ENG7901 - Heat Transfer II
1
Contents
1 Forced Convection: External Flows
1.1
4
Flow Over Flat Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
1.1.1
Non-Dimensional form of the Equations of Motion . . . . . . . . . .
4
1.1.2
Order of Magnitude Analysis for a Boundary Layer Flow . . . . . .
6
1.1.3
Non-Dimensional form of the Energy Equation . . . . . . . . . . . .
7
1.1.4
Order of Magnitude Analysis for a Thermal Boundary Layer . . . .
9
1.1.5
Skin Friction and Heat Transfer Coefficients . . . . . . . . . . . . . .
10
1.1.6
Solutions for Laminar Boundary Layer Flow . . . . . . . . . . . . . .
12
1.1.7
Reynolds-Colburn (or Chilton-Colburn) Analogy . . . . . . . . . . .
15
1.1.8
Turbulent Boundary Layers . . . . . . . . . . . . . . . . . . . . . . .
16
1.1.9
Mixed Boundary Layer Conditions . . . . . . . . . . . . . . . . . . .
17
1.1.10 How to Evaluate Convection Heat Transfer . . . . . . . . . . . . . .
18
1.2
Flow Across Cylinders and Spheres . . . . . . . . . . . . . . . . . . . . . . .
19
1.3
Flow Across Tube Banks . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
2 Forced Convection: Internal Flow
26
2.1
Hydrodynamic Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . .
26
2.2
Thermodynamic Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . .
27
2.3
Internal Flows: Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
2.3.1
Fully Developed Laminar Flow . . . . . . . . . . . . . . . . . . . . .
32
2.3.2
Laminar Flow: Entry Region . . . . . . . . . . . . . . . . . . . . . .
32
2.3.3
Turbulent Flow in Circular Tubes
. . . . . . . . . . . . . . . . . . .
32
2.3.4
Turbulent Flow: Entry Region . . . . . . . . . . . . . . . . . . . . .
34
ENG7901 - Heat Transfer II
2.3.5
Flows in Noncircular Tubes . . . . . . . . . . . . . . . . . . . . . . .
3 Natural Convection
2
34
35
3.1
Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
3.2
Empirical Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
3.2.1
Vertical Flat Plate (Ts = const) . . . . . . . . . . . . . . . . . . . . .
39
3.2.2
Vertical Flat Plate (qs00 = const) . . . . . . . . . . . . . . . . . . . . .
39
3.2.3
Vertical Cylinders . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
3.2.4
Horizontal Cylinders . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
3.2.5
Horizontal Flat Surfaces (Ts = const) . . . . . . . . . . . . . . . . .
40
3.2.6
Horizontal Flat Surfaces (qs00 = const) . . . . . . . . . . . . . . . . .
42
3.2.7
Inclined Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
3.2.8
Spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
44
4 Heat Exchangers
4.1
4.2
4.3
45
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
4.1.1
Types of Heat Exchangers . . . . . . . . . . . . . . . . . . . . . . . .
45
4.1.2
Overall Heat Transfer Coefficient . . . . . . . . . . . . . . . . . . . .
47
Log Mean Temperature Difference . . . . . . . . . . . . . . . . . . . . . . .
48
4.2.1
Parallel Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
4.2.2
Counterflow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
4.2.3
Temperature distributions for Special Cases . . . . . . . . . . . . . .
51
4.2.4
∆Tlm for Other Exchanger Configurations . . . . . . . . . . . . . . .
52
Effectiveness - NTU Method . . . . . . . . . . . . . . . . . . . . . . . . . . .
52
ENG7901 - Heat Transfer II
A Eulerian and Lagrangian Viewpoints
3
57
A.1 Lagrangian View . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
A.2 Eulerian View . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
B Conservation of Mass (Continuity Equation)
58
C Conservation of Momentum (Navier-Stokes Equations)
61
ENG7901 - Heat Transfer II
1
4
Forced Convection: External Flows
1.1
1.1.1
Flow Over Flat Surfaces
Non-Dimensional form of the Equations of Motion
• Consider a two-dimensional, steady state, incompressible flow of a constant property,
Newtonian fluid, with freestream velocity u∞ , past a flat plate of length L.
• The governing equations for this flow are the continuity equation:
∂u ∂v
+
=0
∂x ∂y
(1)
and the Navier-Stokes equations:
∂u
∂2u ∂2u
∂u
∂p
ρu
+ ρv
=−
+µ
+ 2
∂x
∂y
∂x
∂x2
∂y
!
∂2v
∂v
∂p
∂2v
∂v
+ ρv
=−
+µ
+
ρu
∂x
∂y
∂y
∂x2 ∂y 2
!
(2)
(3)
• Define the following non-dimensional variables:
x∗ = x/L u∗ = u/u∞ p∗ = p/ρu2∞
y ∗ = y/L v ∗ = v/v∞
(4)
• Substitution of these non-dimensional variables into Eq. (1) results in the following
form of the continuity equation:
u∞ ∂u∗ u∞ ∂v ∗
+
=0
L ∂x∗
L ∂y ∗
(5)
which can be written in the following non-dimensional form:
∂u∗ ∂v ∗
+
=0
∂x∗ ∂y ∗
(6)
ENG7901 - Heat Transfer II
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• Substitution of the non-dimensional variables into the x-component of the NavierStokes equations, Eq. (2), gives:
ρu2∞ ∂p∗
µu∞
ρu2∞ ∗ ∂u∗ ρu2∞ ∗ ∂u∗
+
=
−
+ 2
u
v
∗
∗
∗
L
∂x
L
∂y
L ∂x
L
∂ 2 u∗ ∂ 2 u∗
+ ∗2
∂x∗2
∂y
!
(7)
Multiplying this equation by L/ρu2∞ gives:
u
∗ ∂u
∗
∂x∗
+
∂u∗
v∗ ∗
∂y
∂p∗
µ
=− ∗ +
∂x
ρu∞ L
∂ 2 u∗ ∂ 2 u∗
+ ∗2
∂x∗2
∂y
!
(8)
But ReL = ρu∞ L/µ, therefore, the non-dimensional form of the x-component of the
Navier-Stokes equations can be written as:
∂u∗
∂u∗
∂p∗
1
u∗ ∗ + v ∗ ∗ = − ∗ +
∂x
∂y
∂x
ReL
∂ 2 u∗ ∂ 2 u∗
+ ∗2
∂x∗2
∂y
!
(9)
• Similarly, the non-dimensional form of the y-component of the Navier-Stokes equations
is:
!
∗
∗
∂p∗
1
∂ 2v∗ ∂ 2v∗
∗ ∂v
∗ ∂v
u
+v
=− ∗ +
+ ∗2
(10)
∂x∗
∂y ∗
∂y
ReL ∂x∗2
∂y
• Equations (6), (9), and (10) are the non-dimensional forms of the equations that
govern the steady state, two-dimensional, incompressible flow of a constant property,
Newtonian fluid, with freestream velocity u∞ , past a flat plate of length L.
• Note: the only parameter in these equations is the Reynolds number, therefore, Re
should appear as a parameter in the solutions for the hydrodynamic boundary layer
on a flat plate.
ENG7901 - Heat Transfer II
1.1.2
6
Order of Magnitude Analysis for a Boundary Layer Flow
• Consider the order of magnitude of the non-dimensional variables that appear in Eqs.
(6), (9), and (10):
O(x∗ ) = 1
O(u∗ ) = 1 O(v ∗ ) =?
O(y ∗ ) = δ/L = δ ∗ 1 O(p∗ ) =? O(ReL ) =?
(11)
• Now, consider the order of magnitude of each term in Eqs. (6), (9), and (10):
∂u∗ ∂v ∗
+
=0
∂x∗ ∂y ∗
(12)
∂p∗
1
=− ∗ +
∂x
ReL
∂ 2 u∗ ∂ 2 u∗
+ ∗2
∂x∗2
∂y
!
∂p∗
1
u
+
v
=
−
+
∂x∗
∂y ∗
∂y ∗ ReL
∂ 2v∗ ∂ 2v∗
+ ∗2
∂x∗2
∂y
!
u
∗ ∂u
∗
∂x∗
∗ ∂v
∗
+
∂u∗
v∗ ∗
∂y
∗ ∂v
∗
(13)
(14)
• Note:
1. From the continuity equation O(v ∗ ) = δ ∗ 1 (since both terms must be of the
same magnitude).
2. For the viscous terms to be the same order as the inertia terms in the xcomponent of the Navier-Stokes equations O(ReL ) = 1/δ ∗2 1.
3. For the pressure term to be the same order as the inertia terms in the xcomponent of the Navier-Stokes equations (to prevent infinite accelerations)
O(p∗ ) = 1.
ENG7901 - Heat Transfer II
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• The non-dimensional forms of the equations governing the flow in the boundary layer
are:
∂u∗ ∂v ∗
+
=0
(15)
∂x∗ ∂y ∗
u
∗ ∂u
∗
∂x∗
+
∂u∗
v∗ ∗
∂y
∂p∗
1
=− ∗ +
∂x
ReL
0=−
• Or in dimensional form:
∂ 2 u∗
∂y ∗2
∂p∗
∂y ∗
0=−
∂p
∂y
(16)
(17)
∂u ∂v
+
=0
∂x ∂y
∂2u
∂u
∂u
∂p
ρu
+ ρv
=−
+µ
∂x
∂y
∂x
∂y 2
!
(18)
!
(19)
(20)
• One term has been eliminated from the x-component of the Navier-Stokes equations.
• The y-component of the Navier-Stokes equations has been reduced to a hydrostatic
pressure distribution.
• By performing the order of magnitude analysis one equation has been simplified, and
the number of equations that must be solved has been reduced by one.
1.1.3
Non-Dimensional form of the Energy Equation
• Consider a two-dimensional, steady state, incompressible flow of a constant property,
Newtonian fluid at freestream temperature T∞ and velocity u∞ past a flat plate of
length L, maintained at a constant temperature Ts .
ENG7901 - Heat Transfer II
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• The energy equation may be written as:
∂2T
∂2T
+
2
∂x
∂y 2
∂T
∂T
ρcp u
+ ρcp v
=k
∂x
∂y
where
∂u
Φ=2
∂x
2
∂v
+2
∂y
2
+
!
+ µΦ
∂u ∂v
+
∂y
∂x
(21)
2
(22)
• Define the following non-dimensional variables
x∗ = x/L u∗ = u/u∞ θ = (T − Ts )/(T∞ − Ts )
y ∗ = y/L v ∗ = v/u∞
(23)
• Substituting these non-dimensional variables into Eqs. (21) and (22) gives:
ρcp u∞ (T∞ − Ts )
∂θ
∂θ
u∗ ∗ + v ∗ ∗
L
∂x
∂y
where
Φ∗ = 2
∂u∗
∂x∗
∂2θ
∂2θ
+
∂x∗2 ∂y ∗2
k(T∞ − Ts )
=
L2
2
+2
∂v ∗
∂y ∗
2
+
∂u∗
∂v ∗
+
∂y ∗
∂x∗
!
+
µu2∞ ∗
Φ (24)
L2
2
(25)
• Simplifying Eq. (24):
∂2θ
∂2θ
+
∂x∗2 ∂y ∗2
∂θ
k
∂θ
+ v∗ ∗ =
u
∂x∗
∂y
ρcp u∞ L
∗
• But
k
k
=
ρcp u∞ L
ρcp u∞ L
µ
µ
µu∞
ρcp L(T∞ − Ts )
=
k
µcp
!
!
µu∞
Φ∗
ρcp L(T∞ − Ts )
+
µ
ρu∞ L
=
µu∞
ρcp L(T∞ − Ts )
=
u2∞
cp (T∞ − Ts )
=
u∞
u∞
!
u2∞
= −
cp (Ts − T∞ )
Ec
= −
ReL
1
1
=
P rReL
Pe
(27)
µ
ρu∞ L
!
(26)
µ
ρu∞ L
(28)
ENG7901 - Heat Transfer II
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• Note:
Reynolds # = ReL =
Prandtl # = P r =
(29)
µcp
ν
rate of diffusion of momentum
= ∝
k
α
rate of diffusion of thermal energy
Peclet # = P e = P rReL ∝
Eckert # = Ec =
ρu∞ L
inertia forces
∝
µ
viscous forces
convective transport of thermal energy
conductive transport of thermal energy
u2∞
kinetic energy/unit volume of flow
∝
cp (Ts − T∞ )
thermal energy/unit volume of flow
(30)
(31)
(32)
• The non-dimensional form of the energy equation can be written as follows:
∂θ
∂θ
1
u
+ v∗ ∗ =
∂x∗
∂y
Pe
∗
∂2θ
∂2θ
+
∂x∗2 ∂y ∗2
!
−
Ec ∗
Φ
ReL
(33)
• Note: the only parameters in the thermal problem are P r, ReL , and Ec.
1.1.4
Order of Magnitude Analysis for a Thermal Boundary Layer
• Similar to the analysis of the hydrodynamic boundary layer, consider the order of
magnitude of each term in Eq. (33):
O(x∗ ) = 1
O(u∗ ) = 1
O(Ec) =?
∗
∗
∗
∗
O(y ) = δT /L = δT 1 O(v ) = δ 1 O(P r) =?
O(θ) = 1
O(ReL ) = 1/δ ∗2 O(P e) =?
(34)
• Expanding all terms of the non-dimensional form of the energy equation:
∂θ
∂θ
1
u
+v ∗ ∗ =
∂x∗
∂y
Pe
∗
!
∂2θ
∂2θ
Ec
+
−
∂x∗2 ∂y ∗2
ReL
∂u∗
2
∂x∗
2
∂v ∗
+2
∂y ∗
2
+
!
∂u∗
∂v ∗ 2
+
∂y ∗
∂x∗
(35)
• Note:
1. For the conduction terms to be of the same order as the convection terms
O(1/P e) = δT∗2 . This is sensible, since O(ReL ) = 1/δ ∗2 , and O(P r) ≈ 1 for
most common fluids.
ENG7901 - Heat Transfer II
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2. For any of the viscous dissipation terms to be of the same order as the remainder
of the equation, the only possibility is for O(Ec) ≈ 1, then the (∂u∗ /∂y ∗ )2 term
will remain
• The non-dimensional form of the energy equation for the thermodynamic boundary
layer on a flat plate (steady state, constant property, Newtonian fluid, incompressible
flow) is:
1 ∂2θ
Ec ∂u∗ 2
∗ ∂θ
∗ ∂θ
u
+v
=
−
(36)
∂x∗
∂y ∗
P e ∂y ∗2 ReL ∂y ∗
But Ec ≥ 1 only at high velocities,
e.g. for air, (cp ≈ 1000 J/kg·o C, (Ts − T∞ ) ≈ 100o C → u∞ ≈ 316 m/s for Ec =
u2∞ /(cp (Ts − T∞ )) = 1. The speed of sound at 300K is 347 m/s.
therefore, at low velocities, viscous dissipation is negligible. Viscous dissipation is very
important at high velocities, e.g. the space shuttle, and SR-71 Blackbird spy plane.
Viscous dissipation gives rise to frictional heating.
• Neglecting viscous dissipation, the equation governing the thermal boundary layer on
a flat plate for steady, two-dimensional, incompressible flow of a constant property,
Newtonian fluid is:
∂T
∂T
∂2T
ρcp u
+ ρcp v
=k 2
(37)
∂x
∂y
∂y
or
∂T
∂T
∂2T
u
+v
=α 2
(38)
∂x
∂y
∂y
1.1.5
Skin Friction and Heat Transfer Coefficients
• We would be interested in the frictional drag due to the hydrodynamic boundary
layer. The frictional drag is due to the shear stress at the plate surface, i.e.:
∂u τs = µ ∂y y=0
(39)
• The shear stress is often written in terms of a skin friction coefficient, Cf :
τs = Cf
ρu2∞
2
(40)
• Since the velocity gradient ∂u/∂y at y = 0 varies with x, the skin friction coefficient
will also be a function of x. Further, the non-dimensional Navier-Stokes equations,
Eqs. (9) and (10) illustrate that the only parameters that would influence the solution
for the velocity gradient at the wall (i.e. y = 0) are Re and ∂p/∂x. But, the pressure
ENG7901 - Heat Transfer II
11
gradient is only a function of x, and it is determined by the geometry of the flow,
therefore, for flows of different fluids past the same geometry, only the Reynolds
number and position on the body should influence the skin friction coefficient:
Cf = Cf (x, Rex )
(41)
• To solve for Cf we need the velocity distribution in the boundary layer, therefore, we
need to solve Eqs. (18) and (19).
• The heat flux at the surface of the plate exposed to the convection environment, qs00 ,
can be written as follows:
∂T 00
(42)
qs = h(Ts − T∞ ) = −k
∂y y=0
Heat is transferred from the wall to the fluid by conduction (since the molecules of
fluid next to the wall have zero velocity relative to the plate).
• The heat transfer coefficient is defined as follows:
h=
−k ∂T
∂y y=0
Ts − T∞
(43)
To determine h we need the temperature gradient at the wall, i.e. the temperature
distribution, therefore, we need to solve the energy equation for the boundary layer,
i.e. Eq. (37), which will require a prior solution for the hydrodynamic boundary layer.
• The non-dimensional form of the energy equation, Eq. (36), illustrates that the only
non-dimensional parameters that should appear in the thermal boundary layer solution are x∗ , y ∗ , P e (or Re and P r), and Ec.
• Instead of working with the heat transfer coefficent, it is common to use a nondimensional variable called the Nusselt number (N u):
N uL =
hL
Actual heat transfer in the presence of flow
∝
k
Heat transfer if only conduction occurs
(44)
hx
k
(45)
or locally:
N ux =
• Since only P r, Re, and Ec are the parameters of the flow:
h = h(P r, Re, Ec)
(46)
N u = N u(P r, Re, Ec)
(47)
• If Ec is small, h and N u are only functions of P r and Re and:
h = h(P r, Re)
(48)
N u = N u(P r, Re)
(49)
ENG7901 - Heat Transfer II
1.1.6
12
Solutions for Laminar Boundary Layer Flow
• Blausius (1908) developed an analytical solution for the hydrodynamic boundary layer
(see Section 7.2.1, Incropera and DeWitt):
5x
δ =
Cf,x
(50)
1/2
Rex
= 0.664Re−1/2
x
(51)
• Polhausen (1921) developed an analytical solution for the thermal boundary layer:
N ux =
hx x
= 0.332Rex1/2 P r1/3
k
P r ≥ 0.6
(52)
• Note:
1. These correlations are for local values, i.e. they are functions of x.
2. The boundary layer thickens at the rate of x1/2 . For a given x, the thickness
decreases with increasing Re (as the influence of viscous forces decreases).
3. The Nusselt number increases at the rate of x1/2 but hx decreases at the rate of
x1/2 (i.e. the thermal boundary layer thickens with increasing x, decreasing the
temperature gradient and the heat transfer rate at the surface of the plate).
4. The expected non-dimensional parameters have arisen in the analytical solutions.
• Often, we are not interested in the local skin friction, or heat transfer rate, but in
the total frictional drag over a surface, or the total heat transfer rate from (or to) the
surface.
• The total frictional drag force, D, would be defined as follows:
Z
L
Z
τs w dx =
D=
0
0
Since ρu2∞ /2 is constant:
D = Cf
where
Cf =
1
L
Z
0
L
Cf,x
ρu2∞
w dx
2
ρu2∞
A
2
(53)
(54)
L
Cf,x dx
(55)
Substituting Eq. (51) into Eq. (55) and performing the integral gives:
−1/2
C f = 1.328ReL
(56)
ENG7901 - Heat Transfer II
13
• The total heat transfer rate from the plate, q, can be evaluated as follows:
L
Z
hx (Ts − T∞ )w dx
q=
(57)
0
but (Ts − T∞ ) is constant for this problem, therefore,
q = hA(Ts − T∞ )
where
1
h=
L
Z
(58)
L
hx dx
(59)
0
Substituting hx from Eq. (52) into Eq. (59) gives:
k 1/2
hL = 0.664 ReL P r1/3
L
P r ≥ 0.6
(60)
P r ≥ 0.6
(61)
or an average Nusselt number can be defined as:
1/2
N uL = 0.664ReL P r1/3
• Note: Cf , N uL , and hL are twice the corresponding value at the position L from the
leading edge (due to the exponent on Rex ).
• The above correlations for h and N u have restrictions on the Prandtl number. Churchill
and Ozoe experimentally obtained the following correlation for laminar flow over a
flat isothermal plate which is valid for all Prandtl numbers:
1/2
0.3387Rex P r1/3
N ux = 1/4
1 + (0.0468/P r)2/3
P ex ≥ 100
(62)
where N uL = 2N uL .
• The correlations developed thus far are based on the assumption that the fluid properties are constant and uniform. When there is a significant difference between the plate
and freestream temperatures, the fluid properties are evaluated at a film temperature:
Tf =
Ts + T∞
2
(63)
ENG7901 - Heat Transfer II
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• When the plate is not heated over its entire length (e.g. heating starts at a location ξ
from the leading edge of the plate) Eq. (52) is modified to give:
3/4 #−1/3
"
N ux =
0.332Rex1/2 P r1/3
ξ
x
1−
(64)
Constant heat flux
• For a laminar boundary layer flow over a flat plate which has a constant and uniform
wall heat flux (qs00 , W/m2 ) the local Nusselt number is:
N ux = 0.453Rex1/2 P r1/3
P r ≥ 0.6
(65)
Note: only the constant has changed between Eq. (52) and Eq. (65).
• If the heat flux is given, we would be interested in the local wall temperature, Ts,x ,
or the mean temperature difference Ts − T∞ :
k
qs00 = hx (Ts,x − T∞ ) = N ux (Ts,x − T∞ )
x
therefore
Ts,x − T∞ =
So:
Ts − T∞ =
1
L
L
Z
0
qs00 x
N ux k
qs00 x
q 00 L/k
dx = s
N ux k
N uL
(66)
(67)
(68)
Using Eq. (65) to determine N ux :
Ts − T∞ =
and
Note:
qs00 L/k
1/2
0.680ReL P r1/3
3
qs00 = hL (Ts − T∞ )
2
(69)
(70)
ENG7901 - Heat Transfer II
15
1. The mean Nusselt number for laminar flow over a flat plate with qs00 = const is:
1/2
N uL |qs00 =const = 0.680ReL P r1/3
(71)
which is only 2% larger than the mean value for the constant Ts boundary condition, Eq. (61), therefore, it is acceptable to use any of the N uL correlations for
constant Ts to determine Ts − T∞ for the constant qs00 boundary condition.
2. The correlation developed by Ozoe and Churchill may also be used for the constant heat flux boundary condition by replacing the constants 0.3387 and 0.0468
in Eq. (62) with 0.4637 and 0.0207, respectively.
1.1.7
Reynolds-Colburn (or Chilton-Colburn) Analogy
• The skin friction coefficient for laminar boundary layer flow over a uniform temperature flat plate is given by Eq. (51). This equation can be rearranged as follows:
Cf,x
= 0.332Rex−1/2
2
(72)
• The expression for the local Nusselt number for a laminar boundary layer on a uniform
temperature flat plate is Eq. (52). Dividing Eq. (52) by Rex P r:
N ux
= 0.332Rex−1/2 P r−2/3
Rex P r
(73)
the left hand side of this equation is called the Stanton number, Stx , therefore:
Stx P r2/3 = 0.332Re−1/2
x
(74)
• Comparison of Eqs. (72) and (74) gives:
Stx P r2/3 =
Cf,x
2
(75)
StP r2/3 =
Cf
2
(76)
or
• This is the Reynolds-Colburn analogy between fluid friction and heat transfer. For
example, if experimental measurements are made of the frictional drag on a body →
C f → St → h → heat transfer rate (or vice versa).
• The Reynolds-Colburn analogy applies for laminar and turbulent boundary layers on
flat plates, and, in a modified form, for turbulent tube flow.
ENG7901 - Heat Transfer II
1.1.8
16
Turbulent Boundary Layers
• Consider a steady, incompressible flow of a Newtonian fluid past a flat plate. The
freestream conditions are constant at u∞ and T∞ , and the plate temperature is a
uniform and constant Ts .
• As the Reynolds number increases, the inertia forces begin to dominate the viscous
forces and instabilities in the flow can no longer be damped out by viscous effects.
The flow will go through a transition from a laminar boundary layer to a turbulent
boundary layer.
• The turbulent portion of the boundary layer is thicker than the laminar portion, and
instead of smooth lamina, it consists of eddies of varying size.
• The effect of these eddies is to give a mean velocity profile that is fuller than in the
laminar portion of the boundary layer. This will result in higher velocity gradients
at the surface of the plate → higher wall shear stress (and Cf,x ) → higher frictional
drag.
• Similarly, the mean temperature profile in the boundary layer becomes fuller, and the
temperature gradient at the wall will increase. Since the temperature gradient at the
wall increases, then the heat transfer rate will increase, therefore, by Eq. (43), the
convection heat transfer coefficient will be higher than for laminar flow.
• In laminar flow, the shear stress is a function of the fluid properties (i.e. µ) and the
velocity gradient. In turbulent flow, the shear stress is a function of µ, the velocity
gradient, and the flow properties. This occurs, because the eddies will cause a
transport of momentum through the boundary layer, and this transport is modelled
as a shear stress:
∂u
τt = ρM
(77)
∂y
where M is the eddy viscosity, which is due to the fluid motion.
ENG7901 - Heat Transfer II
17
• An analytical solution cannot be found for turbulent flows, due to the dependence
of the flow on the flow properties, therefore, experimental measurements are used to
develop empirical correlations for Cf and N u in turbulent flows.
• The local skin friction coefficient is evaluated by the following equation:
Cf,x = 0.0592Re−1/5
x
Rex ≤ 107
(78)
and this equation may be used for Rex < 108 to within 15% accuracy.
• Using the Reynolds-Colburn analogy:
N ux = 0.0296Rex4/5 P r1/3
0.6 < P r < 60, Rex < 108
(79)
• For heating starting at a position ξ from the leading edge of the plate:
1/3
N ux = 0.0296Re4/5
[1 − (ξ/x)9/10 ]−1/9 0.6 < P r < 60, Rex < 108
x Pr
(80)
• For uniform wall heat flux:
N ux = 0.0308Rex4/5 P r1/3
0.6 ≤ P r ≤ 60
(81)
i.e. 4% higher than for the constant wall temperature boundary condition.
1.1.9
Mixed Boundary Layer Conditions
• A laminar boundary layer will eventually go through a transition to a turbulent boundary layer. Define the plate length as L and the position where the critical Reynolds
number, Rex,c , occurs as xc . When 0.95 < xc /L ≤ 1, then the mean laminar convection coefficient, Eq. (60), can be used to determine the total heat transfer rate from
the plate. When xc /L ≤ 0.95 then the laminar and turbulent sections of the boundary
layer should be accounted for when determining the total heat transfer rate from the
plate:
!
Z xc
Z L
1
hL =
hlam dx +
hturb dx
(82)
L
0
xc
where the transition is assumed to occur abruptly at xc .
• Substituting Eqs. (52) and (79) into Eq. (82) gives:
4/5
N uL = (0.037ReL − A)P r1/3
(83)
4/5
1/2
A = 0.037Rex,c
− 0.664Rex,c
(84)
where
is dependent upon the critical Reynolds number.
ENG7901 - Heat Transfer II
18
• The critical Reynolds number for flow over a smooth flat plate is 5 × 105 , therefore:
4/5
N uL = (0.037ReL − 871)P r1/3 0.6 < P r < 60, 5 × 105 < ReL ≤ 108
and
Cf =
0.074
1/5
ReL
−
1742
5 × 105 < ReL ≤ 108
ReL
(85)
(86)
• If the boundary layer is completely turbulent (e.g. it is tripped at the leading edge of
the plate) A = 0:
4/5
N uL = 0.037ReL P r1/3
Cf
−1/5
= 0.074ReL
(87)
(88)
4/5
Note: these equations would also be appropriate when xc /L 1, since A 0.037ReL .
• All of the foregoing correlations are to be used with properties evaluated at the film
temperature.
• These correlations are acceptable for engineering calculations, but they may be up to
25% in error due to freestream turbulence, and surface roughness.
1.1.10
How to Evaluate Convection Heat Transfer
• The calculation of convection heat transfer rates uses the following procedure:
1. Identify the geometry of the flow. All of the convection heat transfer correlations
are dependent upon the geometry involved.
2. Specify the reference temperature, and evaluate all fluid properties at this reference temperature. Usually the film temperature, or a mean bulk temperature is
used, however, there are exceptions.
3. Calculate the Reynolds number. Is the flow laminar or turbulent, or both? Is
there anything to cause the flow to be completely turbulent, e.g. a very rough
surface, or freestream turbulence?
4. Determine the boundary condition on the surface, i.e. constant temperature or
uniform flux.
5. Decide if local or mean values are required.
6. Pick an appropriate correlation based on the previous steps.
ENG7901 - Heat Transfer II
1.2
19
Flow Across Cylinders and Spheres
• The total drag force, FD , acting on a cylinder in a cross flow is a function of a frictional
component, due to the shear stress in the fluid at the surface of the cylinder, and a
component due to the pressure differential acting on the cylinder due to the formation
of the wake. These two drag components are called frictional and form (or pressure
drag), respectively.
• For ReD < 2 the flow remains attached to the cylinder, and the drag is mainly due
to friction.
• As the Reynolds number is increased boundary layer separation and wake formation
become important, and form drag dominates frictional drag.
• As the freestream is brought to rest at the stagnation point on the cylinder, a
maximum pressure is attained. As the flow expands about the cylinder, the pressure decreases, and the boundary layer develops in a favourable pressure gradient
(dp/dx < 0). As the flow passes the maximum height of the cylinder, however, it will
begin to decelerate, and consequently the pressure will increase, producing an adverse
pressure gradient (dp/dx > 0).
• Since the pressure in a boundary layer is constant at any x location, this adverse
pressure gradient will decelerate the flow within the boundary layer. If the flow has
insufficient momentum to overcome the adverse pressure gradient it will separate from
the surface of the cylinder and create a recirculation zone, or a wake. Thereafter, the
pressure cannot increase, and this gives rise to a large pressure differential between
the front and back of the cylinder → high form drag.
ENG7901 - Heat Transfer II
20
• A laminar boundary layer carries less momentum near the surface of the cylinder
than a turbulent boundary layer, therefore, it will separate earlier (θ = 80o ) than the
turbulent boundary layer (θ = 140o ) → higher form drag.
• The transition Reynolds number for a cylinder is approximately 2 × 105 .
ENG7901 - Heat Transfer II
21
• Figure 7.9, Incropera and DeWitt, above, shows the variation of N uD as a function
of angular position from the stagnation point on the cylinder, i.e. 0o . The Nusselt
number decreases from the stagnation point as the laminar boundary layer grows.
Between 80o and 100o N u increases rapidly, due to the transition from laminar to
turbulent flow. The Nusselt number decreases as the turbulent boundary layer is
established. The Nusselt number increases as 140o is reached, due to boundary layer
separation, and increased mixing in the wake region.
• Due to the large changes that occur in the flow over a cylinder, depending on position and Reynolds number, empirical correlations are used to determine overall heat
transfer coefficients for cylinders in cross flow.
• The correlation developed by Hilpert:
hD
1/3
= CRem
P r ≥ 0.6
(89)
DP r
k
is widely used for gases, and liquids. The properties used in this equation are evaluated
at the film temperature, and N uD and ReD are based on the characteristic dimension
of the cylinder, i.e. its diameter. The constants C and m are determined from Table
7.2, Incropera and Dewitt.
N uD =
• Equation (89) may also be used for cylinders with noncircular cross-sections when
Table 7.3, Incropera and DeWitt, is used to define C and m.
• Zhukauskas developed the following correlation:
N uD =
n
CRem
DP r
Pr
P rs
1/4
0.7 < P r < 500, 1 < ReD < 106
(90)
ENG7901 - Heat Transfer II
22
where C and m are obtained from Table 7.4, Incropera and DeWitt, and n = 0.37
for P r ≤ 10 and n = 0.36 for P r > 10. All properties are evaluated at T∞ except for
P rs .
• Churchill and Bernstein have developed the following correlation, which is valid for
ReD P r > 0.2:
1/2
0.62ReD P r1/3
"
N uD = 0.3 + 1/4 1 +
1 + (0.4/P r)2/3
ReD
282, 000
5/8 #4/5
(91)
where all properties are evaluated at the film temperature.
• Note: these correlations may be in error by as much as 20% in engineering calculations.
• The behaviour of fluid flow about a sphere is similar to that about a cylinder, however,
the drag coefficient is reduced due to the three-dimensional nature of the flow.
• McAdams has proposed a correlation for the convection heat transfer coefficient on a
sphere in a gas:
N uD = 0.37Re0.6
17 < ReD < 70, 000
(92)
D
where all properties are evaluated at the film temperature.
• Whitaker has developed the following correlation:
N uD = 2 +
1/2
(0.4ReD
+
2/3
0.06ReD )P r0.4
µ
µs
1/4
(93)
which is valid for 0.71 < P r < 380, 3.5 < ReD < 7.6 × 104 , and 1 < µ/µs < 3.2. All
properties in this correlation are evaluated at T∞ except µs which is evaluated at Ts .
• Ranz and Marshall developed the following correlation for convection heat transfer
from freely falling liquid drops:
1/2
N uD = 2 + 0.6ReD P r1/3
(94)
• Note: Eqs. (93) and (94) reduce to N uD = 2 when ReD → 0, which is the value
obtained for conduction from the surface of a sphere in a stationary infinite medium.
1.3
Flow Across Tube Banks
• Heat exchangers (e.g. boiler, air conditioner) often employ aligned or staggered arrays
of tubes in a cross flow.
ENG7901 - Heat Transfer II
23
• In these figures, D is the outer diameter of the tubes, NT and NL are the number of
rows normal (transverse) and parallel (longitudinal) to the freestream flow, V , and
ST and SL are the transverse and longitudinal spacings or pitches (center to center)
of the tubes, respectively.
• The mean convection coefficient for a tube array can be found from the following
relation:
1/3
N uD = 1.13C1 Rem
(95)
D,max P r
Where the constants C1 and m are given as functions of ST /D and SL /D in Table
7.5, Incropera and DeWitt. This correlation is valid for NL ≥ 10, 2000 < ReD,max <
40, 000 and P r ≥ 0.7.
• The heat transfer coefficient is a function of the position of the tube in the tube bank.
The first tube behaves like a cylinder in a cross flow. The coefficient on the other tubes
is higher, because the first few rows of tubes behave as a turbulence grid, generating
turbulence and increasing heat transfer rates. Usually the flow conditions stabilize
after 10 rows, and there is little further change in the convection coefficient.
• For tube banks consisting of less than 10 rows parallel to the flow a correction factor
C2 , based on the number of rows NL is applied to the Nusselt number.
N uD |NL <10 = C2 N uD |NL ≥10
(96)
Where C2 is obtained from the table below:
NL
Staggered
Aligned
1
0.68
0.64
2
0.75
0.80
3
0.83
0.87
4
0.89
0.90
5
0.92
0.92
6
0.95
0.94
7
0.97
0.96
8
0.98
0.98
9
0.99
0.99
10
1
1
ENG7901 - Heat Transfer II
24
• The Reynolds number, ReD,max , used in Eq. (95) is based on the maximum velocity
in the tube bank.
ρVmax D
ReD,max =
(97)
µ
For the aligned tube bank, the maximum velocity occurs between two tubes, i.e. A1 :
and, from mass conservation:
Vmax =
ST
V
ST − D
(98)
For a staggered tube array, the maximum velocity may occur in one of two locations,
i.e. A1 or A2 , and Vmax is determined from Eq. (98), or:
ST
V
2(SD − D)
Vmax =
where
SD =
SL2
ST
+
2
(99)
1/2
(100)
Note: Eq. (99) should be used when 2(SD − D) < (ST − D).
• All of the properties used in the above relations are evaluated at the film temperature.
• Assuming the surface temperature of the tubes is uniform and higher than the external
fluid temperature, the temperature of the external fluid will increase as it passes
through the tube bank. Since qs00 = h∆T , qs00 will decrease as the fluid passes through
the tube bank. Since the heat transfer rate to the fluid decreases, then the rate at
which its temperature increases will decrease (i.e. the rate of increase is nonlinear).
Using (Ts − T∞ ) or (Ts − (Ti + To )/2) would lead to over and under estimations of
ENG7901 - Heat Transfer II
25
the heat transfer rate. An appropriate mean temperature difference to use is the log
mean temperature difference (derived in Section 2.2):
∆Tlm =
(Ts − Ti ) − (Ts − To )
ln [(Ts − Ti )/(Ts − To )]
(101)
Note: Eq. (101) can be written in the following easy to remember form:
∆Tlm =
∆Tin − ∆Tout
ln [∆Tin /∆Tout ]
(102)
• The heat transfer rate to the fluid can then be written as:
q = hAtot ∆Tlm = ṁcp (To − Ti )
(103)
where, Atot is the total surface area of the tubes exposed to convection:
Atot = (NT × NL )πdL
(104)
and ṁ is the mass flow rate of the freestream fluid:
ṁ = ρ∞ NT ST V L
(105)
• The pressure drop through a tube bank can be evaluated using the following expression:
!
2
ρVmax
∆p = χNL
f
(106)
2
where the appropriate friction factor, f , and correction factor, χ, are obtained from
Figs. 7.13 and 7.14, Incropera and DeWitt. These figures use PL ≡ SL /D and
PT ≡ ST /D as independent variables.
• Note:
1. An alternate correlation has been proposed by Zhukauskas, see Eq. (7.57), Incropera and DeWitt.
2. Tube spacings of ST /SL < 0.7 are undesirable for inline tube arrangements,
because they produce a preferred flow path between the rows of tubes, and much
of the tube surface is not exposed to the main flow. See Fig. 7.12, Incropera and
DeWitt.
ENG7901 - Heat Transfer II
2
26
Forced Convection: Internal Flow
2.1
Hydrodynamic Fundamentals
• Consider steady state flow in a circular tube of radius ro :
• The fully developed velocity profile is parabolic for laminar flow. The profile is flatter
for turbulent flow.
• The friction and heat transfer rate are highest in the developing flow region, and
asymptote to a constant value in the fully developed region.
• The length of the developing flow region (or hydrodynamic entry region), xf d,h , is a
function of the Reynolds number for laminar flows:
xf d,h
D
≈ 0.05ReD
(107)
lam
where
ρum D
(108)
µ
and um is a mean velocity of the flow. The transition Reynolds number, ReD,c , is
2300 for tube flow.
ReD =
• The hydrodynamic entry length for turbulent flow is independent of Reynolds number:
10 ≤
xf d,h
D
≤ 60
(109)
turb
• To derive an expression for the laminar velocity profile in a circular tube of radius
ro , consider the steady, incompressible flow of a constant property Newtonian fluid in
the tube. In the fully developed region v = 0, and ∂u/∂x = 0. The Navier-Stokes
equations written in Polar-Cylindrical co-ordinates can be solved to give:
1
u=−
4µ
"
dp 2
r 1−
dx o
r
ro
2 #
(110)
ENG7901 - Heat Transfer II
27
and the mean velocity is:
um = −
ro2 dp
8µ dx
(111)
• Defining the Moody (or Darcy) friction factor as:
f=
−(dp/dx)D
ρu2m /2
(112)
the pressure drop in a tube of length L can be written as:
∆p = −
Z
L
f
0
ρu2m
L u2
dx = f ρ m
2D
D 2
(113)
where L replaces dx.
• Substituting the expression for the mean velocity, Eq. (111), and the definition of the
Reynolds number into Eq. (112) gives:
64
ReD
f=
(114)
• Experimental data must be used to determine the friction factor for turbulent flows in
rough tubes, i.e. the Moody diagram, Fig. 8.3, Incropera and DeWitt. The following
correlations exist for fully developed turbulent flow in smooth tubes:
f
f
f
2.2
−1/4
ReD ≤ 2 × 104
−1/5
0.184ReD
4
= 0.316ReD
=
(115)
ReD ≥ 2 × 10
−2
= (0.79 ln ReD − 1.64)
(116)
6
3000 ≤ ReD ≤ 5 × 10
Thermodynamic Fundamentals
• Consider a fluid entering a tube with a uniform velocity and temperature.
(117)
ENG7901 - Heat Transfer II
28
• The temperature profile in the fully developed region depends on the boundary condition applied at the tube wall (Ts = const, or qs00 = const)
• The thermal entry length for laminar flow is:
xf d,t
D
= 0.05ReD P r
(118)
lam
i.e. if P r > 1, xf d,h < xf d,t and vice versa. The thermal entry length for turbulent
flow is the same as the hydrodynamic entry length.
• A bulk (or mean) temperature is used in the calculation of heat transfer rates for
internal flows. It is determined from the rate of transport of thermal energy through
a cross-section, Ac :
Z
ρucv T dAc
Ėt = ṁcv Tm =
(119)
Ac
i.e.
R
Ac
Tm =
ρucv T dAc
ṁcv
(120)
• The heat flux is defined as:
qs00 = h(Ts − Tm )
(121)
Note: Tm is a function of x (Tm increases if Ts > Tm and decreases if Ts < Tm ). It will
be shown that h is constant in the fully developed region, therefore, if Ts is constant
then qs00 decreases with x, and if qs00 is constant then Ts − Tm is a constant.
• In the thermally fully developed region ∂T /∂x 6= 0, but:
∂ Ts (x) − T (r, x)
∂x Ts (x) − Tm (x)
therefore:
∂
∂r
=0
(122)
f d,t
Ts − T −∂T /∂r|r=ro
=
6= f (x)
Ts − Tm r=ro
Ts − Tm
But
qs00 = k
(123)
∂T = h(Ts − Tm )
∂r r=ro
(124)
therefore
h
6= f (x)
(125)
k
So in the thermally fully developed region the convection coefficient is constant.
• Since h is constant, the constant qs00 boundary condition gives:
dTs dTm =
dx f d,t
dx f d,t
so (Ts − Tm ) is constant.
(126)
ENG7901 - Heat Transfer II
29
• Applying the 1st Law to an element of fluid flowing in a tube (i.e. an open system of
constant volume fixed in space):
• Neglecting ∆ek , ∆ep , and all forms of work except for flow work:
dqconv = ṁdh
(127)
and if the fluid is assumed to have constant specific heats:
dqconv = ṁcp dTm
(128)
• This equation can be integrated from the inlet to the exit of the tube to give:
qconv = ṁcp (Tm,o − Tm,i )
(129)
• Defining dqconv = qs00 P dx, Eq. (128) can be rearranged to give:
dTm
q 00 P
P
= s =
h(Ts − Tm )
dx
ṁcp
ṁcp
(130)
• For constant wall heat flux Eq. (130) can be integrated to give:
Tm (x) = Tm,i +
qs00 P
x
ṁcp
(131)
so the bulk temperature varies linearly with x. Note: Ts − Tm increases until the fully
developed region is reached (due to the higher h in the entrance region) and (Ts − Tm )
becomes constant.
ENG7901 - Heat Transfer II
30
• For the constant wall temperature boundary condition, Eq. (130) can be rewritten
as:
dTm
d(∆T )
P
=−
=
h∆T
(132)
dx
dx
ṁcp
where ∆T = Ts − Tm . Rearranging this equation, and integrating from inlet to outlet:
Z
∆To
∆Ti
d(∆T )
P
=−
∆T
ṁcp
or
∆To
PL
ln
=−
∆Ti
ṁcp
1
L
Z
L
Z
h dx
(133)
0
!
h dx
0
L
=−
PL
h
ṁcp
(134)
which can be rearranged to give:
∆To
Ts − Tm,o
PL
h
=
= exp −
∆Ti
Ts − Tm,i
ṁcp
!
So the temperature difference (Ts − Tm ) decays exponentially.
(135)
ENG7901 - Heat Transfer II
31
Using Eq. (134):
qconv = ṁcp ((Ts − Tm,o ) − (Ts − Tm,i )) = −P Lh
(Ts − Tm,o ) − (Ts − Tm,i )
ln(∆To /∆Ti )
(136)
or
qconv = ṁcp (Tm,o − Tm,i ) = hA∆Tlm
where
∆Tlm =
∆To − ∆Ti
ln(∆To /∆Ti )
i.e. the log mean temperature difference.
(137)
(138)
ENG7901 - Heat Transfer II
2.3
2.3.1
32
Internal Flows: Correlations
Fully Developed Laminar Flow
• An analytical solution can be obtained for fully developed laminar flow in a circular
tube of diameter D:
N uD =
hD
k
= 4.36
qs00 = const
(139)
= 3.66
Ts = const
(140)
Table 8.1, Incropera and DeWitt, lists N uD values for fully developed laminar flow
in a variety of noncircular cross-sections.
2.3.2
Laminar Flow: Entry Region
• For a thermal entry region occurring in a fully developed velocity field, Hausen obtained the following relation for constant Ts :
N uD =
hD
0.0668(D/L)ReD P r
= 3.66 +
k
1 + 0.04[(D/L)ReD P r]2/3
(141)
which gives a mean h over the entry region. Note: this value asymptotes to the fully
developed value of N uD = 3.66.
• The correlation above is not generally applicable, since it assumes a fully developed
velocity profile. For the combined entry region (i.e. hydrodynamic and thermal entry
region) Seider and Tate obtained the following relation:
ReD P r
N uD = 1.86
L/D
1/3 µ
µs
0.14
(142)
which is valid for Ts = const, 0.48 < P r < 16, 700, 0.0044 < (µ/µs ) < 9.75, and
ReD P r(D/L) > 10.
• The properties in both of these correlations are evaluated at the mean bulk temperature of the fluid, except µs .
2.3.3
Turbulent Flow in Circular Tubes
• The Fanning friction coefficient is defined as:
Cf =
τs
ρu2m /2
(143)
ENG7901 - Heat Transfer II
33
and the shear stress at the wall of a circular tube is:
du
τs = −µ
dr
(144)
r=ro
Using the fully developed (laminar) velocity profile, and the mean velocity obtained
from that profile:
f
Cf =
(145)
4
Using the Chilton-Colburn analogy:
Cf
f
N uD
= = StP r2/3 =
P r2/3
2
8
ReD P r
(146)
and the friction factor, Eq. (117):
4/5
N uD = 0.023ReD P r1/3
(147)
• The Dittus-Boelter correlation is the preferred form of the above correlation:
4/5
N uD = 0.023ReD P rn
(148)
where n = 0.4 when Ts > Tm , and n = 0.3 when Ts < Tm . This correlation is valid for
0.7 ≤ P r ≤ 160, ReD ≥ 10, 000, and (L/D) ≥ 10. All properties should be evaluated
at Tm . This correlation should only be used for moderate Ts − Tm .
• When the temperature difference between the wall and the bulk fluid conditions becomes large, there can be a large variation in properties in the fluid. For these conditions, Seider and Tate have developed the following correlation:
N uD =
4/5
0.027ReD P r1/3
µ
µs
0.14
(149)
which is valid for 0.7 ≤ P r ≤ 16, 700, ReD ≥ 10, 000, and (L/D) ≥ 10. All properties
should be evaluated at Tm except µs .
• Both Eqs. (148) and (149) are valid for constant Ts and qs00 .
• To reduce the errors (which may be as large as 25%) that may be induced by Eqs.
(148) and (149), Petukhov developed the following correlation:
N uD =
(f /8)ReD P r
1.07 + 12.7(f /8)1/2 (P r2/3 − 1)
(150)
which can produce errors of up to 10%. This correlation is valid for 0.5 < P r < 2000,
and 104 < ReD < 5 × 106 , and for constant Ts and qs00 boundary conditions. The
friction factor may be obtained from a Moody diagram, or an appropriate smooth
tube correlation. Fluid properties are evaluated at the fluid bulk temperature, except
for µs .
ENG7901 - Heat Transfer II
34
• Gnielinski modified the Petukhov correlation for use at lower Reynolds numbers:
N uD =
(f /8)(ReD − 1000)P r
1 + 12.7(f /8)1/2 (P r2/3 − 1)
(151)
This correlation is valid for 0.5 < P r < 2000, and 3000 < ReD < 5 × 106 , and for
constant Ts and qs00 boundary conditions. The friction factor may be obtained from
a Moody diagram, or an appropriate smooth tube correlation. Fluid properties are
evaluated at the fluid bulk temperature, except for µs .
2.3.4
Turbulent Flow: Entry Region
• The turbulent entry region is usually small 10 < (xf d,t /D) < 60, and it is often
acceptable to assume that the fully-developed N uD can also be used in the developing
region.
• Nusselt developed the following correlation:
4/5
N uD = 0.036ReD P r1/3
D
L
0.055
10 <
L
< 400
D
(152)
where the properties are evaluated at the mean bulk temperature of the fluid.
2.3.5
Flows in Noncircular Tubes
• The previously listed correlations may be used for tubes of noncircular cross-sections
when the diameter D is replaced by the hydraulic diameter, Dh :
Dh =
4Ac
P
(153)
where Ac is the cross-section of the tube, and P is the wetted perimeter.
• Correlations for flow in a concentric tube annulus are given in Section 8.7, Incropera
and DeWitt, however, the Dittus Boelter correlation, Eq. (148), may be used with
the appropriate hydraulic diameter for fully developed turbulent flow in the annulus.
ENG7901 - Heat Transfer II
3
35
Natural Convection
3.1
Governing Equations
• Natural (or free) convection occurs when a body force acts on a fluid in which there
are density gradients → buoyancy forces.
• Fluid velocities in natural convection are much smaller than for forced convection,
therefore, the heat transfer coefficients are much smaller. Natural convection often is
the largest resistance in multi-mode heat transfer analyses.
• In general, fluid density decreases with increasing temperature (∂ρ/∂T < 0), therefore,
fluids rise when heated.
• The presence of a density gradient, however, does not guarantee the presence of natural
convection.
If T2 is sufficiently larger than T1 , the buoyancy forces become large enough to overcome the viscous forces, and an unstable fluid recirculation develops. When T1 > T2 ,
the flow is stable as the lower density fluid is above the higher density fluid, and the
flow is thermally stratified.
• If a vertical flat plate possessing a uniform temperature Ts is placed in an infinite
quiescent medium at temperature T∞ , where Ts > T∞ , the fluid near the plate will
be heated and begin to rise. This fluid motion will entrain fluid from the quiescent
region, and lead to formation of a boundary layer.
ENG7901 - Heat Transfer II
36
• The velocity of the fluid at the plate and at y = ∞ is zero.
• The boundary layer will initially be laminar, but instabilities in the flow will eventually
overcome the damping effects of viscosity, and the flow will go through a transition
to a turbulent boundary layer (with the expected increase in heat transfer rates).
• The equations governing the fluid motion are the continuity, Navier-Stokes and energy
equations. The equations reduce to the same form as those used for a forced convection
boundary layer on a flat plate, except for a modification in the momentum equation.
• Consider steady, two-dimensional natural convection of a constant property Newtonian fluid driven by a constant temperature (Ts > T∞ ) vertical flat plate. The x
momentum equation is:
u
∂u
∂u
1 ∂p
∂2u
+v
=−
−g+ν 2
∂x
∂y
ρ ∂x
∂y
(154)
The flow is assumed incompressible, however, a variable density must be accounted
for in a buoyancy force term (Boussinesq approximation).
• The flat plate boundary layer approximation illustrated that pressure is constant in
the y direction, therefore, the x pressure gradient inside the boundary layer is the x
pressure gradient (hydrostatic) in the quiescent portion of the fluid.
∂p
= −ρ∞ g
∂x
(155)
• Substituting Eq. (155) into Eq. (154) gives:
u
∂u
∂u
g
∂2u
+v
= (ρ∞ − ρ) + ν 2
∂x
∂y
ρ
∂y
(156)
ENG7901 - Heat Transfer II
37
The first term on the RHS is the buoyancy force term. Defining the volume coefficient
of expansion, β:
1 ∂ρ
(157)
β=−
ρ ∂T p
which can be expressed in the following approximate form:
1
β≈−
ρ
ρ∞ − ρ
T∞ − T
(158)
then
ρ∞ − ρ ≈ ρβ(T − T∞ )
(159)
and the buoyancy force term can be replaced, to give the following form of the x
momentum equation:
u
∂u
∂u
∂2u
+v
= gβ(T − T∞ ) + ν 2
∂x
∂y
∂y
(160)
The dependence of the buoyancy force on the temperature difference is now shown
explicitly in the momentum equation.
• The equations that must be solved for the natural convection boundary layer are the
following forms of the continuity, x momentum, and energy equations.
u
∂u ∂v
+
=0
∂x ∂y
(161)
∂u
∂2u
∂u
+v
= gβ(T − T∞ ) + ν 2
∂x
∂y
∂y
(162)
∂T
∂T
∂2T
+v
=α 2
(163)
∂x
∂y
∂y
Note: the energy and momentum equations must be solved simultaneously, due to the
coupling through the buoyancy force term.
u
• Using the following nondimensional variables:
x
y
x∗ =
y∗ =
(164)
L
L
u
v
T − T∞
u∗ =
v∗ =
θ=
(165)
uo
uo
Ts − T∞
where uo is an arbitrary reference velocity, the x momentum and energy equations
can be written in the following nondimensional forms:
u∗
∗
∂u∗
gβ(Ts − T∞ )L
1 ∂ 2 u∗
∗ ∂u
+
v
=
θ
+
∂x∗
∂y ∗
u2o
ReL ∂y ∗2
(166)
∂θ
1
∂2θ
∗ ∂θ
+
v
=
∂x∗
∂y ∗
ReL P r ∂y ∗2
(167)
u∗
ENG7901 - Heat Transfer II
38
• The dimensionless parameter on the RHS of the momentum equation can be written
in a more convenient form (without uo ), by multiplying it by Re2L . The result is the
Grashof number, GrL :
gβ(Ts − T∞ )L
GrL =
u2o
uo L
ν
2
=
gβ(Ts − T∞ )L3
ν2
(168)
• The Grashof number is the ratio of buoyancy forces to viscous forces, and it plays a
role similar to the Reynolds number in forced convection.
• From the nondimensional form of the governing equations we should expect:
N uL = f (GrL , ReL , P r)
(169)
This would be true when forced and free convection are of similar magnitude, i.e.
GrL /Re2L ≈ 1. When GrL /Re2L 1 natural convection dominates and N uL =
f (GrL , P r). When GrL /Re2L 1 forced convection dominates and N uL = f (ReL , P r).
3.2
Empirical Correlations
• The product GrL P r appears frequently in Natural convection calculations, and it is
defined as the Rayleigh number, RaL :
RaL = GrL P r =
gβ(Ts − T∞ )L3
να
(170)
• The critical Rayleigh number, Rax,c , for transition to turbulent flow on a vertical flat
plate is:
gβ(Ts − T∞ )x3
≈ 109
(171)
Rax,c = Grx,c P r =
να
and as for forced convection, empirical correlations are required to model natural
convection for turbulent flows.
• For most engineering calculations, the correlation for N u in natural convection heat
transfer takes the following simple form:
N uL =
hL
= CRanL
k
(172)
where the constants C and n depend on the flow geometry and the Rayleigh number.
All properties in Eq. (172) are evaluated at the film temperature:
Tf =
T∞ + Ts
2
(173)
ENG7901 - Heat Transfer II
3.2.1
39
Vertical Flat Plate (Ts = const)
• Equation (172) can be used to determine the Nusselt number for a uniform temperature vertical flat plate. The most commonly used constants are:
1/4
RaL < 109
(174)
1/3
RaL > 109
(175)
N uL = 0.59RaL
N uL = 0.10RaL
• The following, more accurate, correlations have been proposed by Churchill and Chu:
1/6
(
N uL =
0.387RaL
0.825 +
[1 + (0.492/P r)9/16 ]8/27
)2
10−1 ≤ RaL ≤ 1012
(176)
1/4
N uL = 0.68 +
0.670RaL
[1 + (0.492/P r)9/16 ]4/9
RaL ≤ 109
(177)
These correlations are valid for a wider range of Rayleigh numbers.
3.2.2
Vertical Flat Plate (qs00 = const)
• A modified Grashof number Gr∗ , may be used in the analysis of constant flux boundary conditions for vertical flat plates.
Grx∗ = Grx N ux =
gβqs00 x4
kν 2
(178)
• For laminar flow:
N ux =
hx x
= 0.60(Grx∗ P r)1/5
k
105 < Grx∗ < 1011
(179)
2 × 1013 < Grx∗ P r < 1016
(180)
and for turbulent flow:
N ux = 0.17(Grx∗ P r)1/4
where all properties are evaluated at the film temperature.
• These correlations are valid for water and air.
• The mean heat transfer coefficient can be determined from these correlations using
the following equation:
5
h = h|x=L
(181)
4
• Equation (177) may also be used for the constant wall heat flux boundary condition
when N uL and ReL are based on the temperature difference at the midpoint of the
plate.
ENG7901 - Heat Transfer II
3.2.3
40
Vertical Cylinders
• It has been determined experimentally that the correlations for free convection from
vertical flat plates, Eqs. (174) through (177), may be used for vertical cylinders when:
D
35
≥
1/4
L
GrL
3.2.4
(182)
Horizontal Cylinders
• The Nusselt number for natural convection from a long isothermal horizontal cylinder
can be determined from:
hD
N uD =
= CRanD
(183)
k
where the constants C and n are determined from Table 9.1, Incropera and DeWitt,
and N uD and RaD are based on the diameter of the cylinder.
• A more accurate (and complicated) correlation valid for a wide range of Rayleigh
numbers was developed by Churchill and Chu:
1/6
(
N uD =
0.387RaD
0.60 +
[1 + (0.559/P r)9/16 ]8/27
)2
RaD < 1012
(184)
which is simplified to:
1/4
N uD = 0.36 +
0.518RaD
[1 + (0.559/P r)9/16 ]4/9
RaD < 109
(185)
for laminar flows.
3.2.5
Horizontal Flat Surfaces (Ts = const)
• The heat transfer coefficient for free convection from horizontal surfaces is strongly
dependent on the orientation of the surface and the sign of the temperature difference
(Ts − T∞ ).
ENG7901 - Heat Transfer II
41
• When a heated surface faces upward, or a cooled surface faces downward, the fluid
is free to move away from the surface, and this helps to promote heat transfer. If
a heated surface faces downward, or a cooled surface faces upward, the fluid has to
travel across the surface before it can rise, or fall, in its preferred direction. These
orientations cause an extra resistance to the fluid flow, and act to decrease heat
transfer.
• The characteristic length to be used in the Rayleigh number is defined by:
L≡
As
P
(186)
where As is the area for heat transfer, and P is the perimeter of the surface.
• For the upper surface of a heated plate, and the lower surface of a cooled plate:
1/4
104 ≤ RaL ≤ 107
(187)
1/3
107 ≤ RaL ≤ 1011
(188)
N uL = 0.54RaL
N uL = 0.15RaL
• For the lower surface of a heated plate, and the upper surface of a cooled plate:
1/4
N uL = 0.27RaL
105 ≤ RaL ≤ 1010
(189)
ENG7901 - Heat Transfer II
3.2.6
42
Horizontal Flat Surfaces (qs00 = const)
• For the upper surface of a heated plate, and the lower surface of a cooled plate:
1/3
N uL = 0.13RaL
1/3
N uL = 0.16RaL
RaL < 2 × 108
(190)
2 × 108 ≤ RaL ≤ 1011
(191)
• For the lower surface of a heated plate, and the upper surface of a cooled plate:
1/5
N uL = 0.58RaL
106 ≤ RaL ≤ 1011
(192)
• The characteristic length is L = As /P .
• The properties in these correlations are evaluated at:
Te = Ts − (Ts − T∞ )/4
(193)
where T s is the mean surface temperature, related to the heat flux by:
h=
qs00
Ts − T∞
(194)
and the Nusselt number is defined as:
N uL =
hL
qs00 L
=
k
(Ts − T∞ )k
(195)
ENG7901 - Heat Transfer II
3.2.7
43
Inclined Surfaces
• Different fluid flows exist depending on which side of an inclined plate is considered.
• For a heated plate facing down, and a cooled plate facing up, it has been found that
Eqs. (176) and (177) can be used to determine N uL when g in the Rayleigh number
is replaced by g cos θ. If θ > 60o the horizontal plate correlations should be used.
• When a heated surface faces upward (or a cooled surface faces downward) parcels of
fluid may rise (or fall) from the surface, depending on the Rayleigh number, and this
results in three-dimensional flow, thinning of the boundary layer, promotion of heat
transfer, and increase in the heat transfer coefficient. This is an area of continuing
research.
ENG7901 - Heat Transfer II
3.2.8
44
Spheres
• Churchill has obtained the following correlation for natural convection from isothermal
spheres:
1/4
0.589RaD
N uD = 2 +
RaD ≤ 1011
(196)
[1 + (0.469/P r)9/16 ]4/9
This correlation is valid for all fluids with P r ≥ 0.7, and the properties are evaluated
at the film temperature.
ENG7901 - Heat Transfer II
4
45
Heat Exchangers
4.1
Introduction
• A heat exchanger is a device used to promote the exchange of heat between two fluids.
Heat exchangers are in common use, e.g. car radiator, air conditioning systems, space
heating, waste heat recovery, boilers, condensers, chemical processes, etc.
4.1.1
Types of Heat Exchangers
• Double pipe (or concentric tube) - parallel and counterflow
– The counterflow exchanger is a better design, because it will maintain a higher
temperature difference between the hot and cold fluids.
• Cross flow - mixed and unmixed (e.g. car radiator)
– In an unfinned tube crossflow exchanger, the tube fluid is unmixed, and the
external fluid is mixed. The temperature of the external fluid is relatively uniform
over the outside of the tube due to this mixing.
ENG7901 - Heat Transfer II
46
– In a finned tube crossflow exchanger, both fluids are unmixed, therefore, a temperature gradient can exist along the length of the tube, this will help to promote
heat transfer, because a larger temperaure difference can be maintained between
the internal and external fluids.
• Shell and tube exchanger (chemical processes, heating system, condenser)
– The tube fluid enters a header and is separated into tubes. These tubes make
two or more passes inside a shell. The shell fluid is forced to flow across the
tubes by baffles inside the shell (to promote heat transfer). By separating the
flow into tubes a large area for heat transfer can be created.
• Plate heat exchanger (industrial processes) - fluids are forced to flow between plates,
which have patterns stamped in them. Very large heat transfer areas can be created,
but large pumping costs result, therefore, they are most often used for gases.
• Compact heat exchangers - exchangers with a very high surface area to volume ratio
(≥ 700 m2 /m3 ). Typically, one fluid is a gas with a low heat transfer coefficient,
ENG7901 - Heat Transfer II
47
therefore, the large area is used to reduce the thermal resitance associated with the gas.
The flow is usually laminar, to reduce the high pumping costs associated with large
flow area. Compact heat exchangers are used in applications where space restrictions
are more important than pumping costs. (e.g. home air/heat exchanger)
• All heat exhangers can be analysed using one of two methods: (1) the Log-Mean Temperature Difference (LMTD) method; and (2) the Effectiveness-Number of Transfer
Units (-NTU) method.
4.1.2
Overall Heat Transfer Coefficient
• Consider a double pipe heat exchanger:
• The overall heat transfer coefficient, U , is defined as:
∆Tlm
q=P
= U A∆Tlm
Rth
(197)
therefore, the overall heat transfer coefficient is:
U A = Ui Ai = Uo Ao =
ln(Do /Di )
1
1
+
+
h i Ai
2πkL
h o Ao
or
1
Ai ln(Do /Di ) Ai 1
+
+
hi
2πkL
Ao h o
−1
Ao 1
Ao ln(Do /Di )
1
Uo =
+
+
Ai h i
2πkL
ho
−1
Ui =
−1
(198)
(199)
(200)
ENG7901 - Heat Transfer II
48
• During operation of a heat exchanger, the heat transfer surfaces of the exchanger
become coated with a film of deposits (chemical, biological, corrosion), and these
deposits add a thermal resistance to the thermal circuit between the two fluids, i.e.
a fouling resistance, Rf . This fouling resistance may exist on both the inner and
outer surfaces of the exchanger. The fouling resistance is defined in terms of a fouling
factor, Rf00 (m2 ·o C/W), which is dependent on the operating conditions, fluids, and
heat exchanger materials. Examples of some fouling factors are given in Table 11.1,
Incropera and DeWitt. Typical fouling factors are of order 10−3 - 10−4 m2 ·o C.
• Inclusion of internal and external fouling factors in U A, Eq. (198) gives:
00
00
Rf,i
1
ln(Do /Di ) Rf,o
1
U A = Ui Ai = Uo Ao =
+
+
+
+
hi Ai
Ai
2πkL
Ao
h o Ao
"
4.2
4.2.1
Log Mean Temperature Difference
Parallel Flow
• Assume:
1. The exchanger is insulated from its surroundings.
2. Axial conduction in the fluids is negligible.
#−1
(201)
ENG7901 - Heat Transfer II
49
3. ∆ek = ∆ep = W = 0
4. Specific heats are constant.
5. U is constant.
• An energy balance on each fluid gives:
dq = −ṁh cp,h dTh = −Ch dTh
= ṁc cp,c dTc = Cc dTc
(202)
Equation (202) can be integrated to give:
q = Ch (Th,i − Th,o ) = Cc (Tc,o − Tc,i )
(203)
Also, the heat transfer rate between the two fluids can be written as:
dq = U ∆T dA
(204)
where ∆T = Th − Tc at any location x, and d(∆T ) = dTh − dTc . Using Eq. (202),
d(∆T ) can be written as follows:
1
1
+
d(∆T ) = −dq
Ch Cc
(205)
Substituting for dq using Eq. (204):
d(∆T ) = −U
1
1
+
∆T dA
Ch Cc
(206)
Rearranging Eq. (206) and integrating from the inlet to the outlet of the exchanger:
o
d(∆T )
1
1
= −U
+
dA
∆T
Ch Cc
i
i
∆To
1
1
ln
= −U A
+
∆Ti
C
Cc
h
Th,i − Th,o Tc,o − Tc,i
= −U A
+
q
q
Z
o
Z
(207)
(208)
(209)
which can be rearranged to give:
q = UA
(Th,o − Tc,o ) − (Th,i − Tc,i )
ln [(Th,o − Tc,o )/(Th,i − Tc,i )]
(210)
or
q = U A∆Tlm,P F
where
∆Tlm,P F =
∆To − ∆Ti
∆Ti − ∆To
=
ln(∆To /∆Ti )
ln(∆Ti /∆To )
(211)
(212)
ENG7901 - Heat Transfer II
4.2.2
50
Counterflow
• Assume:
1. The exchanger is insulated from its surroundings.
2. Axial conduction in the fluids is neglgible.
3. ∆ek = ∆ep = W = 0
4. Specific heats are constant.
5. U is constant.
• An energy balance on each fluid gives:
dq = −ṁh cp,h dTh = −Ch dTh
= −ṁc cp,c dTc = −Cc dTc
(213)
Equation (213) can be integrated to give:
q = Ch (Th,i − Th,o ) = Cc (Tc,o − Tc,i )
(214)
Also, the heat transfer rate between the two fluids can be written as:
dq = U ∆T dA
(215)
ENG7901 - Heat Transfer II
51
where ∆T = Th − Tc at any location x, and d(∆T ) = dTh − dTc . Using Eq. (213),
d(∆T ) can be written as follows:
d(∆T ) = −dq
1
1
−
Ch Cc
(216)
Substituting for dq using Eq. (215):
d(∆T ) = −U
1
1
∆T dA
−
Ch Cc
(217)
Rearranging Eq. (217) and integrating from the inlet to the outlet of the exchanger
(w.r.t. the hot side):
o
o
d(∆T )
1
1
= −U
−
dA
∆T
Ch Cc
i
i
∆To
1
1
ln
= −U A
−
∆Ti
C
Cc
h
Th,i − Th,o Tc,o − Tc,i
= −U A
−
q
q
Z
Z
(218)
(219)
(220)
which can be rearranged to give:
q = UA
(Th,o − Tc,i ) − (Th,i − Tc,o )
ln [(Th,o − Tc,i )/(Th,i − Tc,o )]
(221)
or
where
∆Tlm,CF =
4.2.3
q = U A∆Tlm,CF
(222)
∆To − ∆Ti
∆Ti − ∆To
=
ln(∆To /∆Ti )
ln(∆Ti /∆To )
(223)
Temperature distributions for Special Cases
• For condensation, the temperature of the hot fluid is approximately constant, therefore, Ch → ∞ (a similar temperature distribution is obtained when Ch >> Cc ).
• For evaporation, the temperature of the cold fluid is approximately constant, therefore,
Cc → ∞ (a similar temperature distribution is obtained when Cc >> Ch ).
ENG7901 - Heat Transfer II
4.2.4
52
∆Tlm for Other Exchanger Configurations
• For other heat exchanger configurations, the ∆Tlm,CF is used in conjunction with a
correction factor, F , determined from figures in Section 11S.1 on the Incropera and
DeWitt student website at Wiley.com, i.e.:
q = U AF ∆Tlm,CF
(224)
and F = 1 for evaporation and condensation.
• The LMTD method is very useful in design calculations where the goal of the calculation is the size of the exchanger. For example, given the hot and cold mass flow rates,
the inlet and outlet temperatures of the hot and cold fluids, and the diameters of the
tubes used in a double pipe heat exchanger, one could evaluate U and then calculate
the required area or length of the exchanger.
• In general, the LMTD method is used when all inlet and outlet temperatures are
available, or may be easily evaluated.
• If all inlet and oulet temperatures are not available, the LMTD method becomes
tedious, as an iterative solution is required. For example, a heat exchanger is to be
used at off-design conditions, i.e. a mass flow rate is changed. In this case, the inlet
temperatures, inlet mass flow rates, and geometry of the exchanger would be known.
The desired properties would be the exit temperatures of the two fluids and the overall
heat transfer rate. To determine the exit temperatures iterations would be required.
4.3
Effectiveness - NTU Method
• The Effectiveness - Number of Transfer Units (-N T U ) method can be used for design
(i.e. sizing) and off-design (i.e. performance) calculations, without iterations.
ENG7901 - Heat Transfer II
53
• The heat exchanger effectiveness, , is defined as the ratio of the actual heat transfer
rate between the two fluids to the maximum possible heat transfer rate between the
fluids:
Actual Heat Transfer
=
(225)
Maximum Possible Heat Transfer
i.e. how good is the exchanger at doing its job (2nd Law efficiency).
• The actual heat transfer rate may be calculated from an energy balance on the hot or
cold fluid:
q = Ch (Th,i − Th,o ) = Cc (Tc,o − Tc,i )
(226)
• The maximum heat transfer rate would occur if the fluid with the minimum heat
capacity rate underwent the maximum temperature difference in the exchanger:
qmax = Cmin (Th,i − Tc,i )
(227)
• The minimum fluid may be either the hot or cold fluid, depending on the operating
conditions, therefore, may be defined as follows when Ch < Cc :
=
Th,i − Th,o
Th,i − Tc,i
(228)
=
Tc,o − Tc,i
Th,i − Tc,i
(229)
or if Cc < Ch :
In general:
=
∆Tmin f luid
∆Tmax in the exchanger
(230)
• The heat transfer rate may be determined from knowledge of :
q = qmax = Cmin (Th,i − Tc,i )
(231)
• The goal of the -N T U method is to eliminate iterations in performance calculations,
therefore, an expression of that is not a function of temperature is required. In
general:
= f (N T U, Cr )
(232)
where the number of transfer units:
NTU =
UA
Cmin
(233)
is a reflection of the size of the exchanger, and Cr is a ratio of the heat capacity rates:
Cr =
Cmin
Cmax
(234)
ENG7901 - Heat Transfer II
54
• Consider a parallel flow double pipe heat exchanger where Cc < Ch . Then, from Eq.
(229):
Tc,o − Tc,i
(235)
=
Th,i − Tc,i
and from Eq. (226):
Cr =
Th,i − Th,o
Cc
=
Ch
Tc,o − Tc,i
(236)
Integration of Eq. (206) gives:
Th,o − Tc,o
ln
Th,i − Tc,i
or
1
1
= −U A
+
Ch Cc
UA
Cmin
= −
1+
Cmin
Cmax
= −N T U (1 + Cr )
Th,o − Tc,o
= exp(−N T U (1 + Cr ))
Th,i − Tc,i
(237)
(238)
(239)
(240)
The left hand side of this equation may be written as:
Th,o − Tc,o
Th,o − Tc,i + Tc,i − Tc,o
=
Th,i − Tc,i
Th,i − Tc,i
(241)
Th,o = Th,i − Cr (Tc,o − Tc,i )
(242)
But, from Eq. (236):
therefore, Eq. (241) can be written as:
Th,o − Tc,o
Th,i − Tc,i
(Th,i − Tc,i ) − (Tc,o − Tc,i ) − Cr (Tc,o − Tc,i )
Th,i − Tc,i
= 1 − − Cr = 1 − (1 + Cr )
=
(243)
(244)
Returning to Eq. (240):
1 − (1 + Cr ) = exp(−N T U (1 + Cr ))
(245)
or
1 − exp(−N T U (1 + Cr ))
(246)
1 + Cr
Note: a similar expression would arise if Ch < Cc , see Section 11.4.2, Incropera and
DeWitt.
=
• Expressions for the of other exchanger designs are included in Table 11.3. In design
calculations it is easier to work with:
N T U = f (, Cr )
where these functions are included in Table 11.4.
(247)
ENG7901 - Heat Transfer II
55
• In a condenser or boiler Cr = 0, and for any type of heat exchanger:
= 1 − exp(−N T U )
NTU
= − ln(1 − )
(248)
(249)
• Note:
1. Read section 11.4.2 for information regarding use of Tables 11.3 and 11.4.
2. The expressions in Tables 11.3 and 11.4 are complicated, for certain types of
exchangers, therefore, they are plotted in Figs. 11.10 through 11.15.
ENG7901 - Heat Transfer II
56
Bibliography
Cengel, Y.A., Heat Transfer, A Practical Approach, WCB McGraw-Hill, New York,
1998.
Daily, J.W. and Harleman, R.F., Fluid Dynamics, Addison-Wesley Publishing Company, Don Mills, 1966.
van Dyke, M., An Album of Fluid Motion, Parabolic Press, Stanford, 1982.
Holman, J.P., Heat Transfer, 7th ed., McGraw-Hill Publishing Company, Toronto,
1990.
Incropera, F.P., DeWitt, D.P., Bergman, T.L. and Lavine, A.S., Introduction to Heat
Transfer, 5th ed., John Wiley and Sons, New York, 2007.
Kays, W.M. and London, A.L., Compact Heat Exchangers, 2nd ed., McGraw-Hill,
NewYork, 1964.
Suryanarayana, N.V., Engineering Heat Transfer, West Publishing Company, Minneapolis, 1995.
White, F.M., Viscous Fluid FLow, 2nd ed., McGraw-Hill, Inc., New York, 1991.
Figure References
All scanned figures are from:
Incropera, F.P., DeWitt, D.P., Bergman, T.L. and Lavine, A.S., Introduction to Heat
Transfer, 5th ed., John Wiley and Sons, New York, 2007.
except for the picture of a shell and tube exchanger on page 46 which is from:
Holman, J.P., Heat Transfer, 7th ed., McGraw-Hill Publishing Company, Toronto,
1990.
ENG7901 - Heat Transfer II
A
57
Eulerian and Lagrangian Viewpoints
• The field of motion of a fluid is described in terms of the velocities (~u), and accelerations (~a) of fluid particles at various positions in the fluid space.
• The two methods of studying the motion of groups of particles in a continuum are
the Lagrangian and Eulerian viewpoints.
A.1
Lagrangian View
• The co-ordinates of a moving particle are functions of time, i.e. attention is focussed
on a particle and this particle is followed in space.
x = x(a, b, c, t)
(250)
y = y(a, b, c, t)
(251)
z = z(a, b, c, t)
(252)
Where a, b, and c are used to define the co-ordinates of the particle at an initial time
t0 .
• Since the position of the particle is a function of time only, the velocity and acceleration
of the particle can be defined as follows:
~u =
~a =
dx~ dy ~ dz ~
i+ j+ k
dt
dt
dt
d2 x~ d2 y ~ d2 z ~
i+ 2j+ 2k
dt2
dt
dt
(253)
(254)
• The Lagrangian view is used in Mechanics, Solid Mechanics, and Kinematics.
• The number of particles that would have to be tracked to give an adequate description
of a flow field makes the Lagrangian view unwieldy for fluid flow. Further, one is
usually interested in the velocity, acceleration, temperature, and pressure at certain
locations in a fluid flow field, rather than the motion of a particular particle.
A.2
Eulerian View
• An alternate means of obtaining a description of a flow field is to focus attention on a
fixed point in space, and observe the flow characteristics at that point as particles pass
by. When enough points are used, an instantaneous picture of ~u and ~a throughout
the flow field is obtained.
ENG7901 - Heat Transfer II
58
• In the Eulerian view x, y, and z are independent variables (i.e. the location of a fixed
point in space), whereas in the Lagrangian view x, y, and z are dependent variables
(i.e. the location of a moving particle in space at a time t).
• In the Eulerian view, the velocity and acceleration are functions of position and time:
~u = ~u(x, y, z, t)
(255)
~a = ~a(x, y, z, t)
(256)
• The change in the x-component of velocity, u, is:
du =
∂u
∂u
∂u
∂u
dx +
dy +
dz +
dt
∂x
∂y
∂z
∂t
(257)
Dividing by dt:
∂u dx ∂u dy ∂u dz ∂u dt
du
=
+
+
+
dt
∂x dt
∂y dt
∂z dt
∂t dt
(258)
gives the total, substantial or material derivative:
Du
∂u
∂u
∂u
∂u
=
+u
+v
+w
Dt
∂t
∂x
∂y
∂z
(259)
which is the total rate of change of the x-component of velocity of a particle as it
passes by a fixed point in space.
• The first term on the R.H.S. of Eq. (259), ∂u/∂t, is called the local acceleration, and
the three remaining terms are the convective change, i.e. the rate of change due to
motion of a particle into an area of higher or lower velocity.
• The acceleration may be written in vector notation as follows:
~a =
D~u
∂~u ~ =
+ ~u · ∇ ~u
Dt
∂t
(260)
• A flow is defined to be steady when all local accelerations are zero, i.e. ∂~u/∂t = 0.
~ ~u = 0, which
• A flow is uniform when all convective accelerations are zero, i.e. ~u · ∇
would mean the flow is parallel.
B
Conservation of Mass (Continuity Equation)
• Consider fluid flow through a control volume of dimensions ∆x, ∆y, and ∆z that is
fixed in space.
ENG7901 - Heat Transfer II
59
• Application of the principle of mass conservation to this control volume gives:






Total rate at which
Total rate at which
Net rate of decrease

 
 

mass exits a cv  −  mass enters a cv  =  of mass within the 

across its boundary
across its boundary
control volume
(261)
or
∂(ρv)
∂(ρu)
∆x ∆y∆z − ρu∆y∆z + ρv +
∆y ∆x∆z
∂x
∂y
∂ρ
∂(ρw)
−ρv∆x∆z + ρw +
∆z ∆x∆y − ρw∆x∆y = − ∆x∆y∆z (262)
∂z
∂t
ρu +
ENG7901 - Heat Transfer II
60
Cancelling like terms and dividing by volume gives the continuity equation:
∂ρ ∂(ρu) ∂(ρv) ∂(ρw)
+
+
+
=0
∂t
∂x
∂y
∂z
(263)
which may be written using the total derivative of ρ:
Dρ
~ · ~u = 0
+ ρ∇
Dt
(264)
∂ρ ~
+ ∇ · (ρ~u) = 0
∂t
(265)
~ · (ρ~u) = 0
∇
(266)
or in the following form:
If the flow is steady:
And if the flow is incompressible the continuity equation reduces to:
~ · ~u = 0
∇
(267)
ENG7901 - Heat Transfer II
C
61
Conservation of Momentum (Navier-Stokes Equations)
• The Navier-Stokes equations define the motion of all fluids. They are a direct result
of the application of Newton’s Second Law to fluids.
• The Navier-Stokes equations may be derived using one of the following views of Newton’s Second Law:
1. The acceleration of a fluid particle of mass ∆m results from the application of
body and surface forces.
2. The rate of change in momentum of the flow through a control volume is caused
by body and surface forces.
• Using view (1):
∆m~a = F~surf ace + F~body
(268)
where the acceleration is defined as follows in the Eulerian view:
~a =
D~u
∂~u ~ =
+ ~u · ∇ ~u
Dt
∂t
(269)
∆m
D~u
= F~surf ace + F~body
Dt
(270)
therefore:
• To use view (2), consider the rate of change of momentum as fluid flows through a
control volume of dimensions ∆x, ∆y, and ∆z that is fixed in space.
Note: only fluxes of x momentum are shown.
ENG7901 - Heat Transfer II
62
• The mass of this control volume will be:
∆m = ρ∆x∆y∆z
(271)
• The rate of change in x momentum in the control volume is:

 
 

Rate of change
Total rate at which
Total rate at which

 
 

 momentum exits a cv − momentum enters a cv + of momentum within the 
control volume
across its boundary
across its boundary
(272)
or
∂(ρu)
∆x
u+
∂x
∂(ρv)
+ ρv +
∆y
u+
∂y
∂(ρw)
+ ρw +
∆z
u+
∂z
ρu +
∂u
∆x ∆y∆z − ρuu∆y∆z
∂x
∂u
∆y ∆x∆z − ρvu∆x∆z
∂y
∂u
∆z ∆x∆y − ρwu∆x∆y
∂z
∂(ρu)
+
∆x∆y∆z
∂t
(273)
Cancelling like terms and higher-order terms (e.g. ∆x2 ∆y∆z), and dividing by volume:
∂(ρu)
∂(ρv)
∂(ρw)
∂u
∂u
∂u
∂(ρu)
+u
+u
+u
+ ρu
+ ρv
+ ρw
∂t
∂x
∂y
∂z
∂x
∂y
∂z
(274)
The first term can be written as:
u
∂ρ
∂u
+ρ
∂t
∂t
(275)
Using the continuity equation, Eq. (263), the rate of change of x momentum per unit
volume is:
∂u
∂u
∂u
∂u
Du
+ ρu
+ ρv
+ ρw
=ρ
(276)
ρ
∂t
∂x
∂y
∂z
Dt
and view (2) gives:
ρ∆x∆y∆z
D~u
= F~surf ace + F~body
Dt
(277)
• Note: both views (1) and (2) give the same equation, i.e. Newton’s Second Law.
• The body forces are attributed to the mass of the body, e.g. gravity and coriolis forces
(only gravity forces will be considerd here):
F~body = (ρ∆x∆y∆z)~g
(278)
ENG7901 - Heat Transfer II
63
where
~g = gx~i + gy~j + gz~k
(279)
The x-component of the gravity force is:
Fx,body = (ρ∆x∆y∆z)gx
(280)
• The surface forces are due to the normal and shear stresses on the surfaces of the
control volume:
σx0 = σx +
0
τxy
= τxy +
0
τyz
= τyz +
∂σx
∆x
∂x
∂τxy
∆x
∂x
∂τyz
∆y
∂y
σy0 = σy +
0
τxz
= τxz +
0
τzx
= τzx +
∂σy
∆y
∂y
∂τxz
∆x
∂x
∂τzx
∆z
∂z
σz0 = σz +
0
τyx
= τyx +
0
τzy
= τzy +
• The net surface force acting in the x direction is:
∂σx
=
σx +
∆x ∆y∆z − σx ∆y∆z
∂x
∂τyx
+
τyx +
∆y ∆x∆z − τyx ∆x∆z
∂y
Fx,surf ace
∂σz
∆z
∂z
∂τyx
∆y
∂y
∂τzy
∆z
∂z
(281)
ENG7901 - Heat Transfer II
64
∂τzx
∆z ∆x∆y − τzx ∆x∆y
∂z
∂σx
∂τyx
∂τzx
∆x∆y∆z +
∆x∆y∆z +
∆x∆y∆z
∂x
∂y
∂z
+
=
τzx +
(282)
• Substituting Eqs. (280) and (282) into Eq. (277) and dividing by the volume of the
control volume gives:
ρ
Du
∂σx ∂τyx ∂τzx
=
+
+
+ ρgx
Dt
∂x
∂y
∂z
(283)
Similarly, in the y and z directions:
ρ
Dv
∂σy
∂τxy
∂τzy
=
+
+
+ ρgy
Dt
∂y
∂x
∂z
(284)
ρ
Dw
∂σz
∂τxz
∂τyz
=
+
+
+ ρgz
Dt
∂z
∂x
∂y
(285)
• The stress-rate of strain relations for a Newtonian fluid are defined in Handout 1:
∂u 2 ~
σx = −p + 2µ
− µ(∇ · ~u)
∂x 3
∂v 2 ~
σy = −p + 2µ
− µ(∇ · ~u)
∂y 3
∂w 2 ~
σz = −p + 2µ
− µ(∇ · ~u)
∂z
3
∂v
∂u
τxy = τyx = µ
+
∂x ∂y
∂w ∂v
τyz = τzy = µ
+
∂y
∂z
∂u ∂w
τzx = τxz = µ
+
∂z
∂x
(286)
• Substitution of Eq. (286) into Eq. (283) gives:
ρ
Du
Dt
∂p
∂
∂u
2 ∂
∂u ∂v ∂w
+2
µ
−
µ
+
+
∂x
∂x
∂x
3 ∂x
∂x ∂y
∂z
∂
∂v
∂u
∂
∂u ∂w
µ
+
+
µ
+
+ ρgx
∂y
∂x ∂y
∂z
∂z
∂x
= −
+
(287)
If viscosity is assumed constant (valid for small changes in temperature):
Du
∂p
∂2u ∂2u ∂2u
ρ
=−
+µ
+ 2 + 2
Dt
∂x
∂x2
∂y
∂z
!
+
µ ∂ ~
(∇ · ~u) + ρgx
3 ∂x
(288)
In vector notation:
ρ
D~u
~ + µ∇2 ~u + µ ∇(
~ ∇
~ · ~u) + ρ~g
= −∇p
Dt
3
(289)
If the flow is incompressible, the continuity equation equation gives:
~ · ~u = 0
∇
(290)
ENG7901 - Heat Transfer II
65
therefore, the vector form of the Navier-Stokes equations may be written as follows
for an incompressible flow of a constant viscosity fluid:
ρ
D~u
~ + µ∇2 ~u + ρ~g
= −∇p
Dt
(291)
~
• Defining ~g = −g ∇h,
where h is the vertical direction (measured positive up), the
three components of the Navier-Stokes equations for incompressible flow of a constant
viscosity fluid are:
!
∂u
∂u
∂u
∂u
ρ
+ ρu
+ ρv
+ ρw
∂t
∂x
∂y
∂z
∂p
∂h
∂2u ∂2u ∂2u
= −
− ρg
+µ
+ 2 + 2 (292)
∂x
∂x
∂x2
∂y
∂z
∂v
∂v
∂v
∂v
ρ
+ ρu
+ ρv
+ ρw
∂t
∂x
∂y
∂z
∂p
∂h
∂2v
∂2v ∂2v
= −
− ρg
+µ
+
+
(293)
∂y
∂y
∂x2 ∂y 2 ∂z 2
∂w
∂w
∂w
∂w
ρ
+ ρu
+ ρv
+ ρw
∂t
∂x
∂y
∂z
∂p
∂h
∂2w ∂2w ∂2w
= −
− ρg
+µ
+
+
(294)
∂z
∂z
∂x2
∂y 2
∂z 2
!
!