Drying and Storage Engineering 4(3+1)

Lession–38 Design of Silos

The following terminology is useful for design of storage structures

• Bulk density 

The bulk density of a granular product is the density measured without the influence of any compression on the product. The bulk density depends on particle density, its shape and the manner in which the constituent particles are packed or positioned with respect to each other. Bulk density is also defined as weight of grains per unit bulk volume (including porosity / voids).

Bulk Density,

Where, W = weight of grains,     Vv = void volume and Vg = grain volume

• True density

The true density represents the actual density of particle. It is determined by, liquid displacement method. The liquid generally used is toluene. Toluene is filled in, a measuring cylinder and the volume is recorded. Then, 100 g grain is put in cylinder. The rise in volume of toluene in cylinder is noted. True density is the ratio of weight of grain and the difference in volume of toluene. True density is weight of kernel per unit true volume (excluding porosity/voids).

True Density,

Where, W = weight of grains,  and  Vg = grain volume

• Apparent density

For grains, the bulk density is increased when the grains are compressed. "During this process reorientation and settling of particles take place. As a result air percolates out of the bulk mass and the volume decreases. This increases the bulk density. The density of compressed mass or bulk is called the 'apparent density'. The value of apparent density is higher by 20% than the normal bulk density. In design of a silo, the apparent density of grain is taken into account, because of compaction, the bottom layer or base layer of silo has maximum density whereas the top layer has minimum density.

• External friction

External friction is friction between grain mass of kernels and the wall of bin or a surface.

• Internal friction

Friction of the kernels or grains against each other is known as internal friction.

• Angle of  internal friction

The sliding stress between the grain to grain gives the frictional resistance per unit of surface area of the grain against the grain. This is called the angle of internal friction. Coefficient of friction μ between granular materials is equal to tangent of the 'angle of internal friction' for that material. Higher values of angle of internal friction indicate that the material is cohesive. The lower values of angle of internal friction are indicative of easy flowing products.

• Coefficient of friction

If 'F' is the horizontal force or pull and N is the vertical force caused by the total weight of grain, the coefficient of  friction μ can be given by

The characteristics values of the granular material which cause greatest stresses should be taken into account while determining the lateral.and vertical pressures of storage silo. These values are function of the silo wall nature. For smooth walled silo the greatest vertical and lateral pressure be estimated with maximum values of bulk density and. angle of internal friction while the minimum value of angle of friction of grains on the walls be taken. Whereas for silos with rough walls, the lateral pressure should be calculated with values of minimum bulk density, minimum angle of internal friction and the minimum angle of friction of the grains on the walls. The greatest vertical pressure should be estimated with maximum bulk density, maximum angle of internal friction, and the minimum angle of  friction of the grains on the walls.

38.1 Grain Pressure Theories

The grain pressure in bins were first calculated as being a semi liquid of same density as the grains. The lateral pressure was first calculated using the hydrostatic formula.

Where,

P_l = wh                       ………………………………………………………(1)

P_l = lateral pressure exerted by the grain on the bin wall,  kg/m²

w = density of grain, kg/m³

h = depth of grain from the top of the bin surface, m

This formula has serious deficiencies because many structures buckled under the vertical load arising from the friction of the grain on the walls. The formula (1) was modified to incorporate a factor 'K ' known as Rankine's earth pressure coefficient. The Rinkine's formula is based on the following principle, the resistance to displacement by sliding along a given plane in a loose granular mass is equal to the normal pressure exerted between the parts of the mass on either side of the plane, multiplied by the specific constant. The formula (1)  when multiplied by the factor 'K ' is known as Rankine's earth pressure coefficient.

P_l = Kwh                  

Where,

 K = Rankine's coefficient

    

Φ= angle of internal friction of the grain = natural angle of repose

Therefore,

....................................................... (2)

Fig.38.1 Pressures on an element of soil behind a vertical wall

The whole weight of the grain was assumed to be transferred to the bottom of the beam. The Rankine theory with horizontal soil surface behind a vertical wall is shown in Figure. Consider an element of soil at depth 'h'. The vertical pressure acting on this element Pv and the horizontal pressure Pl is active pressure at failure.

_OR

P_l = K P_v                      …………………………………………..(3)

K = Rankine coefficient = Coefficient of earth pressure

The above Rankine formula (3) is used for determination of pressure induced by granular materials against retaining wall in shallow bins.

• The scientist Airy developed a theory for pressure induced by granular materials against retaining wall or in shallow bins. The lateral pressure exerted by grains in a shallow bin can be given by following Airy's equation

Where,

w = grain bulk density

h = depth of grain to point under consideration

μ = coefficient of friction of grain on grain

    = tan Φ, Φ is angle of internal friction

μ' =  tan Φ', Φ' is the angle of wall friction.

• The scientist Janssen took into account the friction between the grain and bin wall and, proposed the following formula for 'deep bins'.

 

Where,

P_l = lateral pressure

R = hydraulic radius

w = grain bulk density (unit weight of grain)

µ  = coefficient of friction of the grain on the wall

    = tan Φ

h = depth of grain

38.2 Problems of Silo Design

38.2.1  Wheat weighing 900 kg/m3 is loaded in a circular concrete silo of 3 m internal diameter and a clear height of 8 m. The angle of internal friction for wheat is 25° and that for wheat and concrete is 24°. Applying Airy theory, calculate the maximum lateral pressure at the bottom of bin section.

Solution

Lateral pressure against a shallow bin wall is

Where,

  P_l = lateral pressure

  w = bulk density of grain = 900 kg/m³

  h = depth of bin (point) under consideration = 8 m

 μ = coefficient of friction of grain to grain

   = tan Φ, Φ is the angle of internal friction = 25⁰

   = tan 25⁰ = 0.466

μ' =  coefficient of friction of grain to wall

    = tan Φ' , Φ' is the angle of external (wall) friction = 24⁰

   = tan 24⁰ = 0.445

So,

Pl= 2338.5 kg/m²

38.2.2  A RC.C cylindrical grain storage bin has internal diameter of 5 m and is 20 m deep. It is completely filled with paddy weighing 600 kg/m³. The angle of internal friction for paddy can be taken as 35⁰, while the angle of friction between paddy and bin wall is 30°. The ratio of horizontal and vertical pressure intensity k, is 0.4. Calculate the lateral pressure intensity at 4.0 m and 18 m depth using Janssen theory.

Solution

Lateral pressure against a deep bin wall as par Janssen theory is

 

Where,

P_l = lateral pressure

w = grain bulk density (unit weight of grain) = 600 kg/m³

R = hydraulic radius =D/4 = 5/4 =1.25 m

µ  = coefficient of friction of the grain on the wall

    = tan Φ= tan 30⁰ = 0.577

K=  P_l/P_V = 0.4

h = depth of grain = 4.0 m and 18 m

Substituting the values of 4.0 m and 18.0 m depths in above equation, we get