is the material time derivative form.
The virtual mass effect is significant when the discrete phase density is much smaller than the continuous phase density (e.g., for a transient bubble column).
4.2 Interphase exchange coefficients
In multiphase flow, another way of presenting the momentum exchange between the phases is based on the use of the fluid-fluid exchange coefficient Kpq and, for granular flows, the fluid-solid and solid-solid exchange coefficients Kls. Before we discuss about interphase exchange coefficients, we take a short look on momentum equations of fluid-fluid system and fluid-fluid-solid system.
Fluid-Fluid momentum equations
The conservation of momentum for a fluid phase q is
∂
Where−→g is the acceleration due to gravity and other symbols are discussed in equation 36.
Fluid-Solid momentum equations
In the Fluid-Solid momentum equations, the conservation of momentum for a fluid phase is same as equation 52. The momentum equation for sth, solid phase is solved by the following equation,
Whereps is the solid pressure of sth, Kls =Ksl is the momentum exchange coefficient between fluid or solid phasel and solid phases,N is the total number of phases.
4.2.1 Fluid-Fluid exchange coefficient
In general, for fluid-fluid system each secondary phase is assumed to form bubbles or droplets. The exchange coefficient for these types of bubbly, liquid-liquid or gas-liquid mixtures can be written in the following general form as,
Kpq = αqαpρpf τp
(54) In the above equation,f indicates the drag function andτp is the particulate relaxation time, is defined as
τp = ρpd2p
18µq (55)
wheredp is the diameter of the bubbles or droplets of phase p.
Drag functionf depends on a drag coefficient,CD that is based on the relative Reynolds number,Re. It is this drag function that differs among the exchange-coefficient models.
For all these situations, Kpq should tend to zero whenever the primary phase is not present within the domain. To enforce this, the drag functionf is always multiplied by the volume fraction of the primary phase q, from Equation 54.
There are some models for the drag correlation in ANSYS FLUENT 12.1. In the present work, we have used the Schiller and Naumann drag correlation.
f = CDRe
24 (56)
whereCD is calculated with following conditions,
CD =
(24(1 + 0.15Re0.678)/Re Re≤1000
0.44 Re >1000 (57)
In the above equation, Re is the relative Reynolds number. The relative Reynolds number for the primary phaseq and secondary phase p is obtained from,
Re= ρq|−→υp− −→υq|dp µq
(58) The relative Reynolds number for secondary phasep and r is obtained from,
Re= ρrp|−→υr− −→υp|drp
µrp (59)
whereµrp=αpµp+αrµr is the mixture viscosity of the phasesp andr.
There are many correlations for drag force available for fluid-fluid and fluid-solid systems.
Some of them are the Morsi and Alexander model, the symmetric model, etc.[21]. All models are applicable with certain conditions and for certain application.
4.2.2 Fluid-Solid exchange coefficient
The general form of the fluid-solid exchange coefficientKslcan be written in the following form,
Ksl= αsρsf
τs (60)
wheref is the drag function and,τs, is the particulate relaxation time defined as
τs = ρsd2s 18µl
(61) Hereds indicates the diameter of particles of phases. In Equation 60,f include a drag function which is based on the relative Reynolds number (Res). There are some well known correlations for drag function such as the Syamlal-O’Brien model, the Wen and Yu model, the Gidaspow model, etc.[21].
4.2.3 Solid-Solid exchange coefficient
The solid-solid exchange coefficient Kls has very complicated form which is given as follow,
els = coefficient of restitution
Cf r,ls = coefficient of friction between the lth and sth solid-phase particles dl = diameter of the particles of solid l
g0,ls = radial distribution coefficient
5 Two-phase flow simulation in internal-loop airlift reactor
Before two-phase flow simulation for internal-loop airlift reactor is discussed in detail, take a short look for some basic facts of airlift reactor system.
5.1 Airlift reactor morphology
First of all, the term airlift reactor (ALR) covers a wide range of gas-liquid or gas-liquid-solid pneumatic contacting devices that are characterized by fluid circulation in a defined cyclic pattern through channels built specifically for this purpose. In ALRs, the content is pneumatically agitated by a stream of air or sometimes by other gases. In those cases, we can say the name as gas lift reactors. In addition to agitation, the gas stream has the important function of facilitating exchange of material between the gas phase and the medium; oxygen is usually transferred to the liquid, and in some cases reaction products are removed through exchange with the gas phase. The main difference between ALRs and bubble columns lies in the type of fluid flow, which depends on the geometry of the system.The bubble column is a simple vessel into which gas is injected, usually at the bottom, and random mixing is produced by the ascending bubbles.
Figure 10: Different types of ALRs [10].
In the ALR, the major patterns of fluid circulation are determined by the design of the reactor, which has a channel for gas-liquid upflow-the riser-and a separate channel for the downflow. Figure 10 shows the various types of ARL’s
The two channels are linked at the bottom and at the top to form a closed loop. The gas is usually injected near or from the bottom of the riser. The extent to which the gas looses at the top, in the section termed as the gas separator, is determined by the design of this section and the operating conditions. The fraction of the gas that does not disengage, but is entrapped by the descending liquid and taken into the downcomer, has a significant influence on the fluid dynamics in the reactor and hence on the overall reactor performance.
Generally, airlift reactors can be divided into two main types of reactors on the basis of their structure as [10],
• external loop vessels: In which, circulation takes place through separate and distinct ducts.
• baffled (or internal-loop) vessels: In which baffles placed strategically in a single vessel create the channels required for the circulation.
All ALRs, regardless of the basic configuration (external loop or baffled vessel), comprise four distinct sections with different flow characteristics:
• Riser: Riser is the middle part of the reactor. The gas is injected at the bottom of this section, and the flow of gas and liquid is predominantly upward.
• Downcomer: Downcomer is connected to the riser at the bottom and at the top region and is parallel to the riser. The flow of gas and liquid is predominantly downward.
• Base: Usually believed that the base does not significantly affect the overall be-havior of the reactor, but the design of this section can influence gas holdup, liquid velocity, and solid phase flow [10].
• Gas separator: It is a connector of the riser and the downcomer.