EFFECT OF GAS FLOW NONSTATIONARITY ON THE BEHAVIOR OF A GRANULAR MATERIAL BED

A. D. ALEKSEEV

(Submitted by the scientific seminar of the Departments of Applied and Theoretical Mechanics)

In various branches of industry, apparatuses using a fluidized bed have become widespread. The efficiency of such apparatuses is to a considerable extent determined by the aerodynamic regime; therefore, the choice of the regime and its justification are the most important elements in their design.

In the overwhelming majority of apparatuses in operation, the fluidized state of the bed is achieved by blowing a stationary flow of gas mixtures through it. Cases of the use of a nonstationary flow of the fluidizing medium are extremely limited [1], and experience with their operation has not been generalized [2]. Special works reflecting the features of the aerodynamics of nonstationary flows as applied to fluidization processes are lacking.

The present article clarifies the nature of the influence of nonstationary actions of a gas flow on certain parameters of a granular bed.

Theoretical Foundations

At small values of the velocity of an ascending flow passed through a bed of granular material, the gas filters through the bed without causing relative displacement of the particles. In the hydraulic sense, such a bed represents a porous medium, and its state is determined by the regime of gas motion in the pore channels. The latter may be described by the general equations of fluid motion in channels [3, 4].

If it is assumed that the gas motion is one-dimensional, the system is energetically isolated, and friction of the gas particles with one another and with the walls is absent [5], then the general equations of gas dynamics take the form:

\[ \frac{\partial w}{\partial \tau} + w \frac{\partial w}{\partial x} + \frac{1}{\rho}\frac{\partial p}{\partial x} = 0, \tag{1a} \]

\[ \frac{\partial \rho}{\partial \tau} + w \frac{\partial \rho}{\partial x} + \rho \frac{\partial w}{\partial x} = 0, \tag{1б} \]

\[ p = A \cdot \rho^{k}, \tag{1в} \]

where \(w, p, \rho\) are, respectively, the velocity, pressure, and density of the gas flow;

\(x\) is the position of the section under consideration;

\(\tau\) is the characteristic time of the process;

\(k\) is the isentropic exponent;

\(A\) is a constant characterizing the properties of the gases.

The subsequent transformations of equations (1) are carried out by the method of L. I. Kudryashev and A. A. Kopotev [7].

We replace the partial derivatives by total derivatives

\[ \frac{dw}{d\tau}=\frac{\partial w}{\partial \tau}+w\frac{\partial w}{\partial x}, \tag{2} \]

\[ \frac{dp}{d\tau}=\frac{\partial p}{\partial \tau}+w\frac{\partial p}{\partial x}. \tag{3} \]

In equation (1a) we substitute the values of the quantities from (2) and (3), divide by the acceleration of free fall of bodies, \(g\), and reduce it to the form

\[ \frac{dw^{2}}{2g}+\frac{dp}{\rho g} = \frac{\partial p}{\rho\,\partial}\cdot \frac{d\tau}{\partial \tau}. \tag{4} \]

We integrate this equation using (1b) for the case when the change in gas density is substantial [7], and determine the velocity of its motion in an arbitrary section of the bed

\[ w_{2}= \sqrt{ w_{1}^{2} + \frac{2k}{k-1}\cdot \frac{p_{1}}{\rho_{1}} \left[ 1-\left(\frac{p_{2}}{p_{1}}\right)^{\frac{k-1}{k}} \right] }. \tag{5} \]

At low velocities of motion of the fluidizing medium, the effect of gas compressibility may be neglected, and, with accuracy sufficient for practical calculations, one may use simpler relations valid for an incompressible liquid [8]. For such cases

\[ w_{2}= \sqrt{ w_{1}^{2}+\frac{p_{1}-p_{2}}{\rho} }. \tag{6} \]

It follows from formulas (5), (6) that the flow velocity in the pore channels of the bed depends on the state of the gas at the bed inlet and on the completeness of its expansion during motion between the sections under consideration.

The left-hand side of equation (4) characterizes the amount of work that the gas can perform during steady motion over portions of the bed bounded by sections “1” and “2.” The right-hand side of equation (4) determines the magnitude of that additional work which the gas flow can perform during time \(\tau\), if it moves through the section under consideration nonstationarily.

Let us denote

\[ \frac{1}{\partial}\int_{0}^{\tau}\frac{\partial p}{\rho\,\partial \tau}\cdot d\tau = \frac{\Phi(\tau)}{\partial} \tag{7} \]

and express the velocity of nonstationary gas motion in an arbitrary section of the bed

\[ w_{2\text{н}}= \sqrt{ w_{1}^{2} + \frac{2k}{k-1}\cdot \frac{p_{1}}{\rho_{1}} \left[ 1-\left(\frac{p_{2}}{p_{1}}\right)^{\frac{k-1}{k}} \right] + 2\Phi(\tau) } \tag{8} \]

or

\[ w_{2\text{н}}= \sqrt{ w_{1}^{2} + \frac{p_{1}-p_{2}}{\rho} + 2\Phi(\tau) }. \tag{9} \]

Equations (8), (9), upon substituting the values of \(w_2\) from (5) or (6), are reduced to the form

\[ w_{2\mathrm{n}}=\sqrt{w_2^2+2\Phi(\tau)}. \tag{10} \]

In the general case the term \(\Phi(\tau)\) may take any values (positive, negative, zero).

It follows from equations (5–9) that if the ratio \(\dfrac{p_2}{p_1}\) is the same for both flows, and \(\Phi(\tau)>0\), then the instantaneous values of the velocities of motion in the section of the bed under consideration, for nonstationary gas outflow, will be greater than the corresponding velocities of stationary motion. This means that, owing to the nonstationarity of the flow, it is possible to obtain instantaneous gas-velocity values equal to the corresponding velocities of stationary motion at smaller pressure drops \(\dfrac{p_2}{p_1}\).

The relations obtained will be valid for any section of the bed up to the moment at which the particles pass into a mobile state. Since the state of the bed is determined mainly by the instantaneous values of the flow velocities, on the basis of these equations one may expect that, under the action of a nonstationary flow on the bed, particle mobility will set in at lower gas flow rates than under stationary conditions.

Of considerable interest for engineering practice is the magnitude of the instantaneous dynamic impulse, since it characterizes the forces acting on the particles of the material, causing loosening of the bed and its transition from a fixed to a fluidized state. For the cases of outflow under consideration these quantities are determined by the expressions:

\[ P_{\mathrm{n}}=F\rho w_{\mathrm{n}}^{2}, \]

\[ P=F\rho w^{2}, \tag{11} \]

where \(F\) is the free cross section of the channels;
\(P_{\mathrm{n}}, P\) are the values of the impulses for nonstationary and stationary outflow of the gas stream.

We assume that in both cases of gas motion through the bed, the instantaneous values of the density and the total cross sections of the channels remain unchanged, and we transform equation (11), by substituting into it the velocity values, to the following form:

\[ P_{\mathrm{n}}=P\left[1+\frac{2\Phi(\tau)}{w^{2}}\right]. \tag{12} \]

From (12) it is seen that, when the value of the function \(\Phi(\tau)>0\), the magnitude of the impulse of the forces acting on the bed from the nonstationary flow will be greater than under stationary motion of the flow.

This circumstance appears to be exceptionally important from the practical point of view, since it permits the conclusion that, in fluidized-bed installations using nonstationary gas motion, the fluidization process will proceed more intensively than in installations using a stationary flow. It also follows from analysis of the equations that the character of the force action of the flow on the particle bed—and consequently the effectiveness of such action—depends to a considerable extent on the form of the function \(\Phi(\tau)\), and its selection should be approached with particular care.

To obtain quantitative data, an experimental setup was constructed and preliminary investigations were carried out on

of various materials. The layout of the installation is shown in Fig. 1.

The installation consists of compressor 1, receiver 2, pipelines 3, 4 with built-in diaphragms 5 and 6, damping vessel 7, a pulsator with drive 8, pulse line 9, and apparatus 10. The pulsator drive makes it possible to obtain sinusoidal pulses in the frequency range from 0.5–25 cycles/s. Bypass line 13, connecting the damping vessel with the pulse line, was intended for changing the initial shape of the pulse curve by adding an undisturbed flow to the system.

The apparatus includes glass column 14, conical-cylindrical air box 15, and removable upper cover 16. A system of piezometers and differential manometers provides measurement of averaged pressure values at all points of interest. The actual pressure values were measured by a measuring probe,

Fig. 1. Diagram of the experimental installation.

which consists of a pulse tube inserted into the investigated section of the column, a switch, and an indirect-action diaphragm pressure indicator with wire strain gauges. The switch provided for placing the diaphragm under load and, at the same time, disconnecting the differential manometer measuring the averaged pressure values. Four wire resistances were used: one was glued to each side of the diaphragm and to the inner surfaces of the chambers. Two of them detected diaphragm deformation and served as working gauges, while the other two were compensating gauges. Such an arrangement of strain gauges made it possible to measure both excess pressures and vacua with minimal temperature error. All resistances were connected into the bridge circuit of a strain-gauge installation assembled from standard equipment and consisting of a ferroresonant voltage stabilizer, rectifier, multichannel electronic amplifier, and magnetoelectric oscillograph. The material under investigation was loaded into the column onto distribution grid 17.

Procedure and results of the experiment

The assembled apparatus was subjected to careful calibration, as a result of which the laws governing the change in the resistance of individual elements during steady motion of the air and the effect of unsteadiness on them were established. The pressure indicators were calibrated separately. Membrane deformations were produced by the static action on it of an undisturbed gas flow.

During the experiment, portions of material with known characteristics were carefully weighed and loaded into the column. The volume occupied by the material in the column was brought to the volume corresponding to its bulk weight. The height of the bed in all its states was measured by visual observation using measuring rulers installed on two opposite sides of the column. The height was taken as the arithmetic mean of the readings of the measuring rulers. The gas velocity was determined by dividing the flow rate by the area of the free cross-section of the column. Water manometers were used to measure the pressure in the calming

Fig. 2. Dependence of the bed resistance on the air velocity and the frequency of dynamic effects. \(f = 0\). The flow is steady. \(f = 3.35;\ 10;\ 16.7;\ 23.5\)—frequency of dynamic effects, Hz.

vessel, the total resistance of the bed, and the resistance of the apparatus. In the necessary cases the manometers were disconnected, and the pressure at the corresponding points was registered by indicators and recorded on film. By gradually increasing the gas flow rate, the bed was brought to a state of intense fluidization, after which the flow rate was reduced to zero and the bed returned to its initial position.

Figure 2 presents pseudofluidization curves obtained by blowing through crystalline ammonium sulfate with bulk density \(\rho_n = 0.9\ \text{g}/\text{cm}^3\) and effective particle diameter \(d = 0.45\ \text{mm}\). The averaged values of the bed resistance in all regimes were determined from the readings of differential manometers. The pulsation frequency

was set by the attached tachometer and, in the case of recording the pulses on film, was checked by the time marker of the oscilloscope.

Figure 3 presents the regularities of the change in the degree of expansion of the bed for the same material.

Comparison of the curves in Figs. 2 and 3, obtained with unchanged pulse shapes, shows that the regularities of pseudofluidization and the behavior of the particles forming the bed depend not only on the air velocity, but also on the frequency of the dynamic action.

As a result of the experiments it was established that, in the investigated frequency range, the averaged values of the resistance of the pseudofluidized bed under nonstationary action of the gas flow on it have

Fig. 3. Dependence of the degree of bed expansion on air velocity and frequency of dynamic actions. \(h_p\) is the height of the bed in the operating state; \(h_n\) is the height of the bed corresponding to its bulk weight. \(f = 0\): steady flow; \(f = 3.35;\ 10;\ 16.7;\ 23.5\) is the frequency of the dynamic actions, cps.

smaller values than under the action of a steady flow. The difference in the values is the more noticeable, the lower the pulsation frequency. The pseudofluidization stage is preceded by a decrease in the volume of the bed, i.e., its compaction. The degree of such compaction depends on the pulsation frequency and the duration of their action. The expansion of the bed at unchanged averaged air flow rates also depends on the pulsation frequency, and it is the greater, the lower the pulsation frequency. A stable pseudofluidized state of the bed under the action of a nonstationary air flow occurs at flow rates 30–40% lower.

The optimal range of operating frequencies for stable pseudofluidization depends on many factors.

Conclusions

Nonstationarity of the gas flow, created artificially, is an effective means of producing a stable

of the fluidized bed and leads to a substantial reduction in the flow rate of the fluidizing medium.

References

1. Belik Luitfried. Chemie Ingenieur—Technik, No. 32, No. 4, 235—257, 1960.

2. Materials of the 1st International Congress on Chemical Engineering (chisa) in Brno. Engineering Physics Journal, No. 2, 1963.

3. N. I. Syromyatnikov, V. F. Volkov. Processes in a Boiling Bed. Metallurgizdat, 1959.

4. Max Leva. Fluidization. Gostoptekhizdat, 1961.

5. K. P. Stanyukovich. Unsteady Motions of a Continuous Medium, 1955.

6. L. I. Kudryashev, A. A. Kopotev. Theoretical and Experimental Study of the Influence of Nonstationarity on the Process of Efflux in Converging Nozzles. Transactions of the Kuibyshev Aviation Institute, issue XII, 1961.

7. A. Sh. Dorfman, M. M. Nazarchuk et al. Aerodynamics of Diffusers and Exhaust Pipes of Turbomachines. Publishing House of the Academy of Sciences of the Ukrainian SSR, Kiev, 1960.

8. A. S. Yastrzhembskii. Technical Thermodynamics. Gosenergoizdat, 1960.