RU2037458C1 - Method of transportation of loose materials, aerosols and capsules - Google Patents

Abstract

FIELD: transportation of loose materials. SUBSTANCE: method of transportation of loose materials, aerosols and capsules comes to delivery of gas and material to be transported into transportation pipeline in which rarefaction has been previously built. Gas is delivered at a speed not less than local sound velocity, then material to be transported is fed. Pressure level is maintained constant over entire length of pipeline. EFFECT: enlarged operating capabilities. 1 tbl

Description

 The invention relates to a technique for transporting gas solids, aerosols, and also formed capsules. It can be used in various fields of technology, for example, instead of hydraulic transport of mixtures, instead of railway or river transport for the delivery of bulk and dusty materials to unlimited distances when delivering grain from the field to elevators, when transporting flour over long distances, etc.

 There is a known method in which air is supplied between two concentrically arranged pipes, with the material flowing through the inner elastic pipe. An increase in feed range has been achieved. However, the method does not allow transportation over long distances, because it is necessary to apply a parallel supply duct, which is difficult. The speed of transport is limited. Unable to create a transport force field.

 A known method of transporting viscous electrically conductive materials, which has disadvantages: the material must be electrically conductive and viscous, significant energy costs.

·

RT Анализ уравнения и расчеты по нему показывают, что при разумных диаметрах трубопровода и при давлении в конус его, отвечающем уравнению Сен-Венана v

при минимальной скорости истечения газа в соответствии с уравнением v_В.К= C

+ βL $\genfrac{}{}{0ex}{}{\text{2}}{\text{п}}$ _р уже при незначительных расходах воздуха происходит быстрое "запирание" трубопровода, т.е. быстрый рост перепада давлений до значений второго и третьего порядка с увеличением длины и уменьшением диаметра трубопровода. Причиной "запирания" является рост функции коэффициента трения λ_b 0,316 Re^-0,25, где Re V_bD/ν, в связи с многократным увеличением динамической вязкости чистых газов и их смесей с ростом давления по уравнению ( η_m η_m ^o / ζ_m ( 1,08 / [ exp 1,439 ρ_rm exp ( 1,11 ρ_rm ^1,858 ) Увеличение динамической вязкости является также причиной роста внутреннего трения в газе или смесях. Рост давления приводит для некоторых газов к переходу их в критическое состояние, а для других к их насыщению и конденсации. Одновременно в длинном трубопроводе падает скорость газа до значений ниже критической, т. к. возрастает плотность. С ростом давления значительно возрастают капитальные затраты на реализацию задачи. При этом энергетические затраты используются, главным образом, на образование потенциальной энергии.The closest (prototype) is a method of transporting bulk solids, implemented in a suction-discharge unit for pneumatic conveying of bulk materials. Due to the combination of suction and discharge transport methods, the transportation range is increased. The installation has significant disadvantages. The range of the discharge unit is limited to 1-1.5 km due to the impossibility of creating an economical mode. The pressure loss in the pipeline with its large difference is determined by the equation
P $\genfrac{}{}{0ex}{}{\text{2}}{\text{1}}$ -P $\genfrac{}{}{0ex}{}{\text{2}}{\text{2}}$

·

RT Analysis of the equation and calculations on it show that for reasonable diameters of the pipeline and at a pressure in its cone, corresponding to the Saint-Venant equation v

at a minimum gas flow rate in accordance with the equation v _V.K = C

+ βL $\genfrac{}{}{0ex}{}{\text{2}}{\text{P}}$ _p even at low air flow rates there is a quick "locking" of the pipeline, i.e. rapid increase in pressure drop to values of the second and third order with an increase in length and a decrease in the diameter of the pipeline. The reason for “locking” is the increase in the function of the coefficient of friction λ _b 0.316 Re ^-0.25 , where Re V _b D / ν, due to the multiple increase in the dynamic viscosity of pure gases and their mixtures with increasing pressure according to the equation (η _m η _m ^o / ζ _m (1,08 / [exp 1,439 ρ _rm exp (1,11 ρ _rm ^1,858 ) An increase in dynamic viscosity is also the cause of an increase in internal friction in a gas or mixtures. An increase in pressure leads to a transition to a critical state for some gases, and for others to their saturation and condensation. Simultaneously in a long pipeline the gas velocity drops to values below cr cal, t. k. a density increases. With increasing pressure significantly increases the capital cost of implementation of the task. In this case, the energy costs are mainly used for the formation of potential energy.

 Suction transport is even less powerful. The disadvantages of this method are leaks in the system, unorganized input of the gas mixture and material, the inability to create a controlled mode, limited flow rate, and a small transportation distance.

 The purpose of the invention is to increase the efficiency of pipeline transport when moving materials, creating conditions that expand the capabilities of pipeline transport using gas flow energy as a force field. The goal involves expanding the scope of pipeline transport, replacing other modes of transport.

 The goal in the new method is achieved by the following methods. The transport pipeline along its entire length from the place of entry of the material and gas to the place of its unloading is inclusively isolated and sealed from the external environment, then the gas medium in it is pumped to a predetermined vacuum level. By pressure, the gas stream through the Laval nozzle and the substance or capsules through the metering device are separately introduced into the organized (working) space for transportation. Then, the vacuum level specified along the length of the transport pipeline is adjusted and further supported by the pumping devices, the substance, aerosols or capsules are moved under the action of the discharged gas stream in the viscosity, viscosity-molecular or molecular mode in each independently or simultaneously in several in the indicated sequence in any variant.

This solution differs from the known one in the following features:
the presence of insulation and sealing of the working space, i.e. places for entering the gas stream, substance, transport pipeline, connecting pipelines, fittings, space for the deposition of substances, block pumping devices. Insulation is achieved by the design of the device. Sealing at the inlet of the gas stream is achieved by “locking” the pipeline with the gas stream in the critical section of the Laval nozzle due to the flow of gas in it at a speed not lower than the local speed of sound;
separate entry of gas and material into the transport pipeline, a metering device located at a certain distance from the critical section of the Laval nozzle above the transport pipeline. In the space organized for dosing, the vacuum level, taking into account the presence of the dosed material, must correspond to its value in the transport pipeline at the junction of the dosing device. With low levels of rarefaction in the transport pipeline, its sealing from the storage tank for the dosed material can be carried out due to the design of the dosing device and maintaining the material in the storage tank at the required level. Sealing at the exit of the transport pipeline is ensured by attaching to the end of the pipeline an organized space for the separation of gas and substances;
pipeline sealing is provided by its design;
evacuation of a gaseous medium. The tightness of the working space creates the conditions for the possibility of pumping the system to any level set in advance. Entering the calculated gas flow in the presence of the tightness of the system and maintaining the vacuum at the specified calculated level creates a condition for its passage in the transport pipeline at a speed exceeding the Mach number. The range of movement of the gas stream increases with increasing rarefaction and increasing the magnitude of the gas stream. Calculations show that a mode in a pipeline with a length of 150-200 km with an average diameter of 30 cm can be built relatively easily to ensure the entire annual demand of the Nizhny Novgorod region, for example, in cement;
by introducing a gas stream through a Laval nozzle at a speed not lower than the local speed of sound. It is under this condition that it becomes possible to seal the system and at the same time introduce a predetermined settlement flow
separate entry into the transport pipeline of the gas stream and substances. It is a separate input that creates the conditions for locking the pipeline, without which it is impossible to pump out the system or workspace. In existing systems, the combined introduction of a gas stream and a substance into the transport pipeline after the mixing chamber is used. In addition, in existing systems it is also impossible to pump out the mixing chamber, since it provides for uncontrolled aeration of the material; tapering nozzle does not work in shut-off mode "; uncontrolled gas flow through the material supply system is possible.

 As the pipeline is filled with substance and gas mixture, the regime is regulated, which is supported by pumping devices along the length of the transport pipeline. The need to regulate the mode follows from the conditions for obtaining the most economical characteristics of the mixture flow, which depend on the mass of the material being moved, which changes as it is introduced.

The movement of matter is carried out under the action of a rarefied gas stream in a rarefied medium. The gas flow regime is selected from economic considerations as viscous, visco-molecular or molecular, depending on the specific conditions of mass transfer. The peculiarity of using the modes is:
firstly, in the disappearance of friction of the transferred mass on the channel walls, especially in a rarefied medium. The phenomenon increases sharply in rarefied gases due to an increase in the sliding velocity and the transverse velocity gradient; secondly, in decreasing the coefficient of friction of the gas as the flow increases and the vacuum increases, it is proved below; thirdly, in the disappearance of flow turbulence at Re <2200, K _n << 1 the formation of a laminar flow, a decrease in the energy costs of its creation at K _n > 10 ^-2 , fourthly, in a decrease in the absolute value of the gas flow and simplification of separation gas from the substance.

The essence of the method is stated as follows. Isolation and sealing of the working space is ensured by arranging a transport, for example, steel pipeline, to the required length. The input of the substance and its withdrawal from the pipeline are ensured by attaching to it a known device, as previously reported. New is the solution to the issue of introducing a gas stream. Fugitive input of flow and substance is one of the disadvantages of known systems. Inlet flow can be carried out using tapering, expanding and Laval nozzles. The use of a Laval nozzle under certain operating conditions and geometric parameters allows you to "lock" the input section of the pipeline with a gas stream in a critical section. The operation mode of a supersonic Laval nozzle can be built with calculated and non-calculated expansion. The non-calculated expansion of the gas stream is characterized by a backpressure value outside the nozzle exit section P _pr less than a fixed calculated value of P _II . Analysis of the second law of thermodynamics in the annex to the flow in the Laval nozzle to unplanned expansion shows that such a nozzle operation mode corresponds to the case 1st, when the nozzle inlet Mach number <1, and for the critical section M> 1. In this case, P _prI <Р _II ^I , the gathering of rarefaction waves is observed from the edges of the nozzle, and the operation of the nozzle is similar to the case of flow without friction.

> 0,5 2. Отрицательным значением изменения энтропии
S_к-S_н= Rln

Rln

× W_к(1-W $\genfrac{}{}{0ex}{}{\text{2}}{\text{к}}$ )

Rln

·

< 0

3. W_k < W^*, где для из энтропического случая
W_к=

1-

W^*

Основные расчетные характеристики сопла могут быть получены из условия максимальной тяги:

(P_к-P_пр)

0 которое после ингибирования приводится к виду

·

или

·

При ламинарном течении в круглом канале
Re

а для турбулентного течения по каналу круглого поперечного сечения с гладкими стенками определяется уравнением Кармана

2log

Re

-0,8
Условие запирания определяется величиной максимальной приведенной скорости W_н или максимального числа Маха М_н. Течение возможно, если величина А_к/A_t задана и ее значение меньше или равно приведенной площади, вычисленной по уравнению

=

1+

M

Течение возможно, если W_н ≅ (W_н)max или М_н ≅ (М_x)max, где
W

, a U_max=

M²=

·

Условие запирания наступает в недорасширенном сопле, когда скорость потока в суженном сечении принимает значение скорости звука, при этом критическое отношение давлений

зависит от К и выражается Ψ_кр=

Для воздуха K=1,4 ϑ= 0,53. Настоящее состояние течения отвечает случаю 4. Переход в область недорасширенного режима течения, когда значения Р_н ^о и Р_прстановятся независимыми, описан выше тремя условиями. Случай 5 отвечает переходному режиму, когда Р_II Р_пр. В этом случае

0,5; S_к-S_н< 0 и W_к> W^*
Таким образом, задаваясь значением Р_пр и Р_н ^о, а также значением С_fи С_с, определяем из уравнения максимальной тяги величину

. Затем находим значение Р_к. Число Маха определяется для из энтропического течения

= 1+

· μ²=

Далее определяем

C_mX(γ)

C_v(W_к)

Определяем коэффициент тяги

C

W_к)_uз+

(W_к) $\genfrac{}{}{0ex}{}{\text{2}}{\text{u}}$ _з= 1-

По величине C_F
C_F=

+

(I_к) (I_к)_изC_m˙C_v˙C_c где
(I_к)_из= m

·

· M

PA[γM²] где Р Р_к, А А_к, М² М_к ². Находим I_к, а затем A_t По уравнению

P_кA_к

определяем расход газа.An analysis of the operation mode of the nozzle shows that this mode can be fixed by three conditions: 1.

> 0.5 2. The negative value of the change in entropy
S _to -S _n = Rln

Rln

× W _to (1-W $\genfrac{}{}{0ex}{}{\text{2}}{\text{to}}$ )

Rln

·

<0

3. W _k <W ^* , where for from the entropic case
W _to =

1-

W ^*

The main design characteristics of the nozzle can be obtained from the maximum thrust condition:

(P _a -P _etc.)

0 which after inhibition is reduced to

·

or

·

In laminar flow in a round channel
Re

and for turbulent flow along a channel of circular cross section with smooth walls it is determined by the Karman equation

2log

Re

-0.8
The locking condition is determined by the maximum reduced speed W _n or the maximum Mach number M _n . The flow is possible if the quantity A _k / A _{t is} given and its value is less than or equal to the reduced area calculated by the equation

=

1+

M

The flow is possible if W _n ≅ (W _n ) max or M _n ≅ (M _x ) max, where
W

, a U _max =

M ² =

·

The locking condition occurs in the underexpanded nozzle, when the flow velocity in the narrowed section takes the value of the speed of sound, while the critical pressure ratio

depends on K and is expressed Ψ _cr =

For air, K = 1.4 ϑ = 0.53. The current state of the flow corresponds to case 4. The transition to the region of an underexpanded flow regime, when the values of P _n ^o and P _pr become independent, is described above by three conditions. Case 5 corresponds to the transition mode when P _II P _ave . In this case

0.5; S _to -S _n <0 and W _to > W ^*
Thus, given the value of P _CR and P _n ^about , as well as the value of C _f and C _with , we determine from the equation of maximum thrust value

. Then we find the value of P _to . Mach number is determined from the entropic flow

= 1+

Μ ² =

Next we define

C _m X (γ)

C _v (W _to )

Determine the thrust coefficient

C

W _to ) _uz +

(W _to ) $\genfrac{}{}{0ex}{}{\text{2}}{\text{u}}$ _s = 1-

Largest C _F
C _f =

+

(I _k ) (I _k ) _from C _m ˙C _v ˙C _c where
(I _k ) _of = m

·

M

PA [γM ² ] where P P _k , A A _k , M ² M _k ² . We find I _k , and then A _t By the equation

P _to A _to

determine the gas flow.

Таким образом, режим в сопле Лаваля считается построенным, а критическое сечение запертым, что создает условия для откачки газа.A more accurate calculation is based on a detailed analysis of the operation of the Laval nozzle taking into account the change in C _F along the length of the nozzle based on the differential equation

Thus, the regime in the Laval nozzle is considered to be built, and the critical section is locked, which creates conditions for gas pumping.

1+

а для твердых частиц
M $\genfrac{}{}{0ex}{}{\text{2}}{\text{p}}$ =

1+

Уравнения свидетельствуют о том, что в горле обе фазы имеют дозвуковые скорости: смесь газа с твердыми частицами является диссипативной системой даже при отсутствии трения на стенках. С учетом трения на стенках уравнения движения в одномерном случае принимает вид:

(

T)

u

+

u

+ 2

u

Число Маха газовой фазе в горле равно
Mg²

1+

+ 2

u

Клигель и Никерсон теоретически определили число Маха газа в горле, вычисленное по скорости звука в газе, которое оказалось равным 0,8.The condition for blocking the Laval nozzle by the gas flow revealed above is not sufficient to evaluate the phenomenon. No less important for the correct conclusion is the knowledge of the conditions of motion of a multiphase system in the throat. For an adiabatic process without friction on the walls, the speed of sound in the mixture can be written in the form:
Mg ²

1+

and for particulate matter
M $\genfrac{}{}{0ex}{}{\text{2}}{\text{p}}$ =

1+

The equations indicate that in the throat both phases have subsonic speeds: a mixture of gas with solid particles is a dissipative system even in the absence of friction on the walls. Taking into account friction on the walls, the equations of motion in the one-dimensional case take the form:

(

T)

u

+

u

+ 2

u

The Mach number of the gas phase in the throat is
Mg ²

1+

+ 2

u

Kligel and Nickerson theoretically determined the Mach number of the gas in the throat, calculated from the speed of sound in the gas, which turned out to be 0.8.

 The foregoing allows us to make an important conclusion. First, the gas phase mixed with the substance in the Laval nozzle does not reach the speed of sound. Secondly, it is impossible to ensure that the pipeline is blocked by entering through a Laval nozzle, by ejection, or in another way, a gas stream together with the substance. Separate entry of the gas stream and substance is an indispensable condition for sealing the working space.

F(r)dr При расчете крионасосов следует иметь ввиду, что величина скорости адсорбции к чистой поверхности определяется уравнением
S_A= 3,65

·

A, а величина потока из уравнения I 3,65

·

P·A Время пребывания частиц на поверхности описывается уравнением Френкеля:
τ_пр= τ_{пр o}·exp

на основании которого определяется время размораживания. Время полного покрытия поверхности криопанелей мономолекулярным слоем задается выражением:
τ_покр= 0,28·N

·

Равновесие между адсорбцией и десорбцией газов позволяет вычислить число молекул газа, осевших на поверхность в соответствии с зависимостью на 1 см² поверхности:
N₁= 3,5·10²²·τ_пр

·exp

От количества осевших на поверхность молекул газа зависит степень заполнения поверхности θ, которая определяет величину коэффициента прилипания γ.In the presence of sealing the working space, the vacuum level in the system will be determined by the value of the total gas flow along the length of the pipeline. The magnitude of this flow will depend on the role of the bound gases. Adsorption and desorption of gases are the result of weak (van der Waals dispersion forces) and strong (valence forces) interactions of gas particles with a surface, which is characterized by the binding energy according to the equation
W

F (r) dr When calculating cryopumps, it should be borne in mind that the rate of adsorption to a clean surface is determined by the equation
S _A = 3.65

·

A, and the magnitude of the flow from equation I 3.65

·

P · A The residence time of particles on the surface is described by the Frenkel equation:
τ _pr = τ _{pr o} · exp

on the basis of which the defrost time is determined. The time of complete coverage of the surface of cryopanels with a monomolecular layer is given by the expression:
τ _cover = 0.28 · N

·

The balance between the adsorption and desorption of gases allows us to calculate the number of gas molecules deposited on the surface in accordance with the dependence on 1 cm ² surface:
N ₁ = 3.5 · 10 ²² · τ _ol

Exp

The degree of filling of the surface θ, which determines the value of the adhesion coefficient γ, depends on the number of gas molecules deposited on the surface.

-

Диффузия газа в твердых телах по первому закону Фика дает значение потока
I₁= -D

где D D_o·exp

Проникание газа сквозь стенки сосудов может быть оценено коэффициентом проникания
П П_o·exp

где П_о r_oD_o. Поток газа с одного см² в установившемся движении сквозь стенку определяется из уравнения.When taking into account the absorption of gases in solids, the calculation of the flow from diffusion and penetration of gas through the walls is carried out according to the law of Raul, Henry and Siverts in the form n _r r ˙ P ^u . The value of the solubility coefficient is calculated from the equation
rr _o exp

-

The diffusion of gas in solids according to Fick's first law gives the value of the flow
I ₁ = -D

where DD _o · exp

The penetration of gas through the walls of the vessels can be estimated by the penetration coefficient
N _{o o} · exp

where P _o r _o D _o . The gas flow from one cm ² in steady motion through the wall is determined from the equation.

Общее количество газа за время τ, истекающего с единицы поверхности стенки при установившемся движении определяется по второму закону Фика Q₁= 2n

Для неустановившегося движения количество газа определяется как Q

=

I₁dr

(τ-τ_o) τ_o=

Это уравнение при Р ≅ Р₂ и τ ≥ τ_o переходит к виду установившегося движения
Q₁= I_iτ

·τ

·P $\genfrac{}{}{0ex}{}{\text{1}}{\text{2}}$ ·τ
При эксплуатации системы следует ожидать газовыделения и натекания в систему, при этом процесс газовыделения предполагает адсорбционное и диффузионное газовыделение, а также газопроницаемость. Удельная величина газовыделения может быть оценена по уравнению lgq A Bt. Натекание определяется: Q_н К_в ˙ m ˙ Q_теч. Общая величина потока Q_с при определенных условиях может быть сведена к минимуму.I _i = D · r

The total amount of gas over time τ flowing from a unit of the wall surface during steady-state motion is determined by the second Fick law Q ₁ = 2n

For transient motion, the amount of gas is defined as Q

=

I ₁ dr

(τ-τ _o ) τ _o =

This equation for P ≅ P ₂ and τ ≥ τ _o goes over to the form of steady motion
Q ₁ = I _i τ

Τ

· P $\genfrac{}{}{0ex}{}{\text{1}}{\text{2}}$ Τ
During operation of the system, one should expect gas evolution and leakage into the system, while the gas evolution process involves adsorption and diffusion gas evolution, as well as gas permeability. The specific amount of gas evolution can be estimated by the equation logq A Bt. The leakage is determined by: Q _n K _in ˙ m ˙ Q _tech . Under certain conditions, the total flux Q _s can be minimized.

3,64αA

·

1-

и в вязкостном режиме:
S

αA

При значении (1- α) более критического 1-α

скорость потока ниже звуковой и быстроту откачки определяют по уравнению
S_D ^'' α A˙ W, а при скорости выше звуковой
S_D ^''' 3,77 ˙ α X ˙ S^' _т
при X > 2, где X

S

3,64·A

1-

lgP^*

+ BlgT+C+ Величина коэффициента криозахвата определяется расчетным путем для данного вида криопанелей методом Монте-Карло. Быстроту действия двухроторного насоса определяют:
S

а расположение насосов должно быть на расстоянии, определяемым допустимым уровнем концентрации паров масла из уравнения:
q q_o·exp

-

x

Уровень давления в системе зависит от ее объема и времени откачки
P P_o·exp

·

Время откачки находим по уравнению
S [-2,3lg(P_т/760)]

S_τ·v/τ₁
Наиболее ответственным моментом предлагаемого способа является решение задачи создания расчетного режима. Возможны три режима: вязкостный, переходный (вязкостно-молекулярный), молекулярный. Сообщается более дробное определение режимов. Учитывая, что отсутствует практика создания трубопроводов разреженного газа значительной протяженности (например, несколько тыс.км.) при большом диаметре (например 300-500 мм), в настоящей работе приводятся различные подходы решения данной задачи.It was reported above that a vacuum pump or a group of them is attached to the place of deposition of the substance. The most appropriate solution to this problem, taking into account the development of science and technology, will be a solution that includes a combination of a series connection of a cryogenic condensation pump and, for example, a two-rotor vacuum (Roots type pump). Another combination can be chosen, however, the adopted combination will allow us to solve the problem of creating the necessary regime at high gas separation (up to 10 ⁸ l / s). The speed of action of the cryocondensation pump is expressed in the molecular mode by the equation:
S

3.64αA

·

1-

and in viscous mode:
S

αA

If the value (1-α) is more critical, 1-α

flow rate below sound and pumping speed is determined by the equation
S _D ^'' α A˙ W, and at a speed above sound
S _D ^''' 3.77 ˙ α X ˙ S ^' _t
for X> 2, where X

S

3.64 · A

1-

lgP ^*

+ BlgT + C + The value of the cryocapture coefficient is determined by calculation for this type of cryopanel using the Monte Carlo method. The speed of the two-rotor pump is determined by:
S

and the location of the pumps should be at a distance determined by the acceptable level of concentration of oil vapor from the equation:
qq _o exp

-

x

The pressure level in the system depends on its volume and pumping time
PP _o · exp

·

The pumping time is found by the equation
S [-2.3lg (P _t / 760)]

S _τv / τ ₁
The most crucial moment of the proposed method is the solution of the problem of creating a settlement mode. Three modes are possible: viscous, transition (viscous-molecular), molecular. A more detailed definition of the modes is reported. Considering that there is no practice of creating rarefied gas pipelines of considerable length (for example, several thousand km) with a large diameter (for example, 300-500 mm), this paper presents various approaches to solving this problem.

При

≅ 1 имеет место молекулярный режим; для

≥ 100 вязкостный и для 100 ≥ К_н ≥ 1 промежуточный. Основные уравнения, описывающие движение разреженного газа изложены в ряде работ. Наиболее известным является уравнение Паузейля
Q

Pa(P₂-P₁)
Уравнение Паузейля выведено с учетом ряда допущений: газ не сжимаем; течение полностью сформировано; турбулентное течение полностью отсутствует; скорость у стенки равна нулю; течение изотермическое.The type of rarefied gas flow is determined by the Knudsen number
Kn

At

≅ 1 there is a molecular regime; for

≥ 100 is viscous and for 100 ≥ K _n ≥ 1 intermediate. The basic equations describing the motion of a rarefied gas are described in a number of works. The most famous is the Pausale equation
Q

Pa (P ₂ -P ₁ )
The Pausale equation is derived taking into account a number of assumptions: we do not compress gas; the current is fully formed; turbulent flow is completely absent; the speed at the wall is zero; isothermal flow.

уточнено Лангхааром:
P₂-P₁=

U_fa+1,14ρU $\genfrac{}{}{0ex}{}{\text{2}}{\text{f}}$ _a а значение потока задается выражением
Q

Для случая ламинарного течения при Re 1000 поток должен быть меньше, чем 3,72 ˙ 10⁵ Q м тор. л/c для условий воздуха при t 25^оС, где Q в см⁴, Q в м тор.л/с.The equation fairly describes the motion under viscous conditions at velocities corresponding to the value of the Mach number M ≅ 1/3. In this case, the average velocity over the cross section is
U _fa Q / π Q ² P The Pausale equation for the case of a formed flow at
l ≥ 0,304 aRe, where Re

clarified by Langhaar:
P ₂ -P ₁ =

U _fa + 1,14ρU $\genfrac{}{}{0ex}{}{\text{2}}{\text{f}}$ _a and the value of the stream is given by the expression
Q

For the case of laminar flow at Re 1000, the flow should be less than 3.72 ˙ 10 ⁵ Q m torr. l / s for air conditions at t 25 ^о С, where Q in cm ⁴ , Q in m tor.l / s.

Pa мтор·л/с; F 2,84

л/c Увеличение давления при поворотах и изменениях радиуса оценивается значением 1/2ρU_fa ² или

U_fa ². Режимы течения определяются: при
a Р мтор > 500 вязкостный;
a Р мтор < 5 молекулярный;
5 < Р мтор < 500 промежуточный;
a в см, Р в мторах.Assumption 1 is possible: Q <37Q ² P mltor.l / s;
assumption 2: l ≥ 0.304 a Re and assumption 3: Q <3.72 × 10 ⁵ a.m. tor.l / s. For air at t 25 ^о С, η 1.845 ˙ 10 ^-4 pauses, if a and l are expressed in centimeters, and pressure in millimeters, the expression for throughput has the form:
F 2.84

Pa mtor · l / s; F 2.84

l / c The increase in pressure during bends and changes in the radius is estimated by the value 1 / 2ρU _fa ² or

U _fa ² . The flow regimes are determined: at
a P mtor> 500 viscous;
a P mtor <5 molecular;
5 <P mtor <500 intermediate;
a in cm, P in mtors.

где va

а проводимость F

при этом для воздуха значение проводится для длинной трубы постоянного поперечного сечения равно
F

· va или F 19,40

л/c Для цилиндрической трубы радиуса a
F

va
F 30,48

л/с
В промежуточном режиме течения учитывается скольжение газа на границе, при этом пропускная способность для трубы выражается.In molecular mode, the flux is expressed by the Knudsen relation:
Q

where va

and conductivity F

in this case, for air, the value is carried out for a long pipe of constant cross section equal to
F

Va or F 19.40

l / c For a cylindrical pipe of radius a
F

va
F 30.48

l / s
In the intermediate flow regime, gas slip at the boundary is taken into account, while the throughput for the pipe is expressed.

Pa

1+

где ζ

·

·

Наилучшую сходимость с опытом в стеклянных и медных трубах дает зависимость
F F

1+4

-1

Значение f выражается:
4

-1

6,793

Значение A/F_t выражено по зависимости F F

0,1472

+ Z

для значения:
Z

при условии постоянства динамической вязкости η 0,494. Для значения Q/La= 0,323 наблюдается минимум функции, а при

0

1 Величина газового потока в областях молекулярного течения пропорциональна (Р₂ Р₁), а в области вязкостного величине (Р₂ ² -Р₁ ²).F

Pa

1+

where ζ

·

·

The best convergence with experience in glass and copper pipes is given by the dependence
Ff

1 + 4

-1

The value of f is expressed:
4

-1

6,793

The value of A / F _{t is} expressed by the dependence FF

0.1472

+ Z

for value:
Z

subject to a constant dynamic viscosity η 0.494. For Q / La = 0.323, a minimum of the function is observed, and at

0

1 The magnitude of the gas flow in the areas of molecular flow is proportional to (P ₂ P ₁ ), and in the viscosity range (P ₂ ² -P ₁ ² ).

(n_a-n_b)

1-2

W(x)

-2

где выражение в фигурных скобках представляет собой коэффициент Клаузинга W.The movement of the gas stream in the region of molecular flow can be described by the equation:
W πr

(n _a -n _b )

1-2

W (x)

-2

where the expression in braces is the Clausing coefficient W.

≥ 500 W

а молекулярный и массовый расход выражается
W ωπr

(n_a-n_b)
G Wπr

Течение в круглых трубах в широком диапазоне описывается зависимостью:
Q(δ)

2πru(r)dr полученной из решения модельного уравнения. Экспериментальные исследования Донга, Бролея и Б. Т. Породнова удовлетворительно согласуются с теорией. Уравнение приведенного расхода имеет вид:
Q q

где

Δ P P₁ P ₂
a радиус трубы, Т температура стенок, g_Σ объемный расход газа через трубу.For

≥ 500 W

and molecular and mass flow rates are expressed
W ωπr

(n _a -n _b )
G wπr

The flow in round pipes in a wide range is described by the dependence:
Q (δ)

2πru (r) dr obtained from the solution of the model equation. The experimental studies of Dong, Brolley, and B. T. Porodnov are in satisfactory agreement with theory. The equation for the reduced flow rate is:
Q q

Where

Δ PP ₁ P ₂
a is the radius of the pipe, T is the wall temperature, g _{Σ is the} volumetric flow rate of gas through the pipe.

Приведенный расход Q может быть определен из зависимости Кнудсена:
Q

+8,52

При этом режимы течения определяются показателем Кнудсена К_н ≅ 0,01- континуумное; К > 100 свободное молекулярное. Значения К_н задаются:
K_н=

;
σ πd²=

m

;
K_н= 1,27

M/Re;
Re

При гиперзвуковом обтекании:
K_н=

Re_o= ρ_∞u_∞L/μ_o=

C_*·

T_o= T

1+

M

≈

T_∞M² Область континуума может быть определена зависимостью L/l= Rej/M которая с учетом соотношения δ/L

1/

преобразуется к виду δ/L

/M где δ толщина ламинарного пограничного слоя;
L характерный размер тела;
Rey ρ VL/ μ число Рейнольдса для тела;
М число Маха.Sparseness parameter δ

The reduced flow rate Q can be determined from the Knudsen dependence:
Q

+8.52

In this case, the flow regimes are determined by the Knudsen index K _n ≅ 0.01 - continuum; K> 100 free molecular. The values of K _n are set:
K _n =

;
σ πd ² =

m

;
K _n = 1.27

M / Re;
Re

With hypersonic flow:
K _n =

Re _o = ρ _∞ u _∞ L / μ _o =

C _*

T _o = T

1+

M

≈

T _∞ M ² The continuum region can be determined by the dependence L / l = Rej / M which, taking into account the ratio δ / L

1/

transforms to the form δ / L

/ M where δ is the thickness of the laminar boundary layer;
L characteristic body size;
Rey ρ VL / μ Reynolds number for the body;
M is the Mach number.

 The gas dynamics of the continuum should consider problems in which the thickness of the boundary layer is 100 times greater than the average mean free path, i.e.

/M > 100 Соотношение L/l

Rey/M свидетельствует, что концепция континуума не применима для очень больших значений числа Маха и для очень малых чисел Рейнольдса.

/ M> 100 L / l ratio

Rey / M indicates that the concept of continuum is not applicable for very large Mach numbers and for very small Reynolds numbers.

· 4f

=

· 4f

·4f

·4f

·4f

·4f

·4f

·

·4f

При учете сил трения изоэнтропное давление торможения и импульсная функция должны уменьшаться, а энтропия в адиабатических процессах не может уменьшаться, потому значение коэффициента трения f всегда больше нуля.The nature of the change in the flow parameters depends on what values the Mach number takes more or less than unity, since the term (1 M ² ) is present in all differential equations in the denominator:

4f

=

4f

4f

4f

4f

4f

4f

·

4f

When the friction forces are taken into account, the isoentropic braking pressure and the impulse function must decrease, and the entropy in adiabatic processes cannot decrease, therefore, the value of the friction coefficient f is always greater than zero.

 The nature of the change in the flow parameters is determined by the following table.

 It follows from the above that the Mach number tends to unity. A smooth transition from subsonic flow to supersonic flow or vice versa is not possible. Given the initial flow conditions at the inlet of the pipe, its maximum possible length, at which the initial conditions remain unchanged and breaks or jumps do not appear, determines the Mach number, which must be exactly equal to 1.

 Friction contributes to an increase in the speed of subsonic flow and is the cause of the increase in pressure at supersonic speeds.

 In integral form, the equations are as follows and describe the movement of the gas stream at almost any speed.

4f

dm² После интегрирования имеем
4

+

ln

где f средний коэффициент трения, зависящий от длины и определяемый выражением:

fdx Максимальная величина 4

L/D дана для любого исходного значения числа Маха.

4f

dm ² After integration, we have
4

+

ln

where f is the average coefficient of friction, depending on the length and defined by the expression:

fdx Maximum value 4

L / D is given for any initial Mach number.

4

-

4

Из предыдущих уравнений получаем:

dM² Обозначив давление в точке М 1 через Р^*, проинтегрируем последнее уравнение в интервале сечений М М₁, Р Р до М 1, Р Р^* и получим:

M

= lnM²

Величины Р^*, Y^* и т.д. являются параметрами потока для поперечного сечения трубы, в котором М 1.The length of the pipe, which is necessary for the transition of the flow from a state with a Mach number M ₁ to a state with a Mach number M _2, is determined as the difference between these values in the initial and initial states:
4

4

-

4

From the previous equations we get:

dM ² Denoting the pressure at the point M 1 through P ^* , integrate the last equation in the interval of the sections M M ₁ , P P to M 1, P P ^* and get:

M

= lnM ²

Values of P ^* , Y ^* , etc. are the flow parameters for the cross section of the pipe in which M 1.

Наиболее важным моментом в решении задачи является выявление достаточно точного значения коэффициента трения. Для несжимаемого потока он выражается уравнением Кормана-Никурадзе.To determine the parameter of the adiabatic flow in the interval of sections in which the Mach number varies from M ₁ to M ₂ , for example:
P ₂ / P ₁ =

The most important point in solving the problem is to identify a sufficiently accurate value of the coefficient of friction. For an incompressible flow, it is expressed by the Korman-Nikuradze equation.

-0,8+2log₁₀[Rey

] Rey число Рейнольдса для трубы, выраженное через диаметр. Коэффициент трения зависит от шероховатости. Эта зависимость установлена Моуди для дозвуковых потоков. Экспериментальные исследования, полученные Кинаном и Ньюманом, позволили установить, что значение очевидного коэффициента трения для сверхзвукового потока имеет значение в два раза меньше обычного, применяемого для несжимаемых потоков.

-0.8 + 2log ₁₀ [Rey

] Rey Reynolds number for the pipe, expressed in terms of diameter. The friction coefficient depends on the roughness. This relationship has been established by Moody for subsonic flows. Experimental studies by Keenan and Newman have shown that the apparent coefficient of friction for a supersonic flow is two times smaller than the usual value for incompressible flows.

u

где U относительная скорость,
h расстояние между пластинами,
Re число Рейнольдса.In the region of rarefied gases, the reliable value of the coefficient of friction requires clarification. In a first approximation, it can be estimated based on the expression for the resistance force referred to the unit area for the Couette flow:
τ _w =

u

where U is the relative speed,
h is the distance between the plates,
Re Reynolds number.

средняя длина пробега молекулы. По аналогии для круглой трубы постоянного диаметра можно записать:
f

Запишем выражение для местной скорости потока u

где Р ρ RT, тогда
u

а f

Местное значение f зависит от величины потока и с его ростом уменьшается. В континуумной среде местное значение f не может быть менее чем
f

если не учитывать уменьшения динамической вязкости с возрастанием степени разрежения.σ part of the molecules reflected diffusely,

The average path length of the molecule. By analogy for a round pipe of constant diameter, you can write:
f

We write the expression for the local flow velocity u

where P ρ RT, then
u

and f

The local value of f depends on the magnitude of the flow and decreases with its growth. In a continuum environment, the local value of f cannot be less than
f

if you do not take into account the decrease in dynamic viscosity with increasing degree of vacuum.

1+

P

P+

Уравнение можно представить в форме
Q_м=

(P $\genfrac{}{}{0ex}{}{\text{2}}{\text{1}}$ -P $\genfrac{}{}{0ex}{}{\text{2}}{\text{2}}$ )+

(P₁-P₂)

Коэффициент скольжения меняется как средний свободный пробег молекулы, т.е. обратно пропорционально давлению
ζ ζ₁/P; ζ₁= ζP 2C

LP где 0,491 < С < 0,499,
f коэффициент обмена количеством движения.The slip viscosity regime is described by the equation for mass flow:
Q _m =

1+

P

P +

The equation can be represented in the form
Q _m =

(P $\genfrac{}{}{0ex}{}{\text{2}}{\text{1}}$ -P $\genfrac{}{}{0ex}{}{\text{2}}{\text{2}}$ ) +

(P ₁ -P ₂ )

The slip coefficient varies as the mean free path of the molecule, i.e. inversely proportional to pressure
ζ ζ ₁ / P; ζ ₁ = ζP 2C

LP where 0.491 <C <0.499,
f coefficient of exchange of momentum.

1+

P

Для выявления ламинарности или турбулентности режима можно использовать уравнение для Re
Re

Уравнение получено при P₂ 0 и ρ

Последнее выражение справедливо только для изотермического течения.The gas flow in units is recorded:
Q PV

1+

P

To identify the laminarity or turbulence of the regime, one can use the equation for Re
Re

The equation is obtained for P ₂ 0 and ρ

The last expression is valid only for an isothermal flow.

At Re ≅ 1000, a laminar regime takes place;
at Re> 1000 turbulent.

 Turbulent mode in solving practical problems of creating gas flows in main pipelines. Based on experimental studies, it was found that the laminar regime is observed at Re <1200, and the turbulent one at Re> 2200.

(P₁-P₂) При L < R справедливо:
Q_м=

1+

(P $\genfrac{}{}{0ex}{}{\text{2}}{\text{1}}$ -P $\genfrac{}{}{0ex}{}{\text{2}}{\text{2}}$ ) хотя следует отметить, что P ≠

, т.к. давление по длине трубопровода не изменяется по закону прямой, и указанное упрощение может оказаться несправедливым.The basic equation for the molecular regime is
Q _m =

(P ₁ -P ₂ ) For L <R, the following is true:
Q _m =

1+

(P $\genfrac{}{}{0ex}{}{\text{2}}{\text{1}}$ -P $\genfrac{}{}{0ex}{}{\text{2}}{\text{2}}$ ) although it should be noted that P ≠

because the pressure along the length of the pipeline does not change according to the direct law, and the indicated simplification may turn out to be unfair.

см³/с где Р_т среднее давление в барах
α₁=

см³/с·бар
α₂=

Ua 3,0476

см³/с
C₁= 2R

C₂= 2,47R

Функция проводимости выражается:
F

·

+ 3,048·10⁴·Z

М молекулярная масса:
Т температура газа.Knudsen proposes expressions for the transition regime
F α ₁ P _m + α ₂

cm ³ / s where P _t average pressure in bar
α ₁ =

cm ³ / s · bar
α ₂ =

Ua 3.0476

cm ³ / s
C ₁ = 2R

C ₂ = 2.47R

The conductivity function is expressed:
F

·

+ 3.048 · 10 ⁴ · Z

M molecular weight:
T is the temperature of the gas.

L₁ средний свободный пробег молекул газа при рассматриваемой температуре,
Р_т среднее давление в барах. Исходя из этих формул, можно получить уравнения, определяющие приведенный расход (РV), массовый расход и объем при давлении Р и среднюю скорость U
Q_PV= F(P₁-P₂)

+ 3,048·10⁴·Z

(P₁-P₂)
Q_м=

+ 3,048·10⁴·Z

V

+ 3,048·10⁴·Z

U

+ 0,9707·10⁴·Z

Среднее значение давления выражается:
P_м=

Pdx
P_м=

dl,
где где B

P_м=

1-

Окончательно среднее давление получаем
P_м=

P′-

где P¹=

Наложение режимов можно выполнить с помощью формулы
σ l^{- 2 /K}n ^{- 0,2 )} где σ часть молекул, отвечающих закону молекулярного режима
F_n (1 σ )F_c + σ F_м, где F_c проводимость в режиме скольжения;
F_м проводимость в молекулярном режиме. При f ≠ 1 проводимость определяется:
F

F_n Основные уравнения для молекулярного режима могут быть записаны
Q_м=

(P₁-P₂)
Q_pv=

(P₁-P₂)
V

(P₂-P₁)
U

(P₁-P₂) Средняя скорость на выходе
U₂=

-1

При температуре воздуха 17^оС имеем
U₂= 3,0723

-1

см/с см/с. Проводимость
F

или с молекулярной массой М
F

R³ см³/с при написании уравнений предполагалось, что труба имеет бесконечную длину, стенки полностью рассеивают молекулы, газ находится в равновесии и подчиняется закону Максвелла.Z

L _{1 the} average free path of the gas molecules at the considered temperature,
R _{t is the} average pressure in bars. Based on these formulas, it is possible to obtain equations that determine the reduced flow rate (RV), mass flow rate and volume at pressure P and average velocity U
Q _PV = F (P ₁ -P ₂ )

+ 3.048 · 10 ⁴ · Z

(P ₁ -P ₂ )
Q _m =

+ 3.048 · 10 ⁴ · Z

V

+ 3.048 · 10 ⁴ · Z

U

+ 0.9707 · 10 ⁴ · Z

The average pressure value is expressed:
P _m =

Pdx
P _m =

dl
where where is B

P _m =

1-

Finally, the average pressure is
P _m =

P′-

where P ¹ =

Overlay modes can be performed using the formula
σ l ^{- 2 / K} n ^{- 0,2 )} where σ is the part of the molecules that correspond to the law of the molecular regime
F _n (1 σ) F _c + σ F _m , where F _c conductivity in slip mode;
F _m conductivity in molecular mode. For f ≠ 1, the conductivity is determined by:
F

F _n The basic equations for the molecular regime can be written
Q _m =

(P ₁ -P ₂ )
Q _pv =

(P ₁ -P ₂ )
V

(P ₂ -P ₁ )
U

(P ₁ -P ₂ ) Average output speed
U ₂ =

-1

When the air temperature 17 ^{° C} have
U ₂ = 3.0723

-1

cm / s cm / s. Conductivity
F

or with a molecular weight of M
F

R ³ cm ³ / s when writing the equations it was assumed that the pipe has an infinite length, the walls completely disperse the molecules, the gas is in equilibrium and obeys Maxwell's law.

The nature of the regime and the choice of equations depends on the Knudsen number. When K _n ≥ 10, the molecular mode
0.25 ≅ K _n ≅ 10 transient,
10 ^-3 <K _n <0.25 slip mode,
To _n <10 ^-3 mode of classical aerodynamics.

To describe the movement of gas in pipes, the Knudsen number is used, which is expressed as the ratio of the mean free path of a molecule to the radius of the pipe K _n L / R. To characterize the motion of bodies in rarefied gases, the Knudsen number is expressed as the ratio of L to the characteristic size of the body or the thickness of the boundary layer.

Исследования показывают, что для адиабатических условий существует предельная скорость движения разреженного газа:
U²= 2

r_oT

1-

Предельное значение выражается
U $\genfrac{}{}{0ex}{}{\text{2}}{\text{п}}$ _р

В функции средней квадратичной скорости молекул

:
U $\genfrac{}{}{0ex}{}{\text{2}}{\text{п}}$ _р/

= 2γ/3(γ-1) В случае двухатомного газа γ 7/5
U $\genfrac{}{}{0ex}{}{\text{2}}{\text{п}}$ _р/

= 7/3 Для одноатомного газа γ 5/3
U $\genfrac{}{}{0ex}{}{\text{2}}{\text{п}}$ _р/

=

Число Маха можно вычислить
M²=

1-

Вместе с тем опыт показывает, что в коротких трубах можно достичь числа Маха 5, 6 и более. Эта область супераэродинамики характеризуется числом Кнудсена, когда средний свободный пробег становится больше размеров движущегося тела.The viscosity coefficient in the molecular regime changes. The viscosity of the gas can be expressed by the formula:
Z

Studies show that for adiabatic conditions there is a limiting velocity of rarefied gas:
U ² = 2

r _o T

1-

The limit value is expressed
U $\genfrac{}{}{0ex}{}{\text{2}}{\text{P}}$ _R

As a function of the mean square velocity of the molecules

:
U $\genfrac{}{}{0ex}{}{\text{2}}{\text{P}}$ _p /

= 2γ / 3 (γ-1) In the case of a diatomic gas, γ 7/5
U $\genfrac{}{}{0ex}{}{\text{2}}{\text{P}}$ _p /

= 7/3 For a monatomic gas, γ 5/3
U $\genfrac{}{}{0ex}{}{\text{2}}{\text{P}}$ _p /

=

Mach number can be calculated
M ² =

1-

However, experience shows that in short pipes, Mach numbers of 5, 6 or more can be achieved. This area of super-aerodynamics is characterized by the Knudsen number, when the average free path becomes larger than the size of a moving body.

χ″ e

Изменения зависят от значения нормальной скорости пластины или от отношения этой составляющей и наиболее вероятной скорости молекул газа в предложении максвеллового распределения.In this area, changes in the ratio of the number of molecules colliding with the front and rear edges of the plane to the number of molecules hitting the plate from the same sides, but when the stationary plate is in a stationary gas, is characterized by the values:
χ ′ e

χ ″ e

The changes depend on the value of the normal speed of the plate or on the ratio of this component and the most probable speed of the gas molecules in the proposal of the Maxwell distribution.

≥1,6 число молекул, ударяющихся в заднюю кромку равно 1/100 т.е. примерно равно нулю. При этом

M где М число Маха.Known values for κ 'and κ''. At

≥1.6 the number of molecules hitting the trailing edge is 1/100 i.e. approximately equal to zero. Wherein

M where M is the Mach number.

Сопротивление сферы Бассетом записано в форме:
C_f·

R²= F при этом
C_f=

где L 1,25

· η/ρc
С скорость звука при данной температуре.When a solid moves in a rarefied gas at very low Mach numbers, the friction coefficient for a sphere can be expressed by the dependence
C _f =

The resistance of the sphere by Basset is written in the form:
C _f

R ² = F in this case
C _f =

where L is 1.25

Η / ρc
C is the speed of sound at a given temperature.

0,998

-1

L

0,998

-1

1,253

-1

Для дозвуковых и сверхзвуковых скоростей Кейном получена эмпирическая зависимость для сферы:
C_D=

0,97+

1+

где Re₁ значение числа Рейнольдса за ударной волной. Выражение для С_Dполучено для чисел Рейнольдса от 15 до 768.

0,998

-1

L

0,998

-1

1,253

-1

For subsonic and supersonic speeds, Kane obtained an empirical dependence for the sphere:
C _D =

0.97+

1+

where Re ₁ is the Reynolds number behind the shock wave. An expression for C _{D is} obtained for Reynolds numbers from 15 to 768.

Q

где Q коэффициент аккомодации энергии.A characteristic phenomenon in rarefied gases is a jump in speed and temperature at a solid boundary. Temperature jump specified by P. Valander:
ΔT

Q

where Q is the coefficient of accommodation of energy.

κ_т=

·

где σ₁ коэффициент аккомодации тангенциального импульса количества движения
κ_т=

·

где α коэффициент аккомодации энергии. Для многоатомного газа
κ_т=

·

·

Скорость скольжения вдоль поверхности можно найти из соотношения
u_s= κ_м·

κ_v= 2

·

κ_v= 2

где σ₁ коэффициент аккомодации тангенциального импульса движения.The value of ΔT is expressed:
ΔT T _o -T _w = κ _t

κ _t =

·

where σ _{1 is the} coefficient of accommodation of the tangential momentum of the momentum
κ _t =

·

where α is the accommodation coefficient of energy. For polyatomic gas
κ _t =

·

·

The sliding velocity along the surface can be found from the relation
u _s = κ _m

κ _v = 2

·

κ _v = 2

where σ _{1 is the} coefficient of accommodation of the tangential momentum of motion.

·

·l

u $\genfrac{}{}{0ex}{}{\text{°}}{\text{x}}$

·l

K

+ K

где y координата, описываемая от стенки по нормали,
Т_w температура поверхности тела,
Т_о и u_x граничные значения температуры и скорости газа при
K_i (K_i 0,827; K₂ 1,012; K₃ 0,42)
l средняя длина пробега молекулы,
Q_э термический коэффициент аккомодации (не слишком мал по сравнению с единицей).For the conditions of validity of the Navier-Stokes equations, the value of the temperature jump and velocity:
ΔT T ^° -T _w =

·

L

u $\genfrac{}{}{0ex}{}{\text{°}}{\text{x}}$

L

K

+ K

where y is the coordinate described from the wall along the normal,
T _w body surface temperature,
T _about and u _{x the} boundary values of temperature and gas velocity at
K _i (K _i 0.827; K ₂ 1.012; K ₃ 0.42)
l the average path length of the molecule,
Q _{e is the} thermal coefficient of accommodation (not too small compared to unity).

 The most important step in solving the problem of mass transfer in discharged gases is to describe the flow of a mixture of gas with particles. Quite encouraging work results have not yet been found.

ρ_piu_piA

W_gdu_g+

du_pi+AdP_g= 0

[C_pg(T_g-T_go)+

u $\genfrac{}{}{0ex}{}{\text{2}}{\text{g}}$ +

[C_pp(T_pi-T_go)+

u $\genfrac{}{}{0ex}{}{\text{2}}{\text{p}}$ _i] 0
P_g ρ_gRT_g
u_pi(du_pi/dx)

(M_gf_pir^*/m_pr $\genfrac{}{}{0ex}{}{\text{2}}{\text{p}}$ _i)(u_g-u_pi)
U_pi(dT_pi/dx)= -3(M_gg_gir^*/m_pr_pi ²) (C_pg/P_rC_pp)(T_pi-T_g) Низкочастотные бесконечно малые возмущения распространяются с равновесной скоростью звука:

где γ_l=

Так как γ_е < γ, то равновесная скорость звука в смеси газа с частицами меньше неравновесной или замороженной скорости звука в смеси.A description of the flow of a gas-particle mixture is given taking into account a number of assumptions
ρ _g u _g A

ρ _pi u _pi A

W _g du _g +

du _pi + AdP _g = 0

[C _pg (T _g -T _go ) +

u $\genfrac{}{}{0ex}{}{\text{2}}{\text{g}}$ +

[C _pp (T _pi -T _go ) +

u $\genfrac{}{}{0ex}{}{\text{2}}{\text{p}}$ _i ] 0
P _g ρ _g RT _g
u _pi (du _pi / dx)

(M _g f _pi r ^* / m _p r $\genfrac{}{}{0ex}{}{\text{2}}{\text{p}}$ _i ) (u _g -u _pi )
U _pi (dT _pi / dx) = -3 (M _g g _gi r ^* / m _p r _pi ² ) (C _pg / P _r C _pp ) (T _pi -T _g ) Low-frequency infinitesimal perturbations propagate at the equilibrium speed of sound :

where γ _l =

Since γ _e <γ, the equilibrium speed of sound in a mixture of gas with particles is less than the nonequilibrium or frozen speed of sound in the mixture.

1+

+

u

Число Маха газа в горле для многофазных систем, вычисленное по скорости звука в газе, равно 0,8.For multiphase systems, the Mach number of the gas phase in the throat is:
Mg ²

1+

+

u

The gas number of the throat gas for multiphase systems, calculated from the speed of sound in the gas, is 0.8.

)

2(

+1)

1+

(

-1)

P_go/P_g=

1+

(

-1)

ρ_go/ρ_g=

1+

(

-1)

Τ_go/Τ_g= 1+

(

-1)

u_g/u_{g max}=

[(

-1)

]/[2+(

-1)

]

u_g=

(μ_g·r^*/m_pr $\genfrac{}{}{0ex}{}{\text{2}}{\text{p}}$ )[(1-K)/K²]x
u_p= Ku_g, ρ_p= (

/

)(ρ_gK)
T_p=LT_g+(1-L)T_go
L

1+3Pr

·

B

C 1+(

)/(

){K[(1-K)γ+K]+(γ-1)(C_pp/C_pg)BL}

C^1/2M

1+(γ-1)(B/C)
u_{g max}

Градиент осевой скорости есть величина постоянная в соплах с постоянным запаздыванием. Данный вывод, как показали исследования Гильберта, является общим. Карье показал, что в случае движения, когда не соблюдается закон Стокса, имеет место зависимость С_D˙Re/Nu

12,

(μ_gfr^*/m_pr $\genfrac{}{}{0ex}{}{\text{2}}{\text{p}}$ )[(1-K)/K²]
Исследования влияния факторов разреженности, сжимаемости и ускорения свободного потока на коэффициент сопротивления и теплоотдачи показывают, что только разреженность оказывает влияние. Учет осуществляется введением поправки:
C_D= C $\genfrac{}{}{0ex}{}{\text{°}}{\text{D}}$

N_u=

где C_D, N_u коэффициенты сопротивления и теплоотдачи шара при отсутствии разреженности.The solution of the equations for the flow of a mixture of gas with particles under the validity of the Stokes law and particles of the same size has the form:
A ² / A ^{* 2} (

)

2 (

+1)

1+

(

-1)

P _go / P _g =

1+

(

-1)

ρ _go / ρ _g =

1+

(

-1)

Τ _go / Τ _g = 1+

(

-1)

u _g / u _{g max} =

[(

-1)

] / [2+ (

-1)

]

u _g =

(μ _g * r ^* / m _p r $\genfrac{}{}{0ex}{}{\text{2}}{\text{p}}$ ) [(1-K) / K ² ] x
u _p = Ku _g , ρ _p = (

/

) (ρ _g K)
T _p = LT _g + (1-L) T _go
L

1 + 3Pr

·

B

C 1+ (

) / (

) {K [(1-K) γ + K] + (γ-1) (C _pp / C _pg ) BL}

C ^1/2 M

1+ (γ-1) (B / C)
u _{g max}

The axial velocity gradient is a constant value in nozzles with a constant delay. This conclusion, as shown by Hilbert's research, is general. Career showed that in the case of motion, when the Stokes law is not observed, the dependence С _D ˙Re / Nu

12,

(μ _g fr ^* / m _p r $\genfrac{}{}{0ex}{}{\text{2}}{\text{p}}$ ) [(1-K) / K ² ]
Studies of the influence of rarefaction, compressibility and acceleration of the free flow factors on the coefficient of resistance and heat transfer show that only sparseness has an effect. Accounting is carried out by introducing an amendment:
C _D = C $\genfrac{}{}{0ex}{}{\text{°}}{\text{D}}$

N _u =

where C _D , N _{u are} the resistance and heat transfer coefficients of the ball in the absence of sparseness.

lg

которая аппроксимируется уравнением для частиц первой группы 10 мм ≥ dТ≥ 0,75 мм
Re_кр (280 D 5) ˙ 10⁴

^0,4, для частиц второй группы 0,375 мм ≥ dТ ≥ 0,10 мм
Re_кр 53 ˙ 10⁴ ˙ D (dT/D)^0,123 для частиц третьей группы 0,025 мм ≥ dT ≥ 0,009 мм
Re_кр (16 D + 0,16) ˙ 10⁴ ˙ (dТ/D)^-0,173 чему соответствуют критические скорости
V_кр ≥ 0,75 ˙ (56 -1/D)(dТ/D)^0,4,
V_кр ≥ 8 (dТ/D)^0,123,
V_кр ≥ 0,24 (100 + 1/D)(dТ/D)^-0,173.It is impossible to evaluate mass transfer in rarefied gases without establishing a single transfer criterion. Studies of the transportation of materials in dust pipelines made it possible to establish that, at critical gas velocities, no material deposition occurs in the pipeline and mass transfer is ensured. Analytical processing of experimental data made it possible to establish a general functional relationship:
lgRe _cr = f

lg

which is approximated by the equation for particles of the first group of 10 mm ≥ dT≥ 0.75 mm
Re _cr (280 D 5) ˙ 10 ⁴

^0.4 , for particles of the second group 0.375 mm ≥ dT ≥ 0.10 mm
Re _cr 53 ˙ 10 ⁴ ˙ D (dT / D) ^0.123 for particles of the third group 0.025 mm ≥ dT ≥ 0.009 mm
Re _cr (16 D + 0.16) ˙ 10 ⁴ ˙ (dТ / D) ^-0,173 which corresponds to critical speeds
V _cr ≥ 0.75 ˙ (56 -1 / D) (dТ / D) ^0.4 ,
V _cr ≥ 8 (dT / D) ^0.123 ,
V _cr ≥ 0.24 (100 + 1 / D) (dT / D) ^-0.173 .

)(2y/D)^1/7
Значение приданной скорости набегающего потока выражается:
V_н= V_кр(1+1,443

)(dT/D)^1/7 Уравнения относятся к пылевым смесям с плотностью 2500 кг/м³. Для других плотностей вводится поправочный коэффициент

Значение λ принимается с учетом уравнения Л. Д. Альтшулера при относительной шероховатости

4 ˙ 10^-4 1,6 ˙ 10^-4 и скорости воздуха V 17- 50 м/с; Re 0,57 ˙10⁵ 41,7 ˙ 10⁵
λ 0,1/1,46

+100(Re)^0,25 или по формуле Г. А. Мурина для гидравлически гладких труб:
λ=1,01/(gRe)^2,5 для гидравлически шероховатых:
λ

Граничные числа Рейнольдса, Re

= 23/

, при которых начинает влиять шероховатость, и Re

= 560/

при которых начинает действовать квадратичный закон сопротивления. Среднее значение λ 0,018-0,0020.The distribution of speed in pipes under normal or overpressure conditions obeys one seventh law
V _y / V _sr = (1 + 1.433

) (2y / D) ^1/7
The value of the attached velocity of the free stream is expressed:
V _n = V _cr (1 + 1,443

) (dT / D) ^1/7 The equations apply to dust mixtures with a density of 2500 kg / m ³ . For other densities, a correction factor is introduced

The value of λ is taken into account the equation of L. D. Altshuler with relative roughness

4 ˙ 10 ^-4 1.6 ˙ 10 ^-4 and air speed V 17-50 m / s; Re 0,57 ˙10 ⁵ 41,7 ˙ May ¹⁰
λ 0.1 / 1.46

+100 (Re) ^0.25 or according to the formula of G. A. Murin for hydraulically smooth pipes:
λ = 1.01 / (gRe) ^2.5 for hydraulically rough:
λ

Reynolds Boundary Numbers, Re

= 23 /

at which roughness begins to influence, and Re

= 560 /

in which the quadratic law of resistance begins to operate. The average value of λ is 0.018-0.0020.

где при Re < 5 Ψ

1+

Re

а при 3 < Re < 400 по формуле Колячко Ψ

+ 4

Для больших значений числа Re по уравнению Ньютона F 0,055πμ_вd²V_c ², что соответствует ϑ 0,41.When determining the critical mass transfer parameter, it is important to remember that the Stokes law F 3πμdV _{c is} valid for small Reynolds numbers for the particle Rea ≅ 1. For large Reynolds numbers, mass transfer is described by the equations
F

where for Re <5 Ψ

1+

Re

and for 3 <Re <400 according to the formula Kolyachko Ψ

+ 4

For large values of Re according to Newton’s equation, F is 0.055πμ _in d ² V _c ² , which corresponds to ϑ 0.41.

The equations listed above are valid under normal or overpressure conditions and are of little or no validity for rarefied gases. Nevertheless, the dynamic pressure parameter ρ V ^2/2 is the critical determinant in the appointment mode. Taking into account the studies, the critical parameter in rarefied gases should be taken to be the values of the dynamic pressure equivalent to the minimum according to the transfer conditions in the boundary layer, i.e. ρ _in U _n ^2/2 , where ρ _{in the} density of air in the boundary layer; V _n air speed in a free stream.

x erf xdx, где S u_o/

Таким образом, в первом приближении условие обеспечения переноса массы должно быть записано в форме η·ρW·

≥

(1). В разреженных газах условие (1) является необходимым, но не достаточным и связано с несоблюдением закона Стокса в разреженных газах.The probability of collision of molecules with particles in a rarefied medium is taken into account by introducing the parameter η
η S ^-2 erfS + 2S

x erf xdx, where S u _o /

Thus, in a first approximation, the condition for ensuring mass transfer should be written in the form η · ρW ·

≥

(1). In rarefied gases, condition (1) is necessary, but not sufficient, and is associated with non-compliance with the Stokes law in rarefied gases.

·

Значения Rea определяют области режима течения: Rea < 1 стоксов режим; 1 < Rea < 500 переходная область; Rea > 500 ньютоновский режим течения.The study of the minimum transfer rate shows that it depends both on the magnitude of the friction velocity U _w ^* and on the final deposition rate U _t and is characterized by a Reynolds number in the form
Rea const

·

Rea values determine the areas of the flow regime: Rea <1 Stokes mode; 1 <Rea <500 transition region; Rea> 500 Newtonian flow regime.

0,204

где

/

а для любой концентрации

1+2,8

где значение скорости трения U_w ^* определяется

5lgRe-3,90
Значение скорости трения определяется выражением u^*

где

средняя скорость течения газа. Для характеристики ламинарного слоя используется зависимость отношения подъемной силы к силе сопротивления в виде

=

для значений Fx/U << 1. Для развитого течения зависимость записывается в форме

(0,865)·2^1/4·3

при x

Отношение

N_м= (x^*)^-1=

называется критерием межфазного обмена количеством движения. Число идентично числу Рейнольдса газовой фазы
Rex

Приведенный выше критерий переноса массы выражает слабое влияние разреженной среды и не может в полной мере служить мерилом переноса массы. Более полное и точное выражение могло быть получено при характеристике режима отношением суммы подъемной силы, силы сопротивления при оседании смеси, градиента давления по высоте к силе сопротивления при переносе массы. Вопрос требует изучения.It is established that the flow is ensured if the condition for an infinitesimal concentration is met

0.204

Where

/

and for any concentration

1 + 2.8

where the value of the friction velocity U _w ^* is determined

5lgRe-3,90
The value of the friction velocity is determined by the expression u ^*

Where

average gas flow rate. To characterize the laminar layer, the dependence of the ratio of the lifting force to the resistance force is used in the form

=

for values Fx / U << 1. For a developed flow, the dependence is written in the form

(0.865) 2 ^1/4 3

at x

Attitude

N _m = (x ^* ) ^-1 =

called the criterion of interphase exchange of momentum. The number is identical to the Reynolds number of the gas phase
Rex

The above mass transfer criterion expresses the weak influence of a rarefied medium and cannot fully serve as a measure of mass transfer. A more complete and accurate expression could be obtained by characterizing the regime by the ratio of the sum of the lifting force, the resistance force during sedimentation of the mixture, the pressure gradient along the height to the resistance force during mass transfer. The question requires study.

 During mass transfer due to the existence of the triboelectric effect, static electricity is induced on solid granular bodies, which polarize the conducting walls of the pipeline.

The presence of the polarization charge of the conducting wall and the space charge of many particles is manifested in the existence of a drift mass velocity on the pipe walls and a change in the concentration distribution, changes in the transverse velocity V, and the tension E _y . The presence of an electric charge makes a correction to the criterion of mass transfer.

для функции тока
-Ψ

ρdy ρ_pob

y

1+x

+

1+x

+

для распределения концентрации
ρ^* ( 1 + x^*2 ) ^{- 1} + y^*2 ( 1 + x^*2 ) ^{- 4} +. распределение дрейфовой скорости

1-e

y

1+x

+
+ y

/3

1+x

+

где ρ^*

x

=

1+

·e

y^*

Значение

≥ указывает на наличие осажденного слоя. Массовая скорость столкновений частиц с единицей поверхности дна задается выражением [ρ_bV/w при y^* 1
[ρ_pV] _w=

=

u_oRe $\genfrac{}{}{0ex}{}{\text{-1}}{\text{eb}}$ N

1-

-

+

Распределение дрейфовой скорости V/U_o предлагается истолковывать с учетом понятий электрического числа Рейнольдса
Re $\genfrac{}{}{0ex}{}{\text{-1}}{\text{eb}}$

и числа электровязкости E_v:
E_v=

При оценке роли электрических сил при движении пограничного слоя делается допущение о том, что электрические силы достаточно малы по сравнению с гидродинамическими, т.е. их можно принять за возмущения.It is possible to solve the problem with the following data: two-dimensional motion in an electric field (i 1, 2); particle motion does not significantly affect the motion of the continuous phase (it is believed that the particle concentration should not exceed 0.25 kg / m ³ ; all particles have the same size (S 1). Under these assumptions, the values of the main parameters are determined by the equation: for the drift velocity
V -

for current function
-Ψ

ρdy ρ _po b

y

1 + x

+

1 + x

+

for concentration distribution
ρ ^* (1 + x ^{* 2} ) ^{- 1} + y ^{* 2} (1 + x ^{* 2} ) ^{- 4} +. drift velocity distribution

1st

y

1 + x

+
+ y

/ 3

1 + x

+

where ρ ^*

x

=

1+

E

y ^*

Value

≥ indicates the presence of a deposited layer. The mass velocity of collisions of particles with a unit of the bottom surface is given by the expression [ρ _b V / w at y ^* 1
[ρ _p V] _w =

=

u _o Re $\genfrac{}{}{0ex}{}{\text{-1}}{\text{eb}}$ N

1-

-

+

The distribution of the drift velocity V / U _{o is} proposed to be interpreted taking into account the concepts of the electric Reynolds number
Re $\genfrac{}{}{0ex}{}{\text{-1}}{\text{eb}}$

and electroviscosity numbers E _v :
E _v =

When assessing the role of electric forces in the movement of the boundary layer, it is assumed that electric forces are quite small compared to hydrodynamic ones, i.e. they can be taken for indignation.

x^*+

x-

x

Влияние электрического поля пластины незначительно, если:
Re $\genfrac{}{}{0ex}{}{\text{1}}{\text{x}}$ ^/2

≫

Re_ewx Электрический эффект будет несущественен, если:
Re $\genfrac{}{}{0ex}{}{\text{1}}{\text{x}}$ ^/2 ≫ Re $\genfrac{}{}{0ex}{}{\text{-1}}{\text{ex}}$ Характеристический параметр взаимодействия предлагается определить в виде отношения электрической силы к силе вязкости E_v:
E_vwx=

(x/

)
E_vpw=

(g/m)(x²/

) Уравнения принимают вид:
Re $\genfrac{}{}{0ex}{}{\text{5}}{\text{x}}$ ^/2 ≫ E

Re^5/2 ≫ E

или в общем виде Re_x≫ E_vx
Таким образом, изложенное позволяет установить сущность предлагаемого способа.Changes in the thickness of the boundary layer are recorded in the form:
δ _{p ν} δ _1p ^{( o )} + β ₁ δ _p1 ^{( 1 )} + β ₁ δ _p1 ^{( 1 ' )} The first-order perturbation looks like:
δ _p1 =

x ^* +

x -

x

The influence of the electric field of the plate is negligible if:
Re $\genfrac{}{}{0ex}{}{\text{1}}{\text{x}}$ ^{/ 2}

≫

Re _{ewx The} electrical effect will be negligible if:
Re $\genfrac{}{}{0ex}{}{\text{1}}{\text{x}}$ ^{/ 2} ≫ Re $\genfrac{}{}{0ex}{}{\text{-1}}{\text{ex}}$ It is proposed to determine the characteristic interaction parameter in the form of the ratio of electric force to viscosity force E _v :
E _vwx =

(x /

)
E _vpw =

(g / m) (x ² /

) The equations take the form:
Re $\genfrac{}{}{0ex}{}{\text{5}}{\text{x}}$ ^{/ 2} ≫ E

Re ^5/2 ≫ E

or in general Re _x ≫ E _vx
Thus, the foregoing allows us to establish the essence of the proposed method.

The economic efficiency of the transport system of materials, as well as capsules of the proposed method in comparison with other modes of transport is determined by the advantages of the method, which include:
the exclusion of time-consuming loading and unloading operations and the continuity of the transport process;
the possibility of laying the pipeline for the shortest distance between two points;
small areas occupied by transport communications;
the ability to create fully automated and remotely controlled transport systems.

The listed advantages are common to various pipeline transport systems. In relation to the proposed method, it should be noted other, equally important advantages:
the absence of dust formation and losses of the transported material, affecting the environment (Advantage is also inherent in the method of transportation in liquid media);
high speed of material movement. The gas velocity can take values equal to several Mach numbers. Keep in mind the lag of the material velocity from the gas velocity;
reduction of energy costs, which in comparison with the method of transport in liquid media will decrease depending on the ratio of the density of the mixture in the liquid medium to the density of the mixture and gaseous medium. A decrease in energy intensity will also be observed in connection with an increase in the distance of the transportation arm, which is caused by a decrease in the coefficient of friction of the gas with increasing flow and pressure drop, as well as an insignificant effect of dry friction of the material or its complete absence. The latter is especially significant in comparison with the transportation of material using liquid media and gases with overpressure;
the use of rarefied gases and volumetric vacuum pumps will reduce the energy consumption of the pump unit when it enters the operating mode at reduced pressures, which is excluded when using other transport media;
the costs of dewatering the material are excluded, the range of transported materials is expanding. For example, when organizing coal transport, it will be grinded at the mining site and subsequently used in a dust-like state, which is much more effective from the point of view of mechanization of processes and improvement of its combustion;
reduction of one-time costs for laying the pipeline, because the design pressure of the transporting medium in it is much lower, and the abrasive wear of the walls is less at a higher throughput through the material due to significant flow rates;
reduced consumption of compressed gas for material transport; however, its value does not depend on the distance of the transport arm, as is the case in the case of gas transport in excess pressure;
the ability to separate the substance from the gas at almost any predetermined level, moreover, without the use of filters, is especially important in the case of the use of toxic substances;
almost unlimited range of material transportation, which is especially important for the development of the Far East and Siberia;
use in other areas of technology is not excluded.

Claims (1)

 METHOD OF TRANSPORTATION OF BULK SUBSTANCES, AEROSOLS AND CAPSULES, consisting in the form of gas and transported material into a transport pipeline in which a vacuum is previously created, characterized in that the gas is introduced into the pipeline at a speed not lower than the local speed of sound, then the transported material is introduced, and the pressure level along the length of the pipeline support constant.

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