laws; eliminating $N$ shows affinity-scaled points lie on a parabola $h\propto Q^2$ through the origin.
- Operating point definition
- Steady-state flow is the intersection of pump/fan curve and system curve. Affinity scaling moves the machine curve; the system curve is unchanged.
Where it lives: NCEES PE Mechanical Reference Handbook — §3.7.6 Pump Affinity Laws · NCEES PE Mechanical Reference Handbook — §9.3.8.3 Fan Affinity Laws · ASHRAE Handbook—HVAC Systems and Equipment, Ch. 44 (Centrifugal Pumps) and Ch. 21 (Fans) · ASHRAE 90.1 — Energy Standard (variable-speed drive provisions)
Read the Pumps, Compressors & Fans chapterEnergy Recovery4% of the exam
Sensible and total (enthalpy) recovery effectiveness, enthalpy wheels, heat pipes, and run-around loops, with recovered-energy and economizer savings.
Sensible and Total Energy-Recovery Effectiveness
How sensible, latent, and total effectiveness define an HRV/ERV, how to find leaving conditions and recovered energy, and how the supply/exhaust airflow ratio and frost control bound it.
- Sensible effectiveness
- Ratio of actual to maximum sensible recovery. $t_1$ outdoor in, $t_2$ supply out, $t_3$ exhaust in, $t_4$ exhaust out (°F); $C_{min}=\min(\dot m_s c_{ps},\dot m_e c_{pe})$ is the smaller air-capacity rate. Because numerator and denominator share the same $\dot m$ basis, the ratio is dimensionless regardless of whether $\dot m$ is taken in lb/min or lb/h.
- Latent (moisture) effectiveness
- Built on humidity ratio $w$ ($\text{lb}_w/\text{lb}_{da}$). $\dot m_{min}$ = smaller dry-air mass flow (lb/min). Zero for an HRV; nonzero only for vapor-permeable ERV media.
- Total (enthalpy) effectiveness
- Built on moist-air enthalpy $h$ ($\text{Btu/lb}_{da}$). Total recovery equals the sum of sensible and latent recovery.
- Supply leaving temperature
- Conditioned outdoor air handed to the downstream coil. Reduces to $t_2 = t_1 + \varepsilon_s(t_3-t_1)$ when the supply is the smaller (or balanced) stream.
- Exhaust leaving temperature
- Used for the frost check: $t_4$ below 32 °F at the exhaust dew point freezes condensate inside the device.
- Recovered sensible heat
- Btu/h. The $1.08$ coefficient assumes sea-level standard air ($\rho=0.075$ lb/ft³, $c_p=0.24$). $Q_s$ = supply cfm.
- Recovered latent heat
- Btu/h. Use $0.68$ with $\Delta w$ in grains per $\text{lb}_{da}$, or $4760 = 0.68\times7000$ with $\Delta w$ in $\text{lb}_w/\text{lb}_{da}$ — the two embed the same $h_{fg}\approx1058\ \text{Btu/lb}$. Standard air.
- Recovered total heat
- Btu/h. The $4.5$ coefficient ($60\times0.075$) assumes standard air; total = sensible + latent.
- Maximum sensible recovery
- Denominator of $\varepsilon_s$. Built on the smaller airflow $Q_{min}$ and the full inlet temperature difference.
- Capacity rate
- Air-side heat capacity rate (Btu/h·°F) with $\dot m$ in lb/min and $Q$ in cfm; the $60$ converts min$\to$h. The smaller stream caps recovery. In the dimensionless effectiveness ratios the $60$ cancels, so $C_{min}/(\dot m_s c_{ps})$ may be evaluated on either basis.
- Effectiveness decomposition
- Total recovery is sensible plus latent; with $h_{fg}\approx1058\ \text{Btu/lb}$ (the same value behind the $0.68$/$4760$ latent coefficients) this ties $\varepsilon_t$ back to $\varepsilon_s$ and $\varepsilon_L$ at balanced flow.
Where it lives: NCEES PE Mechanical Reference Handbook — §9.2.7 Heat-Recovery Ventilator and §9.2.8 Energy-Recovery Ventilator · ASHRAE Handbook—HVAC Systems and Equipment, Ch. 26 Air-to-Air Energy Recovery Equipment · AHRI Standard 1060 — Performance Rating of Air-to-Air Exchangers for Energy Recovery Ventilation · ASHRAE Handbook—Fundamentals, Ch. 1 Psychrometrics
Read the Energy Recovery chapterCodes, Acoustics, Economics & Electrical10% of the exam
Codes and standards, sound and vibration control (NC, dBA, isolation), life-cycle and payback economics, and motor power, amperage, and heat output.
Life-Cycle Cost, Simple and Discounted Payback
Compare HVAC alternatives on a present-worth basis with the interest factors, and judge energy retrofits with simple and discounted payback.
- Single-payment compound amount (F/P)
- Moves a present sum $P$ forward $n$ periods to its future worth $F$ at periodic rate $i$.
- Single-payment present worth (P/F)
- Discounts a single future sum $F$ (e.g., salvage, a future replacement) back to year 0.
- Uniform-series present worth (P/A)
- Collapses a level annual series $A$ (energy, O&M) over $n$ years into one present worth $P$.
- Capital recovery factor (A/P)
- Spreads a present cost into $n$ equal annual payments; the reciprocal of $(P/A)$. Used for equivalent annual cost.
- Uniform-series compound amount (F/A)
- Future worth of a level annual series; $(A/F)=i/[(1+i)^n-1]$ is its reciprocal (sinking fund).
- Effective interest rate
- $r$ = nominal annual rate, $m$ = compounding periods per year. Convert before using period factors.
- Life-cycle cost
- Present-worth total of ownership: first cost $C_0$, annual costs, less salvage $S_n$. Lowest LCC wins.
- Simple payback
- Years for undiscounted annual savings to repay first cost; screening only — ignores $i$ and post-payback value.
- Discounted payback
- Handbook improved-payback period. Requires $CRF>i$ or the project never pays back.
- Net present worth of a measure
- Decision metric for an efficiency upgrade; $NPW>0$ accept. $\Delta C_0$ = incremental first cost.
- Uniform gradient present worth (P/G)
- Present worth of a cost that increases by a constant amount $G$ each year (e.g., escalating maintenance).
Where it lives: NCEES PE Mechanical Reference Handbook — §1.6 Economic Analysis (factor formulas and factor tables) · ASHRAE Handbook—HVAC Applications, Owning and Operating Costs (life-cycle cost method) · NIST Handbook 135 — Life-Cycle Costing Manual for the Federal Energy Management Program
Read the Codes, Acoustics, Economics & Electrical chapterNow use them on real questions
Ten free PE Mechanical: HVAC and Refrigeration questions with figures and full worked solutions, no account — then the full bank, timed mock exams and the complete study handbook when you are ready.
Equations follow the NCEES PE Mechanical Reference Handbook as NCEES prints it; the on-screen reference NCEES supplies on exam day is the copy that counts, so check the edition for your sitting. Independent study resource; not affiliated with, endorsed by, or sponsored by NCEES.