CHAPTER THREE
METHODOLOGY
The focus of this project is to carry out an analysis on double pipe heat exchangers with passive
heat enhancement techniques and nanofluid being used as the working fluid. The passive
enhancement techniques include extended surface (fins) and twisted tape insert. Three
dimensional (3D) geometry of these double pipe heat exchangers will be design using
Solidworks and Ansys Design modeler and Ansys and Matlab was used to carry out the
simulations and computational analysis. In the first analysis water will be used as the working
fluid. They second analysis will be carried out with nanofluid as the cold fluid in the inner pipe
and water in the annulus using mixture model.
3.1
Mathematical modelling
Represented in the fig. 3.1 are the three geometrical configurations to be considered. A three
dimensional geometry of these configurations of 1.5-meter length were considered. The
dimensions for the various configurations are also tabulated below.
Fig. 3.1: double pipe heat exchanger
Table 3.1: double pipe heat exchanger
Region
Inner tube
Material
Cooper (Cu)
Outer tube
Cooper (Cu)
Part
Inner diameter
outer diameter
Inner diameter
Outer diameter
Dimension (m)
0.00813
0.00953
0.0278
0.0339
Table 3.2: dimensions of double pipe heat exchanger with extended surface (fins)
Region
Inner tube
Material
Cooper (Cu)
Outer tube
Cooper (Cu)
Rectangular fins
Cooper (Cu)
Part
inner diameter
outer diameter
Inner diameter
Outer diameter
Width
Dimension (m)
0.00813
0.00953
0.0278
0.0339
0.001
Height
No of fins
0.006
12
Table 3.3: double pipe heat exchanger with twisted tape inserts
Region
Inner tube
Material
Cooper (Cu)
Outer tube
Cooper (Cu)
Twisted tape insert
Cooper (Cu)
3.2
Part
inner diameter
outer diameter
Inner diameter
Outer diameter
Pitch
Width
Dimension (m)
0.00813
0.00953
0.0278
0.0339
0.375
0.00613
Boundary conditions
In this study the following boundary conditions were used:
Reynold number 𝑹𝒆
Inlet temperature of the cold fluid
Inlet temperature of the hot fluid
Mass flow rate of the hot fluid
3.3
𝟒𝟎𝟎𝟎 < 𝑹𝒆 < 𝟏𝟖, 𝟎𝟎𝟎
15℃
35℃
3Lpm
Governing equations
3.3.1 Mixture model
This model uses a single fluid two-phase approach; it assumes that local equilibrium between the
phases is reached over a short spatial length scale and that there is a strong coupling between the
phases. The mixture model is a simplified multiphase model. It solves the continuity, momentum,
and energy equations. It also solves the volume fraction equation for the particulate phase, and
then it uses an algebraic expression to calculate the relative velocity between the base fluid and
the particle.
The dimensional equations of the mixture model governing equations are stated below [63]:
Continuity,
∇. (𝜌𝑚 𝑣⃗𝑚 ) = 0
(1)
Momentum,
∇. (𝜌𝑚 𝑣⃗𝑚 𝑣⃗𝑚 ) = −∇𝑝 + ∇. (𝜇𝑚 ∇𝑣⃗𝑚 ) + ∇. (∑𝑛𝑘=1 𝑋𝑘 𝜌𝑘 𝑣⃗𝑑𝑟,𝑘 𝑣⃗𝑑𝑟,𝑘 )
(2)
Energy,
∇. [∑𝑛𝑘=1 𝜙𝑘 𝑣⃗𝑘 (𝜌𝑘 𝐻𝑘 + 𝑝)] = ∇. (𝑘∇𝑇)
(3)
And volume fraction
∇. (𝜙𝑝 𝜌𝑝 𝑣⃗𝑚 ) = −∇. (𝜙𝑝 𝜌𝑝 𝑣⃗𝑑𝑟,𝑝 )
𝑣⃗ = ∑𝑛𝑘=1
⃗⃗𝑘
𝜙𝑘 𝜌𝑘 𝑣
(4)
(5)
𝜌
𝜌 = ∑𝑛𝑘=1 𝜙𝑘 𝜌𝑘
(6)
𝜇 = ∑𝑛𝑘=1 𝜙𝑘 𝜇𝑘
(7)
𝑘 = ∑𝑛𝑘=1 𝜙𝑘 𝑘𝑘
(8)
𝐻𝑘 is the sensible enthalpy for phases.
The drift velocity (𝑣⃗𝑑𝑟,𝑘 ) for the secondary phase is
𝑣⃗𝑑𝑟,𝑘 = 𝑣⃗𝑘 − 𝑣⃗𝑚
(9)
The relative or slip velocity is defined as the velocity of the second phase (p) relative to the velocity
of the primary phase (f):
𝑣⃗𝑝𝑓 = 𝑣⃗𝑝 − 𝑣⃗𝑓
(10)
The drift velocity related to the relative velocity becomes
𝑣⃗𝑑𝑟,𝑝 = 𝑣⃗𝑝𝑓 − ∑𝑛𝑘=1
⃗⃗𝑓𝑘 𝜙𝑘 𝜌𝑘
𝑣
𝜌𝑚
(11)
and Manninen et al. [64] and Naumann and Schiller[65] proposed the following respective
equations for relative velocity 𝑣⃗𝑝𝑓 and the drag function 𝑓𝑑𝑟𝑎𝑔 .
𝑣⃗𝑝𝑓 =
𝜌𝑝 𝑑𝑝 2
𝜌𝑝 −𝜌𝑚
18𝜇𝑚 𝑓𝑑𝑟𝑎𝑔
𝜌𝑝
𝑓𝑑𝑟𝑎𝑔 = {
𝑎⃗
(12)
1 + 0.15𝑅𝑒𝑝 0.687 𝑅𝑒𝑝 ≤ 1000
0.0183𝑅𝑒𝑝 𝑅𝑒𝑝 ≥ 1000
(13)
Here the acceleration is determined by
𝑎⃗ = 𝑔⃗ − (𝑣⃗𝑚 . ∇)𝑣⃗𝑚
(14)
And 𝑑𝑝 is the diameter of the nanoparticles of the secondary phases and 𝑎⃗ is the secondary phase
particles acceleration.
The solids shear viscosity is given by the sum of collisional and kinetic parts and the optional
frictional part.
The collisional part is a viscosity contribution due to collisions between particles taken from the
kinetic theory of granular flow of Syamlal et al. [66].
Θ𝑝
4
𝜇𝑝,𝑐𝑜𝑙 = 5 𝜙𝑝 𝜌𝑝 𝑑𝑝 𝑔0,𝑝𝑝 (1 + 𝑒𝑝𝑝 )( 𝜋 )1/2 𝜙𝑝
(15)
while for the kinetic viscosity part the Syamlal et al. [66] model is used to calculate it. The
expression is given as:
𝜇𝑝,𝑘𝑖𝑛 =
𝜙𝑝 𝑑𝑝 𝜌𝑝 √Θ𝑝 𝜋
6(3−𝑒𝑝𝑝 )
2
[1 + 5 (1 + 𝑒𝑝𝑝 )(3𝑒𝑝𝑝 − 1)𝜙𝑝 𝑔0,𝑝𝑝 ]
(16)
and the bulk viscosity is the granular particle’s resistance to compression or expansion. The model
is developed from the kinetic theory of granular flow based on Lun et al. [67].
4
Θ𝑝
3
𝜋
𝜆𝑝 = 𝜙𝑝 𝜌𝑝 𝑑𝑝 𝑔0,𝑝𝑝 (1 + 𝑒𝑝𝑝 )( )1/2
(17)
where, in equations (47-49) 𝑔0,𝑝𝑝 is the radial distribution function and Θ𝑝 is the granular
temperature and 𝑒𝑝𝑝 is the restitution coefficient and 𝜆𝑝 is the bulk viscosity.
3.4
Heat transfer equation
3.4.1 Heat transfer rate
The heat transfer rate 𝑄𝑤 of the hot fluid is calculated by:
𝑄𝑤 = 𝑚̇𝑤 𝐶𝑝𝑤 (𝑇𝑖𝑛 − 𝑇𝑜𝑢𝑡 )𝑤
(19)
Where 𝑚̇𝑤 is the mass flow rate of the hot water and 𝐶𝑝𝑤 is the specific heat capacity of water at
constant pressure.
𝑚̇𝑤 = 𝜌𝑤 𝐴𝑉𝑤
(20)
Where
𝜌𝑤 is the density of water,
𝑉𝑤 is the velocity of water,
𝐴 is the cross-section area of the pipe.
The heat transfer rate (Qnf) of the nanofluid is calculated by:
𝑄𝑛𝑓 = 𝑚̇𝑛𝑓 𝐶𝑝𝑛𝑓 (𝑇𝑖𝑛 − 𝑇𝑜𝑢𝑡 )𝑛𝑓
(21)
Where 𝑚̇𝑛𝑓 is the mass flow rate of the hot water and 𝐶𝑝𝑛𝑓 is the specific heat capacity of water
at constant pressure.
The average heat transfer (Qm) is calculated by:
𝑄𝑚 =
𝑄𝑤 +𝑄𝑛𝑓
2
(22)
𝑄𝑚 is the average heat transfer rate between the nanofluid and the hot water.
3.4.2 Heat transfer coefficient and Nusselt number
The following equations are used to calculate the heat transfer coefficient (hnf) and Nusselt
number (Nunf) of the nanofluid.
ℎ𝑛𝑓 = 𝑇
𝑞𝑚
𝑤𝑎𝑙𝑙 −𝑇𝑛𝑓
𝑁𝑢𝑛𝑓 =
ℎ𝑛𝑓 𝐷
𝑘𝑛𝑓
(23)
(24)
Where
qm is the average heat flux between the nanofluid and the hot water,
𝑇𝑤𝑎𝑙𝑙 and 𝑇𝑛𝑓 are the wall average and bulk nanofluid temperature,
𝐷 is the diameter of the nanofluid and
𝑘𝑛𝑓 is the nanofluid thermal conductivity
3.4.3 Friction factor of nanofluid
The friction factor (𝑓𝑛𝑓 ) of the nanofluid is also calculated as below:
2𝐷∆𝑃𝑛𝑓
𝑓𝑛𝑓 = 𝐿𝜌
2
𝑛𝑓 𝑢𝑚
Where
𝑘∆𝑃𝑛𝑓 is the measured nanofluid pressure drop,
𝐿 is the length of the tube,
𝜌𝑛𝑓 is the nanofluid density and,
𝑢𝑚 is the mean velocity of the nanofluid.
3.5
Thermophysical properties of nanofluid
(25)
The following published correlations are used to calculate the physical properties such as
density, viscosity, specific heat and thermal conductivity of the nanofluid.
3.5.1 Density
The density of the nanofluid is calculated using the equation below as proposed by Pak and Cho
𝜌𝑛𝑓 = 𝜙𝜌𝑝 + (1 − 𝜙)𝜌𝑤
(26)
Where
𝜙 is the volume fraction of the nanoparticles,
𝜌𝑝 is the density of the nanoparticles,
𝜌𝑤 is the density of the base fluid.
3.5.2 Viscosity
The viscosity (𝜇𝑛𝑓 ) of the nanofluid is calculated using the equation below as suggested by
Drew and Passman
𝜇𝑛𝑓 = (1 + 2.5𝜙)𝜇𝑤
(27)
Where
𝜇𝑤 is the viscosity of the base fluid.
This equation is applicable to spherical particles with less than 5% volume fraction and in this
study a very low nanofluid concentration of 0.2% will be used. Hence the above equation can be
applied.
3.5.3 Specific heat
Specific heat ((𝜌𝐶𝑝)𝑛𝑓 ) of the nanofluid can be calculated with the correlation below is
proposed by Xuan and Roetzel
(𝜌𝐶𝑝)𝑛𝑓 = 𝜙(𝜌𝐶𝑝)𝑝 + (1 − 𝜙)(𝜌𝐶𝑝)𝑤
(28)
Where
(𝜌𝐶𝑝)𝑝 is the heat capacity of the nanoparticles and,
(𝜌𝐶𝑝)𝑤 is the heat capacity of the base fluid,
3.5.4 Thermal conductivity
The thermal conductivity (𝑘𝑛𝑓 ) of the nanofluid is calculated using the correlation below known
as Kang model.
𝑎
)(𝑘𝑝 −𝑘𝑤) 𝜙
2.5
𝑎
𝑘𝑝 +2𝑘𝑤 −( )(𝑘𝑝 −𝑘𝑤 )𝜙
2.5
𝑘𝑝 +2𝑘𝑤 +2(
𝑘𝑛𝑓 = [
] 𝑘𝑤
(29)
Where
𝑘𝑝 is the thermal conductivity of the nanoparticles,
𝑎 is the slope of the relative viscosity of the nanoparticle volume fraction.
From the experimental results of Chun et al. a = 15.4150
3.6
Turbulence modelling
Turbulence modeling involves the use of mathematical model to predict turbulent effects.
Turbulent flows governing equations is directly solvable only for simple cases of flow in their
ideal state, but for most real life turbulent flows, computation fluid dynamics (CFD)
simulations are used which uses these turbulent models to predict the evolution of turbulence.
These turbulence models are simplified constitutive equations that predict the statistical
evolution of turbulent flows.
3.6.1 The k–ε (k–epsilon) model
This is the most common model used in computational fluid dynamics (CFD) simulation of mean
flow characteristics for turbulent flow conditions. As a two-equation model it gives a general
description of turbulence by means of two partial derivative equations (PDEs) known as
transport equations. The original incentive for the K-epsilon model was to develop the mixinglength model, as well as to find a substitute to algebraically prescribing turbulent length scales in
moderate to high complexity flows.
The equations for the k–ε (k–epsilon) model as as reviewed by E.J. Onyiruika et al. are defined in
the following equations.
𝜇
𝑑𝑖𝑣(𝜌𝜅⃗⃗⃗⃗)
𝑣 = 𝑑𝑖𝑣 {(𝜇 + 𝜎 𝑡 ) 𝑔𝑟𝑎𝑑 𝜅 } + 𝐺𝑘 − 𝜌𝜀
(30)
𝑘
𝜀2
𝜇
𝑑𝑖𝑣(𝜌𝜀 ⃗⃗⃗⃗)
𝑣 = 𝑑𝑖𝑣 {(𝜇 + 𝜎 𝑡 ) 𝑔𝑟𝑎𝑑 𝜀 } + 𝜌𝐶1 𝑆𝑒 − 𝜌𝐶2 𝜅+
𝑘
√𝑣𝜀
(31)
Where,
𝜂
𝜅
𝐶1 = 𝑚𝑎𝑥 [0.43, 𝜂+5], 𝐺𝑘 = 𝜇𝑡 𝑆 2 , 𝜂 = 𝑆 𝜀 and 𝑆 = √2𝑆𝑖𝑗 𝑆𝑖𝑗
𝐺𝑘 represent the generation of turbulent kinetic energy due to the mean velocity gradients.
𝑆 is the modulus of mean rate-of-strain tensor
𝜎𝑘 and 𝜎𝜀 symbolizes the effective Prandtl numbers for the turbulent kinetic energy and the rate
of dissipation respectively.
𝜇𝑡 is represented as:
𝜇𝑡 = (𝐴0 + 𝐴𝑠
𝜅𝑈 ∗ −1
𝜎𝑘
)
(32)
Where,
𝐴0 and 𝐴𝑠 are the model costants. There values are:
𝐴0 = 4.04 and 𝐴𝑠 = √6𝑐𝑜𝑠𝜙 and
1
𝜙 = 𝑐𝑜𝑠 −1 √6𝑊,
3
Where, 𝑊 =
̅ 𝑖𝑗 Ω
̅ 𝑖𝑗 ,
𝑈 ∗ = √𝑆𝑖𝑗 𝑆𝑖𝑗 + Ω
̅
̅ 𝑖𝑗 − 3ε𝑖𝑗𝜅 ω𝑘
Ω𝑖𝑗 = Ω
𝑆𝑖𝑗 𝑆𝑗𝑘 𝑆𝑘𝑖
𝑆̅ 3
(33)
̅ 𝑖𝑗 is the average rate of rotation tensor with the angular velocity ω𝑘 .
Ω
The values to the constants in the above equations are displayed below:
𝐶1 = 1.44, 𝐶2 = 1.9, 𝜎𝜅 = 1.0 and 𝜎𝜀 = 1.2
3.7
Grid independence
In order to ascertain and justify the precision and stability of the numerical results of this study,
series of calculations was carried out to determine the grid points trusted enough to give the
precise and satisfactory results suitable to define the considered double pipe heat exchangers’
geometry flow and thermal fields. This is known as grid independence study or grid sensitivity
analysis. The analysis involves changing the total number of grid of the geometry and the
combinations are studied to see the stability of the results obtained.
For this analysis various meshes of different element sizes where analyzed. Using water and
Al2O3/water nanofluid at Reynold’s number of 4000 as the working fluids. The grid distribution
study and their metrics are displayed below.
Mesh
Mesh 1
Mesh 2
Mesh 3
Mesh 4
Table 3.1: mesh metrics
Element size (m)
0.0075
0.0035
0.0015
0.00075
Fig 2a shows the variation of temperature along the center of the inner tube of the normal double
pipe heat exchanger water being the working fluid. it can be seen that there is little or no
variation in temperature with change in element size of the deferent meshes
Fig 3.2b shows radial distribution of velocity at point x=0.75m the first three meshes (mesh 1,
mesh 2, mesh3) are almost of no variations except mesh 4. From the analysis as shown in the
figure below mesh 1 and mesh 2 were likely to give an accurate and sensible result.
Fig 3.2a: Grid distribution comparison for
temperature along the center of the inner pipe
for water, Re = 4000
Fig 3.2b: Grid distribution comparison for
radial velocity for water at point x = 0.75m,
Re = 4000
The grid sensitivity analysis was also carried out for double pipe heat exchanger with extended
surface (fin) and double pipe heat exchanger with twisted tape insert with water at Reynold’s
number of 4000 as working fluid. The meshes and their element sizes are tabulated below. The
grid distribution comparison for radial velocity and temperature was carried out at the points
specified above as represented in the figures fig 3.3a and 3.3b below.
Mesh
Element size (m)
Mesh 1
0.0075
Mesh 2
0.0045
Mesh 3
0.001
Table 3.2: Mesh metrics for finned double pipe heat exchanger
From the analysis mesh 1 and mesh 2 were likely to give an accurate and reasonable results.
Fig 3.2a: Grid distribution comparison for
temperature along the center of the inner tube for
finned double pipe heat exchanger, Re = 4000
Fig 3.2b: Grid distribution comparison for radial
velocity for finned double pipe heat exchanger at
point x = 0.75m, Re = 4000
Since there was no significant variation in the values obtained in mesh 1 and mesh 2 in the two
geometries, mesh 1 was selected. Its element size is 0.0075m with 249701 nodes and 192441
elements for the normal double pipe heat exchanger while for the finned double pipe heat
exchanger its number of nodes are 395661 and the number of elements are 260486.