Formula sheet

Building Energy Systems and HVAC Performance Formula Sheet

Building HVAC formulas for envelope loads, ventilation, infiltration, sensible/latent loads, fans, pumps, COP, economizers, heat recovery, demand response, and validation.

Branch: Energy Engineering
Content: Formula sheet
Updated: Jun 23, 2026
Revision: v1.0.0 · reviewed

This formula sheet collects first-pass calculations used to evaluate building energy systems and HVAC performance. It focuses on engineering decisions: load estimates, service boundaries, envelope heat transfer, ventilation, infiltration, air and water flow, fan and pump energy, heat-pump performance, economizers, heat recovery, demand response, commissioning tests, and measurement uncertainty.

Use these equations for screening, design review, commissioning checks, fault diagnosis, and energy-performance validation. Detailed building design also requires climate data, occupancy schedules, equipment data, humidity analysis, indoor-air-quality requirements, envelope details, controls sequences, safety review, maintainability review, and local professional judgment.

Basis and Boundaries

State the boundary before calculating:

whole building, zone, air-handling unit, plant room, hydronic loop, tenant space, or campus;
indoor temperature, humidity, ventilation, pressure, filtration, and comfort requirements;
weather condition, occupancy schedule, internal loads, operating mode, and service constraint;
whether pumps, fans, controls, standby loads, auxiliary heat, heat recovery, or distribution losses are inside the energy boundary;
whether the calculation supports sizing, energy reporting, demand response, commissioning, troubleshooting, or release.

Energy savings are meaningful only if the same useful service is delivered. Lower air flow, lower heating, or lower cooling is not an improvement if it causes poor air quality, discomfort, condensation, pressure imbalance, freeze risk, or process failure.

Symbols

Symbol	Meaning	Typical unit
$\dot{Q}$	heat transfer rate or load	kW
$Q_v$	volumetric flow rate	m3/s
$\dot{m}$	mass flow rate	kg/s
$\rho$	fluid density	kg/m3
$C_p$	specific heat capacity	kJ/(kg K)
$U$	overall heat-transfer coefficient	W/(m2 K)
$A$	area	m2
$\Delta T$	temperature difference	K
$W$	humidity ratio	kg water/kg dry air
$h$	moist-air enthalpy or heat-transfer coefficient depending on context	kJ/kg or W/(m2 K)
$P$	electrical or mechanical power	kW
$\eta$	efficiency	dimensionless
$COP$	coefficient of performance	dimensionless
$U_x$	expanded uncertainty of quantity $x$	same as $x$

Keep units visible. Building calculations often mix airflow, hydronic flow, heat, power, time, weather, and utility data. Unit mistakes can look like energy savings or hidden loads.

Envelope Heat Transfer

Steady conductive heat transfer through an envelope element:

\dot{Q}_{env}=UA(T_{out}-T_{in})

For multiple elements:

\dot{Q}_{env,total}=\sum_i U_iA_i(T_{out}-T_{in})

Envelope conductance:

H_{env}=\sum_i U_iA_i

Then:

\dot{Q}_{env}=H_{env}\Delta T

Use this as a screening relation. Real envelope loads also depend on thermal bridges, solar gain, wind pressure, infiltration, moisture, thermal mass, ground contact, shading, orientation, and construction quality.

Solar and Internal Gains

Solar heat gain through glazing:

\dot{Q}_{solar}=A_{glass}SHGC I_s F_{shade}

where $SHGC$ is solar heat gain coefficient, $I_s$ is incident solar irradiance, and $F_{shade}$ is a shading or availability factor.

Internal sensible load:

\dot{Q}_{internal}=\dot{Q}_{people}+\dot{Q}_{lighting}+\dot{Q}_{equipment}+\dot{Q}_{process}

Lighting power density relation:

P_{lighting}=LPD\cdot A_{floor}

Schedule-weighted internal load:

\dot{Q}_{avg}=\sum_i f_i\dot{Q}_i

where $f_i$ is the fraction of time or operating condition. Peak load and annual energy can be controlled by different cases.

Ventilation and Outdoor-Air Load

Air mass flow:

\dot{m}_{air}=\rho Q_v

Sensible ventilation load:

\dot{Q}_{vent,s}=\rho C_p Q_v(T_{out}-T_{in})

Total outdoor-air load using moist-air enthalpy:

\dot{Q}_{vent,total}=\rho Q_v(h_{out}-h_{in})

Latent load approximation:

\dot{Q}_{lat}\approx \rho Q_v h_{fg}(W_{out}-W_{in})

where $h_{fg}$ is latent heat of vaporization and $W$ is humidity ratio.

Ventilation energy must be interpreted with air-quality and pressure requirements. Reducing outdoor air is not a valid saving if required ventilation, exhaust makeup, humidity control, or pressure relationship is violated.

Infiltration

Air changes per hour:

\displaystyle ACH=\frac{3600Q_{inf}}{V_{zone}}

Infiltration flow from ACH:

\displaystyle Q_{inf}=\frac{ACH\cdot V_{zone}}{3600}

Sensible infiltration load:

\dot{Q}_{inf,s}=\rho C_p Q_{inf}(T_{out}-T_{in})

Infiltration is uncontrolled outdoor air. It can increase heating, cooling, humidity, drafts, pressure instability, and pollutant transport. Air leakage can also make HVAC controls appear weak even when equipment capacity is nominally sufficient.

Heating and Cooling Load Balance

Cooling load screening:

\dot{Q}_{cool}=\dot{Q}_{env}+\dot{Q}_{solar}+\dot{Q}_{internal}+\dot{Q}_{vent}+\dot{Q}_{inf}-\dot{Q}_{useful\ recovery}

Heating load screening:

\dot{Q}_{heat}=\dot{Q}_{env}+\dot{Q}_{vent}+\dot{Q}_{inf}-\dot{Q}_{internal,useful}-\dot{Q}_{solar,useful}-\dot{Q}_{recovery}

Peak design load:

\dot{Q}_{design}=\dot{Q}_{load,case}f_{growth}f_{degradation}f_{reserve}

Keep design load and operating energy separate. A system may need high peak capacity for rare weather while spending most hours at part load.

Degree-Day Energy Screening

Heating degree-day energy estimate:

\displaystyle E_{heat}\approx \frac{H_{loss}\cdot HDD\cdot 24}{\eta_{heat}}

Cooling degree-day energy estimate:

\displaystyle E_{cool}\approx \frac{H_{gain}\cdot CDD\cdot 24}{COP_{cool}}

where $H_{loss}$ or $H_{gain}$ is an equivalent building heat-transfer coefficient on the same temperature basis as the degree days.

Degree-day methods are useful for annual screening and weather normalization. They are weak for humidity, solar gain, occupancy variation, part-load performance, economizers, thermal mass, and control faults unless corrected with measured data.

Air-Side Heat Transfer

Sensible load from airflow:

\dot{Q}_s=\rho C_p Q_v(T_{return}-T_{supply})

Required airflow for a sensible load:

\displaystyle Q_v=\frac{\dot{Q}_s}{\rho C_p\Delta T}

Zone temperature rise under fixed airflow:

\displaystyle \Delta T=\frac{\dot{Q}_s}{\rho C_p Q_v}

This is a heat-balance relation, not a full air-distribution design. Real performance depends on diffuser placement, stratification, bypass, return location, leakage, fan curves, filters, balancing, and controls.

Hydronic Heating and Cooling

Water-side heat transfer:

\dot{Q}=\dot{m}C_p(T_{out}-T_{in})

Volumetric-flow relation:

\dot{m}=\rho Q_v

Required hydronic flow:

\displaystyle Q_v=\frac{\dot{Q}}{\rho C_p\Delta T}

Coil or heat-exchanger heat-balance mismatch:

\displaystyle \epsilon_Q=\frac{\dot{Q}_{air}-\dot{Q}_{water}}{(\dot{Q}_{air}+\dot{Q}_{water})/2}

A large mismatch can indicate sensor error, poor mixing, air leakage, condensate effects, heat loss, wrong properties, bypassing, or unstable operation.

Fan and Pump Power

Fan or pump power:

\displaystyle P=\frac{\Delta P Q_v}{\eta}

where $\Delta P$ is pressure rise, $Q_v$ is volumetric flow, and $\eta$ is total efficiency.

Affinity laws for similar operation:

\displaystyle \frac{Q_2}{Q_1}=\frac{N_2}{N_1}

\displaystyle \frac{\Delta P_2}{\Delta P_1}=\left(\frac{N_2}{N_1}\right)^2

\displaystyle \frac{P_2}{P_1}=\left(\frac{N_2}{N_1}\right)^3

These relations explain why variable-speed control can save energy, but they assume similar system behaviour. Minimum flow, controls, coil performance, valve authority, duct leakage, and static-pressure reset can change the real result.

Heat-Pump and Cooling COP

Coefficient of performance:

\displaystyle COP=\frac{\dot{Q}_{useful}}{P_{input}}

Cooling electric power:

\displaystyle P_{cooling}=\frac{\dot{Q}_{cooling}}{COP_{cooling}}

Heating electric power:

\displaystyle P_{heating}=\frac{\dot{Q}_{heating}}{COP_{heating}}

Auxiliary heat fraction:

\displaystyle f_{aux}=\frac{\dot{Q}_{aux}}{\dot{Q}_{total}}

Report the boundary: compressor only, packaged unit, heat-pump system, plant, or whole building. COP changes with source temperature, sink temperature, part load, defrost, fans, pumps, controls, refrigerant charge, coil fouling, and auxiliary heat.

Economizer Mixed-Air Balance

Mixed-air temperature:

T_{mix}=f_{OA}T_{OA}+(1-f_{OA})T_{RA}

Outdoor-air fraction from measured mixed-air temperature:

\displaystyle f_{OA}=\frac{T_{RA}-T_{mix}}{T_{RA}-T_{OA}}

Outdoor-air fraction required for free cooling to a supply-air target:

\displaystyle f_{OA,req}=\frac{T_{RA}-T_{SA,set}}{T_{RA}-T_{OA}}

Economizer coil-load penalty:

\dot{Q}_{penalty}=\rho C_p Q_v(T_{mix}-T_{SA,set})

Mixed-air calculations are sensitive to sensor placement and stratification, but they are powerful diagnostics when the inferred fraction is far from the commanded damper position.

Heat Recovery Effectiveness

Air-side heat recovery effectiveness for heating:

\displaystyle \epsilon_{HR}=\frac{T_{supply,after}-T_{outdoor}}{T_{return}-T_{outdoor}}

Recovered sensible heat:

\dot{Q}_{HR}=\rho C_p Q_v(T_{supply,after}-T_{outdoor})

Net heat recovery after auxiliary power:

\dot{Q}_{net}=\dot{Q}_{HR}-P_{fan,pump,added}

Heat recovery must be checked for frost risk, leakage or cross-contamination, pressure drop, bypass control, maintenance access, and whether the recovered temperature is useful.

Electrical Demand and Demand Response

Electrical demand:

P_{building}=P_{HVAC}+P_{lighting}+P_{plug}+P_{process}+P_{EV}+P_{other}

Event energy reduction:

E_{event}=\Delta P_{event}t_{event}

Recovery or rebound energy:

E_{recovery}=\Delta P_{recovery}t_{recovery}

Net energy change:

E_{net}=E_{event}-E_{recovery}

Demand-response service margin:

M_{service}=S_{limit}-S_{measured}

where $S$ may represent zone temperature deviation, humidity, pressure, ventilation, or recovery time. Demand response fails if the peak reduction creates unacceptable service degradation or rebound.

Measurement Uncertainty and Commissioning Margins

Heat-duty uncertainty margin:

M_Q=\dot{Q}_{measured}-U_Q-\dot{Q}_{required}

Temperature compliance margin:

M_T=T_{limit}-(T_{measured}+U_T)

Airflow compliance margin:

M_Q=Q_{measured}-U_Q-Q_{required}

Energy savings:

E_{savings}=E_{baseline,normalized}-E_{measured}

Guarded savings:

E_{savings,guarded}=E_{baseline,normalized}-E_{measured}-U_E

Commissioning evidence should state sensor calibration, trend interval, weather, occupancy, operating mode, control sequence, steady-state condition, and whether the result covers peak, part-load, startup, setback, or degraded operation.

Worked Check 1: Envelope and Infiltration Heating Load

A small building zone has:

H_{env}=1.9\ \text{kW/K}

Indoor and outdoor temperatures are:

T_{in}=21^\circ\text{C},\quad T_{out}=-4^\circ\text{C}

Zone volume is:

V_{zone}=1800\ \text{m}^3

Estimated infiltration is:

ACH=0.45\ \text{h}^{-1}

Use:

\rho=1.2\ \text{kg/m}^3,\quad C_p=1.01\ \text{kJ/(kg K)}

Envelope load:

\dot{Q}_{env}=H_{env}(T_{in}-T_{out})=1.9(21-(-4))=47.5\ \text{kW}

Infiltration flow:

\displaystyle Q_{inf}=\frac{ACH\cdot V_{zone}}{3600}=\frac{0.45(1800)}{3600}=0.225\ \text{m}^3/\text{s}

Infiltration sensible load:

\dot{Q}_{inf}=1.2(1.01)(0.225)(21-(-4))=6.82\ \text{kW}

Total screened heating load:

\dot{Q}_{heat}=47.5+6.82=54.3\ \text{kW}

Engineering interpretation: infiltration is about $12.6\%$ of this screened load. Air sealing may reduce heating demand and improve comfort, but the estimate should be validated with pressure testing, trend data, or calibrated model evidence before resizing equipment.

Worked Check 2: Economizer Damper Diagnosis

An air-handling unit reports economizer operation. Measurements are:

T_{RA}=24^\circ\text{C}

T_{OA}=9^\circ\text{C}

T_{mix}=19^\circ\text{C}

The supply-air target is:

T_{SA,set}=14^\circ\text{C}

Actual outdoor-air fraction:

\displaystyle f_{OA}=\frac{T_{RA}-T_{mix}}{T_{RA}-T_{OA}}=\frac{24-19}{24-9}=0.333

Required outdoor-air fraction:

\displaystyle f_{OA,req}=\frac{T_{RA}-T_{SA,set}}{T_{RA}-T_{OA}}=\frac{24-14}{24-9}=0.667

Engineering interpretation: the unit is taking about $33\%$ outdoor air when about $67\%$ is needed for free cooling to the setpoint. If humidity, freeze protection, smoke mode, and pressure overrides are not active, this is evidence for damper, actuator, linkage, sensor, or sequence fault investigation.

Worked Check 3: Fan Speed Reduction

A supply fan draws:

P_1=18\ \text{kW}

at full speed. Static-pressure reset allows a speed reduction to:

\displaystyle \frac{N_2}{N_1}=0.82

Estimate the ideal fan power under affinity-law assumptions.

\displaystyle P_2=P_1\left(\frac{N_2}{N_1}\right)^3

P_2=18(0.82)^3=9.92\ \text{kW}

Ideal power reduction:

\Delta P=18-9.92=8.08\ \text{kW}

Engineering interpretation: the ideal saving is large, but the real saving depends on fan efficiency, drive efficiency, minimum ventilation, terminal-box control, duct leakage, sensor location, and whether all zones still meet temperature and pressure requirements. Trend data should confirm both energy reduction and service quality.

Worked Check 4: Heat-Pump COP Boundary

A heat pump delivers:

\dot{Q}_{heat}=160\ \text{kW}

Compressor input is $42\ \text{kW}$ , indoor and outdoor fans add $8\ \text{kW}$ , and hydronic pumps add $5\ \text{kW}$ .

Compressor-only COP:

\displaystyle COP_{comp}=\frac{160}{42}=3.81

System COP including fans and pumps:

\displaystyle COP_{sys}=\frac{160}{42+8+5}=\frac{160}{55}=2.91

Engineering interpretation: both numbers can be correct, but they answer different questions. Equipment comparison might use compressor or packaged-unit data, while building energy performance should include the system boundary that actually consumes electricity.

Worked Check 5: Demand-Response Net Event

A building reduces HVAC demand by $310\ \text{kW}$ for $2.5\ \text{h}$ during a grid event. Recovery operation adds $95\ \text{kW}$ for $3.0\ \text{h}$ .

Event reduction:

E_{event}=310(2.5)=775\ \text{kWh}

Recovery energy:

E_{recovery}=95(3.0)=285\ \text{kWh}

Net energy reduction:

E_{net}=775-285=490\ \text{kWh}

Engineering interpretation: the peak reduction is useful only if zone temperatures, humidity, ventilation, and recovery time stayed within allowed limits. A good demand-response report includes both electrical demand and service evidence.

Common Mistakes

Common building energy calculation failures include:

comparing energy use without matching weather, occupancy, schedule, and service level;
treating reduced ventilation as savings without validating indoor air quality;
reporting COP without the system boundary;
using fan affinity laws while ignoring minimum flow and zone constraints;
trusting damper command instead of measured mixed-air or blade position;
claiming heat recovery without checking pressure drop, frost, leakage, timing, or auxiliary power;
using annual averages to hide peak-load, humidity, or recovery failures;
reporting savings without uncertainty or commissioning evidence.

Review Checklist

Before using a building energy calculation for a decision, verify:

boundary, service requirement, weather case, occupancy, schedule, and operating mode;
whether the calculation is for peak load, annual energy, demand response, commissioning, or fault diagnosis;
sensor calibration, trend interval, steady-state condition, and uncertainty;
whether ventilation, pressure, humidity, comfort, freeze protection, and safety constraints are preserved;
whether pumps, fans, controls, standby loads, auxiliary heat, and recovery energy are included in the correct boundary;
what action follows: resize, retune, clean, repair, recommission, monitor, or reject the proposed energy claim.

Building energy formulas are most useful when they make assumptions visible and connect calculated savings to measured service quality.

REF

Disciplines