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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 from mechanical similarity between RLFR and a manufacturing equipment 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.

The viscous flow between concentric rotating cylinders, called Taylor-Couette flow (TCF) [

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 [

So far, the studies of the flow in RLFR have not been given due attention. However, for the upside down device of

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. [

The paper is organized as follows: Mathematical formulation is given to 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.

Consider the configuration in

where and denote velocity, density, pressure and kinematic viscosity of the fluid, respectively. denote the top end plate, the bottom end plate, inner and outer cone side, respectively. is the angular velocity of the cones. We set Cartesian coordinate system as in

Let be the unit vector along axes of the cylindrical coordinate system, then with Equation (1) expressed in cylindrical coordinates are as follows:

where

The initial and the boundary conditions are presented as follows:

Consider the initial and boundary value problem (5) - (10), by using the following dimensionless quantities

Equations (5)-(8), by removing “*”, can be nondimensionalized into the following ones:

where

and (the Reynolds number).

By removing “*” the dimensionless initial and bounary conditions (9) and (10) resume to following forms:

with The dimensionless form of boundary condition at the top and bottom end plate remains as

Furthermore, we have

thus.

Removing “*” we have.

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:

Letting, together with, we obtain Together with

the dimensionless solving region becomes:

and the boundary conditions (16) can be written as

According to the symmetry of the boundary conditions the pressure and the velocity of the flow are -independent and together with the above dimensionless analysis the solution of the problem has the following dependencies:

where is the so called aspect ratio. Hence we obtain the following theorem:

Theorem 1 Let be the solution of the initial and boundary value problem (11)-(17), then at any time point and each point of the flow field, are determined by the dimensionless parameters: Reynolds number (Re), aspect ratio, radius ratio and cone inclination angle.

We all know that the size of RLFR in laboratory is very tiny, compared with that of the manufacturing equipment (i.e., the actual object), which is geometrically similar to RLFR. Therefore, one obvious question is weather the data obtained by experiment 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 and dimensionless pressure, as well as corresponding dimensionless parameters: and, where stands for the first flow and the second one. We suppose that the two flows have mechanical similarity, i.e.

where, and satisfy the dimensionless Equations (11)-(14) with the boundary condition (16)-(17). In order to ensure that two solutions are equal, the solving region should be the same one, which leads to .

From and satisfying Equation (13) we obtain

Subtracting two equations and taking account of

we have. Suppose

we have. With the zero-boundary condition of and a result of partial differential equation we obtain in D, which leads to. So it follows that the pressureis independent of. Due to the change of centrifugal forces in the -direction, this is obviously not the fact, hence. Summarizing foregoing analysis, we obtain the necessary 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 and, 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:

The nonlinear and the time dependent Equation (1) together with the boundary conditions (3)-(4) and initial conditions are integrated numerically 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 is used for the diffusion terms which are always secondorder accurate. The temporal discretization involves integrating all the terms in the differential equations with a time step. The integration of the transient terms is implicit by using a second-order formulation. The SIMPLE 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 may refer to [

From Sections 3 and 4, we know that the distribution of the pressure and the velocity of the flow depend on and. In this section, we have chosen and as examples to calculate the distribution of pressure and velocity as well as the streamline of the flows.

. It is clear that the distribution curves of the pressure and the velocity are approximately a straight line at. In this case the streamline of the flow is given in

where.

bution curve of the pressure and the velocity near the bottom end plate (see:

In this work dimensional and mechanical similarity analysis for the flow in rotating liquid film reactor (RLFR) are presented. It is proved that at each point of the flow field the dimensionless quantity of pressure and velocity is completely determined by parameters: and. Moreover, between RLFR and a manufacturing equipment 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 in

In [

The results in this paper imply that the particle size and the particle size distribution depend not only on and (included in) but also on parameters and. How the particle size and particle size distribution depend on and is an interesting problem which is worth studying both experimentally and numerically. Moreover, our analysis reveals that and are not independent, 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 experiment and numerical simulation.