Design of A Thermo-Cell for Studying Optical Characteristics of The Surface Layer During the Heating Process

Аннотация

The article considers the problem of filtration of suspensions with the formation of a nonlinearly compressible sedimentary layer. Based on numerical calculations, the dependence of filtration characteristics on nonlinear effects is established. Keywords: Suspension, filter, moving boundary, filter layer, sediment, pressure. Introduction: Filtration of suspensions refers to complex technological processes and depends on a large number of micro- and macrofactors. Models of filtration of suspensions are based on fundamental equations of mechanics of multiphase, multicomponent media and the theory of filtration consolidation [1,2,3]. The problem of pressure distribution in the sediment layer is reduced to the solution of the classical parabolic equation with moving boundary conditions. Real physical conditions during filtration of suspensions are such that the sediment is heterogeneous in its structure and filtration-capacitive properties and consolidates under the action of pressure and other forces. Thus, in the general case, the consolidation coefficient is variable, depending on the pressure distribution over the sediment thickness. In this paper, the process of sedimentation during filtration of suspensions is European International Journal of Multidisciplinary Research 19 https://eipublication.com/index.php/eijmrms and Management Studies European International Journal of Multidisciplinary Research and Management Studies analyzed taking into account the variability of the distinguished- the filter layer region, and - the sediment consolidation coefficient. region, where is the moving boundary. Due to the listed conditions, the Stefan problem can be formulated for Let there be a flat filter element. The suspension is the filtration process. Let be the pore pressure, - the placed in such a way that it contacts the initially liquid pressure at the initial moment of time, at the existing filter layer of some thickness. Under the action entrance to the sediment layer, at the exit from the filter of the pressure difference, which remains constant layer. over time, the process of separation of the suspension begins, the thickness of the sediment continuously The mathematical model of filtration is written as increases due to the movement of the sediment - follows [1,2] suspension boundary. Thus, two regions can be p 2p =b ( p ) , 0tT , z , (1) t z2 2 p ( z,0 ) = p ( z ), z , (2) 0 1 p ( 0,t ) = p 0tT, (3) 2 p  h ( t ) ,t  = p , 0tT, (4) 1 p dh =l , 0tT, (5) z dt ( ) z p − p r Where h(0)=z , b(p)=b e(p−p 1 ) - consolidation ratio, p = p + 1 2 , 0 z  z l = , r - 0 0 0 2 z 0 u 0 sediment resistivity,  - viscosity coefficient, u- external sedimentation coefficient,  - parameter. Introducing a new variable p d ( ) 1 v =  =a e−p −1 , a = b() −b e−p 1 0 0 we will receive 1 v  p =− ln +1. (6) Then equation (1) is transformed to the form 1 v   v(p) = k(v) , k(v) = − . (7) ( ) t z z  v+a After transforming conditions (2) - (5) we arrive at v| =v ( z ) =a ( e−p 0 (z) −1 ) , (8) t=0 0 ( ) v| =v =a e−p 2 −1 , (9) z=0 2 ( ) v| =v =a e−p 1 −1 , (10) z=h(t) 1 European International Journal of Multidisciplinary Research 20 https://eipublication.com/index.php/eijmrms and Management Studies European International Journal of Multidisciplinary Research and Management Studies v(p) dh k(v) =l . (11) z dt To solve problem (7)-(11) we use the finite difference method [4]. Let's introduce a uniform grid z with a step f =+ = z |z =if,i =0,1,...,m+ j , where in time 1 2 i i we will use a non-uniform grid [4]  = t|t =t =t + ,t =0, j =m+1,m+2,...,N,t =T  j j−1 j m N with variable pitch  0. The time interval step should be selected t ,t so that the moving boundary moves j j j+1 exactly one step along the spatial grid. This approach is known as the grid node front catching method [4]. For the ( ) boundary node of the dynamic stack we use the notation z= f t =i f . j j We approximate equation (7) with a purely implicit scheme vj+1 −vj 1  vj+1 −vj+1 vj+1 −vj+1 i i =  i+1 i − i i−1 , i =1,2,...N −1, (12)  f  i+1 f i f   ( ) ( ) где  =0,5k vj +k vj . i i i−1 Approximation of the initial and boundary conditions (8)-(10) gives ( ) ( ) if p − p v0 =aexpp0 −1 , p0 = p + 1 2 , i =0,1,...m , (13) i i i 2 mf vj+1 = v , (14) 0 2 vj+1 = v , z =i h . (15) i 1 j+1 1 dh f Condition (11) taking into account  we discretize it like this dt  j+1 1 vj+1−vj+1 f ( ) i i−1 +l =0, z =i f . (16) vj+1+a f  j+1 i j+1 Equation (12) is reduced to the form Avj+1 −C vj+1 + B vj+1 = −F , i =1, j , (17) i i−1 i i i i+1 i where A = i , B = i+1 , C =1+ i + i+1 , F =vj . i f 2 i f 2 i f 2 f 2 i i To solve the nonlinear difference problem (12) - (16) at each new time layer t =t iterative processes can be j+1 used. The simplest of these is associated with iterative refinement of the time step  [4]. Let the initial j+1 European International Journal of Multidisciplinary Research 21 https://eipublication.