Recent Advances in Modeling Gas-Particle Flows

Abstract

This chapter is concerned with the mathematical modeling of dense fluidized suspensions and focuses on the so-called Eulerian or multifluid approach. It introduces newcomers to some of the techniques adopted to model fluidized beds and to the challenges and long-standing problems that these techniques present. After introducing the principal approaches for modeling fluid-solid systems, we focus on the multifluid, overviewing the main averaging techniques that consent to describe granular media as continua. We then derive the Eulerian equations of motion for fluidized powders of a finite number of monodisperse particle classes, employing volume averages. We present the closure problem and overview constitutive relations for modeling the granular stress and the interaction forces between the phases. To conclude, we introduce the population balance modeling approach, which permits handling suspensions of particles continuously distributed over the size and any other property of interest.

Appendix

Fluid-Phase Volume Average of Point Variable Spatial Derivatives

We intend to derive an expression for the fluid-phase volume average of point variable spatial derivatives; to this end, we start by considering the derivative:

$$ {\partial}_a\left[\varepsilon \left(\boldsymbol{x}, t\right){\left\langle \zeta \right\rangle}_e\left(\boldsymbol{x}, t\right)\right] $$

(118)

Now, using the definition of fluid-phase volume average given in Eq. 17 and the derivation chain rule, we write the quantity above as:

$$ \begin{array}{l}{\partial}_{x_a}{\int}_{\Lambda_e}\zeta \left(\boldsymbol{z}, t\right)\psi \left(\left|\boldsymbol{x}-\boldsymbol{z}\right|\right) d\boldsymbol{z}={\int}_{\Lambda_e}\zeta \left(\boldsymbol{z}, t\right){\partial}_{x_a}\psi \left(\left|\boldsymbol{x}-\boldsymbol{z}\right|\right) d\boldsymbol{z}=-{\int}_{\Lambda_e}\zeta \left(\boldsymbol{z}, t\right){\partial}_{z_a}\psi \left(\left|\boldsymbol{x}-\boldsymbol{z}\right|\right) d\boldsymbol{z}\\ {}\\ {}\operatorname{}\operatorname{}={\int}_{\Lambda_e}\left[{\partial}_{z_a}\zeta \left(\boldsymbol{z}, t\right)\right]\psi \left(\left|\boldsymbol{x}-\boldsymbol{z}\right|\right) d\boldsymbol{z}-{\int}_{\Lambda_e}{\partial}_{z_a}\left[\zeta \left(\boldsymbol{z}, t\right)\psi \left(\left|\boldsymbol{x}-\boldsymbol{z}\right|\right)\right] d\boldsymbol{z}\end{array} $$

(119)

For the first integral, we can write:

$$ {\int}_{\Lambda_e}\left[{\partial}_{z_a}\zeta \left(\boldsymbol{z}, t\right)\right]\;\psi \left(\left|\boldsymbol{x}-\boldsymbol{z}\right|\right) d\boldsymbol{z}=\varepsilon \left(\boldsymbol{x}, t\right){\left\langle {\partial}_a\zeta \right\rangle}_e\left(\boldsymbol{x}, t\right) $$

(120)

For the second, the Gauss theorem allows writing:

(121)

where ∂Λ _x is the surface bounding the domain containing the mixture and n _a (x, t) is the ath component of the unit vector normal to ∂Λ _x pointing away from the mixture. If the shortest distance from the generic point x ∈ ∂Λ _x is considerably larger than the weighting function radius, the first term of the right-hand side of the equation above is much smaller than the second. Neglecting it, we obtain Eq. 25.

Fluid-Phase Volume Average of Point Variable Time Derivatives

Similarly, to derive an expression for the fluid-phase volume average of point variable time derivatives, we start by considering the derivative:

$$ {\partial}_t\left[\varepsilon \left(\boldsymbol{x}, t\right){\left\langle \xi \right\rangle}_e\left(\boldsymbol{x}, t\right)\right] $$

(122)

Using the definition of fluid-phase volume average given in Eq. 17 and then applying the Leibnitz theorem allows writing this as:

(123)

The integral on ∂Λ _x can be neglected for the same reasons given in Appendix “Fluid-Phase Volume Average of Point Variable Spatial Derivatives.” Now, using the definition of fluid-phase volume average given in Eq. 17, we have:

$$ {\int}_{\Lambda_e}\left[{\partial}_t\zeta \left(\boldsymbol{z}, t\right)\right]\psi \left(\left|\boldsymbol{x}-\boldsymbol{z}\right|\right) d\boldsymbol{z}=\varepsilon \left(\boldsymbol{x}, t\right){\left\langle {\partial}_t\zeta \right\rangle}_e\left(\boldsymbol{x}, t\right) $$

(124)

Obtaining Eq. 26 is then immediate. Note that if Λ _x is time independent, u(z, t) = 0 on ∂Λ _x , and therefore the last integral on the right-hand size of Eq. 123 rigorously vanishes.

Particle-Phase Volume Average of Point Variable Time Derivatives

We intend to derive an expression for the particle-phase volume average of point variable time derivatives; in this case, we consider the derivative:

$$ {\partial}_t\left[{n}_r\left(\boldsymbol{x}, t\right){\left\langle \zeta \right\rangle}_p^r\left(\boldsymbol{x}, t\right)\right] $$

(125)

Now, employing the definition of particle-phase volume average given in Eq. 21, we can express the partial derivative above as:

(126)

From the definition of particle-phase volume average, it is:

(127)

Applying the derivation chain rule yields:

(128)

having used again the partial derivatives commutative property and the definition of particle-phase average. Replacing these last two results in Eq. 126 yields Eq. 39.