4.1 Gas and particle velocities
We have to be careful in the definition of gas and particle velocities
and in the relative velocity between them, the slip velocity. The terms are
often used loosely in the literature and are defined below.
The term superficial velocity is also commonly used.
The superficial gas velocity is defined as:
(Eq. 4) Ufs = [volume flow of gas] /
[cross sectional area of pipe] = Qf / A
The superficial solids (particles) velocity is defined as:
(Eq. 5) Ups = [volume flow of solids] /
[cross sectional area of pipe] = Qp / A
where subscript "s" denotes superficial and subscripts "f" and "p"
refer to the fluid and particles respectively.
The fraction of pipe cross-sectional area available for the flow of gas
is usually assumed to be equal to the volume fraction occupied by gas,
that is, the voidage or void fraction 
.
The fraction of pipe area available for the flow of solids
is therefore (1 - ).
And so, actual gas velocity,
(Eq. 6) Uf = Qf / [A ]
and actual particle velocity,
(Eq. 7) Up = Qp / [A (1 - )]
Thus superficial velocities are related to actual velocities by the equations:
(Eq. 8) Uf = Ufs /
(Eq. 9) Up = Ups / (1 - )
It is common practice in dealing with fluidization and pneumatic transport
to simply use the symbol U to denote superficial fluid velocity. This practice
will be followed in this chapter. Also, in line with common practice,
the symbol G will be used to denote the mass flux of solids,
that is, G = Mp/A, where Mp is the mass flow rate of solids.
The relative velocity between particle and fluid Urel is defined as:
(Eq. 10) Urel = Uf - Up
This velocity is often also referred to as the "slip velocity" USLIP.
It is often assumed that in vertical dilute phase flow the slip velocity
is equal to the single particle terminal velocity UT.
4.2 Continuity
Consider a length of transport pipe into which are fed particles and gas at mass flowrates
of Mp and Mf respectively. The continuity equations for particles
and gas are:
For the particles:
(Eq. 11) Mp = A Up (1 - ) 
p
For the gas:
(Eq. 12) Mf = A Uf f
Combining these continuity equations gives an expression for the ratio of mass
flow rates. This ratio is known as the solids loading:
This shows us that the average voidage 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.
4.3 Pressure drop
In order to obtain an expression for the total pressure drop along a section
of transport line we will write down the momentum equation for a section of pipe.
Fig. 5: Section of conveying pipe:
basis for momentum balance
Consider a section of pipe of cross-sectional area A and length 
L
inclined to the horizontal at an angle 
and carrying a suspension
of voidage .
The momentum balance equation is:
Therefore,
where Ffw and Fpw are the gas-to-wall friction force
and solids-to-wall friction force per unit volume of pipe, respectively.
Rearranging Equation 11 and integrating assuming constant gas density and voidage
Readers should note that Equations 4 to 15 apply in general to the flow
of any gas-particle mixture in a pipe. No assumption has been made as to whether
the particles are transported in dilute phase or dense phase.
Equation 15 indicates that the total pressure drop along a straight length
of pipe carrying solids in dilute phase transport is made up of a number of terms:
(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
Some of these terms may be ignored depending on circumstance. If the gas
and the solids are already accelerated in the line, then the first two terms
should be omitted from the calculation of the pressure drop; if the pipe
is horizontal, terms (5) and (6) can be omitted. The main difficulties are
in knowing what the solids-to-wall friction is, and whether the gas-to-wall
friction can be assumed independent of the presence of the solids;
these will be covered in section 5 below.