TP-07: Thermal Mass

Overview

Thermal mass determines how a building stores and releases heat over time. A heavyweight masonry wall absorbs energy during the heating period and re-emits it as the zone cools, damping temperature swings and shifting peak loads. A lightweight timber-frame wall stores almost nothing: its temperature tracks the air temperature closely.

The HEM represents thermal mass through the areal heat capacity $κ_{m}$ assigned to each building element. This value is distributed across the nodes of the element's discretised model according to BS EN ISO 52016-1:2017, section 6.5.7. The node heat capacities then enter the zone heat balance matrix, coupling current temperatures to those of the previous timestep and producing the implicit (Crank-Nicolson-type) time-stepping scheme defined in section 6.5.6 of the standard.

This page documents how heat capacity is distributed across the node model, how the internal heat capacity of the zone air and furniture is represented, and how the resulting heat balance equations are solved at each timestep. The node geometry and conductance values are documented in TP-05: Fabric Heat Loss; solar gain terms applied to the node equations are documented in TP-08: Solar Gains and Shading.

Inputs

Parameter	Symbol	Unit	Description
Areal heat capacity	$κ_{m}$	J/(m²·K)	Thermal mass per unit area of the building element
Mass distribution class	—	—	One of: I, E, IE, D, M
Element area	$A$	m²	Gross area of the building element
Zone floor area	$A_{z o n e}$	m²	Useful floor area of the thermal zone
Thermal resistance of construction	$R_{c}$	m²·K/W	Total resistance of the material layers
Timestep	$Δ t$	s	Calculation timestep (typically 3600 s for hourly)
Previous node temperatures	$θ_{i}^{t - 1}$	°C	Temperature at each node from the preceding timestep

Calculation

Heat Capacity at Each Node

Each building element is discretised into nodes (five for opaque and ground elements, two for transparent elements). The areal heat capacity $κ_{m}$ is allocated to these nodes according to the mass distribution class, which describes where the thermal mass sits within the construction.

5-Node Model (Opaque Elements)

Opaque elements, elements adjacent to conditioned zones, and elements adjacent to unconditioned zones use five nodes indexed 1 (external surface) to 5 (internal surface). The heat capacity at each node, $k_{i}$ in J/(m²·K), is assigned as follows:

Class	Description	$k_{1}$	$k_{2}$	$k_{3}$	$k_{4}$	$k_{5}$
I	Mass at internal surface	$0$	$0$	$0$	$0$	$κ_{m}$
E	Mass at external surface	$κ_{m}$	$0$	$0$	$0$	$0$
IE	Mass at both surfaces	$κ_{m} /2$	$0$	$0$	$0$	$κ_{m} /2$
D	Distributed evenly	$κ_{m} /8$	$κ_{m} /4$	$κ_{m} /4$	$κ_{m} /4$	$κ_{m} /8$
M	Mass at middle node	$0$	$0$	$κ_{m}$	$0$	$0$

Class I is typical of a plastered masonry wall with insulation on the outside. Class E suits an externally-clad concrete wall. Class D represents a sandwich construction with mass throughout. Class M is used when the heavyweight layer sits at the geometric centre of the element.

2-Node Model (Transparent Elements)

Transparent elements (windows, doors with glazing) have no thermal mass:

$k_{1} = 0, k_{2} = 0$

This is consistent with the physical model: glazing units have negligible heat storage compared with opaque fabric.

3+2 Node Model (Ground Floor Elements)

Ground floor elements use five nodes in a 3+2 arrangement. Three nodes represent the floor construction and two represent the ground layer beneath. The ground layer always carries its own heat capacity at node 2:

$k_{g r o u n d} = d_{g r o u n d} \times c_{v, g r o u n d}$

Where:

$d_{g r o u n d} = 0.5$ m, the thickness of the modelled ground layer (BS EN ISO 52016-1:2017, section 6.5.8.2)
$c_{v, g r o u n d} = 3 \times 1 0^{6}$ J/(m³·K), the volumetric heat capacity of the ground (clay/silt, BS EN ISO 13370:2017, Table 7)

The floor construction's areal heat capacity $κ_{m}$ is then distributed across the remaining floor nodes (nodes 3 to 5) according to the mass distribution class:

Class	$k_{1}$	$k_{2}$	$k_{3}$	$k_{4}$	$k_{5}$
I	$0$	$k_{g r o u n d}$	$0$	$0$	$κ_{m}$
E	$0$	$k_{g r o u n d}$	$κ_{m}$	$0$	$0$
IE	$0$	$k_{g r o u n d}$	$κ_{m} /2$	$0$	$κ_{m} /2$
D	$0$	$k_{g r o u n d}$	$κ_{m} /4$	$κ_{m} /2$	$κ_{m} /4$
M	$0$	$k_{g r o u n d}$	$0$	$κ_{m}$	$0$

Note that for the ground floor, the D-class splits differ from the standard 5-node case: the outer fraction is $κ_{m} /4$ and the inner fraction is $κ_{m} /2$ , reflecting the asymmetric 3+2 topology.

Zone Internal Heat Capacity

The zone air node carries an additional fixed heat capacity representing the air volume and lightweight furniture within the zone:

$C_{in t} = κ_{m, in t} \times A_{z o n e}$

Where:

$κ_{m, in t} = 10, 000$ J/(m²·K), the default areal thermal capacity of air and furniture (BS EN ISO 52016-1:2017, Table B.17)
$A_{z o n e}$ is the useful floor area of the zone in m²

This term ensures that even a zone composed entirely of lightweight elements has some thermal inertia.

Element-Level Heat Capacity

The total heat capacity of a single building element, for reporting purposes:

$C_{e l} = A \times \frac{κ _{m}}{1000} [kJ/K]$

Total Zone Heat Capacity

$C_{T} = \sum_{i = 1}^{n} C_{e l, i} [kJ/K]$

This sum excludes $C_{in t}$ , which is accounted for separately in the zone air node equation.

Node Heat Balance Equations

The heat capacity values enter the heat balance equations described in BS EN ISO 52016-1:2017, section 6.5.6. At each timestep, the model solves a system of linear equations $A \cdot θ = b$ for the unknown node temperatures $θ$ . The heat capacity terms couple the current timestep to the previous one through an implicit time-stepping scheme.

The heat balance equations are presented below in the order: external surface, inside nodes, internal surface, and zone air. The node conductances $h_{pl i}$ referenced here are documented in TP-05: Fabric Heat Loss.

External Surface Node (Node 1)

The heat balance at the external surface node for building element $e l$ follows equation 41 of the standard.

Left-hand side (coefficients in row of matrix $A$ ):

$(\frac{k _{1}}{Δ t} + h_{ce} + h_{r e} + h_{pl i, 1}) θ_{1} - h_{pl i, 1} θ_{2} = b_{1}$

Right-hand side (entry in vector $b$ ):

$b_{1} = \frac{k _{1}}{Δ t} θ_{1}^{t - 1} + (h_{ce} + h_{r e}) θ_{e x t} + α_{so l} (I_{so l, d i f} F_{s h, d i f} + I_{so l, d i r} F_{s h, d i r}) - Φ_{s k y}$

Where:

$k_{1}$ is the areal heat capacity at the external surface node, in J/(m²·K)
$Δ t$ is the timestep in seconds
$h_{ce}$ is the external convective heat transfer coefficient, in W/(m²·K)
$h_{r e}$ is the external radiative heat transfer coefficient, in W/(m²·K)
$h_{pl i, 1}$ is the conductance between nodes 1 and 2, in W/(m²·K)
$θ_{1}^{t - 1}$ is the temperature at this node from the previous timestep
$θ_{e x t}$ is the external temperature (air or virtual ground temperature)
$α_{so l}$ is the solar absorption coefficient of the external surface
$I_{so l, d i r}$ , $I_{so l, d i f}$ are the direct and diffuse solar irradiance on the surface, in W/m²
$F_{s h, d i r}$ , $F_{s h, d i f}$ are the direct and diffuse shading reduction factors
$Φ_{s k y}$ is the thermal radiation to sky, equal to $F_{s k y} \times h_{r e} \times Δ θ_{s k y}$

The term $k_{1} /Δ t$ is the thermal mass coupling: it links the current temperature to the previous-timestep temperature $θ_{1}^{t - 1}$ . When $k_{1} = 0$ (for example, an I-class wall), the external surface has no thermal storage and responds instantaneously to boundary conditions.

Inside Nodes (Nodes 2 to $N - 1$ )

For each inside node $i$ , following equation 40:

$- h_{pl i, i - 1} θ_{i - 1} + (\frac{k _{i}}{Δ t} + h_{pl i, i} + h_{pl i, i - 1}) θ_{i} - h_{pl i, i} θ_{i + 1} = \frac{k _{i}}{Δ t} θ_{i}^{t - 1}$

Inside nodes are coupled only to their immediate neighbours. The right-hand side contains only the thermal mass term. When $k_{i} > 0$ , the node acts as a thermal store, retaining heat from one timestep to the next. When $k_{i} = 0$ , the node temperature is determined entirely by its neighbours within the current timestep.

Internal Surface Node (Node $N$ )

The internal surface node couples to the zone air and to the internal surfaces of all other building elements in the zone via radiative exchange. Following equation 39:

$- h_{pl i, N - 1} θ_{N - 1} + (\frac{k _{N}}{Δ t} + h_{c i} + h_{r i} \sum \frac{A _{e l, k}}{A _{t o t}} + h_{pl i, N - 1}) θ_{N} - \sum_{k} \frac{A _{e l, k}}{A _{t o t}} h_{r i} θ_{s, k} - h_{c i} θ_{ai r} = b_{N}$

$b_{N} = \frac{k _{N}}{Δ t} θ_{N}^{t - 1} + \frac{( 1 - f _{in t, c} ) Φ _{in t} + ( 1 - f _{so l, c} ) Φ _{so l} + ( 1 - f _{h c, c} ) Φ _{h c}}{A _{t o t}}$

Where:

$h_{c i}$ is the internal convective heat transfer coefficient, in W/(m²·K), which depends on the direction of heat flow
$h_{r i} = 5.13$ W/(m²·K) is the internal radiative heat transfer coefficient
$A_{e l, k}$ is the area of building element $k$ and $A_{t o t}$ is the total area of all elements in the zone
$θ_{s, k}$ is the internal surface temperature of element $k$
$θ_{ai r}$ is the zone internal air temperature
$f_{in t, c} = 0.4$ is the convective fraction for internal gains (BS EN ISO 52016-1:2017, Table B.11)
$f_{so l, c} = 0.1$ is the convective fraction for solar gains
$f_{h c, c}$ is the convective fraction for heating/cooling
$Φ_{in t}$ , $Φ_{so l}$ , $Φ_{h c}$ are the total internal, solar, and heating/cooling gains in watts

The radiative fraction of each gain type $(1 - f_{c})$ is distributed uniformly across all internal surfaces by dividing by $A_{t o t}$ .

Zone Air Node

The zone air heat balance follows equation 38:

$(\frac{C _{in t}}{Δ t} + \sum_{k} A_{e l, k} h_{c i, k} + h_{v e} + H_{t b}) θ_{ai r} - \sum_{k} A_{e l, k} h_{c i, k} θ_{s, k} = b_{ai r}$

$b_{ai r} = \frac{C _{in t}}{Δ t} θ_{ai r}^{t - 1} + h_{v e} θ_{s u ppl y} + H_{t b} θ_{e x t, ai r} + f_{in t, c} Φ_{in t} + f_{so l, c} Φ_{so l} + f_{h c, c} Φ_{h c}$

Where:

$C_{in t}$ is the zone internal heat capacity, in J/K
$h_{v e}$ is the ventilation heat transfer coefficient, in W/K (see TP-06: Ventilation and Infiltration)
$H_{t b}$ is the thermal bridge heat transfer coefficient, in W/K
$θ_{s u ppl y}$ is the average ventilation supply air temperature, in °C
$θ_{e x t, ai r}$ is the external air temperature, in °C

The $C_{in t} /Δ t$ term provides the air node's thermal inertia. Without it, the zone air temperature would respond instantaneously to gains and losses.

Implicit Time-Stepping Scheme

The heat balance equations above constitute an implicit (backward Euler) scheme: the unknown temperatures $θ$ at the current timestep appear on the left-hand side of $A \cdot θ = b$ , while the known temperatures from the previous timestep $θ^{t - 1}$ appear on the right-hand side through the $k_{i} /Δ t$ terms.

This scheme is unconditionally stable, meaning the solution does not diverge regardless of the ratio between the timestep $Δ t$ and the thermal time constants of the building elements. An explicit scheme, by contrast, would require very short timesteps when elements have high thermal mass and low resistance.

The matrix $A$ is assembled at each timestep because certain coefficients depend on temperature (specifically $h_{c i}$ , which depends on the direction of heat flow and therefore on the relative magnitudes of the air and internal surface temperatures from the previous timestep). The system is solved using standard linear algebra ( $θ = A^{- 1} b$ ).

Warm-Up Procedure

Before the simulation begins, the model must establish realistic initial node temperatures. The warm-up procedure uses an extended timestep ( $Δ t_{h} = 8760$ hours, one full year) and iterates under steady-state conditions:

Set all node temperatures to the average of the external air temperature and the heating setpoint.
Calculate the space heating demand required to maintain the setpoint at the operative temperature.
Solve the heat balance matrix with the calculated heating gains and the extended timestep.
Compare the resulting temperatures to the previous iteration.
Repeat steps 2 to 4 until all node temperatures converge to within a relative tolerance of $1 0^{- 8}$ .

The large timestep causes the $k_{i} /Δ t$ terms to become very small, effectively driving the system towards its steady-state solution. Convergence typically occurs within a few iterations.

Operative Temperature

The operative temperature used for setpoint comparison is the arithmetic mean of the internal air temperature and the mean radiant temperature:

$θ_{o p} = \frac{θ _{ai r} + θ ˉ _{r a d}}{2}$

Where the mean radiant temperature is the area-weighted average of all internal surface temperatures:

$\overset{ˉ}{θ}_{r a d} = \frac{\sum _{k} A _{e l, k} θ _{s, k}}{A _{t o t}}$

Thermal mass affects the operative temperature directly: heavyweight internal surfaces change temperature more slowly than the air, so the operative temperature is damped relative to the air temperature alone. This influences both the heating demand calculation and occupant thermal comfort.

Outputs

Quantity	Symbol	Unit	Description
Node heat capacities	$k_{i}$	J/(m²·K)	Areal heat capacity at each node of each element
Zone internal heat capacity	$C_{in t}$	J/K	Air and furniture heat capacity of the zone
Element heat capacity	$C_{e l}$	kJ/K	Total heat capacity per element ( $A \times κ_{m} /1000$ )
Total zone heat capacity	$C_{T}$	kJ/K	Sum of all element heat capacities
Node temperatures	$θ_{i}$	°C	Temperature at each node, updated each timestep
Operative temperature	$θ_{o p}$	°C	Mean of air temperature and mean radiant temperature

Assumptions

The areal thermal capacity of air and furniture is fixed at $κ_{m, in t} = 10, 000$ J/(m²·K) per unit floor area (BS EN ISO 52016-1:2017, Table B.17). The model does not distinguish between zones with heavy and light furnishings.
Transparent elements have zero thermal mass. The heat capacity of glazing units and frames is neglected.
The mass distribution class is constant for each element and does not change with temperature or moisture content.
The convective fraction for internal gains is fixed at $f_{in t, c} = 0.4$ and for solar gains at $f_{so l, c} = 0.1$ (BS EN ISO 52016-1:2017, Table B.11). These are not element-specific.
Internal radiative heat exchange is linearised using a fixed coefficient $h_{r i} = 5.13$ W/(m²·K) (BS EN ISO 13789:2017, Table 8). View factors between internal surfaces are not computed explicitly.
The internal convective coefficient $h_{c i}$ depends on heat flow direction (determined by pitch and by the sign of the temperature difference between air and surface from the previous timestep), but it does not depend on the magnitude of the temperature difference. This is a linearisation of the true convective heat transfer.
Ground layer thermal properties are fixed: thermal conductivity $λ = 1.5$ W/(m·K), volumetric heat capacity $c_{v} = 3 \times 1 0^{6}$ J/(m³·K), modelled layer thickness $d = 0.5$ m. These correspond to clay/silt soils.
The implicit time-stepping scheme is unconditionally stable. No Courant or stability conditions are checked.
The warm-up convergence tolerance is $1 0^{- 8}$ (relative). The warm-up timestep is 8760 hours.

Cross-references

TP-04: Space Heating Demand: the heat balance matrix is solved within the space heating demand calculation; thermal mass directly affects demand through the operative temperature
TP-05: Fabric Heat Loss: node geometry, inter-node conductances, and surface resistances that appear in the heat balance equations
TP-06: Ventilation and Infiltration: ventilation heat transfer coefficient $h_{v e}$ that appears in the zone air node equation
TP-08: Solar Gains and Shading: solar irradiance and shading factors applied at the external surface node; transmitted solar gains split between convective and radiative fractions

5-Node Model (Opaque Elements)

2-Node Model (Transparent Elements)

3+2 Node Model (Ground Floor Elements)

External Surface Node (Node 1)

Inside Nodes (Nodes 2 to N−1)

Internal Surface Node (Node N)

Zone Air Node

Inside Nodes (Nodes 2 to $N - 1$ )

Internal Surface Node (Node $N$ )