COMPOSITE REACTANTS OF CALCIUM CHLORIDE COMBINED WITH FUNCTIONAL CARBON MATERIALS FOR CHEMICAL HEAT PUMPS

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1 VI Minsk Intenational Semina Heat Pipes, Heat Pumps, Refigeatos COMPOSITE REACTANTS OF CALCIUM CHLORIDE COMBINED WITH FUNCTIONAL CARBON MATERIALS FOR CHEMICAL HEAT PUMPS Keiko Fujioka, Kensuke Hatanaka, Yushi Hiata Shinsei Cooling Wate System Co., Ltd. 5 Toa bldg., -3- Kyomatiboi, Nishi-ku, Osaka 55-3, Japan Tel / Fax ; kfujioka@shinsei-uwtb.co.jp Division of Chemical Engineeing Gaduate School of Osaka Univesity Toyonaka , Japan Tel / Fax ; hiata@cheng.es.osaka-u.ac.jp Abstact Enhancement of heat tansfe in the eacto bed is one of the most impotant subjects fo developing gas-solid chemical heat pumps. We have developed composite eactants combining calcium chloide with expanded gaphite (EG) and activated cabon fibe (ACF) to pomote the eaction between calcium chloide and woking fluid of methanol and measued vaiations of effective themal conductivity as well as volume and void faction duing eaction cycles fo the beds packed with these composite eactants. Effective themal conductivity of the EG composite bed was lage than 6 % that of unteated calcium chloide bed. Because of the bulky stuctue of expanded gaphite, the oveall void faction of EG composite was lage than.8 fo the bed packed feely without compession. When the oveall void faction was educed below.78 by compession, the effective themal conductivity was shaply inceased up to times highe than that of unteated calcium chloide bed. On the contay, the effective themal conductivity of ACF composite bed was slightly smalle than that of calcium chloide bed and no emakable change was bought about by compession. The volume and oveall void faction of these composite paticle beds vaied only slightly with the amount of methanol eacted, while those of calcium chloide bed wee lagely influenced by eaction. KEYWORDS Heat tansfe enhancement, Effective themal conductivity, Expanded gaphite, Activated cabon fibe. INTRODUCTION The final goal of ou study is to develop chemical heat pumps fo efigeation o ai conditioning by utilizing the eaction of calcium chloide with wate, methanol, methylamine o ammonia. In ode to achieve a high pefomance eacto bed diving the eaction at a ate desied fo commecial use, it is equied that the eactive paticles should have high themal conductivity, lage specific aea and high pemeability. To this aim, we have developed two composite eactants combining calcium chloide with cabon mateials; one is a composite of calcium chloide and expanded gaphite and the othe is that of calcium chloide and activated cabon fibe. These composite eactants ae expected to have high themal conductivity without educing gas pemeability because fine gains of calcium chloide ae deposited inside and outside the cabon matix. The effective themal conductivities of these composite paticle beds wee measued fo the case whee the following fowad o backwad eaction has been halted o completed, CaCl + CH 3 OH CaCl CH 3 OH. () Vaiations of the bed void faction with eaction pogess, which affect the heat tansfe in it, wee also measued. The themal conductivity and the void faction of the composite paticle bed ae compaed to those of unteated calcium chloide bed and the effect of cabon mateial content on heat tansfe enhancement will be discussed. PREVIOUS WORKS Vaious techniques fo enhancing heat tansfe and effective themal conductivity Heat tansfe in a bed packed with eactive paticles fo chemical heat pumps is vey poo. The effective themal conductivity of the beds, which depends on the mateials used in it and the eaction 36

2 system diving the heat pump, is geneally in the ange of.. W/(mK). Theefoe, enhancement of heat tansfe in the eacto bed is one of the most impotant subjects to impove the oveall eaction ate, and vaious methods have been developed fo this pupose. They can be classified oughly into thee categoies: (a) foming a composite eactant of salt o adsobent combined with heat tansfe pomote with high themal conductivity, (b) insetion of metals o cabon fibes into a bed, (c) integation of eactant into a heat exchange. Some of them ae summaized in Table accoding to the kind of heat tansfe pomote. Table. Methods fo enhancing heat tansfe in eacto bed fo chemical heat pumps Heat tansfe pomote Pepaation technique Reaction couple Expanded gaphite Simple mixing CaCl /NH 3 [] Cabon fibe Impegnation of aqueous solution of salt into EG, dehydatation and calcination Impegnation of aqueous solution of salt into compessed EG, dehydatation and calcination Mixing activated cabon with compessed and blocked EG using esin as a binde Impegnation of aqueous solution of salt into fibe and dehydate CaCl /CH 3 OH [] CaCl /CH 3 NH [3] CaCl /CH 3 NH [] MnCl /NH 3 [, 5] CaCl /NH 3 [5]BaCl /NH 3 [5] Activated cabon/co [6] CoCl /NH 3 [7] Insetion of cabon fibe bush into bed MgO/H O [8] Fomation of an intecalation compound MnCl /HN 3 [9] Metal foam (Cu, Ni) Impegnation of a suspension of salt, compession and calcination Zeolite/H O [] Activated cabon/ch 3 OH [] Resin (Polyanilin) Coating paticles with esin netwok Zeolite/H O [] Aluminum hydoxide Mixing, compession and calcination Zeolite/H O [] Metal fin o tube Insetion of fins into bed CaO/H O [3] Integation of eactant with heat exchange by coating fin tubes with an adsobent laye Silica gel/h O [] Zeolite/ H O [5] In method (a), expanded gaphite is nomally used as a mateial fo pomoting heat tansfe. The common pepaation pocedue is to impegnate an aqueous solution of inoganic salt into expanded gaphite paticles, dy and calcine to deposit its salt inside the poes of expanded gaphite []. Using this method one can pepae a eacto bed whose effective themal conductivity is seveal to times lage than that of the bed packed with unteated salt paticles [3]. Expanded gaphite is a vey bulky mateial, and its effective themal conductivity can be impoved emakably by compession. Fo example, some eseaches adopted a method to compess and mold expanded gaphite befoe impegnation of the salt solution and attained extemely high themal conductivity [, 5]. In case of adsobents like activated cabon that is insoluble in wate, esin is used as a binde to bond the adsobent with expanded gaphite [6]. As fo the examples of method (b), insetion of fins with a small void faction of. into a packed bed inceased its themal conductivity by 6 times [3] and insetion of cabon fibe bush with volume faction of.5 inceased it by times [8]. In the study of method (c), it was epoted that eactos integated with adsobent layes packed between fins of heat exchange wee developed and eaction cycle time was educed to almost / [, 5]. Seveal values of the effective themal conductivity, λ eff, ae plotted against the content o weight faction of heat tansfe pomote in composite eacto bed in Fig.. Although the data in each case wee obtained fo diffeent mateials, eaction couples, density, tempeatue o pessue, the effective themal 37

3 conductivity inceases emakably to the weight faction of heat tansfe pomote in the bed. In the compessed and molded beds with high content of heat tansfe pomote shown by open keys have extemely high effective themal conductivity above W/mK, which is seveal hunded times lage than the oiginal packed bed. λ eff [W m - K - ] Weight faction of heat tansfe pomote [-] Expanded gaphite + CaCl /NH 3 (compessed) [8] Expanded gaphite + Activbated cabon/co (compessed) [] Cuppe foam + Zeolite(compessed) [] Expanded gaphite + CaCl /NH 3 (s imlply mixed)[5] Expanded gaphite +CaCl /CH 3 NH [6] Polyanilin + Zeolite [5] Aluminum hydoxide + Zeolite [6] Fig.. Relation between effective themal conductivity and weight faction of heat tansfe pomote Effect of heat tansfe enhancement on eaction ate Effects of impovement in the effective themal conductivity on the oveall eaction ate will be discussed hee using the esults of simulation analysis in ou pevious study. The simulation was pefomed fo the CaCl and CH 3 NH eaction couple [6]. Kunii-Smith equation fo packed bed [7] was modified so that it could be extended to a bed packed with composite paticles of CaCl and expanded gaphite. The eaction pogess was calculated using the effective themal conductivity estimated by the extended Kunii-Smith equation and the ate paametes obtained fom expeimental esults. The aveage eaction ate in the pocess fom X to X. was adopted as a measue to evaluate the eaction pomoting effect, since the eaction in this peiod pogesses elatively fast and is applicable to diving chemical heat 5 pumps. Fo each weight faction of expanded gaphite, the eaction time equied fo CaCl nch 3 NH eaching convesion X. was obtained fom the simulated eaction pogess, fom H.5 mm which the moles of CaCl eacted with CH 3 NH pe unit time and unit volume, N, 3 was obtained as shown in Fig.. Fo each bed height H, N inceased with the effective H mm themal conductivity, but the inceasing ate gadually slows down as the effective themal conductivity becomes lage. H6 mm Simila esults wee epoted fo the expeiments of the eaction of composite eactant of CaCl and expanded gaphite with CH 3 NH. When the effective themal conductivity inceased fom.6 to. W/mK the output of the eacto inceased by.5 times, but when λ eff inceased to about 3 W/(mK) the output was only.5 times lage 38 Moles of CH 3 NH eacted, N [mol m -3 s - ] λ eff [W m - K - ] Fig.. Effect of themal conductivity on aveage eaction ate simulated fo CaCl /CH 3 NH

4 than that at λ eff. W/(mK) []. It is patly because the heat tansfe esistance between the paticles in the bed and eacto wall becomes a dominant facto fo heat tansfe as the effective themal conductivity inceases. Anothe cause may be the eduction in pemeability, which is bought about by the compession of the bed. It should also be noted that the influence of tempeatue change on eaction ate appeas diffeently depending on the woking condition as well as the eaction systems. The effect of the tempeatue change on the diving foce of the eaction has been examined fo seveal eaction systems unde the tempeatue and pessue conditions supposed to be applicable to pactical use. Figue 3 shows the atio of P to P against tempeatue change. P is the diffeence between the vapo pessue and the equilibium pessue at the initial tempeatue of eacto, that is, the oiginal diving foce o pessue diffeence, and P is that when the bed tempeatue has changed by T due to the eaction heat. As shown in Fig. 3, thee exist lage diffeences in the magnitude of eduction in the diving foce depending on the eaction system and woking condition. Absoptin Desoption.8 P/ P [-] T [ ] Abs. Des. MgO/H O (T a, T v 8, T d 35, T c 5 ) CaO/H O (T a 3, T v, T d, T c 3 ) CaCl /H O (T a 3, T v 5, T d, T c 3 ) CaCl /CH 3 NH (T a 3, T v 5, T d 8, T c 3 ) Fig.3. Decease in diving foce of eaction, i.e., pessue diffeence P, with change in bed tempeatue (T a, T d : set tempeatue fo absoption and desoption eactions, T v, T c : evapoato and condense tempeatues) COMPOSITE REACTANTS PREPARED IN THIS STUDY As mentioned above, vaious eactants have been developed in ode to impove the heat tansfe in the bed and some of them attained vey high themal conductivity. Howeve, such high themal conductivity does not diectly lead to lage incease in eaction ate. In addition, a lage content of heat tansfe pomote may not be ealistic fo a eacto bed of chemical heat pump even if it has extemely high themal conductivity. Hence, we have measued effective themal conductivity of beds packed with composite eactants of CaCl and functional cabon mateials in ode to find a simple and adequate method of heat tansfe enhancement. Pepaation pocedue Two kinds of composite eactants have been pepaed using expanded gaphite (EG) and activated cabon fibe (ACF). Expanded gaphite is an extemely bulky mateial with.3 g/cm 3 of bulk density. The shape of an expanded gaphite paticle is like a wom with about -3 µm in diamete and mm in length, having many poes fom seveal to dozen micons sepaated by thin gaphite walls. Activated cabon fibe (AD ALL) is made fom a coal-ta pitch. Each fibe is about µm in diamete and seveal 39

5 millimetes in length. The pepaation pocedue is common to both composite eactants as follows:. Soak and inse the cabon mateial in methanol.. Rinse it with wate and mix it in a calcium chloide aqueous solution in a beake. 3. Set the beake in a vacuum desiccato to concentate the solution.. Geneate CaCl H O paticles by evapoating the concentated solution in a die at 333K fo 3 minutes. 5. Calcine and dehydate CaCl H O paticles in an electic funace at 573K fo one hou. The mass atio of CaCl to ACF of the ACF composite is 5:, 7.5:, 3.8:.9: o.9:. That fo EG composite is 5:, which has been confimed to have the lagest specific suface aea in ou pevious study []. Stuctue of composite eactants SEM photogaphs of composite eactants ae shown in Fig.. In the ACF composite with the mass atio of 5:, membanous coating CaCl is fomed on the suface of ACF and lumps of CaCl deposited between ACF fibes (Fig. (a), (b)). Simila aspect has been obseved in the ACF composite with mass atio of 7.5:. Few such lumps of CaCl ae seen in the ACF composite with the mass atio of 3.8:,.9:, and.9:, whose suface is coveed with fine gains of CaCl. (Fig. (c), (d)). In the EG composite, CaCl gains have bee deposited in the poes of expanded gaphite (Fig. (e), (f)). Compaing these SEM photogaphs in Fig. with each othe, it can be seen that thee is a clea diffeence in the state of CaCl gains in the eactant; in the EG composite aggegates of CaCl gains deposited in the poes ae suounded by gaphite walls and in the ACF composite CaCl gains has deposited outside the gaphite fibe. In the heat tansfe expeiments, ACF composite with mass atio of 3.8: as well as EG composite with that of 5: was used fo futhe investigation. (a) (c) (e) (b) (d) (f) Fig.. SEM photogaphs of ACF composite with mass atio of 5:(a, b), ACF composite with mass atio of.9: (c, d) and EG composite with mass atio of 5: (e, f) MEASUREMENT OF EFFCTIVE THERMAL CONDUCTIVITY Expeimental appaatus and pocedue The eacto vessel used fo themal conductivity measuement is shown in Fig. 5. The expeimental appaatus consists of a gas supply line and two wate bathes. The gas supply line was equipped with a nitogen gas-cylinde and a containe of liquid methanol. The eacto vessel of pessue glass with 7 mm in inne diamete and 5 mm in height with a stainless steel flange was connected to the gas supply line. Inside the vessel, five themocouples of.5 mm in sheath diamete wee placed in such an aangement that the adial distance fom the cente to each position,, was equally spaced. 3

6 Samples used fo measuement wee composite eactants and unteated CaCl paticles. The amount of initially packed in the vessel was 3.g fo EG composite, 3.9 g fo ACF composite and 7.8 g fo unteated CaCl. Each sample was kept at 53 K fo 5 hous to emove wate and then chaged into the vessel. The vessel was connected to the vacuum line and evacuated. Then methanol gas was intoduced into the vessel to pomote the absoption eaction. Afte the absoption eaction was completed, the vessel was evacuated while it was heated at 373 K to pomote desoption eaction. The moles of methanol eacted with CaCl was obtained fom the diffeence between the weight of the eactive solid and that of initial sample packed in the eacto. Afte the eacto Themocouple Coss sectional view of sample bed position of themocouples 5.8mm 7.75mm 3 9.8mm.95mm vessel was shaken so as to make the packed state unifom in the bed and tapped fo seveal times, the bed height was measued. The eacto vessel was connected to the line again and kept at a constant tempeatue T (73 K) in wate bath. Then it was quickly tansfeed to wate bath at a diffeent tempeatue T (93 K). Fom that time on, the outputs of themocouples and pessue gauge wee ecoded at intevals of two seconds. The gas phase in the vessel was nitogen at.3 kpa in all the measuements. 3 5 Fig. 5. Reacto vessel and configuation of themocouples Data eduction The effective themal diffusivity of bed was obtained fom the unsteady tempeatue pofiles. Since the bed height is lage enough compaed with the bed adius, the heat tansfe in the middle of the bed whee the tips of themocouples have been placed can be assumed to take place only in the adial diection. Assuming that the heat capacity of the gas phase can be neglected, the unsteady heat balance in the bed is expessed by dt π[ ( ε ) ρc p ] Td πλeff () θ d whee T is tempeatue, λ eff effective themal conductivity, θ time, ρ density, C p heat capacity and ε void faction. Since the tempeatue pofiles in the adial diection in the bed ae symmetical with espect to the axis, the tempeatue pofile T() can be expessed by Using, Eq. (3) is ewitten as T ( ) T + a + a + T ( ) T + a + a +. (3). () When the values of a i with i ae small compaed with a, T vaies linealy with. The measued tempeatue pofiles of T() become essentially linea to fo lage θ. The linea elation between T and educes the eo consideably in calculating the tempeatue gadient. If the density, the heat capacity and the effective themal conductivity ae assumed to be constant, Eq. () is educed to the following equation dt Td α eff θ (5) d whee λeff αeff. (6) ε ρc ( ) p Integating the tempeatue pofile fom to, and then diffeentiating it with espect to time θ, one can calculate the left hand side of Eq. (5). The value of dt d on the ight hand side of Eq. (5) is obtained by diffeentiating tempeatue pofile with espect to. Then plotting the fome value 3

7 against the latte one can obtain the effective themal diffusivity α eff though the elation of Eq. (5) An example of such a plot to obtain α eff is shown in Fig. 6. RESULTS AND DISCUSSION LHS CaCl ( dtrhs d) [K] Fig.6. An example of plot based on Eq. (5) Vaiations of volume and void faction In gas-solid eactions, the void faction of a bed changes due to the expansion and contaction of eactive solid as it absobs o desobs eactive gas, which affects the heat tansfe in bed. Figue 7(a) shows the vaiations of the atio of bed volume V b to the initial bed volume V b with n duing the fist thee eaction cycles. The volume of bed packed with CaCl paticles expanded ievesibly in the fist absoption eaction; it inceased about two times as lage as the initial volume. On the othe hand, the volume of the ACF composite bed was kept almost constant duing the fist thee eaction cycles. The bed volume of EG composite inceased only slightly with the mole atio, n, of methanol to CaCl and epeated almost the same path. Coesponding to the bed volume, the oveall void faction, ε, also vaied with n as shown in Fig.7(b). The bed volume inceased with n but an incease in solid content with n causes a decease in ε. In the CaCl paticle bed ε vaied fom.55 to.8, while it vaied fom.8 to.9 in the ACF composite bed and.75 to.9 in the EG composite bed. Because of the small changes in bed volume and void faction, it is expected that a steady condition may be kept thoughout the eaction fo the bed of composite eactants. Effective themal diffusivity and effective themal conductivity V b /V b [-] 3 Composite EG ACF CaCl st cycle nd cycle 3d cycle (a) Oveall void faction ε [-].8.6 (b) n [-]. n [-] Fig.7. Vaiations of (a) bed volume and (b) oveall void faction with the mole atio of methanol to CaCl, n 3

8 The effective themal diffusivity α eff was obtained fom the slope of the staight line in the plot of left hand side against ight hand side of Eq.(5) as shown in Fig.6. Then the effective themal conductivity λ eff, was obtained fom α eff by using the elation of Eq. (6). The effective themal diffusivities thus obtained ae plotted against n, which epesents the eactive gas substance eacted with CaCl, in Fig. 8. The effective themal diffusivity as well as the effective themal conductivity depends on void faction that vaies with n. Then λ eff is plotted against the oveall void faction ε in Fig. 9. In the expeiment, both the ACF and EG composite beds wee allowed to expand o contact feely in the fist thee eaction cycles as shown in Fig. 7. Afte the thid eaction cycle was completed, the bed was compessed mildly to educe its volume. α eff and λ eff befoe compession ae shown by open keys and those afte compession ae shown by closed keys in Figs. 8 and 9. In Fig. 8 α eff deceased with inceasing n fo all samples because of the incease in heat capacity of the eactive solid with n. The effective themal conductivity of EG composite inceased up to times lage than unteated CaCl, indicating that the heat tansfe popety of the bed has been impoved. In Fig. 9 the effective themal conductivity of EG composite bed befoe compession is about 6% as lage as that of unteated CaCl. Afte the compession was applied, the effective themal conductivity inceased steeply with the decease in ε ; when the oveall void faction was educed to.7 fom.8, λ eff inceased by thee times and when he oveall void faction was educed to.6 λ eff was nealy five times lage than that at ε.8. On the contay, the effective themal conductivity of ACF composite α eff 6 [m s - ] Unteated CaCl EG composite EG composite (compessed) ACF composite ACF composite (compessed) n [-] Fig. 8. Vaiations of effective themal diffusivity with mole atio of methanol to CaCl, n λ eff [W m - K - ] CaCl nch 3 OH (This study) Unteated CaCl EG composite EG composite (compessed) ACF composite ACF composite (compessed) CaCl nch 3 NH (Pevious study) Unteated CaCl EG composite (Estimated) Oveall void faction ε [-] Fig.9. Relation between effective themal conductivity and the oveall void faction 33

9 was lowe than that of unteated CaCl and no diffeence was obseved in the dependency of λ eff on ε befoe and afte compession. It is obvious fom Fig. 9 that the steep incease in the effective themal conductivity is not solely due to the decease in the void faction, since the dependency of λ eff on ε changed afte compession. The following easoning will be possible: the paticles of EG composite ae linked to each othe foming the continuous path of heat flow by the compession. Reduction of the contact esistance between paticles as well as that between CaCl gain and gaphite wall may contibute the incease in λ eff. On the othe hand, the compession seems to cause only the decease in void space between paticles in the ACF composite. Such a diffeence fom EG composite would be due to the stuctue of ACF composite whee CaCl and cabon fibes ae not mixed micoscopically. In Fig. 9 the effective themal conductivity of CaCl /CH 3 NH obtained in ou pevious study [3] has also been shown. It was obtained so that the simulated time vaiations of convesion and tempeatue fit the expeimental esults measued in a thin packed bed eacto. The estimated λ eff of EG composite/ch 3 NH is a little highe but close to that of the EG composite/ch 3 OH in this study. It indicates that the volume expansion of CaCl due to the absoption of eactive gas bought about the simila effect on the contact state of paticles by compession. CONCLUSIONS Two composite eactants have been developed combining calcium chloide with expanded gaphite (EG) and activated cabon fibe (ACF) to enhance the heat tansfe in a packed bed eacto. The bed volume and the oveall void faction of composite paticle bed vaied only slightly with the amount of methanol eacted, while those of noncomposite o unteated calcium chloide bed wee lagely influenced by eaction. The effective themal conductivity of EG composite bed was lage than about 6 % that of the unteated calcium chloide bed unde fee expansion condition. When the oveall void faction was educed to below.78 by compession, the effective themal conductivity of EG composite paticle bed was shaply inceased. The effective themal conductivity of AFC composite bed was lowe than the unteated calcium chloide bed and no effect of compession was obseved. Refeences Valkov V., Dodelet J. P. Dlleo T., Nikanpou D., Roy J-F. Rondequ L.C. A new themochemical mateial and eacto fo space systems // Poc. of Int. Soption Heat Pump Conf, Shanghai,. Pp Hiata Y., Fujioka K and Fujiki S. Pepaation of fine paticles of calcium chloide with expanded gaphite fo enhancement of diving eaction fo chemical heat pumps // J. Chem. Eng. of Japan, 3. Vol. 37. Pp Fujioka K., Sasaki K., Oido K., Hiata Y. Enhancement of heat tansfe in calcium chloide eacto bed fo chemical heat pumps by use of gaphite composite paticles // Poc. of 5th Minsk Int. Semina Heat Pipes, Heat Pumps, Refigeatos, Minsk, Belaus. 3. Mauan S., Paades P., L Haidon F. Heat and mass tansfe in consolidated eacting beds fo themochemical systems // Heat Recovey Systems & CHP Vol. 3. Pp Han J.H., Lee K.H. Effective themal conductivity of gaphite-metallic salt complex fo chemical heat pumps // Jounal of Themophysics and Heat Tansfe Vol. 3. Pp Goetz V., Guillot A. An open activated cabon/co soption cooling system // Ind.Eng.Chem.Res.. Vol.. Pp Aidom Z., Tenan M. Salt impegnated cabon fibes as the eactive medium in a chemical heat pump: the NH3-CoCl system // Applied Themal Engineeing.. Vol., Pp Nakaso K., Anai M., Sasaki Y., Hamada Y., Fukai J. Impovement of heat tansfe chaacteistics in a solid-gas themochemical eacto // th APCChE Congess, Kitakyushu, Japan.. G-7. 9 Delleo T., Sameo T. D., Touzain Ph. A chemical heat pump using cabon fibes as additive. pat I: Enhancemenst of themal conduction // Applied Themal Engineeing, 999. Vol. 9. Pp Guilleminot J.J., Choisie A., Chalefen J.B., Nicolas S., Reymoney J.L. Heat tansfe intensification in fixed bed adsobes // Heat Recovey Systems & CHP Vol. 3. Pp Hu E.J., Zhu D.S., Sang X.Y., Wang L., Tan Y.K. Enhancement of themal conductivity by using polume-zeolite in solid adsoption heat pumps // Tans ASME (Jounal of Heat Tansfe) Vol. 9. Pp

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