## Abstract

Temperature fluctuations within a building can be attenuated by thermal mass. Adding phase change material (PCM) to thermal mass increases the effective heat capacity during the phase transition. This can anchor the temperature of the mass in a narrow band around the melting point of the PCM, further reducing the temperature swings perceived by occupants of the room. A simple dimensionless model for thermal mass forced by a sinusoidally varying air temperature is developed to calculate the performance of the PCM. The mass temperature satisfies the heat equation, with a temperature-dependent thermal diffusivity, and is solved numerically.

For a given PCM, the energy stored and returned to the room, the surface temperature amplitude and the penetration depth of heat pulses into a hypothetical semi-infinite mass can all be calculated as a function of a single dimensionless parameter. For optimal performance, the latent heat of the PCM should be as large as possible, the melting temperature range should be narrow and the thickness of the mass should exceed the penetration depth. The PCM wallboard is shown to be potentially as effective as conventional concrete, so lightweight buildings could enjoy the benefits of thermal inertia commonly associated with heavyweight structures.

## 1. Introduction

Thermal mass acts as a passive energy store to attenuate temperature fluctuations inside buildings. Heat is withdrawn from the interior at hot times of the day by natural convection and radiation and released back in cold periods. By minimizing deviations from the comfortable temperature range, the need for energy intensive heating and air conditioning can be significantly reduced. Nevertheless, the advantages of a thermally massive building often conflict with practical considerations in the design process. Aesthetics and cost pressures often require modern buildings to be increasingly lightweight. Solutions that increase the thermal mass of a building without increasing the structural weight are therefore particularly desirable.

Phase change materials (PCMs) can be used as thermal mass by microencapsulating paraffin waxes within small polymer spheres, approximately 10 μm in diameter. These are then mixed directly into the building material or facing wallboard. Significant PCM mass fractions, up to at least 30%, are achievable. The PCM is chosen to melt in the working temperature range of the thermal mass. The additional latent heat capacity of the distributed PCM increases the overall effective heat capacity compared with sensible storage alone, potentially allowing improved temperature attenuation from a smaller amount of thermal mass. Useful background is given by Zalba *et al.* (2003) and Khudhair & Farid (2004).

In the absence of PCM, the heat exchanges within a building can be modelled with linear equations, giving analytical solutions for the temperature evolution in the space. Examples can be found in Pratt (1981). In a simple model of a thermally massive building, the interior air temperature is forced by ventilation of exterior air of sinusoidally varying temperature and buffered by convective heat exchange with the thermal mass in the walls. The exterior, interior and thermal mass temperatures will then each be harmonic, but with diminishing amplitudes and increasing phase lags (Holford & Woods 2007).

By contrast, heat transfer problems with latent heat, commonly known as ‘Stefan problems’, are nonlinear and so full analytical modelling of heat transfer in PCM thermal mass is more challenging (Carslaw & Jaeger 1959). The induced temperature cycles in the thermal mass are periodic, but not sinusoidal. While numerical simulations of heat transfer (Feustel & Stetiu 1997; Neeper 1999) and experimental studies of wallboard samples and full-scale rooms (Athienitis *et al.* 1997; Schossig *et al.* 2005; Ahmad *et al.* 2006; Carbonari *et al.* 2006) have confirmed the potential of PCM in reducing interior temperature swings, the thermal response is complex. Analytical solutions are not readily available to demonstrate the sensitivity of the system to varying parameter values. This paper aims to provide a simple analysis of the effectiveness of PCM thermal mass when the thickness, the thermal properties and the amount of PCM change.

In order that a space can be said to be thermally comfortable, the perceived temperature experienced by occupants must fall within a narrow temperature range at about 22°C. The perceived temperature is not only determined by the air temperature, but also depends on diverse factors such as the radiative temperatures of objects in the room, air speed, humidity and occupant dress. However, it is often approximated by the average of the ‘dry bulb’ air temperature and the mean radiative temperature of the surfaces in the room (Fisk 1981).

In a well-insulated building, thermal mass is used to damp the temperature variations due to the ventilation of air from the outside. A minimum ventilation rate is required to provide fresh air and eliminate pollutants. In Holford & Woods (2007), the interior temperature floats freely, driven by heat fluxes from the ventilation and at the mass surface. The aim is then to reduce fluctuations in the perceived temperature, by damping both the interior air temperature and the surface temperature cycles. However, even this simple model for a room depends on four dimensionless variables.

Instead, the major part of this paper uses a similar model to Neeper (1999) supposing that the interior temperature is proscribed, it oscillates around the comfort temperature and the mass exchanges energy with this interior air by convection. It measures the performance of thermal mass by the amount of energy stored and returned to the room (the diurnal energy storage) and the lack of temperature swing at the mass surface. In reality, the comfort temperature varies with the season, making the choice of PCM melting point non-trivial. It is often chosen on the high side, focusing on avoiding the highest temperatures that lead to costly cooling, but the current work considers only the simple symmetric case. Also, it may appear paradoxical that the ability of thermal mass to reduce air temperature amplitude is measured by its response to a fixed interior air cycle. However, this model eliminates the parameters associated with the interior air and the ventilation strength, and so provides a useful tool with which to isolate the change in performance with and without PCM. It is then a reasonable assumption that if the interior air temperature were able to float freely, determined by an energy balance between the ventilation heat flux from the exterior air and convection at the mass surface, a larger diurnal energy storage would lead to a greater attenuation of the interior air temperature. Likewise, the smaller the temperature change at the surface under a proscribed interior temperature cycle, the smaller the fluctuations in the surface temperature ultimately induced by ventilation. At the end of the paper, a short example is used to validate this assertion demonstrating that the large diurnal energy storage and low surface temperatures associated with PCM thermal mass do indeed lead to a significant reduction in the perceived temperature fluctuations in a ventilated room.

As well as altering the diurnal energy storage and maximum surface temperature, PCM has a third benefit: the penetration distance of heat pulses into the mass is smaller. Any mass beyond this depth cannot affect the temperature in the system and can be discarded. Therefore, adding PCM may also allow the thickness and weight of thermal mass to be reduced.

The goal of this paper is to explain the qualitative differences between thermal mass with and without PCM, and to quantify the diurnal energy storage, maximum surface temperature and penetration depths as functions of these parameters. In §2, the dimensionless model is developed. The mass temperature depends on four dimensionless parameters, the convection strength at the mass surface, diffusivity in the mass and the latent heat and melting range width of the PCM. In §3, numerical simulations of the model are used to demonstrate the qualitative differences between normal and PCM thermal mass. Section 4 gives a detailed analysis of the effect of the parameters on the diurnal energy storage, maximum surface temperature and penetration depth. In §5, the dimensionless results are applied to the physical system and the implications for optimal design of PCM thermal mass are discussed. The optimal diurnal energy storage and the maximum surface temperature are calculated as functions of the remaining parameters, as is the mass thickness required to achieve this optimal performance. The method is illustrated with worked examples for wallboard and concrete. In §6, the model is extended to incorporate ventilation from the exterior, and the perceived temperature swing is shown to be considerably reduced by the PCM. Conclusions are drawn in §7. Table 1 shows a summary of the principal notation used in this paper.

## 2. PCM thermal mass model

The thermal mass is modelled as a wall in one-dimensional space, 0<*x*<*L*, with density *ρ*, specific heat *C* and conductivity *K*. The diffusivity *κ* is defined by *K*=*ρCκ*.

The interior air temperature *u*_{i}=*u*_{0} cos *ωt* varies sinusoidally with amplitude *u*_{0} about the comfortable temperature *u*=0. The period of oscillation 2*π*/*ω* is typically a day. The thermal mass is forced by linear convection at the interior wall (*x*=0) with constant coefficient of convection *h*. The exterior wall (*x*=*L*) is assumed to be an insulated zero flux boundary (figure 1).

Temperature *u*(*x*, *t*) in the mass then evolves over time *t* according to the heat equation(2.1)with boundary conditions(2.2)and(2.3)at the interior and exterior walls, and sinusoidal forcing(2.4)The effect of PCM is modelled by replacing the specific heat capacity *C* in equation (2.1) with an ‘effective specific heat’ that increases above *C* during melting and freezing due to the the latent heat of fusion (Carslaw & Jaeger 1959).

Suppose that the wall is impregnated with a distributed mass fraction *ϕ* of PCM with specific latent heat *f*. The PCM melts over a finite temperature range; the function *ψ*(*u*) represents the fraction of PCM in the liquid phase at temperature *u*, so that *ψ*=0 where the PCM is entirely solid, *ψ*=1 where it is entirely liquid and 0<*ψ*<1 at temperatures within the ‘melting range’. At a given temperature, the latent contribution to the effective heat capacity is *ϕf*(d*ψ*/d*u*). Except for latent heat, the PCM (in each of the liquid, solid and partially melted phases) is assumed to have identical properties to the thermal mass substrate in which it is embedded. The system therefore obeys the same equations (2.1)–(2.4) as normal thermal mass except that the specific heat capacity *C* in equation (2.1) is replaced by an effective specific heat defined by(2.5)The effective specific heat at a given temperature depends on the melting profile *ψ*(*u*). Mathematical analysis of the Stefan problem generally assumes that PCM melts isothermally at a discrete temperature, in which case *ψ* takes the form of a delta function (Carslaw & Jaeger 1959). In fact, real PCMs generally melt as some complex function over a finite temperature range. He *et al.* (2004) analyse the melting profile of two paraffin-based PCMs. Simpler models of melting ranges have variously assumed d*ψ*/d*u* to take the form of a step function (Carslaw & Jaeger 1959), a Gaussian (Neeper 1999) and a triangular function (Kedl 1991). Neeper (1999) notes that the exact form of *ψ* matters relatively little, at least beyond some characteristic width of melting range.

This work assumes that *ψ*(*u*) takes a simple form where d*ψ*/d*u* is a step function, taking a fixed positive value 1/2*u*_{m} over a symmetric temperature range −*u*_{m}<*u*<*u*_{m}. The effective specific heat in equation (2.5) then takes only two values, a normal specific heat *C* and an elevated specific heat *C*+(*ϕf*/2*u*_{m}), depending on whether the temperature is inside or outside the melting range.

It is convenient to replace {*x*, *t*, *u*} with non-dimensional variables *X*=*x*/*L*, *T*=*ωt* and *U*=*u*/*u*_{0}. Equations (2.1)–(2.4) can then be written in non-dimensional form as(2.6)(2.7)(2.8)(2.9)and(2.10)There are four key dimensionless parameters.

The dimensionless convection rate .

*H*compares the energy supplied by convection over a period*h*/*ω*with the heat capacity of the mass*ρLC*.

The dimensionless diffusivity .

*Ω*measures the penetration depth of temperature oscillations at the boundary. It is related to the decay distance for thermal mass without PCM (Carslaw & Jaeger 1959; Fisk 1981), by the relation .

The Stefan number

*s*compares the energy required to overcome the latent heat of the PCM*ϕf*with the energy required to increase the temperature of the mass without PCM through the full interior temperature range 2*Cu*_{0}.

The dimensionless melting width

*U*_{m}=*u*_{m}/*u*_{0}.

These can naturally be grouped into a pair of mass parameters {*H*, *Ω*} that are related to the convection, forcing frequency and the thermal properties of the mass without the PCM, and a pair of PCM parameters {*s*, *U*_{m}}.

Since *Ω* is inversely proportional to *C* (through the definition of *κ*), the effective specific heat induces a temperature-dependent effective diffusivity in equation (2.7). The effective diffusivity in the thermal mass decreases from its normal value *Ω* during phase change to take a reduced value as in equation (2.7). This can be seen in figure 2. This *U* dependence in the effective diffusivity destroys the linearity of the system. Note that the *Ω* term in equation (2.8) is unchanged in the melting range, and, unlike in the dimensional heat equation (equation (2.1)), there is no explicit dependence on *C* in the dimensional boundary condition (equation (2.2)).

### (a) Metrics

The nonlinearity induced by the PCM means that the temperature fluctuations of the thermal mass and the deviations of internal energy from the mean (‘energy storage’) are no longer sinusoidal. While they cannot be completely described by a simple harmonic amplitude and phase, they can still be usefully characterized by their maximum value. The diurnal (maximum) energy storage and the maximum surface temperature are of particular interest. A large diurnal energy storage attenuates the dry bulb air temperature by removing more heat from the interior over the warm part of the day. Reducing the maximum surface temperature reduces the radiative temperature of the walls. These metrics are used to measure the performance of PCM thermal mass in attenuating the perceived temperature fluctuations, as discussed in §1. In a full analysis, where the room temperature is forced by ventilation and heat gain within the building, these metrics will be coupled via the induced air temperature. Here, however, by using a fixed air temperature cycle, the diurnal energy storage and the maximum surface temperature provide a useful basis for comparing the performance of thermal mass with different amounts of PCM.

The diurnal energy storage *D* measures the total energy stored and released by the mass over the daily cycle according to(2.11)where the system is assumed to start in steady state at *T*=0. Note that the diurnal energy storage is measured as a dimensionless temperature, normalized by *u*_{0}*ρLC*, the internal energy change of mass with no PCM over the temperature swing of the interior air. The diurnal energy storage is the maximum of the normalized instantaneous energy stored in the mass *E*(*T*), which can be measured in two equivalent ways: by integrating either the normalized local energy storage across the mass width or the instantaneous heat flux at the surface (equation (2.8)) in time. The local energy storage at a given point in the mass is expressed as the sum of two components: *U*, the energy associated with a sensible temperature change (‘sensible energy storage’), and , the energy required to overcome the specific latent heat (‘latent energy storage’). Note that *D* is measured relative to the baseline internal energy of mass at the comfort temperature *U*=0; in particular, the latent energy storage is measured relative to the half-solid/half-melted state *ψ*(0)=1/2.

Without PCM, *s*=0 and the expression for *D* reduces to , the maximum temperature of the mass averaged across its width. Therefore, for PCM thermal mass, *D* represents the average dimensionless temperature that normal thermal mass would reach if the same amount of energy were to be stored within it. Without PCM, the maximum possible diurnal energy storage is 1, occurring when the mass temperature reaches *U*=1 at each point across its width. With PCM, the maximum possible diurnal energy storage becomes *D*=1+*s*, where *s* is the extra energy required to melt the PCM. In dimensional terms, the maximum energy storage, denoted , is given by *Du*_{0}*ρLC* joules per unit surface area of thermal mass.

The maximum surface temperature *M* is defined simply as(2.12)taking values . The dimensional surface temperature deviation from the interior mean is kelvin.

As well as improving performance, PCM has the potential to reduce the amount of mass necessary to achieve this performance, by reducing the distance that heat pulses penetrate into the mass. This introduces a third metric, the penetration distance *P*. It is defined as the maximum distance into the mass at which the PCM fully melts and freezes at some point in the cycle(2.13)taking values . *P* is still a useful measure, even if *s*=0, and there is no actual PCM in the mass; in this case, it measures the depth to which the mass temperature fluctuates beyond the nominal melting range −*U*_{m}<*U*<*U*_{m}. Values of *P*<1 mean that a section at the back of the mass is not accessed by the heat pulses and provides no marginal benefit. The dimensional penetration depth is metres.

In comparing normal and PCM thermal mass, subscripted versions of the metrics are used, e.g. *D*_{norm}, *D*_{PCM}, etc.

### (b) Numerical method

Equations (2.6)–(2.10) were solved numerically using the Crank–Nicolson method with variable diffusivity. Finite-difference schemes are well suited to nonlinear heat transfer problems as the variable diffusion coefficient can be easily incorporated into the algorithm. The effective diffusivity is determined locally at individual grid points. Crank–Nicolson is an implicit scheme; in order to avoid solving a nonlinear system of equations, the implicit value of the diffusivity at the forward time step is assumed to take the current value. To obtain accurate results with this method, it is important that the time step satisfies so that each spatial grid point spends enough time steps in the melting zone over each cycle to experience the local reduction in diffusivity (Hu & Argyropoulos 1996). A relatively fine grid was used: typically, the dimensionless spatial step size was Δ*X*=10^{−3} and the time step was Δ*T*/2*π*=10^{−4}, though these were varied according to the convergence speed at different parameter values. Initially, the thermal mass started with temperature *U*(*X*, 0)=0 throughout, and the system was run through approximately 10 cycles to reach steady state before results were taken. Athienitis *et al.* (1997) report a good agreement between experimental and numerical results using a one-dimensional finite-difference scheme to simulate a room with PCM thermal mass.

## 3. Thermal mass response

This section illustrates the qualitative differences in the behaviour of normal and PCM thermal mass in response to the interior temperature cycle. The system is analysed using the results of numerical simulations of equations (2.6)–(2.10).

### (a) Dynamic response

Figure 3 shows the dynamic evolution of temperature and energy storage in the thermal mass for a series of different diffusivities *Ω* with convection strength *H*=2 fixed. It plots the temperature of normal thermal mass, the temperature of PCM thermal mass and the distribution of energy storage over the mass cross section, at 30 regular times across the day. Values of the diurnal energy storage, maximum surface temperature and penetration depth for the plots in figure 3 are shown in table 2.

Figure 3*a*–*d* is for thermal mass with no PCM (*s*=0). In normal mass, the temperature series at each point is sinusoidal with amplitude decreasing in depth *X*.

The diffusivity increases from left to right. For small values of *Ω*, as in figure 3*a*, heat pulses do not reach the insulated surface *X*=1 and so the mass can be regarded as a semi-infinite width. In this case, the temperature amplitude decreases exponentially in *X* with a dimensionless decay distance , defined as the depth at which the amplitude is attenuated by a factor of e from that at the surface (Carslaw & Jaeger 1959).1 As *Ω* rises, the decay distance increases until heat pulses penetrate the full thickness of the wall, as in figure 3*b*. There are significant temperature fluctuations at all points in the mass, though the amplitude still drops significantly as the depth increases.2

In figure 3*c*,*d*, the diffusivity *Ω* is large and the decay distance *δ*≫1. The mass becomes increasingly isothermal across its width as the diffusivity rises. For large values of *Ω*, the Biot number *H*/*Ω* is small and so the mass can be regarded as isothermal. In this regime, the system is controlled by *H* alone.

From this analysis, three separate regimes can be identified: a low-penetration regime *δ*≪1 in which temperature fluctuations are restricted to a surface boundary layer (figure 3*a*), a full-penetration regime *δ*∼1 in which there are significant temperature fluctuations throughout the mass (figure 3*b*) and an isothermal regime in which *δ*≫1 and the mass temperature is essentially constant across its width (figure 3*d*).

Figure 3*e*–*h* shows the corresponding temperature profiles for PCM thermal mass with *s*=1. Figure 3*i*–*l* shows the local energy storage *N*(*X*, *T*) across the mass. For normal thermal mass, there is no distinction between temperature and local energy storage since energy storage is normalized as a dimensionless temperature (equation (2.11)) and there is no latent heat. Therefore, figure 3*a*–*d* is the reference to which both figure 3*e*–*h* and *i*–*l* should be compared. The instantaneous energy storage is the average of the local energy storage across the mass width. The diurnal energy storage is then the maximum instantaneous energy storage. From the figures and the table, it can be seen that the PCM increases the diurnal energy storage *D*, decreases the maximum surface temperature *M* and decreases the penetration depth *P* for all values of *Ω*. Even though the total energy stored in the mass is greater with PCM, the temperature at each point is typically less than that for normal thermal mass since much of this is latent, rather than sensible, energy storage, which leaves the temperature of the mass unchanged. The maximum surface temperature *M* and the diurnal energy storage *D* are linked since a reduced temperature amplitude at the surface strengthens the convective heat flux (equation (2.8)) and so increases the diurnal energy storage. *M* and *D* therefore tend to move in inverse lock step with both effects being beneficial to the perceived temperature.

For small values of *Ω*, temperature fluctuations are concentrated in a narrower layer than for normal mass (figure 3*e–g*). The low-penetration regime therefore persists for larger values of *Ω*. Rather than declining exponentially, the decrease in amplitude of temperature fluctuations is roughly linear in *X* up to the penetration depth *P*, and the maximum temperatures at points *X*<*P* occur approximately in phase. This is because the temperature at the rear of the thermal mass (*X*>*P*) is effectively pinned in the melting zone by the PCM; for moderate values of *s*, the PCM melts on a longer time scale than that required for diffusion to smooth the temperature gradient in *X*<*P*. Gradients in the fully melted region of the mass are thus approximately linear, inscribing fan-shaped temperature envelopes as seen in figures 3*e*–*h*. It should be noted that while temperature fluctuations are small beyond depth *P*, there can still be considerable latent energy storage. To estimate the width of the region in which this is significant, note that the mass in *X*>*P* has a constant effective diffusivity equal to the reduced value *Ω*_{m}. It can be viewed as normal mass, with reduced diffusivity, forced by a (non-harmonic) temperature amplitude of *U*_{m} at the ‘surface’ *X*=*P*. Therefore, in the region *X*>*P*, both temperature and energy storage decrease exponentially with short decay distance . Comparing figure 3*i*,*j* with figure 3*a*,*b*, it is clear that even paying due regard to this region, the boundary layer in which energy is stored is still much thinner with PCM than without.

As *Ω* increases, the penetration depth *P* reaches the back of the mass. In figure 3*h*, the PCM is overwhelmed and the full-penetration regime sets in. Once this occurs, the temperature gain at the mass surface is no longer limited by the slow diffusion rate during phase change, and the temperature across the mass collapses quickly onto the normal thermal mass solution. In the full-penetration regime, the maximum surface temperature *M*_{PCM} can approach *M*_{norm}, in which case the benefit of PCM would be substantially diminished.

### (b) Surface temperature and energy storage

Figure 4 shows the time evolution of surface temperature and instantaneous energy storage of PCM thermal mass with *s*=3. Without PCM, each of the plots would be sinusoidal with varying amplitude and phase. With PCM, the shapes of the temperature series and the energy storage are irregular with markedly different forms in the low- and full-penetration regimes. Plots are shown for three different values of *Ω*. The *H* values are chosen so that takes the same set of values for each choice of *Ω*. The Stefan number has been increased from figure 3 to make the effects of PCM more clear.

In figure 4*a*,*d*, the diffusivity *Ω*=1. The system is in the low-penetration regime, as seen in figure 3*e*–*g*. The surface temperature is close to sinusoidal, except for a kink in figure 4*a* at , where the surface temperature enters the PCM melting zone and the rate of temperature change is suppressed by the latent heat. Note that there is no such kink upon leaving the melting zone as the temperature is still pegged by the reduced diffusion rate at the back of the mass where the PCM is still changing phase. Figure 4*d* shows the instantaneous energy storage time series. Again, it is approximately sinusoidal though slightly skewed at the point where the surface temperature enters the melting zone. Here, energy storage/release is accelerated due to the widening temperature contrast between the air and the surface. Since the penetration depth *P*<1, a significant fraction of the latent capacity of the PCM is unused, and energy storage falls well short of the theoretical maximum 1+*s*=4.

In figure 4*b*,*e*, the diffusivity *Ω*=5. This is an intermediate case in which the system moves from the low-penetration regime to the full-penetration regime as the convection strength *H* rises. For small values of *H*, the convection strength is insufficient to melt all the PCM and so the temperature series are the same shape as in figure 4*a*. For large values of *H*, there is enough energy to melt the PCM at the back of the wall and the full-penetration regime sets in. As the PCM is fully melted, the limiting diffusion rate across the mass jumps from the reduced value to the normal value, and the surface temperature ‘pops’ up out of the melting zone, leading to the characteristic humps in figure 4*b*. This corresponds to the behaviour seen in figure 3*h*. For larger values of *H*, this point occurs before the surface temperature reaches its maximum. However, for intermediate values, it can occur after the surface reaches its initial maximum temperature, in which case the temperature cycle may be bimodal. In the full-penetration regime, the diurnal energy storage must be at least *D*>*s* since all the PCM changes phase.

In figure 4*c*,*f*, the diffusivity *Ω*=20 is large. For sufficiently small values of *H*, the low-penetration regime still applies, but it breaks down at smaller values as *Ω* rises. The larger diffusivity means the mass is more isothermal than that in figure 4*b*. Consequently, the delay between the PCM fully melting at the front and the back of the mass is shorter. The mass spends more time uninhibited by the PCM, and so the maximum surface temperature is much greater. For this value of *Ω*, the Biot number is small outside the melting zone and the mass is essentially isothermal. However, the effective Biot number is approximately 30 times larger in the melting zone, and the mass is not isothermal during melting. Consequently, there is still a significant difference in the time at which the PCM fully melts at different points across the mass width. In order for the mass to be treated as isothermal in both these regions, the value of *Ω* must be very large indeed.

### (c) Summary

The plots in figures 3 and 4 illustrate the complexity of the PCM thermal mass response to even a simple sinusoidal interior temperature. The regular sinusoidal profiles of normal thermal mass are replaced with irregular complex evolutions that cannot be completely described by amplitude alone. Two qualitatively different regimes have been observed, a low-penetration regime in which *P*<1 and some of the PCM at the rear of the mass does not change phase, and a full-penetration regime in which *P*=1 and the PCM melts across the whole mass width. In the low-penetration regime, the maximum surface temperature is suppressed by the latent heat of the PCM, and this large temperature contrast between the air and the surface drives an increased diurnal energy storage. The boundary layer in which temperature fluctuations are significant is markedly reduced. In the full-penetration regime, all the PCM melts, and so the surface temperature is no longer effectively anchored by the PCM across the whole day, drifting towards the normal mass solution as *H* and *Ω* increase. The mass is significantly less effective at reducing the radiative temperature of the mass surface than in the low-penetration regime.

## 4. Parameter study

This section neglects the detailed structure of the mass response at different depths and at different points in the diurnal cycle. Performance is measured solely by the maximum surface temperature *M*, the diurnal energy storage *D* and the penetration depth *P*, and the response of these metrics to the mass parameters {*H*, *Ω*} is plotted and analysed for a wide range of values. In the low-penetration regime, the system is shown to depend on the mass parameters solely through the value of a low-penetration parameter , whereas in the full-penetration regime, the mass parameters can affect the system independently.

### (a) Low-penetration regime

In §3, a low-penetration regime was identified in which the heat pulses at the surface were unable to penetrate to the back of the thermal mass. Clearly, in this regime, the exact thickness of the mass *L* can have no effect on the behaviour of the physical system. Any controlling parameter must therefore be independent of *L*. Since *H*∝1/*L* and *Ω*∝1/*L*^{2}, the relevant combination is . This parameter *Z* will be called the ‘low-penetration parameter’.

Figure 5 shows a plot of the maximum surface temperature, diurnal energy storage and penetration depth in both normal and PCM thermal mass for varying values of *Ω*, but with *H* also varying in such a way that remains fixed.

For small values of *Ω*<0.1, the penetration depth *P*<1 for both normal and PCM thermal mass and the low-penetration regime applies. The maximum surface temperatures are solely determined by the value of *Z*, and so each is constant in *Ω* with *M*_{PCM}<*M*_{norm}. The diurnal energy storage increases in *Ω* with *D*_{PCM}>*D*_{norm}. In fact, in the low-penetration regime, the diurnal energy storage is proportional to , since the surface temperature cycle is identical for all values of *Ω*, the heat flux at the boundary is proportional to the convection strength *H* (equation (2.8)), which in turn scales with as is fixed.

As *Ω* increases, the penetration depth *P* increases until it reaches *P*=1, first for the normal thermal mass at *Ω*∼0.1 and eventually for PCM thermal mass at *Ω*∼1. At these points, heat pulses reach the back of the mass and the low-penetration regime breaks down. As *H* and *Ω* continue to increase, both the maximum surface temperature and the diurnal energy storage rise as the mass becomes increasingly isothermal and the maximum temperature approaches 1 across the mass width. The maximum surface temperatures *M*_{norm} and *M*_{PCM}→1 and the diurnal energy storages approach their maximum values *D*_{norm}→1 and *D*_{PCM}→1+*s*, respectively. Note that for PCM thermal mass, the maximum surface temperature increases much more rapidly than for normal thermal mass once the PCM is fully melted. As the diffusivity increases further, the benefit of the PCM to the surface temperature is quickly lost. In the full-penetration regime, *M* is no longer a function of a single parameter *Z*, but depends on *H* and *Ω* independently.

Revisiting figure 4, recall that the convection strengths for each value of *Ω* were chosen so that the set of *Z* values were the same. Consequently, whenever the low-penetration regime applies, the surface temperature profiles are the same in each of the three plots. For *Ω*=1, the low-penetration regime applies for all values of *Z*, and so equivalent plots for smaller values of *Ω*<1 would be identical to figure 4*a*.

### (b) Systematic parameter study

Having established that controls the system in the low-penetration regime, it is natural to make a transformation of the mass parameters {*H*, *Ω*} to {*Z*, *Ω*} and to consider the response of the mass to these.

Figure 6 systematically analyses the response of the maximum surface temperature and the diurnal energy storage to a full range of values of {*Z*, *Ω*}. *M* and *D* are plotted as a function of the low-penetration parameter *Z* for a series of different values of *Ω*, for both normal and PCM thermal mass.

Figure 6*a* shows the maximum surface temperature. PCM reduces the value of *M* for all choices of *Z*. For small values of *Z*, the maximum surface temperature for PCM thermal mass is the same for all values of *Ω*, since the system is in the low-penetration regime. This solution is referred to as the ‘low-penetration solution’ and is a function of *Z* alone. For normal mass, there is also a low-penetration solution common to all the plots, but, for *Ω*>0.1, the solutions diverge from this low-penetration solution at smaller values of *Z* than those shown. For both these low-penetration solutions, the values of *M* increase with *Z* as the convection strength *H* increases and the surface responds more to the interior temperature cycle.

As *Z* increases the low-penetration regime may start to break down as the increasing convection strength supplies enough energy for heat pulses that reach the rear surface of the mass. Depending on the value of *Ω*, however, the low-penetration regime may persist for all values of *Z*. This is because no matter how large the convection strength *H*, the surface temperature cycle can only ever approach that of the interior air. The penetration depth is therefore always ultimately limited by *Ω*. For normal thermal mass, the diffusivity must be *Ω*∼0.1 or less in order that the penetration depth *P*<1 for all values of *Z*. For PCM thermal mass with *s*=2, larger values up to *Ω*∼1 will suffice. These orders of magnitude are consistent with the values of *Ω* required to fully melt the PCM in figure 5. The solutions for all values of *Ω* less than these critical values are the same, and they define the low-penetration solutions for normal and PCM thermal mass seen in figure 6.

Suppose instead that *Ω* is larger than this critical value, so that the low-penetration solution does break down as *Z* increases. In the low-penetration regime, the surface temperature cycle is constant at a fixed value of *Z*, but the larger the value of *Ω*, the quicker the heat is supplied from the surface to the back of the mass. Therefore, the larger the value of *Ω*, the smaller the value of *Z* at which this breakdown occurs. For normal thermal mass, the plot gently diverges from the low-penetration solution, while for PCM thermal mass, the maximum surface temperature diverges rapidly, collapsing towards the normal mass solution as *Z* increases. This is consistent with the observations made in figure 4 that once the PCM is fully melted, the mass temperature pops out of the melting zone and some of the benefits of the PCM are lost. As *Z* increases further, the maximum surface temperature *M*→1.

Figure 6*b* shows the diurnal energy storage. PCM increases the value of *D* for each value of *Ω* and *Z*. In the low-penetration regime, *D* is proportional to for fixed *Z*, as in figure 5. As *Z* increases, the energy storage increases as the convective heat flux at the surface strengthens. For large values of *Z*, the surface temperature tends to the interior temperature cycle, and the diurnal energy storage tends to a constant value. This limiting value of *D* is increasing in the diffusivity *Ω*. For normal thermal mass, the maximum value of *D* is 1, whereas for PCM mass, *D* can reach 1+*s*. For those values of *Ω* for which all the PCM melts, the diurnal energy storage must clearly be at least *D*=*s*.

### (c) Summary

In this section, the maximum surface temperature *M* and diurnal energy storage *D* have been evaluated across the mass parameter space {*Z*, *Ω*}. PCM reduces the maximum surface temperature, increases the diurnal energy storage and decreases the penetration depth for all values of the mass parameters.

A low-penetration regime has been identified in which the maximum surface temperature is a function of a low-penetration parameter alone. In the low-penetration regime, the surface temperature tends to be significantly attenuated by the PCM, but the benefit diminishes substantially once the PCM has been fully melted. For sufficiently small values of *Ω*, heat pulses are never able to reach the back of the mass and the low-penetration regime persists for all values of *Z*. With PCM, it persists to larger values of *Ω*.

In §5, these dimensionless results will be applied to the physical system, and used to determine the principles to be followed when designing PCM thermal mass.

## 5. Designing PCM thermal mass

It is important to understand how real thermal mass should be designed to optimize performance in dimensional terms. Many of the dimensional parameters, such as the convection strength *h* and the diffusivity of the wall material *κ*, cannot be easily manipulated in the design process.

Typically, the key parameters that can be altered to improve performance in a real room will be the amount of PCM *ϕ* to be incorporated into the thermal mass and the thermal mass thickness *L*. Altering the value of *ϕ* changes the Stefan number *s*, and changing *L* corresponds to moving along contours of constant .

This section illustrates the optimal way in which *ϕ* and *L* should be chosen in applications. It then evaluates the achievable energy storage and surface temperatures in physical units for PCMs of different latent heats. It also calculates the reduction in mass width that is possible from adding PCM, before applying these results to wallboard and concrete to quantify the scale of the benefit that PCM provides.

### (a) Optimizing PCM thermal mass

Sections 3 and 4 have shown that PCM thermal mass outperforms normal thermal mass. As the amount of PCM increases and the Stefan number rises, performance is expected to further improve. Figure 7*a* shows plots of diurnal energy storage *D*, maximum surface temperature *M* and penetration depth *P* as functions of the Stefan number *s*.

As expected, the performance of the thermal mass improves as the Stefan number increases. The maximum surface temperature *M* drops, the diurnal energy storage *D* rises and the penetration depth *P* falls. With this choice of *Ω* and *H*, all the PCM changes phase for small values of *s*, but as the latent heat increases, there becomes a critical value of *s* at which the low-penetration regime sets in, as the PCM fails to melt at the back of the wall.

It is clear from figure 7*a* that, in preparing PCM thermal mass, it is always best to set the Stefan number as large as possible. As well as choosing a PCM with a large latent heat *f*, the mass fraction *ϕ* of PCM should also be large. However, there will be an upper bound on *ϕ*, beyond which the structural integrity of the thermal mass or wallboard cannot be maintained. This restriction places a limit on the maximum value of *s* that can be achieved.

Different PCMs melt over different melting range widths. It is therefore also important to understand the effect of *U*_{m} on mass performance. Smaller values of *U*_{m} would be expected to lock the mass temperature closer to the mean, leading to smaller values of *M* and larger values of *D*. Figure 7*b* shows a plot of the diurnal energy storage *D* and the maximum surface temperature *M* for varying values of the melting range width *U*_{m}. As *U*_{m} increases, the maximum surface temperature *M* consistently increases. However, the diurnal energy storage remains relatively constant over a reasonably wide range of values of *U*_{m}. Initially, it even increases slightly as *U*_{m} rises. Similar behaviour is observed for other choices of *H* and *Ω*. Note that for melting width values *U*_{m}>*M*, the PCM never fully melts, even at the surface, and as *U*_{m} increases further, performance will inevitably drop off as more of the latent heat is not being accessed. Neeper (1999) found that the diurnal energy storage increases with the melting width, but figure 7*b* suggests that this is not strictly obeyed in all cases. Nevertheless, it broadly confirms that a smaller melting range improves performance.

The next challenge is to establish the optimal width of PCM thermal mass that should be used. Varying the mass width corresponds to moving through contours of constant .

In physical units, the maximum surface temperature, diurnal energy storage and penetration depth must all be constant in *L* in the low-penetration regime. Note that in §4, the diurnal energy storage *D* was not constant in the low-penetration regime, but varied in proportion to . As currently defined, *D* is inappropriate for comparing the true physical energy storage for changing values of *L* because in physical units the diurnal energy storage has an additional *L* dependence. Consequently, changing the width of the mass beyond the limiting penetration depth will affect the value of *D*, even though it has no impact on the physical energy stored within the system. To avoid this problem, it is necessary to normalize *D* by multiplying it by a quantity proportional to *L*. A ‘width-adjusted diurnal energy storage’ is thus defined. In terms of , the physical energy storage has no explicit *L* dependence, so is constant for changing *Ω* in the low-penetration regime and robust for comparing the energy storage under changing mass widths. can be thought of as measuring the diurnal energy storage of an infinite width of mass relative to the energy required to raise the temperature of a reference width by *u*_{0}.

Likewise, a width-adjusted penetration depth is defined. This measures the depth to which the PCM fully changes phase in physically robust units. The dimensional penetration depth is then .

Figure 8 shows the effect of thermal mass thickness on the performance of the mass. The *x*-axis shows , the mass width *L* measured relative to a control mass of thickness . For both normal and PCM thermal mass, the width-adjusted diurnal energy storage *D*′ broadly increases and maximum surface temperature *M* broadly decreases with increasing mass width. At small values of *L*, there is only a very thin layer of mass with little thermal inertia, and so very little energy can be stored, and the temperature at the surface essentially locks onto the interior cycle. For large widths, temperature fluctuations cannot reach the back of the mass and so *D*′ and *M* are insensitive to the exact value of *L*. At intermediate widths, as heat pulses first interact with the insulated boundary at the back of the mass, there is an anomalous region in which *D*′ dips and *M* rises briefly, before tending to their limiting values as the width increases further.

Figure 8 shows that for both normal and PCM thermal mass, the best performance can be obtained by choosing *L* to be sufficiently large that heat pulses cannot reach the back of the mass. Beyond this width, however, any additional thermal mass is unnecessary, providing no improvement in performance. The limiting value of the width-adjusted penetration depth *P*′ shows the relative depths to which heat pulses can penetrate in a semi-infinite section of mass. It is much smaller for PCM thermal mass than for normal thermal mass, so PCM thermal mass can be made much thinner.

In summary, the best PCM mass performance is obtained from choosing *s* as large as is technically possible within the constraints of the available PCMs and the maximum mass fractions of PCM that can be used. The melting range width should also be small, though performance is relatively insensitive to the exact value of *U*_{m}. The mass width should then be chosen to be at least the penetration depth of heat pulses into a semi-infinite section of mass. Any thermal mass beyond this depth cannot be accessed and provides no additional benefit.

### (b) Quantification of optimal performance

Section 5*a* has shown that optimal performance results from choosing a thermal mass thickness *L* larger than the depth to which heat can penetrate. Such a choice guarantees that the system can be analysed entirely in the low-penetration regime and removes the width parameter *L* from the analysis.

With *L* chosen in this way, the intrinsic performance potential of a thermal mass material depends on: (i) the thermal properties of the material: diffusivity *κ*, density *ρ* and specific heat *C*, (ii) the convection strength at the boundary *h*, and (iii) the maximum achievable Stefan number *s*. This section provides complete solutions for the dimensional maximum surface temperature and diurnal energy storage that can be achieved as a function of these physical parameters, as well as the minimum width of mass *L* required to realize this performance.

Figure 9 shows the best achievable maximum surface temperature *M*, width-adjusted diurnal energy storage *D*′ and width-adjusted penetration depth *P*′ as a function of the low-penetration parameter *Z* for different choices of the Stefan number *s*. It assumes that *L* has been chosen in the optimal fashion so that the low-penetration regime applies.

In preparing figure 9, the conditions required for the low-penetration regime must be met. In dimensionless terms, *Ω* must be chosen to be sufficiently small that heat pulses cannot reach the back of the mass; in dimensional terms, the width *L* must be sufficiently large that the same condition is met. Since the penetration depth of the heat pulses is larger the smaller the value of *s*, the case with no PCM (*s*=0) sets the most restrictive condition on *Ω*. In practical terms, any value of *Ω*∼0.01 or less has negligible amplitude at the rear of the mass, no matter how large *H* may be; with *Ω*=0.01, the amplitude at the rear surface would be approximately 10^{−3} as *H*→∞ and the mass surface approaches the interior temperature cycle.

Figure 9*a* shows the variation of the maximum surface temperature *M* with the low-penetration parameter . Each series is a low-penetration solution (figure 6*a*) for different values of *s*. *M* increases with *Z* in each case. As the convection strength *h* increases, the mass receives a greater heat flux at the surface. If *κ* is fixed, this energy cannot be carried away any faster and so accumulates at the surface. *Z* increases and the surface temperature rises. Alternatively, if *κ* increases for fixed *h*, the heat arriving at the surface diffuses more quickly away into the body of the mass and so the surface amplitude decreases. Of course, either of these changes will also affect the width of mass required. For all values of *Z*, the maximum surface temperature decreases significantly as the Stefan number *s* rises and the latent heat of the PCM increases. In dimensional terms, the maximum surface temperature is given by where is the function in figure 9*a*. For an interior temperature amplitude of *u*_{0}=5 K, adding PCM with *s*=10 to mass with *Z*=1 reduces the maximum surface temperature from 2.7 to 0.5 K, and even for PCM with *s*=1, a reduction in surface temperature of 1 K is possible over a wide range of values of *Z*.

Figure 9*b* shows the width-adjusted diurnal energy storage *D*′. *D*′ is also an increasing function of *Z*. In physical units, the diurnal energy storage is joules per unit surface area. For fixed *κ*, increasing the convection strength *h* increases the diurnal energy storage as the amplitude of the convective heat flux into the mass increases. Increasing *κ* for fixed *h* depresses the maximum surface temperature and so strengthens the convective heat flux leading to greater energy storage. Although increasing *κ* reduces *Z* and so decreases the value of *D*′, the dimensional energy storage is proportional to and increases overall. Again as the Stefan number increases, *D*′ increases, more energy can be stored and performance improves. For a moderate value of *Z*=1, the diurnal energy storage increases by approximately 50% after adding PCM with *s*=1.

In summary, increasing *κ* is beneficial to both the maximum surface temperature that is reduced and the diurnal energy storage that is increased. Increasing *h* is detrimental in that it increases the maximum surface temperature, but beneficial in that it increases the diurnal energy storage. Increasing the Stefan number *s* is beneficial on both measures.

In preparing figure 9*a*,*b*, it has been assumed that *L* is larger than the penetration depth. Figure 9*c* can be used to calculate the value of *L* necessary for this assumption to hold. It shows the width-adjusted penetration distance *P*′ as a function of *Z*. The dimensional penetration depth is . Disregarding the negligible energy storage beyond this depth, this is approximately the thickness of mass needed to achieve optimal performance.

For small values of *Z*, the nominal value of *P*′ is zero. This means that there is insufficient energy even to melt the PCM at the surface. Beyond this region, *P*′ is an increasing function of *Z*. As with the diurnal energy storage, the dimensional penetration depth increases in both the convection strength *h* and the diffusivity *κ*. Figure 9*c* is particularly striking for the dramatic decrease in penetration depth as the Stefan number increases. The thermal mass can afford to be many times thinner with PCM than without, even as the performance of the mass increases. Even for *s*=1, allowing sufficient latitude for energy storage in the partially melted region beyond the nominal penetration depth, the thermal mass can be much smaller; for *Z*=1, the penetration depth is approximately five times less with PCM than without.

### (c) Worked examples

In this section, the results of figure 9 will be used to quantify the ability of both construction wallboard and concrete to provide thermal mass, with and without PCM. Wallboard is used to make interior walls and ceilings in frame buildings, but since it is thin and conducts relatively poorly, it typically provides little thermal mass. By contrast, heavyweight concrete and masonry buildings have thicker walls with better conductivity, which are much more effective.

Table 3 gives representative thermal parameter values for wallboard and concrete. The value of the low-penetration parameter *Z* is calculated. Wallboard has a *Z* value of 2.38, whereas concrete has a *Z* value of 0.55. Since the convection strength *h* is held constant, the relative size of these values reflects the differences in conductivity of the materials as well as the relative density and specific heat. In particular, the conductivity of concrete is approximately 10 times that of wallboard. The latent heat *ϕf* is assumed to be constant in dimensional terms in the two cases, but the difference in the specific heats of the materials leads to slightly different Stefan numbers.

Applying these values of *Z* and *s* to the functions plotted in figure 9, the diurnal energy storage, maximum surface temperature and penetration depth can be calculated for a semi-infinite width of wall material. The amplitude of the air temperature cycle was assumed to be *u*_{0}=2.5 K. The resulting temperature at the surface of the hypothetical wallboard would have a relatively large amplitude of 1.88 K, and so the radiative temperature of the walls would provide little relief. By contrast, concrete has a surface temperature amplitude of only 0.9 K. The low conductivity of the wallboard cannot allow heat to diffuse away from the surface as quickly as in concrete, and so heat accumulates in a surface boundary layer that is strongly affected by the interior temperature. The smaller temperature contrast between the air and the mass surface results in energy storage of only 77 kJ m^{−2} compared with 167 kJ m^{−2} for concrete. Note that the energy storage and surface temperatures are quoted relative to *u*=0. They can be doubled for comparison with the peak to peak values that are sometimes given.

It must be emphasized that these results are for a semi-infinite width of mass. The table shows that the thickness of wallboard required to realize this performance would be approximately 12.6 cm, the penetration depth of heat pulses into the semi-infinite width. By contrast, typical wallboard is only approximately 1 cm thick. For this thickness, the amplitude of the surface temperature would differ from the air by less than 1%, and so conventional wallboard provides no thermal mass whatsoever. Instead, the results quoted from table 3 are intended to demonstrate the performance of the wallboard material itself in comparison with concrete, irrespective of the typical thickness found in applications.

Once PCM is added, the performance of the wallboard dramatically improves. The surface temperature amplitude is 0.9 K, and the energy storage increases to 177 kJ m^{−2}. These figures are very similar to those found for conventional concrete. Most importantly, the penetration depth is reduced from 12.6 to 1.1 cm once PCM is added. Beyond this point, energy storage drops off with a decay distance . This means that heat pulses cannot penetrate further than a couple of centimetres and so a standard thickness of wallboard would be sufficient to achieve the optimal performance. PCM wallboard has the potential to replicate the thermal mass of a heavyweight concrete building in a light frame construction.

Adding PCM to concrete also improves performance. The amplitude of the surface temperature cycle is just 0.2 K and the energy storage rises to 237 kJ m^{−2}, providing excellent thermal comfort. The nominal penetration depth is zero, since even the PCM at the surface never fully melts. The effective decay distance in the partially melted region is *δ*=2.6 cm, so energy storage would be significant only in the first few centimetres. The structural requirements of the building walls would dominate in this instance, but PCM needs to be applied only in a thin facing layer to achieve the thermal benefits.

## 6. Exterior air change

In a real room, the interior air temperature is not a proscribed function, but is determined by an energy balance with the heat fluxes from ventilation of exterior air and convective exchange at the mass surfaces.

Following Holford & Woods (2007), a simple model is developed to determine the effect of PCM on the interior air temperature within a building. The exterior air temperature outside the building *u*_{e} is assumed to vary sinusoidally over the forcing period, with(6.1)and this air is ventilated into the room at a constant rate *q*. The interior air also continues to exchange heat with the thermal mass. An energy balance for the interior air temperature gives(6.2)where *V* and *S* are the volume and surface area of the room, respectively, and *ρ*_{a} and *C*_{a} are the density and specific heat capacity of the air, respectively.

The temperature of the thermal mass continues to obey equations (2.1)–(2.3). Casting the new system defined by equations (6.1)–(6.2) and equations (2.1)–(2.3) into dimensionless form gives(6.3)(6.4)(6.5)(6.6)(6.7)and(6.8)where is a dimensionless parameter comparing the energy crossing the wall surfaces over the course of the day to the heat capacity of the air and measures the number of daily air changes.

Equations (6.3)–(6.8) can be simulated for typical values of *A* and *B* to give simulated temperature histories for the interior air and the mass surface. For a 10×5×5 m^{3} room with 10 air changes per hour, the new parameters take values *A*=80 and *B*=38.5 (table 4).

Figure 10 shows graphs of the interior air temperature *U*_{i}, the surface temperature and the perceived temperature , approximated as the mean of these two (Fisk 1981). Each graph is calculated for a 2 cm thickness of wallboard with and without PCM and for a semi-infinite section of concrete.

The 2 cm section of wallboard provides very little thermal inertia. The 2 cm thickness is much less than the penetration depth of 12.6 cm, the heat capacity of the wallboard is overwhelmed and the surface temperature closely tracks the interior temperature. Consequently, the wallboard is ineffective in both removing energy from the interior air by convection and providing radiative relief via a reduced surface temperature.

By contrast, once PCM is added to the wallboard, the penetration depth is just 1.1 cm and the large latent heat is sufficient to reduce the amplitude of the surface fluctuations to just 8% of that of the exterior air. A significant amount of the energy from the exterior is convected into the walls, and the interior, surface and perceived temperatures are all greatly reduced. Again, the PCM wallboard gives a similar performance to a semi-infinite thickness of concrete.

## 7. Conclusions

Thermal mass can buffer temperature oscillations in the interior of a building caused by ventilation from the outside air and heat sources and solar gain within the space. The latent heat of microencapsulated PCM distributed throughout the mass offers the potential for improved performance while requiring less structural weight.

This paper compares the potential of normal and PCM thermal mass to reduce the temperature swing that building occupants perceive. A simple methodology has been developed to confront the nonlinear problem. Performance is measured via the maximum amplitude of the surface temperature of the mass and the diurnal energy storage within the mass. The amount of mass that can be saved by using PCM is measured by the penetration depth of heat pulses into the mass. PCM provides a substantial benefit on all three measures.

In the dimensionless model, the fundamental behaviour of the system depends on four parameters: the diffusivity *Ω* and convection strength *H*, which are properties of the thermal mass material, and the Stefan number *s* and the melting range width *U*_{m}, which are properties of the PCM. A low-penetration regime has been identified in which heat pulses cannot reach the back of the mass and so the physical system is independent of the exact mass width *L*. In this regime, the effect of *H* and *Ω* can be condensed into a single low-penetration parameter .

For a given set of thermal mass parameters, the performance is optimized by choosing the Stefan number *s* as large as possible within the constraints of the mass construction and available PCMs. The mass width *L* should be sufficiently large that heat pulses cannot reach the back of the mass, in which case the mass can be treated as if it were semi-infinite. Solutions for the dimensional energy storage, surface temperature and minimum mass width have been plotted as functions of the maximum achievable Stefan number *s* and the low-penetration parameter *Z*, and can be used to calculate the benefit of PCM thermal mass in a wide range of configurations.

Worked examples have quantified the surface temperature and energy storage of construction wallboard and concrete with and without PCM. As well as improving performance, the PCM significantly reduces the thickness of mass required. Using this simple methodology, the thermal mass potential of typical wallboard impregnated with PCM was shown to rival that of a heavyweight concrete construction.

A full model for a ventilated room with a floating interior temperature was presented, and an example was used to show that the performance benefits suggested by the simplified model carry over to attenuate the perceived temperature experienced by occupants of a ventilated room.

## Acknowledgments

The authors would like to thank BP for funding this research.

## Footnotes

↵When

*Ω*is sufficiently small that the mass can be regarded as semi-infinite, the penetration depth can be expressed as , a simple linear multiple of the conventional decay distance*δ*. In this paper*P*_{norm}, the depth at which the temperature reaches the top of the nominal melting zone*U*_{m}that would exist for PCM thermal mass, is considered in preference to*δ*as this allows the penetration distance with and without PCM to be directly compared.↵Holford & Woods (2007) give a full analytical solution for the temperature of normal thermal mass of finite width forced by a sinusoidal interior temperature.

- Received November 8, 2007.
- Accepted January 2, 2008.

- © 2008 The Royal Society