International Journal of Heat and Mass Transfer 127 (2018) 339?347 Contents lists available at ScienceDirect International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt Transient model of carbon dioxide desublimation from nitrogen-carbon dioxide gas mixture Y.N. Wang a, J.M. Pfotenhauer b, X.Q. Zhi a,?, L.M. Qiu a, J.F. Li a a b Institute of Refrigeration and Cryogenics, Zhejiang University, Hangzhou 310027, China Department of Mechanical Engineering, University of Wisconsin, Madison, WI 53706, USA a r t i c l e i n f o Article history: Received 11 February 2018 Received in revised form 15 June 2018 Accepted 11 July 2018 Keywords: Carbon dioxide capture Desublimation Transient model a b s t r a c t Carbon dioxide (CO2) cryogenic desublimation separation has become an emerging carbon capture method in recent years due to its advantages of a contamination-free process and compactness. So far, there have been few research works on revealing the detailed desublimation characteristics of CO2 associated with the flow as well as mass and energy conservation in the practical cryogenic CO2 capture process. In this study, a transient model for analyzing the CO2 cryogenic desublimating in mixture gas is proposed. The model contains a tube-in-tube counter-flow heat exchanger including three control volumes, the nitrogen (or helium) coolant, the wall with the solid CO2 layer and the mixture. The deposition distribution, capture rate and energy consumption of the dynamic desublimation process under different operation conditions are investigated. The model is verified by some experiment results. Results show that an improved modeling accuracy is obtained by taking the solid CO2 layer into consideration. During the dynamic desublimation process, the deposition rate is the highest near the inlet of gas mixture due to the high mass diffusion there, and a low energy consumption will be obtained at high concentration and low flow velocity of CO2 supply. The theoretical method here provides better understanding of the CO2 desublimation features in annular tube, which will be helpful for conducting an efficient CO2 capture process. г 2018 Elsevier Ltd. All rights reserved. 1. Introduction In recent years, global warming caused by carbon emissions has drawn significant attention [1,2]. It?s noted by the International Energy Agency that the share of CO2 emissions from industry and power plants in the entire carbon abatement industry will increase from 3% in 2020 to 19% in 2050 [3,4]. Therefore, the carbon capture and storage (CCS) technology has become one of the most promising and urgently needed solutions to tackle climate change [5]. Great efforts are being done worldwide on developing high efficiency and low energy consumption CCS technology [6]. Specific methods used to capture CO2 in CCS technology include the solvent absorption method, the membrane separation method, the adsorption method, and the low temperature separation method [7?9]. The low temperature separation method, containing both the liquefaction separation and the desublimation separation, has the advantages of being compact and non-corrosive. And it can be used with the combination of waste cold energy recycle in some special places. The phase diagram of CO2 reveals that the liquefaction sep? Corresponding author. E-mail address: [email protected] (X.Q. Zhi). https://doi.org/10.1016/j.ijheatmasstransfer.2018.07.068 0017-9310/г 2018 Elsevier Ltd. All rights reserved. aration method will require a high pressure (15 MPa or more). While for the desublimation separation, it can be realized directly from the gas state at the pressure below the triple point. For example, the desublimation pressure at 195 K is 0.1 MPa. Therefore, the superiorities of a low pressure system, such as safety and reduced energy consumption are provided by the cryogenic desublimation capture, since it can be realized at atmospheric pressure. Researches have been done on studying the desublimation characteristics of CO2. In the 1930s, W. Giauque and C. Egan gathered many measurements on solid CO2, including the heat capacity from 15 to 195 K, the heat of vaporization at the sublimation point under normal pressure, and the vapor pressure from 154 to 196 K, etc. [10]. In the 1990s, T. Cook and G. Davey measured the thermal conductivity and density of CO2 and nitrogen deposited under cryopumping conditions over a wide range of temperatures and deposition rates [11]. Their work laid a foundation for the study of CO2 desublimation capture. A growing number of researchers have focused on cryogenic CO2 capture technology in recent years. From 2002 to 2004, D. Clodic and M. Younes proposed a new method of creating CO2 frost at atmospheric pressure and made comparisons between different capture processes, focusing mainly on energy consumption referenced to the flue gas mass flow rate 340 Y.N. Wang et al. / International Journal of Heat and Mass Transfer 127 (2018) 339?347 Nomenclature As cp cvh D12 Dh f h hm ht k l m _ m mfs M1, M2 P R r q heat transfer area, m2 specific heat, J/(kgK) convection heat flux, W/m2 mass diffusivity, m2/s hydraulic diameter, m friction coefficient enthalpy, J/kg mass transfer coefficient, m/s heat transfer coefficient, W/(m2K) thermal conductivity, W/(mK) length of the tube, m mass, kg mass flow rate, kg/s mass flux deposited on the wall, kg/(m2s) relative molecular mass, g/mol pressure, Pa radius of the outer tube, m radius of the inner tube, m density, kg/m3 and on the sensitivity of the energy efficiency associated with the initial CO2 concentration [12,13]. In 2006, N. Zhang and N. Lior proposed a near-zero CO2 emission thermal cycle with LNG cryogenic exergy utilization [14]. In 2008, M. Tuinier and M. Annaland proposed a process for cryogenic CO2 capture using dynamically operated packed beds in a cylinder, achieving an effective separation between water vapor, CO2 and the permanent gases due to differences in dew and sublimation points. Their numerical studies showed the required energy to recover >99% CO2 from a flue gas containing 10 vol% CO2 and 1 vol% water vapor is estimated at 1.8 MJ/kg. And lower CO2 concentrations in the inlet and higher initial bed temperatures result in higher required energy per mass of CO2 captured [15]. In 2011, C. Song and Y. Kitumara proposed a cryogenic carbon capture system using Stirling cryocoolers, studying the influence of the operating conditions (vacuum condition, idle operating time, flow rate of gas stream and precooling temperature). The frost layer of CO2 is formed on the cooling fin of the heat exchanger of stirling cryocooler, the system can capture 80% CO2 from flue gas with 3.4 MJ/kg (with a vacuum pump, idle operating time of 4 h, flow rate of 2L/min and precooling temperature of 253 K) [16?18]. In 2014, L. Yuan designed a cryogenic capture process based on a Reverse-Brayton cycle, and explored the influence of the mixture pressure [19]. In 2015, D. Zhang built a desublimation transient model for the ITER cryogenic viscous compressor to separate hydrogen from helium using two concentric tubes, however, he did not consider the deposited solid layer [20]. Although more and more CO2 desublimation capture methods were proposed, the phase transition characteristics during cryogenic capture process remains poorly understood. Such as the proper working condition and required cooling power for desublimation, the dynamic desublimation rate distribution need to be further investigated. Compared to other capture methods, whether the desublimation method is energy-saving or not has not been analyzed and revealed yet. In this study, a transient model is built to solve the phase transition process of CO2 desublimation capture in annular tube. The effects of both considering the internal energy change of the deposited solid layer or not are compared. The required wall condition, energy consumption, frosting speed for the desublimation process and the amount of solid CO2 that can be collected over time are analyzed. t T v u l x th time, s temperature, K velocity, m/s specific internal energy, J/kg dynamic viscosity, kg/(ms) mole fraction thickness, m Subscripts mix mixture w wall s solid carbon dioxide c coolant B bulk v vapor r radial direction z axial direction 2. Experiment setup In order to accurately and quantitatively grasp the mechanism of CO2 desublimation in a flow stream, and obtain the basic data for designing engineered cryogenic carbon capture devices, an experimental set-up was designed and constructed [21]. The entire set-up primarily includes the gas supply and cooling system, vacuum system, data acquisition, and control system. The basic operations after evacuating the system are as follows: two parallel paths of nitrogen, each with a constant flow rate are cooled to 90 K by passing through the liquid nitrogen bath, and then through two heaters H1 and H2 respectively. One flow path enters the inner tube of the transparent channel as the coolant. Upon exiting the concentric tube heat exchanger, it flows through another heat exchanger where it is used to precool the incoming gas mixture stream. The other path of nitrogen is mixed with CO2 in the mixing chamber. The mixture is precooled in the precooling heat exchanger and subsequently enters the annular tube of the visual channel, where the deposition takes place. Since the outer wall of the annular tube is made of glass and the mixture flows through the outer tube instead of inner tube, the desublimation process along the whole tube can be observed using an endoscope, which is fixed on a sliding rail and driven by a vacuum stepper motor. The effective length of the visual channel is 0.7 m, the inner diameter and wall thickness of the stainless steel tube through which the nitrogen coolant flows is 30 mm and 1.5 mm respectively, while the inner diameter and wall thickness of the outer glass tube is 50 mm and 3 mm respectively. 3. Transient model Since many parameters and physical features during CO2 desublimation cannot be obtained by experiment, modelling method is carried out. The transient numerical model described below has been developed to characterize the dynamic interaction between the gas-mixture (CO2 and N2) and coolant flow streams (cold N2 gas), the wall separating them, and the accumulation of solid CO2 on the outside of that wall. The model simulates how the desublimation process varies over time, as well as the precooling process and the rewarming process. In order to simplify the intricate CO2 Y.N. Wang et al. / International Journal of Heat and Mass Transfer 127 (2018) 339?347 341 frosting process, some assumptions are mentioned as follows: (1) there is no radial change of the flux inside the control volumes of the mixture and the coolant, i.e. it is one dimension model; (2) the CO2 molecules near the wall are under saturated state, the temperature of which equals the temperature of the wall; (3) the thermal resistance of the solid layer and the wall is neglected; (4) because the assumption in (1), the thermal diffusion in the control volumes of the mixture and coolant is neglected. 3.1. Physical model simplification and governing equations The modelling region is the visual channel part (cryogenic desublimation tubes) in Fig. 1, in which the desublimation process happens during cooling. It includes the control volume of the mixture, the control volume of the solid CO2 and the wall, and the control volume of the nitrogen coolant. Flow in the annular tube includes both the axial flow of the gas mixture and a radial flow towards to the wall of a portion of the CO2. The whole tube is divided into n nodes, the gas mixture flows from node 1 to node n through the annular tube, while the coolant nitrogen flows reversely from node n to node 1 through the inner tube. The input boundary conditions include the temperature, pressure, velocity and CO2 concentration of the mixture at node 1, the temperature, pressure, and velocity of the coolant at node n, the initial temperature of the wall and the coolant from node 1 to node n (see Fig. 2). Fig. 3. The flow in radial direction and the cross section diagram. Fig. 3 shows the cross section of the cryogenic tubes. For the carbon dioxide-nitrogen mixture in the annular tube, flow along the axial z direction and flow towards the wall of the inner tube in radial r direction exist simultaneously; while for the nitrogen coolant in the inner tube, there is only flow in the axial z direction. The analysis on the control volume of the carbon dioxidenitrogen gas mixture is shown in Fig. 4. Each control volume has a differential length of dz, an inner radius of r and an outer radius Fig. 1. Diagram of the experimental set-up. Fig. 2. Schematic of model nodes and boundary conditions. 342 Y.N. Wang et al. / International Journal of Heat and Mass Transfer 127 (2018) 339?347 Fig. 4. Schematic of the flow in the control volumes. of R. Cz and Cr are the general flux terms and representing the flux of mass, momentum or energy. Cz represents the flux in the axial direction of each control volume and we assume, in the interest of simplicity, that: @C z М0 @r №1о Eq. (1) means there is no radial change of the flux inside each control volume. Actually, it is a one-dimensional model for each control volume. However, there is some mass or energy transfer among the control volumes of the mixture, solid/wall and coolant. For example, when the desublimation pressure at the wall temperature is lower than the partial pressure of CO2 in the bulk, there will be a mass flux in the radial direction between the control volume of mixture and that of the CO2 solid driven by the density difference, as well as momentum and energy flux. So Cr in Eq. (2) represents these radial fluxes between each control volume. The conservation equation for the mixture flux is as follow, the axial rate of change for the flux (notice this is a negative term since flux deposits on the wall continuously) multiplied by the cross section area plus the radial flux multiplied by the perimeter of the inner tube make zero. p№R2 r2 о dC z ў 2prC r М 0 dz №2о Again, C can represent mass, momentum and energy respectively, as for the mass flux, library routines [23]. The radial mass flux is defined by the mass transfer coefficient and the density difference of CO2 between the bulk and the saturated condition at the wall. It is assumed that the temperature of CO2 molecules near the wall equals the temperature of the wall. Nu М Nusselt T fd ў DNurat DNusselt T In which Nusselt T fd М 0:5803 r 2 r ў 6:095 4:456 ў 2:648 R R l 2R Re Pr №8о 1:050 №9о DNurat М 0:6847 ў 0:3153 exp№1:26544559№ln№Prо ln№0:72ооо №Pr > 0:72о №10о For the momentum flux, the pressure loss caused by friction must be considered. The friction factor f is calculated by Eqs. (14)?(17), which are from the model function annularflow_n_local of EES library routines [23]. Uzz М qCO2 v 2CO2 ў P №11о Urz М DPfriction ў Mr;CO2 v CO2 №12о №3о DPfriction М qCO2 f M r;CO2 М hm №qCO2 qCO2 ;w о №4о f М 4fR=Re v 2CO dz 2 2 Dh №13о №14о In which hm Dh D12 №5о 7 D12 М r R DNusselt T М 1:754 exp 0:4028 ln M z;CO2 М qCO2 v CO2 Sh М №7о 1:86 10 T 2 3 №M11 Pr212 X ў №15о plus 1 1 2 о M2 3:44 0:3125 f fd Re 3:44 fR М p????????? ў ў p????????? №2о xplus 4 xplus xplus №1 ў 0:00021x о №6о hm is the mass transfer coefficient, which corresponds to the heat transfer coefficient in a heat transfer equation, Dh is hydraulic diameter and D12 is the mass diffusivity computed from the ChapmanEnskog relation [22]. The Sherwood number (Sh), also called the mass transfer Nusselt number (Nu), is an analogous dimensionless number used in the mass-transfer relation. It represents the ratio of the convective mass transfer to the rate of diffusive mass transport. Since for a given geometry, a heat transfer correlation for Nu in terms of the Reynolds number (Re) and the Prandtl number (Pr) can be used as a mass transfer correlation for Sh by replacing Pr with the analogous dimensionless number Sc (the Schmidt number) for mass transfer. Both Sh and Nu are calculated by Eqs. (7)?(10), which are from the model function annularflow_n_local of EES xplus М f fd М l 2R Re №16о r 2 r 2 64 r 2 R 1ў 2 ln 1 2 1 Re R R R r №17о For the energy flux, 1 _ CO2 hCO2 ў m _ N 2 hN 2 ў m _ mix v 2mix E_ z;mix М m 2 1 E_ r;mix М mfs hCO2 ў v 2CO2 ў cv h 2 №18о №19о 343 Y.N. Wang et al. / International Journal of Heat and Mass Transfer 127 (2018) 339?347 The terms for the solid CO2 and the wall are lumped together in one control volume because of the unknown thermal resistance between the solid and the wall. In a transient model the temperature of the solid and wall, therefore the internal energy of the solid and wall, changes over time. The internal energy change reflects the change of stored energy in the control volume, and is related to the flow of energy into and out of the control volume by the equation: Energystored М Energyin Energyout R1 М 1 ht;mix p№Din ў 2thw ў 2ths оdy №28о R2 М 1 Din ў 2thw ў 2ths ln 2pks dy Din ў 2th №29о R3 М 1 Din ў 2thw ln 2pkw dy Din №30о R4 М 1 ht;c pDin dy №31о №20о The former term may also be called the adsorption energy, and is the sum of the enthalpy difference between the CO2 at the mean flow temperature and that at the phase equilibrium temperature of the wall (Dhbw), the phase change energy (Dhvs) that also depends on the phase equilibrium temperature of the wall [20]; and the convection heat flux (cvh). The kinetic energy is neglected since it is much smaller than the other terms. Q_ b!w М Нmfs №Dhbw ў Dhv s о ў cv h As №21о dU w dU s ў М Q_ b!w Q_ w!c dt dt №22о dU w duw М mw dt dt №23о dU s dT s dms М cps ms ў cps T s dt dt dt №24о Q_ w!c М ht;c №T w T c о As №25о The energy equation of the coolant nitrogen is given as follows, dU c М DE_ c Q_ w!c dt _c DE_ c М m direction. It should be noted that for the cases in which the solid CO2 layer is totally ignored, the internal energy term dUs/dt of the solid layer in Eq. (22) and Eq. (24) will be zero. While for the cases the solid layer is not ignored, the internal energy change will be considered during the solution. №26о 1 1 hc;in ў v 2c;in hc;out ў v 2c;out 2 2 №27о There are four terms contributing to the thermal resistance between the mixture and the coolant that affect the cv h and Q_ w!c , the thermal resistance R1 caused by convection heat transfer between the mixture and the wall; the thermal resistance R2 of the solid CO2 layer; the thermal resistance R3 of the stainless steel wall; the thermal resistance R4 caused by convection heat transfer between the wall and the coolant. The thermal conductivity of the solid CO2 increases with the CO2 layer thickness. According to the study by Song [18], the thermal conductivity is 0.35 W/(mK) when the thickness of the solid layer is 1 mm. Here a relatively small value (0.3 W/(mK)) is used in the early thermal resistances comparison. Moreover, 0.3 W/(mK) is a reasonable value according to Cook T?s measurements [11]. Table 1 shows the value comparison of the four thermal resistances. Among them, R3 is much smaller than R1 and R4 and it can be ignored directly when the wall thickness is small enough. When the thickness of solid layer is less than 10 mm, R2 is less than 10% of the total thermal resistance and can also be ignored. So the thermal resistances of the solid layer and the wall are not considered in the model, and there is no temperature difference inside the solid layer and the wall in the radial 3.2. Calculation for thermal properties under low temperature The desublimation process is influenced by not only the properties of the mixture but also the conditions of the wall. This transient model deals with the viscous regime rather than the molecular regime. In the viscous regime, a large amount of gas molecules will occupy all available sites on the wall quickly. The bulk flow?s interaction with the wall is important and determines the physical properties near the wall. Desublimation occurs when the CO2 saturation pressure associated with the wall temperature is lower than the pressure of the CO2 in the bulk. The energy absorbed by the wall is the sum of the enthalpy difference between the bulk temperature and the wall temperature for the deposited mass, and the desublimation heat, which depends on the wall temperature. The desublimation pressure is one of the most significant properties since it determines whether desublimation occurs and exactly how much mass is deposited. There are many available data sets and empirical formulas for the desublimation pressure due to the widespread use of solid CO2. The correlation equation proposed by Span and Wagner is used in this transient model, since that the uncertainty in this equation is relative small within its scope of application [19]. p ln sub pt Tt М T ( 1:9 2:9 ) T T T ў a2 1 ў a3 1 a1 1 Tt Tt Tt №32о where Tt = 216.592 K, pt = 0.51795 MPa are the temperature and the pressure of the triple point, a1 = 14.740846, a2 = 2.4327015, and a3 = 5.3061778 [24]. According to this equation, when T = 194.6855 K, desublimation can occur at normal pressure, and the saturation pressure falls sharply as the temperature decreases. Either EES or Refprop may provide thermal properties values. However, below the triple point, a fitting formula using data provided by the NIST webbook of the specific heat for CO2 is as follows [25], cp М 107 T 4 0:0001T 3 ў 0:0501T 2 7:4036T ў 1042:8 Table 1 Thermal resistances comparison. №33о where the unit of cp is J/kgK, and the enthalpy difference by definition is Solid layer thickness (mm) R1 (k/W) R2 (k/W) R3 (k/W) R4 (k/W) R2/Rtotal 1.5 10 176.5 118.6 3.343 18.4 0.09945 0.09945 44.9 44.9 0.015 0.101 dh М cp dT №34о The internal energy change of the solid CO2 layer is involved in the transient model, so the specific heat of solid CO2 is needed. 344 Y.N. Wang et al. / International Journal of Heat and Mass Transfer 127 (2018) 339?347 According to the measured data across the temperature range of 15.52?189.78 K provided by Giauque and Egan [10], the correlation equation of the heat capacity of solid CO2 is cps М 106 T 4 ў 0:001T 3 0:2371T 2 ў 28:135T 354:17 №35о According to the measured data across the temperature range of 90?191.5 K provided by O. Mass and W. H. Barnes [26], the correlation equation of the density of the solid CO2 is qs М 0:004T 2 ў 0:1T ў 1679:8 №36о The above density is measured by free/natural frosting of CO2, so it reflects the real density of the porous CO2 solid. The mole weighted average values of enthalpy and specific heat are used as the value of the mixture, however in accordance with the behavior of transport phenomenon, the equations of viscosity and conductivity for the mixture are as follows [27], l l l12 М 1 l21 М 2 l2 l1 M 12 М x12 М lМ №37о M1 M2 M21 М M2 M1 №38о x1 x2 x21 М x2 x1 №39о l1 1 ў x21 №1ўl0:5 M 0:25 21 о 12 2 l2 ў ? p4 №1ўM 12 о0:5 2 1 ў x12 №1ўl0:5 M 0:25 11 о 21 2 №40о ? p4 №1ўM 21 о0:5 2 h i 0:5 2 A12 М 0:25 1 ў №l12 M0:75 21 о №41о h i 0:5 2 A21 М 0:25 1 ў №l21 M0:75 12 о №42о kМ Fig. 5. Temperature profiles of the coolant at various times. k1 k2 ў 1 ў A12 x21 1 ў A21 x12 №43о This transient numerical model is built by EES associated with some thermal properties from Refprop and NIST. It solves not only the changes of the thermal properties and composition of the carbon dioxide-nitrogen gas mixture as it flows through the cryogenic tube, but also how the temperature profiles of the wall and the coolant change over time. The highly coupled non-linear equations can be solved by EES, differential equations are solved by using finite-difference method, the time step is set to be 1 s, the node number is 50, the performance independency of the node number and time node on the above numbers has been examined. When the maximum residual is less than 1e6, the program is considered as convergent. 4. Results and discussions Calculation and experiment data in Figs. 5 and 6 are obtained under following experimental conditions: the inlet volume flow rates of the gas mixture and the coolant are 300 ml/s and 600 ml/s respectively, the inlet temperature of the mixture is 179 K, the mole fraction of CO2 in the mixture is 0.2, the initial and inlet temperatures of the coolant are 96 K, and the initial temperature of the precooled wall is 125 K. Terms identified in the legend as ??solid? mean that the internal energy change of the solid CO2 layer is considered, while those identified as ??no solid? mean the stored energy in the solid is ignored. Figs. 5 and 6 show the temperature profiles of the coolant and mixture at three different moments respectively. It can be seen that the simulated temperature distributions are close to the Fig. 6. Temperature profiles of the mixture at various times. experimental ones with a largest difference less than 2 K, which verifies the validation of the model. As shown in Fig. 5, taking the internal energy change of the solid CO2 layer into consideration causes a lower temperature profile of the coolant. This is because part of the energy from the mixture is stored in the solid CO2 instead of the coolant nitrogen. While for the cases which ignore the solid CO2 layer, the temperature of the coolant nitrogen will be higher. In Fig. 6, the temperatures of the mixture are higher when the internal energy of the solid is ignored, especially at a long time as 50 s. This is because in those cases, the coolant temperature is higher, as shown in Fig. 5. Also because if the solid layer is ignored, the mixture will exchange less heat with the wall as the total heat capacity (compared to the sum heat capacity of wall and solid layer) is decreased. Since the tube is precooled and has initial low temperature at beginning, the temperatures of both coolant and mixture become higher slowly as time goes on, as shown in Figs. 5 and 6. Figs. 5 and 6 convey the same information: taking the experimental results as reference, the model ignoring the internal energy change of the solid layer causes larger deviations for the temperature of the mixture and coolant than the model considering the internal energy change of the solid layer. The deviation becomes larger after a few seconds in the transient CO2 desublimation process. Figs. 7 and 8 show the temperature profiles of the wall and the temperature difference between the mixture (Tmix) and the wall (Tw), respectively. The internal energy of solid layer is considered. Y.N. Wang et al. / International Journal of Heat and Mass Transfer 127 (2018) 339?347 Fig. 7. Temperature profiles of the wall. 345 Fig. 9. Deposition rate distributions, with the initial temperature of the wall at 140 K and 100 K respectively. Fig. 8. Temperature difference between the wall and the mixture. Fig. 10. Solid layer thickness profiles over time. In Fig. 7 the wall temperature is the highest at the entrance of mixture, and it decreases along the whole tube from y/L = 0 to y/L = 1. The decrease of the wall temperature near the mixture inlet location is the fastest, in line with the fastest increase of temperature difference between the wall and the mixture at the same location, as shown in Fig. 8. The temperature difference is the largest near the entrance of coolant (y/L = 1), possibly causing a large heat exchange between the wall and mixture. Over the whole tube, both the increase of Tw and the decrease of the temperature difference are smaller and smaller as time goes on. Finally after 181 s, the wall temperature and temperature difference distribution become almost constant, showing a steady state of desublimation process of CO2. Fig. 9 shows the time dependent profile of mass deposition rate at the wall. The mass deposition rate is defined as the desublimation mass of CO2 per second and per square meter. In Fig. 9, when the initial temperature of the wall changes to 100 K from 140 K, the deposition rate does not increase significantly. This means during a real cryogenic CO2 capture operation, there is no need to precool the wall to a very low temperature since the saturated pressure at 140 K is about 200 Pa which is much lower than the real partial pressure of 30,000 Pa. Fig. 9 also shows that the deposition rate is the highest at the entrance of mixture, it decreases along the tube from y/L = 0 to y/L = 1, especially near y/L = 0 it decreases much fast. Also the deposition rate keeps almost constant within 200 s. Fig. 10 shows how the thickness of the solid layer at different locations increases over time, the slope of the lines is analogous to the deposition rate. One can observe that the solid thickness decreases along the tube from y/L = 0 to y/L = 1, and it decreases much fast near the entrance between y/L = 0 and y/L = 0.2. So as the slope of the lines. Also the slopes of different lines, i.e. representing the deposition rates at different locations along the tube, are constant within 200 s. These phenomena in Fig. 10 are all consistent with those in Fig. 9. Fig. 8 shows the temperature difference between wall and mixture becomes larger from y/L = 0 to y/L = 1, but in Figs. 9 and 10 it shows the desublimation rate becomes smaller from y/L = 0 to y/L = 1. It seems the temperature difference and the heat transfer amount is not the main factor to determine the deposition rate. And this may be explained by the effects of the mass diffusivity on the mass transfer of phase change. The phase change mass flux in the radial direction is defined in Eq. (4) by the mass diffusivity of CO2 in the mixture and the difference between qCO2 and qw, thus there are two reasons for the features displayed in Figs. 9 and 10. Firstly, the mass diffusivity is the largest at the entrance due to highest mixture temperature at that location. Secondly, the mass diffusion is driven by difference between the density of CO2 in the mixture and the density at the 346 Y.N. Wang et al. / International Journal of Heat and Mass Transfer 127 (2018) 339?347 Fig. 11. Experimental photo for the deposition states at different locations (t = 50 s). saturation condition at the wall. Although the wall temperature is lower downstream towards y/L = 1 and cause a lower saturation pressure and lower density, comparing with the density of the CO2 in the mixture, the decrease is too small (for example, decreases from 0.1 kg/m3 at 160 K to 0.003 kg/m3 at 130 K) to make a significant difference in the driving force for mass diffusion. In addition, the density of CO2 in the mixture becomes smaller due to desublimation, leading to a less concentration and density difference downstream. Therefore, the heat transfer associated with temperature difference is not the main driven force to determine the deposition rate, but the efficient mass transfer caused by high mass diffusivity and high concentration of CO2 at high temperature region near the mixture entrance. Besides, as shown in Fig. 6, the mixture temperature decreases very slowly with time. Therefore, the mass diffusivity has little change as time goes on, leading to an almost constant deposition rate. Fig. 11 is a photo of frosting states at two different locations of the annular tube taken by experiment. It also proves that the left part (towards y/L = 0) has faster deposition rate than the right part of the tube (towards y/ L = 1). The fact that the deposition rate only keeps very high at a short part near the mixture entrance also means that it is no need to use a very long desublimation tube in the cryogenic carbon capture process in practice. Fig. 12 shows the influence of mixture velocity on the capture rate and energy consumption. The capture rate is defined as the ratio of total change of mixture mass flow rate (caused by desublimation) to the inlet mixture mass flow rate in the annular tube: Capture rate М №v mix;in qCO2 ;in v mix;out qCO2 ;out о v mix;in qCO ;in №44о 2 Change of mass flow rate М v mix;in qCO2 ;in v mix;out qCO2 ;out Energy consumption М №45о Fig. 12. Influence of mixture velocity on capture rate and energy consumption (t = 2 s). Fig. 13. Influence of CO2 mole concentration on capture rate and energy consumption (t = 2 s). tion rate will be low. Fig. 12 shows that using the capture tube with large diameter or area to decrease the mixture velocity will improve the capture efficiency. Also as stated above, only small part of the tube near the mixture entrance has high deposition rate. Therefore, it can be concluded that in practice, it is better to use short tube with large flow area instead of slender tube for CO2 desublimation capture, which will also have a smaller flow pressure drop. mN2 №hN2 ;273K hN2 ;out о ў №mCO2 ;in mCO2 ;out о№hCO2 ;273K hN2 ;out о ў DmCO2 №Dh273K;w ў hsub о DmCO2 the energy consumption is the heat released by the mixture during the precooling and capture process for capturing per unit mass, i.e. the cold energy costs. As shown in Fig. 12, lower mixture velocity leads to higher capture rate, even close to 1 at velocity below 300 ml/s. And the energy consumption for capturing per unit mass CO2 is lower at lower mixture velocity. This is because when the axial flow velocity is high both the heat transfer and mass diffusion toward the radial direction will be weakened, therefore, the deposi- №46о Fig. 13 shows the influence of CO2 mole concentration in the mixture on the capture rate and energy consumption. The energy consumption is compared with the calculation results of Lingcheng Yuan, who used the same definition [19]. For fixed flow rate, higher inlet concentration leads to less average required energy to capture per unit mass CO2. The liquefaction method is the highest energy consumption method, especially at low concentration. The cryogenic carbon capture method by desublimation has similar Y.N. Wang et al. / International Journal of Heat and Mass Transfer 127 (2018) 339?347 347 with time revealed by the model will be helpful for the conducting of defrost cycle in a real CO2 desublimation capture process. Conflict of interest The authors declared that there is no conflict of interest. Acknowledgements This work was supported by the Nation Key R&D Program of China (2017YFB0603701), and the National Natural Science Foundation of China (Key program, No. 51636007). We are grateful to Lingcheng Yuan, Xiaobo Jiang, and Jianxiong Wang for help building the experimental set-up. References Fig. 14. Change of mass flow rate and energy consumption variation with time. energy consumption with the methods of single compression combining single expansion or double expansion process. The changes of the capture rate due to the concentration increase are very small. In summary, the high concentration and low flow velocity can reduce the energy consumption while ensure a high capture rate. Fig. 14 shows the time dependence of capture rate and energy consumption. The capture rate changes very slowly over time, it almost keeps constant. The energy consumption for capturing per unit mass CO2 drops first, and then rises slowly over time. This may be because at beginning some of the cold energy from the coolant is used to cool the wall and mixture to get smaller and smaller temperature difference between them as shown in Fig. 8, while the cold energy used to desublimation is relatively small. Later, as the wall temperature becomes higher, the heat transfer amount (cold energy) between the wall and the coolant is larger. While at the same time, the capture rate is slowly decreasing, therefore, the energy consumption turn to increase. It shows the defrost operation should be started at least after 70 s when the energy consumption is low, and the defrost time cannot be too late as the energy consumption is slowly increasing. The best defrost moment should be chosen by considering both the capture rate and the energy consumption. 5. Conclusion A transient model is built to study the dynamic desublimation process of CO2 in the annular tubes of cryogenic carbon capture system. Influence of initial wall temperature, mixture flow velocity and concentration of supplied CO2 on the role of dynamic solid deposition process are analyzed. The model has been verified by experiment results. Results show that the model considering the internal energy change of the solid layer has more accurate results, revealing that the solid layer produced by desublimation cannot be ignored during simulation. The initial wall temperature has a little effect on the latter desublimation rate, which means the wall is no need to be precooled to very low temperature before capture started. The largest desublimation rate of CO2 happens near the inlet of supplied gas mixture instead of the inlet of coolant due to the high mass diffusion of CO2 there. 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