NUMERICAL SIMULATION OF LAMINAR FORCED CONVECTION AIR FLOW IN A RECTANGULAR VENTURI CHANNEL

Abstract: In this work, a laminar forced convection air flow in a rectangular vertical venturi has been numerically simulated. Mathematical transformation has been used to transform the irregular profile of the venturi wall into straight line. Transfers equations are solved using finite volume method, Gauss and Thomas algorithms. A computing algorithm has been generated for the problem simulation. Hydrodynamics and thermals effects are investigated in detail. Results are presented as velocity and streamlines patterns, and temperature profiles.


INTRODUCTION
Venturi channels are usually used as scrubbers for particles cleaning in gaseous flow or for gas flow rate measurements.As scrubbers they are widely used for particles and gaseous collection from industrial exhaust [1].The large power requirements for operation and large pressure drop across the device are its main drawbacks.These devices consist of channel with three parts: a convergent section, a throat and a divergent section or diffuser.In air scrubbing process, the polluted gas stream is accelerated in the convergent, reaches its maximum velocity in the throat and finally is decelerated in the diverging section.A liquid or an aqueous solution (generally water) is introduced in the venturi to create a spray of droplets for the particles capture.The droplets that captured particles are separated in a device connected to the venturi and the cleaned air is discharged into the atmosphere [2].Based on the pressure drop across the device, venturi channels can also be used for gas flow rate measurements [3].Their high efficiency, coupled with low construction and maintenance cost, has led to many studies.
Venturi scrubbers' modelling is widely studied by many authors [4][5][6][7] but it is only based on the two phase flow modelling.They don't take account the channel geometry.It is well known that CFD procedure in corrugated channels is complex due to the irregular profile of the wall.It is why recently, Igo et al., 2011 [8] proposed a mathematical transformation of the venturi channel into straight channel to study a laminar forced convection heat and mass transfer in a rectangular venturi tube.They showed that the mathematical transformation facilitates the CFD procedures and the heat and mass transfer is influenced by the flow velocity particularly in the venturi throat where the maximum velocity is observed.
As seen, despite their importance, venturi channels have not received much attention in gas flow simulation.Therefore, the objective of this work is to provide robust computing algorithm for flow simulation in venturi channels.

PROBLEM FORMULATION
The venturi channel of length L is composed of two plates of sections lengths (L1, L2, L3, L4, L5).The distance between the plates is 2R in the entrance region.Temperature T o and downward velocity U o of air are supposed to be uniform in the inlet of the venturi channel (Figure 1).The thermophysical properties of air are depending on temperature.
It is assumed that: -The flow is two-dimensional; -The radial pressure gradient component is neglected; -Dufour and Soret effects are neglected; -Transfers are two-dimensional and axisymmetric.In order to avoid the non-uniformity of the mesh spacing along the venturi plane, we use a mathematical transformation which transforms the irregular surface of the plate into a straight line: (x, z) ( , ) such as: a and b are real numbers.Their calculations can be found in the work of Igo et al. 2011 [8].
The dimensionless equations modeling the air flow through the venturi channel in the referential (o, , ) are (Figure 2): -Continuity equation: where , , and B are defined in Table 1 Dh/ U Re (6) where Dh is the hydraulic diameter.

Half dimensionless physical domain
where Q 0 is the inlet air flow rate.

NUMERICAL SIMULATION
The numerical procedure is based on the CFD methods described by Suhas.V. Patankar [9].

Computing algorithm
The algorithm is based on a line by line scanning of the channel from the inlet to the outlet (Figure 3).
The parameters a and b are then readied at each line corresponding to a discrete number I. The equations ( 5) associated to boundaries conditions (11-14) are discretized using an implicit scheme based on the finite volume method [10].The system of algebraic equation deduced from discretization of the radial momentum equation component and energy equations is for each equation tri-diagonal; so it was solved by Thomas algorithm.The discretization of axial momentum equation leads to an algebraic equation system composed of M equations and (M+1) unknowns variables (U and P).Consequently, it was solved with Gauss algorithm.At each time step, algebraic equations are solved by repeatedly sweeping across a uniform grid until convergence is attained.We note t max the simulation time.The solution was deemed converged when the following criterion was satisfied:

RESULTS AND DISCUSSION
In the present study, calculations were performed for: L1 = 0.1 m, L2 = 0.2 m, L3 = 0.15 m, L4 = 0.3 m, L5 = 0.15 m, R = 0.1 m, T o = 293.15K,T w = 298.15K,t max = 300 s, the Reynolds number in the range of 500 to 1500 and the venturi diameter ratio in the range of 0.25 to 1.     Effects of the venturi diameter ratio (parameter ) on streamlines patterns are presented on Figures 10 -12.equal to 1 related to a classical vertical channel, streamlines are parallels to the channel walls as showing in Figure 10.When decreases, channel is progressively transformed to venturi one.We note that streamlines are very close in the convergent and divergent section, and they are merged in the throat particularly for equal to 0.35.The flow velocity in the throat is of course more important as decreases.
Figure 13 show the effects of on the centerline temperature.The augmentation of the centerline temperature at the throat with the decreasing of parameter is observed.In fact, when decreases, the venturi throat wall is more near to the centerline and the heat exchange between the wall and the centerline is more important.

CONCLUSIONS
A laminar forced convection air flow in a vertical rectangular venturi scrubber was numerically investigated in the present study.Mathematical transformation has been used to transform the irregular profile of the venturi wall into straight line.Transfers equations are solved using finite volume method, Gauss and Thomas algorithms.A computing algorithm has been performed for the problem resolution.The effects of the inlet Reynolds number and the ratio between the throat diameter and the venturi diameter on the flow are been investigated in details.
The major results are: -The inlet Reynolds number has a major influence on the flow intensity particularly at the venturi throat; -The streamlines are influenced by the Reynolds number and the venturi diameter ratio; -The centerline temperature is more important in the venturi throat.
The flow velocity and structure, and the centerline temperature profiles are conforms to the venturi channels flows.The computing algorithm can then be used for others investigations in rectangular venturi channels.

Fig. 2 .
Fig. 2. Physical and numerical domain.-Axialmomentum equation, radial momentum equation and energy equation can be resumed as:

Figures 4 -Figures 7 - 9
show the effect of Reynolds number on the velocity patterns.We note that the velocity increases in the convergent section, reach his maximum in the throat and decreases in the divergent section.In the constants sections the velocity is constant.This result is conform to a classical flow in venturi channels.The increasing of Reynolds number increase the global flow velocity in the venturi channel because the Reynolds number is linearly depending on the inlet velocity U o .

Table 1 .
. Definition of , , and B. Pr are respectively the dimensionless viscosity, the dimensionless thermal conductivity, the dimensionless density, the dimensionless specific heat, the Reynolds number and the Prandtl number.