Determination of the resistance characteristics of pipe ...

Budapest University of Technology and Economics

Determination of the resistance

characteristics of pipe components by

using computational fluid dynamics

models

András Tomor

A short summary of the thesis submitted

for the degree of doctor of philosophy

Supervisor: Dr. Gergely Kristóf

2018

Faculty of Mechanical Engineering Department of Fluid Mechanics

2

Contents

Introduction .................................................................................. 3

Aim and objective of the thesis .................................................... 4

Investigation methods .................................................................. 5

Definition of loss coefficients and the parameter spaces ............. 6

Results and discussion .................................................................. 9

Thesis statements ....................................................................... 11

Publications related to thesis statements .................................... 20

References .................................................................................. 21

3

Introduction

Fluid distribution systems are used in several technical appliances, e.g., in water

and wastewater treatment, swimming pool technology, air engineering, and

polymer processing. The accuracy of models implemented to design hydraulic

networks determines the uncertainty of the design, which is usually

compensated by applying larger overpressures in the systems – at the expense

of energy efficiency. Dividing and combining junctions are major elements of

fluid distribution systems, and their hydraulic resistance involves large

uncertainties. The accuracy of resistance models of the components has a

significant effect on the accuracy of the model used for hydraulic network

design; therefore, it is very important to work with reliable resistance models.

There are several different types of hydraulic components in fluid

distribution systems. Frequently used elements are the junctions, cross-section

transitions, elbows and valves [1–3]. The connection points of more conduits,

i.e., the nodes are of great importance [4, 5]. Junctions and cross-section

transitions, e.g., gradual and sudden expansions and contractions can be

modeled as nodes in network models. In these models, cross-section transitions

can be treated similarly to junctions – one can also model the connection of two

conduits as a node. In my research, I focus on the hydraulic components that

can be modeled as nodes of connecting pipe sections.

In the case of single phase flows of incompressible Newtonian fluids, loss

coefficients of three-way junctions are functions of more geometrical and flow

parameters. Loss coefficients strongly depend on the flow directions and the

ratios of volume flow rates in the conduits, and Reynolds number dependency

can also be observed in many instances. Effects of geometrical changes are also

significant. When ideal geometries with sharp edges are considered, values of

loss coefficients are influenced mainly by the cross-sectional area ratios and the

angles between the conduits. It is important to emphasize that the loss

coefficient of a finite length dividing junction is also affected by the ratio of the

port length and the inner diameter of the main conduit. In the case of geometries

with sharp edges, the loss coefficient of cross-section transitions in cylindrical

pipes is the function of the cross-sectional area ratio, the angle and the Reynolds

number [1–3].

4

Aim and objective of the thesis

The objective of the investigations is the determination of nodal total pressure

losses in hydraulic systems and the loss coefficients of passive hydraulic

components that can be modeled as nodes of connecting pipe sections. In

previous studies of the literature [1–3], there were no possibilities to cover the

mentioned parameter spaces in detail due to the large number of cases;

therefore, a novel approach is used for the determination of the loss coefficients

by applying computational fluid dynamics (CFD) models. Accurate and yet

simple correlations based on new formalisms are found, which are valid in the

most important parameter ranges for engineering practice.

Firstly, I deal with finite length dividing junctions of cylindrical conduits.

The loss coefficient of the port is determined for 40 different geometries by

using the results of more than 1000 three-dimensional CFD simulations. The

new resistance formula is also applied in a discrete model of a simple hydraulic

system. In order to investigate the accuracy of the model, its results are

compared to data of the literature and own experiments.

After elaborating the new resistance formula of finite length dividing

junctions, a novel parametrization of nodal total pressure losses is implemented.

This novel method makes it possible to characterize nodal total pressure losses

independently of flow directions. Previous studies [1–3, 6] provide different

loss coefficient correlations for different combinations of flow directions, i.e.,

different combinations of flow directions are treated as different types of

hydraulic junctions. In changing flow conditions, there can be transitions

between junction types; in the transition point, there is no flow in one of the

three conduits. The loss coefficients are usually defined for the common

channel, and the reference velocity is often calculated differently for different

junction types according to the actual position of the common channel. Therefore, application of these earlier correlations in network models is difficult

when the flow directions change in the hydraulic network.

Recent publications that focus on the determination of the loss coefficients

of three-way junctions do not investigate all possible combinations of flow

directions together [7–10], or the investigation does not cover loss coefficient

values over wide ranges of geometrical parameters [11]. It is difficult to use

these model results for elaborating a general loss coefficient formula that covers

both dividing and combining flow regimes and satisfies the requirements of the

network model. Due to the mentioned deficiencies of the literature, a new

resistance model is elaborated for three-way junctions. Instead of performing

classical experimental investigations, the parameter space is covered by

5

numerous three-dimensional CFD simulations. The resistance model contains a

continuous loss coefficient formula, which is valid for all of the investigated

junction types and flow directions. Consequently, each junction type is treated

in the same way, and there are no discontinuities in the model owing to the

novel interpretation of the reference velocity as well as the application of a

periodic fitting. The introduced general method is also applicable to cross-

section transitions.

Investigation methods

The loss coefficients of the hydraulic components are determined by applying

computational fluid dynamics (CFD) models. Simulations are performed using

the ANSYS FLUENT CFD software. The models solve the Reynolds-averaged

Navier–Stokes (RANS) and continuity equations for incompressible turbulent

steady-state flow based on the two-equation k– shear stress transport (SST)

turbulence model [12]. The geometrical models of the hydraulic components

and the investigated flow directions are shown in Figs. 1 and 2. Models of two

or three connecting conduits are created according to a two-dimensional or

three-dimensional modeling approach, respectively. Benefits of symmetry and

axisymmetry are always utilized. It is important to note that the applied two-

dimensional axisymmetric approach makes it possible to draw conclusions

regarding real three-dimensional problems; hence, the model of cross-section

transitions fits the three-dimensional approach.

A discrete hydraulic model is constructed using the resistance model of the

finite length dividing junction. Volume flow rate distributions in fluid

distribution systems can be calculated with this discrete model. The model

results are validated by means of literature data and own laboratory

experiments.

Fig. 1: Geometrical model of the cross-section transition. Continuous line: expansion;

dashed line: contraction; red line: actual geometry according to the two-dimensional

axisymmetric approach.

6

(a)

(b)

Fig. 2: Geometrical models of hydraulic components that consist of three connecting

conduits and the investigated flow directions: (a) flow manifold segment with one single

port; (b) three-way junction with long branch tube

Definition of loss coefficients and the parameter spaces

The turning loss coefficient of the finite length dividing junction is defined as

follows:

22

2vCp ft

, (1)

7

where Cf is the turning loss coefficient, is the fluid density, v2 is the average flow

velocity at the port outlet, and Δpt is the total pressure drop between an upstream

cross-section in the header and the port outlet – its value does not contain the

friction loss for the header flow. The loss coefficient is determined as a function of

the upstream Reynolds number in the header (Re1) and the ratio of port and header

flow velocities (v2/v1) for different diameter ratios (D2/D1 = 0.2 ÷ 1) and relative port

lengths (L2/D1 = 0.1 ÷ 2).

In the course of the investigations regarding three-way junctions with long

branch tube, three different angles (α = 45°, 60° and 90°) and seven different cross-

sectional area ratios (S2/S1 = 0.1 ÷ 1) for every angle are investigated. Loss

coefficients of three-way junctions for arbitrary flow directions are defined as

follows:

2rms113

2vCp tt

and (2)

2rms223

2vCp tt

, (3)

where Ct1 and Ct2 are the loss coefficients, and vrms is the reference velocity. Total

pressure difference between cross-sections (1) and (3) as well as total pressure

difference between cross-sections (2) and (3) are denoted by Δpt13 and Δpt23,

respectively. Neither of these total pressure differences contains friction losses. The

reference velocity is defined as

3

23

22

21

rms

vvvv

, (4)

in which v1, v2 and v3 are the signed average flow velocities in cross-sections (1), (2)

and (3), respectively. Sign of a velocity value is always positive when fluid enters

the computational domain and flows toward the junction; consequently, signs are

negative when fluid leaves the junction and computational domain.

According to the definition of reference velocity vrms, an equation with

normalized velocities can be written as

3 3 222

2

3

2

2

2

1

ZYX

v

v

v

v

v

v

rmsrmsrms

, (5)

where X, Y and Z are the dimensionless signed average flow velocities in cross-

sections (1), (2) and (3), respectively. The continuity equation for an incompressible

flow can also be written in a normalized form as follows:

0 01

3

1

23

1

32

1

21 ZS

SY

S

SX

v

v

S

S

v

v

S

S

v

v

rmsrmsrms

, (6)

8

in which S1, S2 and S3 are the cross-sectional areas of conduits (1), (2) and (3),

respectively. The solution of the system of equations consisting of Eqs. (5) and

(6) is a circle of radius 3 that is centered at the origin and located on the plane

determined by Eq. (6). According to the new formalism, possible normalized

velocity combinations at a given geometry are located on this circle. Therefore,

every normalized velocity combination can be characterized by an angle, and a

polar coordinate system can be introduced. The polar angle is denoted by , which

can be calculated for each combination of normalized velocities with a general

formula. All physically possible cases and the actual cases covered by this thesis

are illustrated in Fig. 3. The investigated parameter range is restricted to the green

surfaces enclosed by the planes that correspond to the two extreme geometric

cases (S2/S1 = 0.1 and S2/S1 = 1) of the present investigation. Cross-sectional areas

of conduits (1) and (3) are equal in all of the investigated cases.

Fig. 3: All physically possible cases and the investigated parameter range

The introduced method is simpler for cross-section transitions: The sphere that

arises from the definition of the reference velocity becomes a circle, and the

continuity is a line in this case. The polar angle is denoted by this time. The loss

coefficient of the cross-section transition (Ck) is also defined using the new

reference velocity that is the root mean square of the average flow velocities in the

conduits. Two geometrical control parameters are introduced: the cross-sectional

area ratio (S2,k/S1,k = 0.01 ÷ 100) and the angle (= −90 ÷ 90).

9

Results and discussion

The effect of structure and flow conditions on the turning loss coefficient of the

finite length dividing junction is thoroughly scrutinized. An important,

representative result is shown in Fig. 4. Value of the turning loss coefficient

decreases with the increase of the port length to header diameter ratio and

approaches the loss coefficient of a T-junction [1]. The formula for the

calculation of the turning loss coefficient of the finite length dividing junction is

presented in Statement 1.

Fig. 4: Turning loss coefficient as a function of the ratio of the port length and the inner

diameter of the header pipe; Re1 = 3×105

The properties and method that makes possible the cyclic parametrization of

loss coefficients of three-way junctions and the trigonometric loss coefficient

formula are presented in Statement 2 and 3. The effectiveness of the method is

demonstrated in Fig. 5.

Hydraulic loss coefficient of the cross-section transition of cylindrical pipes

can be determined for arbitrary flow directions by using a two-step method. The

method is presented in Statement 4. Figure 6 shows some expressive model

results and their comparison to literature data.

10

Fig. 5: The Ct1 loss coefficient as a function of the polar angle – validation of model results

against correlations of previous studies [1–3, 13]

Fig. 6: Loss coefficient of the cross-section transition as a function of the polar angle –

validation of model results against values from previous studies [1, 14]

11

Thesis statements

Statement 1

The turning loss coefficient of the finite length dividing junction of cylindrical

conduits characterized by the geometrical and operating parameters shown in

Fig. T1.1 and Tables T1.1 and T1.2 can be determined as a function of Re1 and

v2/v1 according to Eqs. (T1.1) and (T1.2) and Table T1.3.

Fig. T1.1: Geometrical and flow properties of the finite length dividing junction

Table T1.1: Nomenclature

A1, A2, A3, B1, B2, B3 constants [-]

Cf turning loss coefficient [-]

D1 inner diameter of the header pipe [m]

D2 inner diameter of the port [m]

L2 port length [m]

Re1 upstream Reynolds number in the header [-]

v1 upstream average flow velocity in the header [m/s]

v2 average flow velocity in the port [m/s]

Δpt total pressure drop between an upstream cross-section in the

header and the port outlet – its value does not contain the

friction loss for the header flow [Pa]

fluid density [kg/m3]

12

Table T1.2: Conditions of the application and the scope of the model

Characteristics of the flow single phase, turbulent, steady-state

Material properties of the fluid constant density and viscosity

Rheology Newtonian fluid

Characteristics of the geometry sharp edges, circular cross-sections, axis of the port

is perpendicular to the axis of the header

D2/D1 0.2 ÷ 1.0

L2/D1 0.1 ÷ 2.0

Roughness of pipe walls hydraulically smooth pipes

Upstream flow conditions fully developed pipe flow

Re1 104 ÷ 3×105

Table 1.3: Constants for the calculation of the turning loss coefficient

D2/D1 L2/D1 A1 A2 A3 B1 B2 B3

0.2

0.1 10.083 0.873 0.939 −0.114 0.031 0.031

0.3 3.633 0.828 0.832 −0.045 0.009 0.013

0.625 12.968 0.643 0.737 −0.168 0.006 −0.002

1.25 23.848 0.926 1.116 −0.224 −0.029 −0.041

2 42.031 1.291 1.626 −0.265 −0.051 −0.067

0.3

0.1 5.880 0.860 0.950 −0.087 0.031 0.031

0.3 5.871 0.612 0.691 −0.087 0.036 0.035

0.625 5.377 0.557 0.635 −0.093 0.018 0.015

1.25 6.257 0.759 0.903 −0.125 −0.017 −0.027

2 9.734 0.987 1.247 −0.165 −0.037 −0.053

0.4

0.1 4.204 0.827 0.942 −0.071 0.029 0.030

0.3 7.123 0.374 0.535 −0.111 0.080 0.064

0.625 5.147 0.423 0.553 −0.092 0.042 0.033

1.25 3.292 0.661 0.777 −0.074 −0.010 −0.016

2 3.770 0.885 1.109 −0.095 −0.035 −0.048

0.5

0.1 3.617 0.798 0.964 −0.068 0.030 0.029

0.3 4.435 0.401 0.551 −0.083 0.075 0.065

0.625 5.143 0.271 0.434 −0.099 0.080 0.058

1.25 2.718 0.526 0.635 −0.059 0.003 0

2 2.484 0.785 0.968 −0.064 −0.032 −0.043

0.625

0.1 3.480 0.554 0.793 −0.075 0.063 0.048

0.3 5.677 0.171 0.377 −0.114 0.149 0.101

0.625 6.079 0.094 0.169 −0.124 0.176 0.149

1.25 2.781 0.338 0.501 −0.062 0.033 0.019

2 1.585 0.892 0.932 −0.025 −0.054 −0.046

0.75

0.1 3.513 0.344 0.646 −0.084 0.105 0.068

0.3 4.299 0.114 0.313 −0.099 0.190 0.123

0.625 3.855 0.136 0.353 −0.091 0.142 0.083

1.25 2.652 0.200 0.390 −0.062 0.074 0.042

2 1.651 0.667 0.798 −0.029 −0.039 −0.037

0.875

0.1 2.832 0.322 0.629 −0.071 0.112 0.073

0.3 2.739 0.201 0.437 −0.068 0.151 0.102

0.625 2.352 0.249 0.464 −0.056 0.099 0.067

1.25 2.688 0.162 0.419 −0.067 0.090 0.037

2 2.141 0.259 0.610 −0.053 0.035 −0.015

1

0.1 2.212 0.386 0.774 −0.055 0.099 0.057

0.3 2.863 0.184 0.481 −0.078 0.176 0.109

0.625 2.356 0.167 0.445 −0.064 0.157 0.088

1.25 2.069 0.150 0.379 −0.051 0.110 0.054

2 1.436 0.425 0.871 −0.025 0.007 −0.037

13

Definition of the turning loss coefficient:

22

2v

pC t

f

. (T1.1)

Formula for the calculation of the turning loss coefficient:

2

1

1213

11

1

212

2

1

211

2

1

1213

11

1

213

2

1

2

23

1

21

23

1

3

ReRe

1Re if ,ReRe

ReRe

1Re if ,Re

BB

BBB

BB

BB

f

AA

A

v

vA

v

vA

AA

A

v

vA

v

v

C .(T1.2)

Related publications: [P1–P5].

Statement 2

Considering single phase flows of incompressible fluids, the following properties

make possible the cyclic parametrization of hydraulic loss coefficients of three-way

junctions for the combinations of flow directions shown in Fig. T2.1:

1) One can define the reference velocity that is the root mean square of the

average flow velocities in the conduits:

3

23

22

21 vvv

vrms

, (T2.1)

where v1, v2 and v3 are the signed average flow velocities in conduits (1), (2) and

(3), respectively. Sign of a velocity value is always positive when fluid flows

toward the junction; consequently, signs are negative when fluid leaves the junction.

2) Any combinations of signed average flow velocities in the conduits

normalized to the reference velocity are located on a sphere of radius 3 in the

space of these dimensionless velocities – the center of the sphere is at the

origin:

3 3 222

2

3

2

2

2

1

ZYX

v

v

v

v

v

v

rmsrmsrms

, (T2.2)

where X, Y and Z are the dimensionless signed average flow velocities in conduits

(1), (2) and (3), respectively.

14

3) In the mentioned space, the continuity equation that contains

dimensionless signed average flow velocities is always an equation of a plane

passing through the origin – the plane is determined by the cross-sectional area

ratio:

0 01

3

1

23

1

32

1

21 ZS

SY

S

SX

v

v

S

S

v

v

S

S

v

v

rmsrmsrms

, (T2.3)

where S1, S2 and S3 [m2]are the cross-sectional areas of conduits (1), (2) and (3),

respectively.

4) According to Eqs. (T2.2) and (T2.3), possible normalized velocity

combinations at a given geometry are located on a circle of radius 3 that is

centered at the origin; therefore, every normalized velocity combination can be

characterized by an angle, and a polar coordinate system can be introduced. The

polar angle can be calculated as follows:

0 if ,

3

arccos2

0 if ,

3

arccos

23

21

13

23

21

13

Y

SS

ZSXS

Y

SS

ZSXS

á

, (T2.4)

Fig. T2.1: Possible combinations of flow directions in three-way junctions

Related publication: [P6].

15

Statement 3

The hydraulic loss coefficients of the three-way junction characterized by the

geometrical and operating parameters shown in Fig. T3.1 and Tables T3.1 and

T3.2 can be determined for arbitrary flow directions according to a four-step

method by using Eqs. (T3.1)–(T3.4) and Tables T3.3 and T3.4.

Fig. T3.1: Geometrical and flow properties of the three-way junction – arbitrary

flow directions


ai0, aif, bif constants [-]

Ct1 loss coefficient of the header flow [-]

Ct2 loss coefficient of the branch flow [-]

D1 inner diameter of the header [m]

D2 inner diameter of the branch [m]

f frequency [-]

i index, i = 1 for calculating Ct1, i = 2 for calculating Ct2 [-]

ReJ Reynolds number of the junction (vrmsD1/) [-]

S2/S1 cross-sectional area ratio [-]

vrms reference velocity [m/s]

X dimensionless signed average flow velocity in conduit (1) (v1/vrms)

Y dimensionless signed average flow velocity in conduit (2) (v2/vrms)

Z dimensionless signed average flow velocity in conduit (3) (v3/vrms)

angle between the main conduit (1) and branch tube (2) [°]

polar angle [rad]

Δpt13 pressure difference between a cross-section of conduit (1) and a cross-

section in conduit (3); its value does not contain friction losses [Pa]

Δpt23 pressure difference between a cross-section of conduit (2) and a cross-

section in conduit (3); its value does not contain friction losses [Pa]

kinematic viscosity of the fluid [m2/s]


16





Characteristics of the geometry sharp edge, circular cross-sections, cross-

sectional areas of two of the three conduits are

equal, and the angle between these two conduits

is 180° (according to Fig. T3.1)

45° ÷ 90°

S2/S1 0.1 ÷ 1.0



ReJ > 105

Loss coefficients of the junction are defined as follows:

2rms

131

2v

pC t

t

and (T3.1)

2rms

232

2v

pC t

t

. (T3.2)

The method consists of the following steps:

1) Determination of the reference velocity vrms that is the root mean

square of the average flow velocities in the conduits.

2) Normalization of the velocities to the reference velocity. Sign of a

velocity value is always positive when fluid flows toward the junction;

consequently, signs are negative when fluid leaves the junction.

3) Determination of the polar angle by using the following formula:

0 if ,6

arccos2

0 if ,6

arccos

YZX

YZX

. (T3.3)

4) Calculation of the loss coefficients as follows:

8

1

0 sincos

f

ififiti fbfaaC . (T3.4)

17

Table T3.3: Constants for the calculation of loss coefficient Ct1

α [°] A2/A1 a10 a11 a12 a13 a14 a15 a16 a17 a18 b11 b12 b13 b14 b15 b16 b17 b18

×103

90

0.1 0 153 0 −36.4 0 −2.86 0 −12.2 0 0 101 0 1.91 0 8.00 0 −0.91

0.2 0 224 0 −87.3 0 7.54 0 −24.6 0 0 175 0 10.1 0 14.4 0 −0.94

0.3 0 298 0 −139 0 19.2 0 −40.6 0 0 229 0 31.2 0 13.8 0 0.015

0.4 0 360 0 −184 0 12.1 0 −32.5 0 0 262 0 53.7 0 15.2 0 2.65

0.6 0 471 0 −294 0 −7.18 0 −59.0 0 0 290 0 107 0 21.9 0 0.964

0.8 0 589 0 −386 0 11.9 0 −54.6 0 0 300 0 140 0 47.9 0 1.76

1 0 669 0 −417 0 −0.94 0 −14.0 0 0 271 0 209 0 41.4 0 1.69

60

0.1 −80.3 154 73.8 −40.0 5.97 −0.47 −4.71 −14.6 3.35 −131 104 18.6 −1.59 11.7 9.84 −4.6 −2.11

0.2 −157 226 150 −88.6 −0.84 10.9 −2.40 −35.8 4.23 −254 178 45.8 5.90 10.5 22.5 −3.91 −8.58

0.3 −216 297 197 −143 9.24 30.0 −4.11 −60.7 −0.64 −358 233 64.2 25.1 3.56 27.4 4.72 −9.65

0.4 −260 363 224 −183 30.7 13.9 −7.56 −48.0 0.833 −430 272 55.6 53.8 6.52 17.9 7.54 1.34

0.6 −368 473 340 −290 18.1 −14.0 −13.2 −59.0 25.3 −576 274 91.8 141 −29.0 −7.12 44.6 4.15

0.8 −430 600 367 −388 42.1 −34.7 −34.6 −13.6 20.0 −662 307 89.1 161 −2.88 16.1 61.9 −11.4

1 −447 694 367 −438 41.8 −35.9 −15.8 23.4 11.0 −730 294 88.0 184 19.6 63.8 34.8 −19.0

45

0.1 −108 159 94.0 −45.5 19.1 1.87 −15.3 −19.6 8.35 −176 110 15.2 −5.80 24.4 12.8 −14.4 −5.27

0.2 −248 218 264 −54.1 −51.5 −43.2 46.8 25.6 −11.9 −416 159 115 49.9 −39.2 −36.3 39.1 24.2

0.3 −349 312 366 −167 −50.6 71.9 32.4 −109 −2.34 −598 254 162 −14.5 −46.6 71.4 24.7 −35.3

0.4 −383 385 331 −208 79.4 13.2 −73.0 −7.51 22.5 −646 299 70.3 38.3 55.9 −13.6 −40.2 36.3

0.6 −488 481 436 −259 47.0 −54.5 −4.17 −53.1 12.3 −780 276 86.6 150 −32.4 −26.7 69.3 2.01

0.8 −595 638 546 −390 29.5 −53.7 −18.8 −61.5 −15.8 −935 309 50.6 142 45.6 −23.1 10.2 8.99

1 −609 757 494 −504 100 7.13 −42.1 −47.0 −20.9 −1121 314 140 203 23.8 −39.0 21.7 36.2

Table T3.4: Constants for the calculation of loss coefficient Ct2

α [°] A2/A1 a20 a21 a22 a23 a24 a25 a26 a27 a28 b21 b22 b23 b24 b25 b26 b27 b28

×103

90

0.1 −329 68.3 −1205 −28.7 227 15.9 −178 −65.7 33.0 1931 72.2 −96.6 −22.0 91.3 43.0 −87.3 −20.8

0.2 −365 108 −1038 −32.0 62.9 −17.0 −68.0 −13.3 −7.54 1800 85.1 55.3 14.0 11.4 −9.82 27.8 15.3

0.3 −258 155 −1194 −48.3 186 −38.6 −140 25.6 −1.64 2072 140 −157 −6.83 174 38.7 −17.4 −24.8

0.4 −315 163 −1046 −10.8 78.5 −111 −150 154 23.4 1950 88.4 −27.4 90.1 148 −73.2 −89.5 37.8

0.6 −348 234 −973 −140 10.7 −13.0 −199 −3.69 17.0 1926 134 14.6 65.5 159 −8.05 −143 2.79

0.8 −422 295 −783 −205 −114 19.5 −118 −81.4 −3.48 1800 178 145 29.8 111 71.9 −74.8 1.98

1 −384 340 −806 −209 −67.7 −19.1 −121 −9.99 −2.31 1759 101 192 134 98.8 26.7 5.04 4.01

60

0.1 −415 −249 −1049 138 133 149 −120 −42.1 11.3 1901 519 −110 −102 48.3 7.77 −38.2 −7.16

0.2 −441 −235 −931 157 46.8 132 −76.3 −38.0 13.6 1778 510 −1.05 −97.4 8.94 46.1 4.29 −27.5

0.3 −511 −218 −758 229 −77.7 43.4 −2.31 63.2 −5.65 1773 533 −4.62 −151 35.6 124 −8.47 −85.4

0.4 −453 −209 −878 298 34.1 −68.6 −85.7 170 0.257 1867 485 −129 −20.4 171 −55.9 −131 0.414

0.6 −555 −173 −709 247 −87.8 16.7 −110 −17.6 −8.83 1716 459 8.53 8.96 125 −31.1 −148 −1.45

0.8 −563 −198 −642 371 −142 −113 −23.6 −12.1 −44.5 1693 388 67.1 67.7 52.9 −30.6 −3.09 25.4

1 −627 −116 −544 266 −120 −45.4 −40.4 −35.6 −14.7 1461 420 170 17.4 78.8 −3.29 −6.99 25.4

45

0.1 −464 −361 −966 181 85.6 206 −106 −24.3 22.6 1936 664 −118 −87.2 1.05 6.29 −41.8 −14.3

0.2 −545 −345 −807 235 −17.9 112 −6.92 31.6 8.80 1736 660 5.70 −23.6 −83.8 −21.2 45.5 −17.8

0.3 −645 −344 −577 297 −175 72.0 73.5 57.2 −10.2 1739 712 −8.30 −170 −34.5 187 4.82 −154

0.4 −577 −425 −640 464 −237 −14.0 166 43.7 −80.2 1838 556 −9.05 −2.85 −54.5 87.2 8.77 −129

0.6 −684 −377 −550 384 −199 53.4 −0.63 −0.15 −36.9 1638 621 63.9 −10.7 96.8 24.6 −177 −6.06

0.8 −721 −392 −459 532 −205 −98.2 18.0 12.4 −26.9 1608 556 96.1 45.7 3.04 −5.49 13.1 15.4

1 −742 −312 −423 427 −140 9.17 −8.63 −65.2 14.7 1543 545 78.9 17.5 103 36.3 −52.3 −25.5


18

Statement 4

The hydraulic loss coefficient of the cross-section transition characterized by

the geometrical and operating parameters shown in Fig. T4.1 and Tables T4.1

and T4.2 can be determined for arbitrary flow directions according to a two-step

method by using Eqs. (T4.1)–(T4.3) and Table T4.3.

Fig. T4.1: Geometrical and flow properties of the cross-section transition –

arbitrary flow directions


Ck loss coefficient of the cross-section transition [-]

d1, d2, d3, d4 constants [-]

D1,k inner diameter of conduit (1) [m]

D2,k inner diameter of conduit (2) [m]

S1,k cross-sectional area of conduit (1) [m2]

S2,k cross-sectional area of conduit (2) [m2]

vrms,k reference velocity that is calculated as the root mean square of the

average flow velocities in conduits (1) and (2) [m/s]

v1,k, v2,k signed average flow velocities in conduits (1) and (2); sign of a velocity

value is always positive when fluid flows toward the transition, and the

sign is negative when fluid leaves the transition [m/s]

angle [°]

Δptk total pressure drop in the cross-section transition; its value does not

contain friction losses [Pa]


polar angle [°]

19





Characteristics of the geometry sharp edges, circular cross-sections, S1,k > 0,

S2,k > 0

0° ÷ 90°

−75° ÷ −15°



Reynolds number Re1 ≥ 2×105 for expansions (if v1,k > 0), Re1 ≥ 105

for contractions (if v1,k < 0)

The loss coefficient of the cross-section transition is defined as

2rms,

2k

tkk

v

pC

. (T4.1)

The method consists of the following steps:

1) Determination of the polar angle by using the following formula

(v1,k ≠ 0):

1sgn if ,arctg

1sgn if ,arctg

,1,2

,1

,1,1

,2

kk

k

kk

k

vS

S

vS

S

. (T4.2)

2) Calculation of the loss coefficient as follows:

432

23

1 ddddCk . (T4.3)

Table T4.3: Constants for the calculation of loss coefficient Ck

β [°] d1 d2 d3 d4

7.5 4.754×10-6 7.934×10-4 0.04253 0.740

15 1.315×10-5 2.179×10-3 0.1163 2.018

30 1.707×10-5 2.901×10-3 0.1574 2.764

45 1.447×10-5 2.657×10-3 0.1512 2.744

60 1.070×10-5 2.305×10-3 0.1424 2.718

90 7.541×10-6 1.964×10-3 0.1310 2.603


20

Publications related to thesis statements

[P1] Tomor, A., Kristóf, G. (2017): Hydraulic Loss of Finite Length Dividing

Junctions. Journal of Fluids Engineering – Transactions of the ASME

139 (3) 031104 1–11.

[P2] Tomor, A., Antal-Jakab, E., Kristóf, G. (2017): Experimental Investigation of a

Finite Length Lateral System in a Dividing-Flow Manifold. Proceedings of the

5th International Scientific Conference on Advances in Mechanical Engineering

(ISCAME 2017). Debrecen, Hungary, October 12–13, 2017, 575–580.

[P3] Tomor, A., Kristóf, G. (2016): Validation of a Discrete Model for Flow

Distribution in Dividing-Flow Manifolds: Numerical and Experimental Studies.

Periodica Polytechnica Mechanical Engineering 60 (1) 41–49.

[P4] Tomor, A., Kristóf, G. (2016): Elosztócsövek menti térfogatáram-eloszlások

meghatározása kísérleti, CFD és diszkrét modellek alkalmazásával. OGÉT 2016:

XXIV. Nemzetközi Gépészeti Találkozó = 24th International Conference on

Mechanical Enginering. Déva, Románia, 2016. április 21–24., 435–438.

[P5] Kristóf, G., Tomor, A. (2015): Loss Coefficient of Finite Length Dividing

Junctions. Proceedings of Conference on Modelling Fluid Flow (CMFF’15). The

16th International Conference on Fluid Flow Technologies. Budapest, Hungary,

September 1–4, 2015, 30 1–8.

[P6] Tomor, A., Kristóf, G. (2018): Junction Losses for Arbitrary Flow Directions.

Journal of Fluids Engineering – Transactions of the ASME 140 (4) 041104

1–13.

[P7] Tomor, A., Mervay, B., Kristóf, G. (2017): Continuous Parametrization of

Hydraulic Losses Caused by Diameter Transition in Cylindrical Pipes.

Proceedings of the 5th International Scientific Conference on Advances in

Mechanical Engineering (ISCAME 2017). Debrecen, Hungary, October 12–13,

2017, 581–587.

21

References

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[11] Gergely, D. Z. (2017): T-idomok hidraulikai analízise. Magyar Épületgépészet

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Engineering Applications. AIAA Journal 32 (8) 1598–1605.

[13] Ito, H., Imai, K. (1973): Energy Losses at 90° Pipe Junctions. Journal of the

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