PE Mechanical: HVAC and Refrigeration formula sheet

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

$$W = 0.62198\,\dfrac{p_w}{p - p_w}$$

$W$ 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

$$W_s = 0.62198\,\dfrac{p_{ws}}{p - p_{ws}}$$

Maximum $W$ at temperature $t$; $p_{ws}$ = saturation pressure of water at $t$ (psia).

Relative humidity

$$\phi = \dfrac{p_w}{p_{ws}(t)}$$

Ratio of actual to saturated vapor partial pressure at the same dry-bulb $t$ and $p$.

Dew point condition

$$p_{ws}(t_d) = p_w$$

$t_d$ = temperature where saturation pressure equals the actual vapor pressure; set by $W$ alone.

Humidity ratio from wet-bulb

$$W = \dfrac{(1093 - 0.556\,t^{*})\,W_s^{*} - 0.240\,(t - t^{*})}{1093 + 0.444\,t - t^{*}}$$

$t$ = 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

$$h = 0.240\,t + W(1061 + 0.444\,t)$$

$\text{Btu/lb}_{da}$; $t$ in °F. First term dry-air sensible, parenthesis vapor latent + sensible.

Moist-air specific volume

$$v = \dfrac{0.370\,(t + 459.67)(1 + 1.6078\,W)}{p}$$

$\text{ft}^3/\text{lb}_{da}$; $t$ in °F, $p$ in psia. Per pound of DRY air, not mixture.

Degree of saturation

$$\mu = \dfrac{W}{W_s(t,p)}$$

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

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)

$$R = \dfrac{L}{k}$$

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

$$R_{\text{total}} = R_o + \textstyle\sum_j \dfrac{L_j}{k_j} + R_i, \quad U = \dfrac{1}{R_{\text{total}}}$$

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

$$U_{\text{avg}} = \textstyle\sum_i f_i\,U_i$$

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

$$q_{\text{heat}} = U A\,(t_i - t_o)$$

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

$$q = U A\,(t_b - t_i)$$

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)

$$q_{\text{cool}} = U A\,(\text{CLTD})_{\text{corr}}$$

Cooling load through a sunlit wall or roof, Btu/h. CLTD (°F) embeds sol-air temperature and mass time lag.

CLTD temperature correction

$$(\text{CLTD})_{\text{corr}} = \text{CLTD}_{\text{table}} + (78 - t_i) + (t_{o,\text{mean}} - 85)$$

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

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)

$$V = \dfrac{Q}{A}$$

$V$ in fpm, $Q$ in cfm, $A$ in $\text{ft}^2$. The first line of nearly every duct calculation.

Velocity pressure (general)

$$p_v = \rho\left(\dfrac{V}{1097}\right)^2$$

$p_v$ 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)

$$p_v = \left(\dfrac{V}{4005}\right)^2$$

Valid only at $\rho = 0.075\ \text{lbm/ft}^3$; $4005 = 1097/\sqrt{0.075}$.

Total pressure

$$p_t = p_s + p_v$$

Total = static + velocity, all in in. wg. Fans and Bernoulli work in total pressure; friction consumes it.

Darcy friction loss

$$\Delta p_f = f\,\dfrac{12L}{D_h}\,p_v$$

$\Delta p_f$ 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

$$D_h = \dfrac{4A}{P}$$

For non-circular duct; $A$ = area, $P$ = wetted perimeter (consistent units). Used inside the Darcy relation.

Circular equivalent diameter

$$D_e = 1.30\,\dfrac{(ab)^{0.625}}{(a+b)^{0.25}}$$

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

$$\dfrac{\Delta p_f}{L} = \text{constant for every section}$$

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

$$\Delta p_{s,\text{regain}} = R\,(p_{v,1} - p_{v,2})$$

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

$$C = \dfrac{\Delta p_t}{p_v}$$

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

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

$$V_{bz} = R_p\,P_z + R_a\,A_z$$

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

$$V_{oz} = \dfrac{V_{bz}}{E_z}$$

Supply outdoor air after correcting for distribution. $E_z$ = zone air-distribution effectiveness (1.0 cooling ceiling supply, 0.8 warm ceiling supply 15 °F or more above space with a ceiling return, 1.2 displacement).

Uncorrected outdoor-air intake

$$V_{ou} = D\sum R_p P_z + \sum R_a A_z$$

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)

$$X_s = \dfrac{V_{ou}}{V_{ps}}$$

Average uncorrected outdoor fraction of the system primary airflow $V_{ps}$ (cfm).

Outdoor-air fraction (critical zone)

$$Z_p = \dfrac{V_{oz}}{V_{pz}}$$

Outdoor fraction required in a zone's primary supply $V_{pz}$. The largest $Z_p$ flags the critical zone.

Zone ventilation efficiency

$$E_{vz} = 1 + X_s - Z_p$$

Appendix-A single-zone estimate. $<1$ when the zone needs a richer outdoor mix than the system average.

System ventilation efficiency

$$E_v = \min\left(E_{vz}\right)$$

Smallest zone efficiency governs; dimensionless, typically 0.6–1.0.

Corrected outdoor-air intake

$$V_{ot} = \dfrac{V_{ou}}{E_v}$$

Actual outdoor air the AHU must draw so the critical zone is satisfied (cfm).

Single-zone intake

$$V_{ot} = V_{oz} = \dfrac{R_p P_z + R_a A_z}{E_z}$$

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

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

$$Re = \dfrac{\rho v D}{\mu} = \dfrac{v D}{\nu}$$

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

$$v = \dfrac{0.4085\,Q}{d^{2}}$$

$v$ 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

$$h_f = f\,\dfrac{L}{D}\,\dfrac{v^{2}}{2g}$$

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

$$f = \dfrac{64}{Re}$$

Exact for $Re < 2000$; independent of roughness.

Swamee–Jain (turbulent f)

$$f = \dfrac{0.25}{\left[\log_{10}\!\left(\dfrac{\varepsilon/D}{3.7}+\dfrac{5.74}{Re^{0.9}}\right)\right]^{2}}$$

Explicit fit to the Colebrook equation; $\varepsilon$ is absolute roughness ($\approx 0.00015$ ft for commercial steel).

Hazen–Williams (water)

$$h_f = 0.2083\left(\dfrac{100}{C}\right)^{1.852}\dfrac{Q^{1.852}}{d^{4.8655}}$$

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

$$h_m = K\,\dfrac{v^{2}}{2g}$$

$K$ 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

$$L_{eq} = K\,\dfrac{D}{f}$$

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

$$H_{pump} = h_{f}+ \sum h_{m} + \sum h_{equip}$$

Closed loop: no static lift, only friction + fittings + equipment drop along the worst (critical) path.

Water horsepower

$$\text{whp} = \dfrac{Q\,h}{3960} = \dfrac{Q\,\Delta P}{1714}$$

$Q$ gpm, $h$ ft, $\Delta P$ psi; constants bake in water density (handbook §3.7.5).

Brake horsepower

$$\text{BHP} = \dfrac{Q\,h\,(SG)}{3960\,\eta_p}$$

Divide hydraulic power by pump efficiency; multiply by specific gravity for non-water fluids. Sizes the motor.

Sensible water heat transport (500 rule)

$$q = 500\,Q\,\Delta T$$

$q$ 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)

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)

$$C_v = Q\sqrt{\dfrac{S_g}{\Delta P}}$$

$Q$ in gpm, $\Delta P$ across the valve in psi, $S_g$ = specific gravity (1.0 for water). Defines the valve size.

Flow from Cv

$$Q = C_v\sqrt{\dfrac{\Delta P}{S_g}}$$

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

$$C_v = \dfrac{Q}{\sqrt{\Delta P}}$$

$S_g = 1$ simplification. $C_v$ equals the gpm of $60^\circ\text{F}$ water at $1\,\text{psi}$ drop.

Equal-percentage characteristic

$$\dfrac{q}{q_{\max}} = R^{\,(x-1)}$$

$x$ = fractional stroke (0–1), $R$ = rangeability (max/min controllable flow). Flow rises by a constant percentage per equal stroke step.

Rangeability

$$R = \dfrac{q_{\max}}{q_{\min}}$$

Ratio of maximum to minimum controllable flow at a stated characteristic; larger is better for turndown.

Valve authority

$$\beta = \dfrac{\Delta P_{\text{valve, open}}}{\Delta P_{\text{branch, open}}}$$

Open-valve drop divided by total variable branch drop (valve + coil + balancing + fittings) at full flow. Target $\beta \ge 0.5$.

Installed characteristic

$$\varphi = \dfrac{f}{\sqrt{\beta + (1-\beta)\,f^{2}}}$$

Maps inherent flow fraction $f$ to installed flow fraction $\varphi$ given authority $\beta$. Lower $\beta$ distorts toward quick-opening.

Valve gain

$$\text{gain} = \dfrac{\Delta q}{\Delta x}$$

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

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

$$q_L = h_1 - h_4 = h_1 - h_3$$

Useful cooling per unit mass (Btu/lb). $h_4 = h_3$ because expansion is isenthalpic.

Compressor work

$$w_c = h_2 - h_1$$

Ideal (isentropic) work per unit mass (Btu/lb); $h_2$ at condenser pressure on the $s = s_1$ line.

Heat rejected

$$q_H = h_2 - h_3 = q_L + w_c$$

Condenser duty per unit mass; energy balance closes the loop.

Capacity from mass flow

$$\dot Q_L = \dot m\,(h_1 - h_4)$$

Evaporator duty (Btu/h) with $\dot m$ in lb/h; divide by 12,000 for tons.

Ton of refrigeration

$$1\ \text{ton} = 12{,}000\ \tfrac{\text{Btu}}{\text{h}} = 200\ \tfrac{\text{Btu}}{\text{min}} = 3.517\ \text{kW}$$

Standard refrigeration capacity unit; use to convert between tons and Btu/h or kW.

Coefficient of performance (refrigerator)

$$\mathrm{COP}_R = \dfrac{q_L}{w_c} = \dfrac{h_1 - h_4}{h_2 - h_1}$$

Cooling delivered per unit compressor work (dimensionless).

COP heat pump

$$\mathrm{COP}_{HP} = \dfrac{q_H}{w_c} = \mathrm{COP}_R + 1$$

Same hardware billed for heat rejected; always one greater than the cooling COP.

EER and kW/ton

$$\mathrm{EER} = 3.412\,\mathrm{COP}, \qquad \mathrm{kW/ton} = \dfrac{3.517}{\mathrm{COP}} = \dfrac{12}{\mathrm{EER}}$$

Mixed-unit restatements of COP. High EER and LOW kW/ton both mean efficient.

Carnot (reversed) COP

$$\mathrm{COP}_{C} = \dfrac{T_L}{T_H - T_L}$$

Thermodynamic ceiling; $T_L$, $T_H$ ABSOLUTE ($^\circ\text{R}$ or K). Real plant 45–60 % of this.

Compressor isentropic efficiency

$$\eta_s = \dfrac{h_{2s} - h_1}{h_{2a} - h_1}$$

Ratio of ideal to actual work; solve for actual exit enthalpy $h_{2a}$ to size real power.

Compressor power

$$\dot W_c = \dot m\,(h_2 - h_1)$$

Btu/h; divide by 2,544 for hp or by 3,412 for kW.

Volumetric flow / displacement

$$\dot V_1 = \dot m\,v_1$$

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

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

$$R = T_{hw} - T_{cw}$$

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

$$A = T_{cw} - T_{wb}$$

Approach (°F): leaving cold water minus entering-air WET-BULB. Fixed by tower size and design wet-bulb; never zero.

Cold-water temperature

$$T_{cw} = T_{wb} + A$$

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

$$\varepsilon = \dfrac{R}{R+A} = \dfrac{T_{hw}-T_{cw}}{T_{hw}-T_{wb}}$$

Dimensionless thermal effectiveness; cold-side limit is the wet-bulb. Real towers ≈ 0.5–0.8.

Water-side heat rejection

$$\dot q_o = 500\,(\text{gpm})\,R$$

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

$$\text{gpm} = \dfrac{\dot q_o}{500\,R}$$

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

$$1\ \text{CT ton} = 15{,}000\ \tfrac{\text{Btu}}{\text{hr}}$$

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

$$\dot q_o = \dot q_{evap}\left(1 + \dfrac{3412\,(\text{kW/ton})}{12{,}000}\right)$$

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

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

$$HHV = LHV + m_{w}\,h_{fg}$$

$m_w$ = 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

$$\eta_{f\text{-}l} = \dfrac{\dot{Q}_{out}}{HHV\cdot \dot{m}_{f}}$$

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. This is the handbook's first §10.3 "thermal efficiency" form (useful heat / heating value of fuel); its alternative $(Q_h-Q_{fl})/Q_h$ form, below, is stack-only.

Combustion efficiency (= handbook §10.3 stack-loss form)

$$\eta_{c} = 100\% - L_{stack} = \dfrac{Q_h - Q_{fl}}{Q_h}\times 100\%$$

What a flue-gas analyzer reports; equals 100% minus the dry-gas and moisture stack losses. The NCEES handbook §10.3 prints this stack-loss form as its alternative ("or") definition of "thermal efficiency," after useful heat / heating value of fuel. Always higher than the fuel-to-load efficiency, which also subtracts radiation and blowdown.

Fuel-input rate from output

$$\dot{Q}_{in} = \dfrac{\dot{Q}_{out}}{\eta}$$

Required fuel energy input (Btu/h). Use the efficiency on the same heating-value basis as the fuel data.

Fuel quantity (mass or volume)

$$\dot{m}_{f} = \dfrac{\dot{Q}_{in}}{HHV}\qquad \dot{V}_{f} = \dfrac{\dot{Q}_{in}}{HHV_{vol}}$$

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

$$\dot{Q} = HHV\cdot VFR\cdot \dfrac{T_s\,P}{T\,P_s}$$

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

$$\eta_{HHV} = \eta_{LHV}\,\dfrac{LHV}{HHV}$$

Convert a lower-heating-value efficiency to the higher-heating-value basis (and inverse). For methane $LHV/HHV = 0.900$.

Boiler horsepower

$$\text{BHP} = \dfrac{\dot{Q}_{out}}{33{,}470} = \dfrac{\dot{m}_{s}}{34.5}$$

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)

$$\dot{Q}_{out} = \dot{m}_{s}\,(h_{s} - h_{fw})$$

$h_s$ = 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)

$$\dot{Q} = 500\cdot \text{gpm}\cdot \Delta T$$

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

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

$$Q = U A\,\Delta T_{lm}\,F$$

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

$$\Delta T_{lm} = \dfrac{\Delta T_1 - \Delta T_2}{\ln\!\left(\Delta T_1/\Delta T_2\right)}$$

$\Delta T_1,\Delta T_2$ are the terminal temperature differences (°F or °R — same value). Reduces to $\Delta T_1$ when the two ends are equal.

Counterflow terminal differences

$$\Delta T_1 = T_{Hi}-T_{Co}, \quad \Delta T_2 = T_{Ho}-T_{Ci}$$

Hot inlet pairs with cold outlet, hot outlet with cold inlet. Use for the LMTD baseline even on multipass units.

Parallel-flow terminal differences

$$\Delta T_1 = T_{Hi}-T_{Ci}, \quad \Delta T_2 = T_{Ho}-T_{Co}$$

Both streams enter the same end. Gives a smaller LMTD than counterflow for identical temperatures (equal only when one stream is isothermal).

First-law duty

$$Q = \dot{m}\,c_p\,(T_{in}-T_{out})$$

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)

$$P = \dfrac{t_o - t_i}{T_i - t_i}$$

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)

$$R = \dfrac{T_i - T_o}{t_o - t_i}$$

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

$$F = \dfrac{\sqrt{R^2+1}\,\ln\!\dfrac{1-P}{1-PR}}{(R-1)\,\ln\!\dfrac{2-P\left(R+1-\sqrt{R^2+1}\right)}{2-P\left(R+1+\sqrt{R^2+1}\right)}}$$

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

$$\Delta T_{lm} = \Delta T_1 = \Delta T_2 \quad(\text{when }\Delta T_1=\Delta T_2)$$

Avoids the 0/0 logarithm; occurs in balanced counterflow ($C_H = C_C$), where $\Delta T$ is constant along the length — not in a condenser, whose ends differ by the other stream's temperature change.

Required area

$$A = \dfrac{Q}{U\,\Delta T_{lm}\,F}$$

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

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

$$\text{COP} = \dfrac{\dot Q_L}{\dot W}$$

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

$$1\ \text{ton} = 12{,}000\ \text{Btu/h} = 3.516\ \text{kW}$$

The conversion that links every chiller rating; from §4 Thermodynamics ($3{,}516\ \text{W}$ per ton).

kW/ton to COP

$$\text{kW/ton} = \dfrac{3.516}{\text{COP}}$$

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

$$\text{EER} = \dfrac{\dot Q_L\,[\text{Btu/h}]}{\dot W\,[\text{W}]}$$

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

$$\text{EER} = \dfrac{12}{\text{kW/ton}} = 3.412\,\text{COP}$$

The 12 is $12{,}000\ \text{Btu/h}$ per ton divided by 1000 W; the $3.412$ is Btu/h per watt.

IPLV, AHRI 550/590 (COP or EER)

$$\text{IPLV} = 0.01\,A + 0.42\,B + 0.45\,C + 0.12\,D$$

Weighted sum of COP or EER (higher is better) at 100/75/50/25% load; this is the AHRI 550/590 definition for efficiencies. Weights are operating-hour fractions. Not printed in the PE Mechanical Reference Handbook.

IPLV, AHRI 550/590 (kW/ton)

$$\text{IPLV} = \dfrac{1}{\dfrac{0.01}{A}+\dfrac{0.42}{B}+\dfrac{0.45}{C}+\dfrac{0.12}{D}}$$

Use when $A$–$D$ are kW/ton (lower is better). It equals $3.516$ divided by the COP-form IPLV. Not printed in the PE Mechanical Reference Handbook.

Condenser approach

$$\text{Approach}_{\text{cond}} = T_{\text{cond,sat}} - T_{\text{cw,leaving}}$$

Saturated condensing temperature minus leaving condenser-water temperature (°F). Widens with fouling.

Evaporator approach

$$\text{Approach}_{\text{evap}} = T_{\text{chw,leaving}} - T_{\text{evap,sat}}$$

Leaving chilled-water temperature minus saturated suction temperature (°F).

Condenser heat rejection

$$\dot Q_{\text{rej}} = \dot Q_L + \dot W$$

First-law balance: heat the condenser must reject equals evaporator cooling plus compressor work (Btu/h).

Condenser water flow

$$\text{gpm} = \dfrac{\dot Q_{\text{rej}}}{500\,\Delta T}$$

$\dot Q_{\text{rej}}$ 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

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

$$\frac{Q_2}{Q_1}=\frac{N_2}{N_1},\quad \frac{h_2}{h_1}=\left(\frac{N_2}{N_1}\right)^{2},\quad \frac{bhp_2}{bhp_1}=\left(\frac{N_2}{N_1}\right)^{3}$$

Fixed impeller, variable speed $N$ (rpm). Flow linear, head/pressure square, power cube. Valid at constant efficiency.

Affinity laws — impeller diameter

$$\frac{Q_2}{Q_1}=\frac{D_2}{D_1},\quad \frac{h_2}{h_1}=\left(\frac{D_2}{D_1}\right)^{2},\quad \frac{bhp_2}{bhp_1}=\left(\frac{D_2}{D_1}\right)^{3}$$

Fixed speed, trimmed impeller diameter $D$. Same 1-2-3 exponents as speed change.

Specific-gravity scaling

$$\frac{bhp_2}{bhp_1}=\frac{SG_2}{SG_1}\quad(\text{at fixed }Q,h)$$

Head in feet of fluid is density-independent; pressure (psi) and brake horsepower scale linearly with $SG$.

Fan law form (pressure)

$$\frac{P_2}{P_1}=\left(\frac{N_2}{N_1}\right)^{2}\left(\frac{\rho_2}{\rho_1}\right)$$

$P$ = 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)

$$\frac{\dot{W}_2}{\dot{W}_1}=\left(\frac{N_2}{N_1}\right)^{3}\left(\frac{\rho_2}{\rho_1}\right)$$

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

$$\frac{P_2}{P_1}\approx\left(\frac{Q_2}{Q_1}\right)^{3}$$

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

$$h_2 = h_1\left(\frac{Q_2}{Q_1}\right)^{2}$$

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

$$h_{pump}(Q)=h_{system}(Q)$$

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)

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

$$\varepsilon_s = \frac{\dot m_s c_{ps}(t_2 - t_1)}{C_{min}(t_3 - t_1)} = \frac{\dot m_e c_{pe}(t_3 - t_4)}{C_{min}(t_3 - t_1)}$$

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

$$\varepsilon_L = \frac{\dot m_s(w_2 - w_1)}{\dot m_{min}(w_3 - w_1)}$$

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

$$\varepsilon_t = \frac{\dot m_s(h_2 - h_1)}{\dot m_{min}(h_3 - h_1)}$$

Built on moist-air enthalpy $h$ ($\text{Btu/lb}_{da}$). Total recovery equals the sum of sensible and latent recovery.

Supply leaving temperature

$$t_2 = t_1 + \varepsilon_s\,\frac{C_{min}}{\dot m_s c_{ps}}\,(t_3 - t_1)$$

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

$$t_4 = t_3 - \varepsilon_s\,\frac{C_{min}}{\dot m_e c_{pe}}\,(t_3 - t_1)$$

Used for the frost check: $t_4$ below 32 °F at the exhaust dew point freezes condensate inside the device.

Recovered sensible heat

$$\dot q_s = 1.10\,Q_s\,(t_2 - t_1) = 60\,\dot m_s c_{ps}(t_2 - t_1)$$

Btu/h. The $1.10$ coefficient assumes sea-level standard air ($\rho=0.075$ lb/ft³, $c_p = 0.24 + 0.45w \approx 0.244$; handbook §9.1.7). $Q_s$ = supply cfm.

Recovered latent heat

$$\dot q_L = 4{,}840\,Q_s\,(w_2 - w_1) = 0.691\,Q_s\,(\text{gr}_2 - \text{gr}_1)$$

Btu/h. Use $4{,}840 = 60(0.075)(1{,}076)$ with $\Delta w$ in $\text{lb}_w/\text{lb}_{da}$ (handbook §9.1.7), or $0.691 = 4{,}840/7{,}000$ with $\Delta w$ in grains per $\text{lb}_{da}$ — the two embed the same $h_{fg}\approx1{,}076\ \text{Btu/lb}$. Standard air.

Recovered total heat

$$\dot q_t = 4.5\,Q_s\,(h_2 - h_1) = \dot q_s + \dot q_L$$

Btu/h. The $4.5$ coefficient ($60\times0.075$) assumes standard air; total = sensible + latent.

Maximum sensible recovery

$$\dot q_{s,\max} = 1.10\,Q_{min}\,(t_3 - t_1)$$

Denominator of $\varepsilon_s$. Built on the smaller airflow $Q_{min}$ and the full inlet temperature difference.

Capacity rate

$$C = 60\,\dot m c_p = 60\,Q\,\rho\,c_p, \qquad C_{min} = \min(C_s, C_e)$$

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

$$\varepsilon_t\,(h_3 - h_1) = \varepsilon_s\,c_p(t_3 - t_1) + \varepsilon_L\,h_{fg}(w_3 - w_1)$$

Total recovery is sensible plus latent; with $h_{fg}\approx1{,}076\ \text{Btu/lb}$ (the value behind the handbook's $4{,}840$ latent coefficient) 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

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)

$$(F/P,i,n)=(1+i)^{n}$$

Moves a present sum $P$ forward $n$ periods to its future worth $F$ at periodic rate $i$.

Single-payment present worth (P/F)

$$(P/F,i,n)=(1+i)^{-n}$$

Discounts a single future sum $F$ (e.g., salvage, a future replacement) back to year 0.

Uniform-series present worth (P/A)

$$(P/A,i,n)=\dfrac{(1+i)^{n}-1}{i\,(1+i)^{n}}$$

Collapses a level annual series $A$ (energy, O&M) over $n$ years into one present worth $P$.

Capital recovery factor (A/P)

$$(A/P,i,n)=\dfrac{i\,(1+i)^{n}}{(1+i)^{n}-1}$$

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)

$$(F/A,i,n)=\dfrac{(1+i)^{n}-1}{i}$$

Future worth of a level annual series; $(A/F)=i/[(1+i)^n-1]$ is its reciprocal (sinking fund).

Effective interest rate

$$i_e=\left(1+\dfrac{r}{m}\right)^{m}-1$$

$r$ = nominal annual rate, $m$ = compounding periods per year. Convert before using period factors.

Life-cycle cost

$$LCC = C_0 + A_{\text{O\&M}}\,(P/A,i,n) - S_n\,(P/F,i,n)$$

Present-worth total of ownership: first cost $C_0$, annual costs, less salvage $S_n$. Lowest LCC wins.

Simple payback

$$n_{\text{simple}}=\dfrac{C_0-S}{A_{\text{savings}}}$$

Years for undiscounted annual savings to repay first cost; screening only — ignores $i$ and post-payback value.

Discounted payback

$$n=\dfrac{\ln[\,CRF/(CRF-i)\,]}{\ln(1+i)},\quad CRF=\dfrac{A_{\text{savings}}}{C_0}$$

Handbook improved-payback period. Requires $CRF>i$ or the project never pays back.

Net present worth of a measure

$$NPW = A_{\text{savings}}\,(P/A,i,n) - \Delta C_0$$

Decision metric for an efficiency upgrade; $NPW>0$ accept. $\Delta C_0$ = incremental first cost.

Uniform gradient present worth (P/G)

$$(P/G,i,n)=\dfrac{(1+i)^{n}-i n-1}{i^{2}(1+i)^{n}}$$

Present worth of a cost that increases by a constant amount $G$ each year (e.g., escalating maintenance).