Determination of Load Dependent Thermal Conductivity of Porous Adsorbents

Transcription

1 Determination of Load Dependent Thermal Conductivity of Porou Adorbent Kraft 1, Gaier 1, Stripf 1, Hee 2 1 Univerity of Applied Science Karlruhe, Baden-ürttemberg, Germany 2 TU Dreden Abtract: Standard meauring technique for thermal conductivity cannot be readily applied to determine the load-dependent thermal conductivity of porou adorbent, becaue the local ad- and deorption inide the pecimen i not conidered. Hence, in thi work a new meauring procedure combining the tranient hot bridge (THB) technique with a finite element model et up in COMSOL Multiphyic oftware for potproceing i preented. An eay to ue COMSOL application i developed to determine the loaddependent thermal conductivity conidering the local ad- and deorption procee and the pecimen geometry. The reult of thi new procedure can be ued for detailed ytem imulation of ad- and deorption procee, e.g. regenerative filter or adorption heat pump. Keyword: Tranient Hot Bridge, thermal conductivity, adorption 1. Introduction The meaurement etup of the THB method i made up of the THB enor which i clamped between two equivalent pecimen. The whole et-up i located inide a chamber in which the vapor preure of the working fluid can be preciely et. The enor itelf conit of a tructured nickel film forming the conducting path between two polyimide heet and i hown chematically in Figure 1a and Figure 1b. The eight meander haped conductor are arranged ymmetrically for an equal-reitance heattone bridge. During meaurement, the contant electric current I B flow through the enor and etablihe an inhomogeneou temperature profile which change the bridge voltage U B(t). The meaured time-dependent bridge voltage i ued to compute the thermal conductivity. IB ground IB Figure 1a. Schematic of the THB-enor polyimide heet Figure 1b. Meaurement etup in the imulation 2. Equation for analytical evaluation To derive an analytical evaluation, Hammerchmidt [1] made everal implification to reduce computational complexity: the enor hot trip are of infinite length and negligible thickne and the ample are conidered infinitely large with homogenou material propertie. ith thee aumption the thermal conductivity can be calculated from k = meandering haped trip adorbent meander haped trip αr 0 2 I B 3 32πL ( U B(t) (ln(t)) ) max (1) UB ymmetry where α i the temperature dependent reitance coefficient of the tructured nickel foil, R 0 i the reference reitance of nickel, I B i the contant current, L i the length of one hot trip and ( U B (t) ) i the maximum derivative of the (ln(t)) max bridge voltage with repect to the logarithm of time. Figure 2 how the qualitative lape of the

2 bridge voltage and it lope which i ued for the evaluation. U B (t) U B (t) (ln(t)) U B (t) (ln(t)) U B (t) ln (t) Figure 2. Qualitative lape of the bridge voltage and it lope 3. Model equation in COMSOL 3.1 Contant Current equation In order to imulate the bridge voltage ignal of the enor, the current conervation equation with ( σ V + e ) = 0, (2) e = I B d HS b HS (3) i ued to pecify the electric current I B a well a the ground for meauring the bridge voltage, where σ i the linearized pecific reitance, V i the electric potential e i the current denity, d HS i the thickne of the nickel film and b HS i the width of the conducting path, where the current denity i applied. In general the linearized pecific reitance can be put in thi form: 3.2 Heat Tranfer The heat tranfer through the enor polyimide foil i decribed with the energy equation (k PI T) = ρ PI c p,pi T t (5) where k PI i the thermal conductivity, c p,pi the pecific heat of polyimide and ρ PI correpond to the denity. To compute the heat tranfer inide the porou pecimen, the propertie of the methanol vapor ρ v c p,v and it velocity field u have to be conidered in the energy equation (k eff T) + Q = (ρc p ) eff T t + ρ v c p,v u T (6) Thereby, Q i a ource term to take care of the adorption heat (ee [2]): Q = h ad X c p,adb (X, T) t. (7) In thi equation h ad decribe the adorption enthalpy and X correpond to the loading. The effective pecific heat, denity and thermal conductivity in equation (5) are mixed material propertie of activated carbon, methanol vapor and adorbed methanol and are decribed with k eff = ε k Adb (X, T) + (1 ε) k v (T) (8) (ρc p ) eff = ε ρ Adb (X, T)c p,adb (X, T) + (1 ε) ρ v (T) c p,v (T) (9) 1 σ = r 0 (1 + α(t T ref )). (4) ρ Adb (X, T) = ρ Adb,tr (T) (1 + X) (10) Here, r 0 repreent the pecific reference reitance and T ref the reference temperature. c p,adb (X, T) = c p,adb,tr (T) + X c p,l (T) (11) The iotropic poroity ε i the ratio between the volume of void and the total volume. It can be meaured with mercury intruion poroimetry

3 and take a value of 0.39 for our pecific pecimen. 3.3 Specie Tranport To imulate the ma tranfer through the porou pecimen, the pecie balance equation ε c t = R i + (D i c i ) (12) i ued. Here, the reaction term R i decribe the ource term for the adorption proce: R i = ρ Adb,tr(T) X ε M t Furthermore the pecie diffuivity (13) D i = D vi + D kn (14) i taken a the um of the mean vicou diffuivity n D vi = ( p ξ i d makro,i ) 32 η v (T) τ vi i=0 and the mean Knuden diffuivity n D kn = ( 4 3 i=0 2 ξ i d makro,i τ kn RT 2πM ) (15) (16) In the equation above d makro i the pore diameter ditribution of the porou pecimen, p i the vapor preure, η v (T) i the temperature dependent dynamic vicoity, M i the molar ma of methanol, τ vi and τ kn are the tortuoitie of the diffuivitie and R i the univeral ga contant. In Figure 3, the pore ize ditribution of the ued activated carbon i hown. Figure 3. Pore ize ditribution of the activated carbon 3.4 Adorption kinetic The partial differential equation n X t = (ξ i kap i (X GG X i )) i=0 (17) i inerted to decribe the adorption kinetic. Thi equation i baed on the linear driving force (LDF) model by uing the time contant kap i = 15 D Ad 2 (18) r p,i to decribe the adorption proce which depend on particle radiu and diffuivity [3]. To conider the particle ize ditribution of the activated carbon raw material, a multitage model i ued. In thee equation ξ i i the volumetric quantity of a particle radiu, X GG i the equilibrium load of methanol at certain preure and temperature, X i i the methanol loading of the particle, D Ad i the adorption diffuivity and r p i the particle radiu. A imilar diffuion coefficient for all particle i aumed. Figure 4 how the particle ize ditribution of the ued activated carbon.

4 number of element, a meh tudy reulted in a meh with domain, boundarie and 5200 edge to imulate the THB meaurement with ufficient preciion. Figure 4. Particle ize ditribution of the ued activated carbon 3.5 Boundary condition At the outer urface a zero flux in ma and heat tranfer i aumed n N i = 0 (19) n q = 0 (20) and additionally there i a ymmetry condition at the ymmetry plane of the tranient hot bridge enor (ee Fig. 1b). 3.6 Initial condition Specimen and enor have the ame initial temperature T 0, i.e., K. The initial loading X 0 for the pecimen i calculated with the initial temperature and preure of the vapor. 4. Ue of COMSOL Multiphyic 4.1 Geometry, Meh and Phyic The geometry of the Senor and pecimen were generated in COMSOL 5.2 and are hown in Figure 5. To meh the 40 µm wide and 2 µm thick nickel trip with the 22 mm wide and 10 mm thick pecimen with a reaonable number of element, the nickel foil i mehed with a mapped meh, which i converted at the contact urface between nickel foil and pecimen into a triangular meh. The triangular meh enable to meh the pecimen tetrahedral. To evaluate the required Figure 5. THB-enor and porou pecimen For the implementation of the equation from chapter three, the COMSOL Phyic: Heat Tranfer in Solid, Heat Tranfer in porou Media, Tranport diolute pecie in porou Media and partial differential equation were ued. 4.2 Pot-proceing with ue of Application Builder COMSOL Application Builder i ued to compute the load dependent thermal conductivity. Therefore the lape of the bridge voltage and it lope are imulated and afterward compared with the meaured one. Since not only the thermal conductivity of the pecimen but alo it denity and heat capacity have an influence on the bridge voltage, the impact of heat capacity and denity were determined. The reult how that both parameter only have an influence within the firt econd. Thu it i poible to adjut the thermal conductivity in the imulation until the lape of the bridge voltage fit to the meaured one. Thereby the lape in the firt econd, where heat capacity and denity have an influence, were not conidered. In order to perform a fat adjutment between the imulated and meaured bridge voltage, the Nelder-Mead algorithm i implemented to minimize the um of leat quare between meaured and imulated value of derivate bridge

5 voltage by altering the effective thermal conductivity of the pecimen [4]. 5. Reult and Dicuion Meaurement were performed in a vacuum chamber with controllable preure of vapor to diverify the loading of the pecimen. The firt tet with the new meauring technique were uccefully performed with the working pair activated carbon and methanol. The computed effective thermal conductivity increae with riing methanol load due to the growing proportion of the thermal conductivity of the adorbed phae. The computed value of the dry carbon complie to meaurement with tandard technique. The Figure 6 preent the analytical and COMSOL evaluation of the load-dependent thermal conductivity. Additionally, the analytical value how a non-continuou behavior at methanol load higher 0.3 /. Due to low amperage, the maximum of the derivative of bridge voltage will not be reached and the determination of thermal conductivity i incorrect. Meaurement at higher current level would lead to methanol diffuion out of the pecimen. In addition to thi effect, the temperature profile would reach the boundarie of the pecimen. Heat and ma tranfer from the boundarie of the pecimen to the tet bench chamber i not conidered in both evaluation method. 6. Concluion It ha been hown for the firt time that it i poible to evaluate the load dependent thermal conductivity with a finite element model for a tranient hot bridge enor. The new technique extend the range of application of the tranient hot bridge method and hed light on the phyic that influence the thermal conductivity of porou media. 7. Reference [1] U. Hammerchmidt and V. Meier, New Tranient Hot-Bridge Senor to Meaure Thermal Conductivity, Thermal Diffuivity, and Volumetric Specific Heat, International ournal of Thermophyic 27.3, S (2006) Figure 6. Thermal conductivity a a function of methanol load Both evaluation method how an increae of thermal conductivity with riing methanol load. In the complete meauring range the analytical value for thermal conductivity are lower than the COMSOL evaluated value, except at zero methanol loading. Thi can be attributed to the aumption which were made for the imple analytical evaluation. In contrat to the new procedure uing COMSOL, all thermal mae of the tranient hot bridge enor are not conidered, internal thermal influence of the temperature profile are not conidered and the local ad- and deorption i neglected. [2] Schnabel, L. (2009): Experimentelle und numeriche Unteruchung der Adorptionkinetik von aer an Adorben-Metallverbundtrukturen. Berlin: Techniche Univerität Berlin. [3] Kat,. (1988): Adorption au der Gaphae, Ingenieurwienchaftliche Grundlagen und techniche Verfahren. VCH Verlaggeellchaft einheim Bael New York [4] Nelder,. A. und Mead, R. (1965): A implex method for function minimization. The Computer ournal 7, S

6 8. Acknowledgement Thi reearch wa upported by the German Federal Minitry of Education and Reearch, reearch grant 03FH023PX4 (OptiSorp). Support from Linei Megeräte GmbH during the conduct of thi work wa greatly appreciated. 9. Nomenclature D Ad D i D kn D vi I B e L M Q Q Ad R R 0 R i A A m mol mol K Ω mol Adorption diffuivity Diffuion coefficient Mean Knuden diffuivity Mean vicou diffuivity Current of heattone Bridge Current denity Length of Tranient Hot Strip Molar Ma of fluid Heat ource term Adorption ource in energy balance Univeral ga contant Reference reitance of nickel Ma ource term T K Temperature T ref K Reference temperature U B V Bridge Voltage V V Electric potential X Load of adorbate Equilibrium load of X GG adorbate b HS m idth of conducting path c c p c p,eff mol K K Concentration Specific heat capacity Effective pecific heat capacity c p,adb Specific heat capacity of K the adorbent c p,adb,tr Specific heat capacity of adorbent without K adorbed fluid c p,l Specific heat capacity of K liquid fluid c p,pi Specific heat capacity of K Polyimid c p,v Specific heat capacity of K fluid vapor d HS m Thickne of nickel film d makro m Pore diameter of adorbent h ad k k Adb k eff λ k v kap 1 Adorption enthalpy Thermal conductivity Thermal conductivity of the adorbent Effective thermal conductivity Analytical evaluated Thermal conductivity Thermal conductivity of vapor Adorption time contant p Pa Preure Specific reference r 0 Ω m reitance r p m Particle radiu t Time m Velocity field for u convection term 1 Temperature dependent α K reitance coefficient ε Poroity of adorbent Dynamic vicoity of fluid η v Pa vapor ξ Relative pore volume ρ ρ Adb ρ Adb,tr ρ eff ρ PI ρ v Denity Denity of adorbent Denity of adorbent without adorbed fluid Effective Denity Denity of Polyimid Denity of vapor

7 σ τ vi τ kn S m Linearized pecific reitance Vicou tortuoity of adorbent Knuden tortuoity of adorbent