vedat s. arpaci and poul s. larsen, prentice-hall...
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1
Convection Heat Transfer
Textbook: Convection Heat Transfer
Adrian Bejan, John Wiley & Sons
Reference: Convective Heat and Mass Transfer
Kays, Crawford, and Weigand, McGraw-Hill
Convection Heat Transfer
Vedat S. Arpaci and Poul S. Larsen, Prentice-Hall Inc
1. Fluid Properties and Conservation Laws
2. External/Internal Laminar Flows
3. External/Internal Natural Convection
4. External/Internal Turbulent Flows
5. High Speed Flows
Content:
Grading: HW (30%)+Midterm (30%) + Final (30%) + Report (15%)
Convection Heat Transfer
2
Wanted
• friction force: 0 cos f
S
F dA
Skin friction coefficient 221
0
UC f
• pressure drag: 0 sin p
S
F P dA
form drag coefficient21
2
pp
FC
U A
Wanted
• heat transfer rate at the surface
no-slip hypothesis at wall : pure conduction adjacent to the wall
Fourier’s law: 0
0
yy
Tkq
• convection heat transfer coefficient: TThq 00
TT
y
Tk
hei y
0
0 ..
• local Nusselt number:k
hxNux
stationary whenratefer heat trans
flow in whenratefer heat trans~
3
Wanted
• averaged convection heat transfer coefficient:
avg
x
avg
xx T
dxqx
T
qh
1
0
00
0
total heat transfer rate over (0,x) = 0 0
0
x
x avgq dx x h T
• overall Nusselt number: k
x
T
q
k
xhNu
avg
xxx
00
0
Analysis Methods
• scaling analysis : qualitative analysis
magnitude of order
related to what parameters and how?
• integral analysis : quantitative analysis
magnitudes with a little errors
related to what parameters and how?
• similarity analysis : exact analysis
under model assumptions
• perturbation analysis : critical analysis
near some critical point
4
Fluid Properties
(1) viscosity coefficient: secm ; secmkg 2
• Newtonian Fluids:
2 usually assumed ( , Landau & Lifschifz, 1959)
3
jiji ji ji ji ji
j i
k
k
uup p
x x
uu
x
Fluid Mechanics
• temperature dependence:
PT ,
r cT T T
as :liquid
as :gases
T
T
5
Fluid Properties
(2) thermal conductivity: secm ; KmW 2pckk
• temperature dependence:
PTkk ,
• Fourier’s Law: Tkq )mW flux,(heat 2
isotropic k usually assumed
as :liquid
as :gases
Tk
Tk
r cT T T
rc
kk
k
6
Dimensionless Parameters
(1) Reynolds number: force viscous
force inertial~Re
UL
(2) Prandtl number: diffusionthermal
diffusion momentum~Pr
(3) Eckert number: massunit per differenceenthalpy
massunit per energy kinetic~
2
Tc
UEc
p
Re Re turbulent flowscr>
timeconvection char.
timediffusion char.~Re
2
UL
L
High speed flows significant viscous dissipation
Ec << 1 negligible viscous dissipation
Time Derivatives
Eulerianderivative
Lagrangian/material derivative
Du
Dt t
total derivative observer
dV
dt t
7
Conservation Laws
massunit per quantity physical some Let
CV
Vd (CV) volumecontrol the withinquantity ofamount total
CS
dAnu
(CS) surface control the throughquantity of rateoutflow
CVCS
VdqdAnu sources Vd
t :requires onConservati
CV
q source per unit time per unit volume
dm dV
Differential Form
CV CV
Conservation law: sources t CV
dV u dV qdV
• Consider an infinitesimal control volume VdCV
Divergence Theorem: Vdx
aVdadAna
V j
j
VS
t
dV u dV qdV
qu
t
CV
Conservation requires: sources t CS CV
dV u ndA qdV
8
Mass Conservation
qu
t
0 t
u
• mass: , no source1 0 q
0t t
Du u u
D
1 1 volume change rate per unit volume
t
D Du
D Dt
vector identi
t
:
y
a a a
CV
total amount of quantity within the control volume (CV) dV
Momentum Conservation
• momentum iu
fluids gsurroundin itsby CV theon acting force source CV
Vdq
= body forces + contact forces
CV
i ji j
CS
X dV n dA CV
jii
jCV
X dV dVx
Divergence Theorem: Vdx
aVdadAna
V j
j
VS
CV
total amount of quantity within the control volume (CV) dV
+ jii
j
q Xx
9
Momentum Conservation
qu
t
j
jiiii x
Xuuu
t
Newtonian fluid: t
jiii i
i j
u pu u X
x x
Newtonian fluid: ji ji jip
0tiu u
• momentum iu
+ jii
j
q Xx
jiji ji
j j
px x
jiji
j j
p
x x
ji
i j
p
x x
Energy Conservation
• total energy: energy) (potential mass)unit per energy (internal 21 Vuue ii
body force X V
• sources
gssurroundin itsby CV theon done work
CV ofinto/out diffusionheat
generationheat exteranl
ji j i
CV CS CS
qdV q ndA n dA u
ijij
ux
qqu
t
ji ij jip
Divergence Theorem: Vdx
aVdadAna
V j
j
VS
j ji ia u
10
• kinetic energy equation: jiii i
i j
Du pu X
Dt x x
Energy Conservation
12 i i i i ji
i j
De u u V q q pu u
Dt x x
• total energy equation:
0i i i i ii
DV V Vu u X u X
Dt t x
jii ii i ji
i i j j
u upp u u
x x x x
upqq Dt
De
• thermal energy equation:
iji
j
u
x
X V
upqq Dt
De
thermal (internal) energy
the time change rate of the internal energy of an infinitesimal
control volume = the heat generation rate
+ the net heat diffusion rate
+ pressure work rate done by surrounding fluid
+ viscous dissipation rate
11
Thermal Energy Conservation
21 1
2 2j ji i
j i j i
ji
j i
uu
x x
u uu u
x x x x
2
2j ji i
j i j i
u uu u
x x x x
iji
j
j
iij x
u
x
u
Newtonian fluid:
jiij
j
i
j
i
i j
u
x
u u
xx x
u
viscous dissipation rate = rate at which kinetic
energy is irreversibly converted to thermal energy by viscosity
j
iij x
u
upqq Dt
De
2
2j ji i
j i j i
u uu u
x x x x
viscous dissipation rate
j
iij x
u
4
2 3j j ji i i
j i j i i j
u u uu u u
x x x x x x
222 23
3 3 31 2 1 2 1 2
, 11 2 3 1 2 1 3 2 3
8 8 8 8
2 3 3 3 3ji
i j j ii j
uu u u uu u u u u u
x x x x x x x x x x x
22 2 2
3 31 1 2 2 1 1
1 1 2 2 1 1 3 3
2223
3 32 2
, 12 2 3 3
4 42 2
3 3
2 42
3ji
i j j ii j
u uu u u u u u
x x x x x x x x
uu u uu u
x x x x x x
22 223
3 31 2 1 2
, 11 2 1 3 3 2
4
2 3ji
i j j ii j
uu u uu u u u
x x x x x x x x
> 0
12
upqq Dt
De
Temperature-based
h e p dh de dp Tds pd dp Tds dp (first law)
p T
s sds dT dp
T p
( , )s T p
( , ) ( , ) ( , )
( , ) ( , ) ( , ) pT
s s T p p
p p T p T p T T
(Maxwell relation)
,1
,
s T
p
p
sdT dp
T
p
sdh T dT T dp
T
Temperature-based
1p
dpdh c dT T
1p
Dh DT Dpc T
Dt Dt Dt
pp
hc
T
p
sT
T
p
sdh T dT T dp
T
,h h T p
upqq Dt
De
13
• enthaphy h e p
upqq Dt
De
1
1 1De Dp p D
Dt D
Dh De Dp pu
t DtDt Dt Dt
Dt
DpTqq
Dt
DTcp
Temperature-based
1 p
DT Dp Dpc T q q p u p u
Dt Dt Dt
1p
Dh DT Dpc T
Dt Dt Dt
Temperature
• Assumptions: (1) negligible compressibility effect 0
(2) no external heat generation 0q
(3) negligible viscous dissipation 1Ec 0
(4) Fourier’s Law: Tkq
TkTut
Tc
Dt
DTc pp
Dt
DpTqq
Dt
DTcp
1 1
p pT T
= thermal expansion coefficient