Ułanowicz Leszek: Properties of homogeneous flow of hydraulic fluid retaining gap. (Właściwości przepływu jednorodnej cieczy roboczej przez hydrauliczną szczelinę oporową.)

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PROPERTIES OF HOMOGENEOUS FLOW OF

HYDRAULIC FLUID RETAINING GAP

WŁAŚCIWOŚCI PRZEPŁYWU JEDNORODNEJ CIECZY

ROBOCZEJ PRZEZ HYDRAULICZNĄ SZCZELINĘ

OPOROWĄ

Leszek Ułanowicz

Air Force Institute of Technology

leszek.ulanowicz@itwl.pl

Abstract: This paper presents the laminar flow of homogeneous liquids in crevices of

smooth hydraulic resistance. The paper presents a theoretical model for the distribution of pressure in the gap and the flow rate through the slot hydraulic. The presented theoretical models for the distribution of pressure and flow in the gap on conventional hydraulic resistance of hydraulic joints, whose shape is related to the errors of their execution. In deriving the theoretical models, by introducing a variable height of the gap in the initial episode, was founded stream velocity profile variability in the hydraulic fluid retaining gap and zero values of local losses at the entrance to the slot. An analysis of the validity of the simplifications adopted in the given formulas for the distribution of pressure and flow in the hydraulic gap and on the basis provides guidelines for estimating the energy losses that occur cracks in the hydraulic resistance.

Keywords: hydraulic resistance gap, the Reynolds number, fluid pressure, fluid

flow, the gap concentric, eccentric gap

Streszczenie: W artykule przedstawiono zagadnienia przepływu laminarnych cieczy

jednorodnych w gładkich hydraulicznych szczelinach oporowych. Przedstawiono wzory teoretyczne na rozkład ciśnień w obszarze szczeliny oraz natężenia przepływu przez szczelinę hydrauliczną. Przedstawione wzory teoretyczne na rozkład ciśnień i natężenia przepływu w szczelinie hydraulicznej dotyczą typowych hydraulicznych szczelin oporowych, których kształt związany jest z błędami ich wykonania. Przy wyprowadzaniu wzorów teoretycznych, przez wprowadzenie zmiennej wysokości szczeliny we wstępnym jej odcinku, założono zmienność profilu prędkości strugi cieczy w hydraulicznej szczelinie oporowej oraz zerowe wartości strat lokalnych na wejściu do szczeliny. Dokonano analizy zasadności przyjętych uproszczeń w podanych wzorach na rozkład ciśnień i natężenia przepływu w szczelinie hydraulicznej i na jej podstawie przedstawiono wytyczne dotyczące szacowania strat energetycznych jakie mają miejsce w hydraulicznych szczelinach oporowych.

Słowa kluczowe: hydrauliczna szczelina oporowa, liczba Reynoldsa, ciśnienie

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1. Introduction

Basic equations describing the motion of Newtonian fluid the hydraulic retaining slot due to three main principles of mechanics [1, 4]: the principle of conservation of mass (continuity equation), the principle of conservation of momentum and angular momentum and the principle of conservation of energy. These equations are not closed system. Therefore, they should be supplemented by additional equations expressing the state of density, viscosity and thermal conductivity as a function of pressure and temperature, as well as on the field of mass forces unit. The general solution of these equations, it is not known, and the designation of their solutions which are functions of four independent variables and subject to specified initial conditions and boundary encounters great difficulties. The data contained in the available literature, the author does not allow for sufficiently accurate calculation of flow parameters of a homogeneous fluid through hydraulic bearing gap. In matters of homogeneous fluid flow through the gap ignores the influence of hydraulic fluid compressibility, the existence of external mass forces associated mainly with the Earth's gravity field, and the impact of changes in viscosity and density of the fluid temperature (the temperature constancy of adoption). For this reason, all derived from theoretical analysis of the results must be verified practical.

2. Pressure distribution and flow rate of liquids working gaps in typical

hydraulic resistance

The flow of hydraulic fluid through a slot bearing the author has treated as a

stationary flow between two flat and parallel plates located from each other

at such a distance that they form a hydraulic bearing gap (see fig.1). As is

known, the length of the initial section of the hydraulic retaining slots in

which there is a stabilization of the flow (velocity distributions in any

cross-sections are identical) is proportional to the Reynolds number and height of

the hydraulic retaining slots. Due to the fact that the initial gap distance of

the liquid velocity profile in the gap is not formed by the reflection on the

speed of the liquid in the initial section of the slot gap height change

introduced a specific symbol



(z) according to fig. 2

Taking into account the change of the flow channel (the hydraulic retaining slots) dependence on the flow of hydraulic fluid in the retaining slot has the form:















_







1 1 0 2 2 3 3 2 2 1 2 2 1 0 2 1 2 1

72

5

4

8

5

l s s h

h

l

h

l

m

p

dxdy

x

l

y

h

l

h

m

p

Q



, (1)

and dependence on the pressure distribution in the gap has the form:

 



 

_{   }

 





   



dx G F F F x F z z z x x dz i E z x p i              

_



2 1 2 1 1 2 1 2 2 2 1 2 1 2 , _         ₍₂₎

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Fig. 1 Schematic hydraulic flat retaining-wall cracks parallel.

Fig. 2 Schematic of the flow channel (the hydraulic retaining slots).

Integral Constants E and G are determined using the continuity of velocity boundary conditions on the stationary walls of the form:

u (1, z) = u (2, z) = 0

(3) w (1, z) = w (2, z) = 0

and pressure p (z) satisfying the conditions:

p (x, 0) = p0 (4) p(x, m) = p1 ls l1 l1

y

z

x

m h h1

(4)

Error of the approximations, with



h ls

, is 16% [1].

Entering into formulas (1) and (2) boundary conditions (3) and (4) was determined depending on the pressure distribution in the gap and the fluid flow through the gap for typical hydraulic retaining slots. Shapes of typical hydraulic retaining slots are associated with the errors of their execution.

For hydraulic flat-wall retaining slots parallel schematically shown in fig. 1 the distribution of pressure in the gap model is described:

 

x,z p0

p  , (5)

and fluid flow in the gap model:

m

p

h

l

Q

s



12

3





. (6)

For hydraulic flat retaining slots of non-parallel walls schematically shown in fig. 3 the distribution of pressure in the gap model is described:

 

p m z h h H m z m z h h H h h H p z x p                                             ₂ 2 2 0 1 2 2 1 , , (7)

                                  3 2 3 4 1 2 3 1 12 h h H h h H h h H m p h l Q s  . (8)

For these patterns can be introduced dimensionless coefficient related to the geometry of the slot (ls, H, h, m) of the form:

h h H

k₁  .

For hydraulic flat retaining slots of non-parallel walls schematically shown in Figure 4 the distribution of pressure in the gap is described by the formula:

 

p m x k k m x m z m z m x k m x k p z x p                            ₂ 2 2 2 2 2 0 1 2 2 2 2 1 , , (9)





          2 2 2 3 2 1 6 k k m p h l Q s



, (10) where: h H h k₂   .

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Fig. 3 Schematic hydraulic flat retaining slots of non-parallel walls.

Fig. 4 Schematic hydraulic flat retaining slots of non-parallel walls

For hydraulic flat retaining slots of non-parallel walls schematically shown in fig. 5 the pressure distribution in the gap model is described:

 

p m z k m x k m z k m x m z k m z k m x k k m x k p z x p                                                              _          ₂ 2 1 2 1 2 1 2 2 1 0 1 2 2 2 2 1 , , (11) ls

z

x

m h H ls

z

x

m h H

(6)

















_                  2 2 1 2 2 2 2 1 1 2 1 2 1 2 1 1 2 2 3 2 16 4 14 2 16 36 ) 2 ( 16 6 4 4 1 16 6 k k k k k k k k k k k k k k m p h l Q s  , where:

h

k

' 1





,

h

H

k

₂





.

Fig. 5 Schematic hydraulic flat retaining slots of non-parallel walls

For the longitudinal annular retaining slot hydraulic schematic shown in fig. 6 the distribution of pressure in the gap model is described:

 

r p0

p  , (12)





m

p

r

Q





6

3 1 2 1







. (13)

Fig. 6 Schematic longitudinal hydraulic annular retaining slot

m H ls

z

x

h h’

(7)

For the transverse annular hydraulic retaining slots (front slit formed by two parallel plates round) schematically shown in fig. 7 the pressure distribution in the gap model is described:

 

_          ₂ 1 2 2 2 2 2 1 3 0 1 1 80 3 ln 4 3 r r h Q r r h Q p r p    _, ₍₁₄₎

 



          1 2 0 1 3 1 1 1 1 1 6 r n n n r m n m p h r Q   _. ₍₁₅₎

F

or the longitudinal eccentric annular retaining slot hydraulic schematic shown in fig. 8 the distribution of pressure in the gap model is described:

 

r p0

p  , (16)





m p r r r Q





6 3 1 2 1    . (17)

Fig. 7 Leading hydraulic resistance gap formed by two parallel circular plates.

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For the longitudinal annular retaining slot hydraulic schematic shown in fig. 9 the distribution of pressure in the gap model is described:

 

_



_

_



_





p m z k m z k m z k k p z p        _ _                     ₂ 2 2 0 cos 1 cos 1 2 cos 1 2 cos 1 ,      , (18)









_                      4 2 125 , 0 75 , 0 25 , 0 5 , 0 3 5 , 0 1 12 2 2 2 2 3 0 0 0 k k k k k k m p d D d Q   , (19) where: 0 0 0 1 0 0 0 1

d

D

d

D

k





.

In these relationships highlights the strong influence that the geometry of the hydraulic parameters of the slot retaining the fluid flow through the gap. In the derivation of these formulas the author assumed a zero value of the loss of local input into the slot. Although these models are designated with the linearized system of equations, using them, small ranges of variation in the flow cross the gap and small Reynolds numbers is warranted.

3. Estimation of energy loss the gap retaining fluid

The length of the initial section of the hydraulic retaining slots in which there is a stabilization of the flow is proportional to the Reynolds number

and the amount of hydraulic retaining slots. At high values of the ratio of length to the height of the gap flow and low Reynolds numbers involved is insignificant pre-cutter and its omission does not lead to significant computational errors.

For short cracks, no stationary faster speed profile causes a pressure drop along the flow may be one reason for the growing discrepancy between the results of experiments and the results obtained from theoretical models. These differences are the greater, the smaller the ratio of length to the height of the gap flow and the higher the value of the Reynolds number. The pressure drop in each slot is the sum of inlet pressure and pressure drop in the length of the slot. For slots, which is an increase in cross-section along the flow pressure is reduced diffuser slot. The pressure drop along the hydraulic flow through the resistive gap is due to local resistance (loss of local) and stands out as so called in the literature. local resistance coefficient ξ. Experimental or theoretical results reported in the literature concerning the loss of pressure in the cracks of various types are not fully compatible with each other, and sometimes divergent. This is due, inter alia, major difficulties with the exact performance measurements within the desired range of parameters. In many cases, lack of practical guidance on the applicability of the given experimental plots.

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Fig. 9 Hydraulic longitudinal annular gap resistance

This is understandable because of the dependence of losses both on the nature of the inlet flow, the actual geometry of the slot, as well as an area in front of her. In the literature [2, 6] is given the formula by which one can calculate the coefficient of local resistance: 2 2 2

2

2 Q

pf

g

v

p

g



m











,

where: v is the average speed of flow of hydraulic fluid surface af retaining slots. In this formula the size of ξ is constant and does not depend on the Reynolds number. In fact, the value of ξ also depends on the Reynolds number. According to the author dependency ratio ξ for the start of local longitudinal annular gap (see Figure 6) as a function of Reynolds number has the form shown in Figure 10, while for the transverse annular gap (see fig. 7) in the range of Reynolds numbers

500 <Re <3000 is in the form shown in fig. 11.

The analysis conducted by the author shows that for the consideration of hydraulic retaining slots:

a) at a flow of liquid when the Reynolds number Re <400 losses may be neglected local ξ,

b) at a flow of liquid at Reynolds numbers 400 <Re <ξ Rekr value strongly depends

on Re,

c) Rekr around ξ value is growing rapidly, which changes the flow,

d) the turbulent flow is constant ξ value and contains the value of 1 ÷ 1.8. D0 d0 D1 d1

r

z

θ

(10)

During the movement of viscous fluid in a hydraulic retaining slot is further dissipation of energy. Referred to as the friction loss factor λ. Dependency ratio λ of the Reynolds number in laminar flow is a hyperbolic, i.e.

Re

A





[2]. For the longitudinal concentric annular gap (see fig. 6) in the range of Reynolds number Re = 100 ÷ 1200 the coefficient λ is derived from experience

Re 100





and set theory

Re 96





. Dependency ratio λ as a function of Reynolds number for a concentric annular gap of the longitudinal Reynolds number Re = 100 ÷ 1200 has the form shown in fig. 12.

Fig. 10 Dependency ratio of local start ξ as a function of Reynolds number for the

longitudinal annular gap

Fig. 11 Dependency ratio of local start ξ as a function of Reynolds number

for 500 <Re <3000 for cross-annular gap

ξ

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Fig. 12 The dependence of friction loss factor λ as a function of Reynolds number for a concentric annular gap longitudinal

For niecentrycznej (eccentric) longitudinal annular gap (see fig. 8) in the range of Reynolds number Re = 100 ÷ 1200 derived factor λ

experience is

Re

57 



and set theory

Re

4 ,

38 



. Dependency ratio λ as a function of Reynolds number for no centric (eccentric) longitudinal annular gap is in the form shown in fig. 13.

Fig. 13 The dependence of friction loss factor λ as a function of Reynolds number for annular gap eccentric longitudinal

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In [5] the author presented the results of experimental studies to determine the formula for the coefficient λ for the transverse slot. The article stated that for the entire laminar flow Re <2000 model gives accurate results

Re 80





. Smaller in this case the value of friction losses can be explained without the influence of the inertia member in the theoretical considerations in deriving the formula

Re 96





. In [3] the author presents a model for friction loss factor λ for the transverse slot of the form

Re 102





.

It should be noted that the practical verification of the friction loss factor λ is assumed constant for the entire length of the hydraulic retaining slots, also within the initial segment.

Based on the results of experiments conducted by the concentric annular gap for the longitudinal course of specifying the nature of the actual pressure distribution in the slot. The nature of the course of the actual pressure distribution in the longitudinal concentric annular gap is presented in figure 14 The solid line b (λ) in fig. 14 is a real pressure distribution, while the dotted line and (λ *) is the theoretical distribution (computing). When the length of the flow of hydraulic thrust m gap tends to zero flow rate does not increase indefinitely, but approaches a constant value.

Based on the results of experiments conducted for the author and not short of concentric ring-centric longitudinal slits hydraulic resistance in laminar flow determined according to the relative expense Q

the relative length of the gap

m

. Under the concept of relative flow Q

we mean the ratio of the observed

flow rate Q to the theoretical value of the flow

QT calculated for long cracks,

i.e.

T

Q



. Under the concept of relative length of the gap

m

we mean the ratio

of flow length to the height of the slot, i.e.

h m

m .

As a result of mathematical data processing of the results of experiments formulated an empirical expression for the concentric annular hydraulic longitudinal slot retaining the laminar flow of the following form:

m Q1,127,2

And for no centric (eccentric) annular hydraulic longitudinal slot of a retaining:

75 , 0

042 ,

0 m

Q



(13)

The dependence of the relative expense Q the relative length of the gap

m

the laminar flow for concentric and no centric (eccentric) annular hydraulic longitudinal slot retaining shown in fig. 15.

Fig. 14 The actual pressure distribution in the longitudinal concentric annular gap

4. Summary

In the presented formulas for calculating pressure distribution and fluid flow in a typical hydraulic retaining slots do not take into account the impact of the initial segment, which forms a laminar velocity profile and the local losses are included only in the specified empirically pressure at the inlet.

0 0,2 0,4 0,6 0,8 1 1,2 1,4 0 20 40 60 80 100 120

Względna długość szczeliny

W zg lę d n y w y d a te k Koncentryczna szczelina pierścieniowa wzdlużna Mimośrodowa szczelina pierścieniowa wzdłużna

Fig. 15 The dependence of the relative expense Q the relative length of the gap

m

concentric and eccentric to the longitudinal hydraulic annular retaining slot

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At high values of the ratio of length to the height of the gap flow and low Reynolds numbers involved is insignificant pre-cutter and its omission does not lead to significant computational errors.

In the case of hydraulic resistance of the small joints of the flow parameters of the flow length is determined by a complicated analysis of the velocity field and pressure in the environment and the same slot. The values of liquid flow in the interstices of a low-resistance flow length calculated from formulas given in this paper only determine the upper limit.

5. Literature

[1] Gryboś R.: Podstawy mechaniki płynów. Część I. Wydawnictwo Naukowe PWN. Warszawa 1998.

[2] Guillon M.: Etude et determination des systemem hydrauliques. Dunod. Paris 1966.

[3] Jędral W.: Badania przepływu przez szczelinę tarczy jako elementu optymalizacji energetycznej układu obciążającego w pompie wirowej. Praca doktorska, Politechnika Warszawska, Warszawa 1977.

[4] Puzyrewski R., Sawicki J.: Podstawy mechaniki płynów i hydrauliki. Wydawnictwo Naukowe PWN. Warszawa 1998.

[5] Wagner W.: Experimentelle Untersuchungen an radial durchstromten Spaltdichtungen. VDIćBerichte, No 193, 1973.