AJMS

1. www.ajms.com 60
ISSN 2581-3463
RESEARCH ARTICLE
On the Choice of Compressor Pressure in the Process of Pneumatic Transport to
Increase Energy Saving
E. L. Pankratov1,2
1
Department of Mathematics, Nizhny Novgorod State University, 23 Gagarin Avenue, Nizhny Novgorod, 603950,
Russia, 2
Department of Mathematics, Nizhny Novgorod State Technical University, 24 Minin Street, Nizhny
Novgorod, 603950, Russia
Received: 25-11-2018; Revised: 15-12-2018; Accepted: 01-02-2019
ABSTRACT
In this paper, we consider a possibility to choose value of pressure of devices for pneumatic transport
to increase energy saving. We also introduce an analytical approach for the prognosis of transport
of free-flowing materials to estimate velocity of the transport and choosing of the required value of
pressure.
Key words: Energy saving, pneumatic transport, prognosis of transport
INTRODUCTION
At present, in many industries, during the processing of various bulk materials, their pneumatic transport
through pipelines[1-4]
is used. Such industries include the production and processing of building materials,
agricultural, food, and pharmaceutical products. For example, in the construction industry, cement,
plaster, and lime are transported through pneumatic conveying systems. The advantages of pneumatic
transport include high productivity, large radii of action, complete absence of losses of transported
material in pneumatic lines, high environmental characteristics of plants, and the possibility of building
branched systems adapted to complete control automation. The disadvantages of pneumatic transport are
the relatively high specific consumption of electricity per unit mass of transported material and the wear
of pneumatic lines due to the abrasive effect.
In the framework of this paper, a pipeline in the form of a straight circular cylinder [Figure 1] is
considered. Through its left end is a put in free-flowing loose material. The main aim of this work
is to choose the pressure value, at which energy saving would increase when pneumatic transport
of this material. The accompanying goal of this work is the formation of an analytical technique for
predicting the transport of bulk materials to assess the transfer rates and the choice of the desired
pressure.
METHOD OF SOLUTION
To solve our aim, we determine the spatiotemporal distribution of the velocity in the pipeline in question
and perform its analysis. The required spatiotemporal distribution of the velocity field was calculated by
solving of the Navier–Stokes equation.[5-7]
∂
∂
∇
( )
( )
− ( ) ∆( ) ( )

 


 

 
v
t
+ v, v =
F z
S h
1
grad P + v +
3
grad div v
ρ ρ
ν
ν
(1)
Address for correspondence:
E. L. Pankratov,
E-mail: elp2004@mail.ru

2. Pankratov: On the choice of compressor pressure in the process of pneumatic transport
AJMS/Jan-Mar-2019/Vol 3/Issue 1 61
Where
 
v = v r,z,t
( ) is the velocity of transport of the considered free-flowing material,
 
F z = P z S k
( ) ( ) ,
S is the pipe section area,

k is the unit vector along the axis Oz, P (z) = Pk
−(Pk
−Pa
) z/h, Pk
is the
compressor pressure, Pa
is the atmosphere pressure, ρ is the gas density, and ν is the kinematic viscosity.
The boundary and initial conditions to these equations could be written in the following form.

v z t
0, ,
( ) ≠ ∞ ,
∂ ( )

v r,z,t
vr
= 0
r=R
,

v r,z, =
0 0
( ) ,
 
v r, ,t = vin
0
( ) ,
 
v r,h,t = vout
( ) ,

v r,z,0 =
( ) 0 (2)
The projections of the velocity defined by Eq. (1) in the cylindrical coordinate system have the following
form,
¶
¶
¶
¶
¶
¶
¶
¶
¶
¶
¶
¶
¶
¶
v
t
+v
v
r
+v
v
z
=
r r
r
v
r
+
r r
r
v
r
r
r
r
z
r r z

1 1


















 ( )






+
3 r r r
rv +
v
z
r
r
 ¶
¶
¶
¶
¶
¶
1
∂
∂
∂
∂
∂
∂
− −
( )






∂
∂
∂
∂
κ
v
t
+v
v
r
+v
v
z
=
S
P P P
z
h
+
v
z
+
v
z
z
r
z
z
z
k 0
2
r
2
2
z
ρ
ν 2
2 z
z a
+
z r r
rv +
v
z
S
P P
h






∂
∂
∂
∂
( )
∂
∂





 −
−
κ
ν
ρ
3
1
(3)
Then, we transform the Eq. (3) to the following integral form,
( ) ( )
( ) ( ) ( )
( ) ( )
2
4
0 0 0 0 0 0 0
2
2
0 0 0 0 0 0 0 0 0
2
2
0 0 0 0
2
3
2
2
t z t z t r z
r
r r r r
t z t z t z t r z
r
r r r
t z t z
z r
v
v = v + r z v dvd r z v v dvd u v dvdu d
r
R
v
r
+r v dvd z v dvd + r z v v dvd z v v dvdu d
r
v v
r
z v dvd +v r z v dvd r z
r r
   ∂
ν
− τ − − τ − τ
  ∂



 ∂
τ − − τ − τ − − τ
 ∂

∂ ∂
− − τ − τ −
∂ ∂
∫∫ ∫∫ ∫∫ ∫
∫∫ ∫∫ ∫∫ ∫∫∫
∫∫ ∫∫ ( ) ( )
( ) ( ) ( )
( )
0 0 0 0 0
2
0 0 0 0 0 0 0
2
0 0 0 0 0
t z t r z
r
t r z t z t z
z
r z z
t t r r z
z z r
v v dvd + z v
v
v dvdu d + z v v dvdu d + r z v dz d r z v v dvd
r
r v d v du d u z v v dvdu

− τ −



∂
× τ − τ − τ − − τ
∂ 

+ τ − τ − − 

∫∫ ∫∫∫
∫∫∫ ∫∫ ∫∫
∫ ∫∫ ∫ ∫
(3a)
Figure 1: Cylindrical piping

3. Pankratov: On the choice of compressor pressure in the process of pneumatic transport
AJMS/Jan-Mar-2019/Vol 3/Issue 1 62
( )
( )
2 2 2 3
4
0 0 0 0 0 0 0
2 3 2 2
0 0 0 0 0 0
2
0 0 0
3
2 6 6 3
t r t r t r z
z
z z r z r
t r z t r z
a r
a z
t r z
z
x
v tS
v v u v du d u v du d u z v v dvdu d r
R u
P P
P v
z
z P P r z tS u vv dvdu d u dvdu d
h h u
v
u dvdu d v
z




   


 

 
   ∂
Θ 
=
+ + − − +
   ∂
 



− ∂
 
× − − − + +

  ∂
  

∂
+ −

∂ 
∫∫ ∫∫ ∫∫ ∫
∫∫ ∫ ∫∫ ∫
∫∫ ∫ ( )
( ) ( )
3
2 2 2
0 0 0 0 0
2 2 3 2
0 0 0 0 0 0 0
3 2 2
0 0 0 0 0 0
1 1
3 2
1
2
18
6 3
t r z r h
z z
t r t r t r h
z
out out k a r
t r h t r h
à r
z
r
t u vv dvdu d z u h v v dvdu
h
t h S
u dud u du d r P P u h v d du d
h h u
P P v
r t S u vv dvdu d u dvdu d u
h u




      


 


− + −


  ∂
− + − + + −
  ∂
 
− ∂
+ − + +
∂
∫∫ ∫ ∫ ∫
∫∫ ∫∫ ∫∫ ∫
∫∫ ∫ ∫∫ ∫
( )
0 0 0
3
2 2 2
0 0 0 0 0
1
3 2
t r h
z
t r h r z
in z z
v
dvdu d
z
r
v t u vv dvdu d u z v v dvdu
h h

 
 
∂
 
∂
 

 
+ + − − 


 
∫∫ ∫
∫∫ ∫ ∫ ∫
Next, we solve this equation by averaging the functional corrections.[6-8]
In the framework of this
method, to obtain the first approximation of the required projections of the velocity of the material under
consideration, we replace them with the yet unknown mean values vr
→αr1
, vz
→ αz1
. Then, the relations
for these approximations take the following form,
v
R
r t
z
z r t r zt r t r t
r r r r r z z
1 1 4 1
2
1
2 2
1 1 1
3 2
= + − − +





 + − −
α
ν
α α α α α α
Θ
r
r r
z
1
3
2
6





 (4)
v
R
t r r z
t S
P P P
z
h
r z
z z r z k a
1 1 4
3
1 1
3 2 3 2
3 2 3
= + +
( )+ − −
( )





 −
α ν α α
ρ
Θ
κ t
t S
P P
h
r z
t z r
h
t r
h
v v t S r
P
a
z
z out in
κ
κ
−
+



× + − +
( )+
−
2
4 2
1
2 2
1
3 2
ρ
α
ν
α
ν 3
3
6 4
3
1
2
2
1
2
3
1
2 3
2
P
v t
r
h
h
r
r zt
h
v t r r t
z
a
in z z
in z
ρ
ν α α
ν
ν α
+ + −






− −
4
4 2
1
2
2
−



αz r
z
The average values of αr1
and αz1
are determined using the standard relations,
α
π
1 2 1
0
0
0
1
r r
L
R
R L
r v d z d r d t
= ∫
∫
∫
Θ
Θ
, α
π
1 2 1
0
0
0
1
z z
L
R
R L
r v d z d r d t
= ∫
∫
∫
Θ
Θ
(5)
Substitution of relations (4) into (5) leads to the following result,
α
α ν
ν
r
z
R
1
1
2
3
=
+
Θ
Θ
; α
ν ν ρ
ν ν
z
E KE 0
v h R v h R P R h S
R h R h
1
2 5 2 5 2 2 2
2 2 2 3
24
15
144 14
=
+ −
−
σ σ
Θ Θ Θ
Θ
Θ Θ
4
4 180 15 5
3
2
3
2 2 3
2
+ + +
+
( )








h
R
R
R
R
ν
ν
ν
Θ Θ
Θ
(6)
The first approximation of the required value of the compressor pressure (at the inlet to the pipeline)
was determined from the condition that the output velocity of the material is equal to zero. In the final
form, the relation for the required value of the compressor pressure in the first-order approximation by
the method of averaging the functional corrections takes the following form,

4. Pankratov: On the choice of compressor pressure in the process of pneumatic transport
AJMS/Jan-Mar-2019/Vol 3/Issue 1 63
P
h S h R
v R h h
R
h R
in z r z z
κ ( ) =
−
( )
−
( ) − +
( ) −
1 2 2 2 1
3 2 2
1 1
2
1
0
18
3
ρ
α ν α α α
Θ
Θ Θ
Θ
Θ
Θ
Θ
Θ
Θ Θ Θ
ν
ν
ρ ρ
α α
12 3
6
2
9 12 6
2
2
1
2 2
1
+



+ − + − −
v
R
h
S P
P R
S
R
h
hR R
h
in
0
0 z z v
v v
R
h
h
in in z
ν
ν α
3 3 12
1
2
+ +









Θ
(6)
The second approximation of the velocity of free-flowing material is defined in the framework of the
standard procedure for the method of averaging of functional corrections.[6-8]
In the framework of this
method, to obtain the first approximation of the required projections of the velocity of the material under
consideration, we replace them by the following sums: vr
→αr2
+vr1
, vz
→αz2
+vz1
, where αr2
and αz2
are the
average values of the second approximation of velocities. Then, the relations for these approximations
assume the following form,
( )( ) ( )( )
( ) ( )( ) ( )
( ) ( )( ) ( )
2 2 1 2 1 2 1
0 0 0 0
1 2 1
0 0 0 0
2 1
2 1 2 1
0 0
1
3 3
t z t z
r r r r r z z
t z t z
r r r r
t z
z
r r r r
v v r z u v du d r v z u du d
r r r r r r
v du d r z u v du d h z
r r r r
v
h z z u v du d z v
r
 
    
 
   
  
   
∂ ∂ ∂ ∂

= + + − + + + −
  
 
∂ ∂ ∂ ∂
   
 
 
∂ ∂ ∂
+ + + − + + −
 
 
∂ ∂ ∂  
 
 
∂
∂
× − − − + − +
∂ ∂
∫∫ ∫∫
∫∫ ∫∫
∫∫ ( )
( )( ) ( )( )
( ) ( ) ( )( )
( )
2 1
0 0 0 0
1
2 1 2 1
0 0 0 0 0
1 1
2 1 2 1 2 1
0 0 0 0
2 1
3
t z t z
r r
h t h t h
z
r r r r
t h t h
z r
r r z z z z
r r
du d u v
u
v z
du d h u v du r h u v du d u
u h r r r
v v
v du d v du d r h u v du d
u u r r
v du d
r r
 

   
     
 
 
− +
  
∂ ∂ ∂

× + − + − − + +
  
∂ ∂ ∂
 


 
∂ ∂ ∂ ∂
× + − + − − +
 
∂ ∂ ∂ ∂
 
∂
× − +
∂
∫∫ ∫∫
∫ ∫∫ ∫∫
∫∫ ∫∫
( )( )
( ) ( ) ( )( )
2 1
0 0 0 0
2
2 1 2 1
0 0 0
1
3
1
t h t h
r r
t h z
r r r r
r h u v du d
r r r
h u v du d z u v du
r h R

 
  
 
 
∂ ∂
− − +
 
 
∂ ∂  
 
 


∂
+ − + − − +
 
∂ 

∫∫ ∫∫
∫ ∫ ∫
(7)
The average values of αr2
and αz2
have been determined standardly.[6-8]
r r r
h
R
h R
r v v d z d r d t
2 2 2 1
0
0
0
2
= −
( )
∫
∫
∫
Θ
Θ
, z z z
h
R
h R
r v v d z d r d t
2 2 2 1
0
0
0
2
= −
( )
∫
∫
∫
Θ
Θ
r r r
h
R
h R
r v v d z d r d t
2 2 2 1
0
0
0
2
= −
( )
∫
∫
∫
Θ
Θ
, z z z
h
R
h R
r v v d z d r d t
2 2 2 1
0
0
0
2
= −
( )
∫
∫
∫
Θ
Θ
(8)
Substituting relation (7) into relation (8) leads to the following result
r
b d
a f
be
a f
b
a
d b
f a
e
f
d c
f a
g
f
c
a
2
2
2
2
4 4
= + + +





 − + − ,
z
bd
a f
e
f
d b
f a
e
f
d c
f a
g
f
2
2
4 4
= + + +





 − + ,

5. Pankratov: On the choice of compressor pressure in the process of pneumatic transport
AJMS/Jan-Mar-2019/Vol 3/Issue 1 64
Where,
( ) ( )
3 4 2 2 3 5
2 2 3
1 1 1 1
3 2 2
2 3
1
1 1
3 5
2 4 1
1 1 3
12 12 16 3 4 3 4
2
3 2 12 6 4 3 4 3
5 4
12 10 5 2 6 3
a
z a r z z
z
z a out a z in
a
z
z r out
P P
h h h h h
a h R S P h
R
h h h h
S P P P P h
h R
P P
h h
h S v
h h

 


    


 
    

 
  

  −
Θ Θ  
= Θ − Θ − + + Θ + Θ −
  
  
 
 
Θ Θ Θ
+ − Θ − − + − − + + × Θ
 
 
−
Θ Θ
− − Θ − + Θ −
( ) ( )
5
2
1
1 1
2
2 2 2
2 3 2
1
2
2 3 5 4 3
2 2 2 3 2
1 1
1 1 1
3 2
2
2
1 1
3 4 8
2
24 3 6 24 6 6
5 4
9 6 3 5 6 24 12 9 8
9 6
z
z z
a
a z a out a
a
z z
z z out z
z z
h
R h
P P
S h S h
P h P P v P P
h h
P P
h h
S v
R h h R h
R

 


  


 
 
 
   


 
 
Θ
− + + ×
 
 

−
Θ Θ
 
− + − Θ − Θ − − Θ + −
  
  
 
 − Θ
Θ Θ Θ
− Θ + + Θ + Θ − − +
  
 

Θ
− Θ +
2
3 5 4 3
2 2 3 2
1 1
1
3 2
4 2 3 5
3 3 2 2 3 2 1
1 1 1 1
3 3
3 3 2 3 2 4
1
5 4
3 5 6 24 12 9 8
6 3 16 24 3 8 12 6
1
3 24 48 3
z
a
z z
out z
a z
z in z a r z
a
in v a
P P
h h
S v
h h R h
P P
h S h h
h v h h P
P P
S
h v h h P
h



 

 


 
    

   

 
 − Θ
Θ Θ
+ Θ + Θ − − +
  
 

−
Θ  
− Θ − Θ + Θ − + − Θ + Θ −
 
 
−
Θ Θ  
× Θ − Θ + + + −
 
 
5 3
2 3
1 1
3 4 2 2 2
3 3 3 3 2 2 2 3 2
1 1 1
2
2 2 2 2
2 2 2 2 2
1 1 1 1 1
2 2 2 2
24 12
53 88
2
20 12 32 8 3 1440 24
36 9 48 24 18 18 4
z
z
z v
in a
r v out z
out in
z z r z z
h h
v P P
h z h R
z R v h S R
R h
v v
R R R R
R R h R
h h h h h h

  
   


        
Θ + Θ +
  −
Θ
× Θ + Θ − Θ + Θ − Θ +
 




Θ Θ
Θ Θ Θ Θ
− − + + + + Θ − 

 
,
b R
h
v v
out in
= −
( )
Θ2 2
3
4
,
( )
( )
( )
4 4 5 2
2 2 2 2 2 2
1 1 1
3 2 5 3 4
2 2 2 4
1 1 1 1 1
2 3 4
4 2 2 2
1 1
1 1
2
4 6 8 24
5 4
48 12 36 4 18 6 12
17 13
12 4 36 18 6 6
r out in r r
r r a z out z z
r z
z a out z out
h h h
c R R R h R
h S h h h
R h P P h
R
h S h
h P P h h
R R


     

       

 
     

Θ
= Θ + − Θ + Θ +
 
Θ Θ Θ
+ Θ − − − Θ − +
 
 
 
Θ Θ Θ
− + − − Θ − Θ + +
 
 
( )
( ) ( ) ( )
( )
( )
2 2
3
2 4 3
2 2 2 3
1
1 1
3 2 2
2 2 2 2 2 2
1 1 1 1
4
3 3
1 1
5 8 5
108 72 1296 144
5 4
4 8 4 18 9 6 6
8
5 4
10368 12 36
in
r
out in a z r a
a
out in r z out z z
a
ê
r z a
R S
h R h R h S
R P P P P
h
P P
h h h h
R R S
h R
P P h h S
R S h P P
 





   
 
 
      


 
 
−
Θ
× − − Θ + − + − Θ
 
− Θ
+Θ × − + Θ + Θ − Θ − +
 
 
+
− × Θ + + − +
( ) ( ) ( )
2
1 1
2
4
3 3
2 3 2 2 3 2 2 4 3 2 2 3 2
1 1 1
12 18 18
8 24 576 1440 8
out
z z
à a
r out out in out in out in z r
h h R h
h P P P
h R
h h h

   
         
 
Θ Θ
− Θ −
 
 
−
Θ
+ Θ + − + Θ − + Θ − − Θ

6. Pankratov: On the choice of compressor pressure in the process of pneumatic transport
AJMS/Jan-Mar-2019/Vol 3/Issue 1 65
( )
( )
2 3 2
2 2 2 4 2 2
1 1 1 1 1 1 1
3 2 3
2 2 2
1 1 1 1 1
2
1 1 1
3 2
8 4 18 18 6 6 8
4
8 36 18 12 12 4 3
5
4 4
9 9 4
r
r z a out z z z r v
z a out z z z r
in z z z
h h S h h R
P P h h h
h R
h S h h
P P h h h
R
P
h
h h R S



 
       


       

     
 
Θ Θ
− Θ × + − − Θ − Θ + − Θ −
 
 
 Θ Θ Θ Θ Θ
× Θ + + − − − − +


−

+ Θ + Θ − + Θ


( )
( )
2
1 1
2
2 5 2 2
2 2 2 2 2
1 1 1 1
3 2
3 2 2 2
1 1 1 1
2
4
18 18 6 12
8 7
12 8 24 6 36 9 18 12
5 4
24 4 36 18 12 12
a
out z z
a
r out in z out z z
out
r z a z z out
P h h
h R
P P
h h h h h h
R S
h R
h h S h h
P P
h h R



   

 
      

  
    

 
Θ
− − + Θ
 
 
 
− Θ
× Θ − Θ − − + − − +
 
 
 
Θ Θ
× Θ − Θ + − − Θ − + −
 
 
( )
( ) ( )
2
2 2
4 3
2 3 3 3 2 3
1 1
6
6 10
16 720 2160 1440
in
out in a
z a a r
P P
h h S R
R P P h S P P h S 
 

 
 
  
−
− −
× Θ − − Θ − Θ − − Θ
,
d h R P P
h S
R
h h
h
v
a z z
o
= + −
( ) − − +
2 2
3
1
2
3
1
2
5
2 2
4
22 47
4320 8 48
Θ
Θ
Θ Θ Θ
ν
ρ
α α
ν
κ
u
ut
z
z z r z
R
h h
h h
192 72
5
144 16 18
2
1
2
3
1
2 3
3
1
2
4
1
3
2
1
+
× + − − −
α
ν
α α α ν α
Θ
Θ
Θ Θ ν
ν ν
Θ Θ
3
2
3
2
27 27
h
v
h
in
−
,
( )
2
2 2 2
1
1 1 1
2
2 3 2 3 3
3 2 2 2 3 2
1 1 1 1
2
3 2
3 2 2 3 3 2 2
1 1 1
2
2
1
5 4
8 16 12 36 18 6 6
2
5
18 9 48 128 8 16
5
288 12 2 9 8
12 4
z
z z a out z
z z z z
a in
r z z
z
R h S
e R P P
h h R
R R h R R h
R R
h h
P P R h
R S R R R
h h h
R h




    

     


   


 
Θ Θ Θ Θ
= + Θ + − − − +
 
 
Θ Θ Θ
+ + Θ + + − Θ − Θ
− Θ
+ Θ + + Θ + Θ − Θ
−Θ +Θ
( )
( )
2
1 1
2 2 2
2 2
1 1 1
2 3
2 3 3 2 2 2
1 1 1 1 1
2 2 3
1
2
5 4 1
18 9 9 2
5 4
48 16 4 18 9 6 6
8 9 18 16 24 32
5
2304 16
a
out z z
z a out z z
r in z z z z
a r
P P
S
h h R
h R h S h
R P P
h R
R h R hR
R R
S P P R
R R
h h



 
  


    


       




 
− Θ
− Θ − + Θ +
 
 
 
Θ Θ Θ Θ
Θ + + − − Θ − +
 
 
Θ
− Θ − Θ − Θ + Θ − − Θ
−
+ Θ + Θ × + Θ
3 2
1
18 12 12
z
in
v
R
R h
h h
 

Θ
+ − Θ
,
f h
R
= 2 2
2
16
Θ ,
g
h
v
h
h
in z r z
z
= − ×





 + +



+
ν ν α α ν ν α
α
Θ
Θ
Θ
2
2
1
2
1 1
1
2
2
2 3
4
3 2
4
3
13
48 2
h
h
R
P P
S
v
h
a out z
2 3 18
5 4 2 3
3
1
2
−





 + −
( ) − +












ν
ρ
ν
α
Θ Θ
κ
,

7. Pankratov: On the choice of compressor pressure in the process of pneumatic transport
AJMS/Jan-Mar-2019/Vol 3/Issue 1 66
( ) ( )
( )
2 3 2
2 2 2 2 3 2 2 2 2
1 1
2 2 4
4 2 2 2 2 2 2 2 2 2 2
1 1
2
3
1
1
5438 3
4 12 2880 16
13
96 3 12 2160 24
5 4
48 2 2 3 3 6 2 3
r out in a r
a
z in out z
z
z a out
h S R
h h R R v v P P h
P P
h R h
R R h R h S R R
h h h S
P P
R h



 
    

      


 
 

Θ
× + Θ + Θ − + − + Θ
−
+ Θ + Θ − Θ + Θ + Θ + Θ
 
 
Θ Θ
× − + − + −

 
  
2
3 2
1
1 1 1
4 4
24 3 3 3 2 4
z
r z x z
h h
R

      
σ
 
 


 


Θ 
+ Θ + × + − − 

 
 
 
The second-order approximation of the required value of the compressor pressure (at the inlet of the
pipeline) was determined from the condition that the output velocity of the free-flowing material is
equal to zero. In the final form, the relation for the required value of the compressor pressure in the
second-order approximation by the method of averaging the functional corrections takes the form,
P =
h S
h R+ P h
S
6
+
h
R
+ P
h S
+ h +
k z z z z
3
3 2
4
3 2
2
2
2 2 2
2
ρ
Θ
α
Θ
ρ
α
ν Θ Θ
ρ
α νΘ α
Θ
α α −




ν − − − −
α Θ ν Θν α
Θ
ρ
α α Θ
ν Θ
ρ
α
α α
r in z z z z
+ +
h
h P
S
6
h
2
h
R
P S
h
+
2 2
2
2
2
2
2
4
3 2 3 2
2
2
2
2
h
Θ 


In the framework of this work, the required velocities of free-flowing material have been determined
in the second-order approximation by the method of averaging the functional corrections. In the same
approximation, the pressure of the compressor is obtained, which makes it possible to increase energy
saving during pneumatic transport. This approximation is usually sufficient to obtain qualitative
conclusions and obtain some quantitative results. The results of the analytical calculations were verified
by comparing them with the results of numerical simulation.
DISCUSSION
In this section, we analyze dependencies of the compressor pressure, which makes it possible to increase
energy saving during pneumatic transport, from certain parameters. Figure 2 shows the dependence
of the optimum value of the compressor pressure on the axial coordinate for different pipe lengths h.
Figure 3 shows the dependence of the optimum value of the compressor pressure on the axial coordinate
Figure 2: Dependences of the optimal value of the compressor pressure from the axial coordinate for different pipe lengths h

8. Pankratov: On the choice of compressor pressure in the process of pneumatic transport
AJMS/Jan-Mar-2019/Vol 3/Issue 1 67
for different diameters of the pipeline R. The increase in the number of the dependence corresponds to
the reduction in the radius of the pipeline (1, 1.5, and 2 m).
CONCLUSION
In this paper, we consider the possibility of determination of a pressure value in devices for pneumatic
transport to increase energy saving. An analytical approach has been introduced for analysis of transport
of free-flowing materials to estimate the transport velocity and the choice of the required pressure value.
The dependencies of the optimum value of the compressor pressure on the parameters were investigated.
REFERENCES
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2. Timoshenko VI, Knyshenko YV. On the application of reasoning. Sci Innov 2013;9:5-17.
3. Krill SI, Chaltsev MN. Appl Hydromech 2010;12:36-44.
4. Chaltsev MN, Bugayev BE. Analytical basic of Therom. Bull Automobile Koad Inst 2016;10:5-11.
5. Levich VG. Physico-Chemical Hydrodynamics. Moscow: Science; 1962.
6. Pankratov EL, Bulaeva EA. On optimization of regimes of epitaxy from gas phase. Some analytical approaches to model
physical processes in reactors for epitaxy from gas phase during growth films. Rev Theor Sci 2015;3:365-98.
7. Pankratov EL, Bulaeva EA. On the basic of detrayed form of mathematics. Multidiscip Mod Mater Struct 2016;12:712-25.
8. Sokolov YD. On the definition of the dynamic forces in mine hoist ropes. Appl Mech 1955;1:23-35.
Figure 3: Dependences of the optimal value of the compressor pressure from the axial coordinate for different diameters of the
pipeline R. The increase in the number of dependence corresponds to the decreasing of the pipeline radius (1, 1.5, and 2 m)