mathematical model of the unsteady fluid flow through tee ... - CiteSeerX

2nd IAHR International Meeting of the Workgroup on Cavitation and Dynamic Problems in Hydraulic Machinery and Systems Timisoara, Romania October 24 - 26, 2007

Scientific Bulletin of the “Politehnica” University of Timisoara Transactions on Mechanics Tom 52(66), Fascicola 6, 2007

MATHEMATICAL MODEL OF THE UNSTEADY FLUID FLOW THROUGH TEE-JUNCTION Jaroslav ŠTIGLER * Victor Kaplan’s Department of Fluid Engineering, Faculty of Mechanical Engineering, Energy Institute, Brno University of Technology

*Corresponding author: Technická 2, Brno, Czech Republic 616 69 Tel.: +42054114 2329, Fax.: +42054114 2347, E-mail: [email protected] ABSTRACT The reason why to deal with mathematical model of tee-junction is to find a way how to treat the teejunction as a pipe-line net element. The current mathematical models Kenji [3] are rather simple based on wrong assumptions and useless for our purpose. The new mathematical model of tee-junction will be showed in this article. This model is possible to use for unsteady fluid flow. The only problem is to determined so-called substitute length of branches of tee junction which appears in unsteady terms of the mathematical model. The first approaching how to set these substitute lengths and first results based on the numerical modeling of fluid flow in the tee-junction will be introduced in this article. KEYWORDS Tee-junction, unsteady fluid flow, mathematical model, substitute length of tee-junction. INTRODUCTION Delivery of water to its destination is a very old problem. The people wanted deliver water to the gardens to water them or to the residences thousands years ago. They made clever channels to water plants or to supply cities with water. When the water is delivered by single pipeline, from one source to the one location, it is simple problem. If there are more sources or more deliver locations, the problem is much more complex and the pipeline has to be branched in that case. Nowadays the large pipeline nets are used for delivering different kind of fluid with different purposes of utilizing. For example, the delivering drinkable

water from water treatment plants to the residences, delivering the hot water from boiler to the heatings in houses. Many kinds of pipeline nets are used in industry and at last we can not forget the vascular system of our body. The pipeline nets consist of pipes, shaped pieces (tee junctions, elbows, contracting and expanding elements) and other elements like pumps, valves, taps, etc. The main purpose of net design is to satisfy the required parameters at the end locations of pipeline net, like flow rate and pressure, with minimum consumed energy and without any element destruction. As regards the safety of pipeline net operation, the study of the unsteady processes is very important. To ensure requirements listed in the previous paragraph the numerical modeling of fluid flow in the net has to be done. It means that mathematical models of all net elements have to be known. One of these pipeline net elements is tee-junction. Tee junction is used to divide flow, to combine flow or for the connection of some pipeline net elements, for example air chamber. The mathematical model of Tee junction for unsteady fluid flow will be subject on this article. MATHEMATICAL MODEL OF TEE-JUNCTION First of all it is important to explain the meaning of the term “mathematical model of tee-junction”. It means set of equations for flow rates and area average pressures over cross-sections in all pipes forming the tee-junction. Contemporary mathematical models are described for example in Kenji [3] and they are based on the ideas of Miller [4].

84

Proceedings of the 2nd IAHR International Meeting of the Workgroup on Cavitation and Dynamic Problems in Hydraulic Machinery and Systems

These models are based on the wrong assumption, that pressure in tee-junction is constant. Some of the coefficients defined for the tee-junction do not have physical meanings. These models are not defined for unsteady flow. A new mathematical model of tee-junction, which shape is depicted in the fig. 1, has been introduced in the Štigler [1]. For the practical calculation is important to know the size of tee-junction. Size of the tee-junction is given by location of the ending cross-sections with area A(a) A(b) and A(c). These cross-sections have to be located in distance, from center of tee-junction, where the disturbances of flow, caused by tee-junction, can be neglected. The minimum size of the tee junction had been found then diameters from the center of the tee-junction. This minimum size was found on the base of numerical modeling for the tee-junction with the same diameters in all branches. This mathematical model consists of three equations. First two of them are derived from the Navier Stokes equation(1). ∂c ∂c ∂p ∂Π ij ρ i + ρ i cj + − = ρg i ∂t ∂x j ∂x i ∂x j

(1)

The power equation is received by multiplying of Navier-Stokes Equation with velocity ci and its subsequent integration over tee-junction area. P( a ) + P( b ) + P( c ) + P( L ) = 0 .

(6)

where P(a) is the power going through the cross-section A(a), P(b) is the power going through the cross-section A(b), P(c) is the power going through the cross-section A(c) and P(L) is the lost power in the tee-junction. The power in our meaning is the energy per time. The power on a given cross-section A(x) (x=a,b,c) is given as a sum of powers P( x ) = P( tx ) + P( kx ) + P( px ) + P( gx ) P( tx ) =

(

1 ∂ ρ ( x ) c (2x ) A ( x ) L ( Px ) 2 ∂t

P( kx ) = α ( x )

ρ ( x ) Q (2x ) 2 A (2x )

)

.Q ( x )

(7) (8) (9)

P( px ) = p ( x ) Q ( x )

(10)

P( gx ) = − g 1ρ ( x ) c ( x ) A ( x ) L ( Mx ) .

(11)

where P(tx) is an unstable power, P(kx) is an kinetic power, P(px) is an pressure power and P(gx) is an potential power. It is assumed that lost power P(L) is proportional to the kinetic power of flow in the branch where the whole flow rate flows through it. P( L ) =

Q (2C ) 1 ρ ( C ) ξ( P ) 2 Q ( C ) 2 A (C)

(12)

Subscript (C) should be replaced with one of branch subscript (a), (b), or (c) where whole flow rate flows through it. The power coefficient ξ(P) is coefficient which show us the lost power in tee-junction. Momentum equation in direction x1 is received by integrating of Navier-Stokes equation over tee-junction area. Momentum equation expresses equilibrium of forces which are affecting on tee-junction.

Fig. 1. Tee-junction Assumptions taken into consideration during derivation are: 9 on the cross-sections A(a) and A(b)

F( t )1 + F( k )1 + F( p )1 + F( g )1 + F( R )1 = 0

(13)

F( t )1 = F( ta )1 + F( tb )1

(14)

F( k )1 = F( ka )1 + F( kb)1

(15)

∂c1 = 0, ∂x 1

(2)

F( p )1 = F( pa )1 + F( pb )1

(16)

c2 = c3 = 0 ,

(3)

F( g )1 = −g1 m

(17)

∂c 2 =0 ∂x 2

(4)

c1 = c 3 = 0 .

(5)

F(t)i is a force affecting tee junction caused by unsteady inertial acceleration of fluid, F(k)i is a force affecting tee-junction caused by advective inertial acceleration of fluid, F(p)i is pressure force, F(g)i is an gravity force. F(T)i is a resultant force affecting the

9 on the cross-section A(c)


tee-junction. For the coordinate system by fig. 1 we can write. F( tx )1 =

∂ (ρ ( x ) c ( x )S( x ) L ( Mx ) ) ∂t

(18)

F( kx )1 = ρ ( x ) c ( x ) Q ( x )

(19)

F( p )1 = p ( x ) n ( x )1S ( x )

(20)

The subscript (x) should be replaced by subscripts (a) or (b). It is assumed that the resultant force F(R)i is proportional to size of force F(kC)i in the branch where the whole flow rate flows through it. F( R )1 = ρ ( C ) .A ( C ) .ξ ( M )1 .

Q (2C ) A (2C )

(21)

Coefficient ξ(M)1 is proportional to the force affected tee-junction in direction x1. Third equation is an equation of continuity. ρ ( a ) .Q ( a ) + ρ ( b ) .Q ( b ) + ρ ( c ) .Q ( c ) = −

∂ ρdV (22) ∂t V∫

where Q( x ) = c( x )i n ( x )i A( x )

ρ ( a ) .Q ( a ) + ρ ( b ) .Q ( b ) + ρ ( c ) .Q ( c ) = 0 .

The friction coefficients are depended on the ratio of flow rates q(ca) in branches (c) and (a), diameters of branches and on the whole flow rate in teejunction. q ( ca ) =

(24)

The power and momentum coefficients appear in the power and momentum equations. Determining these coefficients for steady flow on the base of numerical modeling of flow in the tee-junction is shown in Štigler [1]. Both coefficients can be divided into two parts. ξ( P ) = ξ( PG ) + ξ( PF )

(25)

ξ( M ) i = ξ( MG ) i + ξ( MF ) i

(26)

Part signed with subscript (F) represents the influence of friction. Part signet with subscript (G) represents the influence of tee-junction shape. The friction coefficients ξ(PF) and ξ(MF) are possible to determine theoretically or by measuring in the straight pipes with the constant diameter Štigler [1]. The shape coefficient ξ(PG) and ξ(MG) is possible to determine by measuring or by numerical modeling of flow Stigler [2].

Q (c)

(27)

Q(a )

The shape coefficients of the geometrically similar tee-junction are depended only on the ratio of flow rates q(ca). This is very interesting result. SUBSTITUTE LENGTHS DEFINITION

For unsteady flow the substitute length L(Ma), L(Mb), L(Mc), L(Pa), L(Pb) and L(Pc) appear in the equations forming the mathematical model of tee-junction. Subscript (M) indicates the substitute momentum lengths and subscript (P) indicates substitute power lengths. These substitute lengths say how big part of area, where flow is divided, is included to the given branch. Substitute lengths came from modification of the unsteady term in the Navier-Stokes equation. In accordance with fig. 1 we can write

(23)

These equations (6), (13) and (23) are valid for the unsteady flow of compressible fluid through the tee-junction. For the first approaching we neglected the compressibility of fluid. It is possible to do it, if the lengths of branches of tee-junction are small against to size of pipe-line net. So the equation of continuity transforms into

85

L ( Mx ) = L ( x ) + d ( c ) .δ ( Mx )

(28)

L ( Mc ) = L ( c ) + d ( a ) .δ ( Mc )

(29)

L ( Px ) = L ( x ) + d ( c ) .δ ( Px )

(30)

L ( Pc ) = L ( c ) + d ( a ) .δ ( Pc )

(40)

The subscript (x) can be replaced with (a) or (b). Where L(x) and L(c) are constant lengths given by size of tee-junction, d(c) and d(a) are diameters of branches and δ are the ratios of the part of area lengths and given diameter. For us is important to find these ratios δ. The substitute lengths can be expressed from equations, Štigler [1]. x ( b )1

∫ ρ.c

( av )1

.A (1) .dx 1 =

(41)

x ( a )1

= ρ ( a ) .c ( a )1 .A ( a ) .L ( Ma ) + ρ ( b ) .c ( b )1 .A ( b ) .L ( Mb ) x(c)2

∫ ρ.c ( av )2 .A ( 2) dx 2 = ρ ( c ) .c ( c )2 .A ( c 2) .L ( Mc )

(42)

0

x ( b )1

∫α

( P1)

.ρ.c (2av )1 .A (1) dx 1 =

(43)

x ( a )1

= ρ ( a ) .c

2 ( a )1

.A ( a ) .L ( Pa ) + ρ ( b ) .c

2 ( b )1

.A ( b ) .L ( Pb )

x( c)2

∫0 α ( P 2) .ρ.c (av )2 A ( 2) dx 2 = ρ (c ) .c (sc )2 A ( c2) .L ( Pc ) 2

2

(44)


86

If we take into consideration the equations (28), (29), (30) and (40) and in accordance with fig. 1, it is possible to write δ ( Ma ) = δ ( Mb ) = −

H ( M1a ) .d ( c ) .I' ( M1) − H ( M1b ) .L ( ab )

δ ( Pa ) =

−

d ( c ) .(H ( M1a ) − H ( M1b ) )

H ( M1a ) .d ( c ) .I' ( M1) − H ( M1a ) .L ( ab )

H ( M1a ) .d ( a ) .I' ( M 2 ) d ( a ) .H ( M 2 c )

H ( P1a ) .d ( c ) .I' ( P1) − H ( P1a ) .L ( ab ) d ( c ) .(H ( P1a ) − H ( P1b ) )

H ( P1a ) .d ( a ) .I' ( P 2 ) d ( a ) .H ( P 2 c )

−

d (c)

(45) (46)

(47)

d (a )

d ( c ) .(H ( P1a ) − H ( P1b ) )

δ ( Pc ) =

L ( b)

L(c)

−

H ( P1a ) .d ( c ) .I' ( P1) − H ( P1b ) .L ( ab )

δ ( Pb ) = −

d (c)

−

d ( c ) .(H ( M1a ) − H ( M1b ) )

δ ( Mc ) =

L(a )

−

L (a ) d (c)

−

L(c) d (a )

L ( b) d (c)

(48)

(49)

(50)

where H ( M1x ) = ρ ( x ) .c ( x )1 .A ( x )

x=(a,b)

(51)

x'2 =

x=(a,b)

H ( P 2c ) = ρ ( c ) .c (2c ) 2 .A ( c )

(53) (54)

An integrals I’(M1), I’(M2), I’(P1) and I’(P2) appears in terms for δ(M) and δ(P). The values of these integrals are crucial for setting the δ(M) and δ(P). Dimension of these integrals is [1]. So the integrals have a form x '( b )1

∫

I' ( M1) =

x ' ( a )1 x '( c ) 2

I' ( M 2 ) =

∫ 0

H ( M1) H ( Ma ) H ( M 2) H ( M1a )

x '( b )1

I' ( P1) =

∫

α ( P1) .

x '( a )1

x '( c ) 2

I' ( P 2 ) =

∫

α( P2)

0

.dx '1

(55)

dx ' 2

(56)

H ( P1) H ( P1a )

H ( P2) H ( P1a )

.dx '1

(57)

.dx ' 2

(58)

where x '1 =

x1 d (c)

(59)

(60)

H ( M1) = ρ.c ( av )1 .A (1)

(61)

H ( M 2 ) = ρ.c ( av ) 2 .A ( 2 )

(62)

H ( P1) = ρ.c (2av )1 .A (1)

(63)

H ( P 2 ) = ρ.c (2av ) 2 .A ( 2 )

(64)

Functions H(M1), H(P1), change with dimensionless coordinate x’1 and functions H(M2), H(P2), change with dimensionless coordinate x’2. The interval for integrals (55) and (57) value specifying is x’1∈ for the functions H(M1), H(P1). The interval for integrals (56) and (58) value specifying is x’2∈ for the functions H(M2), H(P2). New variables α(B1) and α(B2) appears in the integrals H(P1), H(P2). These variables can be regarded as a Bousinesque’s numbers which reflect non uniformity of velocity profile. If the velocity profile is uniform the Bousinesque’s number value is 1. α ( P1) =

H ( M 2 c ) = ρ ( c ) .c ( c ) 2 .A ( c ) (52)

H ( P1x ) = ρ ( x ) .c (2x )1 .A ( x )

x2 d (a )

α( P2) =

(ρ.c )

ρ

2 1 ( av ) 2 ( av ) ( av )1

.c

(ρ.c ) ρ

2 2 ( av ) 2 ( av ) ( av ) 2

.c

(65)

(66)

The integrals value can be determined by theoretical way or on the base of numerical modeling of fluid flow in the tee-junction. Evaluating of integrals by experimental way is very difficult, because the setting velocity profiles on the cross-sections A(1) and A(2).

Fig. 2. Cross-sections A(1)

Cross-sections A(1) are received by sections of tee-junction with plane normal to the coordinate axis x1. Cross-sections A(2) are received by sections of tee-junction with plane normal to the coordinate axis x2. Cross-sections A(1) and A(2) are depicted on the fig. 2.

Proceedings of the 2nd IAHR International Meeting of the Workgroup on Cavitation and Dynamic Problems in Hydraulic Machinery and Systems 1

I *( M 2 ) = ∫ 0

1

I *( P1) = ∫ 0

H ( P1) H ( P1a ) 1

I *( P 2 ) = ∫ 0

Fig. 2. Cross-sections A(2) THEORETICAL SUBSTITUTE LENGTHS SETTING

The theoretical way of substitute lengths solution is based on the next assumptions: 9 Values of Bousinesque’s numbers α(P1) and α(P2) are equal 1. 9 The function H(M1) change is a 3-rd order polynomial on the interval with zero tangent at the start and end point of polynomial. Function H(M1) is constant out of this interval. 9 The function H(M2) change is a 3-rd order polynomial on the interval too with zero tangent at the start and end point of polynomial. Function H(M2) is constant out of this interval. 9 The cross-section A(a) = A(b). 9 The fluid is incompressible The integrals (55), (56), (57) and (58) have analytical solution under these assumptions. I'( M1) =

L( a ) d(c)

+ I *( M1) +

I'( M 2 ) = I *( M 2 ) + I'( P1) =

L( a ) d (c)

1

0

H ( M1)

(68)

H ( P1b ) L( b ) H ( P1a ) d ( c )

H ( P 2c ) L( c )

H ( Ma )

q ( ca ) 2

H ( P2) H ( P1a )

+1

q ( ca ) 2

(72)

13 2 .q ( ca ) − q ( ca ) + 1 (73) 35

.dx ' 2 =

13 2 A ( a ) .q ( ca ) . 35 A (c)

(74)

δ ( Pa ) =

δ( Ma ) =

1 2

(75)

δ ( Mb ) =

1 2

(76)

δ ( Mc ) =

1 2

(77)

H ( M 1b ) 13 ⎛⎜ . 1− 35 ⎜⎝ H ( M1a )

⎞ H ( M 1b ) ⎟+ ⎟ H ( M1a ) ⎠ ⎛ ⎞ H ⎜ 1 + ( M 1b ) ⎟ ⎜ H ( M1a ) ⎟⎠ ⎝

δ ( Pb ) =

H ( M 1b ) 13 ⎛⎜ . 1− 35 ⎜⎝ H ( M1a ) ⎛ H ⎜ 1 + ( M 1b ) ⎜ H ( M1a ) ⎝

δ ( Pc ) =

(71)

dx ' 2 =

The dimension of integrals I*(M1), I*(M2), I*(P1) and I*(P2) is [1]. Values of these integrals in dependence on q(ca) are shown in fig. 3. If it is possible to express these integrals I* on the base of numerical modeling of fluid flow in teejunction or even on the base of experiment, they will be suitable for comparison between tee junction with different shape. Now we can found the theoretical ratios δ and than the substitute lengths. If the theoretical solutions of integrals are applied in to equations (45) – (50) then the ratios δ are received.

(69)

(70)

H ( P1a ) d ( a )

.dx '1 = −

.dx '1 =

(67)

Integrals I*(M1), I*(M2), I*(P1) and I*(P2) are integrals only over interval . Values these integrals for the uncompressible fluid are I *( M1) = ∫

H ( M1a )

1−

H ( M 2c ) L( c ) . H ( M1a ) d ( a )

+ I *( P1) +

I'( P 2 ) = I *( P 2 ) +

H ( M1b ) L ( b ) . H ( Ma ) d ( c )

H ( M 2)

87

⎞ ⎟ ⎟ ⎠

13 A ( a ) 35 A ( c )

⎞ ⎟ ⎟ ⎠

(78)

(79)

(80)

The next relationship is valid for the case of uncompressible fluid. H ( M1 b ) H ( M1a )

= q ( ba ) = 1 − q ( ca )

(81)

The dependence δ on the q(ca) is shown in the fig. 4.


88

1 0,9 0,8 0,7

I* [1]

0,6

I*(P1)-teor I*(P2)-teor I*(M1)-teor I*(M2)-teor

0,5 0,4 0,3 0,2 0,1 0 0,0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1,0

q(ca) [1]

Fig. 3. Integrals I*(M1), I*(M2), I*(P1) and I*(P2) in dependence on the q(ca) received by theoretical way. 0,7 0,6 0,5 Delta(Pa) Delta(Pb) Delta(Pc) Delta(Mabc)

δ

0,4 0,3 0,2 0,1 0,0 0,0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1,0

q(ca)

Fig. 4. Ratios δ(Ma), δ(Mb), δ(Mc), δ(Pa), δ(Pb) and δ(Pc), in dependence on the q(ca) received by theoretical way. SUBSTITUTE LENGTHS SETTING ON THE BASE OF NUMERICAL MODELING OF FLUID FLOW IN TEE-JUCNTION

Theoretical results seem to be clear and reasonable. The only problem is with integrals I*(P) and ratios δ(P), because the Bousinesque’s number was neglected in this case. This number has high influence on the value of integral I*(P) and consequently on the δ(P) because the velocity profile in the area of flow division is strongly non-uniform. So now the integrals I* and ratio δ will be determined on the base of numerical modeling of fluid flow in the tee-junction. The numerical modeling was made for four Reynold’s numbers in the inlet branch (a). For every

Reynlolds number in the inlet branch the twenty rates of flow ratio q(ca) was calculated. The ratio q(ca) had been varied from 0 to 1 with step 0,05. Diameter all branches of tee-junction was equal and fluid was assumed as uncompressible. Reynolds numbers for which the solution was made are listed in the table 1. Table 1. List of Reynold’s numbers in inlet branch (a) for which the calculation was made. Re(a) Set number 18 909 01 28 364 02 37 818 03 47 273 04


The Integrals I*(M1) and I*(M2) in dependence on q(ca) are shown in the fig 5. The dependence ratios δ(Ma), δ(Mb) and δ(Mc) on the q(ca) is shown on te fig.6. There is a good agreement between theoretical result and data received from the numerical experiment. The example of Bousinesque’s numbers for the set 01 is showed on the fig 7 and 8. It is apparent that influence of Bousinesque’s number is very high. The values of integrals I*(M) and I*(P) in dependence on the ratio q(ca) for all four computation sets are shown in the fig. 9. The values of ratios δ(Pa), δ(Pa), δ(Pa), obtained from numerical modeling of flow in tee-junction is shown in the fig. 10.

89

RESULT DISCUSSION

Mathematical model of tee-junction, introduced in Štigler [1,2], is possible use for the unsteady flow. The substitute lengths of branches have to be set in this case. The substitute lengths for power equation and momentum equation are different. The aim of this article was to show the method of substitute length setting and bring out first results based on the theoretical solution and numerical modeling of fluid flow in the tee-junction given shape. It is important that these results where carried out under some assumptions. The most important are: 9 change of flow rate ratio q(ca) is very slow, 9 the fluid is uncompressible.

1 0,9 0,8

I*(M) [1]

.

0,7

I*(M1)-01 I*(M1)-02 I*(M1)-03 I*(M1)-04 I*(M2)-01 I*(M2)-02 I*(M2)-03 I*(M2)-04

0,6 0,5 0,4 0,3 0,2 0,1 0 0,0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1,0

q(ca) [1]

Fig. 5. Integrals I*(M1) and I*(M2) received from numerical modeling of fluid flow in tee-junction. 0,6

0,5

Delta(Ma)-01 Delta(Ma)-02 Delta(Ma)-03 Delta(Ma)-04 Delta(Mb)-01 Delta(Mb)-02 Delta(Mb)-03 Delta(Mb)-04 Delta(Mc)-01 Delta(Mc)-02 Delta(Mc)-03 Delta(Mc)-04

δ(M) [1] .

0,4

0,3

0,2

0,1

0 0,0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1,0

q(ca) [1]

Fig. 6. Ratios δ(Ma), δ(Mb) and δ(Mc) received from numerical modeling of fluid flow in tee-junction.


90

50 45 40 35

q(ca)=0,00 q(ca)=0,05 q(ca)=0,20 q(ca)=0,40 q(ca)=0,60 q(ca)=0,80 q(ca)=0,95

α(1) [1]

30 25 20 15 10 5 0 -0,5

0

0,5

1

1,5

2

x1/d(c) [1]

Fig. 7. Bousinesque’s number α(1) in direction x’1, For the set 01. 60

50

α(2) [1]

40

q(ca)=0,05 q(ca)=0,20 q(ca)=0,40 q(ca)=0,60 q(ca)=0,80 q(ca)=0,95 q(ca)=1,00

30

20

10

0 0,0

0,2

0,4

0,6

0,8

1,0

1,2

1,4

1,6

1,8

2,0

x2/d(a) [1]

Fig. 8. Bousinesque’s number α(2) in direction x’2, For the set 01. 3,0

2,5 I*(P1)-01 I*(P1)-02 I*(P1)-03 I*(P1)-04 I*(P2)-01 I*(P2)-02 I*(P2)-03 I*(P2)-04

I*(P) [1]

2,0

1,5

1,0

0,5

0,0 0,0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1,0

q(ca) [1]

Fig. 9. Integrals I*(P) obtained from numerical modeling of flow in tee-junction


91

40

30

δ(P) [1]

20

10

0 0,0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1,0

-10

(P1a)-01 (P1a)-02 (P1a)-03 (P1a)-04 (P1b)-01 (P1b)-02 (P1b)-03 (P1b)-04 (P2c)-01 (P2c)-02 (P2c)-03 (P2c)-04

-20 q(ca) [1]

Fig. 10. Ratios δ(Pa), δ(Pa), δ(Pa), obtained from numerical modeling of flow in tee-junction

These assumptions are rather strong. Therefore these results should be regarded only as the first steps in the modeling of tee-junction. The stetting of substitute lengths by experimental way is very complicated or it is possible to say unrealistic because the functions H(M1), H(M2), α(1) and α(2) have to be found to be able solve substitute lengths. On the other hand inaccuracy of substitute lengths setting is rather small in case that size of tee-junction is big, it means that the distance of cross-sections A(a), A(b) and A(c), is about ten diameters. The accuracy of substitute lengths setting plays high role when the size of tee-junction is decreasing. It is apparent that in substitute momentum length is high agreement between theoretical and numerical results from comparison of fig. 4 and fig. 6. There is only small disturbances in the area of low q(ca) ratio. In case of substitute power length the agreement is worse. It is apparent from comparison of fig.4 and fig.10. The difference is caused due to Bousinesque’s number. This number plays strong role in the substitute power length setting. But it is possible recognize the tendency of ratios δ(Pa) and δ(Pb) go closer to the theoretical results with Reynold’s number increasing from fig. 10. Another way to increase numerical solution is numerical integration of integrals I*(P1) and I*(P2). The integration step is rather rough out of the area flow division. At the end is possible to say that for the substitute momentum lengths are possible use values (75), (76) and (77). The situation is more complicated in case of substitute power length. More investigation has to be done in this case.

The further research will be focused on the numerical solution of unsteady flow and experiment in the tee-junction for better understanding of fluid flow in tee-junction. ACKNOWLEDGEMENTS

The author is grateful to the Ministry of Industry and Trade of the Czech Republic for funding this research under project with registration number 1H – PK2/51. NOMENCLATURE

A [m2] Fi [N] L [m] P [W] Q [m3.s-1] V [m3] ci [m.s-1] d [m] gi [m.s-2] ni [1] p [Pa] q(ca) [1] t [s] x1, x2, x3 [m] x’1, x’2, x’3 [m] α [1] δ [1] ξ ρ [kg.m-3]

Area Force vector Length Power Flow rate Volume Velocity vector. Diameter Gravity acceleration vector. Unit normal vector to the surface Static pressure. Ratio of flow rate Time. Coordinates in given direction Dimensionless coordinates in given direction. Bousinesque’s number Proportional part of division area of tee-junction added to the given branch. Tee-junction coefficients Density

92


Subscript and Superscripts (C) Subscript (C) should be replaced with one of branch subscript (a), (b), or (c) where whole flow rate flows through it. (F) Friction (G) Geometry, shape (L) Losses term (M) Term bounded with momentum equation (P) Term bounded with power equation (a), (b), (c) Sign of branch tee-junction. (av) Average (g) Gravity (k) Kinetic (p) Pressure (t) Unsteady term 1, 2, 3 Vector components in coordinate directions

REFERENCES [1] Štigler, J., 2006, “Tee Junction as a Pipeline Net Element. Part 1. A New Mathematical model.”, Journal of Mechanical Engineering, 57 , pp 249-262. [2] Štigler, J., 2006, “Tee Junction as a Pipeline Net Element. Part 2. Coefficients determination.”, Journal of Mechanical Engineering, 57 , pp 263-270. [3] Kenji, O., Hidesato, I., 2005, “Energy Losses at Tees With Large Area Ratios,” Journal of Fluids Engineering, 127, pp. 110-116. [4] Miller, D., S., 1975, “Internal Flow. A Guide to Losses in Pipe and Duct Systems.”, Hydromechanics Research Association, Cranfiled, Bedford, England, pp. 31-40.