# A brief guide and free tool for the calculation of the thermal mass of building components

In the following text, I will try to provide the most important information on thermal mass calculation for building applications. The second part is a brief guide to understanding and using the Excel-calculator.

## Summary for users not willing to read the whole text…

Putting it in a nutshell, the most relevant application of the tool will be the optimisation (=maximisation) of thermal mass on the interior surfaces of buildings. This will help to reduce the daily temperature swings inside the building. By increasing the internal mass, your wall, floor, or ceiling should be able to absorb most of the solar inputs during the day and release the accumulated heat through natural ventilation during the night.

For this purpose, you will have to maximise the resulting figure “internal areal heat capacity” in the tool. As you will see, this property depends mainly on the internal surface layer – up to a few centimetres or even millimetres below the surface. Therefore in order to achieve high thermal capacity, you will have to choose a material with high thermal conductivity and density of this topmost internal layer.

I regard the other calculation results (time shifts, periodic transmittance…) as being of minor importance. However, to fully understand the topic or for special applications, I still recommend reading the whole text below…

## Introduction

The following calculations are based on the methods described in standard ISO 13786. Without explicitly mentioning it, the standard applies well-known calculation methods used in electrical engineering to describe the behaviour of components in alternating current circuits. The calculations are carried out by utilising matrices of complex numbers.

In order to analytically solve these equations, the boundary conditions (temperatures or heat-fluxes), as well as the resulting variables (temperatures and heat-fluxes), are assumed to be of a sinusoidal shape having a period of 24 hours. Even if this sounds like a severe limitation, it is actually an appropriate and useful assumption. The consideration of a sinusoidal shape is suitable since the actual, average daily temperature swings essentially correspond to sine waves – or have at least a dominant sinusoidal component (see Fourier theorem). The restriction to periodic lengths of 24 hours is also reasonable, as a cyclic temperature variation can only be expected within this 24h time-frame.

The calculation result value thermal admittance describes the ability of a surface to absorb and release heat (energy) upon a periodic sinusoidal temperature swing with a period of 24h. The value represents the amplitude of the heat-flux (=maximum value) caused by a 1 K (°C) temperature swing. The temperature on the opposite side of the wall is assumed to be held constant. Due to the linearity of the underlying equations, you can simply multiply the value with any other temperature amplitudes to get the corresponding heat fluxes. E.g. if you want to estimate the maximum heat flux into/out of your wall caused by an internal temperature swing of 6 °C, and the internal thermal admittance of your wall is 5 W/(m²K), then the maximum heat-flux will be 6 K * 5 W/(m²K) = 30 W/m². Therefore the “response” of this wall to a sinusoidal 6 °C periodic temperature swing will be a sinusoidal heat flux absorbing up to a maximum of 30 Watt per square meter during the day and releasing the same 30 W/m² at night.

The ability of a wall to absorb energy during the day is crucial to avoid summertime overheating – or to reduce cooling costs. The internal thermal admittance can be used to evaluate this ability. However, the internal areal heat capacity, which is almost proportional to this value, is actually more suitable for this job (see below).

## Time-shift – internal thermal admittance

The heat flux caused by the temperature swing is time-shifted, meaning that it does not have its maxima and minima simultaneously. The heat flux usually leads the swings in ambient temperature – whereas the actual surface temperature of the wall will lag. So if the displayed output value for the time shift is “2:00” (like in the example above), the maximum heat-flux in/out of the wall will occur 2 hours earlier than the temperature maximum/minimum.
This time shift is just a “side-effect” of the heat buffering and cannot really be influenced/designed without changing the heat capacity of the wall. It is actually a consequence of the wall’s lagging/trailing surface temperature, as the difference between the surface temperature and the ambient temperature is relevant for the resulting heat flux.

Corresponding to the internal thermal admittance (see above) the external thermal admittance describes the ability to buffer heat upon external temperature swings. Again, it is assumed that the temperature on the opposite side is held constant.
Regarding the significance of this value, please refer to external heat capacity below.

## Time-shift – external thermal admittance

Again, corresponding to the internal time shift, this result value indicates for how long the heat-flux maxima/minima will lead the temperature maxima/minima.

## Periodic thermal transmittance

The output value of periodic thermal transmittance describes the heat flux induced by a temperature swing on the OPPOSITE side of the component, assuming that the ambient temperature on the same side of the wall is held constant. Although it seems that the periodic thermal transmittance, along with its phase shift, is the favourite topic of quite some building scientists and insulation marketing specialists, the effect of the periodic thermal transmittance can nowadays be neglected for most building applications. Based on modern insulation standards (low U-values), the heat flux variations that will actually be induced by temperature swings on the opposite side of the building component will be minor. To illustrate this, we can use the tool to calculate the effect on the periodic thermal transmittance of light-weight insulation vs. heavy-weight insulation. We can demonstrate this with a simple wall (or roof) consisting solely of 20 cm of reinforced concrete and 15 cm of external insulation. A high external temperature variation of +/-15 °C is assumed (=a range of 30°C). Based on these assumptions, we get the following results:

Light-weight insulation (25 kg/m³): temperature swings internal surface: +/- 0.10°C, heat-flux: +/- 0.77 W/m², phase-shift: 7.6 hours

Heavy-weight insulation (250 kg/m³): temperature swings internal surface: +/- 0.04°C, heat-flux: +/- 0.34 W/m², phase-shift: 14.6 hours

This means that effect can be seen very well from a relative point of view. However, the difference is hardly relevant regarding its absolute impact, as the resulting total heat fluxes are minor compared to other heat sources (e.g. unshaded or open windows).

## Time-shift of periodic thermal transmittance

The value describes the lag that the heat wave induced by a temperature swing on the opposite side of the wall will have. In order to stay in line with the other time shift values, the negative sign expresses that the heat flux lags the temperature swings on the other side of the wall. Often it is stated that a time shift of 12 hours should be targeted, as this means that the maximum of the heat waves will arrive on the other side of the wall when the temperatures are lowest (or vice versa). Regarding building components matching modern building standards, this rule can be considered obsolete, as the actual surface temperature swings that are caused by temperature swings on the opposite side of the building component are usually within the range of tenths or even a few hundredths degrees Celsius. The resulting fluxes are, therefore, typically negligible.

## Internal areal heat capacity

The value of the internal heat capacity describes the ability of a building component to buffer heat during a diurnal cycle. The value specifies the amount of heat that can be buffered by one square meter during one day on a temperature swing of 1 degree; therefore, its unit is kJ/m²K. Since the underlying equations are linear, it is possible to multiply this value with any other temperature amplitude to calculate the corresponding amount of heat that can be buffered.

The areal heat capacity is calculated by integrating the heat fluxes described by the thermal admittances for a whole day. Unlike the definitions of the “single” thermal admittances are defined, the internal areal heat capacity considers temperature swings on both sides of the building component. Applying complex-number algebra, it can be calculated based on internal admittance and periodic transmittance. Depending on the actual temporal phase shift of the periodic transmittance, it can either increase – or decrease the capacity compared to a situation with constant external temperatures. However, as mentioned above, the influence of the periodic transmittance will be minor for high insulation standards. For this reason, the internal areal heat capacity is usually largely proportional to the internal thermal admittance.

It is essential to have a sufficiently large internal heat capacity to avoid overheating risks in summer and/or to reduce related cooling costs. On a summer day, the total heat capacity of the interior of the building should be able to absorb the excess heat during the daytime to subsequently discharge it during the nighttime by using natural ventilation at lower outdoor temperatures. The larger the amount of available internal heat capacity is, the lower the internal temperature swings will be. Obviously, on the first hand, daytime heat fluxes into the building should be restricted through optimal shading and by keeping windows and doors closed.

In order to determine the total heat capacity of a room, the specific areal heat capacities of all constructions are multiplied by their actual surfaces (ceiling, floor, wall-1, wall-2,…) and subsequently added. Using the provided tool, you will find out that the areal heat capacity depends primarily on the material of the innermost layer. This material should be sufficiently thermally conductive and exhibit a high heat capacity (mainly determined by its bulk density and conductivity).

This means:  a concrete ceiling will be significantly better than a suspended ceiling, a stone floor will perform better than a parquet floor (or even carpet), a thick gypsum-fibre board will outperform thin gypsum-plaster board, etc.

## External areal heat capacity

Corresponding to the internal areal heat capacity, it describes the ability of a building component to buffer heat on a diurnal temperature cycle on the external surface. Again the heat flux originating from temperature swings on the opposite (now internal) side of the building are also being considered (but usually of minor significance).

# Application notes for the tool

## General

The Excel tool is split into four sheets with different functionalities:

• Calculation-Tool
This is the main sheet where the calculation is performed. Enter your material layers and surface resistance values here to get the results (also on this sheet).
• Interactive Chart
On this page, an interactive diagram illustrates the temperature and heat-flux variations over time. You can set ambient temperature swings for one or both sides of the building component and view the resulting hea -fluxes and temperatures on both surfaces of the component.
• Materials
On this sheet, I have provided typical material data for 200 commonly used materials. You can copy & paste values to the calculation sheet.
• Validation example
On the last sheet, the validation example provided by the standard ISO 13786 is calculated to prove the validity of the algorithm.

## Surface resistances Rsi and Rse

Apart from the material layers, you will have to enter the correct surface resistances for your calculation. These describe the heat transfer from an environment into or out of the surfaces of a building component. They represent a simplified model, as the actual heat exchange results from a combination of the three physical processes (radiation, convection, and conduction). More on the theory and recommended values can be found on the dedicated page.

Please note that for the purpose of standard capacity calculations, it is recommended to use a value of 0,13 m²K/W for all cases when the heat fluxes are predominately caused by the internal temperature swings and there is no, or only little net average heat flux during one day. This means that when you would normally use 0.10 or 0.17 m²K/W for upward or downward heat-flux on U-value calculations for ceilings or floors, it might be more appropriate to use 0,13 m²K/W for either case to calculate heat-capacities. Whenever the major heat-flux caused by the 24h temperature swings is greater than the average net outflow or inflow and, therefore, the total heat-flux changes its direction (sign) two times a day, it will be more suitable to use this value.

## Internal walls, ceilings, floors

Of course, you can also use the tool to calculate the heat capacities of internal building components. In this case, just use the same surface resistance value (usually 0.13 m²K/W) for each side of the component. The labels “internal” and “external” will then only serve as a reference to identify the specific side of the wall.

## Floors with ground contact

You can also use the tool to calculate floors’ (or walls’) internal areal heat capacity with ground contact. For this purpose, I recommend adding a 2m thick layer of soil (e.g. use clay/silt of the material list) to the external side of the building component. Of course, only the internal result values will be of interest in this case. (For the chart, you would then use the monthly or yearly average soil temperature in this depth).

## Chart

The chart will help you to understand the buffering effect of your building component, as well as the occurring phase shifts on both sides.  In order to better understand the effects, you can apply a temperature swing on only one side – or you can apply temperature swings on both surfaces to reflect a more realistic situation. The 24h temperature variations can be defined by specifying an average temperature, a temperature amplitude, as well as a specific time for the maximum temperature.

Of course, the occurring temperature variations would also depend on the resulting heat fluxes going through your component, but they primarily depend on solar inputs and ventilation. Therefore, a full-blown dynamic building performance simulation would be required to precisely determine the actual temperature swings. However, in order to understand the process and estimate the potential range of the surface temperatures and heat fluxes, it is sufficient to apply realistic assumptions for the internal and external temperatures.

## Material list

The tool also includes a list of material parameters for approx. 200 common materials. You can use copy&paste to transfer the appropriate materials as layers to the calculation sheet. For accurate calculations, you should use the precise values that you should usually find on the data sheet of the specific product. If you use our HTflux Software you can use additional materials from the online material database.