CHAPTER 8 Heat Transfer

8/10/2019 CHAPTER 8 Heat Transfer

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INTERNAL

FORCE CONVECTIONPrepared by

Nurhaslina

FKK, UITM


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FLOW CONDITIONS FOR INTERNAL FLOW

For an internal flow, it must be concerned with the existence ofentrance and fully developed regions.

Consider laminar flow in a circular tube, fluid enters the tube with a

uniform velocity

When the fluid makes contact with the tube surface, viscous effectsbecome important. Boundary layer develops with increasing x


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Boundary layer development occurs at the expense of a shrinking

flow region and concludes with boundary layer merger at thecenterline

The distance from the entrance at which this condition is achieved ishydrodynamic entry length, xfd , h

The fully developed velocity profile is parabolic for laminar flow in acircular tube. For turbulent flow, the profile is flatter.


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The Reynolds number for flow in a circular tube is defined as

um = mean velocity

D = tube diameter

In a fully developed flow, the critical Reynolds number is:

* Laminar 2300

ReD , C 2300 * Critical = 2300

* Turbulent 2300 ReD 10,000

Hydrodynamic entry length:

Laminar flow:

Turbulent flow: 10 xfd , h 60

D

Mean velocity, umm = mass flowrate

= fluid density

um= mean velocity

Ac = cross-sectional area of tube

Fully developed

turbulent flow:

10/,

Dxhfd

cmAum

Dhfd Dx Re05.0/,

Du=

Du=Re

mm

D

c

mA

mu

D

m4=Re

D

Ac = D2/4


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VELOCITY PROFILE IN A PIPE

For laminar flow, constant property fluid in the fully developedregion of a circular tube (pipe):

2

2 14

1)(

oo

r

rr

dx

dpru

dx

dpru om

8

2

2

12)(om rr

uru

The maximum velocity is at r = 0, the centerline where u(0) = 2 um

cm

A

mu


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Pressure Gradient and Friction Factor in FullyDevelopment Flow

Friction factor, f :f = - (dp/dx)D

um2/2

Friction coefficient, Cf:

Cf = s = fum

2/2 4

For fully develop laminar flow :

f = 64

ReD

Pressure gradient :

dp = - 64 um2

dx ReD 2D


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For fully developed turbulent flow

f = 0.316 ReD-1/4 ReD 2 x 10

4

f = 0.184 ReD-1/5 ReD 2 x 10

4

f = (0.790 ln ReD1.64)-2 3000 ReD 5 x 10

6

Pressure drop, P for fully developed flow

P = - P1P2 dp = f um2 x1x2 dx = f um2 (x2x1)2D 2D L

Power, P = (P ) V , V = m/


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Moody Diagram


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THERMAL CONSIDERATIONS

If a fluid enters the tube at a uniform temp. that is less than thesurface temp. , convection heat transfer occurs and thermalboundary layer develop.

If the tube surface condition is fixed (Tsis constant) or a uniformheat flux (q

s is constant), a thermally fully developed conditionis

reached.

Thermal entry length:

Laminar flow: xfd , t = 0.05 ReDPr

D

Turbulent flow: 10 xfd , t 60D

)(" mSx TThq

We can write Newtons Law of coolinginside a tube by considering a meantemp. Tminstead of T


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The Energy Balance

mpconv dTcmdq Considering perfect gas, or incompressible liquid:

By integrating:

)(,, imompconv

TTcmq

qconvis related to mean temperatures at inlet and outlet.

Combining equations:

)("

ms

pp

sm TTh

cm

P

cm

Pq

dx

dT

where P = surface perimeter

= pD for circular tube,

= width for flat plate

Ts > Tm, heat is transferred to the

fluid and Tm increases with x

Ts < Tm, heat is transferred from

the fluid and Tm decreases with x


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Constant Surface Heat Flux

where P = surface perimeter

pD for circular tube,

= width for flat plate

)("" LPqAqq ssconv

constqs "

Integrating equation:

xcm

PqTxT

p

simm

"

,)(


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Constant Surface Temperature(Ts= constant)

Thcm

P

dx

Td

dx

dT

p

m

)(

Ts-Tm=T

Integrating from x to any downstream location:

h

cm

Px

TT

xTT

pims

ms

exp

)(

,

For the entire length of the tube:

h

cm

PL

T

T

TT

TT

pi

o

ims

oms

exp

,

,

lmsconv TAhq )/ln( ioio

lmTT

TTT

Asis the tube surface area, As = PL = pDL


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Uniform External Temperature

For heat transfer between fluid flowing over a tube and fluid passing through

the tube, replace Tsby and byT h U

tot

lm

lms

R

Tq

TAUq

=

=

p

s

im

om

i

o

cm

AU

TT

TT

T

T

exp

,

,

-==

totpi,m

o,m

i

o

Rcm

1exp

TT

TT

T

T


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Summary (8.1-8.3)

We discussed fully developed flow conditions for cases involving

internal flows, and we defined mean velocities and temperatures We wrote Newtons law of cooling using the mean temperature,

instead of

Based on an overall energy balance, we obtained an alternative

expression to calculate convection heat transfer as a function of meantemperatures at inlet and outlet.

We obtained relations to express the variation of Tmwith length, for

cases involving constant heat flux and constant wall temperature

)("

mS TThq

)( ,, imompconv TTcmq

xcm

PqTxT

p

simm

"

,)(

h

cm

PL

T

T

TT

TT

pi

o

ims

oms

exp

,

,

T


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Summary (8.1-8.3)

We used these definitions, to obtain appropriate versions of Newtons

law of cooling, for internal flows, for cases involving constant wall

temperature and constant surrounding fluid temperature

lmsconv TAhq

)/ln( io

iolm

TT

TTT

lms TAUq

We can combine equations (8.13-8.16) with (8.9) to obtain values of

the heat transfer coefficient (see solution of Example 8.3)

In the rest of the chapter we will focus on obtaining values of the heat

transfer coefficient h, needed to solve the above equations


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Heat Transfer Correlations for Internal Flow

Knowledge of heat transfer coefficient is needed for calculations

shown in previous slides.

Correlations exist for various problems involving internal flow,

including laminar and turbulent flow in circular and non-circular

tubes and in annular flow.

For laminar flow we can derive h dependence theoretically

For turbulent flow we use empirical correlations

Recall from Chapters 6 and 7 general functional dependence

Pr)(Re,fNu


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Laminar Flow in Circular Tubes

For cases involving uniform heat flux:

constqk

hDNu sD "36.4

For cases involving constant surface temperature:

constTNuD s66.3

1. Fully Developed Region


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Laminar Flow in Circular Tubes

For constant surface temperature condition:

Thermal Entry Length case

3/2

Pr]Re)/[(04.01

PrRe)/(0668.066.3

D

D

LD

LDNu

D

14.03/1

/PrRe86.1

s

D

DLNu

D

Combined Entry Length case (Temperature and velocity profiles develop

simultaneously)

75.90044.0

700,16Pr48.0

s

s constT

All properties, except sevaluated at average value of mean temperature

2

,, omim

m

TT

T

2. Entry Region: Velocity and Temperature are functions of x


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Turbulent Flow in Circular Tubes

For a smooth surface and fully turbulent conditions the Dittus

Boelter equation may be used for small to moderate temperaturedifferences Ts-Tm:

nDDNu PrRe023.0

5/410/

000,10Re

160Pr7.0

DL

D

n=0.4 for heating (Ts>Tm)

and 0.3 for cooling (Ts


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Turbulent Flow in Circular Tubes

For large Reynolds number range, Gnielinski correlation:

Friction factors may be obtained from Moody diagram etc.

For fully developed turbulent flow in smooth circular tubes with constant

surface heat flux, Skupinski correlation:

=+=

4D

2

5D

3

"s

827.0DD

10Pe10

1005.9Re106.3ttanconsqPe0185.082.4Nu

)1(Pr)8/(7.121

Pr)1000)(Re8/(3/22/1

f

fNu DD 6105Re3000

2000Pr5.0

D

For fully developed turbulent flow in smooth circular tubes with constant

surface heat temperature, Seban and Shimazaki correlation:

100PettanconsqPe025.00.5Nu D"s

8.0DD =+=


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Example 1 (Velocity Profile & Pressure Gradient)

Fully developed conditions are known to exists for water

following through a 25 mm diameter tube at 0.01 kg/s and 27c.what is the maximum velocity of the water in the tube? What isthe pressure gradient associated with the flow?

Example 2 (Moody Diagram & Pressure Drop)

What is the pressure drop associated with water at 27cfollowing with a mean velocity of 0.2 m/s through a 600 mlong cast iron pipe of 0.15 m inside diameter?

Example 3 (Thermal and Velocity Entry Length)

Determine the thermal and velocity entry lengths for oil, waterand mercury flowing through a 25 mm diameter tube with a meanvelocity and temperature of um= 5 mm/s and Tm= 27C,respectively.


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Engine oil at a rate of 0.02 kg/s flows through a 3 mm diameter tube30 m long. The oil has an inlet temperature of 60C, while the tube wall

temperature is maintained at 100C by steam condensing on its outer

surface. Estimate the average heat transfer coefficient for internal flow

of the oil and determine the outlet temperature of the oil.

Example 6 - Problem 8.55 (Uniform ExternalTemperature & Turbulent Flow in Circular Tube

Example 7 - Problem 8.56 (Constant Surface

Temperature & Turbulent Flow in Circular Tube

Example 4 - (Constant Surface Temperature &Laminar Flow in Circular Tube

CHAPTER 8 Heat Transfer

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