Calculation of subsonic gas-dispersion flows in curved channels by the particle-in-cell method

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<ul><li><p> ISSN 0040-5795, Theoretical Foundations of Chemical Engineering, 2006, Vol. 40, No. 1, pp. 8088. Pleiades Publishing, Inc., 2006.Original Russian Text I.Kh. Enikeev, 2006, published in Teoreticheskie Osnovy Khimicheskoi Tekhnologii, 2006, Vol. 40, No. 1, pp. 8594.</p><p>80</p><p>Let us consider a flow of a two-phase medium con-sisting of a compressible carrier gas and monodispersedrops in a curved channel of width </p><p>R</p><p> and finite length </p><p>L</p><p>.</p><p>Figure 1 presents a half of the longitudinal section ofthe channel. Let the symmetry axis be the </p><p>ox</p><p> axis of acylindrical coordinate system (</p><p>x</p><p>, </p><p>y</p><p>, </p><p>). Then, one canwrite the equations of the unsteady-state two-dimen-sional flow of the two-phase medium in the channel interms of a model of interpenetrating continua in dimen-sionless form [1, 2]:</p><p>(1)</p><p>Set (1) should be complemented with equations ofstate of the phases; it is assumed that the carriermedium is a perfect gas and the drops are incompress-ible ( = const):</p><p>st-------- </p><p>k svsk( )+ 0,=</p><p>svst------------- </p><p>k svsvsk( )+ </p><p>--- p F,+=</p><p> M02, 1, s 1=</p><p>0, s 2,=</p><p>= =</p><p>sEst-------------- </p><p>k sEs /+( )v sk[ ]+ </p><p>s 1=</p><p>2</p><p> 0,=</p><p>Es es12--- vs vs,( ),+=</p><p>sest------------- </p><p>k sesvsk( )+ q 1/2F v1 v2( ).+=</p><p>20</p><p>(2)</p><p>The dimensionless parameters in Eqs. (1) and (2) canbe represented as</p><p>(3)</p><p>The vector </p><p>F</p><p> of the friction force between the phasesand the interphase heat-transfer intensity </p><p>q</p><p> areexpressed as [1, 2]</p><p>(4)</p><p>To integrate Eqs. (1) and (2), it is necessary toimpose boundary and initial conditions. Let us assumethat the left boundary of the region from which the gasdrop medium flows is far enough. Then, as </p><p>x</p><p>,</p><p>there is a flow without dynamic (velocity) lag or ther-mal (temperature) lag of drops in which the verticalcomponents of the gas and drop velocities are zero. Inthis case, the flow uniformity condition [35]</p><p>p 1( )1e1,=</p><p>e1T1</p><p> 1( )--------------------, e2c2T2</p><p>c 1( )-----------------------.= =</p><p>vsvs( )*U------------, </p><p>s*p10-------, *0</p><p>------,= = =</p><p>EsEs*</p><p>U02-------, es</p><p>es*</p><p>U02------, t</p><p>U0R</p><p>------t*,= = =</p><p>xx*</p><p>R------, y y*R</p><p>------.= =</p><p>F St2 v1 v2( )2, St2</p><p>0U0d2</p><p>18R------------------,= =</p><p>q e1 cc2----e2 2, 12R</p><p>20U0d</p><p>2c</p><p>---------------------.= =</p><p>v 1xx--------- 0,=</p><p>Calculation of Subsonic Gas-Dispersion Flowsin Curved Channels by the Particle-in-Cell Method</p><p>I. Kh. EnikeevMoscow State University of Engineering Ecology, </p><p>Staraya Basmannaya ul. 21/4, Moscow, 107884 RussiaReceived March 1, 2005</p><p>AbstractMultiphase (specifically, two-phase) flows of media in complex-shaped channels are studied by theparticle-in-cell method. The stability of the proposed algorithm is analyzed over a wide range of Mach numbersof an undisturbed flow. The use of the method is exemplified by calculating the characteristics of a gasdropflow in curved channels in which the generatrices of the lateral surface consist of circular arcs.DOI: 10.1134/S0040579506010118</p></li><li><p>THEORETICAL FOUNDATIONS OF CHEMICAL ENGINEERING Vol. 40 No. 1 2006</p><p>CALCULATION OF SUBSONIC GAS-DISPERSION FLOWS 81</p><p>where is the projection of the gas velocity on theaxis x, can be supposed to be met.</p><p>It is also assumed that the flow of the mixture in thisregion is isentropic and isenthalpic, i.e., H0 = const andS = const. The reduced density of the second phase inthis region of the channel is a given value. In the calcu-lations, these boundary conditions are imposed in thesection x = 1.25 (Fig. 1).</p><p>It is supposed that the gas does not penetrate the sidewalls and that drops that have arrived at the channelwalls adhere to them and thus leave the flow. Taken asthe boundary conditions for the gas at the outlet of thechannel are the relations obtained for a one-dimen-sional isentropic gas flow from a nozzle [6]</p><p>v 1x</p><p>v1x</p><p>v1 1( )----------</p><p>MM1-------</p><p>1 0.5 1( )M 1( )2+1 0.5 1( )M2+</p><p>-------------------------------------------- ,=</p><p>As the initial data, the parameters of the undisturbedflow in the section x = 1.25 at F = 0 and q = 0 are taken.The set of Eqs. (1) is integrated by the finite-differencemethod in the curvilinear region M:</p><p>where G(x) and F(x) are the equations of the lower andupper lateral surfaces of the channel. In the integration,it is necessary to replace the continuous curvilinearregion M by a grid. The direct replacement gives rise toirregular nodes or computational cells near the bound-ary of the region. In this case, to impose the boundaryconditions, fractional cells are formed in the layer ofirregular computational cells using the algorithm of theparticle-in-cell method [7]. The practice of calculationsusing fractional cells showed that this algorithm is quitecumbersome, especially in the case where M0 1.Therefore, it is more expedient to introduce such newvariables = (x, y) and = (x, y) that the curvilinearregion becomes rectangular. As was shown [3, 4], if theJacobian I = D(, )/D(x, y) of this transformationexists and is nonzero in any point of the region, then thedivergent form of Eqs. (1) is retained. By changing theindependent variables x = x,</p><p>the curvilinear region M is transformed into the rectan-gular region N:</p><p>Then, in the variables (x, ), the set of Eqs. (1) takes theform</p><p>(5)</p><p>AA 1( )---------</p><p>MM1-------</p><p>1 0.5 1( )M 1( )2+1 0.5 1( )M2+</p><p>-------------------------------------------- .=</p><p>M 0 x L/R G x( ) y F x( ) , { },</p><p> y G x( )F x( ) G x( )------------------------------,=</p><p>N 0 x 1 0 1 , ( ).</p><p>st--------</p><p>sv sxx--------------</p><p>1---</p><p>sUs--------------+ +</p><p>sUs</p><p> G+( )------------------------,=</p><p>sv sxt--------------</p><p>s v sx( )2x---------------------</p><p>1---</p><p>sv sUs---------------------+ +</p><p>= 1</p><p>------ </p><p>px------ G'+( )</p><p>------+</p><p>sv sxUs</p><p> G+( )------------------------ 1( )sFx,+</p><p>sv syt--------------</p><p>sv syv sxx---------------------</p><p>1---</p><p>sv syUs---------------------+ +</p><p>= 1</p><p>-------p------ 1( )</p><p>sFysv s</p><p>yUs</p><p> G+( )------------------------,+</p><p>(c)</p><p>1.0 0.5 0 0.5x/R</p><p>y/R</p><p>0.3</p><p>0.6</p><p>0.9 1.1</p><p>(b)</p><p>(a)</p><p>0.2</p><p>0.4 0.7 0.8</p><p>abcd</p><p>0.040.07</p><p>0.11 0.12</p><p>Fig. 1. Level curves (a) calculated in this work and (b)determined in the literature [5] and the (c) gas and (d) dropstreamlines in a convergent tube at M0 = (a) 0.08, (b) 0.5,and (c) 0.9.</p></li><li><p>82</p><p>THEORETICAL FOUNDATIONS OF CHEMICAL ENGINEERING Vol. 40 No. 1 2006</p><p>ENIKEEV</p><p>If = ' ( + G), then set (5) appears as a set ofequations for the planar case.</p><p>As was shown [7, 8] for flows in rectangular regionsat 0 1, at the Euler step, it is better to use a time-implicit difference scheme for calculating the pressure.</p><p>In this work, this method is generalized to regions ofcomplex shape.</p><p>Let us write the equations describing the gas flow atthe Euler step in the form</p><p>(6)</p><p>In derivation of Eqs. (6), the terms taking intoaccount the axial symmetry and convective transfer andalso interphase interaction are ignored. The region ofintegration is covered with a fixed computational gridwith rectangular cells with sides x and . Let us labelthe centers of the cells with subscripts i and j along thecoordinates x and , respectively, and write Eq. (6) indifference form according to the implicit scheme</p><p>(7)</p><p>where = dt/dx,</p><p>(8)</p><p>sEst--------------</p><p> sEs /p+( )v sxx--------------------------------------------+</p><p>s 1=</p><p>2</p><p>+</p><p>1---</p><p> sUs /p+( )Us---------------------------------------------</p><p>p/ sEs+( )Us G+( )----------------------------------------+ 0,=</p><p>se2t-------------</p><p>2e2v 2xx--------------------</p><p>1---</p><p>2e2U2--------------------+ +</p><p>= q2e2U2</p><p> G+( )------------------------12---F v1 v2( ),+</p><p>Us</p><p>v sy</p><p>'v sx</p><p> G+( ),+=Us v s</p><p>y'v G'+( )v sx, x( ) F x( ) G x( ).= =</p><p>Us</p><p>v sx</p><p>ut------</p><p>1-------</p><p>px------</p><p>' G'+( )-----------------------</p><p>p------,+=</p><p>vt-------</p><p>1----------</p><p>p------,=</p><p>pt------</p><p> 1( )</p><p>---------------- p ux--------- ' G'+( )u</p><p>------------------------------+ .=</p><p>ui j,n 1+</p><p>ui j,n </p><p>i j,n</p><p>------------ pi 1/2 j,+n 1+ pi 1/2 j,</p><p>n 1+( )=</p><p>+ i' j Gi'+( )</p><p>ii j,n</p><p>----------------------------- pi j 1/2+,n 1+ pi j 1/2,</p><p>n 1+( ),</p><p>v i j,n 1+</p><p>v i j,n </p><p>ii j,n</p><p>---------------- pi j 1/2+,n 1+ pi j 1/2,</p><p>n 1+( ),=</p><p>(9)</p><p>In Eqs. (7)(9), the parameters of the gas at theboundaries of cells are approximated by central differ-ences by formulas of the first order of accuracy, e.g.,</p><p>Let us eliminate and at the (n + 1)th timestep from Eq. (9) using Eqs. (7) and (8) and introducethe notation</p><p>pi j,n 1+ pi j,</p><p>n 1( ) pi j,n 1+i</p><p>-------------------------------- i ui 1/2 j,+n 1+</p><p>ui 1/2 j,n 1+</p><p>( )[=</p><p>+ v i j 1/2+,n 1+</p><p>v i j 1/2,n 1+</p><p> i' j 1/2+ ui j 1/2+,n 1+ j 1/2 ui j 1/2,n 1+( ) G' ui j 1/2+,</p><p>n 1+ui j 1/2,</p><p>n 1+( ) ].</p><p>ui 1/2 j,+n 1+ 1</p><p>2--- ui 1 j,+n 1+</p><p>ui j,n 1+</p><p>+( ).=</p><p>ui j,n 1+</p><p>v i j,n 1+</p><p>Ai2 1( )</p><p>i----------------------i j,</p><p>n 1+=</p><p>i 1/2</p><p>i 1/2 j,n</p><p>-----------------</p><p>14---</p><p>Gi' j 1/2+ i'+i j 1/2+,</p><p>n------------------------------</p><p>Gi' j 1/2 i'+i j 1/2,</p><p>n------------------------------ + ,</p><p>Bi2 1( )</p><p>i----------------------i j,</p><p>n 1+=</p><p>i 1/2+</p><p>i 1/2 j,+n</p><p>-----------------</p><p>14---</p><p>Gi' j 1/2+ i'+i j 1/2+,</p><p>n------------------------------</p><p>Gi' j 1/2 i'+i j 1/2,</p><p>n------------------------------ ,</p><p>Ci 1 Ai Bi2 1( )</p><p>i2----------------------i j,</p><p>n 1++ + +</p><p>=</p><p>1 Gi' j 1/2+ i'+( )2+</p><p>i j 1/2+,n</p><p>----------------------------------------------</p><p>1 Gi' j 1/2 i'+( )2+i j 1/2,</p><p>n----------------------------------------------+</p><p>,</p><p>A j*2 1( ) pi j,n 1+</p><p>i----------------------------------</p><p>1 Gi' j 1/2 i'+( )2+ii j, 1/2</p><p>n----------------------------------------------=</p><p>+14---</p><p>Gi 1/2+' ji 1/2+'+i 1/2 j,+</p><p>n---------------------------------------</p><p>Gi 1/2' ji 1/2'+i 1/2 j,</p><p>n--------------------------------------- ,</p><p>B j*2 1( ) pi j,n 1+</p><p>i----------------------------------</p><p>1 Gi' j 1/2+ i'+( )2+ii j, 1/2+</p><p>n----------------------------------------------=</p><p>14---</p><p>Gi 1/2+' ji 1/2+'+i 1/2 j,+</p><p>n---------------------------------------</p><p>Gi 1/2' ji 1/2'+i 1/2 j,</p><p>n--------------------------------------- ,</p></li><li><p>THEORETICAL FOUNDATIONS OF CHEMICAL ENGINEERING Vol. 40 No. 1 2006</p><p>CALCULATION OF SUBSONIC GAS-DISPERSION FLOWS 83</p><p>In this notation, Eq. (9) is written as follows:for the longitudinal sweep,</p><p>(10)</p><p>for the transverse sweep,</p><p>(11)</p><p>Since the sweep coefficients in Eqs. (10) and (11)depend on , these equations are solved by iterationin .</p><p>At each iteration, the values in the expressionsfor Ai, Bi, , , Ci, Cj, Fi, and are replaced bythe respective values obtained at the preceding itera-tion. After solving set (7)(9) with a given accuracy, atthe Euler step, the pressure and velocities of the carrierphase are found. All the parameters of the dispersedphase at the Euler step remain unchanged with time.</p><p>C j* 1 A j* B j*+ +=</p><p>+2 1( )</p><p>i----------------------i j,</p><p>n 1+ i 1/2+'</p><p>i 1/2 j,+n</p><p>-----------------</p><p>i 1/2'</p><p>i 1/2 j,n</p><p>-----------------+ ,</p><p>i*2 1( ) pi j,n 1+</p><p>i----------------------------------=</p><p>Gi 1/2+' ji 1/2+'+</p><p>i 1/2 j,+n</p><p>---------------------------------------</p><p>Gi' j 1/2+ i'+i j 1/2+,</p><p>n------------------------------+ i 1+ j 1+,n 1+</p><p>Gi 1/2+' ji 1/2+'+i 1/2 j,+</p><p>n---------------------------------------</p><p>Gi' j 1/2 i'+i j 1/2,</p><p>n------------------------------+ i 1 j 1,+n 1+</p><p>Gi 1/2' ji 1/2'+i 1/2 j,</p><p>n---------------------------------------</p><p>Gi' j 1/2+ i'+i j 1/2+,</p><p>------------------------------+ i 1 j 1+,n 1+</p><p>+Gi 1/2' ji 1/2'+</p><p>i 1/2 j,n</p><p>---------------------------------------</p><p>Gi' j 1/2 i'+i j 1/2,</p><p>------------------------------+ i 1 j 1+,n 1+</p><p>+i--- i 1/2+ ui 1/2 j,+</p><p>ni 1/2 ui 1/2 j,</p><p>n v i j 1/2+,</p><p>n+[</p><p> v i j 1/2,n G' j 1/2+ ii'+( )ui j 1/2+,n+ G' j 1/2 ii'+( )ui j 1/2,n ],</p><p>Fi pi j,n</p><p> A j* pi j 1,n 1+</p><p> B j* pi j 1+,n 1+</p><p> i,+=</p><p>F j* pi j,n A j pi 1 j,</p><p>n 1+ B j* pi 1 j,+</p><p>n 1+ j*,+=</p><p>i j*.=</p><p>Ai pi 1 j,n 1+ Ci pi j,</p><p>n 1+ Bi pi 1 j,+</p><p>n 1++ i;=</p><p>A j* pi j 1,n 1+ C j* pi j,</p><p>n 1+ B j* pi j 1+,</p><p>n 1++ j*.=</p><p>pi j,n 1+</p><p>pi j,n 1+</p><p>pi j,n 1+</p><p>A j* B j* F j*</p><p>The Lagrange and final steps are performed by theparticle-in-cell method [7].</p><p>Let us consider the stability of the proposed differ-ence scheme using the results [7, 8] of analyzing theparabolic form of the first differential approximation ofthe scheme.</p><p>Since inherent pressure in the medium of particles isabsent, the equations of motion of the dispersed phaseare integrated according to an explicit scheme of theparticle-in-cell method [7]. The scheme is well knownin the literature [7, 8]; therefore, the correspondingequations in the initial set of Eqs. (1) are not consideredhere.</p><p>Let us consider the case where, for several charac-teristic times of the problem, the following relations arevalid:</p><p>Then, the terms describing the intensity of the forceand thermal interphase interaction generally do notaffect the stability of the numerical scheme since theydo not contain higher derivatives. Therefore, the formu-las of the final step for the gas phase ignore the inter-phase interaction forces.</p><p>When max(F, q) 0, an equilibrium scheme ofthe flow of the two-phase medium takes place, in whichthe gas-dispersion flow is regarded as an effective gaswith a changed sound velocity and adiabatic exponent.Therefore, in this case, the interphase interaction forcescan also be ignored. Without loss of generality, the flowis assumed to be plane. Thus, under the assumptionsmade, the equations of the final stage for the gas havethe form</p><p>0 max F</p><p>q</p><p>,( ),</p><p>0 R/U0, F lF/U0, </p><p>q lq/U0.= = =</p><p>i j,n 1+ i j,</p><p>n t2x----------+=</p><p> i 1 j,n</p><p>u i 1 j, u i j,+( ) i j,n u i j, u i j,+( ) -</p><p>+1--- i j 1,</p><p>n U i j 1, U i j,+( ) i j,n U i j, U i j 1+,+( )[ ] </p><p> i j,n U i j,</p><p> t,</p><p>i j,n 1+</p><p>ui j,n 1+</p><p> = i j,n</p><p>u i j,t</p><p>2x---------- i 1 j,n</p><p>u i 1 j, u i 1 j, u i j,+( ) ---</p><p>+</p><p> i j,n</p><p>u i j, u i j, u i 1 j,++( ) 1i--- i j 1,</p><p>nu i j 1, U i j 1, U i j,+( )[+</p></li><li><p>84</p><p>THEORETICAL FOUNDATIONS OF CHEMICAL ENGINEERING Vol. 40 No. 1 2006</p><p>ENIKEEV</p><p>(12)</p><p>Here, it is assumed that + &gt; 0, + &gt; 0, and so on. Obviously, the choice of the flow</p><p>direction does not affect the character of the stability ofthe scheme. Let us substitute expressions (7)(9) for thevelocity components and and the pressure p intoformulas (12) of the final step.</p><p>Supposing that all the functions in the obtained rela-tions are continuously differentiable functions of thevariables t, x, and y, let us expand them in Taylor seriesin the vicinity of the point (tn + 1, xi, yj) and truncatethem after the terms of the second order in t, x, andy. Thus, the parabolic form of the first differentialapproximation of the initial equations is obtained:</p><p>--- i j,n</p><p>u i j, U i j, U i j 1+,+( ) ] i j,n u i j, U i j, t,</p><p>i j,n 1+</p><p>v i j,n 1+</p><p> = i j,nv i j,</p><p>t2x---------- i 1 j,</p><p>nv i 1 j, u i 1 j, u i j,+( ) -</p><p>+</p><p> i j,nv i j, u i j, u i 1 j,++( ) 1i</p><p>--- i j 1,n</p><p>v i j 1, U i j 1, U i j,+( )[+</p><p>-- i j,nv i j, U i j, U i j 1+,+( ) ] </p><p> i j,n v i j, U i j, t,</p><p>i j,n 1+ Ei j,</p><p>n 1+ = i j,</p><p>n E i j,t</p><p>2x---------- i 1 j,n E i 1 j, u i 1 j, u i j,+( ) -</p><p>+</p><p> i j,n E i j, u i j, u i 1 j,++( ) 1i</p><p>--- i j 1,n E i j 1, U i j 1, U i j,+( )[+</p><p>-- i j,n E i j, U i j, U i j 1+,+( ) ] </p><p> i j, E i j, U i j, t,</p><p>U i j,</p><p>u i j, i' i j Gi+( ) iv i j, .+=u i 1 j, u i j, U i j 1,</p><p>U i j,</p><p>u v</p><p>t------</p><p>ux---------</p><p>1---</p><p>U-----------+ +</p><p>= 12--- ux</p><p>1 2 '22+ +( )2</p><p>------------------------------------ p u2 t+ 2</p><p>x2--------</p><p> uU '----------+ </p><p>t</p><p>-----</p><p>2Ux-------------</p><p>+12--- Ux</p><p>t</p><p>-----</p><p>'x--------------</p><p>px------</p><p>1 '22+( )------------------------- p</p><p>U2</p><p>------++</p><p>(13)</p><p>2 p2-------- p,+</p><p>ut--------...</p></li></ul>