Chapter 8:
Internal Forced Convection
Yoav Peles
Department of Mechanical, Aerospace and Nuclear Engineering
Rensselaer Polytechnic Institute
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Objectives
When you finish studying this chapter, you should be able to:
• Obtain average velocity from a knowledge of velocity profile, and
average temperature from a knowledge of temperature profile in
internal flow,
• Have a visual understanding of different flow regions in internal flow,
such as the entry and the fully developed flow regions, and calculate
hydrodynamic and thermal entry lengths,
• Analyze heating and cooling of a fluid flowing in a tube under
constant surface temperature and constant surface heat flux conditions,
and work with the logarithmic mean temperature difference,
• Obtain analytic relations for the velocity profile, pressure drop,
friction factor, and Nusselt number in fully developed laminar flow,
and
• Determine the friction factor and Nusselt number in fully developed
turbulent flow using empirical relations, and calculate the pressure
drop and heat transfer rate.
Introduction
• Pipe ─ circular cross section.
• Duct ─ noncircular cross section.
• Tubes ─ small-diameter pipes.
• The fluid velocity changes from zero at the surface
(no-slip) to a maximum at the pipe center.
• It is convenient to work with an
average velocity, which remains
constant in incompressible flow
when the cross-sectional area
is constant.
Average Velocity
• The value of the average velocity is determined from
the conservation of mass principle
m = ρVavg AC =
∫ ρ u ( r ) dAC
(8-1)
Ac
• For incompressible flow in a circular pipe of radius R
∫ ρ u ( r ) dA
C
Vavg =
Ac
ρ AC
∫
=
R
0
ρ u ( r ) 2π rdr
ρπ R 2
R
2
= 2 ∫ u ( r ) rdr
R 0
(8-2)
Average Temperature
• It is convenient to define the value of the mean
temperature Tm from the conservation of
energy principle.
• The energy transported by the fluid through a
cross section in actual flow must be equal to
the energy that would be transported through
the same cross section if the fluid were at a
constant temperature Tm
pTm = ∫ c pT ( r ) δ m =
E fluid = mc
m
∫ ρ c T ( r ) u ( r )VdA
p
Ac
c
(8-3)
• For incompressible flow in a circular pipe of radius R
Tm =
∫ c pT ( r ) δ m
m
p
mc
∫ c T ( r ) ρ u ( r ) 2π rdr
p
=
Ac
ρVavg (π R 2 ) c p
(8-4)
R
2
=
T ( r ) u ( r ) rdr
2 ∫
Vavg R 0
• The mean temperature Tm of a fluid changes during
heating or cooling.
Idealized
Actual
Laminar and Turbulent Flow in Tubes
• For flow in a circular tube, the Reynolds number is
defined as
ρVavg D Vavg D
(8-5)
Re =
=
µ
ν
• For flow through noncircular tubes D is replaced by
the hydraulic diameter Dh.
4 Ac
Dh =
P
(8-6)
• laminar flow: Re<2300
• Transitional flow: 2300 ≤ Re ≤ 10,000
• fully turbulent flow : Re>10,000.
The Entrance Region
• A fluid entering a circular pipe at a uniform velocity.
• The no-slip condition - the flow in a pipe is divided into
two regions:
– (i) boundary layer region, (ii) irrotational (core) flow region
• The thickness of this boundary layer increases in the
flow direction until it reaches the pipe center.
Irrotational
flow
Boundary
layer
• Hydrodynamic entrance region ─ the region from the
pipe inlet to the point at which the boundary layer merges
at the centerline.
• Hydrodynamically fully developed region ─ the region
beyond the entrance region in which the velocity profile is
fully developed and remains unchanged.
• The velocity profile in the fully developed region is
– parabolic in laminar flow, and
– somewhat flatter or fuller in turbulent flow.
Thermal Entrance Region
• Consider a fluid at a uniform temperature entering a circular tube
whose surface is maintained at a different temperature.
• Thermal boundary layer along the tube is developing.
• The thickness of this boundary layer increases in the flow
direction until the boundary layer reaches the tube center.
• Thermal entrance region.
• Thermally fully developed region ─ the region beyond the
thermal entrance region in which the dimensionless temperature
profile expressed as
(Ts-T)/(Ts-Tm)
remains unchanged.
– Hydrodynamically fully-developed:
∂u ( r , x )
∂x
= 0 → u = u (r )
(8-7)
– Thermally fully-developed:
∂ ⎡ Ts ( x ) − T ( r , x ) ⎤
⎢
⎥=0
∂x ⎢⎣ Ts ( x ) − Tm ( x ) ⎥⎦
(8-8)
− ( ∂T ∂r ) r = R
∂ ⎛ Ts − T ⎞
=
≠ f ( x ) (8-9)
⎜
⎟
Ts − Tm
∂r ⎝ Ts − Tm ⎠ r = R
常數
應為常數
• Surface heat flux can be expressed as
k ( ∂T ∂r ) r = R
∂T
qs = hx (Ts − Tm ) = k
→ hx =
(8-10)
∂
r
T
−
T
r=R
s
m
h 之定義
• For thermally fully developed region From (Eq. (8-9))
( ∂T
∂r ) r = R
Ts − Tm
≠ f ( x)
hx ≠ f ( x )
Fully developed flow
hx = constant Fully developed flow
Heat Transfer coefficient and Friction factor
Developing
region
Lh 與 Lt 何者較長?
Fully
developed
region
Entry Lengths (入口區域)
Laminar flow
– Hydrodynamic
Lh ,laminar ≈ 0.05 Re⋅ D
(8-11)
– Thermal Lt ,laminar ≈ 0.05 Re⋅ Pr⋅ D = Pr⋅ Lh ,laminar (8-12)
Turbulent flow
– Hydrodynamic
Lh ,turbulent = 1.359 D ⋅ Re
(8-13)
– Thermal
(approximate)
Lh ,turbulent ≈ Lt ,turbulent ≈ 10 D
(8-14)
14
Turbulent flow Nusselt Number
• The Nusselt numbers are much
higher in the entrance region.
• The Nusselt number reaches
a constant value at a distance
of less than 10 diameters.
• The Nusselt numbers for the
uniform surface temperature and uniform surface heat
flux conditions are identical in the fully developed
regions, and nearly identical in the entrance regions.
Æ Nusselt number is insensitive to the type of
thermal boundary condition.
General Thermal Analysis
• In the absence of any work interactions, the conservation
of energy equation for the steady flow of a fluid in a tube
p (Te − Ti )
Q = mc
(W)
(8-15)
• The thermal conditions at the surface can usually be
approximated as:
– constant surface temperature, or
– constant surface heat flux.
• The mean fluid temperature Tm must
change during heating or cooling.
• Either Ts= constant or qs = constant at the surface of a
tube, but not both.
Constant Surface Heat Flux
• In the case of constant heat flux, the rate of heat transfer can
also be expressed as
(8-17)
p (Te − Ti ) (W)
Q = qs As = mc
• Then the mean fluid temperature at the tube exit becomes
qs As
(8-18)
Te = Ti +
p
mc
• The surface temperature in the case of constant surface heat
flux can be determined from
qs
(8-19)
qs = h (Ts − Tm ) → Ts = Tm +
h
h 之定義
• In the fully developed region, the
surface temperature Ts will also
increase linearly in the flow direction
• Applying the steady-flow energy
balance to a tube slice of thickness dx,
the slope of the mean fluid temperature
Tm can be determined
dTm qs p
p dTm = qs ( pdx ) →
mc
=
= constant
p
dx mc
• Noting that both the heat flux and h (for
fully developed flow) are constants
由式 (8-19)
微分可得:
dTm dTs
=
dx
dx
(8-21)
(8-20)
• In the fully developed region (Ts-Tm=constant)
∂ ⎛ Ts − T
⎜
∂x ⎝ Ts − Tm
⎞
∂T dTs
1 ⎛ ∂Ts ∂T ⎞
−
=0→
=
⎟=0→
⎜
⎟
Ts − Tm ⎝ ∂x ∂x ⎠
∂x dx
⎠
(8-22)
• Combining Eqs. 8–20, 8–21, and 8–22 gives
∂T dTs dTm qs p
=
=
=
= constant
p
∂x dx
dx mc
(8-23)
• For a circular tube ( p = 2πR = πD )
2qs
∂T dTs dTm
=
=
=
= constant (8-24)
∂x dx
dx
ρVavg c p R
Constant Surface Temperature
• The energy balance on a differential control volume
p dTm = h (Ts − Tm ) dAs
δ Q = mc
(8-27)
• Since the mean temperature of the fluid Tm increases in the flow
direction, the heat flux decays with x.
• The surface temperature is constant (dTm=-d(Ts-Tm)) and
dAs=pdx, therefore,
d (Ts − Tm )
Ts − Tm
hp
=−
dx
p
mc
(8-28)
• Integrating Eq. 6-28 from x=0 (tube inlet
where Tm=Ti) to x=L (tube exit where Tm=Te)
gives
Ts − Te
hAs
ln
=−
p
Ts − Ti
mc
(8-29)
• Taking the exponential of both sides and
solving for Te
p)
Te = Ts − (Ts − Ti ) exp ( − hpL mc
• or
p)
Tm ( x ) = Ts − (Ts − Ti ) exp ( − hpx mc
任何位置 x 之平均溫度
(8-30)
• The temperature difference between the fluid and the surface decays
exponentially in the flow direction, and the rate of decay depends on
the magnitude of the exponent
p
hAs mc
• This dimensionless parameter is
called the number of transfer
units (NTU).
– Large NTU value – increasing tube
length marginally increases heat
transfer rate.
– Small NTU value – heat transfer increases
significantly with increasing tube length.
• Solving Eq. 8–29 for mcp gives
p=
mc
hAs
ln ⎡⎣(Ts − Te ) (Ts − Ti ) ⎤⎦
(8-31)
• Substituting this into Eq. 8–15
p (Te − Ti )
Q = mc
(W)
Q = hAs ∆Tln
(8-32)
where
Ti − Te
∆Te − ∆Ti
∆Tln =
=
ln ⎡⎣(Ts − Te ) (Ts − Ti ) ⎤⎦ ln [ ∆Te ∆Ti ]
∆Τln is the logarithmic mean temperature difference.
(8-33)
Laminar Flow in Tubes
•
•
•
•
•
Assumptions:
steady laminar flow,
incompressible fluid,
constant properties,
fully developed region,
and
straight circular tube.
• The velocity profile u(r)
remains unchanged in
the flow direction.
• no motion in the radial
direction.
• no acceleration.
• Consider a ring-shaped
differential volume element.
• A force balance on the volume
element in the flow direction
gives
( 2π r dr P ) x − ( 2π r dr P ) x+ dx
+ ( 2π r dx τ )r − ( 2π r dx τ )r + dr = 0
(8-34)
• Dividing by 2π dr dx and rearranging
Px + dx − Px ( rτ )r + dr − ( rτ )r
r
+
=0
dx
dr
(8-35)
• Taking the limit as dr, dx → 0 gives
dP d ( rτ )
r
+
=0
dx
dr
(8-36)
• Substituting τ = µ(du/dr) gives
µ d ⎛ du ⎞
dP
⎜r
⎟=
r dr ⎝ dr ⎠ dx
(8-37)
• Rearranging and integrating it twice to give
•
1 ⎛ dP ⎞ 2
u (r ) =
⎜
⎟ r + C1 ln r + C2
4 µ ⎝ dx ⎠
漏印
Boundary Conditions:
(8-38)
– symmetry about the centerline : ∂u/∂r = 0 at r = 0,
– no-slip condition: u = 0 at r = R.
• Eq. 6-38 with the boundary conditions
R 2 ⎛ dP ⎞ ⎛
r2 ⎞
u (r ) = −
⎜
⎟ ⎜1 − 2 ⎟
4 µ ⎝ dx ⎠ ⎝ R ⎠
(8-39)
• Substituting Eq. 8–39 into Eq. 8–2, and performing the
integration gives the average velocity
Vavg
R
R
2
2 R 2 ⎛ dP ⎞ ⎛
r2 ⎞
= 2 ∫ u ( r ) rdr = − 2 ∫
⎜
⎟ ⎜1 − 2 ⎟ rdr
R 0
R 0 4 µ ⎝ dx ⎠ ⎝ R ⎠
R 2 ⎛ dP ⎞
=−
⎜
⎟
8µ ⎝ dx ⎠
(8-40)
• Combining the last two equations, the velocity
profile is rewritten as
⎛
r2 ⎞
u ( r ) = 2Vavg ⎜1 − 2 ⎟
⎝ R ⎠
;
umax = 2Vavg
(8-41)
Pressure Drop
• One implication from Eq. 8-37 is that the pressure drop
gradient (dP/dx) must be constant (the left side is a function
only of r, and the right side is a function only of x).
• Integrating from x=x1 where the pressure is P1 to x=x1=L
where the pressure is P2 gives (速度場完全發展區域)
P2 − P1
dP
= constant =
dx
L
(8-43)
• Substituting Eq. 8–43 into the Vavg expression in Eq. 8–40
∆P = P1 − P2 =
8µ LVavg
R
2
=
32 µ LVavg
D
2
(8-44)
• A pressure drop due to viscous effects represents an
irreversible pressure loss.
• It is convenient to express the pressure loss for all
types of fully developed internal flows in terms of the
dynamic pressure and the friction factor
dynamic pressure
P
f
friction factor
∆PL = ( P1 − P2 ) =
L
⋅ ⋅
D
P
2
ρVavg
(8-45)
2
• Setting Eqs. 8–44 and 8–45 equal to each other and
solving for f gives
– Circular tube, laminar:
64 µ
64
f =
=
ρ DVavg Re
(8-46)
Temperature Profile and the Nusselt Number
• Energy is transferred by mass in the
x-direction, and by conduction in the
r-direction.
• The steady flow energy balance for a cylindrical shell element can be expressed as:
pTx − mc
pTx + dx + Q r − Q r + dr = 0
mc
入
出
• Substituting
入
(8-49)
出
m = ρ uAc = ρ u ( 2π rdr ) = 質量流率
and dividing by 2πr dr dx gives, after rearranging
Tx + dx − Tx
Q r + dr − Q r
1
) = −(
)(
)
ρ c pu (
2π rdx
dx
dr
(8-50)
• Or
∂T
1
∂Q
u
= −(
)
2 ρ c pπ rdx ∂r
∂x
(8-51)
• Since Fourier’s Law (完全發展區域)
∂Q ∂ ⎛
∂T
= ⎜ − k (2π rdx)
∂r ∂r ⎝
∂r
∂ ⎛ ∂T ⎞
⎞
⎟ = −2π kdx ⎜ r
⎟
∂
∂
r
r
⎠
⎝
⎠
(8-52)
Eq 8-51 becomes (能量平衡公式 – Energy Equation)
∂T α ∂ ⎛ ∂T ⎞
=
u
⎜r
⎟
∂x r dr ⎝ ∂r ⎠
;
k
α=
ρcp
(8-53)
Constant Surface Heat Flux
– Laminar Fully Developed flow
• Substituting Eqs. 8-24 and 8-41 into Eq. 8.53
⎛
r2 ⎞
u ( r ) = 2Vavg ⎜1 − 2 ⎟
⎝ R ⎠
2qs
∂T
=
= constant
∂x ρVavg c p R
(8-41)
(8-24)
∂T α ∂ ⎛ ∂T ⎞
u
=
⎜r
⎟
∂x r dr ⎝ ∂r ⎠
4qs
kR
(8-53)
⎛
r 2 ⎞ 1 d ⎛ dT ⎞ (8-55) 常微分
⎜1 − 2 ⎟ =
⎜r
⎟
方程式
⎝ R ⎠ r dr ⎝ dr ⎠
• Separating the variables and integrating twice
qs ⎛ 2 r 4 ⎞
T=
⎜ r − 2 ⎟ + C1r + C2
kR ⎝
4R ⎠
(8-56)
• Boundary conditions
– Symmetry at r = 0:
– At r = R:
∂T ( r = 0 )
∂r
=0
T(r=R) = Ts
qs R ⎛ 3 r 2
r4 ⎞
T = Ts −
⎜ − 2+ 4⎟
k ⎝4 R
4R ⎠
C1=0
C2
(8-57)
• Substituting the velocity and temperature profile relations (Eqs.
8–41 and 8–57) into Eq. 8–4 and performing the integration
∫ c T ( r ) δ m ∫ c T ( r ) ρ u ( r ) 2π rdr
p
p
Tm = 平均流體溫度 =
m
p
mc
=
Ac
ρVavg (π R 2 ) c p
R
2
=
T ( r ) u ( r ) rdr
2 ∫
Vavg R 0
(8-58)
11 qs R
Tm = Ts −
24 k
(8-4)
qs ≡ h (Ts − Tm )
24 k 48 k
k
h=
=
= 4.36
11 R 11 D
D
(8-59)
Constant heat flux (circular tube, laminar)
hD
Nu =
= 4.36 = 常數
k
(8-60)
Constant Surface temperature (circular tube, laminar)
hD
Nu =
= 3.66
k
推導較複雜(研究所)
(8-61)
Laminar Flow in Noncircular Tubes
• The friction factor ( f )
and the Nusselt number
relations are given in
Table 8–1 for fully
developed laminar flow
in tubes of various cross
sections.
Laminar flow
– Hydrodynamic
– Thermal
Lh ,laminar ≈ 0.05 Re⋅ D
Lt ,laminar ≈ 0.05 Re⋅ Pr⋅ D = Pr⋅ Lh ,laminar
Developing Laminar Flow in the Entrance Region
• For a circular tube of length L subjected to constant
surface temperature, the average Nusselt number for
the thermal entrance region (hydrodynamically
developed flow)
Nu = 3.66 +
0.065 ( D L ) Re⋅ Pr
1 + 0.04 ⎡⎣( D L ) Re⋅ Pr ⎤⎦
23
(8-62)
• For flow between isothermal parallel plates
Nu = 7.54 +
0.03 ( Dh L ) Re⋅ Pr
1 + 0.016 ⎡⎣( Dh L ) Re⋅ Pr ⎤⎦
23
(8-64)
Turbulent flow in Tubes
• Most correlations for the friction and heat transfer coefficients in turbulent flow are based on experimental studies.
• For smooth tubes, the friction factor in turbulent flow can
be determined from the explicit first Petukhov equation
f = ( 0.79 ln Re− 1.64 )
−2
3000<Re<5 × 106
(8-65)
• For fully developed turbulent flow the Nusselt number
(Dittus–Boelter equation)
n = 0.4 heating ⎫
0.8
n ⎧ Re > 10, 000
Nu = 0.023Re Pr ⎨
⎬
⎩0.7 ≤ Pr ≤ 160 n = 0.3 cooling ⎭
(8-68)
• Modified correlations are available for/due to :
– liquid metals (Pr<<1),
– large variation in fluid properties due to a large
temperature difference,
– surface roughness,
– flow through tube annulus.
• Original correlations are also approximately
valid for:
– developing Turbulent Flow in the Entrance
Region,
– turbulent Flow in Noncircular Tubes.
Moody Diagram