METHOD OF DETERMINING RATE OF DISCHARGE FROM SILO

István Oldal

Szent István University Hungary Mechanics and Technical Drawing Department

Oldal.Istvan@gek.szie.hu

1. Introduction Discharge is an important question when granular materials are stored in a silo. There are two method of determining rate of discharge granular materials from silo. Johansson’s method is theoretical Beverloo’s one is empirical. We made a theoretical model based on mechanical behaviour of granular materials. Our method better than the others because it gives good results in wider range of parameters. And give an approximation of velocity field. In this paper we present the model and compare it with our experiments.

2. Introduction of a model During running out from a tank velocity of granular materials is constant. It is not agree with fluids where it depends on filling level. The phenomenon is known and the models have assumed it. But it is not any model that describes the causes. It is valid in case of Johansson’s and Beverloo’s models. The first one describes only mass flow another one only funnel flow. Our aim is constructing a model that solve these problems. Introduction of flow out model we take the next assumptions. − Arching effect acts at all times of flowing granular

material through outlets. Arch can be stable or instable.

− This effect is the bottleneck of flowing. − Properties of instable arches are similar to stable

arches (shape, stresses). − Shape of arch can describe as a paraboloid. This

shape depends on material properties of granular media, hopper geometry and wall friction.

There is a surface below outlet where normal stress is zero. Structure of granular material cracks up under such conditions. It gives a yield surface that can called instable arch. Below of this region are free fall conditions irrespectively of the height above. It means in around arch there are constant conditions during running out

We model the granular material flow out like that is forming and breaking of instable arches. That is continous if the conditions of stable arching are ungratified. Granular material is in conditions of free fall below arch. Flow velocity at outlet depends on height of falling only. So the first requirement is satisfied by the model namely discharge velocity is constant over height of bulk. In order to determining quality of mass flow we have to calculate velocity. It depends on height of arch Velocity of granular material at outlet

- f(x,φ) is a function of paraboloid that describes surface of arch in polar coordinate reference frame. Value of this gives the height of falling,

Figure 1. Plane section of arch and velocity field

v(x,φ)

f(x,φ)

∅d

y

x h

yield surface

velocity field

- v(x,φ) is velocity at outlet, - h is height of arch, - d diameter of outlet, - δ=h/d rate of height and diameter of arch.

In case of free fall:

ghxv 2),( =ϕ , ),( ϕxfh = ,

−=

2

21),( d

xhxf ϕ

Velocity field at outlet:

−=

2212),(dxdgxv δϕ

Average velocity:

dgd

dxxddxdg

v

d

δπ

ϕδπ

232

4

212

2

2

0

2

0

2

=

−

=∫ ∫

If we assume density is near constant, the volume of mass flow can be determined from velocity multiply outlet area and density.

AvW ⋅⋅= ρ , 252

62

42

32 d

gddgW ρδππρδ =⋅⋅=

Discharge rate in N/s unit:

25

23

62 dgQ ρδπ

=

This form similar to Johansson’s but instead of using half cone angle (α) parameter δ is used. It gives wider application range. Changing of α between 15 and 30 degrees we measured discharge rate decrease 15% without Johansson’s model gives 50% decreasing. Beverloo’s function is not include similar parameter it only contains density, outlet diameter and empirical constant. 3. Experimental analysis 3.1. Experimental model silo

Our model silo was made of a plastic cylinder of 100 mm in diameter. The hopper was conical and its half cone-angle was 45°. The outlet diameter was variable between 25 and 100 mm. The loads were measured by two load cells wall friction, vertical force on hopper sum of them was mass of material in silo. Experiments were done using grain wheat. Our research was granted by OTKA T 35022. 3.2. Results

Result of a measuring was a mass-time curve. These are linears curves that means volume of mass flow is constant. (This agree our previous experience.) Value

of this can be counted simply. During experiments outlet diameter was changing only. Effect of hopper is not valid in case of outlet diameter d = 90 és 100 mm because it is too little or it is not. Capability of arching is decrease or not. Our model is based on arching so we put out these experiments from examined range.

0

1

2

3

4

5

6

7

8

0 20 40 60 80 100 120

When outlet diameter was less than silo diameter 80% the model shows good approximate to measuring data in case δ = 0.28.

0

0,2

0,4

0,6

0,8

1

1,2

1,4

1,6

0 10 20 30 40 50 60 70 80

4. Conclusions

The problems of known methods were shown. Our model gives a result of them. It shows good agreeing with results of measuring which was had done. The upper limit of usable is higher than real silos occur. Parameter δ can be determined by measuring or FEM analysis but it changes narrow range. 5. References [1] Keppler István, Csizmadia Béla: Néhány gondolat a szemcsés

anyagok természetes boltozódásának modellezési lehetőségeiről, Fiatal Műszakiak Tudományos Ülésszaka V., Kolozsvár 2000.

[2] Johnson, J. R.: Method of calculating rate of discharge from hoppers and bins, Society of Mining Engineers, 1965.

[3] Beverloo, R.: The flow of granular solids through orifices, Chem. Eng. Sci. 15, 262, 1961.

[4] Kézdi-Rétháti: Handbook of soil mechanics. 4. vol., Akadémiai Kiadó, Budapest, 1990.

[5] István Oldal, Ürítés és boltozódás kísérleti vizsgálata silóknál, (Experimental analysis of discharging and arching in silos) OTDK dolgozat, Sopron, 2001.

d[mm]

W[kg/s]

Figure 2. Measured mass flow over diameter

Figure 3. Measured mass flow over diameter

d[mm]

W[kg/s]