**Dilute vs dense Phase**

The boundary between dilute phase and dense phase flow is not clearly defined, and there are no universally accepted text book definitions of dense phase and dilute phase transport.

Dense phase flow from dilute phase flow can be identified using four different definitions

(Konrad, 1986 ‘Dense phase conveying: a review’, Powder Technol., 49, 1–35.):

- on the basis of solids volumes concentration;
- on the basis of solids/air mass flow rates;
- dense phase flow exists when at some point the solids completely fill the cross section of the pipe;
- dense phase flow exists when reverse flow of solids occurs (vertical flow), or when the gas velocity is not sufficient to sustain all particles in suspension mode (horizontal flow).

**Dilute vs dense Phase: choking and saltation velocity**

**Choking velocity**definition is used to set the boundary between dilute phase and dense phase transport in vertical pipelines.**Saltation velocity**is used to set the boundary between dilute phase and dense phase transport in horizontal pipelines.

These definitions are often used in the literature and are quantified through relationships involving gas velocity, solids mass flow rate and pressure drop.

**Fundamentals: Gas and particle velocities mathematic**

**U _{fs} = [volumetric flow rate of gas] / [cross sectional area of pipe] = Q_{g} / A** (1)

The superficial solids (particles) velocity is defined as:

**U _{ps} = [volumetric flow rate of solids] / [cross sectional area of pipe] = Q_{p} / A** (2)

where subscript “s” denotes superficial and subscripts “f” and “p” are referred to the fluid and particles respectively.

The fraction of pipe cross-sectional area available for the flow of gas is assumed to be equal to the volume fraction occupied by gas, i.e., the voidage or void fraction e .

The fraction of pipe area available for the flow of solids is therefore (1 – **ε** ).

Actual gas velocity is then,

**U _{f} = Q_{f} / (A ε )** (3)

and actual particle velocity,

**U _{p} = Q_{p} / [A (1 - ε)]** (4)

**Fundamentals: Gas and particle velocities**

Superficial velocities are related to actual velocities by the equations:

**U _{f} = U_{fs} / ε** (5)

**U _{p} = U_{ps} / (1 – **

The symbol G denotes the mass flux of solids, that is,

**G = M _{p}/A,** (7)

where Mp is the mass flow rate of solids.

The relative velocity between particle and fluid (slip velocity) Uslip can be defined as:

**U _{slip} = U_{f} – U_{p}** (8)

In vertical dilute phase flow it is often assumed that the slip velocity is equal to the single particle asymptotic velocity U_{T}.

**Fundamentals: Continuity**

We can consider a unit length of transport pipe into which particles and gas are fed at constant mass flow rates respectively of M_{p} and M_{f}.

The continuity equation for the particles is:

**M _{p} = A U_{p} (1 – ε) ρ _{p}** (9)

while for the gas it is:

**M _{f} = A U_{f} **

Combining these continuity equations gives an expression for the ratio of mass flow rates. This ratio is known as the **solids loading**:

(11)

The average voidage ε at a particular position along the length of the pipe is a function of the solids loading and the magnitudes of the gas and solids velocities for given gas and particle density.

**Fundamentals: Pressure drop**

Let us compute the total pressure drop along a section of transport using the momentum equation for a section of pipe δL.

**Fundamentals: Pressure drop**

Let us consider a section of pipe of cross-sectional area A and length δL inclined to the horizontal at an angle α and carrying a suspension of voidage ε. The momentum balance equation can be written as: **see figure**

(12)

where F_{FW} and F_{PW} are respectively the gas-to-wall friction force and solids-to-wall friction force per unit volume of pipe.

**Fundamentals: Pressure drop**

Integrating eq. 12 and assuming constant gas density and voidage ε we obtain: see figure 1 (13)

Equation 13 applies in general to any flow of gas-particle mixture in a pipe, as no assumption has been made whether or not the particles are transported in dilute phase or dense phase.

Six different terms determine the total pressure drop along a straight length of pipe carrying solids in dilute phase transport:

**(1) pressure drop due to gas acceleration;****(2) pressure drop due to particle acceleration;****(3) pressure drop due to gas-to-wall friction;****(4) pressure drop related to solid-to-wall friction;****(5) pressure drop due to the static head of the solids;****(6) pressure drop due to the static head of the gas.**

**Fundamentals: Pressure drop**

see figure

Some of these terms can be neglected in some cases:

- a) if the gas and the solids are already accelerated in the line, then the first two terms can be omitted from the calculation of the pressure drop;
- b) if the pipe is horizontal, terms (5) and (6) involving gravity effects can be omitted.

The main unknown in the equations are:

- the solids-to-wall friction coefficient;
- the relevance of the presence of the solids in the gas-to-wall friction.

**Fundamentals: The choking velocity U _{CH} in vertical transport**

- Curve AB represents the frictional pressure loss due to gas only in a vertical transport line without solids.
- Curve CDE represents the frictional pressure for a solids mass flow rate G
_{1}. - Curve FG represents the frictional pressure for a solids mass flow rate G
_{2}> G_{1} - Point C: high gas velocity and low concentration, frictional resistance between gas and pipe wall predominates.
- Decreasing the gas velocity means the frictional resistance decreases, but concentration of the suspension increases and the static pressure head required to support the solids increases.
- Further decreasing the gas velocity (below point D), the increase in static head outweighs the decrease in frictional resistance and as a consequence Δp / ΔL rises again.

**Fundamentals: The choking velocity in vertical transport**

7. In the region DE the decreased velocity results in a rapid increase in solids concentration until the gas can no longer entrain all the solid phase, and a flowing and slugging fluidized bed can appear in the transport line.

8. The phenomenon is known as “choking”, and it is accompanied by large pressure fluctuations. The **choking velocity**, U_{CH} is defined as the lowest velocity at which this dilute phase transport mode can be operated at the solids feed rate G_{1}. Increasing the feed rate to G_{2} will result in a reasonably higher choking velocity.

9. The choking velocity sets the boundary between dilute phase and dense phase in vertical pneumatic transport.

10. Choking can be reached either by decreasing the gas velocity at a constant solids flow rate, or by increasing the solids flow rate at a constant gas velocity.

**Fundamentals: The choking velocity U _{CH} in vertical transport**

Theoretical exact prediction of the choking velocity is not possible, although some correlations for predicting choking velocities are available in the literature. Punwani correlation (Punwani, D. V., Modi, M. V. and Tarman, P. B. (1976), International Powder and Bulk Solids Handling and Processing Conference, Chicago) considers the effect of gas density:

(14) (15)

where:

ε_{CH}is the voidage in the pipe at the choking velocity UCH; ρ_{p} is the particle density; ρ_{F} is the gas density

G is the solids mass flux (= M_{p} / A); U_{T} is the asymptotic free fall or terminal velocity, of a single particle in the gas (Stokes velocity).

Equations 14 and 15 represent the solids velocity at choking under the assumption that the slip velocity U_{SLIP} is equal to U_{T}. Equations 14 and 15 must be solved numerically to find ε_{CH} and U_{CH}.

**Fundamentals: The saltation velocity U _{SALT} in horizontal transport**

The general relationship between gas velocity and pressure gradient Δp / ΔL for a horizontal transport line is shown in the following figure and it is similar to the vertical transport line one.

**Fundamentals: The saltation velocity in horizontal transport**

The curbs define to the following situations:

**AB**= gas only in the conveying line,**CDEF**= solids flux G_{1},**HI**= higher solids feed rate G_{2}>G_{1}.

**point C** : the carrier gas velocity is sufficiently high to carry all the solids in very dilute suspension.

Turbulence effects in carrier gas prevent the solids from settling to the walls of the pipe.

If the gas velocity is reduced at constant solids feed rate, the frictional resistance and Dp / DL will decrease.

The solids will move more slowly and the solids concentration will increase.

**point D** : the gas velocity is insufficient to maintain the solids in suspension and the solids begin to settle out in the bottom of the pipe.

The gas velocity at which this occurs is called the **saltation velocity.**

Further decrease in gas velocity will result in a rapid increase of Δp / ΔL as the area available for the carrier gas is reduced by the presence of settled solids.

**Fundamentals: The saltation velocity in horizontal transport**

In the region E and F some solids may move in dense phase flow along the bottom of the pipe whilst others travel in dilute phase flow in the gas in the upper part of the pipe.

The saltation velocity is defined in a horizontal pipe as the actual gas velocity at which the particles of a homogeneous solid flow will start to fall out of the gas stream, and it marks the boundary between dilute phase flow and dense phase flow in horizontal pneumatic transport. This velocity is used as a basis for choosing the design gas velocity in a pneumatic conveying system.

It is not possible to theoretically predict the conditions under which saltation will occur, but many correlations for predicting saltation velocity are available in the literature.

The saltation gas velocity is usually multiplied by a factor (approximately 1.5) , which is dependent on the nature of the solids, to obtain at a design gas velocity.

Fundamentals: The saltation velocity in horizontal transport

The correlation of Rizk (Rizk, F. (1973) Dr-Ing. Dissertation, Technische Hochschule Karlsruhe), based on a semi-theoretical approach, has an approximate error range of 50% .

- In SI units.
- X
_{s,g}is the solids loading = [mass flow rate of solids] / [mass flow rate of gas] Mp is the mass flow rate of solids U_{SALT}is the superficial gas velocity at saltation. - D is the pipe diameter

d is the particle size.

we can express: see figure.

*1*. Introductory concepts about batch and continuous precess

*2*. Materials in use for food equipments – Part I

*3*. Materials in use for food equipments – Part II

*4*. Equipment for raw material handling: pneumatic systems - Part I

*5*. Equipment for raw material handling: pneumatic systems - Part ...

*6*. Equipment for raw material handling: pneumatic systems - Part ...

*7*. Equipment for raw material handling: pneumatic systems - Part I...

*8*. Size reduction equipment - Part I

*9*. Size reduction equipments - Part II

*10*. Extruders

*12*. Positive displacement pumps

*14*. Cold Chain Equipment - Part I

*15*. Cold Chain Equipment - Part II

*16*. Cold Chain Equipment - Part III

*17*. Separation equipment - Part I

*18*. Separation Equipments – Part II

*21*. Liquid Mixing

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