Conceptual analysis and design of a partitioned multifunctional smart insulation

Applied Energy 114 (2014) 310–319

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Applied Energy journal homepage: www.elsevier.com/locate/apenergy

Conceptual analysis and design of a partitioned multifunctional smart insulation Mark Kimber ⇑, William W. Clark, Laura Schaefer Department of Mechanical Engineering and Materials Science, University of Pittsburgh, 206 Benedum Hall, 3700 O’Hara Street, Pittsburgh, PA 15261, United States

h i g h l i g h t s  Multilayered insulation enables switching between high and low R-values.  Maximum insulation performance is based on air thermal conductivity.  Conductive configuration is achieved by compressing internal wall components.  Proposed design procedure accounts for location specific weather data.

a r t i c l e

i n f o

Article history: Received 2 April 2013 Received in revised form 25 September 2013 Accepted 30 September 2013 Available online 20 October 2013 Keywords: Smart insulation Adaptive insulation Building Energy

a b s t r a c t Building insulation performance in walls and roofs is typically assessed in terms of its R-value, a metric related to its ability to resist heat flow under steady state conditions. Past and present efforts by numerous researchers have resulted in a continued increase in achievable R-values. However, for most climates, there are times during a typical day and throughout a calendar year where it would be advantageous to enable switching between highly insulated and conductive states. A large energy savings potential exists for such an adaptive insulation by decreasing the load imposed on the heating or cooling system; however, practical realizations of adaptive insulation have not been fully developed. A new multifunctional insulation is presented in this paper where thin polymer membranes are positioned within a wall to create layers of air such that the role of natural convection becomes negligible. The heat passing through the wall must therefore travel through alternating layers of stagnant air and polymer membrane. To achieve the low R-value condition, the air is removed and the layers are compressed, essentially leaving only conduction through the polymer membranes. The focus of this paper is the analysis of such a multilayered wall in both the insulated and conductive states. Design strategies are presented for selecting suitable materials and wall geometry. The conceptual analysis presented here provides the framework for future studies focused on fabrication and experimental design of such a multifunctional smart insulation. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction Insulation performance in commercial and residential buildings plays a significant role in dictating the overall energy requirements. Various solutions exist from standard commercially available options (e.g., spray foam or fiberglass batts and blankets) to more state of the art technologies. A few examples of the latter include vacuum insulation panels [1] consisting of small structural pores with the interstitial gas evacuated, and gas insulation panels [2], which are similar in nature to their vacuum counterparts with the exception of allowing the internal pores to be filled with a low conductivity gas. In this case, the pore size is small enough such that the interstitial fluid remains stagnant and convection can be ⇑ Corresponding author. Tel.: +1 (412) 624 8111. E-mail address: [email protected] (M. Kimber). 0306-2619/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.apenergy.2013.09.067

neglected. Another emerging family of insulation focuses on nano-based properties [3–5]. These solutions reduce the scale of the internal pores even beyond those seen in gas insulation panels, thereby decreasing the effective gas thermal conductivity from its bulk value to a molecular level ballistic value. In practice, gas insulation panels and nano-based solutions are easier to fabricate and maintain when compared to the vacuum insulated panels, where the performance can significantly degrade during their lifetime due to the difficulty of maintaining vacuum conditions. The purpose of the aforementioned technologies (both standard and state of the art) is to thermally isolate the inside and outside environments, and to that end, they fulfill their intended purpose. However, for many climates and during numerous months throughout the typical calendar year, it would be advantageous to have a solution which can switch between highly insulating and highly conductive states. The potential for energy conservation

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311

Nomenclature g k kp kw h L Lcond Lins N Nmax Nopt Nud q00rad Rcond Rconv Rd Req Rins Rins,opt Rp Rrad Rw Rad

gravitational acceleration constant (9.81 m/s2) thermal conductivity of air thermal conductivity of partition membrane thermal conductivity of innermost and outermost layers of wall convection coefficient internal thickness of wall available to divide into air layers total thickness of wall in conductive configuration total thickness of wall in insulation configuration number of air layers for insulation configuration maximum number of air layers possible (Nmax = 1 + L/tp) number of layers needed to achieve maximum insulation R-value (Rins,opt) Nusselt number for single layer of air radiation heat flux across a single air layer thermal resistance of wall in conductive configuration convective resistance of a single air layer conductive resistance through single air layer parallel combination of radiation and convective resistances for single air layer thermal resistance of wall in insulation configuration optimum thermal resistance of wall in insulation configuration conductive resistance of a single partition membrane radiation resistance across a single air layer conductive resistance of innermost and outermost wall layers Rayleigh number using air layer thickness as length scale

from passive conditioning was investigated by Neeper and McFarland [6,7] and it was determined that solar insolation on roof and wall exteriors exceeds building heating load demands, even during cold months in harsh climates, like January in Buffalo, NY. Furthermore, it was estimated that passive energy sources and environmental conditions could meet a large percentage of the heating and cooling demands in all climates across the continental US. To illustrate the need for such a concept, consider a warm day during the early summer months when the cooling system of a building is required to maintain a suitable inside environment during the daytime hours. As dusk approaches, the outside temperature begins to drop. However, the cooling system remains engaged to remove the residual heat that has been stored in the building throughout the day. This wasted energy could be saved simply by opening all windows, allowing the outside and inside environment to be thermally connected. However, this is not a desirable course of action when considering factors such as pollution, allergies, or if the occupants are away. Moreover, many commercial buildings are fabricated with permanently sealed windows. It is therefore desirable that this functionality become integrated into the insulation itself. The concept proposed in this paper represents a dual-purpose insulation, namely one that can switch between insulating and conducting configurations. The insulating ability is achieved through the same fundamental principles as the gas insulation panels, namely that the configuration forces the interstitial fluid to be motionless, thereby limiting the effective thermal conductivity to that of the interstitial fluid itself. The primary functional difference is that the proposed concept is able to switch from this insulating state to one of low thermal resistance when conditions exist where energy could be saved. This adjustable insulation has

RaL tp tw Tave Ti Tj Tin Tout DT (DT)d

Rayleigh number using internal wall thickness as length scale partition membrane thickness thickness of innermost and outermost solid layers of smart insulation average temperature of partition or wall temperature of surface i temperature of surface j inside surface temperature outside surface temperature temperature difference between two surfaces temperature difference between two sides of an enclosure

Greek letters a thermal diffusivity b thermal expansion coefficient (for an ideal gas, b = 2/ (T1 + T2)) d thickness of a single air layer dnom air layer thickness needed to reach nominal insulation R-value dopt air layer thickness needed to achieve maximum insulation R-value (Rins,opt) e total, surface emissivity c usage effectiveness factor (c = Rinsk/L) m dynamic viscosity r Stefan Boltzmann constant (5.67  108 W/m2 K4)

the real potential to significantly reduce the loads required from the heating and cooling units of the building. Other investigators have proposed numerous concepts with the intention of reducing the building heating and cooling loads. One such concept which has received a great deal of attention is that of ‘‘breathable walls’’, or ‘‘dynamic insulation’’. These consist of a permeable insulation through which outside air is passed and filtered as a result of a negative pressure differential in the building. The walls therefore act as a heat exchanger of sorts, harvesting the heat normally lost through the walls via conduction. This concept has been the focus of numerous analytical [8,9], experimental [10– 12], and numerical studies [13], with its theoretical basis largely formed by Imbabi and coworkers [14–19]. The technology has proven useful and has received a great deal of consideration in multiple countries throughout the world. This is somewhat similar to a ventilated façade, which has been proposed in numerous forms over the past number of decades [20–24], and consists of removing residual hot or cold air (depending on the season) via ductworklike accommodations built into the walls and/or roof of the building. Another method particularly useful in the summer months is that of evaporative cooling [25]. Each of these methods [8–25] is worthy of a smart insulation label in that they allow the walls and roofs to have multiple purposes. The concept proposed in this manuscript also helps reduce the overall heating and/or cooling loads imposed on the building, but achieves it through physically adjusting the insulation R-value, thereby allowing a greater level of independent control of ventilation rates and heating/cooling conditions. In the remainder of this paper, the conceptual design of the multifunctional insulation is presented, followed by the analytical characterization of the thermal performance in both the insulating

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and conducting configurations. Results and discussion are then provided, covering a test case with historical data from Pittsburgh, Pennsylvania, United States.

2. Design of multifunctional insulation The concept of the smart insulation described in this work is illustrated in Fig. 1. In the insulated state (Fig. 1(a)), the total thickness of the wall (Lins) is comprised of an innermost and outermost layer, each of thickness tw, and the remaining internal space available (L) is separated into N layers of air by thin polymer membranes. Much like a multi-paned window, the thin layers of air impede the effect of thermal convection. Thermal energy is transported through an individual air layer via convection and radiation. These two modes of heat transfer act as resistances in parallel (Rconv and Rrad for convection and radiation, respectively) as shown in the equivalent thermal circuit of Fig. 1(b). The remaining thermal resistance in this equivalent circuit is attributed to conduction through the innermost and outermost wall layers (Rw) and N  1 polymer membranes each with conductive resistance Rp. It should be noted that for a given wall, there exists an upper limit to the achievable R-value. This limit can be determined by assuming infinite radiation resistance and completely stagnant air layers, such that the convective resistance is simply expressed as a conductive resistance through an air layer of thickness d. The maximum overall resistance one could ever hope to achieve using this concept is therefore the ratio of the total length (Lins) to the thermal conductivity of the interstitial fluid, which in this case is air (k = 0.0263 W/ (m K)). As an example, a 10 cm thick wall would yield Lins/ kair = 3.80 K m2/W (21.6 °F ft2 hr/Btu). In reality, this limit could never be reached since some amount of the overall thickness includes components whose thermal conductivity is likely much higher than air. Sectioning off the air into an arbitrary number of layers does not automatically guarantee the heat transfer through the wall will be relegated to conduction only, but as the convection coefficients are analyzed and evaluated for a particular air layer thickness (d), and the overall resistance is calculated, the result can be compared to the performance limit of pure conduction through air. In general, one would design such a wall with the number of air layers (N) to be as high as necessary to render the air motionless. The ability to achieve both insulated and conductive states with this conceptual wall is achieved by physically collapsing the wall

and removing the air. The polymer membranes formerly isolating the internal layers of air from one another are now compressed into a single polymer layer as shown in Fig. 1(c). As a result, the radiation and convective resistances are no longer present and the total wall thickness (Lcond) is comprised of the thickness of the innermost and outermost layers, and the N  1 internal polymer membranes. The analysis of the collapsed wall is straightforward, requiring only a conductive resistance that accounts for the thicknesses (tp and tw) and conductivities (kp and kw) of each material. For the conductive state, a low resistance is desired, and therefore one would want to limit the number of partitions in order to increase the heat conduction through the wall. The analytical approach to fully characterize the performance is discussed next. 3. Performance characterization The wall has two configurations under which it can operate: (i) insulated and (ii) conductive. Although one might attempt to maximize the insulation R-value (Rins) and minimize the conduction Rvalue (Rcond), the absolute value for Rins is likely to carry the greatest significance. It is important to push the Rcond value as low as possible in order to ensure the conductive case can transfer energy quickly enough to take advantage of temporary environmental conditions, but this must not be done at the expense of achieving a value for Rins that is at least as good as conventional insulation. Before detailed design strategies are developed, the thermal resistance in each configuration must be characterized. 3.1. Conductive configuration analysis The simplest analysis is that of the conductive configuration illustrated in Fig. 1(c) with its equivalent thermal circuit in Fig. 1(d). The overall width of the wall decreases significantly and the pathways for thermal transport become limited to conduction only. The individual conductive resistances in this configuration that dictate the overall performance are the resistance through the innermost and outermost layers (Rw) and the resistance through a partition membrane (Rp). Both of these can be expressed as the ratio of their respective thickness (tw and tp) to thermal conductivity (kw and kp), namely:

Rw ¼

tw ; kw

Rp ¼

tp kp

ð1Þ

The overall resistance of the conductive configuration (Rcond) becomes:

Rcond ¼ 2Rw þ ðN  1ÞRp

ð2Þ

Since the air is removed in this configuration, Rcond is not impacted by the properties of the fluid or the overall width of the expanded wall, and is affected only by the properties of the solid materials. 3.2. Insulation configuration analysis In order to quantify the performance of the insulation configuration, analysis must be conducted for the parallel resistances of convection and radiation. Each of these is now discussed in detail followed by characterization of their equivalent combined resistances.

Fig. 1. Illustration of smart insulation with N internal air layers: (a) extended wall (insulated) and (b) collapsed wall (conductive) configurations. The corresponding resistance networks for heat flow analysis are shown in (c) and (d) for insulated and conductive configurations, respectively.

3.2.1. Convective resistance Natural convection exists as a result of the temperature dependent density of air (or any other fluid), thereby creating a buoyancy-driven flow. The parameters that dictate natural convection

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within an enclosure include the size of the enclosure, the temperature difference across the enclosure and the thermophysical properties of the interstitial fluid. The length scale of interest (d) is the distance between the two enclosure walls which are held at different temperatures. The dimensionless number based on this length scale is the Rayleigh number (Rad) defined as:

g  b  ðDTÞd  d3 Rad ¼ ma

ð3Þ

where g is the gravitational acceleration constant (9.81 m/s2), b is the thermal expansion coefficient (for an ideal gas, this is found from the inverse of the average temperature (b = 1/Tave), ðDTÞd is the temperature difference between one side of the enclosure and the other, and m and a are the dynamic viscosity and thermal diffusivity, respectively, of the interstitial fluid. As the Rayleigh number becomes large, the heat transfer coefficient will also increase. Multiple correlations exist (both analytical and experimental) to predict and characterize the thermal behavior under natural convection conditions. For the current study, the analogy to multi-pane windows is leveraged. The design of internal partitions of the wall is such that the distance that defines the air gap is small compared to the other two dimensions (vertically-oriented parallel plate configuration). For window pane studies, this is also the case, and researchers have shown [26] that for an enclosure aspect ratio greater than 40, the aspect ratio itself becomes unimportant (i.e., a one dimensional, parallel plate analysis is appropriate). Wright [26] developed the following high aspect ratio (parallel plate) correlation for natural convection heat transfer in multi pane windows:

Nud ¼ 0:0674ðRad Þ1=3 for Rad > 5  104

ð4Þ

Nud ¼ 0:0282ðRad Þ0:414 for 104 < Rad 6 5  104

ð5Þ

Nud ¼ 1 þ 1:760  1010 ðRad Þ2:298 for Rad 6 104

ð6Þ

where Nud is the Nusselt number, defined as:

Nud ¼

hd k

ð7Þ

and h and k are the convection coefficient and thermal conductivity of the interstitial fluid, respectively. Note that Eqs. (4)–(6)impose no limit on the magnitude of the Rayleigh number. Also, in the expression valid for small Rad (Eq. (6)), the value for Nud approaches unity as Rad ! 0. This is an important behavior to capture, namely because Nud = 1 reveals that h = k/d. In other words, the overall resistance from natural convection is simply equal to that determined from conduction through stagnant air. Because the insulation performance depends on minimizing internal natural convection, the low Rad regime is of primary interest (Eq. (6)). As an example, if Rad = 1000, then Eq. (6) suggests Nud = 1.02. Experiments [26] with an aspect ratio as small as 20 have resulted in Nusselt numbers very close to this value (Nud = 1.07). In other words, even for smaller aspect ratios, the parallel plate correlations can still be used with reasonable accuracy. In any case, Rad < 104 is desirable in order to ensure convection effects are minimized. In terms of characterizing a single air layer in the insulation scenario, the thickness can be defined as

d¼

L  ðN  1Þt p N

of the first partition membrane and outermost surface of the last partition membrane. Substituting these into Eq. (3) yields the following expression for the air layer Rayleigh number:

Rad ¼

g  b  DT  ½L  ðN  1Þt p 3 N4  m  a

Note that as we assume L  (N  1) tp, Rad  RaL/N4 where RaL is the Rayleigh number found with L as the length scale. This assumption can greatly simplify the analysis, but should be taken with some caution, especially when N becomes large. Nonetheless, it is important to recognize the scaling relationship between Rad and N, namely Rad N4. In other words, doubling or tripling the number of partitions would decrease the local Rayleigh number by an approximate factor of 16 and 81, respectively. The convective resistance for a single air layer can be determined by first evaluating Rad from Eq. (9). This is then used in Eq. (6) to determine the magnitude of Nud. The convective resistance (Rconv) for a single air layer can then be found according to:

Rconv ¼

1 d 1 ¼ ¼ Rd h Nud  k Nud

ð10Þ

where the following definition is made:

Rd ¼

d k

ð11Þ

which represents the conductive resistance through an air layer of thickness d. Eq. (10) again reveals the desire for the design to have Nud as close to 1 as possible. For a Nusselt number greater than this, the convective resistance of a single air layer decreases. 3.2.2. Radiation resistance In addition to analyzing the convective resistance, each air layer must also be examined in terms of the radiation heat exchange. This will eventually need to be combined with the convection as a parallel path for heat flow. Each air layer is bounded on either side by two solid surfaces: two partition membranes or a partition membrane and the innermost or outermost wall layer). Here, we assume all solid surfaces to have identical emissivity (e) and zero transmissivity. One-dimensional radiation heat flux (q00rad ) between two parallel flat surfaces (surface i and j) can be determined from [27]:

q00rad ¼

rðT 4i  T 4j Þ 2 e

1



¼

erðT 4i  T 4j Þ ð2  eÞ

ð12Þ

where Ti and Tj are the temperatures of surface i and j and r is the Stefan Boltzmann constant (5.67  108 W/m2 K4). The standard thermal resistance definition can be used in this case as well (i.e., ratio of temperature difference to the heat flux), and is expressed as:

Rrad ¼

ðT i  T j Þ ð2  eÞðT i  T j Þ ð2  eÞ ¼ ¼ q00rad erðT 4i  T 4j Þ erðT 2i þ T 2j ÞðT i þ T j Þ

ð13Þ

This should be calculated independently and will change for each air layer since the bounding temperatures (Ti and Tj) will be slightly different for each layer. However, this can be further simplified by analyzing relative orders of magnitude for the expected temperatures in and throughout the wall. The resistance definition from Eq. (13) could be expressed in terms of the average temperature between surface i and j (Tave = (Ti + Tj)/2) and the temperature difference (DT) according to:

ð8Þ

Furthermore, assuming relatively thin partition membranes, the local temperature difference can be approximated as ðDTÞd  DT/N, where DT is the temperature drop between the innermost surface

ð9Þ

Rrad ¼

h

er ðT av e 

DT 2 Þ 2

ð2  eÞ i  2  þ ðT av e þ D2T Þ ðT av e  D2T Þ þ ðT av e þ D2T Þ ð14Þ

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Since the difference between any two temperatures is much smaller that the absolute temperatures (DT Ti or Tj or Tave), the higher order terms can be neglected which results in the following approximate definition for radiation resistance:

Rrad 

ð2  eÞ

ð15Þ

4erT 3av e

where Tave can now simply be taken as the average temperature of the entire wall (i.e., average of outside and inside surface temperatures). From this expression, it is evident that for a given partition, resistance to radiation for a single air layer is insensitive to the total number of air layers within the wall. As a result, the only design factor one could exploit to enhance performance from the perspective or radiation is the emissivity. 3.2.3. Equivalent resistance for air layers The equivalent resistance for an air layer (Req) is simply the parallel combination of radiation and convection resistances given by: 1 Req ¼ ðR1 rad þ Rconv Þ

1

¼

Rrad Rd Rd þ Rrad Nud

ð16Þ

With an expression for Req, the overall resistance of the insulated configuration can be determined from:

Rins ¼ 2Rw þ ðN  1ÞRp þ N  Req   Rrad Rd ¼ 2Rw þ ðN  1ÞRp þ N Rd þ Rrad Nud

ð17Þ

The first term of Eq. (17) represents the conductive resistance through the innermost and outermost layers of the wall. The second term represents the conductive resistance through the thickness of each of the partition membranes (N  1 resistances in series), and the final term represents the sum of individual convective and radiation resistances for each air layer (N resistances in series). 4. Results and discussion In order to evaluate the local Rayleigh number (Eq. (3)) for an air layer, fluid properties and other parameters provided in Table 1 are assumed. Worth noting is the global temperature difference (DT = Tout  Tin) of 10 K, with an average between the outside and inside of 300 K. All thermophysical properties of the air are evaluated at this temperature and are assumed to remain constant for all air layers. The thermal conductivities for the partition membrane and the innermost and outermost layers are set at reasonable values for polymer and plasterboard values, respectively. 4.1. Performance trends for insulation R-value Since the thickness and material properties of the partition membrane and innermost and outermost layers are fixed during this analysis, the characteristics of the convection and radiation for each layer are of interest. The magnitude of the local Rayleigh and Nusselt numbers are shown in Fig. 2(a) and (b), respectively.

Fig. 2. Partition natural convection and its dependence on the total number of partitions: (a) local Rayleigh and (b) local Nusselt numbers.

As expected, both Rad and Nud decrease as N increases. It should be noted that the global Rayleigh number (RaL) in this case does not depend on N and is simply equal to the first data point (N = 1) in each curve (RaL = 0.9, 3.1, and 7.3  106 for L = 10, 15, and 20 cm, respectively). Notice that Rad continues to decrease indefinitely while Nud approaches a value of unity for large N. Therefore, conditions exist where a further increase in number of partitions, although continually decreasing the local Rayleigh number, no longer provides substantial benefit from the viewpoint of minimizing convection. From Fig. 2(b), it can be seen that this approximately occurs near N = 4, 5, and 6 for L = 10, 15, and 20 cm, respectively. With knowledge of Nud, one can easily evaluate the convective resistance of a single layer from Eq. (10). The result is shown in Fig. 3(a) for L = 10, 15, and 20 cm. Note that the maximum value of each curve does not indicate the best overall performance, since the results shown here must be combined in parallel with the radiation resistance to form an equivalent resistance for each air layer, which is in turn multiplied by the total number of air layers to determine the contribution of the partitions to the overall wall resistance. From an order of magnitude perspective, the single layer convective resistance shown in Fig. 3(a) can be compared to the single layer radiation resistance in Fig. 3(b), which is calculated from Eq. (15). Since the radiation resistance does not depend

Table 1 Values assumed for analysis of insulation configuration. Air properties (evaluated at 300 K) Property Units Value

k W/m K 0.0263

a

m

m2/s 22.5  106

m2/s 16.0  106

Wall and membrane properties b 1/K 3.33  103

tw mm 15

tp mm 2.0

kw W/m K 0.80

Temperatures kp W/m K 0.20

Tin K 295

Tout K 305

Given the properties in Table 1, the only variables left to adjust are the total number of air layers (N), the thickness of the internal space (L) between the innermost and outermost layers, and the membrane emissivity (e). Here we consider three different values for L, namely 10, 15, and 20 cm (note that Lins = 13, 18, and 23 cm for these three cases). The other two parameters (N and e) are allowed to vary in order to comprehensively illustrate important performance characteristics.

M. Kimber et al. / Applied Energy 114 (2014) 310–319

Fig. 3. Single layer thermal resistance for (a) convection and (b) radiation. Radiation resistance is independent of the number of layers (N). Nominal emissivity of untreated polymer taken to be e = 0.85.

on N, it is shown as a function of the emissivity. Comparison between the local resistances of convection and radiation reveals which mode of heat transfer dictates the overall performance. When combined in parallel, the equivalent resistance will always be smaller than the lowest individual component. As an example, consider choosing N = 6 from the L = 20 cm curve, which yields an Rconv of approximately 1.15 K m2/W. If one desires a scenario where radiation can be neglected altogether, then Rrad should be 20–25 times higher than Rconv. For an L = 20 cm wall with N = 6, this requires e  0.01, a somewhat challenging proposition. On the other hand, if one can accept both radiation and convection playing comparable roles (Rconv = Rrad), then e  0.24, a much more reasonable target. Standard polymer materials that could feasibly be used for the membrane would likely have a much higher emissivity (nominal value could be taken as e = 0.85), however careful surface treatments have produced much lower emissivity [28] through deposition of low emissivity coatings, even on plastic substrates. For reference, an untreated polymer with e = 0.85 would result in Rrad = 0.22 K m2/W, a value roughly 5.5 times smaller than Rconv, yielding Req = 0.186 K m2/W. For those conditions, the equivalent thermal performance of a single air layer would predominantly be dictated by the radiation heat transfer, unless the thickness of the air layer becomes small enough such that Rconv becomes more comparable with Rrad. In reality, the resistance in a single partition is only of interest in the context of its influence on the overall resistance of the entire wall (Rins). First we force e = 0 (Rrad ? 1) to illustrate the fundamental behavior in the insulation condition, as shown in Fig. 4(a). Each wall thickness reveals the same trend, namely that Rins increases with N until an optimum condition is reached. This is expected, since a stagnant air condition is achieved as the number of layers is increased. However, adding more layers beyond this condition simply amounts to replacing a fraction of the insulating air with polymer membranes whose thermal conductivity is gener-

315

Fig. 4. Influence of number of layers on the overall insulation performance when neglecting radiation (e = 0) for (a) 1 < N < 15 and (b) 1 < N < Nmax. When N = Nmax, conduction limit is reached (Rins = Rcond).

Fig. 5. Impact of emissivity on the insulation performance using L = 15 cm curve from Fig. 4(a).

ally much higher than that of the air (kp > k). This explains the linear drop in performance beyond the optimal condition, which would continue until N reaches a value Nmax = 1 + L/tp such that d = 0 and the entire internal space L becomes completely filled with polymer membrane, in which case Rins = Rcond since air is no longer present. This is illustrated in Fig. 4(b). For the three walls considered, Nmax = 51, 76, and 101 for L = 10, 15, and 20 cm, respectively. The rate at which the performance drops beyond the optimum value is identical for all three wall widths, and can easily be found from taking the derivative of Eq. (17) with respect to N and letting Nud = 1. The result is the following:

316

   @Rins 1 1 ¼ t  p @N N!Nmax k kp

M. Kimber et al. / Applied Energy 114 (2014) 310–319

ð18Þ

From a design standpoint, one would want to have this quantity be low in order to have a large range of N where near optimum conditions exist. Therefore the partition membrane should be chosen such that tp and kp are as small as possible (if kp and k are nearly equal, Eq. (18) equals zero). For the second of these two design guidelines (low kp), one should also be conscious of the potential effect on Rcond, where the demand is to increase kp. It should be noted that an even more important design factor is the emissivity of these partition membranes. To illustrate this effect, we take the 15 cm curve from Fig. 4(a) and normalize by Lins/ k (Rins will never exceed this value), while activating the additional thermal pathway of radiation by considering non-zero values of emissivity. The result is shown in Fig. 5, revealing radiation’s dual impact of decreasing the overall achievable performance as well as requiring a greater number of partitions in order to reach those optimum conditions. The e = 0 curve suggests the insulation R-value will never exceed 80% of that based on the assumption of a completely stagnant layer of air whose thickness is Lins. Obviously, the wall and partition values assumed in Table 1 affect this result, and one should attempt to make them as thin as possible. The e = 0.10 and 0.25 curves decrease the maximum Rins to roughly 0.65k/Lins and 0.56k/Lins, but also have the positive impact of decreasing the sharpness of the optimum peak. Therefore, when including radiation, there is a greater range of N where optimum or near optimum performance is realized. However, the number of layers needed to reach the optimum range is roughly Nopt = 11 and 17 for e = 0.10 and 0.20, respectively. If blackbody emission were assumed (e = 1), optimum conditions require Nopt  34 and yield Rins  0.35 k/Lins. In gauging conventional insulation R-values, they tend to be between 50% and 75% of a corresponding stagnant air layer of the same thickness. Therefore, although emissivity can significantly degrade the overall performance, it does not preclude

the multi-partitioned concept from achieving similar insulation performance while providing the additional functionality of a highly conductive state. For the geometry considered in Fig. 5, this would allow 0 < e < 0.4. 4.2. Optimum air layer thickness for maximum insulation R-value An important behavior to capture is the optimum insulation performance (Rins,opt) and the number of air layers needed to reach this condition (Nopt). Fig. 6(a) shows the relationship between Nopt and e for the three wall thicknesses being considered, where Nopt can be seen to increase with increasing emissivity. Taking Eq. (8) and defining the optimal conditions in terms of the air layer thickness (dopt) rather than the overall number of layers (Nopt), the three curves from Fig. 6(a) collapse into a single curve as shown in Fig. 6(b). Therefore, the optimal thickness for a single air layer is only dependent on the emissivity for e > 0.1. This suggests that the overall wall thickness and even the temperature difference do not play significant roles, except when e < 0.1. An expression can be found for the optimal air layer thickness by taking the partial derivative of Eq. (17) with respect to d, setting the expression equal to zero, and solving for d. This results in a quadratic equation in terms of d, where only one of the roots is positive. After assuming Rp Rrad (valid for thin partition membranes) and k kp, the following expression is obtained

dopt ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi t p  k  Rrad

ð19Þ

Although this is not applicable for ultra-low values of emissivity, it presents a useful gauge for most practical materials and coatings where e < 0.1. Substituting this into Eq. (17), and employing the same assumptions (Rp Rrad and k kp) the maximum achievable insulation R-value (Rins,opt) can be quantified as:

Rins;opt ¼

L k

rffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 tp þ1 Rrad  k

ð20Þ

Although this expression is a somewhat conservative estimate, it further illustrates the existence of a fundamental ceiling, namely the fact that Rins,max will always be less than L/k. Therefore from a design perspective, one would want to minimize tp and increase Rrad to the extent possible, the latter of which is achieved by driving down the emissivity. An additional design strategy indirectly implied from Eq. (20) is realized from the relationship between L and Lins, namely that Lins = L + 2tw. The designer must work with the available space, and must realize that the portion not occupied with stagnant air can significantly degrade the insulation performance. In other words, by driving down the thickness of the outermost and innermost solid walls (tw), one is able to take advantage of a larger percentage of Lins for implementation of the smart insulation. 4.3. Designing for nominal insulation R-value A logical approach to the design process would likely begin with a fixed wall width and a nominal R-value to target for the insulation configuration. The consequence of this approach (i.e., first establishing an insulation solution that meets or exceeds existing technologies) is to let Rcond initially remain unspecified. Recognizing from Eq. (20) that Rins,max < L/k, we define a usage effectiveness factor c such that:

Rins ¼ c

Fig. 6. Maximum achievable insulation as a function of emissivity: (a) optimum number of air layers and (b) optimum thickness of air layer.

L k

ð21Þ

where c ranges between 0 and 1. Therefore c represents an estimate of how effectively the space available for the air layers is being utilized. Designing for a nominal R-value is achieved by defining this

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Fig. 7. Maps of acceptable design space based with contour levels of c = Rinsk/L: (a) modeled with full analysis from Eq. (17), and (b) modeled with approximate analysis from Eq. (22). For both approaches, L = 15 cm and all other variables are those listed in Table 1.

usage effective factor. Substituting Eq. (21) into Eq. (17) allows us to develop expressions for the air layer thickness (dnom) needed for Rins to meet the desired nominal R-value. Solving the resulting quadratic equation yields an expression for the two intersecting points of the Rins curve for a constant value of c. The result can be expressed as

dnom

2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3  2 1 1c 1c tp 5 4 ¼ Rrad k  4 c c 2 Rrad k

ð22Þ

Only the smaller of the two solutions is truly valid under the Nud  1 approximation, but Eq. (22) still provides a rough estimate on the design requirements to reach a particular R-value. A more comprehensive way to understand the acceptable design space is to create a map of Rins (Eq. (17)) for different values of d and e with contours at constant values of c. This is shown for the L = 15 cm (Lins = 18 cm) wall in Fig. 7(a), while the approximate relationship between d and e as quantified in Eq. (22) is provided in Fig. 7(b). For a particular contour level, the entire region to the left of the curve represents acceptable d–e combinations which either meet (i.e., directly on the contour line) or exceed (i.e., inside the contour) the target R-value for insulation. There is also a unique value of e for each contour, above which no feasible solution exists. For

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example, if the target R-value was such that c = 0.5, then partition membranes with e > 0.7 would never achieve the desired insulation performance, regardless of what is chosen for d. For a number of the lower performing contours (e.g., c = 0.4, 0.3, and 0.2), although mathematically exhibiting the same behavior, the threshold e is above 1 and therefore not physically realizable. Two differences between the exact (Fig. 7(a)) and approximate (Fig. 7(b)) approaches are worth noting. First, the threshold e and its corresponding d are slightly different for each contour level shown. This data is provided in Table 2 and reveals that the error from using the approximate model is greatest when c is large. In each case, the approximate model under-predicts the threshold e, which suggests this approach is conservative by nature. On the other hand, the air layer thickness at the threshold e is consistently over-predicted by the approximate model. However, given the wall properties in Table 1, the analysis here is still reasonably close as a gauge in determining an acceptable design. The second difference between Fig. 7(a) and (b) can be seen at large values of d (top edge of the contour plots). At this point, the Nud  1 assumption on which Eq. (22) is based is no longer valid. Therefore, in order to quantify this trend more accurately, an approach is needed which captures the dependence of additional properties not accounted for in Eq. (22) (e.g., L and DT). To more effectively illustrate this difference between exact and approximate models, additional values for the overall temperature difference are considered, namely DT = 5, 10, 20, and 40 K, while keeping L = 15 cm. The results are shown in Fig. 8 for a single contour (c = 0.6). For a single value of e, there are two possible values of d that exactly yield c = 0.6. The smaller of these two values is completely insensitive to the magnitude of DT, while the larger option for d is very much dependent on DT as long as e < 0.25. The approximate approach on the other hand never accounts for the overall temperature difference, and becomes more valid for the entire design space as DT approaches zero. Because the design space contour generated from a given DT is completely contained within an analogous contour generated from a smaller DT, one should design the wall based on the largest expected temperature difference. Any DT less than this will provide an insulation R-value greater than the original target value. It should also be noted that when considering both viable options for d at a given value of e, the higher option will yield the optimum performance once the wall has transitioned to a conductive state, since fewer air layers would be needed. Taking the design space curves from Fig. 8 and

Table 2 Threshold values for air layer thickness and emissivity from exact and approximate modeling approaches (L = 15 cm with all other properties provided in Table 1). Exact model (Eq. (17))

c = 0.8 c = 0.7 c = 0.6 c = 0.5

Approximate model (Eq. (22))

d (cm)

e

d (cm)

e

1.29 0.78 0.51 0.34

0.082 0.205 0.409 0.705

1.60 0.93 0.60 0.40

0.065 0.180 0.385 0.699

Fig. 8. Design space as a function of overall temperature difference DT with c = 0.6 (fixed). L = 15 cm and all other variables are those listed in Table 1 with the exception of DT.

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Table 3 Air properties for design test case evaluated at extreme summer and winter conditions. Air properties (evaluated at Tave = (Tin + Tout)/2) Property Units Summer Winter

k W/m K 0.0267 0.0238

a

m

m2/s 23.2  106 18.3  106

m2/s 16.4  106 13.1  106

b 1/K 3.28  103 3.73  103

Temperatures Tin K 297.0 293.2

Tout K 312.6 243.2

assuming e = 0.3, then any d between 0.3 cm and 1.25 cm would meet or exceed the performance dictated (in this case, c = 0.6). Either choice of d will provide Rins = 0.6 L/k = 3.42 K m2/W, but Rcond can vary from 0.13 K m2/W when selecting d = 1.25 cm to 0.33 K m2/W for d = 0.3 cm. In one case, the conductive resistance is over 25 times less than the insulation resistance, while in the other case, this ratio is approximately 10. The best conductive results will always be tied to the largest permissible d. 4.4. Design test case As an example, we design a wall based on weather data from the Pittsburgh International Airport [29], where the most extreme summer and winter temperature on record are 312.6 K (103°F) and 243.2 K (22°F), respectively. We choose air as the interstitial fluid and a thickness representing a standard residential exterior wall with L = 14 cm (5.5 in). Assuming the internal wall temperature to be 297.0 K (75°F) during the summer and 293.2 K (68°F) during the winter, the air properties evaluated at average temperatures in each case are shown in Table 3. Initially, we assume the values listed in Table 1 for tw, tp, kw, and kp, so that the remaining space left for the air layers is L = 11 cm.

Traditional off-the-shelf fiberglass insulation of the same overall thickness (Lins = 14 cm) has a nominal R-value of 3.17 K m2/W (18 °Fft2hr/Btu). We choose this as the target value for the insulation performance such that c = (k/L)3.17 K m2/W = 0.769 during the summer and c = 0.686 during the winter due to the difference in the air thermal conductivity at the two extreme conditions. The design space contours for both extreme conditions are shown in Fig. 9(a). The area common to both curves represents the acceptable design space which meets or exceeds the target value under all anticipated conditions. As expected from the higher c during the summer, its design space contour is more demanding than the winter counterpart. Assuming e = 0.1, then a range of d could potentially satisfy the design constraints. However, when accounting for the fact that the total number of air layers must be an integer value, d must therefore assume discrete values. These points are included in Fig. 9(a) for N between 6 and 12. The largest and smallest of these exist outside the contour, and therefore five viable options exist (7 6 N 6 11). For this example, we choose N = 9 (d = 1.56 cm) which would allow a small increase in e while still meeting the target R-value for insulation. This design yields Rins = 3.23 and 3.70 K m2/W during the summer and winter conditions, respectively, and Rcond = 0.118 K m2/W for all conditions, resulting in an R-value that is between 27.5 and 31.5 times smaller than Rins. Note that it is not DT that impacts the design, but the temperature dependence of the air thermal conductivity, which is evaluated at Tave. Since the conductivity of air (and most gases) increases with temperature, the summer extreme conditions will typically present the more strict design requirements. However, both the upper and lower extreme conditions should be evaluated and compared. One of the biggest design challenges for this example and for the concept in general is that of the emissivity. For the design test case, we assumed e = 0.1, which although not completely unreasonable, represents a significant hurdle, especially in terms of preventing degradation of the surface through oxidation and/or dust accumulation, both of which effectively increases e. However, the acceptable design region can be extended simply by changing the geometry of the partition membranes (tp) and innermost and outermost solid layers (tw). The summer curve from Fig. 9(a) is reproduced in Fig. 9(b) (curve for tp = 2 mm, tw = 15 mm) and compared to three modifications driving tw to zero, tp to zero, and then both. All three show a more inclusive design space, and in the case of the two curves with tp = 0, the entire emissivity range can be accommodated. Therefore, if the contours generated yield a design space too restrictive, one can easily extend this by decreasing tp and/or tw. In summary the following design guidelines exist.  Minimize e: The effect of radiation can significantly reduce the performance or require a greater number of air layers for acceptable insulation levels, potentially increasing the complexity of fabrication and assembly.  Minimize tp and tw: In the insulation configuration, any volume of the internal wall not occupied by stagnant air will significantly degrade the performance. Decreasing tp or tw will have a positive impact, opening up a larger design space. This also has the dual benefit of decreasing the conduction R-value.  Maximize kp and kw: Although this can slightly degrade the insulation performance, a large benefit can be realized for the conductive configuration, especially if transitioning to higher thermal conductivity enables one to then decrease the thickness of either material.

Fig. 9. Design space for L = 14 cm with air properties and temperatures found in Table 3 and a target insulation R-value of 3.17 K m2/W (kp and kw found in Table 1): (a) comparison of summer and winter design requirements (tp = 2 mm and tw = 15 mm), and (b) exploring impact of changing tp and/or tw (legend values are in mm).

For the actual design process, one should be conscious of the fabrication limitations both in terms of d and e. Assuming one begins with a fixed width Lins, the following steps outline the method of establishing an acceptable design.

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(1) Determine extreme summer and winter temperatures based on region specific weather data. (2) Decide on interstitial fluid (most likely air). (3) Decide on target value for Rins keeping in mind that c will always be less than unity. (4) Choose materials for solid components of design to specify tp, tw, kp, and kw. (5) Create design space contours to illustrate acceptable d–e regions for both summer and winter conditions. (6) Choose e and d within area common to both summer and winter contours. Round d down to nearest discrete value(s) corresponding to integer number of air layers. If acceptable values for e and/or d are beyond emissivity or fabrication capabilities, decrease tp and/or tw until acceptable design space is extended to a region where design becomes feasible. If at the limit of tp = 0 and tw = 0, no feasible design exists, then the target value chosen in step 3 must be relaxed, or Lins must be increased. (7) Evaluate Rcond and determine if acceptable. If not favorable, one of three things must be done: (i) lower value of e, (ii) change material of solid components to increase kp and/or kw, or (iii) further decrease tp and/or tw. 5. Conclusions The conceptual design described in this work for a switchable insulation is analyzed from a fundamental and analytical perspective. The concept allows the insulation to exist in both a highly insulating and conductive configuration. The performance limit during insulation is based on the interstitial fluid properties and overall wall thickness, while the conduction limit is defined by the thickness and material properties of the innermost and outermost surfaces as well as the partition membranes. Analysis has shown that viable solutions exist which meet or exceed conventional insulation levels, while greatly enhancing the functionality of the wall by enabling switching to a conductive configuration. A straightforward design process is outlined to provide a framework for achieving the desire performance. One of the primary topics worthy of additional investigation is establishing reliable low emissivity coatings for the partition membranes. This factor (e) can under certain circumstances be the determining factor on whether a design is truly feasible or not. There are a number of fabrication challenges which must be addressed before the proposed concept is able to be implemented. Near the top of this list is the actuation method of changing from insulated to conductive states. This could be addressed by routing ductwork from the HVAC system of the building to inflate the wall for insulation and deflate for conduction purposes. Other options might include mechanical actuation, which requires additional energy input and could increase the complexity of the system in general. Additionally, it is not likely aesthetically acceptable to have either the inside or outside surface of a wall actually translate. In reality, the overall width of the wall is likely to remain fixed, in which case the internal partitions of the wall could be compressed against either the outermost or innermost surface, leaving a large air pocket during the conductive state of the wall. Obviously this extra resistance should be accounted for in such a scenario. An additional aspect worth studying is the dynamic response of the proposed concept. This paper focused on the steady state R-values in both the insulating and conductive configurations. In order to truly characterize the concept, additional studies focused on the dynamic response should also be conducted. When the static and dynamic behavior is well understood, models can be incorporated into standard building simulation toolboxes in order to quan-

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