Notation and Conventions

Standard mathematical notation, SI units, and conventions used throughout the HEM technical reference. Covers all symbols from TP-01 through TP-18.

Version note (last verified 25 July 2026). This reference is written against the government's Python reference implementation of the Home Energy Model, at HEM v0.34 with FHS wrapper v0.25. The government's HEM has continued to evolve since; where the current line (HEM 1.0.0a8) diverges materially for this topic, the divergence is noted on this page.

Symbols

The following tables list every mathematical symbol used across the 18 technical pages (TP-01 to TP-18). Symbols are grouped by category. The "Used in" column lists the TP numbers where each symbol appears.

Geometry

Symbol	Description	Unit	Used in
$A$	Area of a building element (gross)	m²	03, 04, 05, 07, 08
$A_{z o n e}$	Useful floor area of a thermal zone	m²	04, 07
$A_{T F A}$	Total floor area of the dwelling	m²	02
$A_{b o d y}$	Total body surface area of occupants	m²	02
$A_{t o t}$	Total area of all building elements in a zone	m²	04, 07
$A_{e l}$ , $A_{e l k}$	Area of building element $k$	m²	04, 07
$A_{l e ak}$	Reference envelope area for airtightness index	m²	06
$A_{f a c a d es}$	Total surface area of vertical facades	m²	06
$A_{r oo f}$	Total surface area of the roof	m²	06
$A_{w, ma x}$	Maximum opening area of a window	m²	06
$A_{w}$	Window opening free area	m²	06
$A_{v e n t}$	Equivalent area of a purpose-built vent	cm²	06
$A_{h e x}$	Heat exchanger surface area (heat battery)	m²	15
$A_{U F H}$	Floor area served by underfloor heating	m²	16
$V$	Volume of a thermal zone	m³	04
$V_{z o n e}$	Internal volume of a ventilation zone	m³	06, 10
$V_{t o t a l}$	Total dwelling volume; or total tank volume (litres, TP-11)	m³ or litres	02, 06, 10, 11
$V_{n}$	Volume of a single tank layer	litres	11
$L$	Length (thermal bridge, pipe, or duct)	m	05, 10
$P$	Exposed floor perimeter	m	03
$P_{in t}$ , $P_{e x t}$	Internal and external duct perimeter	m	06, 10
$P_{d u c t}$	Internal perimeter of a rectangular duct	m	10
$d_{i}$	Internal diameter of a pipe or duct	m	10, 15
$d_{e}$	External diameter of a pipe or duct	m	10
$d_{c a p}$	Capillary tube diameter (heat battery)	m	15
$d_{p i p e}$	Internal diameter of distribution pipework	mm	09
$D_{in s}$	Outer diameter including insulation	m	06, 10
$D$	Depth of an overhang or side fin	m	08
$d$	Gap distance (overhang or side fin to window)	m	08
$t_{in s}$	Insulation thickness	m	06, 10
$H$	Height of a window	m	08
$H_{k}$	Projected height of a surface	m	08
$H_{ba se}$	Base height of a surface above ground	m	08, 18
$H_{z o n e}$	Height of the ventilation zone	m	03
$W$ , $W_{k}$	Width of a surface	m	08
$h_{z}$	Floor-to-ceiling height of a ventilation zone	m	06
$h_{ba se}$	Height of the zone floor above ground level	m	06
$h_{w, f a}$	Height of window free area	m	06
$h_{w, p a t h}$	Mid-height of a window airflow path	m	06
$h_{p an e l}$	Physical height of a PV panel	m	18
$h_{p r o j}$	Projected height of a PV panel	m	18
$w$	Width of a PV panel	m	18
$z_{ba se}$	Height of the base of a ventilation zone above ground	m	03
$z_{0}$	Aerodynamic roughness length	m	03, 06
$z_{min}$	Minimum height for terrain roughness coefficient	m	03, 06

Temperatures

Symbol	Description	Unit	Used in
$T_{ai r}$ , $θ_{e x t}$	External air temperature	°C	01, 03, 04, 06, 12, 14
$\overset{ˉ}{T}_{ai r}$ , $\overset{ˉ}{T}_{ai r, ann u a l}$	Annual mean air temperature	°C	01, 03
$\overset{ˉ}{T}_{ai r, m o n t h}$	Monthly mean air temperature	°C	03
$T_{ai r, d ai l y, min}$	Minimum daily average air temperature	°C	03
$T_{in t, ai r}$ , $θ_{ai r}$ , $θ_{z}$	Internal zone air temperature	°C	03, 04, 06, 07
$T_{o p}$ , $θ_{o p}$	Operative temperature	°C	03, 04, 07
$\overset{ˉ}{T}_{m r}$ , $\overset{ˉ}{θ}_{r a d}$	Mean radiant temperature	°C	03, 04, 07
$T_{s, in t}$ , $θ_{s, k}$	Internal surface node temperature	°C	03, 04, 07
$θ_{i}$ , $θ_{0}$ , $θ_{1}$ , ..., $θ_{n}$	Node temperatures within an element	°C	04, 05, 07
$θ_{i}^{p r e v}$ , $θ_{i}^{t - 1}$	Node temperature from the previous timestep	°C	04, 07
$θ_{ini t}$	Initial node temperature (warm-up)	°C	04
$θ_{se t, h e a t}$	Heating setpoint temperature	°C	04
$θ_{se t, coo l}$	Cooling setpoint temperature	°C	04
$θ_{o p, f r ee}$	Free-floating operative temperature	°C	04
$θ_{s u ppl y}$	Average ventilation supply air temperature	°C	04, 07
$T_{g r o u n d, v i r t u a l}$	Virtual ground temperature for ground floor elements	°C	03
$\overset{ˉ}{T}_{in t, ann u a l}$	Annual mean internal temperature	°C	03
$T_{co l d}$ , $θ_{co l d}$	Mains cold water inlet temperature	°C	09, 11
$T_{h w}$	Hot water system delivery temperature	°C	09, 10, 15
$T_{t a r g e t}$ , $θ_{t a r g e t}$	Target temperature (draw-off or storage)	°C	09, 11
$T_{f l u i d}$	Temperature of fluid inside a pipe	°C	10
$T_{s u r r}$	Surrounding (ambient) air temperature for pipes	°C	10
$θ_{amb}$	Ambient temperature at tank location	°C	11
$θ_{ini t}$	Initial tank layer temperature	°C	11
$θ_{se t, min}$ , $θ_{se t, ma x}$	Thermostat minimum and maximum setpoints	°C	11
$θ_{se t, r e f}$	Reference storage temperature (65 °C)	°C	11
$θ_{amb, r e f}$	Reference ambient temperature (20 °C)	°C	11
$θ_{l a y er}$ , $θ_{n}$	Temperature of a tank layer	°C	11
$θ_{so u r ce}$	Heat pump source-side temperature	°C	12
$T_{o u tl e t}$ , $T_{o u tp u t}$	Heat pump outlet (sink) temperature	K or °C	12
$T_{so u r ce}$	Heat pump source temperature (in kelvin)	K	12
$θ_{r e t u r n, ma x}$	Maximum allowable return temperature	°C	12
$θ_{c u t}$ , $T_{c u t}$	Room temperature above which charging ceases	°C	13, 17
$T_{f l o w}$ , $T_{f l o w, d s g n}$	Flow temperature (design or current)	°C	16
$T_{r e t u r n}$	Return water temperature	°C	14, 16
$T_{f l o w, min}$ , $T_{f l o w, ma x}$	Weather compensation curve flow limits	°C	16
$T_{o u t, min}$ , $T_{o u t, ma x}$	Weather compensation curve outdoor limits	°C	16
$T_{E}$	Mean emitter temperature	°C	16
$T_{E, ma x}$	Maximum achievable emitter temperature	°C	16
$T_{r m}$	Zone room air temperature (in emitter model)	°C	16
$T_{l oc}$	Temperature at the boiler location	°C	14
$T_{r}$	Return water temperature (boiler EBV curve)	°C	14
$T_{z}$	PCM zone temperature (heat battery)	°C	15
$T_{ma x}$	Maximum heat battery temperature (fully charged)	°C	15
$T_{pt, u pp er}$ , $T_{pt, l o w er}$	Phase change transition band upper and lower limits	°C	15
$T_{w, in}$ , $T_{w, o u t}$	Heat exchanger water inlet and outlet temperatures	°C	15
$T_{s p}$ , $T_{s p, min}$ , $T_{s p, ma x}$	Setpoint temperature (current, minimum, maximum)	°C	17
$T_{c u t, cor r}$	Corrected charge cut-off temperature	°C	17
$Δ T_{s k y}$	Sky-to-air temperature difference	K	03, 08
$Δ T_{min}$	Minimum source-sink temperature difference (heat pump)	K	12
$Δ T_{d s g n}$	Design flow-minus-return temperature difference	K	16
$Δ T_{co n d}$	Average condenser temperature difference	K	12
$Δ T_{e v a p}$	Average evaporator temperature difference	K	12
$Δ θ_{t es t}$	Temperature spread at test conditions (heat pump)	K	12
$Δ θ_{e mi tt er}$	Design temperature spread of the emitter system	K	12
$Δ T_{p i p e}$	Temperature drop of pipe contents	K	10

Heat transfer coefficients and thermal properties

Symbol	Description	Unit	Used in
$U$	Thermal transmittance (U-value)	W/(m²K)	03, 05
$R_{c}$	Thermal resistance of the construction layers	m²K/W	05, 07
$R_{se}$	External surface resistance	m²K/W	05
$R_{s i}$	Internal surface resistance	m²K/W	05
$R_{u}$	Thermal resistance of an adjacent unconditioned space	m²K/W	03
$R_{se, e f f}$	Effective external surface resistance (ZTU)	m²K/W	03
$r_{c u r t ain s}$	Window curtain thermal resistance	m²K/W	05
$r_{g r}$ , $r_{f l oor}$	Ground and floor thermal resistances	m²K/W	05
$R_{i}$ , $R_{in s}$ , $R_{e}$	Internal, insulation, and external thermal resistances per unit pipe/duct length	Km/W	06, 10
$R_{t o t a l}$	Total thermal resistance per unit length (pipe or duct)	Km/W	10
$h_{ce}$	External convective heat transfer coefficient	W/(m²K)	03, 04, 05, 07
$h_{r e}$	External radiative heat transfer coefficient	W/(m²K)	03, 04, 05, 07, 08
$h_{c i}$	Internal convective heat transfer coefficient	W/(m²K)	04, 05, 07
$h_{r i}$	Internal radiative heat transfer coefficient	W/(m²K)	04, 05, 07
$h_{se}$ , $h_{se, e f f}$	External surface heat transfer coefficient (standard and ZTU effective)	W/(m²K)	03
$h_{pl i, 0}$ , $h_{pl i, 1}$ , ...	Inter-node conductances between adjacent element nodes	W/(m²K)	04, 05, 07
$h_{i}$	Internal heat transfer coefficient (pipe/duct)	W/(m²K)	10
$h_{e}$	External surface heat transfer coefficient (pipe/duct)	W/(m²K)	10
$h_{h e x}$	Heat transfer capacity of heat battery heat exchanger	kW/K	15
$U A$	Overall heat transfer coefficient (heat battery)	W/K	15
$ψ$	Linear thermal transmittance (thermal bridge)	W/(mK)	03, 05
$χ$	Point thermal transmittance (thermal bridge)	W/K	05
$k_{in s}$	Thermal conductivity of insulation	W/(mK)	06, 10
$λ$	Thermal conductivity (ground)	W/(mK)	07
$κ_{m}$	Areal heat capacity of a building element	J/(m²K)	05, 07
$κ_{m, in t}$	Areal thermal capacity of air and furniture	J/(m²K)	04, 07
$k_{1}$ , $k_{2}$ , ..., $k_{5}$	Node-level areal heat capacities	J/(m²K)	05, 07
$k_{g r o u n d}$	Areal heat capacity of the modelled ground layer	J/(m²K)	07
$c_{v, g r o u n d}$	Volumetric heat capacity of ground (clay/silt)	J/(m³K)	07
$d_{g r o u n d}$	Thickness of the modelled ground layer	m	07
$C_{in t}$	Zone internal heat capacity (air and furniture)	J/K	04, 05, 07
$C_{e l}$	Heat capacity of a single building element	kJ/K	05, 07
$C_{T}$	Total zone heat capacity (sum of all element capacities)	kJ/K	05, 07
$K_{E}$	Thermal mass of the emitter body	kWh/K	16
$κ_{U F H}$	Equivalent specific thermal mass (UFH)	kJ/(m²K)	16
$C_{ab o v e}$ , $C_{d u r in g}$ , $C_{b e l o w}$	Phase change material heat capacities (above, during, below transition)	kJ/K	15
$C_{s t or e}$	Storage capacity of a storage heater	kWh	13
$c_{p}$ , $c_{p, ai r}$	Specific heat capacity (water or air)	J/(kgK)	04, 06, 09, 10, 16
$c_{v}$ , $c_{v o l}$	Volumetric heat capacity of a fluid	J/(litreK)	09, 10
$c_{w}$ , $c_{w}^{'}$	Specific heat capacity of water (various unit forms)	kWh/(kgK) or kJ/(kgK)	10, 15
$ρ$ , $ρ_{w}$	Density of water	kg/litre	09, 10, 11, 15
$ρ_{ai r}$ , $ρ_{a, r e f}$	Density of air (reference, at sea level)	kg/m³	04, 06
$ρ_{a, a l t}$ , $ρ_{a l t}$	Altitude-adjusted air density	kg/m³	03, 06
$ρ_{a} (T)$ , $ρ (T)$	Temperature-adjusted air density	kg/m³	03, 06
$ν$	Kinematic viscosity of water	m²/s	15

Energy and power

Symbol	Description	Unit	Used in
$Φ_{in t}$	Total internal heat gains	W	04, 07
$Φ_{so l}$	Solar gains (transmitted through glazing)	W	04, 07
$Φ_{h c}$	Heating or cooling power	W	04, 07
$Φ_{s k y}$	Thermal radiation to sky per unit area	W/m²	03, 04, 07, 08
$Φ_{in t, s t o}$	Internal gains from recoverable storage losses	W	11
$Φ_{p r ima r y}$	Primary pipework internal gains	W	11
$Φ_{g r o u n d, m o n t h}$	Monthly ground heat flow	W	03
$Q_{h e a t}$	Space heating demand per timestep	kWh	02, 04
$Q_{coo l}$	Space cooling demand per timestep (negative)	kWh	02, 04
$Q_{h c}$	Heating or cooling demand per timestep	kWh	04
$Q_{so l, t r an s}$	Transmitted solar gains through glazing	W	08
$Q_{m e t}$	Metabolic heat gain	W	02
$Q_{e v a p}$	Evaporative heat loss	W	02
$Q_{c w}$	Cold water heat loss	W	02
$Q_{h w}$	Hot water energy demand per timestep	kWh	09
$Q_{e v e n t}$	Energy content of a single hot water event	kWh	09
$Q_{p i p e, in t}$ , $Q_{p i p e, e x t}$	Pipework cool-down losses (internal, external)	kWh	09, 10
$Q_{coo l d o w n}$	Cool-down energy loss from a pipe	kWh	10
$Q_{d i s t}$ , $Q_{d i s t, in t}$ , $Q_{d i s t, e x t}$	Distribution pipework losses (total, internal, external)	kWh	10
$Q_{p r ima r y}$	Primary circuit pipework loss per timestep	kWh	10
$\dot{Q}_{p i p e}$	Steady-state heat loss rate from a pipe	W	10
$\dot{Q}_{d u c t}$	Heat loss rate from a duct	W	06, 10
$\dot{Q}_{d u c t, t o t a l}$	Net heat gain to the dwelling from all ducts	W	10
$\dot{Q}_{g ain s, d h w}$	Internal gains from pipework losses and hot water use	W	10
$Q_{g ain s, u se}$	Fraction of delivered hot water energy becoming internal gain	kWh	10
$Q_{u se}$	Thermal energy withdrawn from storage	kWh	11
$Q_{u nm e t}$	Unmet hot water demand	kWh	11
$Q_{l s}$	Storage tank standing losses per timestep	kWh	11
$Q_{s t o, r b l}$	Recoverable fraction of storage losses	kWh	11
$Q_{pw}$	Primary pipework losses (from storage model)	kWh	11
$H_{s t o, l s}$	Standby losses coefficient of the storage tank	W/K	11
$Q_{s t d, l s, r e f}$	Declared 24-hour standby loss (test conditions)	kWh/24h	11
$H_{e l}$	Fabric heat loss coefficient per element	W/K	05
$H_{T, f ab r i c}$	Total fabric heat loss coefficient for a zone	W/K	05
$H_{t b}$	Thermal bridge heat loss coefficient	W/K	04, 05, 07
$H_{ψ}$	Linear thermal bridge heat loss	W/K	05
$H_{χ}$	Point thermal bridge heat loss	W/K	05
$H_{v e}$ , $h_{v e}$	Ventilation heat transfer coefficient	W/K	04, 06, 07
$H_{p i}$ , $H_{p e}$	Internal and external periodic heat transfer coefficients (ground)	W/K	03
$P_{n}$	Boiler rated output power	kW	14
$P_{in}$	Storage heater charging power	kW	13
$P_{in s t an t}$	Instant backup heater power (storage heater)	kW	13
$P_{r a t e d}$	Rated power of an electric shower	kW	09
$P_{imm}$	Immersion heater rated power	kW	11
$P_{c ha r g e}$	Rated charge power (heat battery)	kW	15
$P_{l oss}$	Maximum standby heat loss rate (heat battery)	kW	15
$P_{f an}$	Fan power (ventilation, storage heater, or fan coil)	W	06, 13, 16
$P_{c i r c}$	Heating circulation pump power	kW	12, 14
$P_{so u r ce}$	Source-side circulation pump power	kW	12
$P_{s t an d b y}$	Standby power consumption	kW	12, 15
$P_{cr ank}$	Crankcase heater power	kW	12
$P_{o f f}$	Off-mode power consumption	kW	12
$P_{ba c k u p, ma x}$	Maximum backup heater power	kW	12
$P_{p u m p}$	Circulation pump power (heat battery)	kW	15
$P_{e l, pl}$ , $P_{e l, f l}$ , $P_{e l, s b y}$	Boiler part-load, full-load, and standby electrical powers	kW	14
$P_{p k}$	PV peak power	kW	18
$P_{in v, D C}$ , $P_{in v, A C}$	Inverter peak DC input and AC output power	kW	18
$E_{f an}$	Fan energy consumption per timestep	kWh	06, 13, 16
$E_{in p u t, H P}$	Compressor electrical energy input	kWh	12
$E_{in p u t, ba c k u p}$	Backup heater electrical energy input	kWh	12
$E_{in p u t, t o t a l}$	Total electrical energy input to heat pump system	kWh	12
$E_{c i r c}$	Circulation pump electrical energy	kWh	12
$E_{so u r ce}$	Energy extracted from the heat source	kWh	12
$E_{a ux}$	Auxiliary energy (standby, crankcase, off-mode)	kWh	12, 14, 15
$E_{d e l i v er e d}$	Energy delivered to zone or service per timestep	kWh	13, 15
$E_{c ha r g e d}$	Energy consumed for charging per timestep	kWh	13, 15
$E_{in s t an t}$	Energy from direct-acting backup element	kWh	13
$E_{s u ppl y}$	Total electrical energy from supply per timestep	kWh	13, 18
$E_{z o n e}$	Total heat delivered to zone per timestep	kWh	13
$E_{r e l e a se d}$	Total energy released by emitters to zone	kWh	16
$E_{r e q}$	Energy required from heat source per timestep	kWh	16
$E_{l i g h t}$	Annual lighting energy consumption	kWh/year	02
$E_{o t h er}$	Annual energy demand for other devices	kWh/year	02
$E_{coo k, e l ec}$	Annual cooking energy demand (electric)	kWh/year	02
$E_{r e g}$	Regulated energy	kWh	02
$E_{e l ec}$	Electrical energy consumed by electric shower	kWh	09
$E_{so l}$	Solar irradiation on PV panel per timestep	kWh/m²	18
$E_{D C}$	DC energy input to inverter per timestep	kWh	18
$E_{A C}$	AC electrical energy produced per timestep	kWh	18
$E_{l os t}$	PV energy lost to conversion and capping	kWh	18
$E_{a cce pt e d}$	Battery energy accepted per charge/discharge event	kWh	18
$Q_{o u t}$	Boiler energy output provided	kWh	14
$Q_{f u e l}$	Fuel consumed (gross CV basis)	kWh	14
$Q_{co mbi}$	Additional combi boiler hot water loss	kWh	14
$Q_{c a p, o p}$	Heat pump thermal capacity at operating conditions	kW	12
$Q_{d e l i v er e d, H P}$	Thermal energy delivered by heat pump compressor	kWh	12
$Q_{d e l i v er e d, ba c k u p}$	Thermal energy delivered by backup heater	kWh	12
$Q_{d e l i v er e d, t o t a l}$	Total thermal energy delivered	kWh	12
$Q_{e mi t}$	Instantaneous emitter power output	kW	16
$Q_{l oss}$	Standby thermal energy loss (heat battery)	kWh	15
$Q_{min}$ , $Q_{ma x}$	Minimum and maximum energy deliverable by storage heater	kWh	13
$\overset{q}{˙}_{so l, o p a q u e}$	Solar flux absorbed at external surface	W/m²	08
$\overset{q}{˙}_{s k y}$	Thermal radiation to sky per unit area	W/m²	08
$q_{in t} (t)$	Internal gain density at timestep $t$	W/m²	04
$q_{a pp} (t)$	Appliance energy supply density at timestep $t$	W/m²	04
$q_{m e t} (t)$	Metabolic gain profile value at timestep $t$	W/m²	02
$C_{n o m}$	Nominal battery storage capacity	kWh	18
$C_{ma x}$	Effective maximum battery capacity	kWh	18
$\dot{Q}_{min}$	Minimum battery charge rate	kW	18
$\dot{Q}_{c h, ma x}$ , $\dot{Q}_{d i s, ma x}$	Maximum battery charge and discharge rates	kW	18

Ventilation and airflow

Symbol	Description	Unit	Used in
$n_{a c h}$ , $A C H$	Air changes per hour	h $^{- 1}$	04, 06
$n_{min}$	Minimum air change rate	h $^{- 1}$	02
$\dot{V}_{min, A}$ , $\dot{V}_{min, B}$	Minimum ventilation rates (methods A and B)	l/s	02
$\dot{V}_{v e}$	Volumetric airflow rate	m³/s	04
$q_{v, O D A, d es i g n}$	Design mechanical ventilation flow rate	m³/h	06
$q_{v, O D A, r e q}$	Required outdoor air flow rate	m³/h	06
$q_{v, S U P}$ , $q_{v, E T A}$	Supply and extract air flow rates at terminal devices	m³/h	06
$q_{v, S U P, d i s}$ , $q_{v, E T A, d i s}$	Actual zone supply and extract flow rates	m³/h	06
$q_{v, h r, s a v in g}$	Heat recovery ventilation saving	m³/h	06
$q_{v, l e ak, p a t h}$	Volume flow rate through a leak path	m³/h	06
$q_{v, v e n t, p a t h}$	Volume flow rate through a vent path	m³/h	06
$q_{v, w, d i v, p a t h}$	Volume flow rate through a window division	m³/h	06
$q_{v, co mb}$	Volume flow rate for combustion air	m³/h	06
$C_{l e ak}$	Overall leakage coefficient	m³/(h Pa $^{n}$ )	06
$C_{v e n t, p a t h}$	Vent airflow coefficient	m³/(h Pa $^{n}$ )	06
$C_{w, p a t h}$	Window airflow coefficient	m³/(h Pa $^{n}$ )	06
$n_{l e ak}$	Flow exponent for leaks (0.667)	n/a	06
$n_{v e n t}$	Flow exponent for vents (0.5)	n/a	06
$n_{w}$	Flow exponent for windows (0.5)	n/a	06
$C_{p}$ , $C_{p, p a t h}$	Wind pressure coefficient	n/a	06
$C_{R}$ , $C_{r g h}$	Terrain roughness coefficient	n/a	03, 06
$C_{T}$ , $C_{t o p}$	Topography coefficient	n/a	03, 06
$K_{R}$	Terrain factor	n/a	03, 06
$C_{D}$ , $C_{D, w}$ , $C_{D, v e n t}$	Discharge coefficient (window, vent)	n/a	06
$p_{z, r e f}$	Internal reference pressure	Pa	06
$p_{e, p a t h}$ , $p_{z, p a t h}$	External and internal path pressures	Pa	06
$Δ p_{p a t h}$	Pressure difference at an airflow path	Pa	06
$Δ p_{l e ak, r e f}$	Reference test pressure (typically 50 Pa)	Pa	06
$Δ p_{v e n t, r e f}$	Reference pressure for vent specification	Pa	06
$q_{v, Δ p, l e ak, r e f}$	Measured air flow rate at test pressure per unit envelope area	m³/(hm²)	06
$R_{v}$	Vent opening ratio (0 = closed, 1 = open)	n/a	06
$R_{w}$	Window opening ratio	n/a	06
$\overset{m}{˙}$ , $\overset{m}{˙}_{in}$ , $\overset{m}{˙}_{o u t}$	Mass flow rates (total, ingoing, outgoing)	kg/s	06, 15
$u_{10}$	Meteorological wind speed at 10 m height	m/s	03, 06
$u_{s i t e}$	Wind speed corrected for terrain and height	m/s	03, 06
$v_{w in d}$	Wind speed from weather data	m/s	01
$θ_{w in d}$	Wind direction from weather data	degrees	01, 03
$\overset{v}{ˉ}_{w in d}$	Annual mean wind speed	m/s	01
$\overset{ˉ}{θ}_{w in d}$	Annual mean wind direction	degrees	01
$η_{h r}$ , $η_{m v h r}$	MVHR heat recovery efficiency	n/a	06, 10
$S F P$	Specific fan power	W/(l/s)	06
$f_{o p, V}$	Mechanical ventilation operational fraction	n/a	06
$f_{c t r l}$	Ventilation control factor	n/a	06
$f_{sy s}$	Ventilation system factor	n/a	06
$E_{v}$	Ventilation effectiveness	n/a	06
$f_{cr oss}$	Cross-ventilation flag (boolean)	n/a	06
$f_{o p, co mb}$	Combustion appliance operation signal	n/a	06
$f_{a s}$	Combustion appliance system factor	n/a	06
$f_{f f}$	Combustion fuel flow factor	n/a	06
$P_{h, f i}$	Combustion appliance fuel input power	kW	06
$h_{a l t}$	Altitude of the building site above sea level	m	03, 06
$\dot{V}$	Volumetric water flow rate (heat battery)	l/min	15
$\dot{V}_{d s g n}$ , $\dot{V}_{min}$ , $\dot{V}_{ma x}$	Design, minimum, and maximum emitter flow rates	l/min	16

Solar

Symbol	Description	Unit	Used in
$I_{b e am}$ , $G_{so l, b}$	Direct beam normal irradiance	W/m²	01, 08
$I_{d i f f}$ , $G_{so l, d}$	Diffuse horizontal irradiance	W/m²	01, 08
$I_{d i r}$	Direct irradiance on a tilted surface	W/m²	08
$I_{d i f, s k y}$ , $I_{d i f, c i r c}$ , $I_{d i f, h or}$	Sky diffuse, circumsolar, and horizon brightening components	W/m²	08
$I_{r e f}$	Ground reflection irradiance	W/m²	08
$I_{so l, d i r}$ , $I_{so l, d i f}$	Calculated direct and diffuse irradiance on a surface	W/m²	04, 07, 08
$I_{so l, t o t a l}$	Total unshaded irradiance on a tilted surface	W/m²	08
$I_{s u r f}$	Total surface irradiance after shading	W/m²	08
$G_{so l, c}$	Extraterrestrial radiation	W/m²	08
$ρ_{g r o u n d}$	Solar reflectivity of the ground	n/a	01, 08
$α_{so l}$	Solar absorption coefficient or solar altitude angle	n/a or degrees	04, 07, 08
$δ_{so l}$	Solar declination	degrees	08
$γ_{so l}$	Solar azimuth	degrees	08
$ω$	Solar hour angle	degrees	08
$ϕ$	Latitude	degrees	01, 08
$λ$	Longitude	degrees	01, 08
$β$	Element pitch (tilt from horizontal)	degrees	03, 08, 18
$γ$	Element orientation (azimuth of surface normal)	degrees	08, 18
$θ$	Angle of incidence of solar beam on a surface	degrees	08
$θ_{z}$	Solar zenith angle	radians	08
$R_{d c}$	Earth orbit deviation	degrees	08
$t_{e q}$	Equation of time	minutes	08
$t_{so l}$	Solar time	hours	08
$t_{s hi f t}$	Time shift (timezone minus longitude correction)	hours	08
$m$	Air mass	n/a	08
$ε$	Dimensionless clearness parameter (Perez model)	n/a	08
$Δ$	Dimensionless sky brightness parameter (Perez model)	n/a	08
$F_{1}$	Circumsolar brightness coefficient	n/a	01, 08
$F_{2}$	Horizontal brightness coefficient	n/a	01, 08
$f_{11}$ , $f_{12}$ , $f_{13}$	Perez circumsolar regression coefficients	n/a	08
$f_{21}$ , $f_{22}$ , $f_{23}$	Perez horizontal regression coefficients	n/a	08
$F_{s k y}$	Longwave sky view factor	n/a	03, 08
$F_{s h, d i r}$ , $f_{s h, d i r}$	Direct shading reduction factor	n/a	04, 07, 08, 18
$F_{s h, d i f}$ , $f_{s h, d i f}$	Diffuse shading reduction factor	n/a	04, 07, 08, 18
$F_{s h, d i f, r e m}$	Remote diffuse shading factor	n/a	08
$F_{s h, d i f, w s}$	Nearby (window-attached) diffuse shading factor	n/a	08
$g$	Total solar energy transmittance (g-value) of glazing	n/a	05, 08
$g_{cor r ec t e d}$	g-value corrected for angle of incidence	n/a	08
$F_{w}$	g-value correction factor (default 0.90)	n/a	08
$F_{f}$	Frame area fraction	n/a	05, 08
$f_{so l, c}$	Convective fraction of solar gains (0.1)	n/a	04, 07, 08
$P_{1}$ , $P_{2}$	Overhang/side fin geometric ratios	n/a	08
$T_{o b s}$	Transparency of a shading obstacle	n/a	08
$H_{o b s t}$ , $L_{o b s t}$	Obstacle height and distance	m	08
$H_{o v h}$ , $L_{o v h}$	Overhang height and distance	m	08
$H_{s ha d e, o b s t}$ , $H_{s ha d e, o v h}$	Shadow heights from obstacles and overhangs	m	08
$W_{s ha d e, f in}$	Shadow width from a side fin	m	08
$H_{k, s u n}$ , $W_{k, s u n}$	Remaining sunlit height and width	m	08
$r_{t r an s}$	Window treatment transmittance reduction factor	n/a	08
$n_{d a y}$	Day of the year (1 to 366)	n/a	08
$n_{h o u r}$	Clock hour of the day (1-based)	n/a	08

Efficiency and performance

Symbol	Description	Unit	Used in
$CoP_{o p}$	Coefficient of performance at operating conditions	n/a	12
$CoP_{C a r n o t}$	Carnot coefficient of performance	n/a	12
$η_{e x er}$	Exergetic efficiency	n/a	12
$N_{e x er}$	Exergy weighting exponent (3.0)	n/a	12
$L R$	Load ratio (fraction of timestep the heat pump runs)	n/a	12
$L R_{o p}$	Load ratio at operating conditions	n/a	12
$L R_{t h eo}$	Theoretical load ratio at a test point	n/a	12
$L R_{co n t, min}$	Minimum continuous load ratio	n/a	12
$L R_{min, l o w}$ , $L R_{min, 55}$	Minimum modulation rates at low and 55 °C flow temperatures	n/a	12
$C_{d, o p}$	Degradation coefficient at operating conditions	n/a	12
$f_{s p r e a d}$	Temperature spread correction factor	n/a	12
$τ_{o n o f f}$	On/off cycling time constant	hours	12
$τ_{ser v i ce}$	Service type time constant	s	12
$P_{in er t ia}$	On/off cycling inertia power penalty	kW	12
$f_{t h r o ug h p u t}$	Exhaust air throughput factor	n/a	12
$r_{o v er v e n t}$	Overventilation ratio (exhaust air HP)	n/a	12
$r_{e x t}$	External air ratio (mixed exhaust air HP)	n/a	12
$η_{f ina l}$	Final boiler efficiency (after all corrections)	n/a	14
$η_{f l, g r oss}$ , $η_{pl, g r oss}$	Full-load and part-load gross efficiencies	n/a	14
$η_{E B V}$	EBV-calibrated efficiency as function of return temperature	n/a	14
$η_{t h eo}$	Theoretical gross efficiency (EBV curve)	n/a	14
$f_{n g}$	Net-to-gross calorific value conversion factor	n/a	14
$β_{min}$	Minimum boiler modulation ratio	n/a	14
$ϕ_{min}$	Fraction of timestep spent firing at minimum rate	n/a	14
$r_{o n / o f f}$	Ratio of boiler off-time to on-time	n/a	14
$f_{g e n}$	Standby heat loss as fraction of current boiler power	n/a	14
$Δ η_{cy c}$	Cycling efficiency adjustment	n/a	14
$Δ η_{l oc}$	Location efficiency adjustment	n/a	14
$c_{5}$ , $c_{6}$	Boiler standing loss coefficients	n/a	14
$n_{s b y}$	Standby heat loss power-law index (1.25)	n/a	14
$Δ T_{s b y}$	Nominal standby loss test temperature difference (30 K)	K	14
$f_{1}$ , $f_{2}$ , $f_{3}$	Combi boiler rejected energy, storage loss, and volume factors	n/a	14
$f_{u}$	Combi daily hot water usage factor	n/a	14
$Δ V$	Combi daily volume factor	litres	14
$η_{w w h r s}$	WWHRS heat recovery efficiency	%	09
$f_{u t i l}$	WWHRS utilisation factor	n/a	09
$f_{h o t}$	Hot water fraction (mixing ratio)	n/a	09, 11
$f_{s t o, m}$	Fraction of tank thermal losses transmitted to room (0.75)	n/a	11
$η_{l i g h t}$	Overall lighting efficacy	lm/W	02
$f_{g ain s}$	Fraction of appliance energy becoming internal gain	n/a	04
$f_{in t, c}$	Convective fraction of internal gains (0.4)	n/a	04, 07
$f_{h c, c}$	Convective fraction of heating/cooling output	n/a	04, 07
$f_{co n v}$	Convective fraction (emitter or storage heater)	n/a	13, 16
$S$	State of charge (storage heater, 0 to 1)	n/a	13
$S_{t a r g e t}$	Target state of charge (storage heater)	n/a	13
$R_{r e t}$	Heat retention ratio (storage heater)	n/a	13
$S o C$	Battery state of charge (0 to 1)	n/a	18
$S oH$	Battery state of health	n/a	18
$η_{r t}$	Round-trip charge/discharge efficiency	n/a	18
$η_{o w}$	One-way charge or discharge efficiency	n/a	18
$η_{in v}$	Inverter DC-to-AC conversion efficiency	n/a	18
$η_{s h}$	Inverter shading efficiency factor	n/a	18
$f_{p er f}$	PV system performance factor	n/a	18
$f_{t e m p}$	Battery temperature capacity factor	n/a	18
$c$	Emitter power coefficient	kW/K $^{n}$	16
$n$	Emitter power exponent	n/a	16
$K_{h}$	UFH system performance factor	W/(m²K)	16
$f_{b y p a ss}$	Bypass recirculation fraction	n/a	16
$A_{h t c}$ , $B_{h t c}$	Empirical heat transfer regression parameters (heat battery)	n/a	15
$R e$	Reynolds number	n/a	15

Time and control

Symbol	Description	Unit	Used in
$t_{s t a r t}$ , $t_{e n d}$	Simulation start and end times	h	01
$Δ t$	Simulation timestep	h or s	01, 04, 06, 07, 14
$Δ t_{h}$	Timestep in hours	h	04, 10
$Δ t_{ser i es}$	Timestep of the weather data series	h	01
$Δ t_{s}$	Timestep of a control schedule	h	17
$Δ t_{hb}$	Sub-timestep for heat battery calculation	s	15
$N_{s t e p s}$	Total number of simulation timesteps	n/a	01
$i$	Zero-based timestep index	n/a	01
$i_{t s}$	Time series index for weather data lookup	n/a	01
$t_{c u r r e n t}$	Current simulation time	h	01
$h_{c u r r e n t}$	Current integer hour	h	01
$h_{d a y}$	Hour of the current day (0 to 23)	h	01
$d_{c u r r e n t}$	Current day number (0-indexed)	n/a	01
$d_{0}$ , $d_{s t a r t}$	Start day of schedule or time series	n/a	01, 17
$t_{d e l a y}$	Time delay before backup heater activates	hours	12
$t_{r u nnin g}$	Cumulative running time within a timestep	hours	12, 15, 16
$t_{a v ai l}$	Time available for a service within a timestep	hours	14, 15
$t_{coo l}$	Emitter cooldown time within a timestep	hours	16
$t_{w a r m u p}$	Emitter warm-up time within a timestep	hours	16
$t_{a d v}$	Advanced start duration	h	17
$t_{o n}$	Daily on-time (cost-minimising control)	h	17
$t_{min}$	Minimum run time (heat battery)	s	15
$f_{c ha r g e}$	Daily target charge proportion (0 to 1)	n/a	15, 17
$Δ T_{c u t}$	Time-varying adjustment to charge cut temperature	K	17
$H D H$	Heating degree hours	degC h	13, 17
$N$	Number of occupants	n/a	02
$N_{v o l}$	Number of equal-volume tank layers	n/a	11
$N_{z o n es}$	Number of PCM thermal zones	n/a	15
$N_{u ni t s}$	Number of installed units (heat battery or fan coil)	n/a	15, 16
$N_{t a p}$ , $N_{t a p s}$	Number of hot water tapping points	n/a	09, 10
$N_{e v e n t s}$	Number of hot water events in a timestep	n/a	09, 10
$n_{b e d}$	Number of bedrooms	n/a	02
$N_{w, d i v}$	Number of window divisions	n/a	06

Other

Symbol	Description	Unit	Used in
$A$	Coefficient matrix of the heat balance	n/a	04, 07
$X$ , $θ$	Vector of unknown node temperatures	n/a	04, 07
$B$ , $b$	Right-hand-side vector of the heat balance	n/a	04, 07
$g$	Gravitational acceleration (9.81 m/s²)	m/s²	06
$K$	Perez model constant (1.014 rad $^{- 3}$ )	rad $^{- 3}$	08
$V_{H W, d ai l y}$	Average daily hot water volume	litres	02, 14
$V_{w a r m}$	Volume of warm water delivered at tapping point	litres	09
$V_{h o t}$	Volume of hot water drawn from heating system	litres	09
$V_{e v e n t}$	Volume of warm water drawn (bath events)	litres	09
$V_{ba t h, ma x}$	Maximum bath capacity	litres	09
$V_{t a pp in g}$	Volume at tapping points (for internal gains)	litres	09
$V_{h w, t o t a l}$	Total volume of hot water demand per timestep	litres	09
$V_{p i p e}$	Volume of fluid in a pipe section	litres	10
$V_{d r a w o f f}$	Total volume of hot water extracted from tank	litres	11
$V_{f i l l}$	Bath fill volume (accounting for body displacement)	litres	02
$V_{d i s p}$	Occupant body displacement volume	litres	02
$\dot{V}_{s h o w er}$ , $\dot{V}_{ba t h}$ , $\dot{V}_{o t h er}$	Hot water flow rates (shower, bath, other)	litres/min	09
$L$	Lumens (annual lighting demand)	lm	02
$F_{ban d}$	DHW decile banding correction factor	n/a	02
$F_{e l ec}$	Electric shower correction factor	n/a	02
$F_{H W}$	Hot water volume calibration factor	n/a	02
$F_{a ppl ian ce}$	Appliance convergence scaling factor	n/a	02
$F E E$	Fabric Energy Efficiency metric	kWh/(m²year)	02
$D E R$	Dwelling Emission Rate	kgCO $_{2}$ /m²	02
$T E R$	Target Emission Rate	kgCO $_{2}$ /m²	02
$D P E R$	Dwelling Primary Energy Rate	kWh/m²	02
$T P E R$	Target Primary Energy Rate	kWh/m²	02
$f_{g l a z, ma x}$	Maximum notional glazing fraction	n/a	02
$c_{0}$ , $c_{1}$ , $c_{2}$	Polynomial regression coefficients (heat pump CoP)	n/a	12
$v_{1}$	Water velocity at 1 l/min (heat battery)	m/s	15
$a$	Battery age	years	18

Units

Unit	Symbol	Quantity
metre	m	Length, height, distance
square metre	m²	Area
cubic metre	m³	Volume
centimetre squared	cm²	Equivalent vent area
millimetre	mm	Pipe diameter
kilogram	kg	Mass
kilogram per litre	kg/l	Density (water)
kilogram per cubic metre	kg/m³	Density (air)
second	s	Time (timestep), duration
hour	h	Time (simulation), duration
minute	min	Duration (DHW events)
year	year	Annual quantities, battery age
degree Celsius	°C	Temperature
kelvin	K	Temperature difference, absolute temperature
watt	W	Power, heat flow rate
kilowatt	kW	Power (heating, charging, pump)
watt per square metre	W/m²	Irradiance, heat flux density
watt per kelvin	W/K	Heat transfer coefficient (aggregate)
watt per square metre kelvin	W/(m²K)	Surface heat transfer coefficient, U-value
watt per metre kelvin	W/(mK)	Linear thermal transmittance, thermal conductivity
square metre kelvin per watt	m²K/W	Thermal resistance (area-based)
kelvin metre per watt	Km/W	Thermal resistance per unit length (pipe/duct)
joule	J	Energy
kilojoule	kJ	Energy (heat battery zone transfer)
kilowatt-hour	kWh	Energy (demand, consumption, generation)
joule per square metre kelvin	J/(m²K)	Areal heat capacity
joule per cubic metre kelvin	J/(m³K)	Volumetric heat capacity (ground)
joule per kilogram kelvin	J/(kgK)	Specific heat capacity
joule per litre kelvin	J/(litreK)	Volumetric heat capacity (fluid)
kilojoule per kelvin	kJ/K	Heat capacity (element, zone, PCM zone)
kilowatt-hour per kelvin	kWh/K	Thermal mass (emitter)
kilojoule per square metre kelvin	kJ/(m²K)	Specific thermal mass (UFH)
pascal	Pa	Pressure, pressure difference
litre	l	Volume (water, tank)
litre per minute	l/min	Volumetric flow rate (water)
litre per second	l/s	Volumetric flow rate (ventilation)
cubic metre per hour	m³/h	Volumetric airflow rate
cubic metre per hour per square metre	m³/(hm²)	Air permeability test result
metre per second	m/s	Wind speed, fluid velocity
square metre per second	m²/s	Kinematic viscosity
inverse hour	h $^{- 1}$	Air changes per hour
lumen	lm	Luminous flux
lumen per watt	lm/W	Luminous efficacy
degree	°	Angle (pitch, orientation, latitude, longitude, wind direction)
radian	rad	Angle (solar zenith, Perez model)
kilogram CO $_{2}$ per square metre	kgCO $_{2}$ /m²	Emission rate
kilowatt-hour per 24 hours	kWh/24h	Standby loss (storage tank)

Conventions

Subscript naming

Throughout the technical pages, subscripts follow a consistent naming scheme:

Element position: $e x t$ = external, $in t$ = internal, $ai r$ = air node, $s$ = surface.
Direction of heat transfer: $ce$ = convective external, $r e$ = radiative external, $c i$ = convective internal, $r i$ = radiative internal.
Component identity: $e l$ = building element, $t b$ = thermal bridge, $v e$ = ventilation, $so l$ = solar, $h c$ = heating/cooling.
Gain type: $in t$ = internal gains, $so l$ = solar gains, $h c$ = heating/cooling gains.
Convective/radiative split: a subscript $c$ on a fraction (e.g. $f_{in t, c}$ ) denotes the convective portion.
Temporal: $p r e v$ or superscript $t - 1$ = previous timestep value.
Service type: $H P$ = heat pump, $ba c k u p$ = backup heater.

Sign conventions

Positive heat flow = heat gain to the zone (energy entering the dwelling).
Negative heat flow = heat loss from the zone (energy leaving the dwelling).
Space cooling demand is negative by convention ( $Q_{coo l} < 0$ ). In the FEE calculation, the subtraction $Q_{h e a t} - Q_{coo l}$ therefore adds the cooling demand to the heating demand.
Battery discharge energy is negative (energy leaving the battery). Battery charge energy is positive (energy entering the battery).
Ventilation outflow is negative (air mass leaving the zone).

Timestep conventions

The simulation timestep $Δ t$ is typically 0.5 h (1800 s) for FHS assessments. Other values (1.0 h, 0.25 h) are supported.
Some equations require $Δ t$ in seconds (heat balance matrix, thermal mass terms); others use hours (energy accounting, demand calculations). The unit is stated explicitly in each TP.
Sub-hourly simulation timesteps use piecewise-constant interpolation from hourly weather data: all sub-steps within the same hour receive the same climate values.

Temperature reference points

External surface resistance uses a fixed combined coefficient $h_{ce} + h_{r e} = 20.0 + 4.14 = 24.14$ W/(m²K), giving $R_{se} = 0.0414$ m²K/W.
Internal radiative heat transfer coefficient is fixed at $h_{r i} = 5.13$ W/(m²K).
Air density reference temperature is $T_{e, r e f} = 293.15$ K (20 °C).
The sky-to-air temperature difference is fixed at $Δ T_{s k y} = 11.0$ K (intermediate climatic region).
Boiler room temperature for internal location adjustment is fixed at 19.5 °C.
Storage tank standby loss reference conditions are $θ_{se t, r e f} = 65$ °C and $θ_{amb, r e f} = 20$ °C.
Mixed hot water delivery temperature at tapping points is fixed at 41 °C.

Node numbering

Building element node models use the following indexing:

External surface node = node 0 (or node 1 in some formulations). This is the node exposed to the external environment.
Internal surface node = node $N$ (the highest-numbered node). This is the node exposed to the zone interior.
Inside nodes = nodes between external and internal surfaces, coupled only to their immediate neighbours in a tridiagonal structure.
Opaque elements use a 5-node model (nodes 1 to 5, external to internal).
Transparent elements use a 2-node model (external surface and internal surface only, with zero heat capacity).
Ground floor elements use a 3+2 node model: two nodes represent the ground layer and three represent the floor construction.

Matrix formulation

The heat balance is formulated as $A \cdot X = B$ , where $A$ is the coefficient matrix of thermal conductances and capacitances, $X$ is the vector of unknown node temperatures, and $B$ is the vector of known boundary conditions and previous-timestep terms. The system dimension is $N + 1$ , where $N$ is the total number of element nodes and the additional entry is the zone air node.

Rounding and tolerances

Warm-up convergence tolerance is $1 0^{- 8}$ (relative) on all node temperatures.
Vent opening solver uses golden section minimisation with a small gradient term ( $1 0^{- 10}$ ) to prevent flat-region stalling.
Mass balance solver (Brent's method) searches a progressively expanding pressure interval up to $\pm 200$ Pa.