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Heat Transfer:

Chapter Eight

Internal Flow:

1

Entrance Conditions

Entrance Conditions

• Must distinguish between entrance and fully developed regions.

• Hydrodynamic Effects: Assume laminar flow with uniform velocity profile at

inlet of a circular tube.

– Velocity boundary layer develops on surface of tube and thickens with increasing x.

– Inviscid region of uniform velocity shrinks as boundary layer grows.

Does the centerline velocity change with increasing x? If so, how does it change?

– Subsequent to boundary layer merger at the centerline, the velocity profile

becomes parabolic and invariant with x. The flow is then said to be

hydrodynamically fully developed.

How would the fully developed velocity profile differ for turbulent flow?

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Entry Lengths (cont.)

– Onset of turbulence occurs at a critical Reynolds number of

, 2300D cRe

– Fully turbulent conditions exist for

10,000DRe

• Hydrodynamic Entry Length

,Laminar Flow: / 0.05fd h Dx D Re

,Turbulent Flow: 10 / 60fd hx D

• Thermal Entry Length

,Laminar Flow: / 0.05fd t Dx D Re Pr

,Turbulent Flow: 10 / 60fd tx D

𝑅𝑒𝐷 = 𝑢𝑚𝐷

𝜈

(8.3)

(8.23)

3

Mean Quantities

The Mean Velocity

• Absence of well-defined free stream conditions, as in external flow, and hence a

reference velocity or temperature , dictates the use of a cross-

sectional mean velocity and temperature for internal flow. u

T

mu mT

or,

,cA cm u r x dA

Hence,

,cA c

m

c

u r x dAu

A

orFor incompressible flow in a circular tube of radius ,

02

2,

or

m

o

u u r x r drr

• Linkage of mean velocity to mass flow rate:

m cm u A

(8.8)

(8.5)

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Fully Developed Flow

Fully Developed Conditions - velocity

• Assuming steady flow and constant properties, hydrodynamic conditions, including

the velocity profile, are invariant in the fully developed region.

v = 0 and 𝜕𝑢

𝜕𝑥 = 0 (8.9)

For the annular differential element of Figure 8.2, the force balance may be

expressed as

−𝑑

𝑑𝑟 𝑟𝜏𝑟 = 𝑟

𝑑𝑝

𝑑𝑥 (8.10)

With y = 𝒓𝒐 − 𝒓, Newton’s law of viscosity,

τr = −μdu

dr (8.11)

Equation 8.10 becomes

μ

r

d

dr r

du

dr =

dp

dx (8.12)

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Equation 8.12 may be solved by integrating twice and subject to the

boundary conditions

u(ro) = 0 and 𝜕𝑢

𝜕𝑟 𝑟=0

= 0

𝑢 𝑟 = −1

4𝜇

𝑑𝑝

𝑑𝑥 𝑟𝑜

2 1 − 𝑟

𝑟𝑜

2 (8.13)

um = −ro

2

8μ

dp

dx (8.14)

𝑢(𝑟)

𝑢𝑚= 2 1 −

𝑟

𝑟𝑜

2 (8.15)

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Fully Developed Flow

Pressure Gradient and Friction Factor in Fully

Developed Flow

The engineer is interested in the pressure drop which will determine pump or

fan power requirements.

• The pressure drop may be determined from knowledge of the friction factor

f, where,

2

/

/ 2m

dp dx Df

u

Laminar flow in a circular tube:

64

D

fRe

Turbulent flow in a smooth circular tube:

2

0.790 1n 1.64Df Re

(8.19)

(8.21)

𝐶𝑓 = 𝜏𝑠

𝜌 𝑢𝑚2 2

, 𝑠𝑖𝑛𝑐𝑒 𝜏𝑠 = −𝜇 𝑑𝑢

𝑑𝑟

𝑟=𝑟𝑜

, 𝑓𝑟𝑜𝑚 𝐸𝑞𝑢𝑎𝑡𝑖𝑜𝑛 8.13, 𝐶𝑓 = 𝑓

4

(8.16)

7

Fully Developed Flow (cont.)

Turbulent flow in a roughened circular tube:

Pressure drop for fully developed flow from x1 to x2:

2

1 2 2 12

mup p p f x x

D

and power requirement

pmP p

(8.20)

1

/ 2.512.0log

3.7D

f

e D

Re f

(8.22a)

(8.22b)

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Entrance Conditions (cont.)

• Thermal Effects: Assume laminar flow with uniform temperature, , at

inlet of circular tube with uniform surface temperature, , or heat flux, .

,0 iT r T

s iT T sq

– Thermal boundary layer develops on surface of tube and thickens with increasing x.

– Isothermal core shrinks as boundary layer grows.

– Subsequent to boundary layer merger, dimensionless forms of the temperature

profile become independent of x. for and s sT q Conditions are then said to

be thermally fully developed.

Is the temperature profile invariant with x in the fully developed region?

9

Mean Quantities (cont.)

• Linkage of mean temperature to thermal energy transport associated with flow

through a cross section:

ct A p c p mE uc T dA mc T

Hence,

cA p c

m

p

uc T dAT

mc

• For incompressible, constant-property flow in a circular tube,

02

2, ,

or

m

m o

T u x r T x r r dru r

Determination of the Mean Temperature

𝑞 = 𝑚 𝑐𝑝 𝑇𝑜𝑢𝑡 − 𝑇𝑖𝑛 (1.12𝑒)

(8.25)

(8.26)

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Mean Quantities (cont.)

• Newton’s law of cooling for the Local Heat Flux:

s s mq h T T

What is the essential difference between use of for internal flow and

for external flow? mT T

(8.27)

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Fully Developed Flow (cont.)

• Requirement for fully developed thermal conditions:

,

,0

s

s m fd t

T x T r x

x T x T x

• Effect on the local convection coefficient:

/

o

o

r rs

s m s mr r

T rT Tf x

r T T T T

Hence, assuming constant properties,

/s

s m

q k hf x

T T k

h f x

Variation of h in entrance and fully developed regions:

Fully Developed Condition

(8.28)

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Mean Temperature

The Energy Balance

• Determination of is an essential feature of an internal flow analysis. mT x

Determination begins with an energy balance for a differential control volume.

conv p m m m p mdq mc T dT T mc dT

Integrating from the tube inlet to outlet,

conv , , (1)p m o m iq mc T T

(8.35) (8.36)

(8.34)

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Mean Temperature (cont.)

A differential equation from which may be determined is obtained by

substituting for

mT x

conv

.s s mdq q Pdx h T T Pdx

2m s

s m

p p

dT q P Ph T T

dx mc mc

• Special Case: Uniform Surface Heat Flux

m s

p

dT q Pf x

dx mc

,s

m m i

p

q PT x T x

mc

Why does the surface temperature vary with x as shown in the figure?

In principle, what value does Ts assume at x = 0?

conv sq q PL T

(8.37)

(8.39)

(8.40)

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Mean Temperature (cont.)

• Special Case: Uniform Surface Temperature

From Eq. (2), with s m

m

p

T T T

d Td T Ph T

dx dx mc

Integrating from x=0 to any downstream location,

,

exps m

x

s m ip

T T x Pxh

T T mc

0

1x x

xh h dx

x

Overall Conditions:

,

,

exp exps m oo s

i s m ip p

T TT h APLh

T T T mc mc

conv s mq hA T

3

1n /

o im

o i

T TT

T T

x

(8.43)

(8.41b)

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(8.42)

Mean Temperature (cont.)

• Special Case: Uniform External Fluid Temperature

,

,tot

1exp exp

m oo s

i m ip p

T TT U A

T T T mc mc R

tot

ms m

Tq UA T

R

Eq. (3) with replaced by .m sT T T

Note: Replacement of by Ts,o if outer surface temperature is uniform. T

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Fully Developed Flow - thermal

Applying the simplified, steady-flow, thermal energy Equation 1.12e to the

annular differential element of Figure 8.9.

𝑢𝜕𝑇

𝜕𝑥=

𝛼

𝑟

𝜕

𝜕𝑟 𝑟

𝜕𝑇

𝜕𝑟 (8.48)

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For constant surface heat flux, negligible net axial conduction,

∂2T ∂x2 = 0.

1

r

∂

∂r r

∂T

∂r =

2um

α

dTm

dx 1 −

r

ro

2 qs

" = constant (𝟖. 𝟒𝟗)

Subject to the boundary conditions

Temperature remains finite at r = 0, T(𝐫𝐨) = 𝑻𝒔

The temperature profile is of the form

T r, x = Ts x − 2um ro

2

α

dTm

dx

3

16+

1

16

r

ro

4

− 1

4

r

ro

2

(𝟖. 𝟓𝟎)

Substitute the velocity profile (8.15) and temperature profile (8.50) into

(8.26) and integrate over r

Tm x = Ts x − 11

48

um ro2

α

dTm

dx (8.52)

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For constant surface temperature, the assumption of negligible axial

conduction is often reasonable. Substitute for the velocity profile from

8.15 and for the axial temperature gradient from 8.33, the energy

equation 8.48 becomes

1

r

∂

∂r r

∂T

∂r =

2um

α

dTm

dx 1 −

r

ro

2

𝑇𝑠− 𝑇

𝑇𝑠− 𝑇𝑚 𝑇𝑠 = constant (𝟖. 𝟓𝟒)

A solution may be obtained by an iterative procedure, which involves

successive approximations to the temperature profile. The resulting

profile is not described by a simple algebraic expression, but the resulting

Nusselt number may be shown to be

NuD = 3.66 Ts = constant (8.55)

Fully Developed Flow

Fully Developed Flow - thermal

• Laminar Flow in a Circular Tube:

The local Nusselt number is constant throughout the fully developed

region, but its value depends on the surface thermal condition.

– Uniform Surface Heat Flux : ( )sq

4.36DhDNuk

– Uniform Surface Temperature : ( )sT

3.66DhDNuk

• Turbulent Flow in a Circular Tube:

– For a smooth surface and fully turbulent conditions , the

Dittus – Boelter equation may be used as a first approximation: 10,000DRe

4/50.023 n

D DNu Re Pr

1/2 2/3

/ 8 1000

1 12.7 / 8 1

DD

f Re PrNu

f Pr

– The effects of wall roughness and transitional flow conditions

may be considered by using the Gnielinski correlation:

3000DRe

0.3

0.4s m

s m

n T T

n T T

(8.53)

(8.55)

(8.60)

(8.62)

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Fully Developed Flow (cont.)

• Noncircular Tubes:

– Use of hydraulic diameter as characteristic length:

4 ch

AD

P

– Since the local convection coefficient varies around the periphery of a

noncircular tube, approaching zero at its corners, Hence in using a circular

tube correlation, the coefficient is presumed to be an average over the

perimeter.

– Laminar Flow:

The local Nusselt number is a constant whose value (Table 8.1) depends on

the surface thermal condition and the duct aspect ratio. or s sT q

– Turbulent Flow:

As a first approximation, the Dittus-Boelter or Gnielinski correlation may be used

with the hydraulic diameter, irrespective of the surface thermal condition.

(8.66)

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Entry Region

Effect of the Entry Region

• The manner in which the Nusselt number decays from inlet to fully developed

conditions for laminar flow depends on the nature of thermal and velocity boundary

layer development in the entry region, as well as the surface thermal condition.

Laminar flow in a

circular tube.

– Combined Entry Length:

Thermal and velocity boundary layers develop concurrently from uniform

profiles at the inlet.

( / )

th Graetze number

D DGz D x Re Pr

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Entry Region (cont.)

– Thermal Entry Length:

Velocity profile is fully developed at the inlet, and boundary layer development

in the entry region is restricted to thermal effects. Such a condition may also

be assumed to be a good approximation for a uniform inlet velocity profile if

1.Pr

• Average Nusselt Number for Laminar Flow in a Circular Tube with

Uniform Surface Temperature:

– Combined Entry Length (Baehr and Stephan):

1

1/3 2/3

1/6 1/6

3.66 0.0499 tanh( )tanh 2.264 1.7

tanh 2.432

D D

D DD

D

Gz GzGz Gz

NuPr Gz

– Thermal Entry Length (Hausen):

2/3

0.06683.66

1 0.04D

D

D

GzNu

Gz

(8.58)

(8.57)

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Entry Region (cont.)

• When determining for any tube geometry or flow condition, all

properties are to be evaluated at DNu

, , / 2m m i m oT T T

• When differences between Tm and Ts correspond to large property

variations, the Nusselt numbers for laminar flow of a liquid can be

corrected as

0.14

, ,D c D c

DD s

Nu Nu

Nu Nu

where the corrected Nusselt numbers are denoted by subscript c.

• Temperature-Dependent Properties:

(8.59)

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Annulus The Concentric Tube Annulus

• Fluid flow through

region formed by

concentric tubes.

• Convection heat transfer

may be from or to inner

surface of outer tube and

outer surface of inner tube.

• Surface thermal conditions may be characterized by

uniform temperature or uniform heat flux . , ,,s i s oT T ,i oq q

• Convection coefficients are associated with each surface, where

,i i s i mq h T T

,o o s o mq h T T

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8.67

8.68

Annulus (cont.)

i h o hi o

h D h DNu Nu

k k

• Fully Developed Laminar Flow

Nusselt numbers depend on and surface thermal conditions (Tables 8.2, 8.3) /i oD D

• Fully Developed Turbulent Flow

Correlations for a circular tube may be used with replaced by . DhD

h o iD D D

𝑁𝑢𝑖 = 𝑁𝑢 𝑖𝑖

1− 𝑞𝑜" 𝑞𝑖

" 𝜃𝑖∗ 8.72

𝑁𝑢𝑜 = 𝑁𝑢𝑜𝑜

1− 𝑞𝑖" 𝑞𝑜

" 𝜃𝑜∗ 8.73

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8.71

For uniform heat flux at both surfaces,