The correct location of level measurement transducers and probes mounted at the top of silos and other vessels will lead to greater accuracy and reliability in product contents measurement. Roger Hicks, Technical Support Manager for level measurement specialists Hycontrol, looks at the mathematical formulae behind finding the optimum position to install equipment.
Processes in a wide range of industries including food, quarry, chemical and construction involve the handling and transportation of millions of tonnes of product every year. During transfer operations of bulk solid material to and from storage silos, the shape of the surface of the material changes. The accurate and reliable measurement of contents in these silos is essential to both safety and profitability, relying on the correct installation of appropriate level measuring systems.
This paper examines the optimum positioning of level measuring transducers in silos to give the most accurate indication of volumetric contents, taking into account the changing shape of the surface. It should be noted that the examples and results in this paper apply for free flowing materials only and ignore any changes in bulk density.
It is assumed in this paper that the level transducers are to be installed on the top of the silo and will measure the distance down to the surface of the material. Examples of technologies used for this purpose include radar, ultrasonic and TDR (Time Domain Reflectometry). Using data for the geometry of the silo, this measured distance from the top can be converted into the volume of material.
The techniques described here pay particular attention to large round silos, which cannot be easily fitted with load cells. Consideration is given to the siting of the transducers in relation to single central loading and single central discharge points. Simple mathematical techniques are used to establish the optimum positions for the transducers to achieve maximum accuracy of volumetric contents.
BASIC GEOMETRY OF THE METHOD
Refer to the diagrams shown in Figure 1 below. Fig. 1(a) shows a conical pile of material of height L and base radius of R. The volume (V1) of such a cone is given by:
V1 = 1/3 πR² L
Fig. 1(b) shows a cylindrical shape having the same base radius R and a height of H. The volume (V2) of the cylinder is given by:
V2 = πR² H
If the cylinder is to have the same volume as the cone (V1 = V2), its height H will be 1/3 x L.
Figure 1 – basic geometry
In Fig. 1(a), the point on the upper surface of the cone which is at height of H, or 1/3 L, above the base is at a radius of 2/3 R. Therefore if the height of a conical pile of material is measured at a radius of 2/3 R, the volume can be found by multiplying this height by the base area. This will give the correct volume irrespective of the angle of repose of the material.
Consider now the conical depression shown in Fig. 1(c). The volume V3 of material in this shape is given by:
V3 = πR^2L – 1/3 πR² L
Simplified: V3 = 2/3 πR² L
Similarly, the cylinder shown in Fig. 1(d) having the same volume as this shape, will have a height H of 2/3 L. The point on the upper surface of the depression of Fig. 1(c) which has this height, is at a radius of 2/3 R. So again if the height of a conical depression is measured at a radius of 2/3 R the volume of the material can be found by multiplying this height by the base area. The correct volume is given independently of the angle of repose.
Therefore we see that for a cone or a conical depression the correct volume can be found by measuring the height at any point that is 2/3rd of the radius of the silo from its centre.
ERRORS IN VOLUMETRIC MEASUREMENT FROM A CENTRALLY-MOUNTED TRANSDUCER
Applying this to a cylindrical silo with free-flow conditions, if the height of material in the silo is measured more or less centrally then significant errors can occur in the volumetric measurement. When the silo is being centrally filled from the top, the surface of the deposited material will be conical in shape and the level reading will be that of the apex of the cone. Conversely, if material is being extracted from the centre of the silo base, the surface of the product will form a conical depression and the level reading obtained will be that of the bottom point of the cone.
To appreciate this, consider the example of a 3 m diameter silo with the conically-heaped material contained within, resting at a 45° angle of repose.
The volumetric error that arises by multiplying the measured level by the cross-sectional area of the silo amounts to either +7.07 m³ or -7.07 m³, arrived at by V3 = 2/3 πR²L, where R and L are each 1.5 depending on whether the silo is being filled or emptied. When an emptying operation follows a filling operation, the change-over from a conical pile to a conical depression causes the error to change from the maximum positive value to the maximum negative value and vice versa. If in this example, the height of the silo were 10 m, the maximum error would be ±7.07 m³ or ±10.0% of the full volume (note that this figure ignores the conical base of the silo). Even if the angle of repose of the material were only 30°, the error would still be ±4.08 m³. These are quite significant errors when dealing with material that is free flowing and behaves in a predictable way.
THE SOLUTION - TRANSDUCER MOUNTED AT 2/3rd RADIUS
The solution is to ensure the transducer is mounted at a 2/3rd radius. As we have previously seen, at this position there is no error in volumetric measurement either when the surface is a perfect conical pile during filling or a perfect conical depression during emptying. However, there will still be errors during the transition phase when emptying follows filling and vice versa. The diagram of Fig. 2(a) shows a typical surface shape during the transition when an emptying operation follows a filling operation. Here, a small depression has occurred in the apex of the previously conical pile. At the stage illustrated, the level of material, Y1, at 2/3rd radius has not changed so there will be a positive error that corresponds to the volume missing from the apex of the cone.
The maximum error occurs when the conical depression has spread to where it just intersects the original conical surface at 2/3rd of the silo radius, as Fig. 2(b). The volume missing from the cone can be expressed as 2(1/3π)(2/3 R)²TAN(Ag):
Emax = 2/3π(2/3R)² TAN(Ag)
(Where Ag is the angle of repose)
Using our example silo of 3 m diameter and 10 m height, the maximum error would be 2.1 m³ of material at a 45° angle of repose or 1.2 m³ with a 30° angle of repose. Thus we can see that siting the transducer at 2/3rd of the silo radius results in a considerable improvement of the accuracy of calculated volumetric contents.
With a perfect conical pile or depression, there will be no error in the level calculation. By locating the transducer at a position 2/3rd of the silo radius the error during the change over from filling to emptying to filling again is typically no greater than 3.0% of the maximum content.
This paper has presented a practical method of deriving the volumetric contents of circular cross-section silos, using transducers which measure the distance of the material surface from the roof of the tank. From this measurement, the depth of the material at that location can be deduced, and in turn, a volume from the known geometry of the silo can be provided.
In silos having single central loading and unloading points and where free flow conditions prevail, good results (better than 2.1 m³ on a 3 m diameter silo) can be achieved with a single transducer placed to measure the depth at a point 2/3rd of the radius of the silo. As has been shown, the error increases significantly when the transducer is placed in the centre of the roof and should be avoided wherever possible.
Please note the results presented in this paper are theoretical and have been arrived at by calculation. However, when mass flow conditions prevail in the silo, errors are theoretically zero, as there will always be a properly-developed cone on the material surface in the silo.
The errors detailed in this paper relate only to materials that have free flowing attributes and does not account for the operational accuracy of the transducers used. However, operators can be confident that following the methodology outlined above will yield more stable and reliable level measurement results.