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4/19/2014
1
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)
5
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)
11
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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7
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)
13
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)
15
(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)
17
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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10
19
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)
21
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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12
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)
23
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
25
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
26
8.71
For uniform heat flux at both surfaces,