Microsoft Word - 476hernandez.docx CHEMICAL ENGINEERING TRANSACTIONS VOL. 43, 2015 A publication of The Italian Association of Chemical Engineering Online at www.aidic.it/cet Chief Editors: Sauro Pierucci, Jiří J. Klemeš Copyright © 2015, AIDIC Servizi S.r.l., ISBN 978-88-95608-34-1; ISSN 2283-9216 A Numerical Solution of a Model for Heat Transfer in Moving Beds Sávio L. Bertoli*, Juscelino de A. Júnior, Rodrigo K. Decker, Laércio Ender, Juliano de Almeida, Aline C. Lovatel, Department of Chemical Engineering, Regional University of Blumenau, Zip Code, 89035-000, Blumenau, Brazil, savio@furb.br This article aims to study the heat transfer in the pyrolysis of oil shale in a diluted and concurrent moving bed reactor. It was developed a numerical solution for a non-linear heat transfer (convective and radiant) model through the orthogonal collocation method. In this method the particle interior temperature profile is approximated by an orthogonal polynomial of an arbitrary degree that is replaced into the model equations generating a system of ordinary differential equations. The results obtained were compared with an existing analytical solution of the quasi-linearized heat transfer model. The comparison shown an excellent agreement in the range of the parameters analyzed. KEYWORDS: Orthogonal collocation method, numerical solution, pyrolysis, heat transfer, oil shale. 1. Introduction Brazil owns the second largest reserves of oil shale in the world. The main oil shale deposits are located less than 100 meters deep and are distributed in layers of variable thickness and distinct levels of kerogen. Generally, the content of organic matter in the deposits of oil shale ranges between 5% and 25%, as in the Irati formation of oil shale at São Mateus do Sul – PR, Brazil, which has a 9% average in oil. Under this perspective of research, aiming new sustainable ways of composing the global energy matrix, the decomposition of oil shale and gas has been studied in various aspects, and largely in the field of Chemical Engineering. Because it is a non-renewable resource and generates hazardous products for to the environment, it has been largely invested in research in order to achieve technologies combining energy efficiency with environmental sustainability. In this way, other works have developed analytical solutions for heat transfer models of pyrolysis in diluted moving beds: Bertoli (1989, 2000) using the Laplace Transform and the Duhamel theorem; Meier et al. (2009) through the self-adjoint operator method. Those solutions incorporate a quasi-linearized radiant heat transfer coefficient between the wall and the particle: ( )( )TpsTwTpsTwhr ++= 22 σερ . (1) With the aim of to evaluate the nonlinear radiant effects, in the present work a numerical solution by the orthogonal collocation method is given. The results obtained are compared with an existent analytical solution (Bertoli, 1989, 2000). 2. Model Equations The main assumptions in the present analysis are the same as described in Bertoli (1989, 2000). The equation of the model differs only by the fact that now the radiant term isn´t linearized. The equations for the fluid and the particle are: DOI: 10.3303/CET1543262 Please cite this article as: Leandro Bertoli S., De Almeida Junior J., Decker R.K., Ender L., De Almeida J., Cristina Lovatel A., 2015, A numerical solution of a model for heat transfer in moving beds, Chemical Engineering Transactions, 43, 1567-1572 DOI: 10.3303/CET1543262 1567 ( ) , θθ βθθ β dτ dθ pffw f 143 −−−=         ∂ ∂ ∂ ∂ = ∂ ∂ r θ r r r τ θ pp 2 2 1 ( ) , θr, θ f p 100 == 00 r θ r p = ∂ ∂ = ( ) ( )414211 1 wpf prp θθβθθβ r θ −+−= ∂ ∂ − = 00 =∂ ∂ =ξξ T ( ) ( )psf fff pppv fw fff pww f - TT uCVρ u A hn - TT uCVρ uAh dt dT −= ; (2)       ∂ ∂ ∂ ∂ = ∂ ∂ ξ T ξ ξ ξ α t T pp 2 2 . (3) With the following initial and boundary conditions: ( ) ( ) ifp TTT == 0,0 ξ ; (4) ; (5) . (6) Defining the dimensionless variables: , R tα , τ R ξ , r T T , θ T T θ p p pi p p i f f 2≡≡≡≡ , k Tσ Rε , β k Rh , β T T θ p ipp p pp i w w ³ 21 ≡≡≡ , α uCVρ R u A hn , β α u CVρ R u Ah β pfff ppppv pfff ppww 2 4 2 3 ≡≡ the following equations are identified: (7) , (8) (9) , (10) . (11) Also, with the variable change, u = r2, it follows that: ( ) ( )1 4 3 pffw f d d θθβθθβ τ θ −−−= , (12) ( ) ( )44 wps p p fps p p Rpξ p TT k σε TT k h ξ T −+−= ∂ ∂ − = 1568 ( ) ( ).2 414211 1 wpf pup θθβθθβ u θ −+−= ∂ ∂ − =       ∂ ∂ + ∂ ∂ = ∂ ∂ u θ u θ u τ θ ppp 2 3 4 2 2 , (13) , (14) (15) 3. Polynomial Approximation The following polynomial approximation is introduced: (16) or ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )211 1 211 1 1 1 1-u , /, /, pj n j j n p Pn uPn buu ττθτθ +=  + = +  . (17) Where: ( ) ( ) ( ) ( ) ( ) ( )( ). , order of polynomial Jacobi a , polynomial inginterpolat Lagrange ju 211 1 211 th /, n /, j Pτ a τb n uPn = = = Now defining, . (18) It follows that: . (19) By the orthogonal collocation method, the collocation points are chosen as the zeros of a Jacobi polynomial; in this case, , 211 )/,( n(u) P then, du dp du pd u NTNT 2 3 2 2 + ( ) ( ) ( ) ( )        +=++ = = n i P / n n .. n, ........ i /, n 11231 10 211 where i = 1, ... n are internal collocation points. ( ) ( ) ( ) ( ) ( )( ) ( ) ( )2/1,11 1 1 1 , , uPnuauuu n j pjj n pp −+=≅  + = + ττθτθτθ  ( ) ( ) )/, ( nNT u P u - up 211 1≡    ( )( ) )/,( n NTNT (u) P/nn du dp du pd u 211 2 2 231 2 3 ++=+ 02 00 =∂ ∂ = ∂ ∂ − == uu u θ u u θ 1569 ( ) ( ) 1 43 - - +−= Npffwf d d θθβθθβ τ θ , ... n i A Bu dτ dθ n j ijiji pi 1 2 3 4 1 1 =    +=  + = ( ) ( ) ( ) =       −− = ++ + =  1 11 1 1 122 u npj n n j uu du d τb- θA  Therefore, from Eq. (2) – (5) we have , (20) , (21) θA Bu dτ dθ pj n j ,jN, j NN pN  + = +++ +     += 1 1 111 1 2 3 4 ( ) ( ) ( )τ b / n n 2314 +++ (22) (23) The term in brackets in the last equation is equal 1. Also, ii udu jd Bij and udu jd Aij 2 2 == The resulting system is: , ... n i θC dτ dθ pj n j ij pi 1 1 1 ==  + = , (24) ( ) ( ) ( )τ b/nn θC dτ dθ pj, jN n j N 23141 1 1 1 +++= + + = +  , (25) ( ) ( )143 += pNf - fw f θθ - β- θθ β dτ dθ , (26) 111 , ... n i θθ fo po +=== . (27) Where:       += ijijiij A Bu C 2 3 4 , (28) ( ) ( ) ( )44 1211 1 1 1 22 wNfNpj n j , j N θθA β θθ β - θ - τa −++ + = + −−=  . (29) Also, the mean temperature of the particle is given by ( ) ( ) ( ) pj n j j n p / p θwduu,τ θuτθ  + = + == 1 1 1 0 121 2 3 (30) Were, jw = weight of the quadrature (Radau quadrature with inclusion of u = 1). More details in the collocation method are given by Villadsen et al. (1978). ( )     + 441 121 w - θpθ β f - θp β θ 1570 4. Results For the integration of the system of Eq. (6)–(9) we choose as initial step, h0 = 1,0 x 10-5 and as precision, ε = 1 x 10-6. Also, the following parameters were obtained from Bertoli (1989, 2000) (thermal test No 5). .4260.009250.0 10663.104823.0 43 4 21 ; β β ; x ; β β - == == For τ = 0, the Jacobian of the system of Eq. (6) – (9) is (N = 2 ; ε = 1 x 10-6 ; ho = 1 x 10-5 ; α = 1 ; β = ½): This Jacobian matrix has the following eingenvalues: 033.071.2022.169 321 −=== ; λ- ; λ - λ 6769.04 -λ = . in this case, the problem is stiff. So, for integration of the system formed by Eq. (6) - (9) we use a third order semi-implicit Runge Kutta method STIFF 3 (Villadsen et al., 1978). The Figure (1) and Figure (2) were generated to demonstrate the variation of the mean temperatures of the particle and of the fluid along the reactor, respectively. The set of conditions were those of the thermal test No 5 presented in Bertoli (2000) (except for the wall temperature and the temperatures at the reactor entrance). The results of the Figures (1) and (2) were obtained at Tw = 500 ° C and Tfi = Tpi = 150 ° C; they shown excellent agreement in the numerical solution of this non-linear model and the analytical solution presented in Bertoli (1989). This agreement is possible because in the region where the greater flow of radiant heat transfer occurs for the particle (near the entrance of the reactor) the quasi-linearization of the radiant contribution is an adequate approximation if the temperature of the particle surface does not vary significantly; that probably is due to the high heat capacity of the solid, associated with the convective heat transfer between the fluid and the particle. Furthermore, in the region where the temperature of the particle surface is high, the temperature difference (Tw - Tps) is small and therefore at this point, the radiation is a small portion of the total heat transferred to the particle in the reactor. It is important to note that although the wall temperature is 500ºC, radiant on the surface of the particles (Qrad / Qtotal)ps flow is in the range 60-80% by analytical solution (Bertoli, 1989). Figure 1: Variation of average particle temperature along the reactor Figure 2: Changes in mean temperature of fluid along the reactor. . 546.0 013.1 0 0 4537.0 05.130 36.34 36.4 0 7.137 33.44 03.20 0 8.700- 9.965 15.70-             − − − − =J 1571 5. Conclusion The agreement between the numerical and analytical solutions demonstrates the capability of the orthogonal collocation method in to handle the problem of heat transfer in moving beds. Also in the present work it was possible to confirm several results from the literature obtained with the analytical solution of the equivalent linearized heat transfer model for moving beds: In Bertoli (1989, 2000) it was pointed that radiant heat transfer between the particle and the wall plays an important role and cannot be neglected in comparison with the fluid–particle convective heat transfer; in Bertoli et al. (2012) a discussion was made relative to the time scales involved in the thermal radiation phenomena; in Meier et al. (2009) it was defined a radiation Biot number where a temperature inversion between the fluid and particle occurs. All these results are now supported considering the non-linear nature of the thermal radiation phenomena, through the numerical solution obtained. The numerical solution is particularly interesting at low values of τ, where the analytical solution has difficulties in convergence (Bertoli, 1989). Nomenclature wA wall differential area, dx2 Rπ wT wall temperature pA surface area of single particle, 2 pdπ= u axial velocity C specific heat at constant pressure x axial distance ph fluid to particle heat transfer coefficient α thermal diffusivity rh radiant heat transfer coefficient between the wall and the particle ε void fraction pε surface emissivity of particles wh wall heat transfer coefficient ξ radial position inside a particle vn number of solid particles in the volume V ρ density pR particle radius σ Stefan-Boltzmann constant T temperature Subscripts psT particle surface temperature f fluid V pipe differential volume, dx2Rπ= p particle References Bertoli S. L., 1989, Radiant and Convective Heat Transfer in Moving Beds, M. Sc. Thesis, COPPE, Rio de Janeiro, Brazil (in Portuguese). Bertoli S.L., 2000, Radiant and convective heat transfer on pneumatic transport of particles: an analytical study, International Journal of Heat and Mass Transfer, 43, 2345-2363, DOI: 10.1016/S0017- 9310(99)00280-X Bertoli, S. L., Valle, J. A. B., Gerent, A. G., Almeida, J., 2012, Heat transfer at pneumatic particle transport — Limit solutions, Powder Technology, 232, 64–77, DOI: 10.1016/j.powtec.2012.07.050 Meier H. F., Noriler D., Bertoli S. L., 2009, A solution for a heat transfer model in a moving bed through the self-adjoint operator method, Latin American Applied Research, 39, 327-336. Villadsen, J., Michelsen, M. L., Solution of Differential Equation Models by Polynomial Approximation, Pearson Education, Limited, 1978. 1572