Formula Sheet: HVAC Systems

Psychrometrics

Properties of Moist Air

Humidity Ratio (Specific Humidity): \[W = \frac{m_w}{m_a} = 0.622 \frac{p_w}{p - p_w}\]

• W = humidity ratio (lb_w/lb_da or kg_w/kg_da)
• m_w = mass of water vapor (lb or kg)
• m_a = mass of dry air (lb or kg)
• p_w = partial pressure of water vapor (psia or kPa)
• p = total atmospheric pressure (psia or kPa)
• Valid for ideal gas behavior

Relative Humidity: \[\phi = \frac{p_w}{p_{ws}} = \frac{W \cdot p}{(0.622 + W) \cdot p_{ws}}\]

• φ = relative humidity (decimal or %)
• p_ws = saturation pressure of water vapor at dry-bulb temperature (psia or kPa)
• 0 ≤ φ ≤ 1 (or 0% to 100%)

Degree of Saturation: \[\mu = \frac{W}{W_s}\]

• μ = degree of saturation (decimal)
• W_s = humidity ratio at saturation

Dew Point Temperature:

• Temperature at which water vapor begins to condense at constant pressure and humidity ratio
• Found from saturation tables where p_ws(T_dp) = p_w
• T_dp = dew point temperature (°F or °C)

Enthalpy of Moist Air

Total Enthalpy: \[h = h_a + W \cdot h_g\] \[h = c_p \cdot T + W(h_{fg} + c_{pw} \cdot T)\] Simplified (common approximation): \[h = 0.240 \cdot T + W(1061 + 0.444 \cdot T)\]

• h = enthalpy of moist air (Btu/lb_da or kJ/kg_da)
• h_a = enthalpy of dry air (Btu/lb_da or kJ/kg_da)
• h_g = enthalpy of water vapor (Btu/lb_w or kJ/kg_w)
• c_p = specific heat of dry air = 0.240 Btu/lb·°F (0.24 Btu/lb_da·°F or 1.006 kJ/kg_da·K)
• c_pw = specific heat of water vapor = 0.444 Btu/lb·°F (1.86 kJ/kg·K)
• T = dry-bulb temperature (°F or °C)
• h_fg = latent heat of vaporization at 0°F ≈ 1061 Btu/lb (2501 kJ/kg at 0°C)

Specific Volume and Density

Specific Volume of Moist Air: \[v = \frac{R_a \cdot T}{p - p_w} = \frac{53.35(T + 460)}{144 \cdot p} (1 + 1.6078W)\]

• v = specific volume (ft³/lb_da or m³/kg_da)
• R_a = gas constant for dry air = 53.35 ft·lbf/lb·°R (287 J/kg·K)
• T = absolute temperature (°R or K)
• For US units: T in °R = °F + 460, p in psia

Density of Moist Air: \[\rho = \frac{1}{v} = \frac{1 + W}{v}\]

• ρ = density of moist air (lb/ft³ or kg/m³)

Wet-Bulb Temperature

Relationship (approximate): \[W_s - W = \frac{c_p(T - T_{wb})}{h_{fg}}\]

• T_wb = wet-bulb temperature (°F or °C)
• W_s = saturation humidity ratio at wet-bulb temperature
• Typically found using psychrometric chart or tables

Cooling and Heating Load Calculations

Sensible Heat Transfer

Sensible Heat Load: \[q_s = \dot{m}_a \cdot c_p \cdot \Delta T = 1.08 \cdot Q \cdot \Delta T\]

• q_s = sensible heat load (Btu/hr or kW)
• ṁ_a = mass flow rate of dry air (lb_da/hr or kg_da/s)
• Q = volumetric flow rate (cfm or L/s)
• ΔT = temperature difference (°F or °C)
• 1.08 = conversion factor for standard air (60 × 0.075 × 0.24)
• For SI units: q_s = 1.23 × Q × ΔT (Q in L/s, result in watts)

Alternative Form: \[q_s = 60 \cdot \rho \cdot Q \cdot c_p \cdot \Delta T\]

• ρ = air density (lb/ft³ or kg/m³)
• 60 = conversion factor (minutes per hour)

Latent Heat Transfer

Latent Heat Load: \[q_l = \dot{m}_a \cdot h_{fg} \cdot \Delta W = 4840 \cdot Q \cdot \Delta W\]

• q_l = latent heat load (Btu/hr or kW)
• ΔW = humidity ratio difference (lb_w/lb_da or kg_w/kg_da)
• 4840 = conversion factor (60 × 0.075 × 1076)
• For SI units: q_l = 3010 × Q × ΔW (Q in L/s, result in watts)

Alternative using moisture addition rate: \[q_l = \dot{m}_w \cdot h_{fg}\]

• ṁ_w = rate of moisture addition (lb/hr or kg/s)

Total Heat Transfer

Total Heat Load: \[q_t = q_s + q_l = \dot{m}_a \cdot \Delta h = 4.45 \cdot Q \cdot \Delta h\]

• q_t = total heat load (Btu/hr or kW)
• Δh = enthalpy difference (Btu/lb_da or kJ/kg_da)
• 4.45 = conversion factor (60 × 0.075)
• For SI units: q_t = 1.2 × Q × Δh (Q in L/s, result in watts)

Sensible Heat Ratio

Sensible Heat Ratio (SHR): \[SHR = \frac{q_s}{q_t} = \frac{q_s}{q_s + q_l}\]

• SHR = sensible heat ratio (dimensionless)
• 0 ≤ SHR ≤ 1
• SHR = 1 indicates purely sensible load
• SHR = 0 indicates purely latent load

Building Load Components

Conduction Heat Gain/Loss: \[q = U \cdot A \cdot \Delta T\]

• q = heat transfer rate (Btu/hr or W)
• U = overall heat transfer coefficient (Btu/hr·ft²·°F or W/m²·K)
• A = area (ft² or m²)
• ΔT = temperature difference (°F or K)

Solar Heat Gain: \[q_{solar} = A \cdot SHGC \cdot SHGF\]

• SHGC = solar heat gain coefficient (dimensionless)
• SHGF = solar heat gain factor (Btu/hr·ft² or W/m²)

Alternative using SC: \[q_{solar} = A \cdot SC \cdot SCL\]

• SC = shading coefficient (dimensionless)
• SCL = solar cooling load (Btu/hr·ft² or W/m²)

Internal Heat Gains:

• People: q = N × q_person (sensible and latent components)
• Lighting: q = W × F_u × F_sa
• Equipment: q = P × η (power × usage factor)

Infiltration/Ventilation Load: \[q_{inf} = \rho \cdot Q_{inf} \cdot c_p \cdot \Delta T + \rho \cdot Q_{inf} \cdot h_{fg} \cdot \Delta W\]

• Q_inf = infiltration air flow rate (cfm or L/s)

Air Mixing Processes

Mixing of Air Streams

Conservation of Mass (Dry Air): \[\dot{m}_{a,3} = \dot{m}_{a,1} + \dot{m}_{a,2}\]

• ṁ_a,3 = mass flow rate of mixed air stream
• ṁ_a,1, ṁ_a,2 = mass flow rates of inlet streams

Conservation of Moisture: \[\dot{m}_{a,3} \cdot W_3 = \dot{m}_{a,1} \cdot W_1 + \dot{m}_{a,2} \cdot W_2\] Conservation of Energy: \[\dot{m}_{a,3} \cdot h_3 = \dot{m}_{a,1} \cdot h_1 + \dot{m}_{a,2} \cdot h_2\] Mixed State Properties: \[W_3 = \frac{\dot{m}_{a,1} \cdot W_1 + \dot{m}_{a,2} \cdot W_2}{\dot{m}_{a,1} + \dot{m}_{a,2}}\] \[h_3 = \frac{\dot{m}_{a,1} \cdot h_1 + \dot{m}_{a,2} \cdot h_2}{\dot{m}_{a,1} + \dot{m}_{a,2}}\] \[T_3 = \frac{\dot{m}_{a,1} \cdot T_1 + \dot{m}_{a,2} \cdot T_2}{\dot{m}_{a,1} + \dot{m}_{a,2}}\] Bypass Factor for Mixing: \[BF = \frac{\dot{m}_{bypass}}{\dot{m}_{total}}\]

• Used in coil analysis where some air bypasses the coil

Cooling Coil Performance

Coil Load Calculations

Total Cooling Capacity: \[q_c = \dot{m}_a(h_1 - h_2)\]

• q_c = total cooling capacity (Btu/hr or kW)
• h₁ = entering air enthalpy
• h₂ = leaving air enthalpy

Sensible Cooling Capacity: \[q_{c,s} = \dot{m}_a \cdot c_p(T_1 - T_2)\] Latent Cooling Capacity: \[q_{c,l} = \dot{m}_a \cdot h_{fg}(W_1 - W_2)\]

Apparatus Dew Point (ADP)

Coil Bypass Factor: \[BF = \frac{T_2 - T_{ADP}}{T_1 - T_{ADP}}\]

• T_ADP = apparatus dew point temperature
• T₁ = entering air temperature
• T₂ = leaving air temperature
• BF represents inefficiency of the coil

Contact Factor: \[CF = 1 - BF = \frac{T_1 - T_2}{T_1 - T_{ADP}}\]

Mean Temperature Difference

Log Mean Temperature Difference (LMTD): \[\Delta T_{lm} = \frac{(T_{a,in} - T_{w,out}) - (T_{a,out} - T_{w,in})}{\ln\left(\frac{T_{a,in} - T_{w,out}}{T_{a,out} - T_{w,in}}\right)}\]

• T_a,in = entering air temperature
• T_a,out = leaving air temperature
• T_w,in = entering water temperature
• T_w,out = leaving water temperature

Heat Transfer: \[q = U \cdot A \cdot \Delta T_{lm}\]

• U = overall heat transfer coefficient
• A = heat transfer surface area

Heating Processes

Heating Coil Performance

Sensible Heating: \[q_h = \dot{m}_a \cdot c_p(T_2 - T_1) = 1.08 \cdot Q \cdot (T_2 - T_1)\]

• q_h = heating load (Btu/hr or kW)
• Humidity ratio remains constant for sensible heating (W₂ = W₁)

Steam or Hot Water Heating Coil: \[q_h = \dot{m}_w \cdot c_{p,w} \cdot \Delta T_w\]

• ṁ_w = water flow rate (lb/hr or kg/s)
• c_p,w = specific heat of water = 1.0 Btu/lb·°F (4.18 kJ/kg·K)
• ΔT_w = water temperature drop

Humidification

Steam Injection (Isothermal Humidification): \[\dot{m}_s = \dot{m}_a(W_2 - W_1)\]

• ṁ_s = steam injection rate (lb/hr or kg/s)
• Temperature remains approximately constant

Adiabatic Humidification (Evaporative Cooling): \[h_2 \approx h_1\]

• Enthalpy remains constant (follows constant wet-bulb line)
• Temperature decreases as humidity increases

Water Spray Humidifier: \[\dot{m}_w = \dot{m}_a(W_2 - W_1)\]

• Process follows constant wet-bulb temperature line if adiabatic

Fan and Duct Systems

Fan Power and Efficiency

Fan Power (Air Power): \[P_{air} = \frac{Q \cdot \Delta p}{6356 \cdot \eta_f}\]

• P_air = fan power (hp)
• Q = volumetric flow rate (cfm)
• Δp = pressure rise across fan (in. w.g.)
• η_f = fan total efficiency (decimal)
• 6356 = conversion constant for US units

Alternative (SI Units): \[P_{air} = \frac{Q \cdot \Delta p}{1000 \cdot \eta_f}\]

• P_air = fan power (kW)
• Q = flow rate (m³/s)
• Δp = pressure rise (Pa)

Motor Power: \[P_{motor} = \frac{P_{air}}{\eta_m}\]

• η_m = motor efficiency (decimal)

Fan Total Efficiency: \[\eta_f = \eta_{static} \times \frac{\Delta p_{total}}{\Delta p_{static}}\]

• η_static = static efficiency
• Δp_total = total pressure rise
• Δp_static = static pressure rise

Duct Pressure Losses

Friction Loss (Darcy-Weisbach): \[\Delta p_f = f \cdot \frac{L}{D} \cdot \frac{\rho \cdot v^2}{2}\]

• Δp_f = friction pressure loss (lbf/ft² or Pa)
• f = friction factor (dimensionless)
• L = duct length (ft or m)
• D = duct diameter (ft or m)
• ρ = air density (lb/ft³ or kg/m³)
• v = air velocity (ft/s or m/s)

Velocity Pressure: \[p_v = \frac{\rho \cdot v^2}{2} = \left(\frac{v}{4005}\right)^2\]

• p_v = velocity pressure (in. w.g.)
• v = velocity (fpm)
• 4005 = conversion constant for standard air

Total Pressure: \[p_t = p_s + p_v\]

• p_t = total pressure
• p_s = static pressure

Dynamic Loss (Fittings): \[\Delta p_d = C \cdot p_v\]

• C = loss coefficient (dimensionless)
• Values from ASHRAE tables for specific fittings

Equivalent Duct Diameter

Rectangular to Round (Equal Friction): \[D_e = 1.30 \frac{(ab)^{0.625}}{(a+b)^{0.25}}\]

• D_e = equivalent round diameter (in. or mm)
• a, b = rectangular duct sides (in. or mm)

Hydraulic Diameter: \[D_h = \frac{4A}{P}\]

• D_h = hydraulic diameter
• A = cross-sectional area
• P = perimeter

Refrigeration Cycles

Vapor Compression Cycle

Coefficient of Performance (Cooling): \[COP_c = \frac{q_e}{W_{comp}} = \frac{h_1 - h_4}{h_2 - h_1}\]

• COP_c = coefficient of performance for cooling
• q_e = refrigeration effect (Btu/lb or kJ/kg)
• W_comp = compressor work (Btu/lb or kJ/kg)
• h₁ = enthalpy at compressor inlet
• h₂ = enthalpy at compressor outlet
• h₄ = enthalpy at evaporator inlet

Coefficient of Performance (Heating): \[COP_h = \frac{q_c}{W_{comp}} = \frac{h_2 - h_3}{h_2 - h_1}\]

• q_c = condenser heat rejection
• h₃ = enthalpy at condenser outlet

Energy Balance Relationship: \[COP_h = COP_c + 1\] Refrigeration Effect: \[q_e = h_1 - h_4\]

• Heat absorbed in evaporator per unit mass of refrigerant

Compressor Work: \[W_{comp} = h_2 - h_1\]

• For isentropic compression: s₂ = s₁

Isentropic Efficiency: \[\eta_{isen} = \frac{h_{2s} - h_1}{h_2 - h_1}\]

• h_2s = enthalpy at compressor outlet for isentropic process
• h₂ = actual enthalpy at compressor outlet

Condenser Heat Rejection: \[q_c = h_2 - h_3\]

Refrigeration Capacity

Cooling Capacity: \[Q_{ref} = \dot{m}_r(h_1 - h_4)\]

• Q_ref = refrigeration capacity (Btu/hr or tons or kW)
• ṁ_r = refrigerant mass flow rate (lb/hr or kg/s)
• 1 ton = 12,000 Btu/hr = 3.517 kW

Compressor Power: \[P_{comp} = \dot{m}_r(h_2 - h_1)\] Energy Efficiency Ratio (EER): \[EER = \frac{Q_{ref}(Btu/hr)}{P_{comp}(W)}\]

• Units: Btu/hr per watt
• Higher values indicate better efficiency

Seasonal Energy Efficiency Ratio (SEER): \[SEER = \frac{\text{Total cooling output (Btu)}}{\text{Total electrical input (Wh)}}\]

Carnot Refrigeration Cycle

Carnot COP (Cooling): \[COP_{Carnot,c} = \frac{T_L}{T_H - T_L}\]

• T_L = absolute temperature of low-temperature reservoir (°R or K)
• T_H = absolute temperature of high-temperature reservoir (°R or K)

Carnot COP (Heating): \[COP_{Carnot,h} = \frac{T_H}{T_H - T_L}\]

Evaporative Cooling

Direct Evaporative Cooling

Saturation Effectiveness: \[\varepsilon = \frac{T_1 - T_2}{T_1 - T_{wb,1}}\]

• ε = saturation effectiveness (decimal)
• T₁ = entering air dry-bulb temperature
• T₂ = leaving air dry-bulb temperature
• T_wb,1 = entering air wet-bulb temperature
• Process follows constant wet-bulb line on psychrometric chart

Water Evaporation Rate: \[\dot{m}_w = \dot{m}_a(W_2 - W_1)\]

Indirect Evaporative Cooling

Wet-Bulb Effectiveness: \[\varepsilon_{wb} = \frac{T_1 - T_2}{T_1 - T_{wb,outdoor}}\]

• No moisture is added to the process air stream
• Temperature reduction without humidity increase

Variable Air Volume (VAV) Systems

VAV Box Control

Supply Air Flow: \[Q = \frac{q_s}{1.08 \cdot (T_s - T_r)}\]

• Q = required supply air flow (cfm)
• T_s = supply air temperature
• T_r = room temperature

Minimum Flow Ratio: \[\text{Min Flow} = \frac{Q_{vent,min}}{Q_{design}}\]

• Q_vent,min = minimum ventilation requirement
• Q_design = design maximum flow

Fan Energy in VAV

Fan Affinity Laws: \[\frac{Q_2}{Q_1} = \frac{N_2}{N_1}\] \[\frac{\Delta p_2}{\Delta p_1} = \left(\frac{N_2}{N_1}\right)^2\] \[\frac{P_2}{P_1} = \left(\frac{N_2}{N_1}\right)^3\]

• Q = volumetric flow rate
• N = fan speed (rpm)
• Δp = pressure rise
• P = power
• Valid for same fan and fluid density

Ventilation and Indoor Air Quality

Outdoor Air Requirements

Ventilation Rate Procedure: \[V_{ot} = \sum_{z} V_{oz}\]

• V_ot = total outdoor air flow required
• V_oz = outdoor air flow for zone z

Zone Outdoor Air Flow: \[V_{oz} = R_p \cdot P_z + R_a \cdot A_z\]

• R_p = outdoor air flow rate per person (cfm/person)
• P_z = zone population
• R_a = outdoor air flow rate per unit area (cfm/ft²)
• A_z = zone floor area (ft²)

Air Change Rate

Air Changes per Hour: \[ACH = \frac{60 \cdot Q}{V}\]

• ACH = air changes per hour (hr⁻¹)
• Q = volumetric flow rate (cfm or m³/min)
• V = room volume (ft³ or m³)
• 60 = minutes per hour

Contaminant Dilution

Steady-State Concentration: \[C_{ss} = \frac{G}{Q} + C_o\]

• C_ss = steady-state concentration
• G = contaminant generation rate (mass/time)
• Q = ventilation flow rate (volume/time)
• C_o = outdoor air contaminant concentration

Transient Concentration (Decay): \[C(t) = C_o + (C_i - C_o)e^{-Qt/V}\]

• C(t) = concentration at time t
• C_i = initial indoor concentration
• t = time

Economizer Operation

Economizer Control Strategies

Dry-Bulb Economizer:

• Use 100% outdoor air when T_outdoor <>_return
• Simple control but may bring in high humidity

Enthalpy Economizer:

• Use 100% outdoor air when h_outdoor <>_return
• More energy-efficient control considering both temperature and humidity

Mixed Air Temperature: \[T_{mixed} = \frac{Q_{oa} \cdot T_{oa} + Q_{ra} \cdot T_{ra}}{Q_{oa} + Q_{ra}}\]

• Q_oa = outdoor air flow rate
• Q_ra = return air flow rate
• T_oa = outdoor air temperature
• T_ra = return air temperature

Outdoor Air Fraction: \[X = \frac{Q_{oa}}{Q_{sa}} = \frac{T_{mixed} - T_{ra}}{T_{oa} - T_{ra}}\]

• X = outdoor air fraction (decimal)
• Q_sa = supply air flow rate

Heat Recovery Systems

Heat Exchanger Effectiveness

Sensible Effectiveness: \[\varepsilon_s = \frac{T_{oa,leaving} - T_{oa,entering}}{T_{ea,entering} - T_{oa,entering}}\]

• ε_s = sensible effectiveness
• T_oa,leaving = outdoor air temperature leaving heat exchanger
• T_oa,entering = outdoor air temperature entering heat exchanger
• T_ea,entering = exhaust air temperature entering heat exchanger

Latent Effectiveness: \[\varepsilon_l = \frac{W_{oa,leaving} - W_{oa,entering}}{W_{ea,entering} - W_{oa,entering}}\]

• ε_l = latent effectiveness

Total (Enthalpy) Effectiveness: \[\varepsilon_t = \frac{h_{oa,leaving} - h_{oa,entering}}{h_{ea,entering} - h_{oa,entering}}\] Recovered Heat: \[q_{recovered} = \varepsilon \cdot \dot{m}_a \cdot c_p \cdot (T_{ea} - T_{oa})\]

Cooling Towers

Cooling Tower Performance

Range: \[\text{Range} = T_{w,in} - T_{w,out}\]

• T_w,in = hot water inlet temperature
• T_w,out = cold water outlet temperature
• Measure of tower load

Approach: \[\text{Approach} = T_{w,out} - T_{wb,air}\]

• T_wb,air = entering air wet-bulb temperature
• Measure of tower effectiveness

Cooling Tower Effectiveness: \[\varepsilon_{ct} = \frac{T_{w,in} - T_{w,out}}{T_{w,in} - T_{wb,air}} = \frac{\text{Range}}{\text{Range} + \text{Approach}}\] Heat Rejection: \[q_{ct} = \dot{m}_w \cdot c_{p,w} \cdot (T_{w,in} - T_{w,out})\]

• q_ct = heat rejection rate (Btu/hr or kW)
• ṁ_w = water flow rate (lb/hr or kg/s)

Evaporation Rate: \[\dot{m}_{evap} \approx 0.001 \times \dot{m}_w \times \text{Range}\]

• Approximate formula for evaporation loss
• ṁ_evap = evaporation rate (lb/hr or kg/s)
• Range in °F (multiply by 0.0008 for °C)

Makeup Water: \[\dot{m}_{makeup} = \dot{m}_{evap} + \dot{m}_{drift} + \dot{m}_{blowdown}\]

• ṁ_drift = drift loss (typically 0.1-0.2% of circulation rate)
• ṁ_blowdown = blowdown for water quality control

Liquid-to-Air Ratio

L/G Ratio: \[\frac{L}{G} = \frac{\dot{m}_w}{\dot{m}_a}\]

• L = liquid (water) flow rate
• G = gas (air) flow rate
• Affects tower size and performance

Chillers and Heat Pumps

Chiller Performance

Chiller Capacity: \[Q_{chiller} = \dot{m}_{chw} \cdot c_{p,w} \cdot \Delta T_{chw}\]

• Q_chiller = chiller cooling capacity (Btu/hr or tons or kW)
• ṁ_chw = chilled water flow rate (lb/hr or kg/s)
• ΔT_chw = chilled water temperature difference (typically 10-12°F or 5-7°C)

Alternative Form: \[Q_{chiller} = 500 \times GPM \times \Delta T_{chw}\]

• GPM = gallons per minute (water flow)
• ΔT_chw = temperature difference (°F)
• Result in Btu/hr (divide by 12,000 for tons)

Kilowatts per Ton (kW/ton): \[\text{kW/ton} = \frac{P_{comp}(kW)}{Q_{chiller}(tons)}\]

• Lower values indicate better efficiency
• Typical range: 0.5-0.8 kW/ton for centrifugal chillers

Condenser Heat Rejection: \[Q_{cond} = Q_{evap} + P_{comp}\]

• Q_cond = condenser heat rejection
• Q_evap = evaporator cooling load

Part Load Performance

Part Load Ratio (PLR): \[PLR = \frac{Q_{actual}}{Q_{design}}\]

• Used in performance curves

Integrated Part Load Value (IPLV): \[IPLV = 0.01A + 0.42B + 0.45C + 0.12D\]

• A, B, C, D = EER or COP at 100%, 75%, 50%, 25% load
• Weighted average efficiency metric

Pumping Systems

Pump Power and Head

Pump Head: \[H = \frac{\Delta p}{\rho \cdot g}\]

• H = pump head (ft or m)
• Δp = pressure rise (lbf/ft² or Pa)
• ρ = fluid density (lb/ft³ or kg/m³)
• g = gravitational acceleration (32.2 ft/s² or 9.81 m/s²)

Pump Power (Water Horsepower): \[WHP = \frac{Q \cdot H \cdot \rho}{33000}\]

• WHP = water horsepower (hp)
• Q = flow rate (ft³/min)
• H = total head (ft)
• 33,000 = conversion constant (ft·lbf/min per hp)

Alternative Form (GPM): \[WHP = \frac{GPM \times H \times SG}{3960}\]

• GPM = gallons per minute
• SG = specific gravity

Brake Horsepower: \[BHP = \frac{WHP}{\eta_p}\]

• BHP = brake horsepower
• η_p = pump efficiency (decimal)

Motor Power: \[P_{motor} = \frac{BHP}{\eta_m}\]

System Head Curve

Total Head: \[H_{total} = H_{static} + H_{friction}\] \[H_{total} = H_{static} + K \cdot Q^2\]

• H_static = static head (elevation difference)
• H_friction = friction head loss
• K = system resistance coefficient

Pump Affinity Laws

For Same Impeller Diameter: \[\frac{Q_2}{Q_1} = \frac{N_2}{N_1}\] \[\frac{H_2}{H_1} = \left(\frac{N_2}{N_1}\right)^2\] \[\frac{P_2}{P_1} = \left(\frac{N_2}{N_1}\right)^3\] For Same Speed: \[\frac{Q_2}{Q_1} = \frac{D_2}{D_1}\] \[\frac{H_2}{H_1} = \left(\frac{D_2}{D_1}\right)^2\] \[\frac{P_2}{P_1} = \left(\frac{D_2}{D_1}\right)^3\]

• D = impeller diameter

Pipe and Piping System Losses

Friction Losses in Pipes

Darcy-Weisbach Equation: \[h_f = f \cdot \frac{L}{D} \cdot \frac{v^2}{2g}\]

• h_f = head loss due to friction (ft or m)
• f = friction factor (dimensionless)
• L = pipe length (ft or m)
• D = pipe diameter (ft or m)
• v = flow velocity (ft/s or m/s)
• g = gravitational acceleration

Hazen-Williams Equation: \[h_f = \frac{4.52 \times L \times Q^{1.85}}{C^{1.85} \times D^{4.87}}\]

• C = Hazen-Williams coefficient (120-140 for typical water pipes)
• Q = flow rate (gpm)
• D = pipe diameter (in.)
• h_f = head loss (ft)
• Valid for water flow only

Velocity in Pipe: \[v = \frac{Q}{A} = \frac{4Q}{\pi D^2}\] For GPM and inches: \[v = \frac{0.4085 \times GPM}{D^2}\]

• v = velocity (ft/s)
• D = diameter (inches)

Minor Losses

Loss Coefficient Method: \[h_m = K \cdot \frac{v^2}{2g}\]

• h_m = minor head loss (ft or m)
• K = loss coefficient (dimensionless, from tables)

Equivalent Length Method: \[L_e = K \cdot \frac{D}{f}\]

• L_e = equivalent length of straight pipe
• Add to actual pipe length for total friction calculation

Thermal Energy Storage

Storage Capacity

Sensible Heat Storage: \[Q_{storage} = m \cdot c_p \cdot \Delta T\]

• Q_storage = storage capacity (Btu or kJ)
• m = mass of storage medium (lb or kg)
• ΔT = temperature change (°F or K)

Latent Heat Storage (Ice): \[Q_{storage} = m_{ice} \cdot h_{fusion}\]

• m_ice = mass of ice (lb or kg)
• h_fusion = latent heat of fusion = 144 Btu/lb (334 kJ/kg)

Ton-Hours: \[Q_{storage}(\text{ton-hr}) = \frac{Q_{storage}(\text{Btu})}{12000}\]

• 1 ton-hr = 12,000 Btu

Discharge Rate

Available Cooling: \[\dot{Q}_{discharge} = \frac{Q_{storage}}{\Delta t}\]

• Δt = discharge time period

Energy Recovery and Efficiency

Annual Energy Usage

Annual Energy Consumption: \[E_{annual} = P_{avg} \times h_{annual}\]

• E_annual = annual energy consumption (kWh or Btu)
• P_avg = average power consumption
• h_annual = annual operating hours

Equivalent Full Load Hours: \[EFLH = \frac{E_{annual}}{P_{design}}\]

• EFLH = equivalent full load hours
• P_design = design (peak) power

Heat Pipe Effectiveness

Effectiveness: \[\varepsilon = \frac{T_{supply,leaving} - T_{supply,entering}}{T_{exhaust,entering} - T_{supply,entering}}\]

• For air-to-air heat pipe heat exchanger

Boilers and Steam Systems

Boiler Efficiency

Combustion Efficiency: \[\eta_{comb} = \frac{Q_{output}}{Q_{fuel}} = \frac{\dot{m}_{steam}(h_{steam} - h_{feedwater})}{\dot{m}_{fuel} \times HHV}\]

• η_comb = combustion efficiency
• Q_output = useful heat output
• Q_fuel = heat input from fuel
• HHV = higher heating value of fuel

Stack Loss: \[Q_{stack} = \dot{m}_{flue} \times c_{p,flue} \times (T_{stack} - T_{ambient})\]

• Major component of boiler heat loss

Steam Properties and Flow

Steam Flow Rate for Heating: \[\dot{m}_{steam} = \frac{Q}{h_{fg}}\]

• For steam heating applications where condensate forms

Condensate Return: \[\dot{m}_{condensate} = \dot{m}_{steam}\]

• Mass balance for closed steam systems

Control Systems and Sequences

Proportional Control

Proportional Band: \[PB = \frac{100}{\text{Gain}}\]

• PB = proportional band (%)
• Range of input change for full output change

Control Output: \[CO = K_p \times (SP - PV) + CO_{bias}\]

• CO = control output
• K_p = proportional gain
• SP = setpoint
• PV = process variable
• CO_bias = bias or offset

Reset Control

Supply Air Temperature Reset: \[T_{sa,reset} = T_{sa,design} + K \times (T_{oa} - T_{oa,design})\]

• T_sa,reset = reset supply air temperature
• K = reset ratio (slope)
• T_oa = outdoor air temperature

Piping System Design

Expansion and Contraction

Thermal Expansion: \[\Delta L = \alpha \times L \times \Delta T\]

• ΔL = change in length (ft or m)
• α = coefficient of linear thermal expansion (in/in·°F or m/m·K)
• L = original length (ft or m)
• ΔT = temperature change (°F or K)
• For steel: α ≈ 6.5 × 10⁻⁶ in/in·°F

System Fill and Expansion Tank

Expansion Tank Volume (Closed System): \[V_t = \frac{V_s \times \Delta v}{1 - \frac{P_1}{P_2}}\]

• V_t = expansion tank volume
• V_s = system water volume
• Δv = change in specific volume of water
• P₁ = fill pressure
• P₂ = maximum operating pressure

Net Positive Suction Head (NPSH): \[NPSH_a = h_s + h_p - h_{vp} - h_f\]

• NPSH_a = available net positive suction head
• h_s = static head on suction
• h_p = pressure head on liquid surface
• h_vp = vapor pressure head
• h_f = friction head loss in suction piping
• Must have NPSH_a > NPSH_r (required) to avoid cavitation