Adobe, one of the earliest earthen building materials, consists of sun-dried mud bricks layered with clay mortar to form thick walls [38,40,74]. While adobe wall thickness is typically dictated by structural rather than thermal needs, the resulting 30–60 cm (12–24 in.) constructions greatly stabilize interior temperatures in both summer and winter [59,71,72,75].

Adobe derives its thermal properties from its primary substrate, clay, as modified by silt, sand, organic content, and moisture [39] (Table 1). Since adobe bricks tend to shrink and crack during drying, organic fibers (straw, rice husk, bagasse) are typically added for dimensional stability and tensile strength [38]. These diminish the heat capacity and thermal conductivity of adobe proportionately [40,57]. In contrast, moisture uptake tends to increase adobe’s thermal conductivity (and to erode the adobe itself); as a result, adobe walls are typically protected from moisture by stone foundations, stone parapets, lime plasters, or stuccos [74,76], or by mixing lime, cement, or bitumen (also known as asphalt) into the adobe itself [38]. The latter also increase the compressive strength of the bricks [39].

The compressive strength of earth can be increased greatly by compaction. To produce “rammed earth” or pise, layers of earth are built up within wooden or metal formwork, using a manual or pneumatic tamp to compress each layer. The earth mixture usually has greater sand and lower clay content than adobe and is typically stabilized with cement, resulting in a denser, more water-repellent product with greater thermal conductivity [39,72,77] (Table 1).

Adobe, rammed earth, and related earth materials such as cob and straw clay are gaining popularity in contemporary building for their beauty, local availability, low embodied energy, and high thermal inertia [38,39,71,78,79]. Thermally, the common perception that earth provides insulation [72,78] is refuted by experimental data [39,77], but the existence of this misperception highlights a valued property of external massive walls: the ability to dampen internal temperature swings [39,61]. This effect results not from insulation (i.e., low thermal conductivity), as the thermal conductivities of adobe and rammed earth are more than ten-fold greater than those of foam or fiber insulation materials (Table 1), but from the ability to store daytime heat for release during cooler nighttime hours. In coastal climates, additionally, moisture absorbed by earth walls at night is evaporated during the day, increasing the cooling effect [71].

The special durability of burned clay was well-known among ancient peoples, and fifty centuries after the first known adobe, Mesopotamians devised a controlled way to “fire” clay bricks in kilns, raising their temperatures above ~1000 °C for about a day and then cooling them slowly to waterproof them and increase their strength [73]. During firing, brickclay becomes a metamorphic rock: clay minerals release water, yielding a mixture of quartz and other minerals, particularly including mullite (3AlO₃∙SiO₂), which grows in thin strands and binds brick together like straw binds adobe. Iron also oxidizes to hematite, giving most bricks a reddish color [80,81].

Thermally, brick is quite similar to adobe and rammed earth, consistent with its basis in raw materials of sand and clay (Table 1). Bricks fired longer or at higher temperatures, however, become denser and more thermally conductive; high metal oxide content also increases thermal conductivity [82]. The practice of coring bricks to facilitate firing evenness, in contrast, lowers density and thermal conductivity, and large cores must be filled for passive solar applications [16]. While a great range of brick thermal properties is therefore achievable in theory, contemporary brick-making in developed countries is standardized, minimizing variation [16,82].

Just as adobe bricks are set in mud, bricks are set in mortar, and mortar can form up to 30% of the volume of a brick wall [82]. Early lime mortars were more porous than fired bricks, but contemporary Portland-cement mortars closely resemble bricks in specific heat capacity, thermal conductivity, and density, with the result that passive solar applications can reasonably assume uniform wall properties [16,73,82].

In passive solar applications, adobe, rammed earth, and brick exterior walls (Section 2, Figure 2a) absorb solar energy through exterior surfaces and deliver part of that energy, many hours later, to the interior. This effect is illustrated in Figure 4a, showing adobe walls of 30, 45, and 60 cm thickness (12, 18, and 24 in. respectively) exposed to outdoor conditions of a typical sunny January day in Colorado chosen to illustrate patterns of thermal mass behavior clearly. (While these and all other patterns shown are characteristic of the respective materials and configurations, the quantities of heat stored and released are specific to the hourly solar radiation and temperature values of the day simulated (Section 7)). In this and all other simulations, indoor conditions were kept from falling below 68 °F (20 °C) by supplemental heat (Section 7).

Sun-driven heat fluxes into the three walls are similar, as are nighttime losses to the outdoors. Heat fluxes to the interior, however, vary dramatically with wall thickness: for the thinnest wall, peak heat delivery occurs at 10:00 p.m.; for the middle wall, 3:00 a.m., and for the thickest wall, no net heat at all is delivered to the interior under the conditions shown. The magnitude of heat delivery to occupied space varies greatly as well, with 10 W/m² for the thinnest wall, 2 W/m² for the middle wall, and −5 W/m² for the thickest. At the same time, the thickest wall loses less total heat from the interior over the 24-hour period.

The paradox above illustrates the designer’s dilemma in external mass wall sizing: should heat delivery during occupied hours be maximized (the thinnest wall), or should total heat loss from the interior be minimized (the thickest wall)? Standard evaluation methods of massive wall performance always favor the latter, because they assume constant indoor temperatures and they evaluate net gains and losses using monthly average indoor/outdoor temperature differences [7,13,15,31,32]. This approach is problematic for two reasons, however: first, it fails to account for thermostat setbacks during unoccupied hours, over-estimating the value of heat delivered passively during those hours. Second, it neglects warmer indoor temperatures created during some hours of heat delivery that diminish rates of that delivery, risking general over-estimation of delivered heat.

The approach presented here, in contrast, highlights daily patterns of heat uptake and release so that thermal storage designs can be matched to daily space-heating (or cooling) needs. While the current simulations omit thermostat setbacks for the sake of pattern clarity, their addition would simply improve performance of a design already well-matched to occupant needs. These simulations do, however, explicitly model internal warming during heat delivery, which limits the total extent of that delivery and which is not incorporated into conventional methods (Section 3 and Section 7).

The designer’s dilemma is therefore resolved by Figure 4a: for evening heating, the thinnest wall is best, and thermostats should be set back during the day to limit heat losses from the interior into the wall. For daytime heating, none of the three perform well, and alternative solutions should be investigated. For cooling, however, the thicker options are both worth investigating further through experimentation or simulations with summer weather data.

With few exceptions, the mineral grains in building stones are small enough that many are traversed along the primary direction of heat flow, for example over the thickness of an external wall. Fourier’s law [Equation (2)] predicts that a significant temperature difference across the short distance of a grain diameter produces a large heat flux, and so tends to even out the temperatures of adjacent mineral grains. For this reason, adjacent grains approach a state of thermal equilibrium, with their temperatures evolving in sync as heat is transported over larger distances. This reasoning implies that the effective volumetric heat capacity ρc can be approximated by the average product of component densities ρ_i and specific heat capacities c_i, weighted by their volumetric fractions ψ_i, so that [49,51]: (6)

The specific heat capacities of minerals are inversely correlated with their densities [17,65], so typical volumetric heat capacities of building stones vary over a relatively modest range as long as the porosity is low. Mineral conductivities vary widely, with extremely high values in some rare cases (i.e., native metals, diamond), but a much more modest range exists for less exotic constituents. The relatively high thermal conductivity of quartz, combined with its common abundance, ensures that effective conductivities of many stones vary in proportion to quartz content. This leads for example to the relatively high conductivity listed for quartzite in Table 1. By contrast, the relatively low thermal conductivities of the feldspar minerals cause a general decrease in effective conductivity with feldspar content [18]. The presence of feldspars in granites lowers their conductivities relative to quartzite, whereas the higher quartz content in granite compared to basalt produces a higher thermal conductivity for the former stone (Table 1, Figure 3).

In addition to the mineral content, a stone’s effective thermal conductivity is also sensitive to the geometrical relationships among its constituents, as well as the predominant direction of heat flow [18]. In the isotropic arrangements typical of granite and basalt, the effective thermal conductivity can be approximated by the geometric mean so that, (7) when heat flow is parallel to layering, as in sandstone and slate, the arithmetic mean gives a better estimate, (8) whereas the harmonic mean describes the effective conductivity perpendicular to layering, (9)

These considerations ensure that conduction across layering, as described by Equation (9), is always slower than conduction parallel to layering [Equation (8)] since the latter arrangement allows for preferential heat flow through more conductive pathways, whereas in the former arrangement an equal heat flux must traverse each layer. Less ordered arrangements of minerals that are best described by Equation (7) have effective conductivities that are intermediate between the two layered extremes.

Air has much lower thermal conductivity and density than any solid constituent of stone (Table 1, Figure 3). Rocks that form deep below the Earth’s surface, like granite and quartzite, normally have very low porosities, so their constituent minerals control the thermal behavior. Volcanic rocks like basalt can form from bubble-rich magmas that leave behind many air-filled pores, but in most building stones these still occupy a relatively small fraction of the total volume (i.e., <10%). By contrast, rocks like sandstone that form by cementing clastic sediments together can contain many air-filled pores, and porosities of 20%–35% are not uncommon. At high values, porosity controls thermal performance.

Air causes an appreciable reduction to both the volumetric heat capacity and the thermal conductivity of stone, lowering the effusivity, and usually the diffusivity as well. However, stones with low porosities are typically preferred for building use because of the susceptibility of more porous materials to damage by the action of moisture and frost [83,84,85].

Structural needs are more easily satisfied by stone, with its high compressive strength, than by adobe, allowing wall constructions to be thinner; here, granite illustrates thermal storage and release behavior of stone walls in 10, 20, and 30 cm (4, 8, and 12 in.) constructions (Figure 4b). Solar heat gains by the granite walls are much higher than those of the adobe walls, peaking at about 450 W/m² on the day shown in accord with granite’s higher thermal effusivity (Table 1). The granite walls also deliver heat to occupied space much more rapidly, and in much greater quantity, than the adobe walls do: peak heat fluxes range from 40 W/m² to 110 W/m² and occur from 2:00 to 5:00 p.m., with heat delivery completed in the evening in all cases. Conversely, the granite walls lose much more heat from the interior overnight, in this illustration, than do adobe walls. This strikingly different heat delivery pattern illustrates granite’s much higher thermal diffusivity (Table 1).

In the 1980s, black-coated aluminum and copper foils were investigated as “selective surfaces” or “selective coatings” that, when applied to the sun-facing mass surface, improved heat storage even more dramatically [26,87,90], although they fell short of creating the “one-way heat valves” enthusiastically described by some designers [87]. High thermal conductivity of these coatings allowed them to admit solar gain readily, while low emissivity (Table 1) allowed them to diminish radiative heat loss to the outdoors (Section 3.2).

Trombe walls are typically described as having optimal thicknesses, in the range of 25–35 cm (8–14 in.), that minimize indoor-air temperature swings and maximize annual heating benefit [7,15,16,32]. Trombe walls are also described as having long “lag times”, usually in the realm of 8–12 h, referring to the time between peak solar gain and peak heat delivery to the interior, but frequently interpreted as the time until any substantial heat is delivered [12,13,16,32]. Third, a Trombe wall’s performance is typically evaluated according to its “solar savings fraction” (SSF), or proportion of total energy that it saves in a building, evaluated in terms of aggregate monthly heating need [12,15,26,91].

These three parameters—thickness, lag time, and SSF—currently govern Trombe wall design and deployment, but work remains to advance design sensitivity to the needs of a particular space. Heat (or cooling) is most useful at certain hours: during the day for an office, school, or shop; during the early morning or evening for a residence; all night for a greenhouse (Figure 1). Timing the heat delivery (or absorption) of a passive system is therefore at least as important as maximizing it, if not more so.

To illustrate the thermal behavior of an unvented Trombe wall in passive solar heating, the exposed mass walls of the previous models (Section 4, Figure 4) were replaced with Trombe walls using concrete mass (Table 1) with varying external layers. For comparability, simulations employed the same building under the same conditions; in addition, Trombe Wall algorithms were called upon to simulate natural convection in the airspace between glass and mass [87,88] (Section 7).

The effectiveness of even a single pane of clear glass is evident in Figure 5a,b: for the illustrative day, nighttime heat loss is approximately half of that lost by the unglazed wall, while heat gain to the interior is positive for every hour. Heat gain during collection is even improved slightly. The basis for this improvement lies partly in the protection of the wall from forced convection (e.g., wind; Section 3.6) and partly in the opacity of glass to the long-wave radiation emitted by the warm mass wall [92]. The air between the two has insulating properties, as well (Table 1): with low effusivity, motionless air transfers little heat as it equilibrates with a warmer or cooler environment, although natural convection can increase its effective effusivity substantially (Section 3.6). In the single-glazed Trombe wall shown (Figure 5b), therefore, benefitting from its layers of glass and air, heat delivery to interior space rises noticeably only a few hours after heat collection begins. Heat delivery peaks by 5:00 p.m., afterward declining gradually such that midnight heat delivery is still over half of peak delivery. In contrast, heat delivery by the unglazed wall reaches zero by midnight.

Double clear glazing, on the sunny cold day shown, further reduces Trombe wall heat loss and increases delivery to occupied space, maintaining the same hourly pattern observed with single glazing; in this case, solar heat gain is not noticeably diminished (Figure 5c). Addition of a selective coating representing black oxide-coated copper, however, has an even stronger effect than addition of a second glass layer (Figure 5d), increasing both heat gain and heat delivery in the characteristic pattern.

Trombe walls are often touted for “long thermal lag times” and “slow, even nighttime” heat delivery [12,13,16,32]. In contrast, however, the characteristic patterns of Figure 5 show that Trombe walls made of concrete, the most common material, transmit solar heat gains to the interior fairly rapidly; concrete’s fairly high thermal diffusivity (Table 1) is responsible. Although a peak in heat delivery may be delayed until evening, appreciable heat is delivered for many hours before that. This pattern illustrates the problem that occurred in the Zion Visitors Center in Utah: an 8–10 h thermal lag was assumed for an 8-inch thick grouted CMU wall, with properties similar to concrete [93], but investigation of the completed building soon revealed that the wall was delivering heat hours earlier than expected, with unintended consequences [94]. Nevertheless, effective workplace heating by Trombe walls appears quite promising, given their characteristic heating patterns, as long as the hourly behaviors of the walls are considered.

To contain water for thermal storage, designers have experimented with rigid tubes, plastic bags within rigid tubes, stackable glass chambers, corrugated metal culverts, steel tanks, and 55-gallon drums [12,16,32,95]. Plastic containers may be left translucent, allowing visible light to enter [11,95], but most are painted black or coated with selective surfaces [12,16]. Water chambers are usually placed behind external glazing, such that they resemble Trombe walls [16,32] or fully internal mass (Section 6.2).

Like other massive storage walls, water walls are typically evaluated by a monthly heating contribution, estimated from monthly average indoor-outdoor temperature differences, rather than by daily matching of heat delivery to occupant needs [12,15,26,91]. In contrast to solid Trombe walls, however, for which design guides prescribe specific thicknesses beyond which performance declines (Section 5.1), water walls are viewed as offering ever-increasing heat delivery with volume, diminishing returns notwithstanding [15,26], and to have no maximum thickness beyond which performance declines [7,26,32]. The basis for this perception is understandable, but heat delivery patterns of water walls are, in fact, highly tunable by varying their thicknesses. Illustration of this through further simulations (Section 5.2.3), however, required external calculations of convective heat transfer for incorporation into EnergyPlus models (Section 7).

To model convective heat transfer, we make use of published correlations between the Nusselt number Nu and the characteristic Rayleigh number Ra (Section 3.6). Of particular interest for the water wall (and the PCM-Trombe wall discussed below) is the effective thermal conductivity across a fluid-filled horizontal gap. At high Ra, Nu is approximately proportional to Ra^1/3 and does not depend strongly on the aspect ratio between the gap height and its width (e.g., [69,70]). In principle, Nu should also be affected by the ratio of kinematic viscosity to thermal diffusivity, Pr = v / α. However, correlations for the controls on Nu along a single vertical plate suggest that the heat transfer is relatively insensitive to Pr, especially as it attains higher values for more viscous fluids [96]. Accordingly, we adopt the heat transfer correlation proposed by Wright [69] for the typical operating range with Ra > 5 × 10⁴ and take (10) So that the heat transport is described by Fourier’s law, Equation (2), with k replaced by an effective thermal conductivity of k_e= Nu·k. This predicts a linear relationship between k_e and gap width L, for example reaching 9.7 W/(m K) (Nu = 16) when L = 10 cm (~4 in.) and the temperature contrast across the wall is ΔT ≈ 1 °C.

From Equation (5) Ra is proportional to ΔT, so Equation (10) implies that Nu and the effective thermal conductivity k_e are even higher when the temperature contrast is larger because convection becomes more vigorous and so tends to diminish temperature contrasts more rapidly. By the same reasoning, smaller temperature contrasts are characterized by lower k_e. However, since the Nusselt number correlation described by Equation (10) implies that k_e is proportional to the 1/3 power of ΔT, changes in k_e are proportionally much more modest than changes in ΔT. Accordingly, for each wall thickness we adopt a single value of k_e that corresponds to the expected value of Nu for a nominal temperature contrast of ΔT ≈ 1 °C. This simplification should not influence the overall patterns of performance that are the focus of this work.

To illustrate a water wall’s performance in passive solar applications, the massive storage walls of the previous models were replaced with water walls in both exterior wall and Trombe wall configurations; in the exterior wall configuration, only a single layer of clear glass separated the water from the outdoors (Figure 6a).

Three main results follow from water’s natural convection: first, outside surfaces remain cooler during the day, allowing water walls to gain more heat than solid walls of equivalent total capacity [Equation (1)]. Second, water walls begin delivering heat only a few hours after heat gain begins, and third, outside surfaces stay warmer at night, making water walls more vulnerable to nighttime heat losses (remove paragraph break here). These trends are prominent in the exterior water walls (Figure 6a), which achieve the greatest passive solar heat capture of the materials examined to this point (see adobe, granite, and concrete in Figure 4). The exterior water walls lose more heat at night, however, due to the same high effective thermal conductivity that enabled such extensive solar gain. As a result, the net heat delivery of an exterior water wall is not substantially different from that of an exterior concrete wall of equivalent thickness, under the conditions shown. Note, however, that concrete’s thermal lag increases substantially as thickness increases (Figure 4c), while a water wall’s thermal lag does not (Figure 6a), due to increases in natural convection (and effective thermal conductivity) with wall thickness (Section 5.2.2).

In a Trombe wall configuration, water’s convective heat transport combined with the heat retention capabilities of the confined air space yield dramatic results (Figure 6b): in the thinnest wall, appreciable heat flux begins just hours after insolation and continues in a sustained broad peak through the afternoon, without net heat loss at any hour. The intensity and duration of this delivery can then be modulated by wall thickness: doubling the thickness to 20 cm (8 in.) substantially increases nighttime delivery at the expense of peak afternoon delivery, while increasing the thickness further to 30 cm (12 in.) results in virtually constant 24 h delivery (Figure 6b). Comparison of 8in water and concrete Trombe walls, in turn, highlights water’s higher effective thermal effusivity and diffusivity in increasing solar heat collection and extending heat delivery into early morning hours (Figure 5c and Figure 6b). Convection causes the effusivity to increase above water’s inherent effusivity listed in Table 1 by a factor of , while the effective diffusivity increases by a factor of Nu. As a result, convective heat transport may be used to further advantage in conditioning entire spaces through cold nights, as illustrated below (Section 6).

PCMs most suitable for indirect-gain systems possess phase-change temperatures in the range of 20–40 °C (68–104 °F): they must freeze above the desired indoor temperature, so that heat released by freezing is useful. However, they must not require overly high temperatures to melt, because the phase change might then occur only rarely. Ideal PCMs also have high latent heats of fusion, ease of prolonged containment, low susceptibility to supercooling, low toxicity, and low cost. While numerous organic phase-change materials melt and freeze in the desired temperature range, they are somewhat expensive and they tend to possess relatively low latent heats [19,20,21]. Salt hydrates are generally affordable, in contrast, but problems of supercooling, thermal expansion, formation of sharp crystals, and incongruent melting can compromise their performances. Nevertheless, a number of salt hydrates exhibit advantageous melting transitions and are fairly tractable in building applications: CaCl₂∙6H₂O, or antarcticite; Na₂SO₄∙10H₂O, or mirabilite; and Zn(NO₃)₂∙6H₂O [19,20,21,97].

Phase-change materials are particularly attractive in building applications for their compact thermal storage [13]. Because substantial heat capacity can be realized in less than an inch (2.5 cm) of thickness, applications can take exceptionally thin forms, such as PCM-packed ceiling tiles [12] and PCM-impregnated gypsum board [19]. In addition, PCMs are promising as low-volume replacements for water in Trombe-style walls (Section 5.2), encapsulated in thin tubes or pouches, and in internal storage tanks (Section 6) [56,98,99,100].

Phase changes are accompanied by changes in density and therefore in volume. While the large volume changes associated with sublimation and boiling are prohibitive for building thermal storage applications, several strategies have been developed to accommodate the modest (i.e., 10%) volumetric expansion that occurs upon melting. Rigid tanks may simply be oversized; long, thin tubes may be assembled into arrays, and deformable plastic pouches may be used [21,97,100]. Although sharp dendrites can cause punctures in such pouches [101], these crystal forms develop under a restricted range of conditions, most commonly when the liquid is supercooled before freezing. Nucleating agents can be added to avoid supercooling [97], although these may cause other problems, including the formation of meta-stable phases in some systems [102]. Micro-encapsulation techniques enable PCMs to be incorporated within concrete, wallboard, and plaster, potentially offering another solution to volume expansions, but these have found limited applicability to salt hydrates [19].

As melting or freezing takes place, the difference between the solid and liquid densities leads to small changes in specific volume. As a result, a portion of any heat transfer involves pressure-volume work and is therefore not available to change the system’s internal energy. The “specific enthalpy” is therefore useful for comparing energy storage among different PCMs because a change in enthalpy accounts for both the change in internal energy and the work associated with volume changes at constant pressure. On either side of the melting temperature, a PCM’s enthalpy is directly proportional to temperature through the specific heat capacity c, as illustrated by the sloping solid lines for three salt hydrates in Figure 7a. At the phase change temperature, however, the enthalpy of the solid is higher than the enthalpy of the liquid, with the difference given by the latent heat of fusion L. This is the energy that can be stored during melting or released during freezing, and it gives rise to an abrupt step in specific enthalpy at the melting temperature. For each of the aqueous solutions and solid hydrates shown in Figure 7a, the size of the jump in enthalpy provides a quantitative measure of the amount of energy that is stored when the solid hydrate melts completely.

The fourth curve in Figure 7a shows the enthalpy changes for a 33% by weight aqueous solution of sodium sulfate, which does not have an abrupt phase change, but instead shows a more gradual change in enthalpy at temperatures below 32 °C (90 °F) [54,55,56,102] (remove paragraph break here). Just as the solidification and melting of igneous rocks and sea ice occur over a range of temperatures as the residual liquid composition varies (e.g., [103,104]), salt hydrates can form mushy solid-liquid mixtures that broaden the phase transition so that it is no longer isothermal, and the step-change in enthalpy disappears (Figure 7a, Section 5.3.4).

Such behavior can be avoided if the salt solubility is high enough to match the composition of the solid hydrate. This is the case for calcium chloride hexahydrate, also known as the mineral antarcticite, which melts at 29 °C (84 °F) to form an aqueous solution with a calcium chloride concentration of 50% by weight [54,105]. Likewise, a super-eutectic aqueous solution can be formed, with 64% zinc nitrate by weight, that freezes at 36 °C (97 °F) to form solid zinc nitrate hexahydrate [55,56,106]. In their fluid states, both of these PCMs are able to convect, which helps them to maintain a near-uniform temperature throughout.

The solubility of sodium sulfate in water reaches a maximum of approximately 33% by weight near the melting temperature of its decahydrate form, the mineral mirabilite, which is 44% sodium sulfate by weight (see Figure 7b) [55]. Heating pure crystalline mirabilite to its melting temperature of 32 °C (90 °F) produces an aqueous solution, but the solubility of sodium sulfate (33% by weight) is lower than the 44% sodium sulfate content of mirabilite, and the remaining 16% precipitates as Na₂SO_4(s): pure sodium sulfate. In consequence, the effective latent heat of the two-phase mixture is diminished to 84% of its tabulated value [55]. The solid line in Figure 7a for a solution of 44% sodium sulfate corresponds to a case where the sodium sulfate crystals remain in close proximity with their saturated solution so that the entire assemblage can recombine to form solid mirabilite upon cooling to 32 °C (90 °F).

If the sodium sulfate crystals that precipitate when mirabilite melts are able to settle, a process that can be limited by addition of thickening agents, the remaining aqueous solution is too dilute to refreeze completely upon subsequent cooling [55]. Instead, when freezing commences near 32 °C (90 °F), the concentration in the remaining liquid is depleted, which causes its phase change temperature to decrease gradually along the liquidus curve (see Figure 7b). Just beyond the onset of crystallization, the liquid concentration drops by approximately 0.8% for every 1 °C decrease in temperature. Further cooling produces more and more solid mirabilite as the residual liquid is diluted progressively; this gradual freezing can continue until an ice/mirabilite solid solution forms at the eutectic temperature of approximately −1 °C (30 °F). Figure 7a shows how the enthalpy of a 33% aqueous sodium sulfate solution evolves as the temperature is decreased below 32 °C(90 °F) and mirabilite gradually accumulates.

The striking features of PCM performance, in comparison with single-phase materials, are the extent of heat gain and steadiness of heat delivery that can be achieved. When the 10 cm (8 in.) concrete of the previously-described passive solar Trombe wall is replaced by a salt hydrate PCM, comparable heat gain can be achieved in a much thinner wall: calcium chloride hexahydrate, or antarcticite, accomplishes the same heat gain in half the thickness (4 in.; 10 cm), while zinc nitrate hexahydrate and 44% sodium sulfate decahydrate, or mirabilite, need only one-quarter (2 in.; 5 cm) (Figure 8a–d).

External wall temperatures show that each of the three high-performing salt hydrates (Figure 8b–d) melted fully during the day, climbing above its respective phase-change temperature and gaining heat sensibly, but remained warm enough to avoid fully freezing (and entering sensible cooling) at night. As a result, overnight heat delivery is virtually constant by each of the three PCMs under the conditions shown, with each phase change temperature controlling the external and internal wall temperatures and, therefore, rates of heat delivery to the interior. Antarcticite’s phase change temperature of 29 °C (Figure 7a) is the lowest; as a result, its wall maintains the lowest temperatures and delivers slightly less heat to the interior than the others (Figure 8b). Mirabilite, with a phase-change temperature of 32 °C, maintains warmer temperatures and delivers slightly more heat (Figure 8d), while zinc nitrate melts at 36 °C, maintains that temperature at the inner wall surface, and delivers the most heat accordingly (Figure 8c).

The thermal behavior shown for mirabilite (Figure 8d) represents ideal behavior of the pure substance, as discussed above, in which no solid Na₂SO₄ precipitates from the system during repeated cycles of freezing and thawing. Figure 8e, in contrast, illustrates the consequence of such precipitation and loss, which effectively reduces the concentration of the mixture to 33% Na₂SO₄∙10H₂O by weight. In this case, as the system cools, it separates into an aqueous solution of Na₂SO₄, with concentration given by the curved liquidus line of Figure 7b and a solid phase of Na₂SO₄∙10H₂O. While the system can fully melt, it can no longer fully freeze at the temperatures of interest, diminishing its effective latent heat, and its phase change takes place over a much greater temperature range (Figure 7a). As a result, wall surface temperatures are no longer kept near the nominal phase-change temperature, and heat delivery no longer exhibits the characteristic PCM constancy (Figure 8e).

Thin PCM walls designed to freeze and thaw fully under the conditions of interestas well as thick water walls (Figure 6b), therefore exhibit heat uptake and delivery patterns well-suited to steady, all-night heating of greenhouses, sunspaces, barns, sleeping quarters, and other spaces in which early-morning heat or constant temperatures are of paramount importance.

The priority of thermal stability during solar collection is unique to direct-gain systems and persists even in design of spaces that are not occupied during all hours [7,12,13,14,15,34,107]. Residential direct-gain systems, for example, might easily tolerate high mid-day temperatures, when occupants are away, as well as low early-morning temperatures, when occupants are asleep. Nevertheless, the perceived necessity to maintain all-day comfort has led designers to value steady indoor temperatures over heat delivery targeted to useful hours. Beginning with the influential work of Balcomb and colleagues [32,34,107], design guides have consistently recommended minimizing temperature swings in direct-gain spaces, usually by providing large amounts of thermal storage mass in floors, described below, or water tanks, considered subsequently (Section 6.2) [12,13,14,15].

Floors are popular locations for thermal storage mass in direct-gain systems: underfoot, stone and concrete provide durable, waterproof, often beautiful surfaces of substantial area, within reach of solar gain, that support themselves structurally. Recommendations of floor thermal storage mass thickness, accordingly, usually resemble those for structural slab-on-grade thickness: 10–15 cm (4–6 in.) [13,14,15,16]. Direct-gain mass “sizing” is then typically reduced to selecting an area, based on the area of solar-collecting glass, from tabulated “mass-to-glass” ratios [12,13,14,15,91]; these, in turn, are derived from the venerable calculations that use monthly average indoor-outdoor temperature differences to estimate solar savings fractions [33,34,35,36]. As we have seen, however, the thickness of a stone, earth, or concrete wall greatly influences the amplitude, timing, and duration of heat delivery (Figure 4); analogous variation in thickness or material of floor-level thermal mass could similarly be recruited to match heat delivery patterns to heating needs.

While the interior surface of a direct-gain floor contacts only air, collecting and returning heat through radiation and convection, the exterior surface may contact soil, sand, or gravel and lose significant heat by conduction [108,109,110,111]. The problem is intensified if the soil is moist, since soil thermal conductivity increases substantially as pore spaces fill with water [111,112,113]; moreover, fine-grained soils can wick moisture upward from the water table, increasing soil moisture even in locations (e.g., under buildings) that receive no direct precipitation [114].

Perplexingly, however, heat loss through massive thermal storage floors to underlying soils is often addressed (in the U.S.) only at the perimeter [7,12,13,107]: the ASHRAE Handbook of Fundamentals, citing observations that slab-on-grade heat loss is closely related to perimeter length for sufficiently small, unheated slabs, has long recommended only perimeter insulation for all slabs unless radiant heating tubes are buried in the floor [52,115,116]. Massive floors can warm significantly during solar gain, however, creating large enough temperature differences over underlying soil to promote substantial heat loss [24]. Moreover, most building energy simulation tools, including EnergyPlus, use a single “standard” soil thermal diffusivity value from a given weather file, and the common TMY3 files currently provide an unrealistically low soil thermal diffusivity of 0.02 mm²/s [117]. As a result, designers are vulnerable to the belief that perimeter insulation is, indeed, sufficient, when illustrative results below clearly indicate the opposite.

To illustrate the risk of heat loss from direct-gain massive floors to soil, the thermal storage walls of the previous examples were replaced by double clear glazing and a concrete thermal storage floor (Figure 2c). The resulting direct-gain system was simulated, on the same sunny cold day in Denver, Colorado, using a range of representative soil thermal diffusivities. Dry peaty soils have the lowest values likely to be found at building sites, at 0.1 mm²/s; a wide range of silty and loamy soils is represented by the middle value, 0.5 mm²/s; and moist clays have the highest values near 1 mm²/s [109,118].

While the floor over the lowest-diffusivity soil returns more heat overnight than any of the storage walls (Figure 9a, top), even with an equivalent collector area (Section 7), the floor over the medium-diffusivity soil returns less than half that amount (Figure 9b, top), and the floor over the moist clay loses nearly all of its heat to the soil below (Figure 9c, top). Adding perimeter insulation to the latter case, however, only improves it to the performance of the middle case (Figure 9c, middle); full under-slab insulation, equivalent to 2 in. (5 cm) of rigid foam, is necessary to realize heat gains equivalent to those of the initial case (Figure 9c, bottom).

This result illustrates a general benefit of insulation that can be applied, in movable form, to each of the thermal storage walls described above, as well as to the direct-gain glazing itself. After a material, wall thickness, and/or glazing suited to the solar resource and occupant heating needs are selected, movable insulation on tracks or rollers can be deployed to diminish nighttime heat losses to the exterior [7,15,16] or, although seldom mentioned in design guides, to fine-tune heat delivery to the interior.

Sunspaces are usually constructed with tilted glazing (Figure 2d), increasing solar access and creating brightly daylit spaces with high potential for heat gain. Occupants often fill their sunspaces with plants and comfortable furniture, valuing winter daylight, warmth, and greenness as much as heat delivery to occupied space [24,29,119]. While tilted glazing is essential to both experience and performance, therefore, particularly in cloudy winter climates [24], it increases the risk of overheating on sunny days, which can damage plants [12,29]. Sunspaces with plants have, in effect, no “unoccupied” period during which either overheating or freezing is acceptable, although plants tolerate much wider temperature ranges than humans do. As a result, sunspaces, unlike workplaces, schools, or residences, have valid needs for thermal stability.

An additional vital function of sunspaces, particularly in northern, coastal, and other overcast climates, is the “thermal buffering” of the attached building from heat loss to the outdoors. Simply by maintaining a warmer environment than the outdoors, a passively-heated sunspace diminishes the temperature difference across its common wall(s) and reduces heat loss from those walls accordingly. Sunspaces often enclose entry doors, as well, acting as vestibules to reduce heat loss from conditioned interior space, and they can provide tempered ventilation air to the interior [120,121]. For these purposes, sunspace temperatures are most profitably kept warmest when the temperature difference across the common wall is greatest.

Sunspaces are often built with massive floors, but like direct-gain floors, they lose heat readily to damp soils (Section 6.1.3, [24]). An effective alternative to under-floor insulation is fully internal mass, i.e., mass that cannot lose heat to the outdoors. Internal mass may take numerous configurations: potted plants, rainwater barrels, fish tanks, stone tables, and earthen fireplaces are all effective, for example [11,12,24]. Among these materials, water is particularly useful because it can collect heat in compact volumes and, despite its thickness, can release heat evenly over many hours (Section 5.2.3).

To illustrate the potential for internal mass to protect sunspace plants from freezing and to buffer conditioned space from heat loss to the outdoors, a 6 m² (64 sf) sunspace was attached to the building model used in previous examples (Figure 2d) and equipped with three, six, or twelve 208 L (55-gallon) drums of water. The common wall between the building and sunspace was insulated, in this example, and the sunspace temperature was allowed to float freely. Note that the sunspace received greater solar gain than the other configurations, due to its tilted glazing, but that exterior conditions were identical, as the model was simulated for the same sunny January day in Denver, Colorado, and heat flux values were normalized to the total south wall area for comparability with other systems (Section 7).

Plants would have been in danger of frost damage on the night shown, with outside air temperatures dipping below freezing for several hours in the early morning (Figure 10a). Addition of three water drums keeps overnight temperatures 7–10 °C (13–18 °F) warmer overnight, averting the frost problem, but does not prevent daytime overheating, given the relatively low mass of the space. Despite the low mass, however, heat flux from the water drums remains remarkably constant through the night and into the coldest early morning hours (Figure 10a), illustrating the usefulness of internal convection to internal mass performance.

Addition of three further drums, for a total of six, diminishes afternoon high temperatures by 5 °C (9 °F) and increases overnight air temperatures by about the same amount, 5 °C (Figure 10b); another doubling of water storage, to twelve barrels, further reduces afternoon overheating and further warms nighttime hours, increasing heat delivery by about 30% (Figure 10c). This pattern is generally applicable to internal water storage, whether in direct-gain systems or sunspaces: increasing volumes yield increasingly stable temperatures and increasingly great heat return, up to the point where further solar collection is no longer possible, with minimal heat losses to the outdoors.