Engineering Mathematics  Control Systems  Control of FlowTransport System of Concrete Mixing Plants
Control of the flowtransport system
Let's consider a problem of calculation of processes of supply of aggrigates of concrete to account bunkers of concrete plants. We'll assume,
that process of dozing of aggrigates at concrete mixing plants has a stable character. Therefore as the model describing the basic strategy of
control by concrete mixing plants facilities, we accept a periodic model with a constant updating, applied usually in conditions of stable
demand [1]. Possible random deviations in the system which dispersion is insignificant, should be compensated in special control mode – a
correction mode.
Problem of control is maintenance of uninterrupted consumption of concrete aggrigates from account bunkers of concrete mixing plants. Therefore
the condition of maintenance of a material stock in account bunkers in the given limits is entered. Duly loading of bunkers and inclusion of
necessary mechanisms of the system (feeders in storehouses of aggrigates, the mobile reversive conveyors, dumping carriages, switching centres,
etc.) should exclude impact of various flows of aggrigates on conveyor lines.
Input data are: the rated expenses of feeders and dozing devices, transport delay for each device of the system (time of moving of a
material from a storehouse up to an appropriating arrangement), the top and bottom levels of a material stock in bunkers, initial material stocks
in bunkers, structure of the conveyor transport scheme.
Control variables are instants and duration of inclusion of feeders in storehouse of aggrigates.
At control of the bunkers loading system one of the following modes is realized:
– the stationary (periodic) mode of loading appropriating a stable demand for aggrigates;
– a transitive (starting) mode of loading;
– the correction mode necessary for the compensation of mistakes cumulative in system, leading to a deviation of actual materials stocks
in bunkers from the rated ones.
Let's consider consistently each of these modes.
Let's enter following designations:
k – a number of bunker ;
j(k) – a number of storehouse, from which an aggrigate moves into the kth bunker;
p_{j(k)} – an expense of feeders of the j(k) th storehouse, t/min;
q_{k} – an expense of dozing device of the kth bunker, t/min;
V_{sk} and V_{mk} – accordingly the top and bottom limits of a material stock in the kth bunker, t;
τ_{k} – transport delay for the kth bunker, min;
σ_{ij} – transport delay between the ith and the jth storehouses, min;
t_{pk}– duration of loading of the kth bunker in a stationary mode, min;
t_{t} – technological interval on the conveyor tape between adjacent portions of materials, min;
T – the cycle time, min;
V_{0k} – an initial material stock in the kth bunker, t;
t_{k}^{(m)} – an instant of the mth inclusion of feeder at supply of aggrigate into the kth bunker, min;
V_{hc} – useful volume of the heatingcooling bunker, t;
s_{pk} – duration of loading of the kth bunker in a transitive (starting) mode, min;
V_{k} (t) – a current value of a material stock in the kth bunker, t;
Δp_{rk} – an error in expense of a feeder of the kth bunker during the rth cycle, t/min;
Δq_{rk} – an error in expense of a dozing device of the kth bunker during the rth cycle, t/min;
Δt_{pk} – an error in duration of feeding of the kth bunker, мин;
θ_{k} – an increment of duration of feeding of the kth bunker in a correction mode, min.
The stationary (periodic) mode
Construction of algorithm of control is based on a structure of the scheme of conveyor lines [2]. Each technologically independent site of the
scheme represents either a junction when flows of materials from storehouses pass all through one main conveyor, or a network,
when flows of materials are distributed by the branched system of conveyor lines.
For junction with n account bunkers at stationary process of unloading the cycle time is
(1)
that is T is defined by the bunker, at which own time of a cycle (size in square brackets) is minimal as only in this case for all bunkers
a fulfillment of the following condition:
(2)
is ensured.
Fig.5. Variation of material stock in a bunker in the transitive
(starting) and stationary (periodic) modes.
On Fig.5 the graph of stationary process of variation in time of a material stock in a bunker V_{k} (t) is resulted. It
is accepted, that taking of material away from a bunker goes on uninterruptedly. From the mode stationarity follows, that an increase of a material
stock in a bunker during its loading is equal to the stock reduction during unloading:
(3)
Then duration of bunker loading
(4)
At implementation of stationary mode of supply of aggrigates on the main conveyor with technological intervals t_{t} between
adjacent portions
(5)
From equality (5) with account of (4)
(6)
For switching intermediate executive mechanisms (mobile reversive conveyors, tripper devices, distributive funnels, etc.) some minimally admissible
interval t_{tm} between adjacent portions should be provided, it is necessary to fulfill of the condition
(7)
From expression (6) in view of (7) follows, that for realization of a considered mode of supply of aggrigates the inequality should take place:
(8)
The value t_{tm} is defined by the mechanism with the minimal speed.
Let's define now instants of inclusion and shutdown of feeders in a stationary (periodic) mode.
Fig.6. To the conclusion of the equation of the graph of inclusions of feeders:
а – sequence of supply j – i; б – sequence of supply i – j.
Let the kth bunker is loaded from the jth storehouse, and the lst bunker – from the ith storehouse. If the
fraction of aggrigate, acting from the ith storehouse, on a tape of the conveyor follows fraction of a aggrigate from the jth
storehouse (Fig.6, а), then feeders of the ith storehouse should be included so that in an instant of shutdown of feeders of the
jth storehouse between portions of fillers on the conveyor tape the interval t_{t} has been provided. As the distance
between the ith and jth storehouses (or difference of their distances up to a central point in which flows of materials are united)
may be expressed by transport delay σ_{ij}, then instants of inclusion of feeders are connected by dependence:
(9)
If sequence of supply of fractions of aggrigates from the ith and jth storehouses is return (Fig.6, b), then
(9а)
Having united (9) and (9а), it is possible to receive the following recurrent relation:
(10)
where –
elements of the skewsymmetric matrix S of transport delays between storehouses:
(11)
Here
An instant t_{1}^{(m)} of the mth inclusion of feeders of the 1st bunker is determined by the
condition of implementation of the (m –1)th cycle, and an instant t_{1}^{(1)} of the 1st inclusion depends on
the initial state of the system.
After calculation of duration of loading of the kth bunker it is necessary to specify the lower limit of the material stock:
(12)
where Δ_{k} – an increment of the lower limit of the material stock in the kth bunker.
Let's consider now the scheme site, being a network. We'll notice, that a network can be considered as set of junctions technologically connected
among themselves. Calculation of a site of such structure is carried out basically on the same dependences, but requires some additions.
First of them consists that for definition Т the expression in square brackets in the formula (1) is minimized on all bunkers entering
into a network. Then all junctions making the given network and technologically connected among themselves, will work in a general periodic mode.
Other addition to calculation consists in coordination of technological intervals calculated for various junctions, entering in a network. It is
necessary for the coordination of instants of supply of aggregates with objective of exception of a possibility of impact of various fractions in
reloading points of a network (that is there where there is an association of various flows). For this purpose it is possible, for example, having
calculated t_{t} for each junction under the formula (6) and having defined a minimum, to accept it as a technological interval on
all junctions of a network.
If in system it is stipulated warming (cooling) of materials then the material portion volume which is given out from a storehouse, should not
surpass useful volume V_{hc} of the heatingcooling bunker:
(13)
The transitive (starting) mode
Let the system initial state is characterized by a stock of material V_{0k} in the kth bunker (Fig.5). Then duration of
loading s_{pk} of the kth bunker in a transitive mode is defined from the equation:
(14)
where λ – a quantity of the heatingcooling bunkers containing to rated point in time a material, submitted to the kth bunker;
V_{a} – a volume of material in the аth heatingcooling bunker, t; N – a number of loadings by
duration s_{pk} in the transitive mode; t_{k} – a rated instant of inclusion of feeders of the kth
bunker, received under the assumption, that right after startup of the system the stationary control mode begins, min; μ – a
quantity of passed loadings of the kth bunker in the transitive mode.
Instants t_{k}^{(1)} of inclusion of feeders in the transitive mode are defined by recalculation t_{k} :
(15)
First the equation (14) is solved at N = 1.
The transitive (starting) mode with duration of loading s_{pk} is considered realizable if the following conditions are satisfied:
(16)
Besides it is necessary, that the material stock in the bunker in an instant of beginning of its loading was not below a critical level, that
is, otherwise, would not be the bunker emptying. Default of any of these restrictions entails an indispensability either increasing N, or
preliminary loading of the bunker at the termination of batching of fillers at concrete plants.
The correction mode
During transportation of materials from storehouses in bunkers there will be deviations from rated parameters in the system, that finally will lead
to a deviation of actual stocks of materials in bunkers from the rated ones. These errors result from a lot of the reasons:
– nonuniformity of supply of materials by feeders;
– elimination of a part of materials as a result of sorting in department of control screening;
– possible losses of a material on the way;
– nonuniformity of expense of dozing devices;
– mistakes in duration of inclusion of feeders, etc.
As a result mistakes in the system will be accumulated and at achievement maximum permissible deviations it is necessary to compensate them in a
socalled correction. This mode represents the "deformed" stationary mode in which duration of supply of a material in the bunker is a little bit
changed, but within the limits of performance of conditions (16). Thus, variation of duration of bunkers feeding at correction is carried out either
due to reduction of a technological interval up to minimally admissible, or due to reduction of feeding duration. Clearly, that in a similar way
it is possible to carry out the compensation of only insignificant deviations in the system.
Fig.7. Variation of material stock in a bunker in the correction mode.
Let to instant of beginning of loading of the kth bunker in the correction mode (Fig.7) a total error for the past М cycles of
the stationary mode has made
(17)
where ε_{k} – an admissible deviation of bottom level of material in the kth bunker;
– an accumulating error in the kth bunker during the rth cycle.
Thus
(18)
For simplicity we suppose, that a mistake of duration of bunker feeding for each cycle is constant and equal to Δt_{pk} .
The basic equation of the correction mode can be received proceeding from the condition, that after carrying out of correction the beginning of
next loading (in a stationary mode) corresponds to the bottom level of material stock in a bunker (a point А on Fig.7).
Then
(19)
where
In view of that ,
after carrying out of some transformations we'll receive finally the dependence of increment of loading duration of the kth bunker
θ_{k} in the correction mode
(22)
Thus, the developed method of preparation of algorithm for calculation of complex flowtransport system of loading bunkers allows to design in some
optimum way the scheme of conveyor transport and to choose its parameters. Executed on an example of the actual scheme of concrete mixing plants
facilities calculations have allowed to estimate possibilities of control of the transportation system on various modes.
