Prediction of Pressure Drop in Vertical Pneumatic Conveyors

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1 Journal of Alied Fluid Mechanics, Vol. 10, No., , 017. Available online at ISSN , EISSN DOI: /acadub.jafm Prediction of Pressure Dro in Vertical Pneumatic Conveyors S. M. El-Behery, A. A. El-Haroun and M. R. Abuheazy Mechanical Power Enineerin Deartment, Faculty of Enineerin, Menoufyia University, Shebin El-Kom Eyt Corresondin Author (Received July, 015; acceted October 16, 016 ABSTRACT This aer resents a steady state one-dimensional two-fluid model for as-solid two-hase flow in a vertical riser. The model is solved usin conservative variable aroach for the as hase, and fourth order Rune- Kutta method is used for the solid hase. The model redictions for ressure dro are comared with available exerimental data and with Eulerian-Laranian redictions, and a ood areement is obtained. The results indicate that the ressure dro increases as the solid mass flow rate, article size, and articles density increase. In addition, the model redictions for minimum ressure dro velocity are comared with exerimental data from literature and the mean ercentae error. MPE for minimum ressure dro velocity is -9.89%. It is found that the minimum ressure dro velocity increases as the solid mass flow, article size and article density increase, and decreases as the system total ressure increases. Keywords: Gas-solid; Pneumatic conveyin; Two-fluid model; Pressure dro. NOMENCLATURE A C Cd D d f H h k m Mr P ie cross-sectional area secific heat dra coefficient ie diameter article diameter friction coefficient ravity acceleration enthaly heat transfer coefficient thermal conductivity mass of a sinle article ratio of mass flow rate of articles to the mass flow rate of as total as ressure R as constant T temerature u velocity x distance alon the ie α void fraction λ ratio of mass flow rate of articles to the total mass flow rate ρ density µ viscosity Subscrits article as hase 1. INTRODUCTION Gas-solid flows systems are imortant in many industrial alications such as chemical rocesses, neumatic conveyin, dryin, rains and metal owders. To imrove these systems, it is imortant to understand the as-solid flow reimes. This understandin is considered a reat challene due to the comlexity of this flow. Many arameters includin article size, density and shae are still mysterious althouh many researches had been done in this field. The ressure dro in the system is a vital desin issue. Zenz (1949 was the first one to suest flow reime diaram (also called the state diaram for neumatic conveyin systems. This diaram resents the ressure dro er unit lenth of the conveyin ie as a function of the suerficial fluid velocity at a constant solid mass flow rate. The state diaram reflects a oint of minimum ressure dro. Many other investiators had confirmed the results of Zenz (1949 (see for examle Narimatsu et al. 007; Konno and Saito 1969; Sinh 198; Rautiainen et al. 1999; El- Behery et al. 01, 013 and many others. All of these investiations showed that the ressure dro

2 S. M. El-Behery et al. / JAFM, Vol. 10, No., , 017. increases as the solid mass flow rate increases. In addition, Narimatsu et al. (007 also investiated the heat transfer coefficient usin lass shere and alumina articles transorted in a vertical conveyor. They found that the maximum heat transfer coefficient occurs at the same velocity of minimum ressure dro. Therefore, it can be concluded that oeratin the vertical neumatic conveyin at velocities close to the minimum ressure dro velocity reduces the consumed ower and enhances the heat transfer rate. The effect of article size on the ressure dro was investiated by El-Behery et al. (01, Xiao-in et al. (009, Nieuwland et al. (1979, Hariu and Molstad (1949 and Plasynski et al. (1994. The common conclusion of these studies was the increase of ressure dro with the increase in article size. On the other hand, Chun et al. (001 found that the ressure dro decreases as the article size increases. Narimatsu and Ferreira (001 found that the article size is more sinificant at low conveyin velocities. Henthorn et al. (005 investiated the effect of Reynolds number, mass loadin, and article shae and size on ressure dro in a vertical assolids neumatic conveyin line. They found that the ressure dro increases as the article diameter increases at Reynolds number of 15,000. While at Reynolds number of 0,400 the ressure dro first decreases as article size increases, and then increases with increasin article size. Generally, as-solid flow can be modeled usin either Eulerian-Eulerian or Eulerian-Laranian aroaches. A detailed descrition of these aroaches can be found in the comrehensive reviews by Enwald et al. (1996, Gouesbet and Berlemont (1998 and Balachandar and Eaton (010. Realized and sohisticated Eulerian- Eulerian and Eulerian-Laranian models are used when detailed local flow arameters and variables are required at a microscoic level (see Tamayol et al. 008; Mehdizadeh et al. 008; El- Behery et al. 01, 013; Jamaleddine and Ray 011; Patro et al. 014; Liu et al However, many industrial alications such as on-line control and automation require a robust model caable of redictin macroscoic variables (Narimatsu et al The onedimensional model based on the two-fluid theory is one such model. Desite the hue simlified assumtions emloyed in the one-dimensional two-fluid models, they are caable of redictin the flow variable at macroscoic level with fair accuracy (see Narimatsu et al. 007; Nieuwland et al. 1997; Arastooour and Gidasow 1979; Solsvik et al. 01, 014. The resent aer investiates the effect of different oeratin arameters on the as-solid ressure dro in vertical ies. The analysis is carried out for as-solid flow in a vertical ie usin a one-dimensional steady-state model based on two-fluid theory. The model is caable of modelin comressible as-solid with heat transfer. However, the results resented in the current aer concentrate on the ressure dro in isothermal as-solid flow.. MATHEMATICAL MODELING To formulate the suested model, a quasi-onedimensional situation has been considered. This model is concerned with two-hase flow of as and articles throuh a vertical ie under the followin assumtions: 1. The flow is one-dimensional and steady.. The articles are sherical in shae. 3. All articles interaction is inored in the model. This imlies that any momentum transfer between articles is neliible comared to the momentum transfer between the articles and the as stream. 4. The model assumes that the solid will be conveyed as discrete articles..1 Governin Equations Based on the recedin assumtions the overnin equations for the as and disersed hases are derived accordin to the basic laws of fluid mechanics as follows: The mass balance equation for the as hase may be written as: d ( u A S (1 mass Whereas the momentum equation for the as hase can be exressed as: d d ( u A A A F w S mom ( The total enery equation for the as hase can be formulated as: d [ u A( H 0.5u ] Qwall Senery (3 where, Smass, Smom and Senery are mass, momentum and enery coulin source/sink terms, resectively. In the current study the mass transfer between faces as well as the heat transfer throuh the ie wall are nelected (i.e. Qwall = Smass = 0.0 The equation of motion for a article in a as is iven by: du ( u u (1 / 4 f u u / D (4 where, β reresents the interhase momentum transfer coefficient due to dra. Accordin to Gidasow (1994 the factor β is calculated by: 50

3 S. M. El-Behery et al. / JAFM, Vol. 10, No., , u u d d Cd u u 4 d The dra coefficient, Cd is calculated by: C d ( Re Re 0.44 Re Re where, Re d u u / is the article Reynolds number. The equation for article temerature, assumin uniform temerature throuhout the article, can be written as: u m C dt d h( T T (5. Coulin Between Phases An imortant concet in the analysis of two-hase as-solid flow is to consider the mutual effect between the two hases. The number of articles er unit volume, N, can be exressed as: N 3 6 /( d (6 d The momentum coulin source term due to the reverse effect of articles can be written as: S mom N Am u u /( (7 ( The enery coulin source term for the total enery equation evolves convective heat transfer and the work due to article dra (first and second terms on the riht-hand side, resectively (Mofreh 005. S enery N Ah d ( T T S u (8.3 Friction Force mom The friction force er unit lenth between the ie wall and the as hase was estimated by, Fw 0.5Df ( u (9 The total friction factor, f is the sum of as and articles friction factors which is exressed as iven by Han et al. (000 as: f f f 1 (10 The as friction factor, f can be calculated from the well-known Blasius formula while, the friction factor between articles and the wall of the ie can be calculated as iven by Han et al. (000 as: f Fr (11 where, Fr u / d is the article Froude number..4 Heat Transfer The convective heat transfer coefficient h is calculated from Nusselt number, Nu hd / k which is exressed as a function of Reynolds number Re. Various emirical correlations can be used to calculate the heat transfer coefficient. Baeyens et al. (1995 correlation was used in the resent study. Nu 0.15Re (11.5 Sulementary Equations In order to solve the above set of equations, the followin sulementary equations are required. The volume fraction equation: 1 (1 Density of as stream: /( RT (13 3. SOLUTION PROCEDURE The system of Eqs. (1 to (5 is solved numerically usin the conservative variable formulation for the as hase with the hel of auxiliary and sulementary equations (Crowe et al The fourth order Rune-Kutta method is used for the disersed hase. The conservative variable formulation is a cell by cell iterative rocedure in which the as hase variables are secified at the cell inlet and are souht at the cell exit. The averae values of the as hase variables are then used to calculate the solid hase velocity and temerature. The source terms and void fraction are then evaluated and new flow variables at the cell exit can be calculated. The rocedure is continued until the as velocity no loner chanes with continued iteration. Once the solution is obtained for one cell, the exit conditions are taken as the start condition for the adjacent cell, and the rocedure is reeated. 4. MODEL VALIDATION The models redictions are verified usin exerimental data reorted by Hariu and Molstad (1949 and Henthorn et al. (005. The exerimental data of Hariu and Molstad (1949 includes the ressure dro in the acceleration reion while the data of Henthorn et al. (005 were for fully develoed flow. The effect of solid mass flow rate on the total ressure dro at different articles diameters and as velocities is resented in Fi. 1. The fiure indicates that the ressure dro increases linearly with the solid mass flow rate in both the redicted results and measurements. It can also be seen that the articles size has a small effect at these flow conditions. The fiure also shows ood areement between redicted and measured ressure dro. In the second test case, the effect of article 51

4 S. M. El-Behery et al. / JAFM, Vol. 10, No., , 017. diameter on the ratio between two-hase and sinlehase ressure dros is shown in Fi.. The Eulerian-Laranian redictions of El-Behery et al. (011 are also included for comarison. Desite the simlifyin assumtions of the one-dimensional model, it redicts the ratio between two-hase and sinle-hase ressure dros fairly ood as comared with the realized Eulerian-Laranian model. In addition, the model redicts the linear variation of ressure dro with mass loadin ratio very well. In eneral, the model accuracy is quite accetable and can be used for ressure dro redictions in vertical neumatic conveyin. this rane of conveyin velocities, the solid concentration increases and the contribution of weiht and as-to-articles friction forces in the total ressure dro is dominated. The dilute neumatic conveyin is assumed to be at velocities hiher than Um. In this rane of conveyin velocities, the solid concentration is low, and the contribution of mixture friction force with ie wall is dominated. The standard Zenz diaram is resented in Fi. 3. It can be seen from this fiure that the ressure radient increases as the solid mass flow rate increases. This can be attributed to the increase of solid holdu and weiht force as the solid mass flow rate increases. In addition, the minimum ressure dro velocity increases with solid mass flow rate which is in areement with the findin of Zenz (1949 and Rautiainen et al. (1999. a Effect of article size a Re = 0400 b Effect of as velocity Fi. 1. Comarisons between redicted ressure dro and exerimental data of Hariu and Molstad ( RESULTS AND DISCUSSION A arametric study on the effects of as velocity, solid mass flow rate, article size and density on the ressure radient in the fully develoed flow was carried out, as shown in Fis In eneral, the results indicate that the ressure radient decreases with increasin as velocity u to certain value then the ressure dro increases. The oint of minimum ressure radient aears because the contributions of weiht force, as-to-articles friction force and mixture friction force with ie wall chane with conveyin velocity. The dense hase neumatic conveyin is considered to occur at velocities less than the minimum ressure dro velocity, Um. In b Re = Fi.. Comarisons between redicted ressure dro and measured data of Henthorn et al. (005 and Eulerian-Laranian simulations of El-Behery et al. (011. The effect of article size on the ressure radient in vertical neumatic conveyin is shown in fiure 4. The article size has two contra effects on the article s dra force. Increasin the article size results in a hiher article mass which increases the sli velocity between the two hases. Therefore, the dra force increases to overcome weiht force. On the other hand, as the article size decreases the number of articles increases and the total surface area increases as a result. Therefore, the total dra 5

5 S. M. El-Behery et al. / JAFM, Vol. 10, No., , 017. force increases as the article size decreases. It can be seen from the results resented in Fi. 4 that the ressure radient increases as the articles size increases. Since the contribution of article dra and weiht force is more sinificant in dense hase, the ressure radient increased sinificantly in the dense hase. Similar observation was reorted by Mastellone and Arena (1999 and Narimatsu and Ferreira (001. The fiure also shows that the minimum ressure dro velocity increases as the article size increases. This can be attributed to the increase in articles terminal velocity as the article size increase (Mastellone and Arena dense hase neumatic conveyin due to the hiher solid concentration. Fi. 5. Effect of article density on the ressure o s radient ( m = 0.5 k/s, d = 500 µm. Fi. 3. Effect of solid mass flow rate on the ressure radient (d = 500 µm, ρ = 500 k/m 3. Fi. 4. Effect of article diameter on the ressure o s radient ( m = 0.5 k/s, ρ = 500 k/m 3. Fiure 5 deicts the effect of article density on the ressure dro. The articles density has similar effects to the articles size. Thus, the article mass increase as the articles density increases. On the other hand, the number of articles and the total surface area increases as the article density decreases. This fiure indicates that the ressure radient increases as the article density increases and bein more sinificant at low conveyin velocities (dense hase. The minimum ressure dro velocity increases as the article density increases due to the increase in terminal velocity. In eneral, it can be concluded that the article roerties (size and density are more sinificant in Accordin to Plasynski et al. (1994, the velocity corresondin to the minimum ressure dro, Um for the system is the most souht-after information when desinin a system. Many correlations have been reorted in the literature for the minimum ressure dro velocity. However, none of these correlations can be alied to all systems. For instance, when Narimatsu and Ferreira (001 alied the correlation develoed by Rizk (1986 to their exerimental results they found hih discreancies between measured and estimated values. They develoed a new correlation for the minimum dro velocity usin their exerimental results for lass articles of different diameters. However, they reorted that the new correlation did not rovide ood redictions for olyroylene articles. Another examle can be found in Plasynski et al. (1994. When they alied the correlation develoed by Knowlton and Bachovchin (1975 they found that it cannot fit exerimental data and a new correlation was develoed. Therefore, it is concluded that the validation of the resent model for redictin the velocity corresondin to the minimum ressure dro is necessary. Several test cases by different investiators at different oeratin conditions are comared with resent redictions. Table 1 lists the condition for these test cases. Fi. 6 resents a comarison between redicted and measured velocity at minimum ressure dro. It can be seen from this fiure that most of the redictions lay in error rane of ±30%. Althouh this error is relatively hih, it is quite accetable in as-solid flows where many uncontrolled arameters act. The mean ercentae error, MPE was found to be %. The MPE is calculated by: MPE where, N 1 U mmeas U m meas and U N mest / U mmeas 100 (1 U m est are the measured and estimated velocities at minimum ressure dro, resectively, and N is the number of data oint. 53

6 S. M. El-Behery et al. / JAFM, Vol. 10, No., , 017. Table 1 Geometric and oeratin conditions for minimum ressure dro velocity test cases Author(s Particle material Particle density ρ (k/m 3 Particle diameter d (µm Total ressure Pt (kpa Solid mass flow rate, (k/s Pie diameter, D (mm Rae seeds Zenz (1949 Glass Sand Salt Glass Plasynski et al. (1994 Glass Coal Coal Glass Narimatsu and Ferreira (001 Glass Glass Glass Polyroylene Mok et al. (1989 Sand lass lass lass Costa et al. (004 lass lass lass lass lass lass Fiure 7 shows a comarison between redicted minimum ressure dro velocity and measured values of Mok et al. (1989. The fiure indicated that the minimum ressure dro velocity increases as the solid mass flow rate increases, in areement with many authors such as Rautiainen et al. (1999 and Rizk (1986. In addition, the fiure shows that the areement between redicted and measured minimum ressure dro velocity is fairly ood. Fi. 7. Effect of solid mass flow rate on the minimum ressure dro velocity: comarison with exerimental data of Mok et al. (1989. The effect of total (oeratin ressure on the velocity at minimum ressure dro is resented in Fi. 8. It can be seen from this fiure that the minimum ressure dro velocity decreases as the total ressure increases. This can be attributed to the increase of the as density as the total ressure increases which in turn increases the ressure dro due to friction with the ie wall. 54

7 S. M. El-Behery et al. / JAFM, Vol. 10, No., , 017. The fiure also shows that the areement between the resent redictions and measured value is fairly ood. An imortant arameter in as-solid is the article roerties (size and density. The effects of these arameters are resented in Fis. 9 and 10. It can be seen from these fiures that the minimum ressure dro velocity increases as article size or article density increase. This can be attributed to the increase of solid holdu and the increase of weiht force as a result. conveyor with a ood accuracy. In addition, the model redicted the linear variation of ressure with mass loadin ratio very well. It was also found that the ressure dro increases as the solid mass flow rate, article diameter, and article density increase. The transition from dense hase to dilute hase occurs at the oint of minimum ressure dro. The redicted minimum ressure dro velocity was comared to exerimental measurements for a wide rane of oeratin conditions, and a ood areement was obtained. The velocity at minimum ressure dro increases as the solid mass flow rate, article diameter, and article density increases and decreases as the system total ressure increases. Fi. 8. Effect of total oeratin ressure on the minimum ressure dro velocity: comarison with exerimental data of Plasynski et al. (1994. Fi. 10. Effect of article density on the minimum ressure dro velocity: comarison with exerimental data of Narimatsu and Ferreira (001. REFERENCES Fi. 9. Effect of article diameter on the minimum ressure dro velocity: comarison with exerimental data of Narimatsu and Ferreira ( CONCLUSION The as-solid flow in vertical neumatic conveyor is numerically simulated usin a steady state onedimensional two-fluid model. The model was validated aainst exerimental data and twodimensional Eulerian-Laranian simulations. The comarisons showed that the model is caable of redictin the ressure dro in vertical neumatic Arastooour, H. and D. Gidasow (1979. Vertical neumatic conveyin usin four hydrodynamic models. Industrial and Enineerin Chemistry Fundamentals 18(, Baeyens, J., D. V. Gauwberen and I. Vinckier (1995. Pneumatic dryin: the use of larescale exerimental data in a desin rocedure. Powder Technoloy 83(, Balachandar, S. and J. K. Eaton (010. Turbulent disersed multihase flow. Annual Review of Fluid Mechanics 4, Chun, J. S., K. Lee, A. Tischler and G. Yarim (001, January. Effect of article size and Concentration on ressure radient in twohase vertically uward transort. In Proceedin of 4 th ISOPE Ocean Minin Symosium. International Society of Offshore and Polar Enineers. Costa, I. A., D. C. Ferreira and J. T. Freire (004. Analysis of reime transitions and flow instabilities in vertical conveyin of coarse articles usin different solids feedin 55

8 S. M. El-Behery et al. / JAFM, Vol. 10, No., , 017. systems. The Canadian Journal of Chemical Enineerin 8(1, Crowe, C., M. Sommerfeld and Y. Tsuji (1998. Multihase flow with drolets and articles. CRC Press, Florida, USA. El-Behery, S. M., W. A. El-Askary, M. H. Hamed and K. A. Ibrahim (013. Eulerian- Laranian simulation and exerimental validation of neumatic conveyin dryer. Dryin Technoloy 31(1, El-Behery, S. M., W. A. El-Askary, M. H. Hamed and K. A. Ibrahim (01. Numerical and exerimental studies of heat transfer in article-laden as flows throuh a vertical riser. International Journal of Heat and Fluid Flow 33(1, El-Behery, S. M., W. A. El-Askary, M. H. Hamed and K. A. Ibrahim (011. Hydrodynamic and thermal fields analysis in as solid two-hase flow. International Journal of Heat and Fluid Flow 3(3, Enwald, H., E. Peirano, and A. E. Almstedt (1996. Eulerian two-hase flow theory alied to fluidization. International Journal of Multihase Flow, Gidasow, D. (1994. Multihase flow and fluidization. Academic Press, San Dieo, USA. Gouesbet, G. and A. Berlemont (1998. Eulerian and Laranian aroaches for redictin the behaviour of discrete articles in turbulent flows. Proress in Enery and Combustion Science 5(, Hamed, M. H. (005. Choked as-solid two-hase flow in ies. Journal of Enineerin and Alied Science 5, Han, T., A. Levy and Y. Pen (000. Model for dilute as-article flow in constant-area lance with heatin and friction. Powder Technoloy 11(3, Hariu, O. H. and M. C. Molstad (1949. Pressure dro in vertical tubes in transort of solids by ases. Industrial and Enineerin Chemistry 41(6, Henthorn, K. H., K. Park and J. S. Curtis (005. Measurement and rediction of ressure dro in neumatic conveyin: effect of article characteristics, mass loadin, and Reynolds number. Industrial and Enineerin Chemistry Research 44(14, Jamaleddine, T. J. and M. B. Ray (011. Dryin of slude in a neumatic dryer usin comutational fluid dynamics. Dryin Technoloy 9(3, Knowlton, T. M. and D. M. Bachovchin (1975, June. The determination of as-solids ressure dro and chokin velocity as a function of as density in a vertical neumatic conveyin line. In Proceedin of the International Conference on Fluidization, Pacific Grove, CA, USA. Konno, H. and S. Saito, (1969 Pneumatic conveyin of solids throuh straiht ies. Journal of Chemical Enineerin of Jaan (, Liu H., J. B. Hao, G. Yu, H. F. Yan, L. W. Wan and Z. T. Han (016. A Numerical Study on Metallic Powder Flow in Coaxial Laser Claddin. Journal of Alied Fluid Mechanics 9(5, Mastellone, M. L. and U. Arena (1999. The effect of article size and density on solids distribution alon the riser of a circulatin fluidized bed. Chemical Enineerin Science 54(, Mehdizadeh, A., B. Firoozabadi and B. Farhanieh (008. Numerical Simulation of Turbidity Current Usin v -f Turbulence Model. Journal of Alied Fluid Mechanics 1(, Mok, S. L. K, Y. Molodtsof, J. F. Lare and M. A. Berounou (1989. Characterization of dilute and dense hase vertical uflow as-solid transort based on averae concentration and velocity data. The Canadian Journal of Chemical Enineerin 67(1, Narimatsu, C. P. and M. C. Ferreira (001. Vertical neumatic conveyin in dilute and dense-hase flows: Exerimental study of the influence of article density and diameter on fluid dynamic behavior. Brazilian Journal of Chemical Enineerin 18(3, 1-3. Narimatsu, C. P., M. C. Ferreira, and J. T. Feire (007 Dryin of coarse articles in a vertical neumatic conveyor. Dryin Technoloy 5(, Nieuwland, J. J., J. A. M. Kuiers and W. P. M. Van Swaaij (1997. An enineerin model for dilute riser flow. Powder Technoloy 90(, Patro, P., B. Patro, and S. Muruan (014. Prediction of two-hase heat transfer and ressure dro in dilute as solid flows: A numerical investiation. Dryin Technoloy 3(10, Plasynski, S., G. Klinzin and M. Mathur (1994. Hih-ressure vertical neumatic transort investiation. Powder Technoloy 79(, Rautiainen, A., G. Stewart, V. Poikolainen and P. Sarkomaa (1999. An exerimental study of vertical neumatic conveyin. Powder Technoloy 104(, Rizk, F. (1986. Princiles of Pneumatic Conveyin. Encycloedia of Fluid Mechanics. Gulf Publishin Comany, Houston, Texas. Sinh, B. (198. Analysis of ressure dro in vertical neumatic conveyin. A eneralized aroach for as-article and liquid-article systems. Powder Technoloy 3(, Solsvik, J., Z. Chao and H. A. Jakobsen (01. A 56

9 S. M. El-Behery et al. / JAFM, Vol. 10, No., , 017. one-dimensional two-fluid as solid model alied to fluidized bed reactors: The SMR and SE-SMR rocesses. Procedia Enineerin 4, Solsvik, J., Z. Chao, and H. A. Jakobsen (014. Numerical evaluation of a one dimensional two fluid model alied to as solid cold flows in fluidised beds. The Canadian Journal of Chemical Enineerin 9(3, Tamayol, A., B. Firoozabadi and G. Ahmadi (008. Determination of settlin tanks erformance usin an Eulerian-Laranian method. Journal of Alied Fluid Mechanics 1(1, Xiao-in, C., Z. Yun, L. Cai, M. Qin-min, L. Pen, P. Wen-hao and X. Pan (009. Effect of roerties of ulverized coal on dense hase neumatic conveyin at hih ressure. Journal of Physics: Conference Series 147(1. Zenz, F. A. (1949. Two-hase fluid-solid flow. Industrial and Enineerin Chemistry 41(1,