^{1}

^{*}

^{1}

^{1}

The simplified momentum equations of the two-phase flow have been adopted as the basic assumptions in this study. For vessels of small diameter, the shear stress becomes important and the friction pressure drop proposed by Ergun considers this effect by involving the wall effect. By replacing the Ergun pressure drop and the first order velocity term for particles drag model in the momentum equations, the relation for the drag coefficient versus the volume fraction is obtained. The calculus of variations is used with certain restriction for extremization of this drag coefficient. An analytical correlation for the drag coefficient is obtained depending on the volume fraction of “fluid particles”. The drag function obtained in previous studies does not match with the empirical data in the bed volume fraction range of [ 0.45 to 0.59 ] . Therefore, the function is modified and the results are better adjusted with the empirical data.

Fluidized bed systems are used in many commercial processes; including mixing particles, particle separation, solids coating, power generation through combustion or gasification and particle drying. A proper understanding of the hydrodynamic behavior of the fluidized bed system is needed for the design and to scale up the new efficient reactor [

New insights have been proposed in recent literature based on the combination of CFD method and empirical correlations. Sun Liyan et al., [

The approach in which effective correlations for fluidization process were obtained analytically was introduced for the first time by the novel study of Grbavcic et al., [

The equation obtained by Grbavcic et al., [

To fix this problem in this study, the pressure drop equation has been replaced by Ergun’s correlation. For vessels of small diameter, the wall effect becomes important and influences the bed voidage. The frictional pressure drop, proposed by Ergun, is proper [

An acceptable adjustment between the analytical and experimental data is obtained by considering the combination of the first and second order terms of polynomials in the drag equation. The process of obtaining relations in this study includes dividing the fluidization process into two intervals of voidage of [0.45, 0.59] and [0.59, 1] and defining two functions with different criteria for these two intervals. To achieve the required quantities in the first range of voidage, the governing momentum equations for the two-phase of the fluid-solid are written, and the drag term is replaced with the first degree velocity term. Linear velocity term is removed from the drag equation to find the required criteria in the second range of the voidage. In this case, for the first range of voidage, a new equation is introduced but in the second range the equation obtained in the studies of Littman and Morgan, [

To obtain the required parameters in the fluidization process, the conservation equations of mass and momentum in the vertical transport of fluids and solids with the assumption of no-acceleration motion are used. Also, the motion of fluid and solids is considered to be one-dimensional, laminar and uniform in the steady state condition with no wall shear stress.

The drag coefficient is a dimensionless quantity which is used to calculate the drag force acting on the particles suspended in the fluidized bed. This factor depends on voidage of the particles in the fluidization process.

During the fluidization process, by accelerating the fluid, the voidage of the solids increases and it changes the drag force acting on the solid particles. Therefore, the relationship between the drag coefficient and the voidage of particles is the first important parameter in this research and will be found subsequently.

To find this relationship and for simplification, the one-dimensional conservation equations of mass and momentum for vertical transport of fluid and particles by assuming no acceleration in movement should be expressed and the relation for drag force should be replaced. The conservation equations of mass for two-phases of fluid and solid particles will be generally in form of relations 1) and 2). In these equations, the air moves as the continuous phase and suspended particles in the fluidized bed are described by the Eulerian approach. Accordingly, conservation equations of mass and momentum for the two-phase flow of the particle and air are written in the form of Eulerian-Eulerian, outlined as follows:

Fluid phase d d x [ ε ρ f u f ] = 0 (1)

Solidphase d d x [ ( 1 − ε ) ρ s u s ] = 0 (2)

Momentum equations representing the motion of the fluid and solid phases are written as relations (3) and (4):

Fluid : ε ρ f u f d u f d x = − ε d p d x − ε ρ f g − 4 τ w f D b e d + DragForce (3)

Solid : ( 1 − ε ) ρ s u s d u s d x = − d σ d x − ( 1 − ε ) ( ρ s − ρ f ) g − 4 τ w s D b e d − Drag Force (4)

Regardless of the wall shear stress and ignoring the stress of particles on each other, or in other words, assuming the two-way coupling, the momentum equation is reduced and rewritten as follows:

Fluidphase ε ρ f u f d u f d x = − ε d P d x + DragForce (5)

where the average normal pressure is introduced in the form of Equation (6):

P = p + ρ g z (6)

In the present study, the drag force is considered a linear function (see Equation (7)). In the research of Grbavcic et al., [

Davidson [

Drag Force = β B ( u f − u s ) Current study (7)

Drag Force = β B ( u f − u s ) 2 Grbavcic et al. [

Now, it is desirable to detect a relation for the drag coefficient from the momentum equation. In order to achieve this aim, the pressure drop equation should be placed in Equation (5). In this study, the pressure drop term is replaced with Ergun’s equation. Ergun’s equation for pressure drop of the solid-fluid phase is introduced in Equation (9). In the research of Grbavcic et al., [

∂ P ∂ x = k 1 ( 1 − ε ) 2 ε 3 ( u f − u s ) + k 2 ( 1 − ε ) ε 3 ( u f − u s ) 2 Current study (9)

∂ P ∂ x = ( 1 − ε ) ( ρ s − ρ f ) g Grbavcic et al. [

where k 1 and k 2 are constants.

The relation for the drag coefficient in terms of voidage is obtained (Equation (12)) by replacing Ergun’s equation (Equation (9)) in the momentum equation

(Equation (5)) for non-accelerating beds ( d u f d x = 0 ). Similarly, by putting the

Darcy correlation (Equation (10)) in the momentum equation (Equation (5)), the relationship between the drag coefficient and the voidage in the research of Grbavcic et al., [

β ( ε ) = k 1 ( 1 − ε ) 2 ε 2 + k 2 U ( 1 − ε ) ε 3 Current study (11)

β ( ε ) = ε 3 ( 1 − ε ) ( ρ p − ρ f ) g U 2 Grbavcic et al. [

where superficial velocity is defined as:

U ( ε ) = ε ( u s − u f ) (13)

After deriving the drag coefficient, it is necessary to extremize this function. In the following sections, the drag coefficient is extremized, the relationship between fluidization velocity and voidage is determined, and the results are discussed.

In order to be able to extremize this drag coefficient function (in terms of voidage using the calculus of variations), it is necessary to change it to the shortest path problem. In the fluidization process, the physics of the problem dictates that for different voidages, the drag force acting on the suspended solids should be at their least magnitude. It is physically equivalent to a rope that is attached from both ends to two points. In this situation, its potential energy will stand at the lowest possible level.

To formulate this physical sense mathematically, it is necessary to define the drag coefficient and voidage as dimensionless parameters and then normalize them in the interval of [0, 1]. Doing this on the drag coefficient and voidage leads to emergence of two quantities x and y, in the form of relations (14) and (15).

x = ε − ε m f 1 − ε m f (14)

y = 1 − β β m f (15)

Since the fluidized bed is addressed between the minimum and maximum fluidity, it is necessary to consider the drag coefficient in the minimum fluidity as the beginning of the fluidization process. This quantity is obtained by considering the voidage coefficient of the minimum fluidity in Equation (11) Equation (12).

β m f = k 1 ( 1 − ε m f ) 2 ε m f 2 + k 2 U m f ( 1 − ε m f ) ε m f 3 Current study (16)

β m f = ε m f 3 ( 1 − ε m f ) ( ρ s − ρ f ) g U m f 2 Grbavcic et al. [

Now, after the definition of the normalized quantities, it is necessary to determine the required functional of the calculus of variations. It can be perceived from a mathematical viewpoint that, among all the curves available for the

normalized drag coefficient (y), the area under the curve of I = ∫ y d x is

constant and this relation is considered a restriction for the problem. Then:

I = ∫ 0 1 y d x = 1 1 − ε m f ∫ ε m f 1 ( 1 − β β m f ) d ε (18)

The next step is to determine the conditions in which the least possible

lengths of the normalized drag curves y = 1 − β β m f occur. This condition is

mathematically defined as Equation (19):

∫ 0 1 1 + y ′ 2 d x = min (19)

The drag coefficient curve will change the maximum between the first and end point. Therefore, the functional of this problem is defined as Equation (20):

F ( y , y ′ ) = [ 1 + ( y ′ ) 2 ] 1 2 + λ y (20)

In Equation (20), the parameter λ is called the Lagrange multiplier.

Now, after definition of the functional of the problem, its extremum is obtained using the calculus of variations. Referring to the approach of the calculus of variations of Gelfand and Fomin, [

d d x ( ∂ F ∂ y ′ ) − ∂ F ∂ y = 0 (21)

By substituting the relation F , which was obtained in Equation (20), into Equation (21) and solving it, the extremum value of y is obtained as:

y = c 2 − 1 λ [ 1 − ( λ x + c 1 ) 2 ] 1 2 (22)

In the relation λ , c 1 and c 2 are constant values and for detecting them, three boundary conditions are required. Relations (23), (24) and (25) are referred to as boundary conditions.

y ( 0 ) = 0 (23)

y ( 1 ) = 1 (24)

y ′ ( 1 ) = − ( 1 − ε m f β m f ) d β d ε | ε → 1 (25)

The boundary condition expressed in Equation (25) will be in the form of the following relations by using Equations (14) and (15):

y ′ ( 1 ) = k 2 ε m f 3 U t k 1 ε m f ( 1 − ε m f ) + k 2 U m f Current study (26)

y ′ ( 1 ) = ( U m f / U t ) 2 ε m f 3 Grbavcic et al. [

where U t describes terminal velocity when ε → 1 . Using these boundary conditions, the constant values of c 2 , and c 1 are obtained:

λ = 1 − c 1 2 − c 1 (28)

λ = 1 − c 1 2 − c 1 c 2 = 1 − c 1 2 1 − c 1 2 − c 1 (29)

And,

c 1 = [ 1 + ( k 2 ε m f 3 U t k 1 ε m f ( 1 − ε m f ) + k 2 U m f ) 2 ] − 1 2 Current study (30)

c 1 = [ 1 + ( U m f 2 U t 2 ε m f 3 ) 2 ] − 1 2 Grbavcic et al. [

By substituting the values of y and x into the relation (22), the extremum of the drag coefficient equation in terms of voidage is obtained as Equation (32) [

β β m f = ( 1 − c 2 ) + 1 λ 1 − ( λ ( ε − ε m f 1 − ε m f ) + c 1 ) 2 (32)

This dimensionless drag coefficient is the function of the bed fluidization velocity in the balanced bed situation.

As mentioned before, another important parameter in the fluidization process is the fluidization velocity for different voidages. To find this parameter, the relations obtained for the drag coefficient in terms of voidage are used. After simplifying relations (11) and (12), the connection between the fluidization velocity and voidage is obtained as Equations (33) and (34):

U ( ε ) = β ε 3 k 2 ( 1 − ε ) − k 1 ε ( 1 − ε ) k 2 Current study (33)

U ( ε ) = U m f ε 3 ( 1 − ε ) ε m f 3 ( 1 − ε m f ) β m f β Grbavcic et al. [

In this section, the relations which were mentioned previously are validated with the empirical data. To achieve this purpose, the empirical research data of Grbavcic et al., [

The data obtained from the experiment of Grbavcic et al., [

As can be seen in

Bed voidage

In the voidage range of [0.49, 0.59], the relation derived in the present study represents a closer approximation of the empirical data and is specified in

In the voidage range of [0.45, 0.59], the measured pressure gradients in the research of Grbavcic et al., [

The assumptions of Ergun pressure drop and the linear term of superficial velocity in the drag model are more consistent with reality. However, in the voidage range of [0.59, 1], the assumptions of Darcy pressure drop and non-linear term of superficial velocity, which are used in the investigation of Grbavcic et al., [

Nazif, H.R., Javadi, A.H. and Fallahnezhad, N. (2018) Predicting the Two-Phase Liquid-Solid Drag Model Using the Calculus of Variation. Journal of Applied Mathematics and Physics, 6, 103-113. https://doi.org/10.4236/jamp.2018.61010