DEVELOPMENT OF GAS DYNAMIC LINEAR SYSTEMS FOR SETTING LOW PRESSURES

Development of modern technologies (microelectronics, medicine, biotechnology), expansion of gas analytical systems, in particular, for determining atmospheric air pollution, as well as conducting scientific research, will require high precision systems of stabilization of gas flow parameters, first of all, pressures, as they determine the flow rate in such gas-dynamic systems [1, 2]. Industrial setting devices (stabilizers) are generally designed to maintain relatively high pressures while gasdynamic systems, in particular, gas analytical, often also need small excess pressures (pressure drops) – at the levels of hundreds, tens and less pascal [3, 4]. To prepare complex multicomponent mixtures (for example, with microand nano concentrations of components) by the gas-dynamic method, substantially different pressure drops at the mixer’s capillaries for various components should be provided [5]. That requires a whole system of various means to stabilize the pressure of components. For example, to prepare synthetic natural gas with 13 components, a corresponding number of means is required. This makes the system cumbersome and imperfect because it is significantly affected by obstacles (pressures of source components, barometric pressure, temperature, etc.). A similar situation exists in the chromatographs with the task of flow rate of gas-carrier, where the error of the flow rate determining may be up to 5 % [6]. Thus, the development of high-precision means of simultaneous generation of the pressures different in magnitude (drops), including very low ones, is relevant.


Introduction
Development of modern technologies (microelectronics, medicine, biotechnology), expansion of gas analytical systems, in particular, for determining atmospheric air pollution, as well as conducting scientific research, will require high precision systems of stabilization of gas flow parameters, first of all, pressures, as they determine the flow rate in such gas-dynamic systems [1,2].
Industrial setting devices (stabilizers) are generally designed to maintain relatively high pressures while gasdynamic systems, in particular, gas analytical, often also need small excess pressures (pressure drops) -at the levels of hundreds, tens and less pascal [3,4].
To prepare complex multicomponent mixtures (for example, with micro-and nano concentrations of components) by the gas-dynamic method, substantially different pressure drops at the mixer's capillaries for various components should be provided [5]. That requires a whole system of various means to stabilize the pressure of components. For example, to prepare synthetic natural gas with 13 components, a corresponding number of means is required. This makes the system cumbersome and imperfect because it is significantly affected by obstacles (pressures of source components, barometric pressure, temperature, etc.).
A similar situation exists in the chromatographs with the task of flow rate of gas-carrier, where the error of the flow rate determining may be up to 5 % [6].
Thus, the development of high-precision means of simultaneous generation of the pressures different in magnitude (drops), including very low ones, is relevant.

Analysis of publications and the problem statement
To ensure the quality of any gas-dynamic system (e. g., gas analyzers, chromatographs, setting devices of low rates), the parameters (obstacles) that affect their work should be stabilized. For example, with this purpose a block of stabilization of input parameters is used in the control system of composition of flue gases [7] because, if feeding the analyzed gas with unstabilized parameters directly to the measuring transducer, it will result in a large error of its composition information at the output.
At the same time, designers of gas analytical systems do not always pay sufficient attention to reduce the impact of obstacles, and sometimes consider it sufficient to initially adjust gas rate by throttle [8].
Traditionally, to reduce the impact of external factors, the pressures (pressure drops) are stabilized at the input of gas-dynamic system and its individual elements or gas rate.
Maintaining different pressures in one system is performed by installing several stabilizers, often different in the principle of performance, design, range and the type of stabilized pressures (excessive or absolute). For example, in the devices for dynamic preparation of gas mixtures, both typi-DEVELOPMENT OF GAS DYNAMIC LINEAR SYSTEMS FOR SETTING LOW PRESSURES І . D і l a y cal reducers and pressure regulators of the type "to oneself" are used [9]. Particularly indicative is this situation in the gas-dynamic mixers of complex multicomponent gas mixtures with micro-and macro concentrations of components [10][11][12]. Due to nonidentity of characteristics of stabilizers by the value of the stabilized pressures in such gas-dynamic systems, the obstacles (e. g., pressure at the input and output of the system, barometric pressure, temperature) produce different effects both in magnitude and, sometimes, in direction. Similar situation occurs with stabilization of the gas rate. Thus, the automatic gas mixing system is equipped with multiple, independently working, gas mass rate controllers with the tasks of the rate, different by magnitude, -20 and 500 ml/min [13].
Research found that these factors cause errors of defining the parameters of different gas-dynamic systems, for example, error of gas mixture generators is 2-3 % [14]. In this regard, the actual task is compensation for the effects of these obstacles.
In addition, well known tools for stabilizing low pressures do not always meet modern requirements. For example, low pressure stabilizing sometimes is carried out using a fluid manostat, which is actually a laboratory instrument [15].
Thus, designing setting devices of the pressures different in magnitude, which could enable compensation for the impacts of obstacles and thereby could enable reducing errors of determining parameters of different gas-dynamic systems, should be considered promising. This, in particular, will enable solving the problem of changing stabilizers in the gas-dynamic system by pressure repeaters, which reproduce inter-throttle pressures of individual throttle circuit, which is not connected by gas flows with the main gas dynamic system [5,6]. This can be, for example, a subsystem of setting stabilized pressures, built on the basis of serial and parallel connection of capillaries.

The aim and objectives of the study
The aim of the work is to improve characteristics of gas-dynamic systems by improving the accuracy of stabilization, particularly, of low pressures.
The tasks of the work are: -research and construction of linear pressure dividers for their use in gas-dynamic systems of setting the pressures different in magnitude (drops); -development of schemes to ensure a substantial increase in coefficients of pressure division compared to a separate divider; -building a high-precision four-decade discrete device for setting absolute pressures (pressure drops), which provides their linearity at the change in supply pressure.

Linear capillary pressure dividers
Pressure divider is a serial connection of throttles Tr j (j=1,…, m), the gas-dynamic resistance of which causes a certain reduction in inter-throttle pressure (Р m-1 >…>Р j-1 >…>Р 1 ) due to throttling the flow (Fig. 1). Mass rate through the throttles of a divider equals G=G m =…=G j =…=G 1 and the pressure drops on each of them equal DP j =Р j -Р j-1 .

Fig. 1. Multithrottle pressure divider
To build pressure dividers one can use throttle elements that provide both laminar character (Re≤2320) of the throttle gas flow -capillary tubes (metallic or glass) and turbulent flow -diaphragms, nozzles, watch stones, etc. However, for gas-dynamic schemes, including pressure dividers, the most promising is the use of glass capillary elements CE. Such elements (capillaries) have stable consumption characteristics, they practically do not change geometrical dimensions (diameters d and lengths l) of passable channels at temperature change. Due to a smooth reduction of the length of the channel (e. g., grinding the end face) of the capillary, a precision matching of its gas-dynamic resistance is provided [6]. Due to the specified advantages, the capillaries are used as throttles in the designed schemes.
To increase the number of different values of inter-throttle pressures in the divider, instead of individual throttles Tr j one can install a capillaries package Pс j (parallel connection) or as block, formed by a combination of serial and parallel connections of capillaries. Package Pс j of capillaries is made as a package of alternating gas dynamic resistance and the required capillaries are engaged by installing solenoid valves at their outputs.
The dependency of the mass gas rate G through the capillary CE j of the divider will take the form [16].
where ( ) 1 j l j j A a l 4 l − = = π µ ξ is the coefficient of rate; μ is the coefficient of dynamic viscosity of gas; ξ is the coefficient of end effects; Y j =K j X is the complex of the channel (pass) dimensions, parameters of type of gas and temperature; is the constructive complex; d j , l j are the diameter and length of the capillary channel; DP j, j-1 =P j -P j-1 is the pressure drop in the capillary; X is the parametric gas complex, X=(512R g Tμ 2 ) -1 ; R g =R/M is the gas constant; R is the universal gas constant; Т is the absolute temperature of gas; M is the molecular weight. Dimensions of passable channels of capillaries, which are used in gas-dynamic devices, are limited due to design considerations and are included in the When building the pressure dividers, it is important to ensure linearity of change in inter-throttle pressures on the change in pressure Р m at the input of divider. But this is possible only when all the elements of the divider are linear while consumption characteristic of a capillary in general is nonlinear. The linearity of consumption characteristic of a separate capillary can be provided at a certain ratio of its length and diameter of its channel and absolute constant temperatures and gas pressure at the output. As shown in Fig. 1, the specified requirements can be easily provided for the throttle Tr 1 in the pressure divider (capillary CE 1 ).
The condition of linearity of consumption characteristic of the capillary CE 1 , obtained from ( ) Dependency of gas rate for linear capillary CE 1 , taking into account (2), will take the form ( ) Pressure dividers, particularly the capillary ones, in general are nonlinear systems, but choosing the design dimensions of their capillaries, as shown below, can provide the linearity of inter-throttle pressures Р j when changing the pressure at the input of divider P m . Linear dividers open up a prospect of improving characteristics of gas-dynamic systems, in particular, for preparation of complex mixtures due to compensation of the change in components consumption at the change of pressures at the inputs of dosing capillaries [5,16]. In this regard, further we will consider only linear capillary pressure dividers and the schemes based on them.
A necessary condition for building such dividers is the linearity of consumption characteristic (1) of capillary CE 1 , which, in compliance with the condition ∂ 2 P 1 /∂P 2 2 =0 that means equality to zero of the curvature characteristic Р 1 =f(Р 2 , Р 0 ), provides linearity of the divider.
Dependency Р 1 =f(Р 2 , Р 0 ), obtained for the two-element divider with linear capillary CE 1 , will take the form (4), with regard to the condition ∂ 2 P 1 /∂P 2 2 =0, the expression is obtained that connects dimensions of capillaries of the divider and which, along with the dependency (2), is the condition of linearity of a two-capillary pressure divider After substituting the first equation of the system (5) to the expression (4), we obtain a dependency of inter-throttle pressure P 1 on the input pressure Р 2 for a linear two-capillary divider in the form The defining characteristic of a two-capillary divider is the coefficient c 1 of pressure division ( ) ( ) where ∆P 2 , ∆P 1 is the pressure drop in the divider and in the first capillary, respectively.

2. A multi-capillary divider
To set a larger number of different drops (absolute pressures) by a single divider, one can build multi-element dividers (Fig. 2), which, due to the appropriate dimensions of passable channels of capillaries, can provide necessary values of inter-throttle pressures Р j (j 1,m 1 = − ), as well as their linearity from the change Р m , and thus constant coefficients of the division c j , which are determined according to the dependency ∆ is the pressure drop accordingly in the whole divider and in j capillaries, respectively (at the output).
A necessary condition for building a linear multi-capillary pressure divider, as well as a two-capillary one, is the linearity of consumption characteristic of capillary CE 1. To determine the rest of the dimensions (d j and l j for j 2,m = ) of passable channels of linear m-capillary divider of pressures, a successive substitution is performed of m-l pairs of {CE j ; CE (j-1) } of neighboring capillaries by an equivalent linear capillary CE (j) based on the dependencies (5) and (6), where CE (j-1) is the conditional sign of the equivalent linear capillary, which is a substitution of the sequence CE 1 ,…, CE j-1 of capillaries of the divider by an imaginary capillary with diameter d (j-1) and length l (j-1) (Fig. 2).
For any pair of capillaries, for example {CE j ; CE ( j-1) }, with the set pressures P j , P j-1 and P 0 , the dependency (6) will take the form Taking into account that ( ) , from (9) we receive a dependency to determine the length l j of a passable channel of capillary CE j.
For the above mentioned pair of capillaries {CE j ; CE (j-1) }, based on the first equation of the system (5), we receive the dependency where d (j-1) =d j /d (j-1) , and from which -the expression for the diameter d j of capillary CE j . As a result, the system of equations for determining the dimensions (d j , l j ) of passable capillary channels will take the form [16]: where j 2,m = ; ( ) According to results of the research into linear pressure dividers, it was found that the maximum value of division coefficient χ max as a rule does not exceed 30 and may be reached in a two-capillary divider only while it is slightly lower for multi-capillary ones [5].

3. Scheme of cascade connection of pressure dividers
To set the pressure drops, different in magnitude (at the level of several orders of magnitude), and ensure constant division coefficients, a scheme of cascade connection of linear multi-capillary dividers (i=1,…, n) is proposed, with arbitrary number of capillaries CE i, j , where j=1,…, m i [5,16].
The maximum value of division coefficient C max χ of cascade scheme division, in the case of applying in each of n cascades of linear two-capillary dividers of pressure, equals n n C max max 30 . χ = χ ≈ The scheme has one input channel, where the flow divides into n pressure dividers. At the input of each divider, the appropriate pressures are set: at the first -with a stabilizer of absolute or excess pressure, on the rest of dividers, the pressures are created by repeaters Rp k (k=2,…, n), each setting section of which is connected to the first (from the gas outpit) inter-throttle section, formed by capillaries CE k-1, 2 and CE k-1, 1 of the previous k-1-th divider. The outputs of all pressure dividers are connected to the output channel (flow combiner), the pressure of gas P 0 in which is maintained constant with the help of stabilizer of absolute pressure [17]. The pressure stabilizers at the output and input of the scheme (Fig. 3) are not shown.
Represented below is a system of recurrent dependencies, built on the basis of the system (11), for determining the dimensions (d i, j and l i, j ) of all passable channels of capillaries CE i, j of a cascade connection of linear pressure dividers The advantage of this scheme is that it remains linear when switching the sections of setting the repeaters Rp to other inter-throttle sections of dividers of higher pressure. Due to this, without modification of capillaries design of each divider, the scheme may provide a significant increase in the number of set inter-throttle pressures P i,j .
However, this scheme is limited in the value of the lower limit of the set pressure that is associated with errors of pressure reproductions by repeaters. Thus, for the repeaters of individual manufacture and adjustment of their pair "nozzle-valve", the absolute error of reproduction (repetition) of pressure may amount to a few tens of pascal and for the industrial repeaters it is higher by an order of magnitude [18].
The error of pressure reproduction can be significantly reduced when using a separate node of pressure reproduction. This node is a system of automatic control with negative feedback, which contains a high-sensitive null indicator to detect the pressure differences DP NI on the node's output of pressure reproduction and in the chamber of a repeater setting, to which a controlled throttle is connected, by changing the gas-dynamic resistance of which DP NI is eliminated.

Scheme with a binary ramification of dividers
Promising for building the systems of setting the pressures, different in magnitude, is the scheme of binary ramification of dividers, shown in Fig. 4. In general, each of its n pressure dividers is multi-element and contains, accordingly, m i (i=1,..., n) of capillaries. With this scheme, in contrast to the cascade one, one can obtain a value of inter-throttle pressures at the level of particles of pascal.
Gas under pressure of 1 1,m P .is supplied to the input of the scheme, which is also the input of the first divider. In the flow chamber of this divider, formed by capillaries CE 1,1 and CE 1,2 , the flow branches off to the input of the second divider. Then every subsequent i-th divider, which is connected to the corresponding inter-throttle chamber of the і-1-th divider, formed by capillaries CE i-1,1 and CE i-1,2 , branches off a part of the flow.
As a result of this connection of n pressure dividers, n-1 of nodes form in the scheme, in Fig. 4  To implement a linear function of the change in inter-throttle pressures P i,j = =f(P 1,m 1 ), an algorithm was developed for determining dimensions of passable channels of capillaries of the scheme, presented in [5].
Based on the laws of conservation of mass and Kirchhoff, using the dependency (1) for a scheme of binary ramification of pressure dividers, its mathematical model is received is the mass rate of gas through the capillary CE i,j (i 1,n = is the number, n is the number of branches; i j 1,m = is the number of capillary in the branch from the output to the input of gas, m i is the number of capillaries of the i-th branch); i,j
The scheme of the setting device (Fig. 5) is based on the principal scheme with binary ramification of the pressure dividers shown in Fig. 4. The setting device is created by four branches, each of which contains a pressure divider of 10 capillaries. The scheme is fed with air, the pressure of which at the input Р 1,10 =115 kPa and at the output Р 0 =105 kPa of a throttle scheme is maintained stable by, respectively, a stabilizer of absolute pressure Sp 1 and stabilizer Sp 0 . All scheme's elements are placed to the thermostat Тs, which keeps the temperature at T=313 K.
For alternate setting of inter-throttle pressures P i,j, the corresponding chambers with the help of electromagnetic valves Vl i,j are connected to the pneumatic switch Ps, the output channel of which is connected with a consumer by the valve Vl. Setting a certain sequence (separate value) of pressures is controlled by MCU, which switches on corresponding electromagnetic valves.
All inter-throttle pressures P i,j , as well as the input Р 1,10 and the output Р 0 pressures of the setting device, can be reproduced simultaneously without using the switch Ps.
A number P k ∆ of different drops of pressures in the case of setting within the i-th divider of equal drops of pressures on each of its capillaries (Fig. 4), that is, Thus, the setting device shown in Fig. 5 can set simultaneously 523 different pressure drops.
To ensure linearity and decade division of pressures in each of the scheme's dividers, the dimensions of passable channels are defined by the algorithm, presented in [5].
and is the system of 36 nonlinear equations relative to inter-throttle pressures P i,j , where i 1; 4; j 1; 9. = = Modeling the operation of a setting device scheme, presented in Fig. 5, demonstrated that the change in the input pressure does not lead to the change in the division coefficients c i,j . However, the change of the pressure Р 0 causes a slight change of c i,j , for example, at the unidirectional change in the supply pressure (P 1,10 , Р 0 ) by ±50 Pa, the maximum deviation of division coefficients c i,j does not exceed 0.005 %. The temperature change by up to ±5 K causes a deviation of division coefficients by the magnitude of up to 0.2 %.

Discussion of results of the research into linear dividers and development of devices for setting the pressures, different in magnitude
The results of the study of two-and multi-capillary dividers of pressures allowed the development of cascade connection schemes and binary ramification of dividers. The use of devices for setting pressures, built on the basis of these dividers, enables compensation of uncontrollable effects of external factors in different gas-dynamic systems, in which pressures should be stabilized. Thus, for example, in gas -dynamic synthesizers of gas mixtures, proportional changes in pressures at the inputs of dosing capillaries are achieved due to them, and, therefore, the rates of all dosed components are provided so that their concentrations in the mixture remain unchanged. This effect is virtually impossible when using certain independent pressure stabilizers for each component of the synthesized mixture.
Along with the mentioned advantages of the designed setting devices, one should admit the need of using a stabilizer of absolute pressure in them at the output, though the nonlinearity of divider without it is not substantial.
Research into pressure dividers revealed the prospects of their use in the devices for setting pressures, different in magnitude, i. e., for simultaneous setting of a large number of pressures (drops), significantly different in the value of absolute pressure. Such setting devices are particularly promising for the construction of synthesizers of gas mixtures with micro-concentrations of the components. In this regard, two schemes were designed -cascade and the scheme with binary ramification of pressure dividers. The advantage of the first one is the possibility of its adjustment for setting other pressures by switching the chambers of setting of repeaters to other inter-capillary chambers of dividers. The disadvantage is the need to use pressures repeaters to build the scheme, which limits the coefficient of pressures division. The advantage of the second scheme is its functioning without pressure repeaters, which removes the restrictions for setting any low pressures, which occur in gas-analytical practice. The disadvantage of this scheme is the inability of its adjustment to other inter-throttle pressures (drops) without replacing of all capillaries.
The use of such schemes in the devices for setting pressures opens up the prospect of constructing high precision gas-dynamic tools, in particular, to obtain complex mixtures with micro-concentrations of the components, setting low and micro rates of gas, testing micromanometers.
Quite a variety of tasks in the practice of scientific research into the area of gas-analysis requires the development of tools for providing the deployment of pressures by the dependencies that are different from the linear ones. Therefore, a solution of this problem requires separate study in future of capillary pressure dividers.

Conclusions
1. The possibility of providing the linearity of changing the inter-throttle pressures of capillary pressure dividers was shown, and thus the constancy of the coefficients of pressure division.
2. Based on the connection of linear pressure dividers, the schemes were built that provide the coefficients of division at the level of several orders of magnitude.
3. With the help of the scheme with binary ramification of capillary pressure dividers, a linear four-decade device for setting absolute pressures (drops) in the range of 105-115 kPa (0-10 kPа) was proposed. The setting device is almost insensitive to the effects of supply pressures on the coefficients of division and their deviation is less than ±0.2 % at the change in temperature within ±5 K.