Calculation and Adjustment of the Activation Temperature of Switchable Heat Pipes Based on Adsorption

Abstract

Recently, thermal regulators based on adsorption in a heat pipe have been proposed. The advantage of these so-called “switchpipes” over similar approaches is their low on state thermal resistance. In this paper, we propose a methodology to calculate and adjust the activation temperature of such switchpipes. For this purpose, we use a mass balance-based model that considers both the heat transfer properties of the heat pipe itself, which depend on the amount of working fluid, and the adsorption equilibrium of the adsorbent used. This model can be used not only to describe the activation behavior of a given heat pipe but also to optimize the configuration of a heat pipe for specific operating conditions and to select appropriate adsorbents. In this paper, we also propose definitions for basic indicators of the activation properties of the heat pipe, such as the activation temperature and the activation temperature span. Finally, a simplified calculation method is presented that allows the selection of the correct adsorbent among all adsorbents with Type IV and Type V adsorption isotherms.

1. Introduction

Autonomous thermal regulators, which can be used to automatically control heat flow from a device to its environment in dependence of the operating conditions, are already in use e.g., for thermal control in space systems [1], and could be useful in several additional applications. For this purpose, different concepts have already been introduced in the literature and are summarized in a recent review paper by Wehmeyer et al. [2]. An important distinction according to Wehmeyer et al. is that, in addition to thermal regulators, there are also thermal switches. Thermal switches can be triggered actively and independently of the system temperature. In contrast, thermal regulators are passive systems that change their heat transfer properties from the non-conducting off to the conducting on state when the system temperature exceeds the switching temperature $T_s$. Other important characteristics of thermal regulators are the switching ratio r, which is the ratio of the conductance in the on and off states and the maximum heat transfer rate in the on state ${\dot{Q}}_{max}$. Recently, a new approach for a thermal regulator and an improved version of this thermal regulator have been presented by Winkler et al. [3,4,5,6,7]. This thermal regulator is realized as a switchable heat pipe, also referred to as a switchpipe (The term “switchpipe” is a suggestion by the authors that has not been used in the literature for this type of heat pipe to date), and is illustrated schematically in Figure 1. In its first version [3,4], it consisted of a heat pipe with an adsorbent inserted into the evaporation zone. The evaporation zone is then connected to the system whose temperature is to be controlled. In a later version [5,6,7], the adsorbent was placed outside the heat pipe in a separate adsorbent reservoir. In this way, either a thermal regulator can be realized if the temperature of the adsorbent reservoir is coupled to the evaporator temperature or a thermal switch can be realized if the temperature of the adsorbent reservoir is controlled separately.

We note that the mentioned approach to realize thermal regulation or switching is novel and has not been proposed previously by other authors. As outlined in the review paper by Wehmeyer et al. [Wehmeyer], there are several working principles to realize thermal switches or regulators with heat pipes. The published works exhibit a variety of design methods and challenges that are different for each working principle. For example, for a variable conductance heat pipe (VCHP), the amount of non-condensable gas must be precisely defined. In contrast, switchpipes use an adsorbent loaded with working fluid as the element that enables thermal switching or regulation. Since the working principle is new, the design methods for the switchpipes are also new and have to be specifically developed. This work marks an important step towards the adaption of the necessary adsorbents (type, amount, etc.) to the boundary conditions imposed by the application.

In general, heat pipes can transport large amounts of heat by evaporation and condensation of a working fluid in an evaporation and condensation zone, respectively [8]. Conventional heat pipes are always in the on state when the operating temperature is at least slightly above the triple point of the working fluid used. For this reason, Winkler et al. [3,4,5,6,7] used an adsorbent to realize thermal regulators and switches. Depending on the operating conditions, some of the working fluid is adsorbed or released by the adsorbent, which affects the thermal transfer ability of the heat pipe. When most of the working fluid is adsorbed, the thermal resistance of the heat pipe increases because there is not enough working fluid available to transport heat by evaporation and condensation. In this way, a thermal regulator with a switching ratio of approx. 10 [3,4] and a thermal switch with a switching ratio of approx. 50 [5,6] could be realized. An advantage of the presented heat pipe-based thermal regulators and switches is their high conductance in the on state, which is comparable to that of conventional heat pipes [5,6,7]. However, to date, only switchpipes with a switching temperature of approx. 60–65 °C have been realized using TAPSO-34 as adsorbent and the effect of ambient temperature on the switching temperature has not yet been studied. The purpose of this paper, therefore, is to address the following challenges:

• Develop, for the first time, a method for calculating the heat transfer properties of switchpipes depending on the configuration and operating conditions. In this context, heat pipes with a wick structure for operation in against-gravity orientation shall also be considered.
• Introduce basic parameters to describe the thermal activation behavior of switchpipes and investigate the influence of adsorbent type and boundary conditions on these parameters.
• Provide methods for designing and optimizing switchpipes for specific applications. In this context consider simple methods as well as more sophisticated methods for a detailed optimization process.

2. Materials and Methods

The method developed in this paper for calculating the activation properties of switchpipes is based on a steady-state mass balance model, the working principle of which is described in detail at the end of this chapter in Section 2.3. The model considers the heat transfer properties of the heat pipe as a function of the available mass of working fluid, e.g., water. This heat transfer capability is measured in a first step without the use of any adsorbent and is called the activation function of the heat pipe. The presence of the adsorbent is then taken into account in the next step by calculating how much of the total available working fluid is adsorbed and therefore not available for the heat transfer process in the heat pipe. To do this, we calculate the amount of adsorbed working fluid as a function of the operating conditions of the heat pipe, e.g., the temperature of the adsorbent and the vapor pressure of the working fluid. A modified version of Dubinin’s potential theory of adsorption is used for this purpose. With the help of this method, a so-called characteristic curve of adsorption is calculated from the experimental data of the adsorption equilibrium. This characteristic curve can then be used to calculate arbitrary adsorption equilibrium data: if the adsorption characteristic curve for an adsorbent is known, we can calculate how much working medium is adsorbed as a function of the operating conditions of the heat pipe. In this work, we use characteristic curves of adsorption that are directly available in the literature as well as characteristic curves calculated from adsorption equilibrium data measured in this work. For this purpose, adsorption isobars of two activated carbons (ENV 208C 4X8 and BPL 4X10, Chemviron, Seneffe, Belgium), a metal–organic framework (MOF) (aluminum fumarate, type Basolite A520, BASF, Ludwigshafen am Rhein, Germany), and a Y-zeolite (binder-free granules, type NaYBFK, Chemiewerke Bad Köstritz, Bad Köstritz, Germany) are measured using a gravimetric sorption analyzer.

The results calculated with the mass balance-based model and the resulting design and optimization methods for switchpipes are then discussed in Section 3: Results and Discussion. The results include basic parameters for describing the activation behavior of a given switchpipe, such as the temperature at which the heat transfer capacity increases drastically and how suddenly this activation of the switchpipe occurs when the activation temperature is exceeded. In the next step, we introduce a cost function that returns a lower numerical value. The more exactly the calculated activation temperature corresponds to the desired activation temperature, the faster the heat pipe is activated when this temperature is exceeded, and the less adsorbent is required for this process. By minimizing this cost function using a numerical optimization procedure, the optimal configuration of a heat pipe can be determined for a given adsorbent and application. The optimal adsorbent can then be selected by comparing the optimal results of different adsorbents. Finally, the effect of changing the cold side temperature of the heat pipe on the activation temperature is analyzed and a simplified method for selecting the appropriate adsorbent is derived.

2.1. Adsorption Equilibria

2.1.1. Description of Adsorption Equilibria

A modified version of Dubinin’s potential theory of adsorption, which has recently been published by one of the authors [9], is used to calculate adsorption equilibria in this paper. The main idea of Dubinin’s original potential theory of adsorption [10] is that for many adsorption equilibria, all experimental data coincide at a single characteristic curve of adsorption in the so-called W-$\varphi$ space. To calculate this characteristic curve of adsorption, the negative Gibbs free energy of adsorption ${-\Delta g}_{ads}$ is calculated for each measured adsorption equilibrium point using Equation (1), which is also called the adsorption potential $\varphi$. In Equation (1), $p_a$ is the partial pressure of the adsorptive and $p_s\left(T\right)$ the equilibrium vapor pressure of the liquid at the same temperature.

$$\varphi =-{\mathsf{\Delta }g}_{ads}=-RT\ln \left({\displaystyle \frac{p_a}{p_s\left(T\right)}}\right)$$

(1)

The filled pore volume $W\left(\varphi \right)$ is calculated from the adsorbent’s loading $x\left(\varphi \right)$ using an adsorbate density model to estimate ${\rho }_{ads}\left(T\right)$.

$$W\left(\varphi \right)={\displaystyle \frac{x\left(\varphi \right)}{{\rho }_{ads}\left(T\right)}}$$

(2)

Different adsorbate density models are also briefly summarized in Ref. [9]. If the described transformation results in a single characteristic curve $W\left(\varphi \right)$, the characteristic curve of adsorption is temperature invariant. In some cases where a single characteristic curve is not obtained using Equations (1) and (2), it can be useful to replace the adsorption potential with the modified adsorption potential ${\varphi }^{*}$ in Equation (3)

$${\varphi }^{*}=\varphi +\mathsf{\Delta }{h}_{add}\left({\displaystyle \frac{T_{adsorbent}}{T_{ref}}}-1\right)$$

(3)

with the reference temperature $T_{ref}$ and the additional phase transition enthalpy $\mathsf{\Delta }{h}_{add}$. By convention, $T_{ref}$ is set to 273.15 K. The characteristic curve given in terms of $W\left(\varphi \right)$ or $W\left({\varphi }^{\mathrm{*}}\right)$ is then fitted using a mathematical procedure that is also described in Ref. [9]. From this fit of the characteristic curve, the loading of the adsorbent can be calculated as a function of the temperature of the adsorbent $T_{adsorbent}$ and the pressure of the adsorptive $p_a$. In Ref. [9], such characteristic curve fits of about 30 different adsorbents for water vapor adsorption have been documented.

One way to classify adsorption equilibria is by the shape of their adsorption isotherms. These can be either measured or calculated e.g., using the methods mentioned above. Depending on the shape of the isotherms, adsorption equilibria can be classified into different groups according to Ref. [11] (see Figure 2).

2.1.2. Experimental Investigation of Adsorption Equilibria

In addition to the existing adsorption equilibrium data from the literature, we experimentally investigated the adsorption equilibrium of four adsorbent materials. These are two activated carbons (ENV 208C 4X8 and BPL 4X10, Chemviron), aluminum fumarate (Basolite A520, BASF), and binder-free granules of Y-zeolite (NaYBFK, Chemiewerke Bad Köstritz). All materials are available in granular form (see Figure 3). The samples were provided free of charge for this study by the manufacturers.

For the experimental investigation of the adsorption equilibria, a gravimetric sorption analyzer (IsoSORP, Rubotherm) was used. The experimental setup is shown in Figure 4. A bubbler–condenser setup, operating at a high bubbler temperature $T_{bubbler}$ was used to generate a humidified nitrogen stream with a defined water partial pressure $p_a$. A two-stage condenser was then used to adjust the water vapor partial pressure by condensing excess water vapor at the condenser temperature $T_{condenser}$ with an accuracy of approx. 0.2 K. The humidified nitrogen stream was then fed into the sorption analyzer where the sample was weighed with an accuracy of 20 µg. The sample temperature $T_{adsorbent}$ is increased stepwise from slightly above $T_{condenser}$ to either 100 °C (ENV 208C 4X8, BPL 4X10 and Basolite A520) or 140 °C (NaYBFK). The experimental conditions used for the investigation of different sample types are summarized in

Table 1. Experimental conditions for adsorption equilibria measurements.

Material	${\mathit{T}}_{\mathit{p}\mathit{r}\mathit{e}}$ (°C)	${\mathit{p}}_{\mathit{a}}$ (kPa)	${\mathit{T}}_{\mathit{c}\mathit{o}\mathit{n}\mathit{d}\mathit{e}\mathit{n}\mathit{s}\mathit{e}\mathit{r}}$ (°C)	${\mathit{T}}_{\mathit{a}\mathit{d}\mathit{s}\mathit{o}\mathit{r}\mathit{b}\mathit{e}\mathit{n}\mathit{t}}$ (°C)
ENV 208C 4X8	150	2.339	20	23–100
BPL 4X10	150	2.339	20	23–100
Basolite A520	150	2.339	20	25–100
NaYBFK	200	1.228	10	40–140
	200	4.247	30	40–140

. Prior to each measurement, the samples were dried for 6 h in a dry nitrogen stream at the pretreatment temperature $T_{pre}$ according to Ref. [12].

2.2. Thermal Resistance of Commercial Heat Pipe (without Adsorbent)

The main objective of the investigation of the thermal resistance of the heat pipe is to obtain its activation function $R_{th}\left(m_{adsorptive,hp}\right)$, i.e., the relationship between its thermal resistance and the free mass of the working medium. To this end, a commercial copper heat pipe supplied by SITUS Technicals GmbH, Wuppertal, Germany with a sintered wick, pre-filled with approx. 4.4 mL of water as working fluid, an outer diameter of 10 mm, a wall thickness of 0.5 mm, and a total length of 40 cm was used (item number HP-N10-100-900SA). From the average wick thickness of approx. 0.87 mm and an estimated porosity of 50%, the free pore volume of the wick was estimated to be 4.4 mL. One end of the heat pipe was cut off (cut-off length approx. 10 mm) and the heat pipe was heated until all the water evaporated. Afterward, a vacuum valve (novotek Vakuumtechnik, item number 3032) was attached to the open end of the heat pipe to evacuate the heat pipe and degas the working fluid. The required amount of water was then filled in with a syringe. This was followed by evacuation and a degassing sequence, i.e., freezing the water at the bottom of the heat pipe using liquid nitrogen and removing non-condensable gases (NCG) multiple times. The degassing sequence was repeated (typically 3–4 times) until the remaining amount of NCG was deemed negligible, i.e., when the pressure rise due to the presence of NCG was below 0.2 mbar when the working fluid was in a frozen state and the vacuum valve to the turbo pump was closed.

The experimental setup used is shown in Figure 5. A wire heater (Constantan heater wire, wire diameter of 0.2 mm) was attached to the closed end of the heat pipe, forming an evaporation zone with a length of 10 cm. Adjustable electrical power supplies were used to provide the electrical heating power $P_{el}$. In order to minimize parasitic heat flow to the environment, the heater and the adiabatic zone were insulated using a tubular insulation sleeve made of HT/Armaflex (elastomeric foam with a thermal conductivity of approx. 0.04 W/(m·K) and an insulation thickness of 35 mm). Thus, we assume that $P_{el}$ corresponds to the longitudinal axial heat flow $\dot{Q}$ through the heat pipe. The heater temperature was measured with four type T fine-wire thermocouples from TC Mess- und Regeltechnik GmbH, Mönchengladbach, Germany with a wire diameter of 80 µm and accuracy class 1, fixed with Kapton tape. The thermocouples were positioned 0, 3, 6, and 9 cm above the bottom of the heater and the corresponding temperatures were acquired with a data logger (Graphtec GL840 WV, Graphtec Europe, Hoofddorp, The Netherlands). For each heating power and after thermal equilibrium was established, the temperature at the center of the heater $T_H$ was calculated from the temperatures determined using the thermocouples in the heater area according to Ref. [4] and used for further calculations. The condensation zone was cooled by a water-cooling block with a length of 12 cm and water was provided by a circulating chiller (Huber Unichiller 007 Olé, Huber Kaltemaschinenbau, Offenburg, Germany) with a constant temperature $T_C$ of 21 °C. The resulting length of the adiabatic zone of 10 cm was given by the heat pipe section between the heater and condenser.

Altogether, the thermal resistance $R_{th}$ of the heat pipe is given by the temperature difference between the center of the heater and the condenser, divided by the heating power $P_{el}$:

$$R_{th}={\displaystyle \frac{T_H-T_C}{P_{el}}}$$

(4)

Determination of thermal resistance was carried out for three different orientations, namely, vertical orientation with the heater at the bottom (90°), horizontal orientation (0°), and a tilt angle of −16° from the horizontal line in against-gravity orientation i.e., with the heater above the condenser. For each orientation, measurements with 0, 1.1 g, 2.2 g, 3.3 g, and 4.4 g of water as working fluid were performed. At least five different ascending heating powers were applied for each combination of orientation and water quantity. The heating powers were chosen to cover the optimal operating range for the heat pipe i.e., the range of minimum thermal resistance below the “dryout” limit, which is characterized by a sudden increase in thermal resistance with increasing heating power. Altogether, heating powers between 1 W (heat pipe with no working fluid) and 60 W (heat pipe with full fluid filling of 4.4 mL) were applied, and the heater temperature did not exceed ca. 80 °C during the experiments. Thermal resistance was measured separately for each applied heating power. From these thermal resistances, the lowest thermal resistance $R_{min}$ was selected as the reference thermal resistance for the corresponding combination of orientation and water amount, respectively. In the next step, a correction was applied to all $R_{min}$ to account for the radial parasitic heat flow, i.e., the convective heat flow from the heat pipe wall through the tube insulation to the environment. The procedure used for this is described in our previous work—Ref. [4], Section 2.4.

The determination of measurement uncertainty was described in detail in our previous work [4] (Appendix A.2). The temperature uncertainty of the used thermocouples was determined as ±0.11 K from calibration data (standard deviation of actual temperatures taken by the thermoelements vs. reference temperatures). With the mentioned temperature uncertainty, the measurement uncertainty of $R_{min}$ was determined as described in Ref. [4], Appendix A.2.2.

2.3. The Mass Balance-Based Switchpipe Model

The main concept of the steady-state heat pipe model is to use a mass balance to calculate the amount of free working fluid in the heat pipe. From the mass of free working fluid, the thermal resistance of the heat pipe can then be calculated using the thermal resistance of the heat pipe measured as in Section 2.2. Figure 1 gives a schematic overview of the heat pipe model used in this work. The heat pipe is divided into four zones, namely, the condensation zone (I), the adiabatic mass transfer zone (II), the evaporation zone (III), and the adsorbent reservoir (IV) located in the evaporation zone. The heat pipe is a hermetically sealed system, so the total amount of water in the heat pipe $m_{adsorptive,tot}$ is constant. The amount of freely available working fluid $m_{adsorptive,hp}$ in the heat pipe is then the total amount of working fluid reduced by the adsorbed amount:

$$m_{adsorptive,hp}=m_{adsorptive,tot}-m_{adsorbed}$$

(5)

The amount of working fluid adsorbed is

$$m_{adsorbed}=m_{adsorbent}·x_{eq}\left(T_{adsorbent},p_a\right)$$

(6)

with the mass of the adsorbent $m_{adsorbent}$ and its equilibrium loading $x_{eq}\left(T_{adsorbent},p_a\right)$ (Of course, $m_{adsorbed}$ cannot be bigger or equal to $m_{adsorptive,tot}$ since some working fluid always remains in the gaseous phase. If $m_{adsorbed}$ approaches $m_{adsorptive,tot}$, the water vapor pressure in the heat pipe is reduced, leading to a lower $x_{eq}\left(T_{adsorbent},p_a\right)$). It is assumed that the temperature of the adsorbent reservoir (zone IV) $T_{adsorbent}$ is equal to the temperature in the evaporation zone $T_H$ (zone III) and that the adsorptive pressure is given by the vapor pressure of the pure working fluid at the condensation temperature $T_C$ (zone I). When considering a thermal switch with an external adsorbent reservoir, the temperature of the adsorbent may differ from the temperature of the evaporation zone. The equilibrium loading of the adsorbent is calculated using the methods described in Section 2.1.1. Using the amount of free working fluid and the heat pipe activation function $R_{th}\left(m_{adsortpive,hp}\right)$, which is assigned to the adiabatic mass transfer zone (II) in the model, the thermal resistance of the heat pipe can then be calculated for different configurations and operating conditions. The model parameters can be summarized as follows:

• configuration of the heat pipe:

  ○

  Type of adsorbent and its adsorption equilibrium $x_{eq}\left(T_{adsorbent},p_a\right)$

  ○

  Amount of adsorbent $m_{adsorbent}$

  ○

  Total amount of working fluid $m_{adsorptive,tot}$

  ○

  Design-related activation function of the adiabatic mass transfer zone (II) $R_{th}\left(m_{adsorptive,hp}\right)$

• boundary conditions:

  ○

  Evaporation zone (III) temperature $T_H$

  ○

  Condensation zone (I) temperature $T_C$

Other important assumptions are that there are no radial heat losses in the adiabatic zone of the heat pipe and that the presence of the adsorbent does not affect the activation function of the heat pipe.

3. Results and Discussion

The results of the experimental investigation of the adsorption equilibria and the heat transfer properties of the commercial heat pipe as a function of the working fluid are described in the first two sections of this chapter. We now use the results from the switchpipe model described in Section 2.3 to introduce basic indicators to characterize the switching behavior of adsorption-based heat pipe thermal regulators in general. Further conclusions are then drawn from these results. First, the influence of the adsorbent type on the performance indicators is investigated. Then, an optimization procedure is presented to design and fine-tune switchpipes according to the specific requirements of a given application. Finally, the influence of the cold side temperature on the switching temperature is studied and a simplified method for selecting suitable adsorbents with type IV and type V isotherms is proposed.

3.1. Experimental Investigation of Adsorption Equilibria

The measured adsorption isobars for ENV 208C 4X8, BPL 4X10, Basolite A520, and NaYBFK are shown in Figure 6, Figure 7, Figure 8 and Figure 9. The experimental data are used to calculate the characteristic curve for each corresponding adsorbent. These characteristic curves are then used to calculate the theoretical isobars shown as dashed lines in Figure 6, Figure 7, Figure 8 and Figure 9. In general, there is good agreement between the experimental and calculated isobars. The more hydrophilic the adsorbent, the higher the temperature required to desorb the bound water at the same water vapor pressure. Accordingly, the adsorbents can be sorted by decreasing hydrophilicity: NaYBFK, Basolite A520, ENV 208C 4X8, and BPL 4X10. It is expected that the switching temperature of the heat pipe decreases with decreasing hydrophilicity. Therefore, it is likely that relatively low switching temperatures can be achieved using Basolite A520 and the two activated carbons.

For validation, the characteristic curve of data for Basolite A520 and NaYBFK calculated from the experimental data in this work are compared to the characteristic curve of data calculated from literature-derived data. Adsorption equilibrium data for aluminum fumarate were extracted from Teo et al. [13]. The data from Ref. [13] can be explained by an additional phase transition enthalpy ${∆h}_{Add}$ of 145 kJ/kg. This additional phase transition enthalpy is also applied to Basolite A520, which is a pelletized aluminum fumarate, whereas in Ref. [13], aluminum fumarate powder is investigated. This results in good agreement of the characteristic curve between the literature data for aluminum fumarate and the Basolite A520 data from this work (see Figure 10). A small exception is the region with low loadings ($W<0.06{\displaystyle \frac{\mathrm{c}{\mathrm{m}}^3}{\mathrm{g}}}$), where Basolite A520 shows slightly higher loading than aluminum fumarate powder. This could be due to the process of pelletizing Basolite A520. For NaYBFK, reference data are taken from Velte [14] and are in good agreement with the data from this work (see Figure 11). For ENV 208C 4X8 and BPL 4X10, no reference data for water vapor adsorption could be found.

The characteristic curves can also be used to calculate adsorption isotherms, resulting in type V isotherms for ENV 208C 4X8 and BPL 4X10, where the adsorption occurs at high relative pressures. Basolite A520 also shows a type V isotherm with adsorption at lower relative pressures than the activated carbons. NaYBFK shows type I adsorption isotherms.

3.2. Thermal Resistance of the Heat Pipe

The thermal resistance of the heat pipe as a function of the mass of the working fluid (water) is shown in Figure 12. The less working fluid contained in the heat pipe, the greater its thermal resistance. With no water inside, the thermal resistance of the heat pipe is approx. $R_{th,off}=23{\displaystyle \frac{\mathrm{K}}{\mathrm{W}}}$. After full saturation, the thermal resistance is approx. $R_{th,on}=1{\displaystyle \frac{\mathrm{K}}{\mathrm{W}}}$. It is known that the working fluid in a heat pipe should fully saturate the wick to achieve a low thermal resistance [8]. For the selected heat pipe, this corresponds to a fluid volume of approx. 4.4 mL. As shown in Figure 12, high minimum thermal resistance $R_{th,min}$ results when the wick of the heat pipe is “undersaturated”. In the vertical orientation, the heat pipe is less amenable to undersaturation than in the horizontal or against-gravity orientations. This is because gravity aids in transporting the liquid fluid phase back to the evaporation zone at the bottom of the heat pipe, leading to a stronger saturation of the wick in this area compared to other heat pipe areas. Thus, even if the average saturation of the wick along the whole heat pipe length is not full, there is enough working fluid in the evaporator to maintain efficient heat pipe operation.

The heat pipe activation function $R_{th}\left(m_{adsorptive,hp}\right)=R_{th}\left(m_{H2O,hp}\right)$ shown in Figure 12 is obtained from the experimental data using a straight line fit to the 0° $R_{th,min}$ data in the range 0 $<m_{H2O}<$ 4.4 g. There are no measurement data for $m_{H2O}>$ 4.4 g. For simplification, we assume that the heat pipe keeps its minimum resistance from the on state if slightly more working medium is added, so in this case, $R_{th}=R_{th,on}$. Although this assumption is correct to a first approximation in the case of our previous investigations [4], it should be further investigated in the future.

3.3. The Switchpipe Model

3.3.1. Parameters for Describing the Activation Behavior of the Heat Pipe

Using the basic model described in the previous chapter, the thermal resistance of a given heat pipe, i.e., with defined adsorbent type, adsorbent and working fluid masses, and known activation function can be calculated at defined boundary temperatures $T_H$ and $T_C$, respectively. This can be used to derive basic parameters that describe the activation behavior of the heat pipe. For this purpose, the thermal resistance of an example heat pipe with the heat pipe activation function described in Section 3.2 and using 15 g of Basolite A520 as adsorbent and 6 g of water as working fluid is calculated at constant $T_C$ and plotted against $T_H$. The result is shown as the blue line in Figure 13. Then, the dimensionless thermal resistance

$$\psi ={\displaystyle \frac{R_{th}-R_{th,on}}{R_{th,off}-R_{th,on}}}$$

(7)

is introduced. For $\psi =1$, $R_{th}=R_{th,off}$ and for $\psi =0$, $R_{th}=R_{th,on}$. The activation temperature

$$T_{act}=T_H\left(\psi =0.5\right)$$

(8)

of the heat pipe is the temperature of the evaporation zone where the dimensionless thermal resistance is 0.5. The activation temperature span

$${\mathsf{\Delta }T}_{act}=T_H\left(\psi =0.1\right)-T_H\left(\psi =0.9\right)$$

(9)

describes how quickly the thermal resistance of the heat pipe decreases when the activation temperature is exceeded. The smaller the ${\mathsf{\Delta }T}_{act}$, the faster the heat pipe will be activated when the temperature rises.

Under given boundary conditions, the heat pipe can only be fully activated and deactivated if both temperatures $T_1\left(\psi =0\right)$ and $T_2\left(\psi =1\right)$ can be found. All possible combinations of $m_{adsorbent}$ and $m_{adsorptive,tot}$ that satisfy this criterion are called the working field of the heat pipe (For numerical reasons, $\psi =0.01$ and $\psi =0.99$ are chosen to calculate $T_1$ and $T_2$). The working field of the example heat pipe discussed in Figure 13 is shown in Figure 14 as the gray area. The minimum amount of working fluid is the amount of working fluid required to completely wet the wick structure. The minimum amount of adsorbent is the amount of adsorbent required to adsorb all of this minimum amount of working fluid in the heat pipe. Within this working field, the activation temperature and the activation temperature span of a given heat pipe at a given $T_C$ can be calculated. The results of these calculations for the example heat pipe are shown in Figure 14 and Figure 15. The activation temperature is in the range of 313–316 K in the main area of the working field using Basolite A520 as adsorbent. In this area, the activation temperature is only slightly dependent on the adsorbent and the adsorptive masses. Activation temperature spans ${∆T}_{act}$ of less than 2 K can be achieved if enough adsorbent, i.e., more than 25 g, is used. This is due to the step-like shape of the adsorption isotherm of Basolite A520.

A switchable heat pipe using NaYBFK as the adsorbent shows a completely different activation behavior (see Figure 16 and Figure 17). First, the shape of the working field is different because the maximum amount of water adsorbed by NaYBFK is smaller than that of Basolite A520 (Figure 16). Therefore, the working field is shifted towards larger amounts of adsorbents and less amounts of working fluid per mass of adsorbent. Second, the activation temperature is generally higher because NaYBFK is more hydrophilic than Basolite A520 and higher temperatures are required to desorb the bound water at the same water vapor pressure (see Figure 16). There is also no region of nearly constant activation temperature due to the non-step-like adsorption isotherms of NaYBFK. Therefore, the activation temperature of the heat pipe can be tuned by selecting the correct adsorbent and adsorptive masses, respectively. However, this results in a larger activation temperature span (refer to Figure 17).

3.3.2. Optimizing a Heat Pipe with Fixed Adsorbent Type

Using the methods from the above section, $T_{act}$ can be calculated for a heat pipe with a given activation function $R_{th}\left(m_{adsorptive}\right)$ and adsorbent type as a function of $m_{adsorbent}$, $m_{adsorptive,tot}$, and $T_C$, so $T_{act}=T_{act}\left(m_{adsorbent}, m_{adsorptive,tot},T_C\right)$. Usually, $T_C$ and a desired activation temperature $T_{act, target}$ are defined by the application. The challenge now is to find the best combination of $m_{adsorbent}$ and $m_{adsorptive,tot}$ to construct a heat pipe with $T_{act}\approx T_{act,target}$. This can be done by minimizing the cost function

$$cost\left(m_{adsorbent}, m_{adsorptive,tot}\right)=\left|{\displaystyle \frac{{T_{act}\left(m_{adsorbent}, m_{adsorptive,tot},T_C\right)-T}_{act,target}}{\left[K\right]}}\right|.$$

(10)

We propose a more general cost function (Equation (11)) when designing switchpipes that also takes the activation temperature span $\mathsf{\Delta }{T}_{act}$ and the mass of adsorbent $m_{adsorbent}$ into account.

$$cost={\left|{\displaystyle \frac{C_1·\left({T_{act}-T}_{act,target}\right)}{\left[K\right]}}\right|}^{n_1}+{\left({\displaystyle \frac{C_2·{∆T}_{act}}{\left[K\right]}}\right)}^{n_2}+{\left({\displaystyle \frac{C_3·m_{adsorbent}}{\left[g\right]}}\right)}^{n_3}$$

(11)

Alternatively, $m_{adsorbent}$ can also be replaced by $m_{adsorbent}+m_{adsorptive,tot}$ or be divided by the bulk density to account for the required volume of the adsorbent as the cost factor. In this work we use $n_1=n_2=n_3=1$, $C_1=C_2=1$ and $C_3=0.2$. For real applications, the parameters should be adjusted according to the importance of the different parameters. The cost function is calculated for an example case with $T_{act,target}=320 \mathrm{K}$ and $T_C=293 \mathrm{K}$ and the results are visualized in Figure 18. It has a single global optimum that is calculated using a simplex algorithm to obtain the best combination of $m_{adsorbent}$ and $m_{adsorptive,tot}$.

The activation behavior of the optimized heat pipe is shown in Figure 19. Compared to the configuration from Figure 13, the heat pipe has a lower activation temperature span. However, the activation temperature of the heat pipe can only partially be adjusted to the desired 320 K. This is due to the fact that the activation temperature of adsorbents with type V isotherms can only be adjusted to a limited extent by changing the adsorbent and adsorptive mass (see also Figure 14). Therefore, the next chapter presents a method for selecting the most suitable adsorbent for a particular application.

3.3.3. Identification of the Best Adsorbent for a Given Use Case

The calculations described in Section 3.3.2 can be repeated for any adsorbent with sufficient adsorption equilibrium data to identify the best adsorbent for a particular application. For illustrative purposes, this calculation was performed for the example $T_C=293 \mathrm{K}$ and $T_{act,target}=320 \mathrm{K}$ using the adsorption equilibrium data from this work. The results are shown in Figure 20. The contributions from different terms in the cost function are also shown. A heat pipe using Basolite A520 as the adsorbent is best suited to meeting the requirements in this example, and the optimal adsorbent and adsorptive masses are 20.90 g and 7.49 g, respectively.

Finally, this optimization process can be repeated with the adsorption equilibrium data given in Ref. [9] in order to identify the best adsorbent among a larger number of candidates. To this end, the adsorption equilibrium data obtained in this work are added to the database developed in Ref. [9].

3.3.4. Influence of the Cold Side Temperature on the Activation Temperature

So far, all calculations have been performed with a fixed cold side temperature $T_C$. However, in real applications, $T_C$ may vary. Since most technical systems are designed to operate at a constant temperature, the influence of $T_C$ on $T_{act}$ is investigated. To this end, $T_{act}$ is first calculated as a function of $T_C$ for a heat pipe with the given activation function $R_{th}\left(m_{adsorptive,tot}\right)$ and using Basolite A520 as the adsorbent. The activation temperature $T_{act}$ and the activation temperature span $∆T_{act}$ of this heat pipe at $T_C=293 \mathrm{K}$ have already been shown in Figure 14 and Figure 15. Both variables are relatively constant in the middle part of the working range, so we choose $m_{adsorbent}=30 \mathrm{g}$ and $m_{adsorptive,tot}=10 \mathrm{g}$ in the beginning and calculate the dependence of $T_{act}$ on $T_C$ for this configuration. The results are shown in Figure 21. In this case, $T_{act}$ is almost a linear function with respect to $T_C$ and a slope of 1. This means that an increase in $T_C$ will result in almost the same increase of $T_{act}$.

The difference between $T_{act}$ and $T_C$ is referred to as the working temperature difference

$$∆T_{work}=T_{act}-T_C.$$

(12)

$∆T_{work}$ depends mainly on the adsorbent used and can be calculated for any adsorbent with sufficient adsorption equilibrium data. However, the calculation of $∆T_{work}$ requires the assumption of reasonable values for $m_{adsorbent}$ and $m_{adsorptive,tot}$. To this end, we calculate a representative heat pipe configuration using the cost function given in Equation (11) with $T_C=293 \mathrm{K}$ but set $C_1$ to 0. $∆T_{work}$ is then calculated as the mean value in the $T_C$—interval from 275 K to 315 K and is listed in

Table 2. $∆T_{work}$ for adsorbents investigated in this work.

Material	Isotherm	${\mathit{m}}_{\mathit{a}\mathit{d}\mathit{s}\mathit{o}\mathit{r}\mathit{b}\mathit{e}\mathit{n}\mathit{t}}$ (g)	${\mathit{m}}_{\mathit{a}\mathit{d}\mathit{s}\mathit{o}\mathit{r}\mathit{p}\mathit{t}\mathit{i}\mathit{v}\mathit{e}}$ (g)	$∆{\mathit{T}}_{\mathit{w}\mathit{o}\mathit{r}\mathit{k}}$ (K)	$∆{\mathit{T}}_{\mathit{a}\mathit{c}\mathit{t}}$ (K)
ENV 208C 4X8	Type V	24.7	7.8	10.5	4.6
BPL 4X10	Type V	23.1	7.6	5.0	4.1
Basolite A520	Type V	20.8	7.5	20.8	2.8
NaYBFK	Type I	81.0	20.4	60.8	15.2
TAPSO-34	Type V	49.4	8.3	36.2	4.7

for the four adsorbents studied in this work and for TAPSO-34, which was used in the previous studies. The adsorption equilibrium data used for this purpose were taken from Ref. [9].

It can be shown that the working temperature difference increases with increasing hydrophilicity, as originally postulated. By using less hydrophilic adsorption materials such as BPL 4X10, thermal switches or regulators with low working temperature differences can be realized. Note that $∆T_{work}$ is a constant only for materials with Type IV and Type V isotherms in the first place. Adsorption equilibrium materials, which are characterized by less step-like isotherms, have a larger activation temperature span and a more inhomogeneous working range in which the activation temperature varies significantly. In Ref. [4], the measured activation temperature of the heat pipe using TAPSO-34 as adsorbent was approx. 56–58 °C, while the cold side temperature was held constant at 21 °C, so $∆T_{work}\approx 36 \mathrm{K}$. Although the activation function of the used thermosyphon differs from the activation function of the heat pipe used in this work, $∆T_{work}$ is in good agreement with the data presented in Table 2, proving the robustness of the simplified calculation method.

4. Conclusions

A new class of heat pipe-based thermal regulators and switches has recently been presented by our group, using an adsorbent loaded with the required working fluid to achieve a thermal switching effect. In this work and for the first time, a method is outlined to select and optimize the type and amount of adsorbent to be used with respect to the given boundary conditions.

In the first part of this paper, adsorption isobars of four different adsorption materials are measured. Three less hydrophilic adsorption materials, BPL 4X10, ENV 208C 4X8 (both Chemviron), and Basolite A520 (BASF), and a much more hydrophilic material, binder-free Y-zeolite granules (NaYBFK, Chemiewerke Bad Köstritz), were selected as adsorbents. While BPL 4X10, ENV 208C 4X8, and Basolite A520 show an adsorption behavior that is characterized by step-like type V adsorption isotherms, NaYBFK shows non-step-like type I adsorption isotherms. All adsorption equilibria could be well explained by the modified potential theory of adsorption. In the next step, we investigated a commercial heat pipe for its heat transfer capability as a function of the amount of working fluid inside. In the horizontal orientation, the heat pipe has a thermal resistance of about 23 K/W in the absence of working fluid. As the amount of working fluid is increased to 4.4 g of water, the thermal resistance decreases linearly to approximately 1 K/W until saturation is reached and the thermal resistance cannot be further decreased by increasing the amount of working fluid. This activation behavior of the heat pipe was described by the so-called heat pipe activation function.

Subsequently, a mass balance-based heat pipe model was developed that uses the modified adsorption potential theory to calculate the amount of free working fluid in a given heat pipe, which is then used to calculate the thermal resistance of the heat pipe. Then, basic parameters for describing the activation behavior of the heat pipe, such as the activation temperature and the activation temperature span, were defined. Further, the influence of the adsorbent type, the amount of adsorbent, and the total amount of working fluid on these parameters was investigated. It turns out that there is a working range of possible parameter combinations that allow full activation and deactivation of the heat pipe. Within this working range, adsorbents with type V isotherms show almost the same activation temperature at a constant cold side temperature of the heat pipe, while adsorbents with type I isotherms show activation temperatures that can be adjusted within a wide range by proper selection of the amount of working fluid and adsorbent. In addition, most adsorbents with step-like adsorption isotherms show lower activation temperature spans than adsorbents with non-step-like adsorption isotherms.

In the next step, we used optimization techniques to find the optimal configuration in terms of adsorbent and working fluid amounts of a heat pipe with given adsorbent type and boundary conditions. This is done by minimizing a cost function that considers the deviation of the activation temperature from the desired activation temperature, the activation temperature span, and the amount of adsorbent used. By performing this optimization for all adsorbents with available adsorption equilibrium data, the best adsorbent for a given application and the corresponding heat pipe configuration can be identified. Finally, we presented a simplified adsorbent selection technique based on the assumption that the activation temperature of adsorbents with step-like adsorption isotherms depends only slightly on the amount of adsorbent and adsorptive in the main region of the working field. Then, an effective working temperature difference can be calculated, which is the almost constant difference between the cold side temperature of the heat pipe and its activation temperature in the first place.

So far, we have only considered water as the working fluid in the heat pipe, but all the calculations presented in this paper could also be performed with other working fluids. In the future, we plan to validate the heat pipe model through measurements with different heat pipe configurations. A crucial part that also needs to be investigated is the thermal stability of the adsorbents used. Currently, we only consider the adsorption equilibrium, but not whether a given adsorbent can be immersed in liquid water or whether there is rapid degradation of the adsorbent under the given hydrothermal conditions.

5. Patents

A patent was filed for the concept of realizing a thermal regulator based on an adsorption material in a heat pipe (European Patent EP20180665.0).