Dimensional and Mechanical Similarity Analysis of the Flow in Rotating Liquid Film Reactor

A rotating liquid film reactor (RLFR) is a device of two coaxial rotating conical cylinders with the inner cone rotating and the outer one stationary. A complete mathematical model for the flow between the conical cylinders is built and a dimensional analysis is carried out. It is proved that at each point of the flow field the dimensionless pressure and velocity of the flow are determined by parameters: Reynolds number (Re), aspect ratio (Γ), radius ratio (η) and wall inclination angle (α). Furthermore, a sufficient and a necessary condition are derived for mechanical similarity between RLFR and a manufacturing equipment being geometrically similar to RLFR. Finally, a numerical simulation for the distribution of pressure and velocity is performed. The results may provide a theoretical basis for experiment method and numerical simulation of the flow in a RLFR-like device.


Introduction
The viscous flow between concentric rotating cylinders, called Taylor-Couette flow (TCF) [1], has been studied by numerous research workers for over 300 years (see [2] and the references therein).This is a classical system to investigate properties of flow driven by rotation.So far, an abundance of experimental and numerical results on TCF are available.
Recently, another configuration related to rotating body has aroused the interest of researchers.A few chemists have constructed a precipitation reactor, called rotating liquid film reactor, which is used as a reactor to prepare new functional nano-particles.It is found that, compared with conventional precipitation reactors, the particles produced in the RLFR are smaller in size and narrower in size distribution [3].The RLFR consists of two coaxial conical cylinders with the inner cone rotating and the outer one stationary, as displayed in Figure 1.The walls of cones are parallel giving a constant width of the gap for RLFR.The gap between two cones is filled with reactants which are usually considered as a viscous incompressible fluid.In order to understand the effect of the RLFR on precipitation, it is necessary to investigate the properties of the flow in the gap.This can be done by experimental and numerical simulation.However, the pro-perties of the fluid flow depend on many parameters, e.g. (rotor's angular velocity),  (viscosity), H (cone's height),  (inclination angle of the cone), 1 (bottom radius of the inner cone), 2 ( bottom radius of the outer cone).In order to reduce the dimension of the problem, it is necessary to conduct a dimensional analysis for this problem.The dimensional analysis is also a basis for experimental and numerical simulation.Moreover, the RLFR is very tiny, compared with the manufacturing equipment.The reactor has a height of 17 mm, upper diameter of 40.8 mm, bottom diameter of 50 mm and inclination angle of 75 deg.The gap width can be adjusted in the range of 0.1 mm -0.5 mm by moving the stator (outer cone).The rotor (inner cone) can rotate at variable speeds up to 5000 rpm by a adjustable-speed motor.One may ask that, to what extent, the flow properties obtained by experimental and numerical simulation for the RLFR could reflect that of the real flow in manufacturing equipment.This is a problem of mechanical similarity.Try to reduce the dimension and to derive mechanical similarity of this problem are the motivations of our paper.As far as we know, there is no paper dealing with this problem.

R R
So far, the studies of the flow in RLFR have not been given due attention.However, for the upside down device of Figure 1, the properties of the flow in the gap with the inner one rotating and the outer one at rest have been experimentally studied by Wimmer [4,5] and numerically by Noui-Mehidi et al. [6,7], Xu et al. [8,9] and Li et al. [10].In [4,5] Wimmer studied the stability of basic flow and the transition to Taylor vortices, as well as the occurrence of Taylor vortex at different geometries.Noui-Mehidi et al. [6,7] investigated the effect of wall alignment on the flow and the stability of the helical flow, as well as the transition to turbulence.Xu et al. [8] studied the dependence of the velocity and the pressure magnitude on the cone inclination.In [9] Xu et al. showed that the behavior of the flow is dominated by a competition between the meridional flow and the radial flow.Li et al. [10] discussed the local maximum value of velocity and the local maximum of pressure, as well as the transition to Taylor vortices.
The paper is organized as follows: Mathematical formulation is given Section 2. Section 3 and 4 are devoted to dimensional analysis and mechanical similarity analysis, respectively.A numerical simulation for the pressure and the velocity is implemented in Section 5. Finally, the conclusions and some discussion are made in Section 6.

Mathematical Formulation
Consider the configuration in Figure 1 in which the inner cone and the outer one have the same inclination angle  .The gap between two cones is filled with a viscous incompressible fluid.The inner cone rotates at angular velocity  and the outer one is at rest.It is assumed that the top and the bottom end plate are rigid and the boundary condition at the cone side is no-slip.Then the governing equations with the initial and the boundary conditions are as follows:  is the angular velocity of the cones.We set Cartesian coordinate system as in Figure 1.Let z  be the unit vector along axes of the cylindrical coordinate system, then with where The initial and the boundary conditions are presented as follows:

Dimensional Analysis of the Model
Consider the initial and boundary value problem ( 5) - (10) by using the following dimensionless quantities 5)-( 8), by removing "*", can be nondimensionalized into the following ones: where (the Reynolds number).
By removing "*" the dimensionless initial and bounary conditions ( 9) and (10) resume to following forms: The dimensionless form of boundary condition at the top and bottom end plate remains as Furthermore, we have Removing "*" we have 1 The behavior of fluid in the RLFR is governed by the Equations ( 11)-( 14) with the initial and the boundary conditions (15)-(17).The solving region is: the dimensionless solving region becomes: and the boundary conditions (16) can be written itions, According to the symmetry of the boundary cond the pressure and the velocity of the flow are  -independent and together with the above dimensi analysis the solution of the problem has the following de onless pendencies:

Mechanical Similarity Analysis
We all know that the size of FR in laboratory is very tiny, compared with that of th RL e manufacturing equipment lly similar to whether the (i.e., the actual object), which is geometrica RLFR.Therefore, one obvious question is data obtained by experimental and numerical simulation for the flow in RLFR can characterize the properties of real flow in the actual object.That is the question we like to answer in this section.We now consider the necessary condition for mechanical similarity of two flows.Assuming that there are two flows with dimensionless velocity   , , .Summarizing foregoing analysis, we obtain ry condition for the mechanical similarity of the two flows as following: Moreover, with the uniqueness of the solution the sufficient condition for mechanical similarity of the two flows is at hand.Now we formulate our results in following theorem: Theorem 2 Assuming that we have two flows with parameters , , , then the necessary condition for mechanical similarity of the two flows is: Moreover, if the Equations (11)-( 14) have unique solution, the sufficient condition for mechanical similarity of the two flows is:

A Numerical Simulation of Pressure and Velocity Distribution
The nonlinear and the time dependent Equation (1) tol The initial conditions for the two flows are the same, The boundary conditions for two flows are the same.

Re Re
gether with the boundary conditions ( 3)-( 4) and initia conditions 0 0 m u are integrated numericall using the finite volumes method.For the convection terms in equations, a second-order upwind scheme is used to interpolate the face values of the various quantities from the cell center values.Central difference quotient which is always second-order accurate is used for the diffusion terms.The temporal discretization involves integrating all the terms in the differential equations with a time step t  .The integration of the transient terms is implicit by using a second-order formulation.The SIM-PLE algorithm is used to link pressure and velocity.The discretized equations are then solved sequentially using a segregated solver.For the justification of the numerical method, one ay refer to [8].
From Sections 3 and 4, we know that the distribution of the pressure and the velocity of the flow depend on , , Re   and  .In this section, we have chosen  the pr sure and the velocity for some parameistributio nd the velocity is calculated along a line from the midpoint of the bottom end plate to the midpoint of th ate in the xoy -plane for 0 x  .In cylindrical coordinate system, the line is given by It is clear that the distribution curves of the pressure and the velocity are approximately a straight line at 12.5 Re  . In this case the streamline of the flow is given in

Conclusions and Discussion
In this work dimensional and mechanical similarity analysis for the flow in rotating liq the dimensionless quantity of pressur completely determined by parameters: , , Re   and  .
Moreover, between RLFR and a manufacturing equipment being geometrically similar to RLFR, a necessary condition and a sufficient condition for mechanical similarity are derived.Finally, as examples, numerical simulation for some parameters is implemented.The distribution of pressure for 12.5 Re  in Figure 2(a) looks like a horizontal straight line, and the actual calculation result is not the case.The reason is that the pressure amplitude for 192.5 Re  is much larger than the one for 12.5 Re  , and the pressure in this figure is nondimensionalized by the largest pressure.
It is showed in [3] that increasing the speed of the rotor    in the RLFR or increasing the rotor-stator gap   d lted in a decrease in particle size and narrower particle size distribution.The experiment in [3]  R .The results in this paper imply that the particle size and the particle size distribution depend not only on  and d (included in Re ), but also on parameters ,   and  .How the particle siz and particle size distributio e n depend on ,   and  is an interesting problem which is worth studying both experimentally and nu-merically.Moreover, our analysis reveals that  and d are not indep ent, and they may substitute each other, at least from qualitative point of view.This assertion requires experimental verification.
The results obtained in this paper provide a theoretical basis for further study of the reactor by experimental and numerical simulation.

Figure 1 .
Figure 1.The sketch of the coaxially rotating liquid film reactor.
density, pressure and kinematic viscosity of the fluid, respectively.the top end plate, the bottom end plate, inner and outer cone side, respectively.
Equation (1) expressed in cylindrical coordinates are as follows: the distribution of pressure and ve bution of es ters of the system.The d n of the pressure a e top end pl locity as well as the streamline of the flows.

Figure 2
illustrates the numerical results of the distri-

Figure 3 .
Figure 3. Front view of streamlines of the flow for ., .  12 5 0 8   and , (a) at Re = 12.5 Taylor vored; (b) at 112.5 the first Taylor vortex appears; air of vortices.thevelocity ne ottom turbulence had big effect on particle size and par icle size distribution.The turbulent effects were d tly related to  and d .There was no dis-cussion on the effects of parameters end G.I. Taylor, "Stability of a Viscous Liquid ed between Two Rotating l Transactions Royal 223, No .