Master of Science in Mechanical Engineering Structural Mechanics January 2021
Heat Transfer Analysis on the Applications to Heat Exchangers
Degree Project for Master of Science in Engineering with an Emphasis on structural Mechanics
Jagadeesh Reddy Medapati
Raghu Lakshman Gundra
This thesis is submitted to the Faculty of Engineering at Blekinge Institute of Technology in partial fulﬁlment of the requirements for the degree of Master of Science in Mechanical Engineering Structural Mechanics. The thesis is equivalent to 20 weeks of full time studies.
The authors declare that they are the sole authors of this thesis and that they have not used any sources other than those listed in the bibliography and identiﬁed as references. They further declare that they have not submitted this thesis at any other institution to obtain a degree.
Contact Information:
Author(s):
Jagadeesh Reddy Medapati Email: jamd19@student.bth.se Raghu Lakshman Gundra Email: ragu19@student.bth.se
University advisor:
Wureguli Reheman
Department of Mechanical Engineering
Faculty of Engineering Internet : www.bth.se
Blekinge Institute of Technology Phone : +46 455 38 50 00
Abstract
A heat exchanger is a device used to transfer thermal energy between two or more ﬂu ids, at diﬀerent temperatures in thermal contact. They are widely used in aerospace, chemical industries, power plants, reﬁneries, HVAC refrigeration, and in many in dustries. The optimal design and eﬃcient operation of the heat exchanger and heat transfer network plays an important role in industry in improving eﬃciencies and to reduce production cost and energy consumption. In this paper, signiﬁcance of shape of inner pipe of double pipe heat exchanger was analyzed with respect to triangular, hexagonal and octagonal shaped inner pipes. The performance of double pipe heat exchangers was investigated with and without dent pattern using CFD analysis in ANSYS and eﬃcient heat transfer results are identiﬁed from CFD outputs. On ba sis of literature review, few factors inﬂuencing the eﬃciency of heat exchanger and method to improve the eﬃciency are discussed.
Keywords: CFD anlysis, Heat exchangers, Heat transfer analysis, Heat equation temperature, Temperature equilibrium.
Acknowledgments
We express our warmest thanks to our supervisor Wureguli reheman, mechanical en gineering over the entire duration of the thesis for her valuable supervision, patience, suggestions, and exceptional guidance. We wish to express our deep gratitude to our parents, Naganna dora, Padmavathi and Raghavareddy, Padmaja, and to our roommates for their enduring, unparalleled love and continual support. Finally, I want to thank all my friends who were standing next to us. It helped us a lot to complete our thesis through my good and bad times and made our thesis journey so good.
Contents
Abstract i
Acknowledgments iii
1 Introduction 3
1.1 Introduction . . . 3
1.1.1 Tubular heat exchanger . . . 5
1.2 Literature Review . . . 8
1.3 Aims and Objectives . . . 10
1.4 Strategies . . . 11
1.5 Research Questions . . . 11
1.6 Hypothesis . . . 11
1.7 Tools . . . 11
2 Theory and related works 13 2.1 Principle of Heat Transfer . . . 13
2.1.1 Heat Transfer . . . 13
2.2 Heat Generation . . . 14
2.3 OneDimensional Heat Equation . . . 15
2.4 ThreeDimensional Heat Conduction Equation . . . 17
2.5 MultiDimensional Heat Transfer . . . 19
2.6 Fluid kinematics . . . 19
2.6.1 Continuity equation in cardesian coordinates. . . 19
2.6.2 Reynold’s number. . . 21
2.7 Computational ﬂuid dynamics . . . 23
2.7.1 Concept of computational ﬂuid dynamics . . . 23
2.7.2 Navier stokes equation. . . 23
2.7.3 Grids . . . 26
3 Numerical Method 28 3.1 Numerical Methods in Heat Conduction . . . 28
3.1.1 Importance of Numerical Methods . . . 28
3.1.2 Limitations . . . 29
3.1.3 Framework . . . 29
3.1.4 Flexibility . . . 29
3.1.5 Modelling . . . 29
3.1.6 Boundary Conditions and Meshing . . . 32
4 Results 34 4.0.1 Inﬂuence of heat transfer in the presence of triangular inner pipe. 34 4.0.2 Inﬂuence of heat transfer in the presence of Hexagonal inner
pipe. . . 35 4.0.3 Inﬂuence of heat transfer in the presence of octogonal inner pipe. 37 4.0.4 Inﬂuence of heat transfer in the presence of dent to the inner
pipe of heat exchanger. . . 39
5 Conclusions and Discussions 43
6 Future works 45
A Supplemental Information 48
List of Figures
1.1 Heat exchanger ﬂow conﬁguration. . . 4
1.2 Shell and tube heat exchanger with one tube pass.[2] . . . 5
1.3 Shell and tube heat exchanger with two tube pass.[2] . . . 5
1.4 Tube conﬁguration used in shell and tube heat exchanger.[2] . . . 6
1.5 Single phase baﬄe.[2] . . . 6
1.6 Double phase baﬄe.[2] . . . 6
1.7 Rod baﬄe exchanger supported by four rodes.[2] . . . 7
1.8 Tubes with triangular layout. [2] . . . 7
1.9 Assembly of double pipe heat exchanger. [2] . . . 7
1.10 Classiﬁcation of heat transfer. . . 9
2.1 Graph illustrating sign convention for conduction heat ﬂow. . . 14
2.2 Direction of heat conduction. . . 14
2.3 Onedimensional heat conduction through a volume element in a large plate wall. . . 15
2.4 Threedimensional heat conduction through a rectangular volume el ement. . . 17
2.5 Three dimensional ﬂuid elements.[20] . . . 20
2.6 Flow chart of computational ﬂuid dynamics.[4] . . . 23
2.7 Grid domain[26] . . . 26
2.8 Structured grid and Unstructured grid. . . 26
3.1 Diagrammatic representation of dimension from numerical modelling. 30 3.2 Diagramic representation of dimension from dent model. . . 31
3.3 Meshing for octagon and hexagon inner pipes of the double pipe heat exchanger. . . 32
3.4 Meshing for triangular inner pipe of the double pipe heat exchanger. . 33
3.5 Meshing for dent model. . . 33
4.1 Temperature distribution along the length of the triangle shaped inner pipe. . . 34
4.2 Pressure distribution along the length of the triangle shaped inner pipe. 35 4.3 Temperature distribution of ethanol for triangular inner pipe. . . 35
4.4 Temperature distribution of water for triangular inner pipe. . . 35
4.5 Temperature distribution along the length of the hexagonal inner pipe. 36 4.6 Pressure distribution along the length of the hexagonal inner pipe . . 36
4.7 Temperature distribution of ethenol for hexagonal inner pipe . . . 37
4.8 Temperature distribution of water for hexagonal inner pipe . . . 37
4.9 Temperature distribution along the length of the octogonal inner pipe. 38 4.10 Pressure distribution along the length of the octogonal inner pipe. . . 38 4.11 Temperature distribution of ethenol for octagonal shaped inner pipe. 38 4.12 Temperature distribution of water for octogonal shaped inner pipe . 39 4.13 Temperature between various pipe shapes. . . 39 4.14 temperature with dent along the length of the pipe. . . 40 4.15 Pressure distribution with dent along the length of the pipe. . . 40 4.16 Temperature distribution for dent pattern along the length of the pipe. 40 4.17 Temperature distribution without dent along the length of the pipe. . 41 4.18 Pressure distribution without dent along the length of the pipe. . . . 41 4.19 Temperature distribution without dent pattern along the length of the
pipe. . . 41 4.20 Comparison of temperature distribution between with and without dent. 42
List of Tables
3.1 Geometric properties for numerical model. . . 30
3.2 Geometric properties for numerical model. . . 31
3.3 Material properties of numerical model for copper. . . 31
3.4 Material properties of numerical model for brass. . . 32
3.5 Material properties of numerical model for Ethanol. . . 32
List of Abbreviations
q_{k} Rate of conduction.
T Temperature.
dT
dx Temperature gradient.
a area.
k Thermal conductivity.
˙G Rate of heat genearation per unit volume.
˙g Heat generation.
Q˙_{x} Rate of heat conduction along xaxis Δx Thickness of the element.
ΔE Rate of change of energy in element.
ρ Density.
c Speciﬁc heat.
ΔT Change in time.
˙v_{element} Volume of the element.
˙Q_{x} = ˙Qy =
˙Q_{z} Magnitudes of heat transfer in respective directions.
σ_{xx} Normal stress along xaxis.
σ_{yy} Normal stress along yaxis.
σ_{zz} Normal stress along zaxis.
_{xx} Normal strain along xaxis.
_{yy} Normal strain along yaxis.
_{zz} Normal strain along zaxis.
γ Shear strain.
u Deﬂection.
Strain tensor.
P Pressure.
r Radius.
σ_{H} Hoop stress.
d Inside diameter.
t Thickness.
Q Rate of ﬂow.
Re Reynold’s number.
μ Dynamic viscosity.
Dh Hydraulic diameter of the pipe.
Pe Peclet number.
Sc Schmidt number.
Pr prandtl number.
Chapter 1
Introduction
1.1 Introduction
Industrial heat exchangers are industrial equipment that is designed to exchange heat from one medium to another. The primary purpose of heat exchangers is heating the element or cooling it down[22]. Within the industrial sector, cooling has a more important function to prevent equipment from overheating. There are many types of heat exchangers, each has its advantages and drawbacks. Heat exchangers have a broad range of industrial applications[24]. They are used as air conditioning com ponents in various cooling systems and heating systems. In general, many industrial processes need to be operated at certain degree of heat. For this, Great care must be taken to maintain these processes at optimum temperature. Within industrial plants, heat exchangers are highly required to keep machinery, chemicals, water, gas and the other substances within a safe operating temperature. Heat exchangers are also used to capture the excessive heat or steam, that is released as a byproduct during the operation, So that heat can be put to better use elsewhere, thereby eﬃciencies are improved.[16].
Diﬀerent types of heat exchangers function in diﬀerent ways, use diﬀerent ﬂow ar arrangements, equipment, and design features. One common thing in all heat ex changers is that, they all function to directly or indirectly expose a warmer medium to a cooler medium. Heat exchangers are usually accomplished with a set of tubes within some type of casing.
Heat exchangers are generally classiﬁed by following ways,
• Nature of the heat exchange process.
• The physical state of the ﬂuid.
• Heat exchangers ﬂow arrangements.
The heat exchanger classiﬁcation method depends on, whether or not the substance between which the heat is being exchanged come into direct contact with each other or not, whether they are separated by a physical barrier such as the walls of their tubes. In direct contact heat exchangers, the hot and cold ﬂuid comes into direct contact with each other within the tubes rather than on radiant heat or convection.
Direct contact is an extremely eﬀective means of transferring heat since the contact is direct, These Direct contact heat exchangers must function in a safe environment.
Direct contact heat exchangers are suitable if the hot and cold ﬂuid have slight tem perature variation. When it comes to the indirect contact heat exchanger, the hot and cold ﬂuids are physically separated from each other. In general, an indirect con tact heat exchanger will keep hot and cold ﬂuids in a diﬀerent set of pipes instead of radiating energy and convection to exchange the heat. This is done to prevent contamination of one ﬂuid by the other.
Heat exchangers may also be classiﬁed based on the physical state of the hot and cold ﬂuid such as,
• Liquid→ Gas.
• Liquid→ Solid.
• Gas→ Solid.
In some situations, immersible liquids may also exist that will not blend. Eg: Oil and water.
The arrangement of ﬂuids ﬂow within the heat exchanger is another important way of classifying heat exchangers. The three major categories are parallel ﬂow, Counterﬂow, and Crossﬂow. In the parallel ﬂow heat exchanger, the hot and cold ﬂuid moves into the heat exchanger from the same end and ﬂow parallel to each other in the same direction. Although, these arrangements result in lower eﬃciency than a counterﬂow arrangement, it allows achieving greater thermal uniformity across the walls of the heat exchanger. In the counterﬂow heat exchanger, the hot and cold ﬂuid enters the heat exchanger from the opposite direction and ﬂow towards each other.
The most commonly employed conﬁguration of the ﬂow is counter ﬂow arrangements.
This arrangement exhibits the highest eﬃciency as it allows a greater amount of heat transfer between ﬂuids. In the crossﬂow heat exchanger, ﬂuids ﬂow perpendicular to one another. The eﬃciency of the heat exchanger which employs this ﬂow type falls between that of countercurrent and cocurrent heat exchanger[5][25].
Figure 1.1: Heat exchanger ﬂow conﬁguration.
[25]
1.1.1 Tubular heat exchanger
Tubular heat exchanger is mostly build with circular tubes, elliptical or rectangular or some other complex shape twisted tubes depending upon the application used. These kind of heat exchangers are ﬂexible in design and geometry and can easily change the tube diameter, length and arrangement of ﬂow. That’s why these heat exchangers are considered over other heat exchangers. Tubular heat exchangers are preferred for applications related to high pressure, where the pressure diﬀerence between the ﬂuids are high, these tubular heat exchangers are popularly used for liquid to liquid and liquid to phase change heat transfer application. These heat exchangers are popularly classiﬁed as shell and tube, double pipe and spiral tube exchangers. All primary surface of heat exchangers are same, but ﬁnns may be constructed outside or inside the tube.
Shell and tube heat exchangers.
These heat exchangers are generally built by bendling the round tubes in a cylindrical shell in which tubes are aligned parallel to that of a axial to shell. One ﬂuid ﬂows through the inner pipe and other ﬂuids ﬂow across and along the tube. The major components of this heat exchanger are tube, shell, rare and head, front and head, baﬄes and tube shells etc. Depending upon the desire heat transfer and pressure drop performance, diﬀerent internal construction is used in shell and tube exchanger.
Mostly, these exchangers are used to reduce thermal stress to prevent leakage and ease of cleaning operating pressure and temperature to control corrosion.
Figure 1.2: Shell and tube heat exchanger with one tube pass.[2]
Figure 1.3: Shell and tube heat exchanger with two tube pass.[2]
Tubes
Round tubes in diﬀerent shapes are used in shell and tube heat exchanger. Mostly, the tubes are straight and Ushaped used in power industries. However, other shape
of tubes like signwave bend, Jshaped, Lshaped and inverted hockey stick are also used in advance nuclear exchangers to tolerate large thermal expansion of the tubes.
Figure 1.4: Tube conﬁguration used in shell and tube heat exchanger.[2]
Baﬄes
Baﬄes are commonly classiﬁed as transverse and longitudinal baﬄes. The purpose of longitudinal baﬄes is to control the overall ﬂow. Transverse baﬄes are used to support the tube during assembly and operation and to direct a ﬂuid in their tube bundle approximately at right angle of the tube to achieve higher heat transfer coeﬃcients.
Figure 1.5: Single phase baﬄe.[2]
Figure 1.6: Double phase baﬄe.[2]
Tubes sheets
Tube sheets are used to hold tubes at the end. These tube sheets are designed in such a way that a round metal plate with holes drilled through, for further desire tube pattern, holes for the tight rods and bolt holes for ﬂanging to the shell and channel. To prevent the leakage of the shell at the tube sheets a clearance between the tube holes and tube is used.
Figure 1.7: Rod baﬄe exchanger supported by four rodes.[2]
Figure 1.8: Tubes with triangular layout. [2]
Double pipe heat exchanger
Double pipe heat exchanger is one of the simplest type of heat exchanger. In this heat exchanger one ﬂuid ﬂows inside a pipe and the other ﬂuid ﬂows between the pipe and other pipe that surrounds the inner pipe. Figure 4.9 shows the concrete structure of the double pipe heat exchanger. In double pipe heat exchanger generally contour ﬂow direction is used for the higher performance at the surface area. These are very easy to maintain as it is very easy to disassemble them for cleaning. These heat exchangers are mostly suited when one or both of the ﬂuids are at high temperature.
Double pipe heat exchanger is generally used for small capacity application where the total heat transfer surface area required is50m^{2}. These heat exchangers are bundled with Ushaped tubes in a pipe so that, they can be used as segmental baﬄes and hair pin or jacketed Utube exchanger. In this thesis, we have concentrated on this double pipe heat exchanger because of its vast usage in oil and gas industries and in some other industrial sector. In this thesis, we had investigated various types of tubes and dent pattern on the surface of inner tube to get proper idea on how heat transfer rate varies from inlet to outlet. This idea was simulated using numerical modelling in CFD[2].
Figure 1.9: Assembly of double pipe heat exchanger. [2]
1.2 Literature Review
A heat exchanger is a device that is used to pass thermal energy in between two or more ﬂuids, inbetween ﬂuid and solid, or between solid particles and ﬂuid parti cles, at diﬀerent temperatures at thermal contact. In General, applications of heat exchangers include heating, cooling of the ﬂuid stream, and evaporation or conden sation of various types of the ﬂuid stream[24] . Heat exchangers are placed to reject heat or distill, concentrate, crystallize, or control the ﬂuid process.
In heat exchangers, heat transfer takes place in between ﬂuid and separating wall or into the outside of the wall in a transient manner. Fluids are separated by a heat transfer surface and ideally, they do not mix or leak. These types of heat exchangers are termed as direct heat transfer heat exchanger. The exchangers in which intermittent heat exchange in between hot and cold ﬂuid by transferring the thermal energy and releasing it through the heat exchange surface are termed as in direct heat transfer heat exchanger. These types of heat exchangers generally have problems of ﬂuid leakage from one ﬂuid to another ﬂuid due to the phenomena of valve switching or matrix rotation and due to pressure diﬀerences. In some cases the ﬂuids are not immeasurable, the separating wall may be eliminated, these types of heat exchangers are known as direct contact heat exchangers.
Some of the commonly used heat exchangers are namely Shell and tube heat ex changers, Automobile radiators, Airpre heaters, Cooling towers, Evaporators, and Condensers. A heat exchanger can be constructed for various applications depend ing upon the requirements. For example, there could be internal sources of thermal energy like electric heaters or nuclear fuel elements. In heat exchangers like boilers and ﬁred heaters, chemical reactions namely combustion is used for heat exchange.
Where as in inscribed surface exchangers and stirred tank reactors, mechanical de vices are used to exchange the heat. In conduction or heat pipe heat exchanger, heat transfer is in between separating walls. Here the pipe does not act as a separating wall but, allow the transfer of heat by conduction, evaporation, and conduction of working ﬂuid inside the heat pipe.
A new design of heat exchanger was proposed by Josua P.Meyer and Hilde van der vyver regarding the increase in heat transfer area signiﬁcantly using fractals.
The obtained results are investigated analytical, numerical and experimental meth ods. The results from fractal heat exchanger has a higher heat transfer to overall volume rather than a coventional tube heat exchanger[15]. Hesham G. Ibrahim [11]
had analyzed forced convection with turbulent ﬂow using imperical models and also used numerical solutions from CFD analysis to examine turbulent ﬂow pattern and heat transfer from air walls in a horizontal pipe. Even though, we didn’t concen trate turbulant ﬂow in our thesis, this paper helped us in understanding imperical correlation of Nusselts number. A study was conducted by Mehrain Hashemian and Samad Jafarmadar by introducing conical tubes instead of cylindrical tubes in order to improve geometry of double pipe heat exchanger. These conical tubes are ex amined with nine diﬀerent arrangements corresponding to diﬀerent ﬂow directions.
In this study eﬀect of hydraulic, geometrical and thermodynamic characteristics are studied. Finally, they concluded that there is a increase of ﬁftyﬁve percent in ef fectiveness and increament of forty percent in heat transfer[9]. A comprehensive review on double pipe heat exchanger was done by Mohamad omidi, Mohamad ja
Figure 1.10: Classiﬁcation of heat transfer.
fari. In this study, role of change in geometry shapes was studied in relation with heat transfer rate and various heat transfer enhancement method are reviewed and importance of nano ﬂuid in double pipe heat exchanger was discussed in detail and ﬁnally, mathematical correlation of nusselt’s number in pressure drop coeﬃcient are presented[17]. A study has been conducted by Amit rao and SV dingare regard ing the enhancement of heat transfer rate by introducing dimple on surface of heat exchanger and various heat transfer argumentation techniques are studied[21]. An experiment has been conducted by Ganesh V.wafelkar and Dr.L V kamble on triple concentric heat exchanger with reference to double pipe heat exchanger. Triple pipe heat exchanger provided larger heat transfer area as compared to double pipe heat exchanger. To enhance eﬀectiveness of heat exchanger, dimples have been introduced onto the middle tube. Experimental investigation was carried on diﬀerent ﬂow rates of hot and cold ﬂuid and relationship between Nusselt number, friction factor and heat exchanger eﬀectiveness were studied[7]. An experimental study was conducted by Pooja patil, Padmakar deshmukh on heat transfer coeﬃcients in a circular tubes fabricated with almond type dimples on the surface, the ﬁnal results are compared with basic plain tube. And results showed that dimple in circular tube has66 percent greater thermal performance factor than normal tube[19]. Heat transfer and pres sure drop are important parameters during the construction and they greatly eﬀect the performance of plate heat exchangers. The research performed by Vinay Patel on a plate heat exchanger where CFD(computational ﬂuid dynamics) computational ﬂuid dynamics applications are used to design the optimization of pasteurizer plant.
In this research CFD test results and analytical results are validated. Using these results temperature distribution, ﬂow combination and comparison of material ther mal conductivity are studied [18]. In Shell and tube heat exchangers design play a crucial role in the performance of heat ex changers. A research was conducted by Santosh k regarding the numerical study of heat transfer enhancement in shell and
tube heat exchangers with the help of CFD analysis. Here the CFD analysis is car ried with baﬄes and without tabulator’s in the initial stages and design incorporated with diﬀerent designs of baﬄes and semicircular tabulators in later stages. Finally, the pressure drop, outlet temperatures, and heat transfer coeﬃcients are validated against theoretical results[10]. The double pipe heat exchanger is the common type of heat exchanger, which are widely utilized in industrial applications the eﬃciency of heat exchangers is examined by enhancement techniques in research performed by Anton Gonez. Passive heat transfer enhancement methods are used to investigate thermal performance is impacted. The inner pipe of the heat exchanger is modiﬁed with a crosssection following the Koch snowﬂake fractal pattern. The performance of the fractal heat exchanger is compared with the double pipe heat exchanger [8].
The borehole heat exchanger is popular in a wide range of industrial applications because of its high eﬃciency, environmental conservation, and low maintenance cost so many researches are conducted on borehole heat exchangers. Yong li, Jinfeng Mao investigates the heat transfer among the legs of pipe presented inside of BHE (Bore hole Heat Exchanger). The transferred heat can either be exretracted or rejected by the subsurface of twoleg pipes namely DLP (Downward leg of pipe) and ULP (upward leg of pipe) presented inside the vertical BHE. The very small diameter of the borehole(0.11 m to 0.22 m) may lead to the temperature diﬀerence between DLP and ULP, which may eventually cause a thermal shortcircuiting. The obtained shortcircuit was investigated with the 2D model and then, A bestﬁt expression of shortcircuiting thermal resistance in dimensionless form is presented[14]. The other author has done a heat transfer analysis of ground heat exchangers consisting of inclined boreholes, this research was done by Ping Cui, Hongxing yang. In this research, the author had analyzed the heat exchange using multiple inclined bore holes. First, the authors constructed a transient threedimensional heat conduction model that describes the heat exchange between the ground and the bore while deal ing with a single inclined line source. Then the authors studied the heat exchange corresponding to multiple boreholes by superimposing the temperature ex change resulted from individual boreholes[3]. The design of heat exchangers is an important role in smooth running of heat exchangers, so the construction of heat exchangers should be based on experimental analysis using which optimum size of the parts like coil pitch, coil diameter, and ﬂow rate can be obtained. On this path, one research was done by N Jamshidi, M Farhadi in which experimental apparatus and Taguchi method is used to investigate the ﬂuid ﬂow eﬀect and change in heat transfer because of the geometrical parameters. In optimum condition, overall heat transfer coeﬃ cient of the heat exchangers are found. The contribution ratio obtained by Taguchi method says that the shellside ﬂow rate, coil diameter, and tube side ﬂow rate, and coil pitch are the important design parameters in coil heat exchangers.[12].
1.3 Aims and Objectives
This thesis aims to investigate heat transfer analysis on applications to Double pipe heat exchangers. Heat exchangers are widely used in refrigeration applications, Aerospace, the Petrochemical industry, and various other ﬁelds. Diﬀerent shapes of inner pipe are analyzed during the operation of double pipe heat exchanger, where
the volume of inner ﬂuid remains constant. Other aspect of the double pipe heat exchanger is that we analysed dent on the external surface to the inner pipe. Here, the temperature and pressure are analyzed with and without dent.
1.4 Strategies
The strategy of this master thesis is as follows
1. Objective results (data) without analysis and interpretation.
2. Objective analysis/interpretation of the results, that is based solely on the collected data.
3. Interpretation of the results and analysis within the context of the body of knowledge (external to your thesis).
1.5 Research Questions
1. How, the shape of the inner pipe inﬂuence the heat transfer in double pipe heat exchanger?
2. How implementing of dent to the inner pipe of the heat exchanger inﬂuence the heat transfer?
3. what factors inﬂuence the eﬃciency of the heat exchangers and what method we can be used to improve them?
1.6 Hypothesis
1. Numerical modelling has been done in ANSYS, to estimate the transfer of heat in double pipe heat exchanger with respect to diﬀerent shapes for inner pipe(Triangle, Hexagon, Octagon).
2. Numerical modelling is done to investigate the role of dent to the double pipe heat exchanger.
3. The intensive literature review has been done by referring to various scientiﬁc papers, articles, and journals. Various industrial applications were studied to get a proper understanding of the factors inﬂuencing the eﬃciency of the heat exchangers.
1.7 Tools
• ANSYS.
• MATLAB.
• NUMERICAL METHODS.
Chapter 2
Theory and related works
2.1 Principle of Heat Transfer
Heat transfer analysis is highly necessary to operate diﬀerent mechanisms like Con vection, Conduction and Radiation. The principals of heat transfer were followed while designing of heat exchangers and energy conversion systems. For our thesis, we have considered the basic principles of heat transfer and some simple applications.
2.1.1 Heat Transfer
Conduction is a type of heat transfer mode in which heat is transferred in the opaque solid medium under a certain temperature gradient, that exists in the particular body. In general, heat is transferred from the hightemperature region to the lower temperature region. The rate at which heat is transferred through conduction is proportional to the temperature gradient times the area through which the heat is transferred.
q_{k} ∝ AdT
dx (2.1)
Whereas,
q_{k} = rate of conduction, T=Temperature,
dT
dx = temperaturegradient,
The actual rate of heat ﬂow depends on the thermal conductivity, a physical property of the medium. Therefore, the heat transfer expression is expressed as
q_{k}= −kAdT
dx (2.2)
Figure 2.1: Graph illustrating sign convention for conduction heat ﬂow.
[13]
Figure 2.2: Direction of heat conduction.
[13]
2.2 Heat Generation
Heat conduction analysis studies the medium in which electrical, nuclear or chemical energy is converted into thermal energy. These conversion process is termed as heat generation. The rate of heat generation in a medium will varies with respect to the time as well as with respect to the position within the medium time. When the variation of heat generation with the position is known, the total rate of heat generation in a speciﬁc volume can be expressed as
˙G =
v
˙gdV (2.3)
Where ˙G is the constant rate of heat generation per unit volume.
2.3 OneDimensional Heat Equation
Let us consider, a metal plate in which heat conduction in onedimensional geometry by neglecting other directions (rectangular coordinate system). Suppose, consider a thin element of thickness Δx in a rectangular plate. Assume, the density of the plate is ρ, Speciﬁc heat is C and the area of plate to the direction of heat transfer is A.
The energy equation can be expressed as
Figure 2.3: Onedimensional heat conduction through a volume element in a large plate wall.
[1]
Rate of heat conduction at x,y,z

Rate of heat conduction at x+Δ X
+
Rate
of heat generation inside the element
=
Rate of change of energy content of the
element
or
Q˙_{x}− ˙Qx+Δx+ ˙Gelement = ΔE
Δt (2.4)
The rate of energy generation within the element is expressed as,
ΔEelement = Et+Δt− Et= mC(Tt+Δt − Tt) = pCAΔx(Tt+Δt − Tt) (2.5)
˙G_{element} = ˙gVelement= g ˙AΔx (2.6)
By substituting the above equations,
˙Q_{x}− ˙Qx+Δx+ g ˙AΔx = pCAΔxT_{t+Δt}− Tt
Δt (2.7)
Dividing by AΔx,
−1 A
˙Q_{x+Δx}− ˙Q_{x}
Δx + ˙g = pCT_{t+Δt}− Tt
Δt (2.8)
Taking limits as Δx → 0 and Δt → 0 gives, 1
A
∂
∂x
kA∂T
∂x
+ ˙g = pC∂T
∂t (2.9)
Appling Fourier law of heat conduction
lim_{Δ}_{x} ˙Q_{x+Δx}− ˙Q_{x}
Δx = ∂ ˙Q
∂x = ∂
∂x
− kA∂T
∂x
(2.10) As the area is constant for plate,
Variable conductivity
∂
∂x
k∂T
∂x
+ ˙g = pC∂T
∂t (2.11)
By reducing the above equation the thermal conductivity in practical application is expressed as,
Constant conductivity
∂^{2}T
∂x^{2} + ˙g k = 1
α
∂T
∂t (2.12)
where α=_{pc}^{k}, thermal diﬀusity.
Reducing the above equation, we get
∂^{2}T
∂x^{2} + ˙g
k = 0 (2.13)
Transient, no heat generation (˙g = 0).
∂^{2}T
∂x^{2} = 1 α
∂T
∂t (2.14)
Steady state, no heat generation (_{∂t}^{∂} = 0, ˙g = 0).
d^{2}T
dx^{2} = 0 (2.15)
2.4 ThreeDimensional Heat Conduction Equation
Major heat transfer applications are approximated as being onedimensional. In re quality, Complex material such as anisotropic as discussed above, where the property of the material may undergo more than onedimensional heat conductions. In these conditions, we have to consider the heat transfer in all directions as well. In such cases, heat conduction is said to be multidimensional.
Let us consider, a small rectangular element length Δx, width Δy height Δz.
Figure 2.4: Threedimensional heat conduction through a rectangular volume ele ment.
[1]
Assume density of body as ρ, speciﬁc heat is c, the heat balance on element during a smalltime interval Δt can be expressed as,
Rate of heat conduction at x,y and z

Rate of heat conduction at x+Δ x,y+Δ
y and z+Δ z
+
Rate of heat generation inside the element)
=
Rate of change
of the energy content of the element)
or
˙Q_{x}+ ˙Qy+ ˙Qz− ˙Qx+Δx− ˙Qy+Δy − ˙Qz+Δz+ ˙Gelement = ΔEelement
Δt (2.16)
As we are considering cubical element, the volume of the element is v_{element}=ΔxΔyΔz, the rate of heat generation within the element can be expressed as,
ΔEelement = Et+Δt− Et= mC(Tt+Δt− Tt) = pCAΔxΔyΔz(Tt+Δt− Tt) (2.17)
˙G_{element} = g ˙Velement = ˙gΔxΔyΔz (2.18)
By substituting the above equation, we get
Q˙_{x}+ ˙Q_{y}+ ˙Q_{z}− ˙Qx+Δx− ˙Qy+Δy− ˙Qz+Δz+ ˙gΔxΔyΔz = pCΔxΔyΔzT_{t+Δt}− Tt
Δt (2.19) By dividing by Δx Δy Δz we get,
− 1
ΔyΔz
˙Q_{x+Δx}− ˙Qx
Δx − 1
ΔxΔz
˙Q_{y+Δy} − ˙Qy
Δy − 1
ΔxΔy
˙Q_{z+Δz}− ˙Qz
Δz + ˙g = pCT_{t+Δt}− Tt
Δt(2.20) Consider the heat transfer area of the element for heat conduction in the x,y,z di rection are taken as Ax = Δy,Δz , Ay = Δx Δz and Az = Δx Δy. taking limit as Δx,Δy,Δz and Δt→0 gives,
∂
∂x
k∂T
∂x
+ ∂
∂y
k∂T
∂y
+ ∂
∂z
k∂T
∂z
+ ˙g = pC∂T
∂t (2.21)
By applying Fourie’s law of heat conduction we get
lim_{Δx→0} 1 ΔyΔz
˙Q_{x+Δx}− ˙Qx
Δx = 1
ΔyΔz
∂Q_{x}
∂x = 1 ΔyΔz
∂
∂x
−kΔyΔz∂T
∂x
= ∂
∂x
k∂T
∂x
(2.22) lim_{Δy→0} 1
ΔxΔz
˙Q_{y+Δy}− ˙Qy
Δy = 1
ΔxΔz
∂Q_{y}
∂y = 1 ΔxΔz
∂
∂y
−kΔxΔz∂T
∂y
= ∂
∂y
k∂T
∂y
(2.23) lim_{Δz→0} 1
ΔxΔy
˙Q_{z+Δz}− ˙Qz
Δz = 1
ΔxΔy
∂Q_{z}
∂z = 1 ΔxΔy
∂
∂z
−kΔxΔy∂T
∂z
= ∂
∂z
k∂T
∂z
(2.24) The general heat conduction equation in rectangle cordinate system, on reduction of above equation we get,
∂^{2}T
∂x^{2} +∂^{2}T
∂y^{2} + ∂^{2}T
∂z^{2} + ˙g k = 1
α
∂T
∂t (2.25)
Where α=k/ρ c , thermal diﬀusivity Reducing the above equation, we get Steady state (poison’s equation)
∂^{2}T
∂x^{2} + ∂^{2}T
∂y^{2} +∂^{2}T
∂z^{2} + ˙g
k = 0 (2.26)
Transient, no heat generation (diﬀusion equation )
∂^{2}T
∂x^{2} +∂^{2}T
∂y^{2} + ∂^{2}T
∂z^{2} + ˙g k = 1
α
∂T
∂t (2.27)
Steady state, no heat generation ( Laplace equation )
∂^{2}T
∂x^{2} + ∂^{2}T
∂y^{2} +∂^{2}T
∂z^{2} + ˙g
k = 0 (2.28)
2.5 MultiDimensional Heat Transfer
In general, heat transfer applications are divided into three types namely  dimen sional, Twodimensional, and Threedimensional, on basis of relative magnitudes of heat transfer in two diﬀerent directions. In the general case, the temperature distri bution along the medium at a certain time and heat transfer rate at a speciﬁc location can be termed in three coordinate systems namely x,y,z in a rectangle. Whereas, in a polar coordinate system the cylindrical coordinates at time and location are speciﬁed as r,ψ and z.
The rate of heat conduction along with the speciﬁc medium in a speciﬁc direction is proportional to the temperature diﬀerence across the medium and the area perpen dicular to the direction of heat transfer, but inversely proportional to the distance in the direction. This is expressed in the diﬀerential form by Fourier’s law of heat conduction for onedimensional heat conduction is,
Q_{cond} = −kAdT
dx (2.29)
Generally, heat is transferred in the direction of decreasing temperature, and such, the temperature gradient is negative when the heat is conducted in the positive x direction. The negative sign from the above equation speciﬁes the heat transfer in the positive xdirection as a positive quantity. In practical applications , the medium for temperature distribution is 3dimensional. The rate of heat conduction at a speciﬁc point can be expressed with Fourier’s law as,
Q_{n} = −kAdT
dx (2.30)
In the rectangular coordinate system, the heat conduction vector can be expressed as a component.
Q˙_{n}= ˙Q_{x}−→
i + ˙Q_{y}−→
j + ˙Q_{z}−→
k (2.31)
Here i, j, k are unit vectors and Q_{x}, Q_{y}, Q_{z} are magnitude of heat transfer in respec tive directions.
The majority of engineering materials come under isotropic, which means they consist of the same property in all directions. For, these kinds of materials, we no need to consider all directions as the properties are the same. Whereas, there is another type of material namely anisotropic material such as the ﬁbrous or composite materials, these materials may change their property by the direction. In such cases, the thermal conductivity must be expressed as a tensor quantity concerning the direction.
2.6 Fluid kinematics
2.6.1 Continuity equation in cardesian coordinates.
The continuity equation states that("If no ﬂuid is added or removed from the pipe in any length then the mass passing across diﬀerent sections remains same"). This
equation is based on principle of conservation of mass.
Let us consider, a ﬂuid element in a control volume where, ﬂuid element is in the form of parallelopiped with sides dx,dy and dz as shown below,
Figure 2.5: Three dimensional ﬂuid elements.[20]
Let us take,
ρ= mass density at a particular instant,
u,v,w= components of velocity at three surfaces of the parallelopiped ﬂuid element.
Rate of mass of ﬂuid at the entrance(ABCD)=ρ*area of ABCD*velocity at x direc tion.
Rateof massof f luidattheentrance(ABCD) = ρudydz (2.32) Rate of mass of ﬂuid at the exit(EFGH)
= ρudydz ∂
∂x(ρudydz)dx (2.33)
Mass accumulated per unit time due to ﬂow in xdirection
= ρudydz −
ρu+ ∂
∂x(ρu)dx
dydz (2.34)
Mass accumulated per unit time in Ydirection
= − ∂
∂y(ρv)dxdydz (2.35)
Mass accumulated per unit time in zdirection
= − ∂
∂z(ρw)dxdydz (2.36)
Net gain in ﬂuid mass per unit along three dimensional coordinate system
= −
∂
∂x(pu) + ∂
∂y(pv) + ∂
∂z(pw)
dxdydz (2.37)
Rate of change of mass in a control volume.
= ∂
∂t(ρdxdydz). (2.38)
substituting equations 2.37 and 2.38 we get,
= −
∂
∂x(pu) + ∂
∂y(pv) + ∂
∂z(pw)
dxdydz = ∂
∂t(ρdxdydz). (2.39) On rearrangement of terms, above equation can be reduced as below
∂
∂x(ρu) + ∂
∂y(pv) + ∂
∂z(pw) + ∂ρ
∂t = 0 (2.40)
This general equation of continuity in threedimensional coordinate system is appli cable to any type of ﬂow.
For steady ﬂow
∂ρ dt = 0
, for incompressible ﬂuid ρ is constant, the equation can be reduced to
∂u
∂x +∂v
∂y + ∂w
∂y = 0 (2.41)
For two dimensional ﬂow, the above equation(2.41) can be reduced to
∂u
∂x + ∂v
∂y = 0, (w = 0) (2.42)
For onedimensional ﬂow, we can reduce above equation
∂u
∂x = 0, (v = 0, w = 0) (2.43)
On integration with respect to x, we obtain u=constant,
The rate of ﬂow under area a is Q=a.u= Constant for steady ﬂow[20].
2.6.2 Reynold’s number.
Reynold’s number is a dimensionless quantity in ﬂuid mechanics, which is helpful in estimating the ﬂow pattern in various ﬂuid ﬂow situations[20]. At low reynold’s number, ﬂow tends to move in a laminar pattern like a sheet. The ﬂow is said to be turbulent when the reynold’s number is high. Reynold’s number is also useful in scaling of diﬀerent sizes ﬂow situations, such as between an aircraft model and calculating ﬂuid behaviour on bigger scale namely global air or water moments, me treological eﬀects.
The applications of reynold’s numbers with respect to laminar and turbulent ﬂow al lows scaling factor to be developed. Laminar ﬂow occurs while the reynold’s number is low and domination of viscous forces takes place. Whereas, turbulent ﬂow occurs while the reynold’s number is high and dominated by initial forces and results in
producing of insteabiliity and inﬂow.
Reynold’s number is deﬁned as
Re= uL
v = ρuL
μ (2.44)
Where as,
Density of ﬂuid =ρ, Flow speed = u,
characteristic linear dimension =L, Dynamic viscosity = μ,
Kinematic viscosity = v.
Reynold studies the condition, where the ﬂow of ﬂuid in pipe changes from lam inar ﬂow to turbulent ﬂow with the help of experiment. In this experiment, the behaviour of water ﬂow is examined under diﬀerent ﬂow velocities using a small stream of dyed water introduce into the centre of clean water in a large pipe.
For ﬂuid in a pipe or tube, the reynold’s number is deﬁned as Re= uD_{H}
v = ρuD_{H}
μ = ρQD_{H}
μA = W D_{H}
μA (2.45)
Where as,
Hydraulic diameter of the pipe =D_{H}, Volumetric ﬂow rate =Q,
Velocity of ﬂuid=u, Dynamic viscosity =μ, Density of ﬂuid= ρ, Mass ﬂow rate = W.
Reynold’s number relationship to other dimensionless parameter.
Peclet number is a dimensionless number, which is refers to the transport phe nomenon in a continuum. The ratio of the rate of advection of physical quantity by the ﬂow to the rate of diﬀusion of the same quantity is derived by an approximate gradient.
The peclet number is derived as
P e= advectivetransportrate
dif f usivetransportrate (2.46) Schmidt number(Sc) is a dimensionless parameter which is deﬁned as ratio of mo mentum diﬀusivity and mass diﬀusivity. This parameter is used mostly when ﬂuid ﬂows with simultaneous momentum and mass diﬀusion convection processes.
P e_{L}= Lu
D = ReLS_{c}. (2.47)
Sc= v D = μ
pD = viscousdif f usionrate
molecular(mass)diffusionrate (2.48)
The Prandtl(Pr) deﬁned as the ratio of momentum diﬀusivity to thermal diﬀusivity
P r= v
α = momentumdif f usivity thermaldif f usivity =
μ ρ k cpρ
= c_{p}μ
k (2.49)
2.7 Computational ﬂuid dynamics
2.7.1 Concept of computational ﬂuid dynamics
The simulation of ﬂuid engineering systems using combination of mathematical mod elling and numerical modelling is known as computational ﬂuid dynamics. CFD en ables engineer and scientists to perform various experiments in a virtual laboratory.
These numerical simulations of ﬂuid ﬂow helps in various sectors such as meteoro logical phenomenon, environmental hazard, combustion in automobile engine and complex ﬂow in furnace heat exchangers and chemical reactors. CFD gives an in sight into various ﬂow pattern that are diﬃcult, expensive, in few cases impossible to study using experimental techniques. When compared with experiments simu lation are cheaper and results can be obtained faster and simultaneously various phenomenons could be studied and monitored at same instant of time. The process of computational ﬂuid dynamics are shown in the ﬁgure below
Figure 2.6: Flow chart of computational ﬂuid dynamics.[4]
2.7.2 Navier stokes equation.
Navier stokes equations are generally considered as governing equation in compu tational ﬂuid dynamics. These equations are based on conservation law of physical properties of ﬂuids. Fluids have important properties such as velocity, pressure, tem perature and viscosity. The density of ﬂuid is deﬁned as mass per unit volume. If the ﬂuid is incompressible then we can express density of ﬂuid as below.
P = M V
kg m^{3}
(2.50)
Other important properties of ﬂuids is viscosity, it oﬀers resistance to ﬂow this vis cosity can be expressed as
μ=
N s m^{3}
= [P osie] (2.51)
After the conservation of mass momentum and energy equation, we can derive con tinuity equation and momentum equation as below
Continuity equation
Dρ
Dt + ρ∂U_{i}
∂x_{i} = 0 (2.52)
Momentum equation ρ∂U_{j}
∂t + ρUi
∂U_{j}
∂x_{i} = −∂P
∂x_{j} − ∂τ_{ij}
∂x_{i} + pgj (2.53)
Where
τ_{i}j = −μ
∂U_{j}
∂x_{i} + ∂U_{i}
∂x_{j}
+2
3δ_{ij}μ∂U_{k}
∂x_{k} (2.54)
Change in energy with time=ρ^{∂U}_{∂t}^{j}. Momentum convection =ρU_{i}^{∂U}_{∂x}^{j}
i. Surface force = −_{∂x}^{∂P}_{j}.
Molecular−dependent momentum exchange =^{∂τ}_{∂x}^{[}^{ij]}_{i} . Mass force= pg_{j}.
Energy equation ρc_{μ}∂T
∂t + ρcμU_{i}∂T
∂x_{i} = −P∂U_{i}
∂x_{i} + λ∂^{2}T
∂x^{2}_{i} + −τ_{[}i_{j}]∂U_{j}
∂x_{i} (2.55)
Where
Change in energy with time=ρc_{μ}^{∂T}_{∂t} Convective term=ρc_{μ}U_{i}^{∂T}
∂xi
Pressure work =−P^{∂U}_{∂x}_{i}^{i} Heat ﬂux =λ^{∂}_{∂x}^{2}^{T}_{2}
i.
Irreversible transfer of mechanical energy into heat. =−τ_{[}i_{j}]^{∂U}_{∂x}_{i}^{j} For compressive ﬂuid we can simplify aboove equation as follows
Continuity equation
∂U_{i}
∂x_{i} = 0 (2.56)
Momentum equation ρ∂U_{j}
∂t + ρUi
∂U_{j}
∂x_{i} = −∂P
∂x_{j} − μ∂^{2}U_{j}
∂x^{2}_{i} + pgj (2.57)
The Navier stoke equation can be rewritten in simpliﬁed form or the general form as below
∂(ρΦ)
∂t + ∂
∂x_{i}
ρU_{i}Φ − ΓΦ∂Φ
∂x_{i}
= qΦ (2.58)
On applying integration to Navier stoke equation, we obtain the following equation
V
∂
∂x_{i}ΦdV =
S
Φ.nidS (2.59)
We multiply the volume and the value of centre of the control volume to approximate the volume integral. The mass and momentum of control volume is approximately
as below
V
∂ρΦ
∂t dV +
S
ρU_{i}Φ − Γ∂Φ
∂x_{i}
.n_{i}dS =
V
q_{Φ}dV. (2.60)
m =
V
ρdV ≈ ρpV, mu=
V
ρ_{i}μ_{i}dV ≈ ρpμ_{p}V (2.61) The surface integral is approximated with pressure poles as below,
S
P dS ≈ ΣkP_{k}S_{k}, k = n, s, e, w (2.62) The variables are showed at the centre of control volume, So we need to interpolate them to obtain P_{k}, which are located at the surface of control volume. Genrally there are two kinds of interpolation namely upwind interpolation and central interpolation Upwind interpolation
U_{e} =
U_{p}if(U.n)e >0
(2.63)
U_{e} =
U_{E}if(U.n)e <0
(2.64)
Central interpolation
U_{e} = UEλ_{e}+ Up(1 − λe)λ = x_{e}− x_{p} x_{E} − xp
(2.65) Navier stoke equations are analytical equations. Computer cannot understand them So, we have to transfer these analytical equations into discreteized form. General discreteization methods are ﬁnite diﬀerence, ﬁnite element and ﬁnite volume method.
If we use ﬁnite diﬀrence and ﬁnite element approach, we must control conservation of mass, momentum and energy in manual manner, so ﬁnite volume method is preferred to discreteize navier stokes equations. If navier stoke equation satisﬁed with respect to control volume, then it will be automatically satisﬁes entire domain as well. That’s why ﬁnite volume is preferred in computational ﬂuid dynamics.
Figure 2.7: Grid domain[26]
2.7.3 Grids
There are mainly three types of grids used in computational ﬂuid dynamics they are
• Structured grids
• Unstructured grids
• Block structured grids
These type of grids are highly useful for simple domain, where all nodes contain same number of elements around them, these grids are termed as structured grids. Un structured grids are widely used for complex domains where ﬁne grids are required at the complex shape region. Unstructured grids are suitable for all geometries. This type of grids are widely used in CFD analysis. Block structured grids are the combi nation of structured and unstructured grids. At the beginning the domain is divided into several blocks and then diﬀerent structured grids are used in diﬀerent blocks.
Figure 2.8: Structured grid and Unstructured grid.
Boundary conditions.
While solving the equation system, we need few boundary conditions, the common boundary condition that are used in CFD are inlet, oulet boundary condition, Noslip boundary condition and periodic boundary conditions.
Chapter 3
Numerical Method
3.1 Numerical Methods in Heat Conduction
While we are considering simple heat conduction applications that consist of simple geometries with simple boundary conditions, which are easier to solve analytically.
But in practical daytoday applications, we come across various complex geometries and complex boundary conditions or variable properties which are very diﬃcult to solve analytically in such cases accurate approximate solutions can be obtained by using computers with numerical methods. Analytical solution methods are applied by solving the governing diﬀerential equation concerning the boundary conditions these results in solution function for the temperature at every point of the medium.
Whereas, on another hand numerical method are applied by replacing diﬀerential equations by a set of n algebraic equations for the unknown temperature at n selected points in the medium, solutions of these equations results in temperature values at those discrete points. There are various types of numerical methods for solving heat conduction applications such as the ﬁnite diﬀerential method, ﬁnite element method, the boundary element method, and energy balance method, etc. these methods have their advantages and disadvantages in practical usage.
3.1.1 Importance of Numerical Methods
The advancement of highspeed computers and powerful software had a major im pact on engineering education. Engineers in the olden days must rely on analytical skills to solve speciﬁc engineering problems. They must undergo complex training in mathematics. But these days engineers have access to a tremendous amount of computing power at their ﬁngertips. In the analytical method, we have to follow a pattern to solving the engineering problem and various geometries in asymmetric but highly mathematical by deriving the governing diﬀerential equation, expressing the boundary condition in proper mathematical form, and solving the diﬀerential equation, and applying the boundary condition to determine the integration con stants. For example, the mathematical formulation of a onedimension study state condition in a sphere of radius r_{0} where the outer temperature is T1 with uniform heat generation at a rate of ˙g_{0}
1 r^{2}
d dr
r^{2}dT
dt
+ ˙g0
k = 0 (3.1)
dT(0)
dr = 0andT (r_{0}) = T_{1} (3.2)
T(r) = T_{1}+ ˙g_{0}
6k(r^{2}_{0}− r^{2}) (3.3)
With the help of the above equation, we can determine the temperature at any point within the sphere by substituting the r coordinate of the point into the analytical solution function. The analysis above didn’t require any mathematical complexity beyond simple integration. These instructions show clearly the functional dependence of temperature and heat transfer on the independent variable r.
3.1.2 Limitations
We can apply analytical solution methods for highly simpliﬁed applications in simple geometries, the geometries must be in such a way that enters surface must be math ematically in a coordinate system by setting the variables equal to the constants.
Even minor complications in geometry make analytical solution methods diﬃcult to apply even in simple geometry heat transfer applications cannot be solved analyt ically unless the thermal conditions are simple. Analytical solutions are limited to problems that are simple or can be simpliﬁed with reasonable approximations.
3.1.3 Framework
When we observe realworld problems, the solutions we obtain are solved using math ematical modeling, the degree of applicability depends on the accuracy of the model.
An approximate solution of the realtime model of a physical application is usually more accurate than the exact solution of a mathematical model. While deriving an analytical solution from a physical problem, we try to oversimplify the problem to make a mathematical model simple over an analytical solution. A mathematical model is applied for a numerical solution is mostly represents the actual problem better. Therefore, the numerical solution of engineering applications has a better solution than expecting analytical solutions.
3.1.4 Flexibility
While observing engineering problems, we often require more parametric study to understand some variables on the solutions to obtain the right set of variables. This is an interactive process that would consume more time if we work by hand. Com puter and numerical methods are suitable for such calculations and many related problems can be solved by minor modiﬁcations in input parameters or code. Today any signiﬁcant optimization studies in engineering are carried out with the help of the power and ﬂexibility of computers in numerical methods.
3.1.5 Modelling
Geometry of Finite Element Model
The main aim of this ﬁnite element model in ansys is to investigate the the heat transfer in a double pipe heat exchanger. We have constructed a double pipe heat exchanger, where the ethanol is ﬂowing in the inner pipe of heat exchanger and the
water is ﬂowing from the external pipe. To understand the heat transfer analysis, we have considered three types of inner pipes namely triangular, hexagonal and octagonal. Flow rate of 0.1 kg/sec and turbulent intensity of 5 percent of turbulent viscosity ratio of 10 was considered, to investigate heat transfer.
Geometric Properties
The material is build using the below geometric properties for double pipe heat ex changer.
S.no Parameters Values/Specimen SI units
1 Metal Copper Cu
2 Pipe external di
ameter 100 mm
3 Pipe internal di
ameter 90 mm
4 Thickness of the
pipe 10 mm
5 Length of the
pipe 3250 mm
Table 3.1: Geometric properties for numerical model.
Figure 3.1: Diagrammatic representation of dimension from numerical modelling.
The model is build using the above geometric parameter for dent model.
S.no Parameters Values/Specimen SI units
1 Metal Copper Cu
2 Length of inner
tube 750 mm
3 Dent size 2 mm
4 Diameter of in
ner pipe 16.5 mm
5 Length of exter
nal pipe 450 mm
Figure 3.2: Diagramic representation of dimension from dent model.
Table 3.2: Geometric properties for numerical model.
Material properties
The specimen used to build this numerical model is copper, below are the material properties of the specimen.
S.no Parameters Values SI units
1 Density(copper) 8960 kg/m^{3}
2 Speciﬁc
heat(copper) 376.812 j/kgk
3
Thermal conductiv ity(copper)
394 w/mk
Table 3.3: Material properties of numerical model for copper.
S.no Parameters Values SI units
1 Density(Brass) 8730 kg/m^{3}
2 Speciﬁc
heat(Brass) 920 j/kgk
3 Thermal con
ductivity(Brass) 109 w/mk
Table 3.4: Material properties of numerical model for brass.
S.no Parameters Values SI units
1 Density(Ethanol)
790 kg/m^{3}
2 Speciﬁc
heat(Ethanol) 2470 j/kgk
3
Thermal conductiv ity(Ethanol)
0.182 w/mk
4 Mass ﬂow 0.1 kg/sec
Table 3.5: Material properties of numerical model for Ethanol.
3.1.6 Boundary Conditions and Meshing
The numerical model is constructed by considering few boundary conditions and parameters as discussed in above tables 3.1, 3.2, 3.3, 3.4. For this double pipe heat exchanger, the external tube is made up of brass and the internal tube is made up of copper. The ethanol ﬂows from the inner pipe of the heat exchanger and the external pipe is supplied with water. The temperature condition of ethanol was considered as 78^{0} at inlet and the water temperature was considered as10^{0} at the inlet. Diﬀerent types of inner tubes are considered such as triangle, hexagon, octagon. These are considered to investigate temperature change across the pipe length. Fine meshing was done to produce accurate results.
Figure 3.3: Meshing for octagon and hexagon inner pipes of the double pipe heat exchanger.
Figure 3.4: Meshing for triangular inner pipe of the double pipe heat exchanger.
Other numerical model is constructed by employing dent to the surface of external part of inner pipe in a double pipe heat exchanger. Heat transfer was analyzed between, with dent and without dent.
Figure 3.5: Meshing for dent model.