HANDBOOK OF EQUATIONS, TABLES AND CHARTS FOR ME3122/ME3122E HEAT TRANSFER
Department of Mechanical Engineering National University of Singapore November 2014
CONDUCTION HEAT TRANSFER
1st law of thermodynamics:
dU Q W
Conduction: Convection: where 5.67 108 Wm-2K -4
Radiation: Control Volume:
Surface:
Heat Conduction Equation:
Cartesian:
Cylindrical:
Spherical:
1
One-Dimensional Walls
Fin Equations: d 2 dx
2
m2 0 where m hP / kA which has the general solution C 1e mx C 2emx .
Fin Efficiency: Fin Effectiveness: Overall Surface Efficiency:
o
qt qmax
qt hAt 0
2
1
NA f At
1 where A NA f
t
f
Aunfinned .
Lumped Capacitance Method:
,
,
,
Other Equations (Thermal Properties):
Solids: Free electrons: Gases: Joule heating:
g I 2 R E
Interfaces: Heat wave speed: Two semi-infinite solids touch:
3
,
CONVECTION HEAT TRANSFER
All symbols have their usual meaning. Constants Gravitational acceleration: g = 9.81 m/s2
Specific gas constant for air: R = 287 J/kgK Definitions Kinematic viscosity, /
Thermal diffusivity, k / c p Volumetric thermal expansion coefficient,
Newton's Law of Cooling, q hT s T Ideal gas law : pv RT
AcV Mass flow rate, m
u y
c pT Thermal energy flux through a section m
Dimensionless Groups Reynolds Number, Re L VL / VL / Prandtl Number, Pr / Nusselt Number, Nu L hL / k g T s T L
3
Grashof Number, Gr L
2
Rayleigh Number, Ra L Gr L Pr
Stanton Number, St x
Nu x
h x
Re x Pr c p u
4
1
for an ideal gas. T p T
General
Shear stress,
1
u v 0 x y
2D Continuity Equation:
2u 2u u u p v 2 2 X u x y x x y
2D x -Momentum Equation:
2D Energy Equation:
2T 2T k 2 2 q y x
T T v c p u y x
2 2 u v 2 u v 2 y x x y
where viscous dissipation,
2D Boundary Layer Equations:
x-Momentum Equation:
u u 2u v 2 u x y y
Energy Equation:
T T 2T v 2 u x y y d
Integral Momentum Equation:
dx
d dx
Integral Energy Equation:
u y y0
u(u u )dy
0
t
0
Forced Convection Over External Surfaces
Generally, Nu C Re Pr m
n
Forced Convection Over a Flat Plate:
For constant
,
.
Mean heat transfer coefficient, h
1
1
L
h dA L h dx
A A
x
x
0
5
T y y0
uT T dy
Uniform Surface Temperature (Isothermal): For laminar flow ( Re x 5 10 ): 5
1 2
5 x Re x
;
t 1
C f , x
Pr 1 3 ; 1
Nu x 0.332 Re x 2 Pr 3 ;
Nu L
h L k
s , x
0.664 Re x ; 1
2
u / 2
0.664 Re L
1
12
C f , L 1.328 Rex
2
1
Pr 3
2
For turbulent flow ( Re x 5 10 ): 5
1
C f , x 0.0592 Re x 1 5 ;
turb 0.37 x Re x 5 ;
Nu x 0.0296 Re x4 5 Pr 3 1
For mixed boundary layer conditions ( Re L 5 105 ): C f , L 0.074 Re L1 5 1742 Re L1 ;
h L
Nu L
k
Pr (0.037 Re L0.8 871) 1
3
Uniform Surface Heat Flux (Isoflux): Nu x 0.453 Re x 2 Pr 3
For laminar flow ( Re x 5 10 ):
1
5
T s T
1 L
L
0T s T dx
1 L
For turbulent flow ( Re x 5 10 ): 5
L
q s
0
h x
dx
1 L
1
L
q s x
0
k Nu x
dx
q s L 1
1
0.680k Re L2 Pr 3
Nu x 0.0308 Re x4 5 Pr 3 1
For Unheated Starting Length, xo, with laminar flow for both isothermal and isoflux conditions: Nu x Nu x x
o 0
3 4 1 3
1 x / x o
Forced Convection Across Long Cylinders: Nu D
h D k
C Re Dm Pr 1 3
where C and m are given by
Re D
C
m
0.4-4
0.989
0.330
4-40
0.911
0.385
40-4000
0.683
0.466
4000-40,000
0.193
0.618
40,000-400,000
0.027
0.805
6
Forced Convection Across Spheres: 14
μ Nu D 2 0.4 Re D1 2 0.06 Re D2 3 Pr 0.4 k μ s h D
where all properties are evaluated at the free-stream temperature, except μ s , which is evaluated at the surface temperature of the sphere.
Forced Convection Across Non-Circular Cylinders Nu D
h D k
C Re Dm Pr 1 3
where C and m are given by
Forced Convection Across Tube Banks 14
Nu D C 1 Re
m D ,max
Pr Pr Pr s 0.36
where all properties, except Pr s, are evaluated at the average of the fluid inlet and outlet temperatures, Re D,max is based on the maximum fluid velocity, and C 1 and m are given in the table below for number of tube rows for various aligned and staggered arrangements of tubes.
7
(a) Aligned tube rows
For below:
:
Nu
D N L 20
(b) Staggered tube rows
C 2 Nu D N 20 where C 2 for various L
8
is given in the table
Forced Convection in Tubes and Ducts
Friction factor,
f
Hydraulic Diameter, Dh
dp / dx D um2 / 2
2
or
Δ p
f
L ρum
D 2
4 Cross - sectional Area Wetted Perimeter
For thermally fully-developed condition:
T s ( x) T (r , x) 0 x T s ( x) T m ( x)
Laminar Flow ( Re D 2300): Fully developed velocity profile:
u(r ) um
where mean fluid velocity, um
r 2 21 2 r 0
m r 02
r 02 dp 8 dx
Friction factor, f = 64/ Re D Nu and f for Fully-Developed Laminar Flow in Tubes of Various Cross-Sections
9
Turbulent Flow ( Re D > 2300): For smooth tubes and ducts, the Dittus-Boelter equation: Nu Dh 0.023 Re Dh Pr with n = 0.4 for heating of fluid, and n = 0.3 for cooling of fluid 45
n
2
Friction factor for smooth tubes: f 0.790 ln Re D 1.64
Friction factor for rough tubes of roughness e : f 1.325 lne / 3.7 D 5.74 / Re D0.9
2
Reynolds-Colburn Analogy
St x . Pr 2 3 C f , x / 2 ;
For flow over a flat plate:
23
St . Pr
For flow in a tube or duct:
St L . Pr 2 3 C f , L / 2
f / 8
FREE CONVECTION
Generally,
Nu L C Gr L Pr C Ra L m
m
with m 1 4 for laminar flow, and m 1 3 for turbulent flow.
Laminar Free Convection on an Isothermal Vertical Plate:
Boundary layer momentum equation:
u u 2u u v g T T 2 x y y
Integral Momentum Equation for Free Convection BL:
u d 2 0 u dy y s dx
Boundary layer thickness,
g T T dy 0
3.93 x Pr 1 2 0.952 Pr Gr x 14
Critical Ra = 109 . Free Convection from an Isothermal Sphere Nu D
h D k
2 0.43Gr D Pr 1 4 for 1 Gr D 105 /
10
1 4
Free Convection from Isothermal Planes and Cylinders
Nu L C Gr L Pr C Ra L m
Geometry
m
Gr L Pr
Vertical plane and cylinder
C
104 – 109
0.59
1/4
10 – 10
0.10
1/3
0.68
0.058
10- – 10
1.02
0.148
102 – 104
0.85
0.188
104 – 109
0.53
1/4
109 – 1012
0.13
1/3
104 – 107
0.54
1/4
107 – 1011
0.15
1/3
105 – 1011
0.27
1/4
10-10 – 10-2 Horizontal cylinder
Hot surface facing up or cold surface facing down Hot surface facing down or cold surface facing up
m
Characteristic Length
Height
Diameter
Area/Perimeter Area/Perimeter
Free Convection from a Vertical Plate with Constant Surface Heat Flux Laminar :
Nu x
h x x k
0.60Gr x*. Pr
Turbulent : Nu x 0.17Gr x* Pr
where
1
1
5
for 2 1013 Gr x* Pr 1016
4
Gr x* Gr x .Nu x
for 105 Gr x* Pr 1011
g qs x 4 k ν 2
11
RADIATION HEAT TRANSFER
Solid angle: An / r 2 ,
d sin d d where 5.67 108 Wm-2K -4
Radiation: q" rad hr T s T sur
2 T s T sur hr T s2 T sur
Spectral directional Intensity:
Diffuse emitter:
Blackbody:
E b (T ) T
4
Spectral black body emissive power
E ,b( ,T )
C 1 exp( C 2 / T ) 1 5
where C 1 3.742 108 W. m4 /m2 and C 2 1.439 104 m.K
Wein’s displacement law:
maxT 2898 m.K
Emissivity of real surfaces: E (T ) (T ) E b (T ) T 4 Absorptivity of surface: Gabs G
Semitransparent medium: 1
12
(W/m 2 . m )
Black Body Radiation Functions
13
View factors:
F 2 ,31
A2 F 2 1 A3 F 31 A2 A3
Radiation exchange between black-body surfaces:
Radiation network approach: q
E b J
1 / A
q12
J 1 J 2 1 / A1 F 12
where
1 / A
surface resistance
whe re 1 / A1 F 12 spatial resistance
Radiation Exchange Network for a Two-Surface Enclosure
q12
T 14 T 24 1 1 1 1 2 1 A1
A1 F 1 ,2
2 A2 14
View factor for aligned parallel rectangles
View factor for coaxial parallel disks
15
View factor for perpendicular rectangles with common edge
HEAT EXCHANGERS
Log Mean Temperature Difference, T lm
q
T A T B R R 1 1 fi Rw fo hi Ai Ai Ao ho Ao
Effectiven ess,
T 2 T 1 lnT 2 / T 1 where R fi and R fi are fouling factors.
Actual heat transfer rate, q Max possible heat transfer rate, qmax
q C min T hi T ci
ΔT (minimum fluid) Max temper ature difference in heat exchanger
Capacity Rate Ratio, C r
m cmin C min m cmax C max
UA / C min NTU (Number of Transfer Units)
c p , is infinite for a condensing or boiling fluid. Capacity rate, C m
16
Correction Factor Charts
q UAF ΔT lm
Correction Factor for Heat Exchanger with One Shell Pass and Two (or Multiples of Two) Tube Passes.
Correction Factor for Heat Exchanger with Two Shell Passes and Four (or Multiples of Four ) Tube Passes.
17
Correction Factor for Single Pass Cross-Flow Heat Exchangers with the Shell Side Fluid Mixed , and the Other Fluid Unmixed .
Correction Factor for a Single Pass Cross-Flow Heat Exchanger with Both Fluids Unmixed .
18
-NTU Charts for Heat Exchangers
Effectiveness of parallel flow heat exchangers
Effectiveness of counterflow heat exchangers
Effectiveness of Heat Exchangers with Two Shell Passes and Four (or Multiples of Four ) Tube Passes.
Effectiveness of Heat Exchangers with One Shell Pass and Two (or Multiples of Two) Tube Passes.
19
Effectiveness of Single-Pass Cross-Flow Heat Exchangers with Both Fluids Unmixed .
Effectiveness of Single-Pass Cross-Flow Heat Exchangers with One Fluid Mixed , and the Other Unmixed .
20
Heat Exchanger Effectiveness Relations
Heat Exchanger NTU Relations
Use the above two equations with
21
