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APPLIED INDUSTRIAL ENERGY AND ENVIRONMENTAL MANAGEMENT
Z. K. Morvay, D. D. Gvozdenac
Part III:
FUNDAMENTALS FOR ANALYSIS AND CALCULATION OF ENERGY AND
ENVIRONMENTAL PERFORMANCE
1
Applied Industrial Energy and Environmental Management Zoran K. Morvay and Dusan D. Gvozdenac © John Wiley & Sons, Ltd
Toolbox 9
HEAT TRANSFER IN A THERMAL APPARATUS
DIMENSIONLESS NUMBERS
1. Referent temperatures for calculations. It is very important for proper calculation that
temperature and other fluid properties, especially pressure, are defined precisely. The temperature
definitions relevant to calculations are as follow:
t1 = temperature related to fluid at the inlet conditions, [oC]
t2 = temperature related to fluid at the outlet conditions, [oC]
tw = temperature of the wall, [oC]
tc (t∞) = temperature of the fluid flow core, [oC]
tf = temperature of the fluid calculated as: 2
ttt 21
f , [oC]
tm = temperature of the fluid calculated as: 2
ttt5.0t 21
wm , [oC]
2. The Archimedes Number (Ar) states the ratio of gravitational force and viscous force:
fs2
3s Lg
Ar (9.1)
Where:
g = gravity acceleration, [m/s2]
L = characteristic length, [m]
= dynamic viscosity, [Pa s]
f = fluid density, [kg/m3]
s = solid density, [kg/m3]
3. The Biot Number (Bi) states the ratio of internal thermal resistance and surface film resistance:
k
xhBi (9.2)
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HEAT TRANSFER IN A THERMAL APPARATUS 2
Where:
x = mid-plane distance, [m]
h = heat transfer coefficient, [W/(m2 K)]
k = fluid thermal conductivity, [W/(m K)]
4. The Colburn-Chilton Factor (jh) is used in heat transfer in general. It is equivalent to (St Pr2/3
):
3/2
p
p
h
3/2
p
p
hk
c
mc
hjor
k
c
wc
hj (9.3)
Where:
h = heat transfer coefficient, [W/(m2 K)]
cp = isobaric specific heat, [kJ/(kg K)]
= fluid density, [kg/m3]
w = fluid velocity, [m/s]
k = fluid thermal conductivity, [W/(m K)]
= dynamic viscosity, [Pa s]
m = fluid mass velocity, [kg/(m2 s)]
5. The Euler Number (Eu) states the ratio of friction head and velocity head:
2w
pEu (9.4)
Where:
p = pressure drop, [Pa]
= fluid density, [kg/m3]
w = velocity, [m/s]
6. The Fanning Friction Factor (f) states the ratio of sheer stress at pipe and conduit wall as a
number of velocity heads:
L
p
w2
dfor
Lw2
22 (9.5)
Where:
d = characteristic length, [m]
p = pressure drop, [Pa]
= fluid density, [kg/m3]
w = velocity, [m/s]
L = length, [m]
p/ L = pressure drop per unit length, [Pa/m]
7. The Fourier Number (Fo) is used in unsteady heat transfer calculations. It characterizes the speed
of temperature field changes during unsteady heat transfer:
22p L
FoorLc
kFo (9.6)
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HEAT TRANSFER IN A THERMAL APPARATUS 3
Where:
k = fluid thermal conductivity, [W/(m K)]
= thermal diffusivity, [m2/s]
= time, [s]
= fluid density, [kg/m3]
cp = isobaric specific heat, [kJ/(kg K)]
L = characteristic length, [m]
8. The Froude Number (Fr) states the ratio of inertial force and gravity force:
Lg
wFror
La
wFr
22
(9.7)
Where:
a = acceleration, [m/s2]
g = gravitational acceleration, [m/s2]
L = characteristic length, [m]
w = velocity, [m/s]
9. The Galileo Number (Ga) states the ratio of gravity force and viscous force:
2
23dgGa (9.8)
Where:
g = gravitational acceleration, [m/s2]
d = characteristic length, [m]
= dynamic viscosity, [Pa s]
= fluid density, [kg/m3]
10. The Grätz Number (Gz) states the ratio of thermal capacity and convective heat transfer:
p
p
cwd
k
d
LGzor
Lk
cMGz (9.9)
Where:
M = mass flow rate, [kg/s]
d = diameter, [m]
w = fluid velocity, [m/s]
= fluid density, [kg/m3]
cp = isobaric specific heat, [kJ/(kg K)]
k = fluid thermal conductivity, [W/(m K)]
L = characteristic length, [m]
11. The Grashof Number (Gr) states the ratio of buoyancy force and viscous force:
2
3
2
23 TgLGror
TgLGr (9.10)
Where:
L = characteristic length, [m]
Part III – Toolbox 9:
HEAT TRANSFER IN A THERMAL APPARATUS 4
= fluid density, [kg/m3]
g = gravitational acceleration, [m/s2]
= coefficient of volume expansion of fluid, [1/K]
T = temperature difference, [K]
= dynamic viscosity, [Pa s]
= kinetic viscosity, [m2/s]
12. The Lewis Number (Le) is used in combined heat and mass transfer calculations:
DLeor
cD
kLe
p
(9.11)
Where:
k = fluid thermal conductivity, [W/(m K)]
= fluid density, [kg/m3]
cp = isobaric specific heat, [kJ/(kg K)]
k = fluid thermal conductivity, [W/(m K)]
= thermal diffusivity, [m2/s]
D = diffusivity, [m2/s]
13. The Mass Transfer Factor (jm) is used in mass transfer calculations:
3/2
cm
Dw
kj (9.12)
Where:
D = diffusivity, [m2/s]
kc = diffusion rate, [m/s]
w = fluid velocity, [m/s]
= fluid density, [kg/m3]
= dynamic viscosity, [Pa s]
14. The Nusselt Number (Nu) states the ratio of total heat transfer and conductive heat transfer:
k
dhNu (9.13)
Where:
d = characteristic length, [m]
h = heat transfer coefficient, [W/(m2 K)]
k = fluid thermal conductivity, [W/(m K)]
15. The Peclet Number (Pe) states the ratio of bulk heat transfer and conductive heat transfer:
a
wdPeor
k
cmdPeor
k
cwdPe
pp (9.14)
Where:
d = characteristic length, [m]
w = fluid velocity, [m/s]
Part III – Toolbox 9:
HEAT TRANSFER IN A THERMAL APPARATUS 5
= fluid density, [kg/m3]
cp = isobaric specific heat, [kJ/(kg K)]
k = fluid thermal conductivity, [W/(m K)]
m = fluid mass velocity, [kg/(m2 s)]
a = thermal diffusivity, [m2/s]
16. The Prandtl Number (Pr) states the ratio of momentum diffusivity and thermal diffusivity:
k
cPr
p (9.15)
Where:
cp = isobaric specific heat, [kJ/(kg K)]
k = fluid thermal conductivity, [W/(m K)]
= dynamic viscosity, [Pa s]
17. The Reynolds Number (Re) states the ratio of inertial force and viscous force:
mdReor
wdRe (9.16)
Where:
d = characteristic length, [m]
w = fluid velocity, [m/s]
= dynamic viscosity, [Pa s]
= fluid density, [kg/m3]
m = fluid mass velocity, [kg/(m2 s)]
18. The Schmidt Number (Sc) states the ratio of kinetic viscosity and molecular diffusivity:
DSc (9.17)
Where:
= kinetic viscosity, [m2/s]
D = diffusivity, [m2/s]
19. The Sherwood Number (Sh) states the ratio of mass diffusivity and molecular diffusivity:
D
LkSh c (9.18)
Where:
L = characteristic length, [m]
kc = diffusion rate, [m/s]
D = diffusivity, [m2/s]
Part III – Toolbox 9:
HEAT TRANSFER IN A THERMAL APPARATUS 6
20. The Stanton Number (St) states the ratio of heat transfer and thermal capacity of fluid:
mc
hStor
wc
hSt
pp
(9.19)
Where:
h = heat transfer coefficient, [W/(m2 K)]
w = fluid velocity, [m/s]
cp = isobaric specific heat, [kJ/(kg K)]
= fluid density, [kg/m3]
m = fluid mass velocity, [kg/(m2 s)]
21. The Weber Number (We) states the ratio of inertial force and surface tension force:
22 md
Weorwd
We (9.20)
Where:
d = characteristic length, [m]
w = fluid velocity, [m/s]
= fluid density, [kg/m3]
m = fluid mass velocity, [kg/(m2 s)]
= surface tension, [N/m]
CORRELATIONS FOR CONVECTIVE HEAT TRANSFER
1. Forced Convection Heat Transfer. Forced convection implies a process in which heat is
transferred from a flowing fluid to a solid surface which represents one of its boundaries in space. A
device producing pressure difference causes the flow. In addition to geometrical factors, thermo
physical properties and the ratio between fluid temperature and solid surface, the speed of undisturbed
fluid core can be considered as a condition for producing uniform solutions of general differential
equations for energy conservation and momentum equations.
2. Forced Convection Flow over a Flat Plate
Laminar flow (Ref < 5×105 and 0.6 < Prf < 10). The geometrical dimension is the length of
the flat plate in flow direction. Wall temperature is constant.
n
w
f33.0f
5.0ff
Pr
PrPrRe664.0Nu (9.21)
n = 0.25 for fluid heating and n = 0.19 for fluid cooling.
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HEAT TRANSFER IN A THERMAL APPARATUS 7
Laminar flow (Ref < 5×105 and 0.6 < Prf < 10). The geometrical dimension is the length of the
flat plate in flow direction. Heat flow is constant.
n
w
f33.0f
5.0ff
Pr
PrPrRe906.0Nu (9.22)
n = 0.25 for fluid heating and n = 0.19 for fluid cooling.
Fully developed turbulent flow (Ref > 5×105 and 0.7 < Prf < 200). The geometrical dimension
is the length of the flat plate in flow direction.
Liquids: 25.0
w
c43.0c
8.0cc
Pr
PrPrRe037.0Nu (9.23)
Gases: 78.0
mmm PrRe057.0Nu (9.24)
3. Forced Convection Flow inside Circular Tubes and Ducts
Laminar viscous and gravitational flow within straight smooth pipes (Ref < 2300)
n
w
f1.0f
43.0f
33.0flf
Pr
PrGrPrRe15.0Nu (9.25)
n = 0.25 for fluid heating and n = 0.19 for fluid cooling.
Table 9.1: φl for Laminar Flow
l/d 1 2 5 10 15 20 30 40 50
φl 1.98 1.70 1.44 1.28 1.18 1.13 1.05 1.02 1.00
Transient flow within straight smooth pipes (2300 < Ref < 10,000)
25.0
w
f43.0fof
Pr
PrPrKNu (9.26)
Table 9.2: Ko for Transient Flow
Re 2100 2300 2500 3000 3500 4000 5000 6000 7000 8000 9000 10 000
Ko 1.9 3.3 4.4 6.0 10.0 12.2 15.5 19.5 24.0 27.0 30.0 33.0
Turbulent flow in straight smooth pipe or duct (10,000 < Ref < 5,000,000 and 0.6 < Prf <
2,500)
25.0
w
f43.0f
8.0flf
Pr
PrPrRe021.0Nu (9.27)
Part III – Toolbox 9:
HEAT TRANSFER IN A THERMAL APPARATUS 8
Table 9.3: φl for Turbulent Flow
Ref l/d
1 2 5 10 15 20 30 40 50
10 000 1.65 1.50 1.34 1.23 1.17 1.13 1.07 1.03 1.00
20 000 1.51 1.40 1.27 1.18 1.13 1.10 1.05 1.02 1.00
50 000 1.34 1.27 1.18 1.13 1.10 1.08 1.04 1.02 1.00
100 000 1.28 1.22 1.15 1.10 1.08 1.06 1.03 1.02 1.00
1 000 000 1.14 1.11 1.08 1.05 1.04 1.03 1.02 1.01 1.00
Defining the geometrical dimension is diameter (d) for pipes and equivalent diameter (P
A4deq ; A
= flow cross sectional area and P = flow section perimeter regardless of what part of the perimeter
participates in heat transfer).
For bent tubes, the value of the heat transfer coefficient (h) has to be multiplied by the coefficient
ζ taking into account the relative curvature of the coil:
D
d54.31 (9.28)
where d is diameter of a tube and D is diameter of a coil turn.
4. Forced Convection Flow inside Concentric Annular Ducts, Turbulent Flow (Re > 2300)
Di
Do
Figure 9.1:Concentric Annular Pipe
ioh DDD
mh wDRe
k
DhNu h
Fluid properties have to be calculated for mean bulk temperature.
(a) Heat transfer occurs at the inner wall. The outlet wall is insulated.
16.0
i
o
TUBE D
D86.0
Nu
Nu (9.29)
(b) Heat transfer occurs at the outlet wall. The inner wall is insulated.
6.0
o
i
TUBE D
D14.01
Nu
Nu (9.30)
(c) Both walls participate in heat transfer and the wall temperatures are the same.
Part III – Toolbox 9:
HEAT TRANSFER IN A THERMAL APPARATUS 9
o
i
6.0
o
i
84.0
i
o
TUBE
D
D1
D
D14.01
D
D86.0
Nu
Nu (9.31)
The characteristic length for non-circular ducts is defined as follows (if not specified otherwise):
P
A4d c
e (9.32)
Where:
P = wetted perimeter of the duct (length of boundary), [m]
Ac = flow cross-sectional area, [m2]
5. Forced Convection Flow Perpendicular to Banks of Smooth Tubes
Re < 1,000
D
sL
sW
um
Figure 9.2: Square Pitch
D
sL
sW
um
Figure 9.3: Triangular Pitch
Ref < 1000 25.0
w
f36.0f
5.0ff
Pr
PrPrRe56.0Nu
Ref > 1000 25.0
w
f33.0f
65.0ff
Pr
PrPrRe26.0Nu
Ref > 1000 25.0
w
f33.0f
6.0ff
Pr
PrPrRe41.0Nu
The flow velocity is defined for the narrowest bank section.
Table 9.4: Coefficient φ (Angle of Attack)
Angle 90 80 70 60 50 40 30 20 10
φ 1 1 0.98 0.94 0.88 0.78 0.67 0.52 0.42
Part III – Toolbox 9:
HEAT TRANSFER IN A THERMAL APPARATUS 10
Figure 9.4: Shell & Tube Heat Exchanger
When equations from this paragraph are used for
shell & tube heat exchangers with baffles (Fig.
9.4) the coefficient φ is 0.6.
6. Heat Transfer in Flow around Bank of Transverse Finned Pipes (Fig. 9.5)
)8.4t/d3;000,25Re500,2(PrRet
h
t
dCNu f
4.0f
nf
14.054.0
f (9.33)
Where:
d = external diameter of the tube, [m]
D = diameter of the fin, [m]
t = pitch of the fin, [m]
h = height of the fin, [m]
Square pitch banks: C = 0.116; n = 0.72
Triangular pitch banks: C = 0.25; n = 0.65
This formula gives the heat transfer coefficient (h) which has to be multiplied by 0.73 to get the
so-called reduced heat transfer coefficient (hred). This reduced heat transfer coefficient is used in the
equation for overall heat transfer coefficient which is related to the total external heat transfer area.
The overall heat transfer coefficient is as follows:
w
in
ex
inred
RA
A
h
1
h
1
1K
(9.34)
Where:
Aex = total external surface and finned tube per unit length
Ain = internal tube surface
h2 = heat transfer coefficient of the fluid inside the tube, W/(m2 K)
wr = sum of thermal resistance of the wall and existing scaling
D
d t
h
δ
Figure 9.5: Transverse Finned Tube
7. Heat Transfer in Stirring Liquids with Agitator. The characteristic correlation for determination
of the heat transfer coefficient in an apparatus with a coil (Fig. 6) or jacket (Fig. 7) and an agitator can
be calculated by the equation:
Part III – Toolbox 9:
HEAT TRANSFER IN A THERMAL APPARATUS 11
1
k
w
mnm
mmm SPrReCNu (9.35)
Where
d
DS;
dnRe;
k
DhNu
m
2m
mm
m
D = inside diameter of vessel, m
d = diameter of the circle covered by the agitator, m
n = speed of the agitator, rpm
The values of the remaining physical properties have to be taken at the average liquid temperature
in the vessel.
Figure 9.6: Coil Type
D
d
Figure 9.7: Jacket Type
The mean values of exponents for turbulent flow are m = 0.67, n = 0.33, k = 0.14. For high
viscosity fluids (non-Newtonian) these values are: m = 0.50, n = 0.24 - 0.33, k = 0.14 – 0.24.
The constant C depends mostly on the type of agitator and its geometrical characteristics.
a. Coil Type of Vessel:
C = 0.87 – 1.0
b. Jacket Type of Vessel:
C = 0.73 – 0.74 (Ref > 400)
(turbine agitator with side bumper)
C = 0.54 – 0.64
(propeller agitator without side bumper)
C = 0.38 – 0.55
(frame agitator)
Part III – Toolbox 9:
HEAT TRANSFER IN A THERMAL APPARATUS 12
C = 0.36 - 0.40
(agitator with flat paddle)
C = 0.40 - 0.53
(turbine closed type agitator with angled blades)
8. Natural Convection. The flowing field in natural convection is determined exclusively by
temperature field. Fluid motion occurs as a consequence of the different densities of certain
macroscopic fractions arising from the difference in their temperatures. Heat transfer correlations do
not contain velocity in this case. As a condition for the same solution of general differential equations
including boundary conditions on a solid surface, there is also the shape and a size of body, the
distribution of temperature along its surface, a temperature difference between a surface and
undisturbed fluid, the thermal and physical properties of fluid and gravitational acceleration.
In the majority of cases, the thermal and physical properties of fluid refer to the middle fluid
temperature calculated as 2
ttt fw
m , [oC]. When defining the Nusselt number, the coefficient of
heat transfer (h) is considered to be the middle one for the total heat transfer surface area.
Heat transfer outside horizontal tubes (cylinders). The characteristic length is the tube’s
(cylinder) outside diameter. The equations are valid for diameters of less than 200 mm and for
gases and liquids.
25.0
mmm PrGr53.0Nu 9m 10Pr)Gr(
(9.36) 3/1
mmm PrGr10.0Nu 9m 10Pr)Gr(
Vertical flat plate and cylindrical surfaces. The characteristic length is the height of the
plate or cylinder which is valid for gases and liquids.
45.0Num 3m 10Pr)Gr(
(9.37)
8/1mmm PrGr18.1Nu 500Pr)Gr(10 m
3
25.0mmm PrGr54.0Nu )10(Pr102Pr)Gr(500 m
7m
25.0mmm PrGr65.0Nu )10(Pr102Pr)Gr(500 m
7m
3/1mmm PrGr129.0Nu 7
m 102Pr)Gr(
The defining dimension is the diameter for the horizontal tube and the height for the vertical
surface.
Part III – Toolbox 9:
HEAT TRANSFER IN A THERMAL APPARATUS 13
Horizontal plate – heat transfer surface facing up
(a)tf
tw > tf
tw
(b)
tw
tf
tw < tf
nmmm PrGrCNu (9.38)
mPrGr C n
105 - 2·10
7 fw tt 0.54 0.25
2·107 - 3·10
10 fw tt 0.14 1/3
105 - 2·10
7 fw tt 0.25 0.25
If the plate is rectangular, the characteristic length (L) is the shorter side (if L < 0.6 m). If L > 0.6
m, increased plate dimensions do not influence the heat transfer coefficient.
Horizontal plate – heat transfer surface facing down
(a)tf
tw < tf
tw
(b)
tw
tf
tw > tf
mPrGr C m
105 - 2·107 fw tt 0.25 0.25
105 - 2·107 fw tt 0.54 0.25
2·107 - 3·1010 fw tt 0.14 1/3
If the plate is rectangular, the characteristic length (L) is the shorter side (if L < 0.6 m). If L > 0.6 m,
increased plate dimensions do not influence the heat transfer coefficient.
9. Heat Transfer in Thermal Radiation of Solids
Heat passing from a hotter body to a colder one by radiation is defined as follows:
42
4121radiation TTAWQ (9.39)
where:
Qradiation Heat energy transmitted by radiation, W
A Area of radiating surface, m2
W1-2 Power of radiation, W/(m2 K
4)
T1 Hotter body surface temperature, K
T2 Colder body surface temperature, K
φ Dimensionless angular coefficient
Part III – Toolbox 9:
HEAT TRANSFER IN A THERMAL APPARATUS 14
The power of radiation depends on (1) mutual arrangement and (2) emissivity of radiating
surfaces having temperatures T1 and T2.
If a body with radiating surface A1 is totally within a body with radiating surface A2 then
A = A1 and angular coefficient φ = 1 and the power of radiation is as follows:
bb12
1
1
21
W
1
W
1
A
A
W
1
1W
(9.40)
where:
W1 = ε1·Wbb Radiating power of smaller body
W2 = ε2·Wbb Radiating power of larger body
ε1 and ε2 Emissivities of the surface of the smaller and larger bodies,
respectively. The emissivity for some materials is given in the Table
below.
Wbb Radiating power of a black body, = 5.6687·10-8
W/(m2 K
4)
Material Emissivity
(ε)
Material Emissivity
(ε)
Material Emissivity
(ε)
Aluminum 0.05–0.07 Iron, galvanized 0.27 Varnish 0.80–0.98
Asbestos 0.96 Iron (steel), oxidize 0.74–0.96 Varnish, aluminum 0.4
Brick masonry 0.93 Iron (rough), oxidize 0.85 Water 0.93
Copper 0.30–0.87 Lead 0.28 Wood, dressed 0.90
Glass 0.94 Oil paint 0.78–0.96
Gypsum 0.78–0.90 Plaster 0.93
If area A2 is much bigger compared to area A1, the ratio A1/A2 in equation (9.39) above is
close to zero, and the radiation power becomes W1-2 = W1.
If A1 = A2, the radiation power is as follows:
bb11
21
W
1
W
1
W
1
1W
(9.41)
The summary coefficient of radiation and convection heat transfer is:
conrad hhh (9.42)
where
21
42
4121
21
radiationrad
TT
TTW
ATT
Qh (9.43)
and hcon (coefficient of convection heat transfer) is defined by the relevant equation.
The following approximate formula is used to calculate the heat losses of apparatus in closed
premises when the temperature of the apparatus surface is less than 150 oC:
t07.074.9h (9.44)
Part III – Toolbox 9:
HEAT TRANSFER IN A THERMAL APPARATUS 15
h Coefficient of heat transfer by radiation and convection, W/(m2 K)
Δt Temperature difference of the apparatus surface and surrounding air, K
References Dimic, M. (1973) Kriterijalne jednacine za konvektivni prelaz toplote i materije (Correlations for
Convective Heat and Mass Transfer), Novi Sad (in Serbian).
Nashchokin, V.V. (1979) Engineering Thermodynamics and Heat Transfer, Mir Publishers
Moscow.
Pavlov, K.F., Romankov, P.G., Noskov, A.A. (1976) Examples and Problems to the Course of Unit
Operations of Chemical Engineering, Mir Publishers, Moscow.
Spang, B. Correlations for Convective Heat Transfer, http://www.cheresources.com.
Thomas, L.C. (1992) Heat Transfer, Prentice Hall.
www.processassociates.com/process/dimen/dn_all.htm