Internal Incompressible Viscous Flow

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Internal Incompressible Viscous Flow

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Contents

Part A Fully Developed Laminar Flow

Part B Flow in Pipes and Ducts

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Laminar versus Turbulent Flow

𝑅𝑒 = 𝜌 ത𝑉 𝐷 𝑅𝑒_{𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙} ≅ 2300𝜇

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Laminar versus Turbulent Flow

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Laminar versus Turbulent Flow

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Flow Regions in a Pipe

The Entrance Region

𝐼𝑛𝑣𝑖𝑠𝑐𝑖𝑑 𝐶𝑒𝑛𝑡𝑟𝑎𝑙 𝑅𝑒𝑔𝑖𝑜𝑛

𝐸𝑛𝑡𝑖𝑟𝑒𝑙𝑦 𝑉𝑖𝑠𝑐𝑜𝑢𝑠

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Flow between Parallel Plates

Fully Developed Laminar Flow Between Inﬁnite Parallel Plates

نایرج داجیا لیلاد :

1 - تاحفص اب یزاوم راشف نایدارگ

2 - تاحفص یبسن تکرح

3 - تاحفص اب یزاوم نزو یورین دننام یمجح یورین

4 - دراوم بیکرت

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Flow between Parallel Plates

Both Plates Stationary

Steady State + Fully Developed + ∆𝒑

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Flow between Parallel Plates

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For a Newtonian Fluid: 𝝉_𝒙𝒚 = 𝝁^𝒅𝒖

𝒅𝒚

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Boundary Conditions:

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Flow between Parallel Plates

Shear Stress Distribution 𝝉_𝒙𝒚 = 𝒂 𝝏𝒑

𝝏𝒙 [𝒚

𝒂 − 𝟏 𝟐] Volume Flow Rate:

For a depth l in the z direction,

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Upper Plate Moving with Constant Speed, U Steady State + Fully Developed + ∆𝒑 + Plate Movement

𝑼

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Flow between Parallel Plates

تسا هدرک رییغت یزرم طیارش طقف ،یلبق لح و تلاداعم نامه .

New Boundary Conditions:

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Flow between Parallel Plates

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Gravity-Driven Flow on a Plate

Example 8.3 LAMINAR FILM ON A VERTICAL WALL

A viscous, incompressible, Newtonian liquid flows in steady, laminar flow down a vertical wall. The thickness, δ, of the liquid film is constant. Since the liquid free surface is exposed to atmospheric pressure, there is no pressure gradient. For this gravity-driven flow, apply the momentum equation to differential control volume 𝒅𝒙 𝒅𝒚 𝒅𝒛 to derive the velocity distribution in the liquid film.

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Gravity-Driven Flow on a Plate

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Flow in a Pipe

Fully Developed Laminar Flow

in a Pipe

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Flow in a Pipe

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Flow in a Pipe

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Flow in a Pipe

Boundary Conditions: 𝒂𝒕 𝒓 = 𝑹: 𝒖 = 𝟎 &

𝒂𝒕 𝒓 = 𝟎: 𝒖: 𝒇𝒊𝒏𝒊𝒕𝒆

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Flow in a Pipe

Shear Stress Distribution

Volume Flow Rate:

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Flow in a Pipe

Hagen-Poiseuille

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Flow in a Pipe

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Flow in a Pipe: Laminar or Turbulent

Flow in Pipes and Ducts

راشف تفا یلصا لماع هس :

1 - عطقم حطس شهاک

2 - لااب ور بیش Upward slope

3 Friction -

For Laminar Flow

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Flow in a Pipe: Laminar or Turbulent

Shear Stress Distribution in Fully Developed Pipe Flow

Turbulent Flow?

𝑚𝑎𝑥. 𝑠ℎ𝑒𝑎𝑟 𝑠𝑡𝑟𝑒𝑠𝑠:

𝑢^′& 𝑣^′: randomly ﬂuctuating velocity components in x-dir & y-dir

Reynolds Stress Viscous Shear Stress

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Flow in a Pipe: Laminar or Turbulent

𝒏𝒆𝒂𝒓 𝒘𝒂𝒍𝒍: 𝒎𝒂𝒙 𝒚

𝑹 → 𝟎 , 𝒏𝒆𝒂𝒓 𝒕𝒉𝒆 𝒄𝒆𝒏𝒕𝒆𝒓: 𝒎𝒊𝒏 (𝒚

𝑹 → 𝟏)

سیو یشرب شنت هباشم زوک

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تسا رارقرب هراوید یور نانچمه شزغل مدع طرش .

𝒂𝒕 𝒘𝒂𝒍𝒍: 𝒖^′& 𝒗^′ = 𝟎

دوش یمن هدید لبق لکش یور قافتا نیا .

راوید دوخ یور

𝒚

𝑹 = 𝟎 لایس هتیزوکسیو زا یشان طقف شنت :

: وکسیو شنت ز

یحاون ریاس :

زوکسیو شنت +

زدلونیر شنت

میوش رت کیدزن هراوید هب هچ ره :

زدلونیر شنت

↑

The velocity proﬁle for turbulent ﬂow through a smooth pipe may also be approximated by the empirical power-law equation:

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𝑛 = 7

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Flow in a Pipe: Energy Considerations

Energy Considerations in Pipe Flow

فرصم رب زوکسیو یاهورین ریثأت یسررب یژرنا

زوکسیوریغ ضرف اب 𝑬𝑮𝑳 :

تسا تباث .

هتیزوکسیو نتفرگ رظنرد اب 𝑬𝑮𝑳 ↓ :

یکیناکم یژرنا هدنروخ هتیزوکسیو

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Flow in a Pipe: Energy Considerations

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(1) 𝑊ሶ_𝑠 = 0, ሶ𝑊_{𝑜𝑡ℎ𝑒𝑟} = 0.

(2) 𝑊_{𝑠ℎ𝑒𝑎𝑟} = 0 (although shear stresses are present at the walls of the elbow, the velocities are zero there, so there is no possibility of work).

(3) Steady ﬂow.

(4) Incompressible ﬂow.

(5) Internal energy and pressure uniform across sections 1 and 2 .

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we have not assumed the velocity to be uniform at sections 1 and 2 , since we know that for viscous ﬂows the velocity at a cross-section cannot be uniform.

رض زا هدافتسا اب نیگنایم تعرس فیرعت بی

یشبنج یژرنا حیحصت 𝜶

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Flow in a Pipe: Energy Considerations Kinetic Energy Coefﬁcient

For laminar ﬂow in a pipe 𝛼 = 2

From n=5 to n=6 for high Reynolds numbers, α varies from 1.08 to 1.03;

For n=7, α=1.06.

Use the approximation α=1 in pipe ﬂow calculations.

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Flow in a Pipe: Head Loss

Head Loss

Using the deﬁnition of α, the energy equation:

÷ 𝑑𝑚:

𝑅𝑒𝑎𝑟𝑟𝑎𝑛𝑔𝑖𝑛𝑔:

Mechanical energy per unit mass at a cross section

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Flow in a Pipe: Head Loss

نایم یکیناکم یژرنا فلاتا 1

و 2 ( مرج دحاو رب :)

𝑢₂ − 𝑢₁ :

ینورد یژرنا رییغت

𝛿𝑄

: 𝑑𝑚

زا یشان یترارح یژرنا لکش هب ریذپان تشگزاب فلاتا Friction

𝑖𝑓 𝛼 = 1 & ℎ_{𝑙 𝑇} = 0:

𝐵𝑒𝑟𝑛𝑜𝑢𝑙𝑙𝑖 𝐸𝑞.

𝑳^𝟐/𝒕^𝟐

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Flow in a Pipe: Head Loss

÷ 𝑔: [𝑳]

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Calculation of Head Loss

ℎ_{𝑙 𝑇} ൝ 𝑀𝑎𝑗𝑜𝑟 𝑙𝑜𝑠𝑠𝑒𝑠: 𝐹𝑟𝑖𝑐𝑡𝑖𝑜𝑛𝑎𝑙 𝐸𝑓𝑓𝑒𝑐𝑡𝑠: ℎ_𝑙

𝑀𝑖𝑛𝑜𝑟 𝑙𝑜𝑠𝑠𝑒𝑠: 𝑓𝑖𝑡𝑡𝑖𝑛𝑔𝑠, 𝑏𝑒𝑛𝑑𝑖𝑛𝑔𝑠, 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑎𝑟𝑒𝑎: ℎ_𝑙𝑚

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Calculation of Head Loss

Major Losses: Friction Factor

For fully developed ﬂow through a constant-area pipe

ℎ_𝑙𝑚 = 0 𝒉_𝒍

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Calculation of Head Loss: Laminar Flow

رد یراذگیاج :

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Calculation of Head Loss: Turbulent Flow

ن دوجو راشف تفا هبساحم یارب یلیلحت هطبار هتفشآ نایرج یارب دراد

.

زا یداعبا زیلانآ مینک یم هدافتسا

:

تباث عطقم حطس اب یقفا هلول کی رد هتفای هعسوت نایرج یارب :

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دهد یم ناشن ار تیعبات هوحن طقف یداعبا زیلانآ .

دیاب قیقد ریداقم ندروآ تسد هب یارب جیاتن زا

experiment دوش هدافتسا

.

Experiments show that the nondimensional head loss is directly proportional to L/D

𝐿𝐻𝑆 ÷ 1 2

Kinetic Energy per Unit Mass

Darcy Friction factor, 𝒇

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دیآ یم تسد هب یبرجت تروص هب کاکطصا بیرض .

Moody Diagram

لوهجم 𝑯_𝒍

؟ 1 - هبساحم ،𝑹𝒆

2 - یربز ندناوخ ،لودج زا (𝒆)

3 - زا هدافتسا اب

𝒆

𝑫 & 𝑹𝒆 𝒇 ،

زا

دیآ یم تسد هب رادومن .

4 𝑯_𝒍 -

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Moody Diagram

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Calculation of Head Loss: Turbulent Flow

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Calculation of Head Loss: Laminar Flow

For Laminar Flow:

𝒇_{𝒍𝒂𝒎𝒊𝒏𝒂𝒓} = 𝟔𝟒 𝑹𝒆

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To avoid having to use a graphical method for obtaining f for turbulent ﬂows, various mathematical expressions have been ﬁtted to the data. The most widely used formula for friction factor is from Colebrook.

Wall Shear Stress Eq.:

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Calculation of Head Loss

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Calculation of Head Loss: Minor Loss

Minor Losses

𝑀𝑖𝑛𝑜𝑟 𝑙𝑜𝑠𝑠𝑒𝑠 (ℎ_𝑙𝑚)

𝑖𝑛𝑙𝑒𝑡 𝑎𝑛𝑑 𝑜𝑢𝑡𝑙𝑒𝑡

𝑒𝑛𝑙𝑎𝑟𝑔𝑒𝑚𝑒𝑛𝑡 𝑎𝑛𝑑 𝑐𝑜𝑛𝑡𝑟𝑎𝑐𝑡𝑖𝑜𝑛 𝑝𝑖𝑝𝑒 𝑏𝑒𝑛𝑑

𝑣𝑎𝑙𝑣𝑒 𝑎𝑛𝑑 𝑓𝑖𝑡𝑡𝑖𝑛𝑔

دراد دوجو یئزج فلاتا هبساحم یارب شور ود :

ℎ_𝑙𝑚 = 𝐾 ത𝑉²

2 ℎ_𝑙𝑚 = 𝑓𝐿_𝑒

𝐷 ത𝑉²

2

𝐾: 𝐿𝑜𝑠𝑠 𝐶𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 𝐿_𝑒: 𝐸𝑞𝑢𝑖𝑣𝑎𝑙𝑒𝑛𝑡 𝐿𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑠𝑡𝑟𝑎𝑖𝑔ℎ𝑡 𝑝𝑖𝑝𝑒

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Calculation of Head Loss: Minor Loss

Minor Losses: Inlets and Exits

negligible

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Calculation of Head Loss: Minor Loss

Minor Losses: E𝑛𝑙𝑎𝑟𝑔𝑒𝑚𝑒𝑛𝑡s 𝑎𝑛𝑑 C𝑜𝑛𝑡𝑟𝑎𝑐𝑡𝑖𝑜𝑛s For circular ducts

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Losses caused by area change can be reduced somewhat by installing a nozzle or diffuser between the two sections of straight pipe.

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Losses in diffusers: Pressure Recovery Coefficient

دوش یم هبساحم راشف تفایزاب بیرض مان هب یبیرض زا هدافتسا اب رزویفید رد فلاتا .

تسا هدش راشف شیازفا هب لیدبت یدورو یشبنج یژرنا زا نازیم هچ هکنیا اب لداعم .

= شیازفا راشف کیتاتسا راشف یکیمانید یدورو

1 2

𝒉_𝒍𝒎?

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Calculation of Head Loss: Minor Loss

Ideal (frictionless) pressure recovery coefﬁcient:

𝐶_{𝑝𝑟𝑒𝑎𝑙} < 𝐶_{𝑝𝑖𝑑𝑒𝑎𝑙}

𝒉_𝒍𝒎?

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𝒉_𝒍𝒎?

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For real fluid flow:

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Minor Losses: Pipe Bends

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Calculation of Head Loss: Minor Loss

Minor Losses:

Valves and Fittings

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Pumps, Fans, and Blowers in Fluid Systems

Driving force for maintaining the ﬂow against friction: a pump (for liquids)/ a fan or blower (for gases)

We generally neglect heat transfer and internal energy changes of the fluid (we will incorporate them later into the definition of the pump efficiency)

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Pumps, Fans, and Blowers in Fluid Systems

∆ℎ_{𝑝𝑢𝑚𝑝} = 𝑊ሶ_{𝑝𝑢𝑚𝑝}

ሶ

𝑚 = ∆𝑝_{𝑝𝑢𝑚𝑝} 𝜌

When applying the energy equation to a pipe system, we may sometimes choose points 1 and 2 so that a pump is included in the system. For these cases we can simply include the head of the pump as a “negative loss”.

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Noncircular Ducts

Hydraulic diameter

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Internal Viscous Flow

Example 8.5 PIPE FLOW INTO A RESERVOIR: PRESSURE DROP UNKNOWN

A 100-m length of smooth horizontal pipe is attached to a large reservoir. A pump is attached to the end of the pipe to pump water into the reservoir at a volume ﬂow rate of 0.01 ^𝑚

3

𝑠 . What pressure (gage) must the pump produce at the pipe to generate this ﬂow rate? The inside diameter of the smooth pipe is 75 mm.

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Internal Viscous Flow

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Internal Viscous Flow

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