Dr. Roman N. Tunkel
m - correction factor;
f - area the cross section of pneumoresistance;
P o , P 1 - absolute values of air pressures before and behind pneumoresistance;
To , g , r - temperature and specific gravity of air bbefore the pneumoresistance;
R - universal constant for gas;
g - gravitational acceleration ;
Z=P i+1 /P i - ratio the pressure of gas in the volume flow in to the pressure of gas in the volume flow out;
k - factor of adiabatic;
Wi ,W i+1 - velocities in the section flow in and in the section flow out;
Y - the function of consumption.
Most of problems designed air - distribution systems by pneumatic and mechanic manual bore-drive and cutting tools result in dependence between the gas flow and overfall of pressure on elementary part of pneumoline. If the gas motion with the velocity is small (w < 50 m/s ) without turbulence the flow parameters in a pipe of varying cross section - pneumoresistance - usually are calculated by means of following equations (deducted for the fluid incompressible): Veisbach [1], Zelenetskiy [12] etc.. So Turnquist R.O. [9] it was received experimental equation for weight gas flow per second, for the small overfall of pressure and the small compressibility:
If the gas motion with the velocity more than 50 m/s then necessary to take into account the thermodynamic parameters of the gas flow and the compressible of gas. In most of cases for the adiabatic turbulence flow out the weight gas flow calculated in accordance with Saint-Venan B. - Wantzel L. equation [8], deducted for flow out from unlimited volume through the short pipe (pneumoresistance) into volume with smaller pressure:
where: Z > 0.528
Because of expression (2) is very complicated many authors was proposed simplificat, approximate equations for the gas flow. These expressions make for the undercritical flow out of gas and to differ from expression (2) not more than 5 percent. For example, according to Berezovets [2], it was received the following experimental equation:
In accordance with Hertz and Krejnin [4] was received the equation for the isothermal flow out:
According to Lobyntsev [5], it was received the following experimental equation for the adiabatic process:
where: D P = P o - P 1 For comparison these equations it will be separating the part of the expression independ from (signed A):
Then variable the part of equations (1) - (5), dependent from Z , named the function of consumption Y can be define it in the following way:
Calculated values the function of consumption defined after equations (7) - (11) are illustrated in Fig.1. As it can be seen from Fig 1, values of the function Z = Z cr , (critical) suitable to maximum of the function of consumption Y and of the gas flow, exchange within from 0 to 0.607. Maximal (critical) value of the gas flow to fix in the time of full sound blocking up of the pneumo- resistance in its minimal the cross section. However, in experiments, the full sound blocking up takes place by anthers overfalls of pressure P an values Z cr . For example, for the round hole in the thin wall [3] Z cr has values from 0.1 to 0.26 and for pipes of Venturi [6],[10] from 0.6 to 0.8. It has been demonstrated that equations (7)-(11) can not produce high accuracy. For promotion the accuracy determining the gas flow by equation of the function of consumption to introduce the correction factor m . It is determined by ratio of the practically measured the gas flow - G exp ,to theoretical - Gteor , calculated by equations (1) - (5). Its the numerical value must not to exceed 1.0 [4], [11], that is not always observed in the experiment.
There are two reasons of this the theoretical and experimental discrepancies. First, deducing to above the gas- hydraulic equations can be used only for the flowing pneumoresistances with smoothly changing forms. Second, equation (2) and its simplifications were deduced for of flow out the gas from unlimited volume and account the velocity of gas motion has been equal to 0 before the pneumoresistance. This article is the attempt to determine of the gas flow through the pneumoresistance with of account the velocity of gas. For that we consider the motion of gas through ideal pneumo- resistance, which satisfies the following conditions:
- the gas in non-viscous, that is, therre is no internal friction force between adjacent gas layers;
- the gas motion is steady, meaning that thhe velocity, temperature and pressure at each point in the gas do not change in time;
- the gas moves without turbulence.
It is necessary to decide a system of differential or integral equations [7] -conservation of energy, equation of state and equation of continuity for determine relation between the gas flow parameters and its velocity. Then we may write Bernoulli-Saint-Venan's equation for termo-insulating gas flowing through pneumoresistance in its two cross sections - before and after the pneumoresistance:
We may write also for average in cross by section meaning of parameters:
- the equation of continuity gas flowiing
- and the equation of state
Transforming equations (12) and (14) we can receive by equation for W i+1:
where: Z i+1 = Pi+1 / Pi.
Or aid equation (13) and using the value of velocity W i+1 we have the expression for the weight gas flow per second:
- ratio cross of sections, before and after flowing change.
From (15) we separate the function of consumption:
For to define the maximum of this function we must to find the first derivative of the equation (16) and to equate its to 0:
This equation can be rearranged to the more famillar-looking form:
After to substitute for variable
and to consider (17), we will receive the algebraically expression as polynom six power of a number:
The graphical analysis by equation (18) it shows that:
-if e < 1 (the pneumoresistance is narrow ), the values of roots are in the interval 0.83 < x < 1.00;
-if e = 1 (the cross of section is constant) there is single root x = 1;
-if e > 1 (the pneumoresistance is widen ) - don't exist roots.
It is impossible to solve the equation (18) in general form.
Therefore, we used the linear of approximation on e4 :
This equation to allow us to find the approximate value of Z cr and maximum the function of consumption (16).
Better approximation can be received by more complicated expression:
It will be also, that in case e = 0 (fi = oo - this is flow out from unlimited volume ) equation (16) to transform into equation (8), that tell us about its more universal character, thanks to introducing parameter of the pneumoresistance " e".
Now, we stand on the interpretation receiving of the equation (16). For this purpose we consider three main fields of changing the parameter e (Fig.2):

field 1: 0 <= e < 1; field 2: e = 1; field 3: e > 1.
Field 1.
Supposing fi+1 is the same for every 0 <e <1 and using the equation (15) we receive the curves of the gas flow (Fig.3). The analysis of Fig.2 and Fig.3 it shows promotion of maximum the gas flow with increasing "e " and removing the numerical value of Zcr into the direction 1. This circumstance to illustrate the results of experiments, when Zcr has got values from 0.6 to 0.8 and it was m> 1. Really, for m > 1 it's enough in order to the decrease of gas flow occasioned by the compressed stream will be less than difference between the values theoretical of gas flow, calculated from using equations (2) and (15). Thus if m'= Gexp / Gsv > 1 , then m = Gexp / Ge <= 1 and Z cr decreasing up to 0.1 - 0.3 , this may be occasioned by the next cause.
Any the pneumoresistance is a system usually including of several elementary pneumoresistances are connecting with each other. For such system the total's value of Z cr always is less than the elementary value Zcr el ,calculated with equation (19) and equal:
where i - ordinal number the pneumoresisitance of system.
For example, if n = 2, Z1 = 0.8 and Z2 = 0.6 then Zo = 0.48, that is just less than Zcr = 0.528 , calculated or aid Saint - Venan's equation. In accordance with (20), as soon as even though at one the pneumoresistance will be Zel<Zcr then for the system at whole will be Zo <Zcr . In spite of <<Z cr for the system at whole we have possible to find the undercritical order flow out through the elementary pneumoresistance, and, in case of the overcritical order flow out even though at one the pneumoresistance we will have less than Zcr for the system at whole.
Field 2.
The situation is corresponds to flow out the gas in pipe through thin diaphragm (Fig.2). When Z i+1 =1 (the overfall of pressure and the diaphragm are absent) - the gas flow is the maximum. When Z i+1 = 0 (the diaphragm is closed of full) - the gas flow is equal 0. When 0<Z i+1<0 to observed the overfall of pressure and the gas flow to change.
Field 3.
The situation is corresponds for flow out the gas through the short widening pipe - pneumoresistance (Fig.2). In this case Pi<Pi+1 and Zi+1 >1 , therefore the numerator and the denominator in expression (16) is negative and in the same time all expression under the square root remains by positive. With "e " increase the maximum of gas flow abruptly fall but with increase it monotonously increases since when e > 1 there is not maximum of function (16). Thus we elucidate a question how velocity approaching of gas to the pneumoresistance influence on its capacity. Also, we received equations (15) and (16), given relations between the ratio of pressures and the ratio of sections areas "e " (before and after flow change) and the weight gas flow per second Ge.
Authors conclusions was experimentally verificated on blowing off test bench of enterprise "PNEUMATICA" (St.Peterburg, Russia). It were investigated pneumoresistance "types" of abrupt expansion and "abrupt narrow spot" with following values of "e " : 1.079, 1.332, 2.025, 4.000, 9.467, 18.778 & 0.055, 0.106, 0.205, 0.494, 0.751, 0.927. During the experiment it was registered following parameters: input pressure, pressure and temperatures before and after abrupt changing of flow, usual the compressed air consumption.
As far as during the experiment we measured the volume compressed air consumption
transformed to the form:
Then we compared the values of Ytheor, calculated according to formula (16), using known Z and e with Y exp:
Result of calculations and their analyses are presented in tabl.1:
The parameters of statistic Results of dispersion analyses
e 0.106 0.494 0.927 1.332 9.467
sum(dY) -1.401 -0.802 0.035 0.228 0.100
dYav -0.108 -0.057 0.030 0.016 0.007
sum(dY)2 0.193 0.082 0.001 0.007 0.001
0.059 0.053 0.008 0.016 0.005
61.284 -4.050 1.340 3.765 4.930
T tabl
2.179 2.179 2.160 2.160 2.160
dY -difference the values of consumption function , calculated through formula (16) and experimentally determened;
dYav -average of the differences;
Sd -corrected mean square error;
T -calculated values of Student's criterion;
T tabl - values of Student's criterion, received from standard tables on mathematic statistic.
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