DEFORMATION OF THE VELOCITY PROFILE OF A TWO-PHASE FLOW UNDER PNEUMATIC TRANSPORT CONDITIONS

M. E. Dotsin, V. P. Lebedev

(Presented by the Department of Industrial Heat Power Engineering)

The study of the mechanism of motion of a solid suspension in a pneumatic-transport pipeline is closely connected with concepts concerning the velocity distribution of a two-phase flow. The insufficient study of the velocity distribution in the cross section of a pneumatic-transport pipeline causes considerable difficulties and contradictions in theoretical investigations.

For the most part, determination of the velocity profile has been carried out under hydraulic-transport conditions. For pneumatic transport, experimental data on the velocity distribution are given only in the work of M. Kalinushkin [1], on the basis of his measurements during the transport of sawdust in a horizontal pipeline with \(D = 270\) mm at a velocity of \(10 \div 17\) m/s and a mass concentration \(0.046 < \mu < 0.184\).

In contrast to hydraulic transport, under pneumatic-transport conditions a substantial deformation of the velocity profile is observed, owing to the presence in the flow of a solid suspension whose specific weight differs substantially from the specific weight of air. Moreover, the degree of deformation depends on the loading of the flow, on the properties of the material being transported, and also on the characteristics of the pipeline.

In the present work an assessment is given of the influence of the indicated parameters on the character of the velocity distribution along the vertical of the cross section of a horizontally positioned pipe, in experiments on the pneumatic transport of wheat and quartz sand.

The experimental investigations were carried out on a test stand, a detailed description of which is given in the authors’ work [2].

The experimental setup consisted of devices for forcing and measuring the air flow rate, a device for uniform feeding of the loose material and determining its flow rate, a straight horizontal section of pipeline 12.5 m long and with diameter \(D = 124.8\) mm, and a device for collecting the transported material.

At the end of the pipeline, at a distance of more than 10 m from the loading device, a device was installed for obtaining the velocity profile in the cross section of the flow. The adopted distance considerably exceeded the length of the acceleration section and, consequently, the velocity measurements were made in a section with established motion of the two-phase flow.

The device for taking velocity profiles consisted of a pneumometric tube and an arrangement for its smooth displacement along the vertical diameter of the pipeline (Fig. 1).

The pneumometric tube could be set at any vertical distance from the axis of the pipe.

The position of the tube was fixed from the measuring scale of the device with an accuracy of up to 0.1 mm.

In view of the fact that a standard pneumometric tube rather quickly became clogged with particles of the material being conveyed, especially quartz sand, a special pneumometric tube was made for this purpose; it differed from the standard one by a considerably smaller diameter ($d = 1$ mm) of the frontal opening receiving the dynamic head. Behind the opening in the tube nozzle there is a certain space for trapping particles of material that may slip through the opening. This part of the nozzle is made detachable so that it can be cleaned of accumulated material. Experience in operating the tube showed its reliable performance. Clogging of the tube did not occur, and cleaning was required extremely rarely.

Calibration was performed against a standard Prandtl tube in a stream of clean air.

Measurement of the pressure drop in the pneumometric tube was carried out with an MMN-type micromanometer of the Teploavtomat plant. In addition, at the point where the velocity profile was taken, the static pressure and the air temperature were recorded.

The flow velocity at each measured point of the pipe cross section was calculated from the formula

\[ w = \sqrt{\frac{2k\Delta p}{\rho}}, \]

where $k$ is the calibration coefficient of the nozzle,

$\Delta p$ is the pressure drop according to the micromanometer,

$\rho$ is the density of the air in the pipe.

Fig. 1. Device for taking velocity profiles: 1—pipe; 2—fitting; 3—nut; 4—rubber seal; 5—guide post; 6—pneumometric tube; 7—post; 8—scale; 9—gear wheel; 10—rack.

To construct the velocity profile, measurements were made at seven points along the vertical diameter of the pipe, including the point on the pipe axis. The adopted number of measurements made it possible to fully reveal the character of the velocity profile for different transport regimes.

Each experiment to determine the velocity profile was preceded by a preliminary adjustment of the test stand to the prescribed transport regime. The adjustment consisted in filling the feeder bunker with the corresponding material, establishing a definite weight concentration of the transported mixture, and setting a definite transport velocity.

To monitor and maintain the specified parameters during each experiment, constant observations were made from the instruments of the stand of the consumption of material and air, as well as of other characteristic parameters of the two-phase flow.

To estimate the influence of the weight concentration of the transported mixture $\mu$ on the deformation of the velocity profile of the two-phase flow, Fig. 2 presents the results of determining the profile at a constant mean air velocity $w_{\text{av}} = 22.0\ \text{m/sec}$ and at different loadings of the flow with wheat. On the same graph the velocity profile for clean air $(\mu = 0)$ is plotted.

Fig. 2. Velocity profiles in relative coordinates during wheat transport at a constant mean transport velocity $w_{\text{av}} = 22.0\ \text{m/sec}$ and different concentrations $\mu$.

From comparison of the profiles shown it follows that, in the transport of loose materials, deformation of the velocity profile occurs, and it is the more substantial the higher the weight concentration $\mu$. The symmetry of the profile, characteristic of clean air, is increasingly disturbed as the loading of the flow with the solid phase increases. The upper part of the profile is stretched out in the direction of motion of the flow, while the lower part lags behind and straightens. The maximum velocity is displaced from the pipe axis into its upper half, and the higher it is, the greater the concentration of material in the flow.

With increasing weight concentration, the velocity gradient decreases considerably at the bottom of the profile, near the lower wall of the pipe, and increases at the top and in the core of the flow.

The deformation of the velocity profile considered is explained by the greater retardation of the flow in the lower part of the pipe. A two-phase flow under pneumatic-transport conditions is characterized by a nonuniform vertical distribution of material in the cross section of the pipe [3]. The higher the weight concentration, the more the material is transported in the lower part of the pipeline and the greater the velocity of “slip” of the air relative to the solid phase.

Figures 3a and 3b show profiles of absolute and relative velocities, recorded at a constant value of the weight concentration but at different mean transport velocities.

At the subcritical velocity \(w_{\mathrm{av}} = 12\ \text{m/sec}\), the transported material partly moves along the bottom of the pipe and causes additional

Fig. 3. Velocity profiles in wheat transport at constant concentration \(\mu = 1.5\): a) in absolute velocity values; b) in relative velocity values. \((---)\) clean air; \((—)\) pneumatic transport.

retardation of the flow in its near-bottom part. In this case the lower half of the profile undergoes substantial deformation. The entire profile becomes elongated along the pipe. The velocity gradient in the core of the flow increases considerably.

When the flow velocity is above the critical value (\(18\text{–}25\ \text{m/sec}\)), all the material is transported in suspension. As the velocity increases, the material moves from the lower half of the pipe to the upper half, and a more uniform distribution of material is achieved over an ever larger part of the pipe cross section. Only in the upper part of the flow is the amount of transported material substantially smaller than in the rest of it. Therefore, in this region the flow is less retarded and the velocity maximum shifts into the upper half of the pipe.

The dependence of the relative displacement of the velocity maximum,

\[ \frac{h_1}{r}, \]

where \(h_1\) is the distance from the pipe axis to the position of \(w_{\max}\), and \(r\) is the pipe radius, on the weight concentration \(\mu\) and on the mean flow velocity is presented in Figs. 4a and 4b.

As follows from the graphs, the relative displacement of the maximum,

\[ \frac{h_1}{r}, \]

initially increases in direct proportion to the transport velocity, while at velocities above the critical value the increment of

\[ \frac{h_1}{r} \]

slows down; the depen-

the dependence $\dfrac{h_1}{r}=f(v)$ at $w=\mathrm{const}$ may be regarded as linear in the range of the measured concentrations when the material moves in a suspended state.

It is necessary to note a characteristic feature in the change of the magnitude of the maximum velocity in the transverse section of the pipe. In all the experiments carried out, the values of the maximum velocity during transport of the material and during the motion of clean air, for one and the same flow rate, remain the same and do not depend on the weight concentration $\mu$ (Fig. 2 and Fig. 5, c).

Fig. 4. Dependence of the relative displacement of the velocity maximum:
a) as a function of the mean transport velocity;
b) as a function of the mean concentration of the flow.

The decisive influence on the degree of deformation of the velocity profile under pneumatic-transport conditions is exerted by the character of the distribution of the material in the transverse section of the flow. This proposition is fully confirmed by experiments on recording the velocity profile during pneumatic transport of quartz sand.

In work [3] it was established that, during pneumatic transport of quartz sand, a fairly uniform distribution of the material in the transverse section of the flow is observed. Moreover, the influence of velocity, concentration, and particle size on their distribution is insignificant.

Under such conditions, a considerably smaller distortion of the symmetry of the velocity profile should be observed. Experiments with quartz sand for particles of two sizes ($d_{\mathrm{e}}=0.42$ and $d_{\mathrm{e}}=0.715$ mm) at $\mu=1.5$ and $w_{\mathrm{av}}=25.5$ m/sec, shown in Fig. 5, a, confirmed the indicated assumption.

In Fig. 5, b, a comparison is given of the dimensionless velocity profiles for sand and wheat under the same transport conditions.

For the transported material to which, under these conditions, the greatest nonuniformity of distribution corresponds, a more substantial deformation of the velocity profile is obtained.

All the experiments presented were performed in a pipe with stable wall roughness. However, the roughness also has a significant influence on the deformation of the velocity profile during pneumatic transport.

In [4], in studying the influence of pipe roughness on the magnitude of hydraulic resistance under pneumatic-transport conditions, it was indicated that the increase in hydraulic resistance with increasing roughness is accompanied by a more uniform distribution of solid particles over the cross section of the pipe. Such a change in the character of the distribution of the material should lead to a restoration of the symmetry of the velocity profile as the roughness of the walls increases. The experiments presented below illustrate this proposition.

Fig. 5.
a) Velocity profiles for quartz sand \(d_3 = 0.42\) mm, \(d_3 = 0.715\) mm, and clean air at concentration \(\mu = 1.5\) and mean transport velocity \(\overline{w}_{\mathrm{tr}} = 25.5\) m/sec;
b) comparative profiles of dimensionless velocities for sand \(d_3 = 0.715\) mm and wheat under the same transport conditions: \(\mu = 1.5\) and \(\overline{w}_{\mathrm{tr}} = 25.5\) m/sec.

In Figs. 6a and 6b comparative graphs are given of the profile of absolute and dimensionless velocities during the transport of wheat for a “smooth” and a rough pipe at constant concentration \(\mu = 1.5\) and mean velocity \(\overline{w}_{\mathrm{sr}} = 22\) m/sec. The relative roughness of the pipe was

\[ \frac{\Delta}{D} = (0.6 \div 0.7)\,10^{-2}. \]

The roughness of the inner walls of the pipe was produced artificially by the generally accepted method [4, 5].

As follows from these figures, for the rough pipe the symmetry of the velocity profile proves to be less deformed than under the same conditions in the “smooth” pipe. The velocity gradient in the core of the flow of the rough pipe has a smaller magnitude than in the “smooth” one. However, in the near-wall region the velocity gradient in the rough pipe is greater, owing to the decrease in transported material in this region.

If, during the transport of such a material as wheat, which in a “smooth” pipe is distributed rather nonuniformly over the cross section of the flow, roughness has a substantial influence on the restoration of the symmetry of the velocity profile, then during the transport of sand the influence of wall roughness proves to be smaller.

It was shown above that deformation of the velocity profile during the transport of sand occurs to a lesser degree, owing to a more uniform distribution of solid particles over the cross-section of the “smooth” pipe. Therefore, when the roughness of the pipeline walls is increased, no substantial change in the curve occurs. However, as follows from the graph in Fig. 6, c, in this case as well the influence of roughness on the velocity profile manifests itself in the same way as in the transport of wheat, but is less perceptible.

It is also of interest to compare velocity curves obtained during the transport of wheat and quartz sand in rough pipes (dashed curves in Figs. 6, b and 6, c). Because of a certain difference in the transport velocities, such a graph is not presented, but from an external comparison one may conclude that, as roughness increases, the dependence of the nature of the velocity-profile development on the characteristics of the transported material becomes less significant.

Fig. 6. Comparative velocity curves for smooth (——) and rough (— — —) pipes: a) absolute-velocity profiles during the transport of wheat, $\rho = 1.5$, $w_{\mathrm{tr}} = 22$ m/sec; b) the same in relative velocity values; c) relative-velocity profiles during the transport of quartz sand: $d_{\mathrm{e}} = 0.42$ mm, $\rho = 1.5$, and $w_{\mathrm{tr}} = 25.5$ m/sec.

Analysis of the experimental data on velocity curves at various flow rates and concentrations, for different characteristics of the transported material and of the pipeline, shows that the nature of the velocity profile is formed depending on the distribution of the solid suspension over the height of the cross-section of the two-phase flow.

REFERENCES

1. M. I. Kalinushkin. Pneumatic Transport. The Influence of Mechanical Impurities in Air on Losses in a Pipeline. Publishing House of the Central Aero-Hydrodynamic Institute named after Prof. N. E. Zhukovsky, 1936.

2. M. E. Login, V. P. Lebedev. Investigation of resistances in pneumatic transport in a horizontal pipeline. Proceedings of the Moscow Power Engineering Institute, vol. 28, 1960.

3. M. E. Login, V. P. Lebedev. Experimental data on the distribution of a solid suspension in a horizontal pipeline. Proceedings of the Moscow Power Engineering Institute, vol. 33, 1961.

4. M. E. Login, V. P. Lebedev. Pipe roughness under conditions of pneumatic transport. Izvestiya Vuzov, Thermal Power Engineering, No. 7, 1962.

5. F. A. Shevelev. Investigation of the Basic Hydraulic Laws of Turbulent Motion in Pipes. State Publishing House of Literature on Construction and Architecture, 1953.