HEAT TRANSFER
There are three modes of heat transfer: conduction,
convection, and radiation.
Conduction Through a Cylindrical Wall
Conduction
Fourier’s Law of Conduction
Qo =- kA dT , where
dx
oQ = rate of heat transfer (W)
k = the thermal conductivity [W/(m•K)]
A = the surface area perpendicular to direction of heat transfer (m2)
T2
r2
Cylinder (Length = L)
2rkL ^T1 - T2h
Qo =
r
ln d r2 n
1
Qo = hA _Tw - T3i, where
h = the convection heat transfer coefficient of the fluid [W/(m2•K)]
A = the convection surface area (m2)
Tw = the wall surface temperature (K)
T∞ = the bulk fluid temperature (K)
Radiation
The radiation emitted by a body is given by
Qo = fvAT 4, where
ε = the emissivity of the body
σ = the Stefan-Boltzmann constant
= 5.67 × 10-8 W/(m2•K4)
A = the body surface area (m2)
T = the absolute temperature (K)
Critical Insulation Radius
k
rcr = insulation
h3
Thermal Resistance (R)
k
Qo = DT
Rtotal
Fluid 1
T∞1 T1
h1
Q
- kA ^T2 - T1h
Qo =
, where
L
kA
Fluid 2
T∞2
h2
kB
T2
Q
= wall surface area normal to heat flow (m2)
= wall thickness (m)
= temperature of one surface of the wall (K)
= temperature of the other surface of the wall (K)
1
h1 A
Q
HEAT TRANSFER
k insulation
Composite Plane Wall
T2
84
r insulation
Plane Wall Conduction Resistance (K/W): R = L , where
kA
L = wall thickness
r
ln d r2 n
1
Cylindrical Wall Conduction Resistance (K/W): R =
,
2rkL
where
L = cylinder length
Convection Resistance (K/W) : R = 1
hA
Conduction Through a Plane Wall
A
L
T1
T2
h∞
Resistances in series are added: Rtotal = RR, where
CONDUCTION
L
r1
k
Convection
Newton’s Law of Cooling
T1
Q
T1
BASIC HEAT TRANSFER RATE EQUATIONS
T∞1
T1
LA
LB
LA
kA A
LB
kB A
T2
T3
1
h2 A
T3
T∞2
To evaluate Surface or Intermediate Temperatures:
T2 - T3
T - T2
Qo = 1
RA = RB
Transient Conduction Using the Lumped Capacitance
Method
The lumped capacitance method is valid if
Steady Conduction with Internal Energy Generation
The equation for one-dimensional steady conduction is
o
d 2T + Qgen = 0, where
2
k
dx
Qo gen = the heat generation rate per unit volume (W/m3)
For a Plane Wall
Biot number, Bi = hV % 1, where
kAs
h = the convection heat transfer coefficient of the fluid
[W/(m2•K)]
V = the volume of the body (m3)
k = thermal conductivity of the body [W/(m•K)]
As = the surface area of the body (m2)
Fluid
h, T∞
Body
T(x)
Ts1
Ts2
k
Q gen
Q"1
Q"2
x
−L
T ^ xh =
As
0
L
2
Qo gen L2
T
T
T
T
d1 - x 2 n + c s2 - s1 m b x l + c s1 - s2 m
L
2
2
2k
L
Qo 1" + Qo 2" = 2Qo gen L, where
Qo " = the rate of heat transfer per area (heat flux) (W/m2)
Qo 1" = k b dT l and Qo 2" = k b dT l
dx - L
dx L
For a Long Circular Cylinder
Constant Fluid Temperature
If the temperature may be considered uniform within the body
at any time, the heat transfer rate at the body surface is given
by
Qo = hAs ^T - T3h =- tV ^cP h b dT l, where
dt
T
T∞
ρ
cP
t
= the body temperature (K)
= the fluid temperature (K)
= the density of the body (kg/m3)
= the heat capacity of the body [J/(kg•K)]
= time (s)
The temperature variation of the body with time is
T - T3 = _Ti - T3i e-bt, where
hAs
b=
tVcP
Ts
ρ, V, c P, T
where b = 1x and
x = time constant ^ sh
The total heat transferred (Qtotal) up to time t is
Qtotal = tVcP _Ti - T i, where
Q gen
Ti = initial body temperature (K)
r0
Q'
o
1 d b r dT l + Qgen = 0
r dr dr
k
T ]r g =
2
Qo genr02
1 - r 2 p + Ts
f
4k
r0
Qo l = rr02Qo gen, where
Qo l = the heat transfer rate from the cylinder per unit length of
the cylinder (W/m)
HEAT TRANSFER
85
Variable Fluid Temperature
If the ambient fluid temperature varies periodically according
to the equation
T3 = T3, mean + 1 _T3, max - T3, min i cos ^~t h
2
The temperature of the body, after initial transients have died
away, is
b ; 1 _T3, max - T3, min iE
T= 2
cos =~t - tan- 1 c ~ mG + T3, mean
b
~2 + b2
Fins
For a straight fin with uniform cross section
(assuming negligible heat transfer from tip),
Qo = hPkAc _Tb - T3i tanh _ mLci, where
h = the convection heat transfer coefficient of the fluid [W/(m2•K)]
P = perimeter of exposed fin cross section (m)
k = fin thermal conductivity [W/(m•K)]
Ac = fin cross-sectional area (m2)
Tb = temperature at base of fin (K)
T∞ = fluid temperature (K)
hP
kAc
m=
Terms
D = diameter (m)
h = average convection heat transfer coefficient of the fluid
[W/(m2•K)]
L = length (m)
Nu = average Nusselt number
c n
Pr = Prandtl number = P
k
um = mean velocity of fluid (m/s)
u∞ = free stream velocity of fluid (m/s)
µ = dynamic viscosity of fluid [kg/(s•m)]
ρ = density of fluid (kg/m3)
External Flow
In all cases, evaluate fluid properties at average temperature
between that of the body and that of the flowing fluid.
Flat Plate of Length L in Parallel Flow
ReL =
tu3 L
n
NuL = hL = 0.6640 Re1L 2 Pr1
k
3
NuL = hL = 0.0366 Re0L.8 Pr1
k
3
_ReL < 105i
_ReL > 105i
Cylinder of Diameter D in Cross Flow
A
Lc = L + c , corrected length of fin (m)
P
ReD =
Rectangular Fin
tu3 D
n
NuD = hD = C Re nD Pr1 3, where
k
T∞ , h
P = 2w + 2t
Ac = w t
t
Tb
CONVECTION
w
L
C
0.989
0.911
0.683
0.193
0.0266
ReD
1–4
4 – 40
40 – 4,000
4,000 – 40,000
40,000 – 250,000
n
0.330
0.385
0.466
0.618
0.805
Flow Over a Sphere of Diameter, D
Pin Fin
T∞ , h
P= π D
D
Tb
86
HEAT TRANSFER
L
Ac =
πD 2
4
NuD = hD = 2.0 + 0.60 Re1D 2 Pr1 3,
k
^1 < ReD< 70, 000; 0.6 < Pr < 400h
Internal Flow
ReD =
tumD
n
Laminar Flow in Circular Tubes
For laminar flow (ReD < 2300), fully developed conditions
NuD = 4.36
(uniform heat flux)
NuD = 3.66
(constant surface temperature)
For laminar flow (ReD < 2300), combined entry length with
constant surface temperature
NuD = 1.86 f
1 3
ReDPr
L p
D
d
nb
n
ns
0.25
tl2 gh fg D3
H
NuD = hD = 0.729 >
k
nlkl _Tsat - Tsi
0.14
, where
L = length of tube (m)
D = tube diameter (m)
µb = dynamic viscosity of fluid [kg/(s•m)] at
bulk temperature of fluid, Tb
µs = dynamic viscosity of fluid [kg/(s•m)] at
inside surface temperature of the tube, Ts
Turbulent Flow in Circular Tubes
For turbulent flow (ReD > 104, Pr > 0.7) for either uniform
surface temperature or uniform heat flux condition, SiederTate equation offers good approximation:
n
NuD = 0.027 Re0D.8 Pr1 3 d nb n
s
0.14
Non-Circular Ducts
In place of the diameter, D, use the equivalent (hydraulic)
diameter (DH) defined as
DH = 4 # cross -sectional area
wetted perimeter
Circular Annulus (Do > Di)
In place of the diameter, D, use the equivalent (hydraulic)
diameter (DH) defined as
DH = Do - Di
Liquid Metals (0.003 < Pr < 0.05)
NuD = 6.3 + 0.0167 Re0D.85 Pr0.93 (uniform heat flux)
NuD = 7.0 + 0.025 Re0D.8 Pr0.8 (constant wall temperature)
Condensation of a Pure Vapor
On a Vertical Surface
0.25
ρl
g
hfg
L
µl
kl
Tsat
Ts
Outside Horizontal Tubes
tl2 gh fg L3
H
NuL = hL = 0.943 >
k
nlkl _Tsat - Tsi
, where
= density of liquid phase of fluid (kg/m3)
= gravitational acceleration (9.81 m/s2)
= latent heat of vaporization [J/kg]
= length of surface [m]
= dynamic viscosity of liquid phase of fluid [kg/(s•m)]
= thermal conductivity of liquid phase of fluid [W/(m•K)]
= saturation temperature of fluid [K]
= temperature of vertical surface [K]
Note: Evaluate all liquid properties at the average temperature
between the saturated temperature, Tsat, and the surface
temperature, Ts.
, where
D = tube outside diameter (m)
Note: Evaluate all liquid properties at the average temperature
between the saturated temperature, Tsat, and the surface
temperature, Ts.
Natural (Free) Convection
Vertical Flat Plate in Large Body of Stationary Fluid
Equation also can apply to vertical cylinder of sufficiently large
diameter in large body of stationary fluid.
hr = C b k l RaLn, where
L
L = the length of the plate (cylinder) in the vertical direction
gb _Ts - T3i L3
RaL = Rayleigh Number =
Pr
v2
Ts = surface temperature (K)
T∞ = fluid temperature (K)
β = coefficient of thermal expansion (1/K)
2
(For an ideal gas: b =
with T in absolute temperature)
Ts + T3
υ
= kinematic viscosity (m2/s)
Range of RaL
104 – 109
109 – 1013
C
0.59
0.10
n
1/4
1/3
Long Horizontal Cylinder in Large Body of Stationary Fluid
h = C b k l Ra nD, where
D
RaD =
gb _Ts - T3i D3
Pr
v2
RaD
10 – 102
102 – 104
104 – 107
107 – 1012
C
1.02
0.850
0.480
0.125
–3
n
0.148
0.188
0.250
0.333
Heat Exchangers
The rate of heat transfer in a heat exchanger is
Qo = UAFDTlm, where
A
= any convenient reference area (m2)
F
= heat exchanger configuration correction factor
(F = 1 if temperature change of one fluid is negligible)
U
= overall heat transfer coefficient based on area A and
the log mean temperature difference [W/(m2•K)]
∆Tlm = log mean temperature difference (K)
HEAT TRANSFER
87
Heat Exchangers (cont.)
Overall Heat Transfer Coefficient for Concentric Tube and
Shell-and-Tube Heat Exchangers
D
ln d o n
R
R fo
Di
fi
1
1
1 , where
UA = hiAi + Ai + 2rkL + Ao + hoAo
2
Ai = inside area of tubes (m )
Ao = outside area of tubes (m2)
Di = inside diameter of tubes (m)
Do = outside diameter of tubes (m)
hi = convection heat transfer coefficient for inside of tubes
[W/(m2•K)]
ho = convection heat transfer coefficient for outside of tubes
[W/(m2•K)]
k = thermal conductivity of tube material [W/(m•K)]
Rfi = fouling factor for inside of tube [(m2•K)/W]
Rfo = fouling factor for outside of tube [(m2•K)/W]
Log Mean Temperature Difference (LMTD)
For counterflow in tubular heat exchangers
DTlm =
_THo - TCi i - _THi - TCoi
ln d
THo - TCi
n
THi - TCo
For parallel flow in tubular heat exchangers
DTlm =
∆Tlm
THi
THo
TCi
TCo
_THo - TCoi - _THi - TCi i
T - TCo
n
ln d Ho
THi - TCi
, where
= log mean temperature difference (K)
= inlet temperature of the hot fluid (K)
= outlet temperature of the hot fluid (K)
= inlet temperature of the cold fluid (K)
= outlet temperature of the cold fluid (K)
Heat Exchanger Effectiveness, ε
Qo
actual heat transfer rate
f= o
=
maximum possible heat transfer rate
Qmax
f=
C _T - TCi i
CH _THi - THoi
or f = C Co
Cmin _THi - TCi i
Cmin _THi - TCi i
where
C = mc
o P = heat capacity rate (W/K)
Cmin = smaller of CC or CH
Number of Transfer Units (NTU)
NTU = UA
Cmin
88
HEAT TRANSFER
Effectiveness-NTU Relations
C
Cr = min = heat capacity ratio
Cmax
For parallel flow concentric tube heat exchanger
f=
1 - exp 8- NTU ^1 + Cr hB
1 + Cr
NTU =- ln 81 - f ^1 + Cr hB
1 + Cr
For counterflow concentric tube heat exchanger
f=
1 - exp 8- NTU ^1 - Cr hB
1 - Crexp 8- NTU ^1 - Cr hB
f = NTU
1 + NTU
NTU = 1 ln c f - 1 m
Cr - 1 fCr - 1
NTU = f
1-f
^Cr< 1h
^Cr = 1h
^Cr< 1h
^Cr = 1h
RADIATION
Types of Bodies
Any Body
For any body, α + ρ + τ = 1 , where
α = absorptivity (ratio of energy absorbed to incident energy)
ρ = reflectivity (ratio of energy reflected to incident energy)
τ = transmissivity (ratio of energy transmitted to incident energy)
Opaque Body
For an opaque body: α + ρ = 1
Gray Body
A gray body is one for which
α = ε, (0 < α < 1; 0 < ε < 1), where
ε = the emissivity of the body
For a gray body: ε + ρ = 1
Real bodies are frequently approximated as gray bodies.
Black body
A black body is defined as one which absorbs all energy
incident upon it. It also emits radiation at the maximum rate
for a body of a particular size at a particular temperature. For
such a body
α=ε=1
Shape Factor (View Factor, Configuration Factor)
Relations
Reciprocity Relations
One-Dimensional Geometry with Thin Low-Emissivity Shield
Inserted between Two Parallel Plates
Radiation Shield
Q12
AiFij = AjFji, where
Ai = surface area (m2) of surface i
Fij = shape factor (view factor, configuration factor); fraction
of the radiation leaving surface i that is intercepted by
surface j; 0 ≤ Fij ≤ 1
ε3, 1
Summation Rule for N Surfaces
A1 , T1,
ε1
N
! Fij = 1
j=1
Qo 12 =
Qo 12 = the net heat transfer rate from the body (W)
ε
= the emissivity of the body
σ = the Stefan-Boltzmann constant
[σ = 5.67 × 10-8 W/(m2•K4)]
A = the body surface area (m2)
T1 = the absolute temperature [K] of the body surface
T2 = the absolute temperature [K] of the surroundings
A1 , T1 , ε1
Qo 12 = A1F12 v `T14 - T24j
Generalized Cases
A1 , T1 , ε1
v `T14 - T24j
1 - f3, 1 1 - f3, 2
1 - f1
1 - f2
1
1
f1A1 + A1F13 + f3, 1A3 + f3, 2A3 + A3F32 + f2A2
Reradiating Surface
Reradiating Surfaces are considered to be insulated or
adiabatic _Qo R = 0i .
Net Energy Exchange by Radiation between Two Black
Bodies
The net energy exchange by radiation between two black
bodies that see each other is given by
Net Energy Exchange by Radiation between Two DiffuseGray Surfaces that Form an Enclosure
A2 , T2 ,
ε2
A3 , T3
Net Energy Exchange by Radiation between Two Bodies
Body Small Compared to its Surroundings
Qo 12 = fvA `T14 - T24j, where
ε3, 2
Q12
AR , TR , εR
A2 , T2 , ε2
Qo 12 =
v `T14 - T24j
1 - f1
1 - f2
1
-1 + f A
f1A1 +
2 2
A1F12 + =c 1 m + c 1 mG
A1F1R
A2F2R
A2 , T2 , ε2
Q12
Q12
A1 , T1 , ε1
A2 , T2 , ε2
Qo 12 =
v `T14 - T24j
1 - f1
1 - f2
1
f1A1 + A1F12 + f2A2
HEAT TRANSFER
89