Chapter 9- Flow in Pipes & Annuli

1

Chapter 9 – Flow in Pipes & Annuli

306

•Discuss fluid flow:-both Newtonian & non-Newtonian,-in pipes & concentric annuli,-for laminar & turbulent regimes

•Assumption: flow is steady (time-independent)•Calculate frictional pressure drop gradient as a function of flow rate•Formulas are valid for horizontal conduits, but can be used for inclined & vertical conduit•Sometimes it is advisable to approximate annulus (with Di/Do>=30%) by an slot. In “open” slot, only upper & lower surfaces are subjected to shear stresses

2

307

Flow in Pipes & Annuli

General fluid model (shear stress as a function of shear rate)

τ = f (γ̇)

½τ = K (γ̇)n γ̇ ≥ 0τ = −K (−γ̇)n γ̇ < 0

τ = μ γ̇

⎧⎨⎩ τ = τy + μp γ̇ γ̇ > 0τ = −τy + μp γ̇ γ̇ < 0γ̇ = 0 −τy ≤ τ ≤ τy

Newtonian

Ostwald

Bingham plastic

308


Field (equilibrium) Equations: Independently of rheological model,this shows the relationship between the forces acting on fluid flowassociated directly with the geometry of conduit

Annulus

Pipe

Slot

τ =1

2

dpf

dLr

τ =1

2

dpf

dLr +

c

r

τ =dpf

dLy

: Shear stress at any radial distance r (or distance y for slot) from conduit (slot) axis c: constant to be determined

3

309


Shear rate defined

shear stress:

shear rate: γ̇ =dγ

dt=d

dt

dx

dy=d

dy

dx

dt=dv

dy

τ = f (γ̇)

310


Shear rate (velocity profile) for pipes, annuli, and slots

γ̇ = −dvdy

γ̇ = −dvdr

axis

4

311


Flow rate and continuity

q =

ZA

v dA

v̄ =q

A

dq = v dA

Ave. velocity profile

•Ave. velocity is representative if just want to calculate kinetic energy of fluid

•Real fluid velocity varies with position across section

312


Flow rate and continuity

Pipe q = 2π

RZ0

v r dr

Annulus

Slot

q = 2π

RoZRi

v r dr

q = w

t/2Z−t/2

vdy

q = −πRZ0

r2µdv

dr

¶dr

q = −πRoZRi

r2µdv

dr

¶dr

q = −wt/2Z−t/2

y

µdv

dy

¶dy

5

313

Pipes (Poiseuille's Eq.) Annuli (Lamb's Eq.) Slot approximation (for annuli)

Newtonian

Pipes Annuli

Bingham Plastic Model

Pipes Annuli

Power-law model

Non-Newtonian

Fluid Flow- Laminar


Pipes (Fanning Eq.)

Hydraulic Radius,Pipe analogy &Slot annuli to pipe analogymodels

Annuli Slot approximation (for annuli)

Newtonian

Pipes Annuli

Bingham Plastic Model

Pipes Annuli

Power-law model

Non-Newtonian

Fluid Flow-Turbulent

314

Flow of Newtonian fluids in pipes (laminar)Poiseuille’s Eq.

τ = μ γ̇

Purpose: find the relationship between pressure drop (gradient) and flow rate

γ̇ = −dvdr

τ =1

2

dpf

dLr

q = −πRZ0

r2µdv

dr

¶dr

constitutive equation (Newtonian fluid) field equation

continuity

definition

dpf

dL=8μv̄

R2=32μv̄

D2

dpf

dL=

μv̄

1500 D2

Combining constitutive & field equations produces a flow model

Filed units:

dpf /dL (psi), (cP), ft/s), D (in)

6

315

Flow of Newtonian fluids in annuli (laminar)Lamb’s Eq.


γ̇ = −dvdr

constitutive equation field equation

velocity

definition

τ =1

2

dpf

dLr +

c

rτ = μ γ̇

v(r)Z0

dv = −rZ

Ri

µ1

2μ

dpf

dLr +

C

r

¶dr

v(r) = − 1

4 μ

dpf

dL

¡r2 −R2i

¢− C lnr

Ri

v(Ro) = 0 → C = − 1

4μ

dpf

dL

R2o − R2iln RoRi

316

Flow of Newtonian fluids in annuli (laminar)Lamb’s Eq.

continuity

C = − 1

4μ

dpf

dL

R2o − R2iln Ro

Ri

q = −πRoZRi

r2µdv

dr

¶dr

dv

dr= −

µ1

2μ

dpf

dLr +

C

r

¶

dpf

dL=

8μv̄

R2o +R2i − R2

o−R2i

ln RoRi

=32μv̄

D2o +D2i − D2

o−D2i

ln DoDi

dpf

dL=

μv̄

1500

µD2o +D

2i − D2

o−D2i

lnDoDi

¶Field units:

7

317

Flow of Newtonian fluids – example

318


8

319

Flow of Newtonian fluids – slot approx.

τ = μ γ̇


constitutive equation field equationdefinition

τ =dpf

dLyγ̇ = −dv

dy

t: Thickness of slot is equal to radial clearance of annulus

t=(Ro-Ri)

=(Ro+Ri)

Thickness & width of approximating slot

320

Flow of Newtonian fluids – slot approx.

τ = μ γ̇

constitutive equation field equation

continuity

definition

τ =dpf

dLyγ̇ = −dv

dy

q = −wt/2Z−t/2

y

µdv

dy

¶dy

dpf

dL=

12μv̄

R2o −R2i=

48μv̄

(Do −Di)2

dpf

dL=

μv̄

1000 (Do −Di)2Field units:

9

321


322

Flow of non-Newtonian fluids (laminar)

Pressure Drop Gradient for Bingham-Plastic Fluids in Pipes

Pressure Drop Gradient for Bingham-Plastic Fluids in Annuli (Slot Approximation)

dpf

dL=

μp v̄

1500 D2+

τy

225 D

dpf

dL=

μp v̄

1000 (Do −Di)2+

τy

200 (Do −Di)

Filed units:

Filed units:

y (lbf/100 ft2)

10

323

Flow of non-Newtonian fluids (laminar)

Pressure Drop Gradient for Power-Law Fluids in Pipes

dpf

dL=

K v̄n

143, 640 Dn+1

µ243n+ 1

n

¶nPressure Drop Gradient for Power-Law Fluids in Annuli (Slot Approximation)

dpf

dL=

K v̄n

143, 640 (Do −Di)n+1µ482n+ 1

n

¶n

Field units:

Field units:

K (in equivalent cP)

324

Flow of non-Newtonian fluids – example

11

325

Turbulent Flow in Pipes & Annuli

Laminar flow• particles move in orderly

layers (“laminae”), the velocities may change between layers

• continuous velocity profile

• no eddies or streamline crosses

Turbulent flow• particles follow irregular

paths, with large change in velocity & flow direction compared with other near particles

• although steady state in average, local velocity may change

• eddies and vortices may occur

326


• Flow regimes:– laminar

– Transition (flow is turbulent in central portion of flow section, but may remain laminar close to boundaries)

– turbulent

• Factors that affect the flow regime:– fluid velocity

– fluid density

– conduit size, shape, roughness, etc.

12

327


328


• boundaries between laminar and turbulent flow

• Reynolds (1883) extensive tests with water in glass pipes

• boundaries related to a dimensionless group now called Reynolds Number (Re)

Re =ρ v̄ D

μRe = 928

ρ v̄ D

μ

in general the flow will be laminar if Re < 2000 and turbulent if Re>3000,& transitional if in between

pipe diameter, D (cm),

ave. velocity (cm/s),

density, (g/cm3),

dynamic viscosity, (P)

pipe diameter, D (in),

ave. velocity (ft/s),

density, (lbm/gal),

dynamic viscosity, (cP)

13

329

Turbulent Flow – example

330

Turbulent Flow in Pipes (Newtonian Fluids)

• Several (thousands!) of experimental results for Newtonian fluids in pipes

• Colebrook, Moody, Fanning, Blasius, etc

• Friction factor: dimensionless group relating shear stress in flow to specific kinetic energy:

Purpose: determine pressure drop gradient in pipes for turbulent flow (Newtonian fluids)

f =τ

ek

14

331


f =τ

ek

τ =1

4

dpf

dLD

ek =Ek

V=1

2ρ v̄2

dpf

dL=2 ρ v̄2

Df

dpf

dL=

ρ v̄2

25.8 Df

Fanning equation

2

4DAp

Pipe cross

section area

LDAc Lateral area of fluid element

Field units:

332


if friction factor can be determined, pressure drop gradient can be calculated

for laminar flow, we compare this expression with Poiseuile’s equation

dpf

dL=2 ρ v̄2

Df

dpf

dL=32μv̄

D2

f =16 μ

ρ v̄ D=16

Re

Valid for laminar flow! Determined experimentally for turbulent flow

15

333


if friction factor is related to Reynolds number for laminar flow, it is reasonable to suggest that they should also be related for turbulent flow

f =16

Re

dpf

dL=2 ρ v̄2

Df

Fanning equation friction factor for laminar flow

334


friction factor can be read directly from the chart above (Stanton chart)

/d: Pipe relative roughness (usually < 0.0004),

absolute roughness

16

335


Colebrook obtained an implicit formulation to relate friction factor with Reynolds number and the relative roughness of the surface

1√f= −4 log10

µ²/D

3.7+1.255

Re

1√f

¶

1√f= −4 log10

µ1.255

Re

1√f

¶(hydraulically smooth)

336

TurbulentFlow in Pipe

–example

17

337

Turbulent Flow in Pipes (Newtonian Fluids)Alternative expressions for the friction factor

Blasius (2,100 < Re < 100,000 – hydraulically smooth)- covers most drilling conditions

f =0.0791

Re0.25

Using this expression in Fanning Equation:

Field units:

338


Swamee–Jaine (5,000 < Re < 108,000 – 10-6 < /D < 10-2)

f =1h

4 log10

³²/D3.7

+ 5.74Re0.9

´i2

18

339

Turbulent Flow in Pipes – example

340


19

341


342

Turbulent Flow in Pipes – criterion

•Calculate friction factor using laminar relation & one of turbulent factors (Colebrook, …),

•Use the larger to obtain frictional pres. drop gradient using Fanning formula

20

343

Turbulent Flow in Annuli (Newtonian Fluids)Purpose: determine pressure drop gradient in annuli for turbulent flow (Newtonian fluids)

Several empirical procedures exist to apply results for pipes to other geometries. This needs to determine an equivalent pipe diameter Deq, of flow cross section geometry. Then use this to determine the Reynolds number and to be used in the Fanning equation. Three ofthese criteria are:

• Hydraulic radius

•Exact annulus to pipe analogy

•Slot annulus to pipe analogy

344

Turbulent Flow in Annuli – hydraulic radius

By definition, the hydraulic radius of a conduit is the ratio of the area to the wet perimeter

circular pipe: (rH)pipe =π R2

2 π R=R

2=D

4

annulus: (rH)annulus =π¡R2o −R2i

¢2 π (Ro +Ri)

=Ro −Ri

2=Do −Di

4

assumption: cross sections with the same hydraulic radius are equivalent for the calculation of the Reynolds number and to be used in the Fanning equation

equivalent diameter of an annulus: Deq = Do −Di

21

345

Turbulent Flow in Annuli – exact analogy

circular pipe:

annulus:

assumption: equivalent cross sections give the same pressure drop gradient for the same average velocity

equivalent diameter of an annulus:

Equivalence is obtained for annuli comparing the exact expressions for pressure drop in pipes and in annuli (for laminar flow) & equating them

dpf

dL=32μv̄

D2

dpf

dL=

32μv̄

D2o +D2i − D2

o−D2i

lnDoDi

Deq =

sD2o +D

2i −

D2o −D2

i

ln Do

Di

346

Turbulent Flow in Annuli – slot analogy

circular pipe:

annulus:

assumption: equivalent cross sections give the same pressure drop gradient for the same average velocity

equivalent diameter of an annulus:

Equivalence is obtained for annuli comparing the expressions forpressure drop in pipes and in annuli (slot approx.) (for laminar flow)

dpf

dL=32μv̄

D2

dpf

dL=

48μv̄

(Do −Di)2

Deq =

r2

3(Do −Di)

22

347

Turbulent Flow in Annuli (Newtonian Fluids)

These criteria are used to calculate an equivalent diameter to determine the Reynolds number and to be used in the Fanning equation, which were derived for pipes only

using hydraulic radius: Deq = Do −Di

using exact formulas:

using slot approximation:

Deq =

sD2o +D

2i −

D2o −D2

i

ln Do

Di

Deq =

r2

3(Do −Di)

348

Turbulent Flow in Annuli – example

23

349

Turbulent Flow in Annuli – example

350

Turbulent Flow for non-Newtonian fluidsHow to determine the Reynolds number and the criterion for turbulence?

Pressure Drop Gradient for Bingham-plastic Fluids in Pipes

Reynolds number depends on the dynamic viscosity, which is not defined for non-Newtonian fluids

parameters are the plastic viscosity p and yield stress y

an “apparent” viscosity (a), which causes the same pres drop for laminar flow can be obtained comparing the expressions for pressure drop gradient in pipes (Newtonian & Bingham)

dpf

dL=

μv̄

1500 D2

dpf

dL=

μp v̄

1500 D2+

τy

225 D

μa = μp + 6.66τy D

v̄a is used to determine Reynolds number, friction factor & pressure drop gradient (Fanning eq.)

24

351

Turbulent Flow for non-Newtonian fluids

Pressure Drop Gradient for Bingham-plastic Fluids in Annuli

An “apparent” viscosity can be obtained comparing the expressions for pressure drop gradient in annuli. Only slot approximation formulation should be used

a and Deq are used to determine Reynolds number, friction factor and pressure drop gradient

dpf

dL=

μp v̄

1000 (Do −Di)2+

τy

200 (Do −Di)dpf

dL=

μv̄

1000 (Do −Di)2

μa = μp + 5τy (Do −Di)

v̄

Deq =

r2

3(Do −Di) (equivalent diameter – slot approx.)

352

non-Newtonian fluids – example

25

353


354

Turbulent Flow for non-Newtonian fluidsPressure Drop Gradient for Power-law Fluids in Pipes

Parameters are the consistency index K and behavior index n

An “apparent” viscosity can be obtained comparing the expressions for pressure drop gradient in pipes

a is used to determine Reynolds number, however the following formula is needed to determine the friction factor (Colebrook & … don’t provide accurate calculation)

dpf

dL=

μv̄

1500 D2

dpf

dL=

K v̄n

143, 640 Dn+1

µ243n+ 1

n

¶nμa =

K v̄n−1

95.9 Dn−1

µ243n+ 1

n

¶n

Dodge & Metznerformula

1√f=

4

n0.75log10

¡Re f1−

n2¢− 0.395

n1.2-

-

-

26

355

Turbulent Flow for non-Newtonian fluidsPressure Drop Gradient for Power-law Fluids in Annuli

an “apparent” viscosity can be obtained comparing the expressions for pressure drop gradient in annuli

dpf

dL=

K v̄n

143, 640 (Do −Di)n+1µ482n+ 1

n

¶ndpf

dL=

μv̄

1500 D2

a and Deq are used to determine Reynolds number, friction factor gradient using the Dodge-Metzner formula, and pressure drop using the Fanning equation

Deq =

r2

3(Do −Di) (equivalent diameter – slot approx.)

μa =K v̄n−1

143.9 (Do −Di)n−1µ482n + 1

n

¶n

356


27

357


358

Chapter 9- Flow in Pipes & Annuli

Documents

Transcript of Chapter 9- Flow in Pipes & Annuli