Some Further Considerations and Development of the Pressure Loss Equation

The pressure loss equation (14.113) was derived from the general equation (14.102) by modeling the forces Fmx and /"sw.

The force Fim was obtained with the aid of the virtual power method by assuming that the floating power arises only from the force Fint. This kind of approach has been used by Weber.3

Modeling of the force /"sw was actually based on the force balance of a single solid particle; extending the result to cover a larger set of solid particles, we obtained Eq. (14.105). This method of modeling is adequate for a homoge­neous flow where the particles move separately, not as a new kind of solid­phase structure. Weber3 has written the friction force in the form

^sw = D |^c2’ (14.121)

Where A! is a friction factor that is analogous to the friction factor AG. Equa­tion (14.121) brings an additional parameter to the pressure-loss equation that has to be determined experimentally. The advantage of this approach is that it enables us to handle the different types of pneumatic flows discussed in Section 14.2.2, whereas the model of Eq. (14.105) is restricted to dilute flows.

The first term on the right-hand side of Eq, (14.113) comes from the iner­tial forces. Because of the pressure drop the density of gas decreases in the di ­rection of the flow and therefore, on the basis of mass balance of gas flow, the velocity V increases along the flow. If the pipe is isolated, then the flow can be treated as adiabatic, which on the basis of energy balance implies that along the flow we have

1 1

H + ^v~ = constant, {14.122}

Where B is the specific enthalpy of the gas (J/kg).

On the other hand, when the gas is modeled as an ideal gas, then H = B(T) (enthalpy does not depend on the pressure), and since the velocities are quite low, we deduce from Eq. (14.122) that

T= const.

Along the flow. For instance, if the velocity of air changes from zero to 60 m/s, it decreases the temperature only by

At /*2. 60 /2 m“/s“ Q o/"’

Cp l000J/kgoC c

So that the approximation of constant temperature is accurate. Hence the flow, can be treated as an isothermal process provided that the pipeline is isolated.

In the dilute flow, a good approximation is 4> si. For instance, if C/ V Ј 0.5 (very common), ps/pG ‘ 1000, and P. Ј 10, we get from Eq. (14.17)

D> = ——— ———- ‘ Z——— 77;—I:—, = 0.980 .

^ „ V pG 1 + 10 • 2 • 0.001

1 + it ’ — “ ———

C Ps

The approximation <f> = 1 means that the partial density Pg (= 4> pG) can be re­placed by PG in Eq. (14.113). Using p? = pG the mass balance of gas flow is

PGv = const.,

From which it follows that



подпись: pm

Dv = ,.dPG = D

‘ dx dx dx


P dx

Where we have also used the ideal gas law and T = const. Since Dp/dx < 0, we see from Eq. (14.124) that Dv/dx > 0, as expected.

The velocity of material, C, also increases as a function of x. By differen­tiating the relation

Some Further Considerations and Development of the Pressure Loss Equation


Which follows from Eqs. (14.118)-(14.119) and from the ideal gas law com­bined with Eq. (14.123), we obtain

,-iadp p dx’

Consequently, combining this with Eqs. (14.124) and (14.123) we obtain

De _ V + Cаp Dx 2 p dx

подпись: de _ v + cаp dx 2 p dx’(14,125)

Which shows that Dc/dx >0 because the pressure drops (dpldx < 0).

Substituting Eqs. (14.123) and (14.125) into Eq. (14.113) and replacing Pe by Pc; (the approximation $ = 1), we obtain

1 _ PGiв _ = 2


D 2 + M-Pa

If 2 ? V

—cos o+-




C c



V ✓


+ pGg sin S

1 , V 1 + P.-


Some Further Considerations and Development of the Pressure Loss Equation



Equation (14.126) is our final result; it can be used for calculating the pressure loss in the pipe flows.

In ejectors and tube bends the most important part of the pressure loss conies from the acceleration of solid particles. In a bend the velocity of the particles is reduced due to the friction and the pressure loss is caused by the reacceleration of the particles after the bend.

The pressure loss due to the acceleration of solids is obtained from

Eq. (14.113):

Dp dc

~Tx = PgV^Tx’


Where we have replaced pe by PG (4> A 1). Integrating Eq. (14.127), we get

P2-pi=-pcvp(c1-cl), (14.128)

Where Cj is the initial and C2 the final velocity of the solids.

Equation (14.128) can be used for calculating the pressure drop due to the acceleration of solid particles provided that the velocity change c2 — C1 can be estimated. In addition to the acceleration pressure loss we have the “normal” pressure drop


подпись: (14.129)AP = Щpv2,

Where Ј is the pressure drop coefficient of the bend.


подпись: (14.130)If the material ejector in the pipe is accelerated from a velocity of zero to C then the corresponding pressure drop is

A P = — pGvpc.

In the next two sections we present how these formulas are used in some ap­plications.