# 13.8.1: Solid Particles with Heavier Density \(\rho_S>\rho_L\)

Solid–liquid flow has several combination flow regimes. When the liquid velocity is very small, the liquid cannot carry the solid particles because there is not enough resistance to lift up the solid particles. The force balance of spherical particle in field viscous fluid (creeping flow) is

\[

\overbrace{ \dfrac{\pi\, D^3 \,g\,(\rho_S - \rho_L)}{6}}

^{\text{ gravity and buoyancy forces} } =

\overbrace{ \dfrac

```
Callstack:
at (Textbook_Maps/Map:_Fluid_Mechanics_(Bar-Meir)/13:_Multi–Phase_Flow/13.8:_Solid–Liquid_Flow/13.8.1:_Solid_Particles_with_Heavier_Density_\(\rho_S%3E\rho_L\)), /content/body/p[1]/span, line 1, column 4
```

^{\text{ drag forces} }

\label{phase:eq:forceSP} \tag{47}

\]

Where \({C_D}_{\infty}\) is the drag coefficient and is a function of Reynolds number, \(Re\), and \(D\) is the equivalent radius of the particles. The Reynolds number defined as

\[

Re = \dfrac{{U_L}\, D\,\rho_L}{\mu_L}

\label{phase:eq:Re} \tag{48}

\]

Inserting equating (48) into equation (47) become

\[

\overbrace{f(Re)}^{{C_D}_{\infty}({U_L})} {U_L}^2 =

\dfrac{ 4 \, D\,g\,(\rho_S - \rho_L)}{3\,\rho_L}

\label{phase:eq:Uparticle} \tag{49}

\]

Equation (49) relates the liquid velocity that needed to maintain the particle "floating'' to the liquid and particles properties. The drag coefficient, \({C_D}_{\infty}\) is complicated function of the Reynolds number. However, it can be approximated for several regimes. The first regime is for \(Re<1\) where Stokes' Law can be approximated as

\[

{C_D}_{\infty} = \dfrac{24}{Re}

\label{phase:eq:Stokes} \tag{50}

\]

In transitional region 1\(<\)Re\(<\)1000

\[

{C_D}_{\infty} = \dfrac{24}{Re}

\left( 1+ \dfrac{1}{6}\,Re^{2/3} \right)

\label{phase:eq:transitionalCD} \tag{51}

\]

For larger Reynolds numbers, the Newton's Law region, \({C_D}_{\infty}\), is nearly constant as

\[

{C_D}_{\infty} = 0.44

\label{phase:eq:last} \tag{52}

\]

In most cases of solid-liquid system, the Reynolds number is in the second range. For the first region, the velocity is small to lift the particle unless the density difference is very small (that very small force can lift the particles). In very large range (especially for gas) the choking might be approached. Thus, in many cases the middle region is applicable. So far the discussion was about single particle. When there are more than one particle in the cross section, then the actual velocity that every particle experience depends on the void fraction. The simplest assumption that the change of the cross section of the fluid create a parameter that multiply the single particle as

\[

\left.{C_D}_{\infty}\right|_{\alpha} =

{C_D}_{\infty} \, f(\alpha)

\label{phase:eq:CDepsilon} \tag{53}

\]

When the subscript \(\alpha\) is indicating the void, the function \(f(\alpha)\) is not a linear function. In the literature there are many functions for various conditions. Minimum velocity is the velocity when the particle is "floating''. If the velocity is larger, the particle will drift with the liquid. When the velocity is lower, the particle will sink into the liquid. When the velocity of liquid is higher than the minimum velocity many particles will be floating. It has to remember that not all the particle are uniform in size or shape. Consequently, the minimum velocity is a range of velocity rather than a sharp transition point.

*Fig. 13.8 The terminal velocity that left the solid particles.*

As the solid particles are not pushed by a pump but moved by the forces the fluid applies to them. Thus, the only velocity that can be applied is the fluid velocity. Yet, the solid particles can be supplied at different rate. Thus, the discussion will be focus on the fluid velocity. For small gas/liquid velocity, the particles are what some call fixed fluidized bed. Increasing the fluid velocity beyond a minimum will move the particles and it is referred to as mix fluidized bed. Additional increase of the fluid velocity will move all the particles and this is referred to as fully fluidized bed. For the case of liquid, further increase will create a slug flow. This slug flow is when slug shape (domes) are almost empty of the solid particle. For the case of gas, additional increase create ``tunnels'' of empty almost from solid particles. Additional increase in the fluid velocity causes large turbulence and the ordinary domes are replaced by churn type flow or large bubbles that are almost empty of the solid particles. Further increase of the fluid flow increases the empty spots to the whole flow. In that case, the sparse solid particles are dispersed all over. This regimes is referred to as Pneumatic conveying (see Figure 13.9).

*Fig. 13.9 The flow patterns in solid-liquid flow.*

One of the main difference between the liquid and gas flow in this category is the speed of sound. In the gas phase, the speed of sound is reduced dramatically with increase of the solid particles concentration (further reading Fundamentals of Compressible Flow'' chapter on Fanno Flow by this author is recommended). Thus, the velocity of gas is limited when reaching the Mach somewhere between \(1/\sqrt{k}\) and \(1\) since the gas will be choked (neglecting the double choking phenomenon). Hence, the length of conduit is very limited. The speed of sound of the liquid does not change much. Hence, this limitation does not (effectively) exist for most cases of solid–liquid flow.

### Contributors

Dr. Genick Bar-Meir. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or later or Potto license.