## Solid–Liquid Flow

**Introduction: **Solid–liquid system is simpler to analyze than the liquid-liquid system. In solid–liquid, the effect of the surface tension is very minimal and can be ignored. Thus, in this discussion, it is assumed that the surface tension is insignificant compared to the gravity forces. The word “solid” is not really mean solid but a combination of many solid particles. Different combination of solid particle creates different “liquid.” There for, there will be a discussion about different particle size and different geometry (round, cubic, etc.). The uniformity is categorizing the particle sizes, distribution, and geometry. For example, analysis of small coal particles in water is different from large coal particles in water.

The density of the solid can be above or below the liquid. Consider the case where the solid is heavier than the liquid phase. It is also assumed that the “liquids” density does not change significantly and it is far from the choking point. In that case there are four possibilities for vertical flow:

1. The flow with the gravity and lighter density solid particles.

2. The flow with the gravity and heavier density solid particles.

3. The flow against the gravity and lighter density solid particles.

4. The flow against the gravity and heavier density solid particles.

All these possibilities are different. However, there are two sets of similar characteristics, possibility, 1 and 4 and the second set is 2 and 3. The first set is similar because the solid particles are moving faster than the liquid velocity and vice versa for the second set (slower than the liquid). The discussion here is about the last case (4) because very little is known about the other cases.

**Solid Particles with Heavier Density: **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. A particle in a middle of the vertical liquid flow experience several forces. The force balance of spherical particle in field viscous fluid (creeping flow) is: 1. gravity and buoyancy forces 2. Drag forces

Where CD1 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

Inserting equating (1) into equation (2) become

Equation relates the liquid velocity that needed to maintain the particle “floating” to the liquid and particles properties. The drag coefficient, CD1 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

In transitional region 1 < Re < 1000

For larger Reynolds numbers, the Newton’s Law region, CD1, is nearly constant as

In most cases of solid-liquid system, the Reynolds number is in the second range9. 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.