*The purpose of the work* . The study of practical methods for determining the amount of bulk cargo in stacks of *regular geometric shape* .

*General instructions* . Bulk cargoes are stored in piles of *various* geometric *shapes* , but the most common are piles of *regular geometric shapes* .

On fig. 26 shows the appearance of bulk cargo *piles* in the form of *regular geometric shapes* ( *cone* (a), *pyramid* (b), *prism* (c), *wedge* (d), *obelisk* (e)).

Rice. 26. Types of stacks of bulk cargo

*All* stacks, except for the *prism* and *obelisk* , are obtained by *free dumping of* cargo. *A prism* is formed when a stack is formed between two *vertical walls* . *The obelisk* is a truncated wedge, and the stack in the form of an obelisk is formed *by limiting the height of the* storage of cargo. *The limitation of the storage height* may be due to *the technical capabilities* of the storage space (load) and handling equipment or *the transport characteristics of the* cargo.

*The cone* , *pyramid* and *prism* are *elementary* geometric figures, while the *wedge* and *obelisk* are *composite* . A *wedge* is defined as the *sum of* the volumes of a *prism* (length ℓ and width A) and a *pyramid* with base side A (Fig. 26, d). *An obelisk* is defined as the *sum of* the volumes of a *parallelepiped* (length ℓ, width b and height H), *two prisms* (width A, one length ℓ, the other length b) and a *pyramid* with base side A (Fig. 26, e).

In *practice,* the dimensions of the *base of the wedge* (length L and width A) and the *obelisk* (length L and width B) can be directly determined by measurements. Determining *the height of the* obelisk H is not difficult, since it *is known in advance,* since the *filling* (formation of the stack) was carried out to this height. Then the volume of the *wedge* is defined as the *difference* between the volumes of the *prism* (length L and width A) and *the pyramid* (with base side A). The volume of *an obelisk* is the *difference* between the volume of a *parallelepiped* (length L, width B and height H) and *two prisms* (width A, one length L, the other length B), *plus* the volume of the *pyramid* (with base side A). The volume of the pyramid *must be added* , since when subtracting the volumes of one and then the other prism, the volume of the pyramid *was subtracted twice* .

When loading vehicles, the question often arises about the *amount of cargo* in the port. That is, it becomes necessary to decide whether it is *necessary* or not to *bring* additional cargo to the port to ensure the full loading of vehicles.

In the port, the *initial data* for calculations are obtained by *measuring* the *dimensions of* the base of the stacks with a tape measure, and *the angle of repose* a with a goniometer. *Bulk weight* is determined according to *regulatory documents* or using a *measuring box* . Then, according to the data obtained, the *volume of the stack* V is determined and, using the bulk mass g, *the amount of cargo* Q is calculated.

*The volume of* a stack of regular geometric shape can be determined using a *nomogram* or by *calculation* , using formulas known in geometry.

*The nomogram* (Fig. 27) allows you to *quickly* , *simply* and with sufficient accuracy determine *the volumes of* geometric bodies of the correct shape. The nomogram *consists* of 9 logarithmic scales arranged on 5 axes:

the first scale is *the diameter of the cone* ; *the length* of the side of the *pyramid* ; *the width of the prism* (wedge) – denoted by 1 and A;

the second scale – the circumference of the *cone* – is denoted by 2 and S;

the third scale – the volume of the *pyramid* – is indicated by 3 and V _{p} ;

the fourth scale – the volume of the *cone* – is indicated by 4 and V _{to} ;

the fifth scale – the *cross-sectional area of u200bu200bthe prism* – is indicated by 5 and C;

the sixth scale – the volume of the *prism* – is indicated by 6 and V _{pr} ;

the seventh scale – *the height* of the stack – is indicated by 7 and H;

the eighth scale – the length of the *prism* and *the tangent of the angle* of repose – is indicated by 8, L and tga;

the ninth scale – *the angle* of repose – is indicated by 9 and a.

*A line* drawn *through two points* on any two *axes* makes it possible to determine *all other* elements.

When the value of the initial data *is greater* than on scales 1 and 2, then they *decrease* by 10 times, and the result ( *volume* ) *increases* by 1000 times.

The exception is the *prism* , when *reduced* by a factor of 10 on scale 1, and *the volume is increased* by a factor of 100. This is due to the fact that *the volume* is *not obtained immediately* , but through the intermediate result C (area). If the input data *is greater* than on a scale of 8, it is *reduced* by a factor of 10 or 100, and the result (volume) *is increased* by the corresponding number of times.

*The principles of* reduction and increase can be formulated as follows:

if *the one-dimensional value (length) decreased* on a scale of 1 and 2, and a *three-dimensional value* (volume) was immediately obtained, then the volume of the stack *increases* by the reduction *factor* (number) to the *third degree* ;

if *the one-dimensional value decreased* on a scale of 1, and an intermediate *two-dimensional value* (area) was immediately obtained, then the volume of the stack *is increased* by the reduction *factor* to the *second degree* ;

if a *one-dimensional value was reduced* on a scale of 8, then the volume of the stack (another value) *is increased* by this reduction *factor* .

In the work, *separately for each type of stack* , when determined by *nomograms* , the following is given:

name of the stack type;

the initial data necessary for the calculation;

value (volume) taken from the nomogram, magnification factor (if needed), final result (volume);

all intermediate values taken from the nomogram (if any).

*Values* from the nomogram are taken with the *highest* possible *accuracy* .

*The calculation* method *is more accurate* , but it requires *complex* (using computer technology) and labor- *intensive* calculations and *knowledge of the formulas* for calculating the volumes of geometric shapes.

In the work, *separately for each type of stack* , when determined by the *calculation method* , the following is given:

name of the stack type;

the initial data necessary for the calculation;

final calculation formula and result (volume).

The value of the *tangent* of the angle of repose a is taken from scale 8 *opposite* the value a of scale 9 with *maximum accuracy* .

*Work order* . In accordance with the given *option* , we determine *the angle* of repose (a) of the stacks, their *name* (type), *dimensions* and bulk *mass* .

Rice. 27. Nomogram for determining the volume of stacks

We determine *the volumes of* stacks using a *nomogram* (Fig. 27).

*cone* . We set aside S on the scale 2 and a on the scale 9, connect these points with a straight line (with a ruler or other even object). At the intersection of this straight line with the scale 4, we take the value of the volume V _{to} , m ^{3} .

*Pyramid* . Set aside A on scale 1 and a on scale 9, connect these points with a straight line. At the intersection of this straight line with the scale 3, we take the value of the volume V _{p} , m ^{3} .

*Prism* . Set aside A on scale 1 and a on scale 9, connect these points with a straight line. At the intersection of this straight line with scale 5, we remove the intermediate value C. We connect this value With a straight line with the value L on scale 8 and on scale 6 we determine (remove) the value of the volume V _{pr} , m ^{3} .

*Wedge* . We find, according to the previously given algorithms, the volumes of a prism with a width A and a base length L (V _{pr} ), and a pyramid with a base A (V _{p} ). The volume of the wedge is defined as the difference between these volumes, m ^{3} :

V _{class} u003d V _{pr} – V _{p} .

*Obelisk* . We set aside H on the scale 7 and a on the scale 9, connect these points with a straight line. At the intersection of this straight line with scale 1, we remove the value A. Next, we find, according to the previously given algorithms, the volumes of a prism with a width A and a base length L (V _{pr} ^{L} ), a prism with a width A and a base length B (V _{pr} ^{B} ) and a pyramid with base A (V _{p} ).

The volume of the obelisk is determined from the following expression, m ^{3} :

V _{about} u003d L × B × H – V _{pr} ^{L} – V _{pr} ^{B} + V _{p} .

We determine *the volumes of* stacks of goods by *calculation* , m ^{3} :

*Cone* V _{k} u003d 0.00423 × S ^{3} × tga;

*Pyramid* V _{p} u003d 1/6 × A ^{3} × tga;

*Prism* V _{pr} u003d 1/4 × A ^{2} × tga × L;

*Wedge* V _{class} u003d 1/4 × A ^{2} × tga × L – 1/6 × A ^{3} × tga;

*Obelisk* V _{vol} u003d L × B × H – 1/4 × A ^{2} × tga × (L + B) + 1/6 × A ^{3} × tga,

where A = 2 × H / tga.

We calculate *the mass of cargo* in *each* type of *stack* for *each calculation method* , t:

Q _{pcs} = V _{pcs} × g.

At the end of the work, we determine the *errors* of the methods. To do this, we calculate *the deviation of the mass of* cargo DQ in a stack *of each type* , determined by the *first* Q _{1} and by the *second* method Q _{2} (t). We determine *the maximum weight of the* cargo Q _{max} (t) and calculate the *error* D,%:

DQ u003d ½Q _{1} – Q _{2} ½;

Qmax = _{max} {Q _{1} ; _{Q2} };

D = DQ × 100 / _{Qmax} .

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