classification of flowsvvr145 vatten the equation 5) engineering answers are not usually required to...

1

14. Grundläggande begrepp, Grundläggande ekvationer I (4.2-4.4)

Stationär och icke-stationär strömning, strömlinjer, strömtuberj ,En-, två- och tre-dimensionell strömningLaminär och turbulent strömningReynolds talKontrollvolymK i i k i

VVR145 VattenVVR145 VattenVVR145 VattenVVR145 Vatten

Kontinuitetsekvationen

Övningstal: I1, I2, I4, I5

CLASSIFICATION OF FLOWS

Flow characterized by two parameters – time and distance.

Division of flows with respect to time:Division of flows with respect to time:Steady (stationär) flow – time independentUnsteady (icke-stationär) flow – time dependentQuasi-steady flow – slow changes with time

Division of flows with respect to distance:

VVR145 VattenVVR145 Vatten

Division of flows with respect to distance:Uniform (likformig) flow – constant section area along flow pathNon-uniform (olikformig) flow – variable section area

2

Examples of types of flow:

Steady uniform flow: flowrate (Q) and section area (A)

Steady non-uniform flow: Q = constant, A = A(x).

are constant


Steady = time independent (stationär) Uniform = constant section (likformig)

Unsteady uniform flow: Q = Q(t), A = constant

Unsteady non-uniform flow: Q = Q(t), A = A(x).


Steady = time independent (stationär) Uniform = constant section (likformig)

3

VISUALIZATION OF FLOW PATTERNSStreamline:

a curve that is drawn in such a way that it is tangential to the velocity vector at any point along the curve. A curve that is representing the direction of flow at a given time No flowrepresenting the direction of flow at a given time. No flow across a stream line.


(Strömlinje)

Streamtube: A set of streamlines arranged to form an imaginarytube. Ex: The internal surface of a pipeline


(Strömningstub)

4

Potential flow: Flow that can be represented by streamlines.


Streakline (Stråklinje) = path made by injected colour in a flow field

Exempel strömlinje och stråklinje

Ett flödesfält är periodiskt på så sätt att strömlinjemönstret upprepas med bestämda intervaller. Under den första sekunden rör sig vätskan uppåt åt höger med 45o vinkel och under den andra

k d ö i ä k då å hö d 45 li Fi )sekunden rör sig vätskan nedåt åt höger med 45o etc enligt Fig. a). Flödeshastigheten är konstant = 10 m/s. Efter 2.5 s ges partikelbanan för en partikel som släppts vid punkten A vid tiden noll av Fig. b). Om färg injiceras kontinuerligt från punkt A från tiden 0 hur ser den resulterande stråklinjen ut efter tiden 2.5 s?


5

TWO WAYS OF DESCRIBING FLUID MOTION

Lagrangian view: the path, density, velocity and other h t i ti f h fl id ti l i fl i t dcharacteristics of each fluid particle in a flow is traced.

Eulerian view: study the flow characteristics (velocity, pressure, density, etc.) and their variation with time at fixed points in space.


LAMINAR AND TURBULENT FLOW

Laminär strömning : Turbulent strömning:

Flow along parallel pathsShear stress proportional to velocity gradient (τ = μ⋅du/dy)Disturbances in the flow are rapidly damped by viscous action

Fluid particles moves in a random manner and not in layersLength scales >> molecular scales in laminar flowRapid continuous mixingInertia forces and viscous forces of importance


importance

6

Reynolds experiment

Small velocities ⇒ line of dye intact, movement in parallel layers ⇒ laminar flow

Q and P variables

layers ⇒ laminar flow

High velocities ⇒ rapid diffusion of dye, mixing ⇒turbulent flow

Critical velocity ⇒ line of dye b i t b k t iti

(Stråklinje)


begin to break-up, transition between laminar and turbulent flow

Reynolds´ number

Reynolds generalized his results by introduction of y g ya dimensionless number (Reynolds number):

Re VD VDρν μ

= = ν= μ/ρ, V=Q/A

e


ν = kinematic viscosityμ = dynamic viscosityD = diameter (for pipes)

(Re= Reynolds tal)

7

Reynolds numbers for pipe flow

Laminar flow: Re < 2000Transitional flow: Re = 2000 to 4000Turbulent flow: Re > 4000Turbulent flow: Re > 4000

Two thresholds:Upper critical velocity – transition of laminar flow to

turbulent flow


turbulent flowLower critical velocity – transition of turbulent flow to

laminar flow

The critical Reynolds number, RcDefining the division between laminar and turbulent flow, is very dependent on the

geometry of the flow

1. Parallel walls: Rc ≅ 1000 (using mean velocity V and spacing D)

2. Wide open channel: Rc ≅ 500 (using mean velocity V and depth D)


p )3. Flow around a sphere: Rc ≅ 1 (using approach velocity

V and sphere diameter D)

8

I1:When 0.0019 m3/s of water flow in a 76 mm pipeline at 20°C, is the flow laminar or turbulent?



9

FLUID SYSTEM AND CONTROL VOLUME

Fluid system: Specified mass of fluid within a closed surface (Fluidsystem: specifierad massa av en fluid inomsurface (Fluidsystem: specifierad massa av en fluid inomen sluten yta) (Lagrange baserad)

Control volume: Fix region in space that can’t be moved or change shape. Its surface is called control surface. (Kontrollvolym: fix region i rummet som inte flyttas


(Kontrollvolym: fix region i rummet som inte flyttaseller ändrar form. Dess gränsyta kallas kontrollyta)

(Euler baserad)

CONTINUITY EQUATION

Steady flowρ1⋅V1⋅A1 = ρ2⋅V2⋅A2 (m1 = m2) m1 m2

Incompressible flowV1⋅A1 = V2⋅A2 or Q1 = Q2(Q = V⋅A)

Control volume

Fluid system volume


V: Average velocity at a section (m/s)A: Cross-section area (m2) Q: Flow rate (m3/s)

10

Continuity equation applied to changing pipe diameter

V1⋅A1 = V2⋅A2 or Q1 = Q2Q = constant, A =(x)

1 1 2 2 Q1 Q2(Q = V⋅A)

Q1 Q2


Q1=V1 ⋅ A1 Q2=V2 ⋅ A2

Control volume

Flow in a pipe junction Channel flow (unsteady)

d(Vol)/dt = Q1– Q2


Q1 + Q2 + Q3 = 0 or V1⋅A1 + V2⋅A2 + V3⋅A3 = 0

Vol: Volume of water in channel between section 1 and 2

11

I2:What is the maximum speed at which a spherical sand grain of diameter 0.254 mm may move through water (20°C) and the flow regime be laminar?


I4:Water flows in a pipeline composed of 75 mm and 150 mm pipe. Calculate the mean velocity in the 75 mm pipe when that in the 150 mm pipe is 2 5 m/s What is its ratio to the mean pipe is 2.5 m/s. What is its ratio to the mean velocity in the 150 mm pipe?


12

I5:Using the Y joint and the control volume in the fig. find the mixture flowrate and density if freshwater (r1= 1000 kg/m3) enters section 1 at 50 l/s, while saltwater (r2 = 1030 kg/m3) enters section 2 at 25 l/s.( 2 g/ ) /


1

15. Grundläggande ekvationer II (4.7-4.8)

Bernoullis ekvation (energiekvationen)Kinetisk energi, potentiell energi och tryck för vätska i rörelseEnerginivå: tryckhöjd, geometrisk höjd, hastighetshöjdBernoullis ekvation, applikationer


Övningstal: I9-I10, I16

BERNOULLI’S EQUATION

Bernoulli’s equation is the energy equation for an ideal fluid (friction and energy losses assumed negligible).

Bernoulli’s equation may, however, be used with satisfactory accuracy in many engineering problems and have the advantage of providing valuable insight about energy conditions in fluid flow


2

BERNOULLI’S EQUATION

2.

2

p Vz H const

gγ+ + = = Quantity Name Measure of

Bernoulli’s equation is a useful relationshipbetween pressure, p, velocity, V, andgeometric height, z, above a referenceplane (datum).

H: energy head (m)z: elevation head above datum (m)

gγH Energinivå Total energi

P/γ Tyckhöjd “tryckenergi”

Z Geometrisk höjd Lägesenergi

V2/(2g) Hastighetshöjd Rörelseenergi


( )V: velocity (m/s) g: gravity acceleration (m/s2)p: pressure (Pa)γ: weight density for the flowing fluid

(N/m3)LineGradeHydraulicLGH

orheadcpiezometrizp

=

=+

..γ

=>Trycknivå

γ = w = ρ g

1 2


3

Validity criterias of Bernoulli’s equation

1) Along a streamline (Kan användas even till större1) Along a streamline (Kan användas even till störretvärssektionsytor om strömlinjerna är raka o parallela

2) For an ideal fluid (ingen inre friktion energiförlusterförsummas); annas kan ej användas

3) Stationär strömning4) Inkompressibelt flöde


Relation mellan P och V (Venturi meter)


När V ökar, P minskar !(“head loss” p g v P gradient, turbulens, eddy, etc)

4

I9:If crude oil flows through this pipeline and its velocity at A is 2.4 m/s, where is the oil level in the open tube C?

ZZ


KINETIC ENERGY CORRECTION FACTOR, α

For a real fluid, friction will cause a non-uniform velocity distribution ⇒the velocity head have to be corrected before use of the Bernoulli equation.

Th l ki ti i bt i d b i t ti th ti d iThe real kinetic energy is obtained by integration over the section area and isthen expressed in terms of the mean velocity, V, and a correction coefficient, α.The corrected velocity head becomes

g

V

2

2α Eqn, 4.26: α = Σ(v3dA)/V3A


Some values of α (table 4.2 text book):α = 2 (laminar pipe flow)α ≈ 1.06 (turbulent pipe flow)α ≈ 1.05 (turbulent flow in wide channel)

5

WHY THE ENERGY CORRECTION COEFFICIENT α (MOMENTUM COEFFICIENT β) OFTEN MAY BE OMITTED

1) Most engineering pipe flow problems involve turbulent flow in ) g g p p pwhich α is only slightly more than unity.

2) In laminar flow where α is large, velocity heads are usually negligible when compared to the other Bernoulli terms

3) The velocity heads in most pipe flows are usually so small compared to the other terms that inclusion of α has little effect

4) The effect of α tends to cancel since it appears on both sides of


the equation5) Engineering answers are not usually required to an accuracy

which would justify the inclusion of α in the equation.

I10*:Water is flowing The

11

Water is flowing. The flow picture is axisymmetric. Calculate the flowrate and manometer reading. 3322 ZZ


5544

6

I16* Channel and gate are 1 m wide (normal to the plane of the paper). Calculate q1, q2, and Q3.

11

22ZZ


33

L14L15

classification of flowsvvr145 vatten the equation 5) engineering answers are not usually required to...

Documents