Essay: Design and Analysis of a Ranque-Hilsch Vortex Tube to Maximise Cooling

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Abstract
The primary aim of this project is to design and analyse a Ranque-Hilsch Vortex Tube (RHVT). To get the project up and running research was undertaken to investigate the thermodynamics of the RHVT. As the RHVT obeys the first and second laws of thermodynamics, the laws are explained to gain a basic knowledge of thermodynamics. It was soon discovered that thermodynamics is the backbone to the operation of the RHVT. The conservation of mass and energy must also be obeyed during the operation of a RHVT. Using the conservation of mass, a mass balance was carried out for a RHVT, where a non-dimensional equation was formed. Using the conservation of energy, an energy balance was also carried out for a RHVT, where another non-dimensional equation was formed. These non-dimensional equations are useful when carrying out analyse of a RHVT. An Entropy balance was then carried out, which involved a great deal manipulating to equations, to form a transcendental equation. This transcendental equation allowed for the analysis of an isentropic RHVT and a commercial RHVT. After comparing the RHVTs, as predicted the isentropic vortex tube caused the greater amount of temperature separation.
Finally a description of the future work is discussed followed by a conclusion to summarise the project to date.
Symbols
Ac cross-sectional area m2
C specific kinetic energy J
COP coefficient of performance
COPHP heat pump ‘ coefficient of performance
COPR refrigeration – coefficient of performance
Cv constant heat capacity at constant volume J/kg.K
Cp constant heat capacity at constant pressure J/kg.K
E total energy J
Ek kinetic energy J
Ep potential energy J
Esystem total energy of system J
F force applied N
gz specific gravitational potential energy m2/s2
h specific enthalpy J/kg
m mass kg
mcv mass of control volume kg
min mass entering control volume kg
mout mass exiting control volume kg
m ?? mass flow rate kg/s
m ??_cv mass flow rate kg/s
m ??_in mass flow rate kg/s
m ??_out mass flow rate kg/s
m ‘? energy of a flowing fluid J/s
P pressure Pa
Patm atmospheric pressure Pa
Q ?? rate of heat energy transfer J/s
q ?? rate of specific heat J/s/kg
R gas constant J/kg.K
rc compression ratio
s displacement of applied force m
s specific entropy J/kg.K
S entropy J/K
Sgen entropy generated J/K
S ?? rate of entropy J/K/s
S ??_gen rate of entropy generated J/K/s
T absolute temperature K
U internal energy J
u specific internal energy J/kg
V Volume m3
v specific volume m3/kg
V average velocity unless otherwise stated m/s
Vavg average velocity m/s
Vn flow velocity normal to the area m/s
v ?? volumetric flow rate m3/s
W work energy J
w specific work energy J/kg
Wnet net-work energy J
W ?? rate of work energy transfer J/s
w ?? rate of specific work energy transfer J/s/kg
Greek Symbols
?? fluid mass fraction
?? dimensionless temperature
?? fluid density kg/m3
?? ratio between heat capacities
?? variation
Subscripts
atm atmosphere
C cold air exit
CV control volume
H hot air exit
in entering a system
O ambient air inlet
out exiting a system

Glossary
CCTV Closed-circuit Television
COP Coefficient of Performance
ITW Illinois Tool Works
MS Microsoft
RHVT Ranque-Hilsch Vortex Tube
UK United Kingdom
US United States
Table of Contents
1 Introduction 9
1.1 Introduction 9
1.2 Aims & Objectives 9
1.2.1 Aims 9
1.2.2 Objectives 9
1.3 Motivation 9
1.4 Project Layout/Plan 9
2 The Ranque-Hilsch Vortex Tube 9
2.1 Introduction 9
2.2 Maxwell’s Demon 9
2.3 Ranque’s Contribution 10
2.4 Hilsch’s Contribution 11
2.5 Types of Vortex Tubes 11
2.6 Principle of Operation of the Counter-Flow Vortex Tube 12
2.7 Commercial Vortex Tubes 13
2.7.1 Spot Cooling Applications 14
2.7.2 Spot Heating Applications 16
2.7.3 Experimental Ranque-Hilsch Vortex Tube Equipment 17
2.7.4 Personal Air Conditioners 17
3 Literature Review 18
3.1 Introduction 18
3.2 Thermodynamics 18
3.3 The First Law of Thermodynamics 18
3.4 The Second Law of Thermodynamics 19
3.5 Conservation of Mass 19
3.6 Conservation of Energy 21
3.7 Mass Balance 24
3.8 Energy Balance 25
3.9 Compressor Work 26
3.9.1 Kelvin Planck Statement 26
3.9.2 The Clausius Statement 26
3.9.3 Compressor Work 27
3.10 Entropy Balance 30
3.11 Analysis of an Isentropic Vortex Tube 33
3.12 Analysis of a Commercial RHVT 37
4 Future Work 41
5 Conclusion 41
6 References 41

Table of Figures
Figure 2 1: Vortex Tube Operation (Carrasca & Sala Lizarraga, 2013) 11
Figure 2 2: Illustration of Hilsch’s Tube (Slager, 1959) 12
Figure 2 3: Counter-flow vortex tube (De Vera, 2010) 13
Figure 2 4: Uni-flow Vortex Tube (De Vera, 2010) 13
Figure 2 5: Exploded view of Counter-flow Vortex Tube (Denniston, 2013) 14
Figure 2 6: Cold Air Gun by ITW Vortec (PennTool, 2000) 15
Figure 2 7: Cooling CCTV Camera Housing by Nex Flow (NexFlow, n.d.) 16
Figure 2 8: Control Cabinet RHVT by Exair (Exair, 2002) 17
Figure 2 9: Explanation of Control Cabinet Cooling RHVT by Exair (Exair, 2002) 17
Figure 2 10: Experimental RHVT Equipment by P.A. Hilton Ltd (Hilton Ltd, 2011) 18
Figure 2 11: Personal Air Conditioner by ITW Vortec (Vortec, 2012) 18
Figure 3 1: Steam Boiler under Steady-flow Operation (Oklahoma, n.d.) 24
Figure 3 2: Mass Balance for RHVT 26
Figure 3 3: Force Applied to a Piston (Dmic001, n.d.) 30
Figure 3 4: Isentropic & Adiabatic Process (Amirault, n.d.) 35
Figure 3 5: MATLAB Fx Function File for ??c 36
Figure 3 6: MATLAB Fx Function File for ??H 36
Figure 3 7: MATLAB Newton Function File Generated for ??c 37
Figure 3 8: MATLAB Newton Function File Generated for ??H 38
Figure 3 9: Isentropic RHVT relationship of Xc vs. ??c and Xc vs. ??H 39
Figure 3 10: Data of a Commercial RHVT by Nex Flow (NexFlow, n.d.) 40
Figure 3 11: Commercial RHVT relationship of Xc vs. ??c and Xc vs. ??H 41
Figure 3 12: Isentropic RHVT vs. Commercial RHVT 42
Table of Tables
Table ‘2 1: Commercial RHVT Data Generated using MS Excel 41

Introduction
Introduction
In this chapter, a descrption of the project is given. The aims and objectives of the project are mentioned to ensure the project is kept on track and delivered on time. They are also listed in numerical order, from beginning to end. The reasons for choosing the project are mentioned, to highlight the benefits of carrying out such a project. Finally, a table has been added containing important dates for the project deliverables.
Aims:
To design a Ranque-Hilsch Vortex Tube working model to maximize cooling
To test the working the working model using the in-house air compressor
To analyse the performance of the working model
To compare the performance of the working model against the performance of a commercial model
To increase the performance of the model using different methods
To gain a better understanding of the devices operation
Objectives:
To gain a knowledge of the history and background of the device
To look at various applications where the device is used in industry, primarily focusing on cooling
To view the device as a form of a heat pump/refrigerator and identify the similarities
To research the thermodynamics behind the operation of the device
To analyse an isentropic vortex tube using equations manipulated from the research of the thermodynamics
To analyse a commercial vortex tube using data provided by the manufacturer
To compare the performance of the isentropic vortex tube and the commercial vortex tube using Microsoft Excel graphs
To choose appropriate dimensions provided by a manufacturer for the design of the working model
Design a working model using rapid prototyping methods
To analyse the performance of the working model and compare results against the commercial vortex tube
To identify opportunities to increase the performance of the working model
Motivation
The design and manufacture of a RHVT was chosen as a final year project as it covers a large aspect of the level 7 and 8 Energy Engineering course content in Galway Mayo Institute of Technology. The project will cover a wide range of the modules that have been undertaken by the author such as Machine Design, Electrical Machines, Computer Aided Engineering, Plant Engineering, Heat Transfer, Thermofluids, Thermodynamics Systems, Computer Aided Design, Mechanics and Dynamics of Machines, Thermodynamics, Manufacturing Engineering, Mechanics and Properties of Materials, Statics & Dynamics and Fluid Mechanics.
The project was also chosen, as the device is a percular device which can provide a large amount temperature separation for a relativetly small amount of work input. There is no actual definition out there to how the device actually works other than over the years a number of people have published their concept of the devices operation.
The project is quite technical as it covers a vast amount of engineering topics such as mathematics, thermodynamics, thermofluids, engineering design and engineering manufacture. In the near future the projects technical aspect will benefit the author when seeking employment in the field of engineering.
Project Plan
The idea of a project plan to is to ensure the project achieves its aims and objectives within an accurate time frame, so the project can be successful. Below in numerical order is the aims and objectives of the project.
To begin the literature review by investigating the historical background of the device.
To look at varios applications in industry where the device is being used.
To investigate the design parameters for optimising the performance of the device.
Investigate the study of thermodynamics which occurs during the devices operation.
From the study of thermodynamics, solve techniques for the analysis of an isentropic process.
Analyse the operational limits from an isentropic process.
Analyse a commercial device based upon manufacturers data.
Design a device viable for in-house operational conditions, such as, compression ratio, compressor power output and flow rate.
Manufacture a device once the design is successful.
Analyse the performance of the manufactured device and record the results.
Plot the recorded results and compare the performance of the device against the commercial device.
Identify opportunities to optimise the performance of the manufactured device.
In Table 1 1 below is the project plan with tasks and key dates for the project throughout the academic year.
Table 1 1: Project Plan
The Ranque-Hilsch Vortex Tube
Introduction
The vortex tube was invented by a French physicist named George J. Ranque in 1931. The concept was not a breakthrough in engineering due to its apparent inefficiency. The concept was abandoned for several years until 1947, when a German engineer Rudolf Hilsch made a number of changes to the design, improving its efficiency (Vera 2010).
The Ranque-Hilsch vortex tube (RHVT) is a mechanical device that absorbs compressed air through an inlet and releases high temperature air at one outlet and low temperature air at another outlet (Vera 2010).
The air is injected into the vortex tube at extremely high speeds by means of an air compressor and creates a vortex. The flow and temperature are totally controllable. Some of the air is forced to spin inward to the centre of the tube and travels up a long tube where a valve turns the spinning column of air inside itself. The inside column of air gives its heat to the outside column (Co., n.d.).
Maxwell’s Demon
Maxwell’s Demon is a thought experiment which observes the splitting of air into two flows at different temperatures. The experiment is named after a physicist James Clerk Maxwell. Maxwell proposed that the molecules in a vessel full of air at equal temperature move with speed that are not uniform. He divided the vessel into two parts A and B which are separated by a small hole, which opens and closes. The faster moving molecules are only allowed to pass from A to B, and the slower ones to pass from B to A. Therefore the faster moving molecules will be separated from the slower molecules. This states the temperature of B is raised and the temperature of A is lowered. This statement is incorrect as the experiment violates the second law of thermodynamics. Maxwell’s Demon is similar to what occurs inside the vortex tube as the air flow is separated into two flows, hot and cold. (Carrasca & Sala Lizarraga, 2013)
The device which was initially called the Maxwell’s Demon Tube, has become known as the RHVT. The cross sectional schematic shown in Figure 1 illustrates that the RHVT is a device that separates the flow of a gas into two streams, greatly hotter and cooler than the inlet temperature, despite no moving parts or the input of work, chemically or electrically. The absence of a conventional energy source is the main reason why there has been so much interest in the device throughout the years. (Ronan, 2008)
Figure 2 1: Vortex Tube Operation (Carrasca & Sala Lizarraga, 2013)
Figure ‘2 1 shows the direction of the air flow throughout the vortex tube. The origin of the RHVT is accredited to two men, French man Georges Joseph Ranque and a German called Rudolf Hilsch. Their contributions to the RHVT will now be discussed (Ronan, 2008).
Ranque’s Contribution
The origin of the vortex tube can be traced back to French man Georges Joseph Ranque. The first record of the RHVT occurred on 12th December 1931, when Ranque filed for a French patent. He also filed the same docket for a US patent and was awarded on 27th March 1934. (Technologies, 2005)
In the patent Ranque showed that the tangential entrance may contain a single nozzle, a number of nozzles, or a set of blades. Ranque also explained that by adjusting the orifice size of the cold air or the restriction at the hot end, a small quantity of cold air may be obtained. He stated that the temperature of the hot end is a maximum when the hot tube is fully closed, and that the greater the supply pressure is applied, the colder the air will be. (Ronan, 2008)
Hilsch’s Contribution
Not long after World War 2 had ended the US had gotten word that the Germans had developed a simple device which could achieve very low temperatures. The device which Ranque created is said to have been found by the German Army when France was occupied. The work was passed over to a German physicist named Rudolf Hilsch, who was already working on low temperature devices. Hilsch made improvements to Ranque’s model. (Slager, 1959)
Figure 2 2: Illustration of Hilsch’s Tube (Slager, 1959)
Figure 2 2 shows an exploded view of Hilsch’s Tube. Hilsch observed that by increasing the size of the hot pipe, inlet nozzle and diaphragm, lower temperatures could be achieved with the same inlet pressure and the efficiency of the device increased.
Despite the performance of the Hilsch vortex tube, its efficiency was quite low. The best efficiency Hilsch achieved with the larger pipe was 20%. (Slager, 1959)
Types of Vortex Tubes
There are 2 types of vortex tubes, the counter-flow vortex tube and the uni-flow vortex tube. Both of these are used in the industry, with the counter-flow vortex (Figure 2 3) tube being more popular.
In the counter-flow vortex tube, hot air is exhausted from the end of the tube with the longest section, which is controlled by a valve. The cold air exhausts through an orifice at the opposite end of the tube next to the inlet. (Kar, et al., 2012)
Figure 2 3: Counter-flow vortex tube (De Vera, 2010)
Unlike the counter-flow vortex tube, the uni-flow vortex tube (Figure 2 4) does not have cold air exhausting next to the inlet. Instead the cold air is exhausted through a concentrically located exit in the control valve. The uni-flow vortex tube is used where space and equipment cost are of great importance. The efficiency of the uni-flow vortex tube is often lower than that of the counter-flow vortex tube. (Kar, et al., 2012)
Figure 2 4: Uni-flow Vortex Tube (De Vera, 2010)
The type of vortex tube being focused on in the project will be the counter-flow vortex tube.
Principle of Operation of the Counter-Flow Vortex Tube
Air which is compressed to a high pressure enters the inlet of the vortex tube through tangential nozzle which accelerates the air flow. The air has a large velocity and rotates at very high speeds. Therefore the air has a vortex motion which spirals down the tube. The inner stream gives off kinetic energy in the form of heat to the outer stream. A cone shaped valve, which controls the pressure in the system, causes a reversal of the central core of air. A washer is fitted at the end of the cold pipe, which has a diameter half that of the pipe. Washers with different diameters can be used to adjust the performance of the system. Thus in Figure 2 5, cold air exits at the left side of the vortex tube and hot air exits at the right side through the valve. (Prabakaran & Vaidyanathan, 2010)
Figure 2 5: Exploded view of Counter-flow Vortex Tube (Denniston, 2013)
Commercial Vortex Tubes
A vortex tube provides a low cost, reliable and maintenance free solution for a range of applications. The temperature, flow rate and refrigeration are easily adjusted using the control valve, located on the hot end of the tube. The vast majority of vortex tubes are constructed of stainless steel, due to its resistance to corrosion and oxidation, therefore providing years of reliable and maintenance free operation. (Exair, 2002)
There is a range of RHVT manufacturers who have put theory into cooling applications for industrial use. These include manufacturers such as Exair, Arizona Vortex and STREAMTEK, which are based in the United States. For the Irish market, Vacuum Spares Ireland (VSI) who is located in Cobh, Cork manufacture and supply vortex tubes. Flowtech, based in the UK, supply Exair RHVTs. (Ronan, 2008)
These companies market their products having the following advantages:
‘ No moving parts
‘ No electricity or chemicals
‘ Small, lightweight
‘ Low cost
‘ Maintenance free
‘ Instant cold air
‘ Durable – stainless steel
‘ Adjustable temperature
‘ Interchangeable generators (Exair, 2002)
Spot Cooling Applications
RHVTs are used in a wide range of applications for spot cooling on machines, processes and assembly lines.
Figure 2 6: Cold Air Gun by ITW Vortec (PennTool, 2000)
Shown in Figure 2 6 is a cold air gun, which can be used in various industrial applications such as the cooling of metals, plastics, wood, rubber and other materials during machining. Cold air guns also increases feed rates and extend tool life. (VorTech, 2015)
Figure 2 7: Cooling CCTV Camera Housing by Nex Flow (NexFlow, n.d.)
Shown in Figure 2 7 is a Vortex Tube used to cool down CCTV camera housing. The benefits of using Vortex tube cooled housings include lower costs and a return line is not necessary.
Other spot cooling applications include:
Cool electronic and electrical controls, as illustrated in Figure 2 8 and Figure 2 9
Cooling soldered parts
Cooling gas samples
Cooling heat seals
Cooling environmental chambers
Figure 2 8: Control Cabinet RHVT by Exair (Exair, 2002)
Figure 2 9: Explanation of Control Cabinet Cooling RHVT by Exair (Exair, 2002)
Spot Heating Applications
The hot air exhaust applications include:
Heating solders and adhesives to speed up processes
Drying ink on bottles and labels
Pre-heating materials
Experimental Ranque-Hilsch Vortex Tube Equipment
Experimental equipment can be purchased for analysing the performance of a RHVT, which would be ideally situated in thermodynamic and fluid dynamic laboratories. This type of equipment is available from P.A. Hilton Ltd. Shown below in Figure 2 10 is a picture of the equipment. (Hilton Ltd, 2011)
Figure 2 10: Experimental RHVT Equipment by P.A. Hilton Ltd (Hilton Ltd, 2011)
Personal Air Conditioners
People working in extreme temperatures can wear an air vest, which is connected to a RHVT, to provide heating or cooling depending on the environment. Shown in Figure 2 11 is a picture of a product marketed by ITW Vortec. (Vortec, 2012)
Figure 2 11: Personal Air Conditioner by ITW Vortec (Vortec, 2012)
Design Parameters for Performance Optimisation

Thermodynamics of a Ranque-Hilsch Vortex Tube
Introduction
In this chapter, research was undertaken to investigate the thermodynamics of the RHVT. As the RHVT obeys the first and second laws of thermodynamics, the laws are explained to gain a basic knowledge of thermodynamics. The conservation of mass and energy must also be obeyed during the operation of a RHVT. Using the conservation of mass, a mass balance was carried out for a RHVT, where a non-dimensional equation was formed. Using the conservation of energy, an energy balance was also carried out for a RHVT, where another non-dimensional equation was formed. These non-dimensional equations will be useful when analysing a RHVT. An Entropy balance was then carried out, which involved a great deal manipulating to equations, to form a transcendental equation. This transcendental equation allowed for the analysis of an isentropic RHVT.
Thermodynamics
A thermodynamic air-standard cycle is used for the RHVT to provide thermodynamic analysis for setting operating limits according to the conservation laws of mass and energy, as well as the second laws of thermodynamics. The study uses a control volume approach for establishing working equations for predicting the performance of a RHVT (Xue, 2013).
Conservation of Mass
The amount of mass flowing through a cross section per unit time is called the mass flow rate (‘). A fluid flows in or out of a control volume through pipes or ducts. The mass flow rate of a fluid is proportional to the cross sectional area (Ac), the fluid density (??) and the component of the flow velocity normal to the area which we call Vn and is expressed as
‘ = ??.Vn.Ac
Velocity is never constant over a cross section of a pipe because the fluid sticking to the pipe surface and therefore having zero velocity at the wall. The velocity varies from zero at the walls to a maximum value at the centre of the pipe. The average velocity Vavg is defined as the average value of Vn across the entire cross section.
For incompressible flow or for compressible flow where ?? is uniform, mass flow rate is expressed as
‘ = ??.Vavg.Ac
For simplicity, the subscript on the average velocity is dropped. Unless otherwise stated, V represents the average velocity in the flow direction.
The amount of volume flowing through a cross section per unit time is called the volumetric flow rate (v ??) and is expressed as
v ?? = Vavg.Ac = V.Ac
In fluid mechanics, Q is used instead of v ?? for volumetric flow rate. v ?? is used to avoid confusion with heat transfer. The mass and volumetric flow rates are related by
‘ = ??.v ?? = v ??/v
Where v is the specific volume of the fluid.
The conservation of mass principle for a control volume can be expressed as, the net mass transfer to or from a control volume during a time interval is equal to the net change in the total mass within the control volume. It is expressed as
min ‘ mout = ??mcv
It can also be expressed in rate form as
‘in – ‘out = ‘??cv
These equations are referred to as the mass balance and can be applied to any control volume undergoing a process.
For a steady flow process, the total amount of mass in a control volume does not change with respect to time. Therefore the conservation of mass principle states that the amount of mass entering a control volume equals the amount of mass leaving the control volume. For a steady flow system with multiple inlets and outlets, the conservation of mass principle can be expressed in rate form as
‘in = ‘out
Engineering devices such as nozzles, diffusers, compressors, turbines and pumps having only one inlet and one outlet, the inlet is denoted by the subscript 1 and the outlet by the subscript 2. But for a device like a RHVT having one inlet and two outlets, the inlet is denoted by the subscript 1 and the outlets by the subscripts 2 and 3. The mass balance equation for the vortex-tube becomes
‘1 = ‘2 + ‘3
Therefore
??1.V1.A1 = ??2.V2.A2 = ??3.V3.A3
(Cengel & Boles, 2010)
Mass Balance
For a steady-flow process, the total mass contained within a control volume does not change with time. The conservation of mass principle states that the total amount of mass entering a control volume must equal to the total amount of mass leaving it. Applying mass conservation to the vortex tube, results in the following:
Figure 3 2: Mass Balance for RHVT
m ??_C+m ??_H=m ??_O
If the mass balance equation is divided by the total mass flow rate, the cold mass flow rate and the hot mass flow rate can be expressed as fractions. These are called mass fractions as the sum of the cold and hot mass flow rates must equal 1. The mass balance equation becomes
m ??_C/m ??_O +m ??_H/m ??_O =1
Allowing ?? become the symbol for mass fractions, the cold mass fraction and hot mass fraction are expressed as
??_C=m ??_C/m ??_O
??_H=m ??_H/m ??_O
therefore
??_C+??_H=1
This is a dimensionless equation.
The First Law of Thermodynamics
The first law of thermodynamics states that energy cannot be created or destroyed, but can be changed from one form to another. The first law of thermodynamics, also known as the conservation of energy principle, allows for the studying of the relationships between various forms of energy and energy interactions. During a process, every quantity of energy should be accounted for.
Consider a stone at some elevation. At this elevation the stone has some potential energy, and as the stone falls the potential energy is converted into kinetic energy. The decrease in potential energy is equal to increase in kinetic energy when the air resistance is negligible. This confirms the conservation of energy principle.
A process where there is no heat transfer across the boundaries of a control volume is called an adiabatic process. If a system undergoes a number of adiabatic processes from one state to another state, these processes do not involve any heat transfer, but may involve work interactions. For all adiabatic processes transferring from one state to another in a closed system, the net work done is the same.
A major part of the first law is the property, total energy (E). If the net work is the same for all adiabatic processes of a closed system between two states, the amount of net work depends on the end state of the closed system only, and it must react to a change in a property of the system. The first law states that the change in the total energy during an adiabatic process is equal to the net work done. (Cengel & Boles, 2010)
From discussing the first law, the conservation of energy principle can be expressed as, the increase or decrease in the total energy of the system during a process is equal to the difference between the total energy entering the system and the total energy leaving the system. In other words,
Ein ‘ Eout = ??Esystem
Conservation of Energy
A large number of engineering devices operate for long periods of time under the same conditions once the start-up period is completed. These are called steady flow devices. These devices can be represented as a somewhat ideal process, called the steady-flow process. A steady-flow process is described as a process during which a fluid flows through a control volume steadily.
No intensive or extensive properties within the control volume change with time. Therefore the volume V, the mass m and the total energy content E of the control volume remain constant. As a result, the boundary work is zero for steady-flow systems, and the total mass or total energy entering the control volume must be equal to the total mass or total energy leaving the control volume. The fluid properties at the inlet or outlet remain constant during a steady-flow process. (Cengel & Boles, 2010)
Therefore the amount of energy entering a control volume, whether its heat, work or mass, it must be equal to the amount of energy leaving it. The energy balance equation is written as
E ??_in = E ??_out
Mentioned previously that the energy can be transferred by heat, work and mass, the energy balance equation for a steady-flow system can also be as
Q ??_in + W ??_in + ‘?? = Q ??_out + W ??_out + ‘??
Q ?? = the rate of heat energy transferred
W ?? = the rate of work energy transferred
‘?? = the energy of a flowing fluid of mass
Consider, for example, a steam boiler under steady-flow operation as shown in Figure 3 1.
Figure 3 1: Steam Boiler under Steady-flow Operation (Oklahoma, n.d.)
Cold water with a mass flow rate is continuously flowing into the steam boiler, and hot water of the same mass flow rate is continuously flowing out of it. The steam boiler is losing energy to drive a shaft at a rate of W ??out, and is gaining energy by means of heating the water, at a rate of Q ??in. On the basis of the conservation of energy principle, the total energy supplied to the control volume is equal to the energy needed to drive the shaft, plus any energy lost to the surrounding (Cengel & Boles, 2010).
The energy balance relation is easy to use when the quantity and direction of heat and work transfers are known. When carrying out a problem that involves unknown heat or unknown transfers, the direction for heat or work transfers need to be assumed. To try solving these problems, it is common practice to assume heat to be transferred into the system at a rate of( Q) ??, and work produced by the system at a rate ofW ??. The energy balance relation for a general steady flow system becomes
Q ??-W ??=m ??_out (h+ C^2/2+gz)- m ??_in (h+ C^2/2 gz)
If a negative value for Q ?? or W ?? is obtained, the assumed direction is wrong and should be reversed. The terms appearing in the energy balance equation are as follows:
Q ?? = rate of heat transfer. When a control volume is gaining heat, Q ?? is positive. If there is no heat loss across the boundaries of the control volume, Q ?? is equal to zero.
W ?? = power. W ?? represents the forms of work per unit time. For many steady-flow devices W ?? is the shaft power. If a steady-flow device is crossed by electrical wires, W ?? represents the electrical work done per unit time.
h = enthalpy. The enthalpy of a fluid can be determined by reading the enthalpy values from the tables.
C = kinetic energy. The kinetic energy can be neglected at low velocities. When a fluid in a steady-flow device enters and leaves at the same velocity, the change in kinetic energy is close to zero. At high velocities caution should be taken as small changes in velocities may cause large changes in kinetic energy.
gz = potential energy. For most industrial devices like turbines and compressors the difference in elevation between the inlet and outlet is minimal. Therefore for these devices the potential energy is neglected. The potential energy is important when a process has a big difference in elevation between the inlet and outlet. (Cengel & Boles, 2010)
When the mass flow rates entering and exiting a system obey mass conservation and there are negligible changes in kinetic and potential energy of a fluid, the energy balance equation can be expressed as
Q ??-W ??=m ??(h_2-h_1)
or
q ??-w ??=h_2-h_1
(Cengel & Boles, 2010)
Energy Balance
The energy balance equation for a steady-flow process is
Q ??-W ??=m ??_out (h+C^2/2+gz)-m ??_(in ) (h+C^2/2+gz)
This equation can be applied to the vortex tube when carrying out an energy balance. As there is no net-work or net heat carried out during the process, these terms can be neglected. The energy balance equation becomes
m ??_O (h_O+(C_O^2)/2+gz)=m ??_C (h_C+(C_C^2)/2+gz)+m ??_H (h_H+(C_H^2)/2+gz)
Neglecting potential and kinetic energy, the equation becomes
m ??_O (h_O )=m ??_C (h_C )+m ??_H (h_H)
For an ideal gas,
h = Cp.T
Therefore
m ??_O (Cp.T_O )=m ??_C (C_P.T_C )+m ??_H (C_P.T_H)
The specific heat capacities for the inlet and two outlets are equal.
Therefore
m ??_O.T_O=m ??_C.T_C+m ??_H.T_H
Divide across by m ??_I
T_O=m ??_C/m ??_O .T_C+m ??_H/m ??_O .T_H
From knowing the mass fractions, these can be subbed into the equation
T_O=??_C.T_C+??_H.T_H
Divide across by inlet temperature TI
1=??_C.T_C/T_O +??_H.T_H/T_O
Let ?? (zeta) represent the temperature fractions of the inlet and outlet temperatures. Therefore
??_C.??_C+??_H.??_H=1
Knowing ??_C+??_H=1
??_H=1-??_C
Therefore
1=??_C.??_C+(1-??_C ).??_H
This is a non-dimensional equation and will be useful when carrying out tests on the vortex tube. If the temperature is measured at the inlet and the hot and cold outlet, using the equation will help in finding the mass fractions for the two outlets and for graphing results.
Compressor Work for Heating/Cooling
In order to work out the entropy balance for the process, the work input from the air compressor must be defined. The vortex-tube will be analysed and compared to an ideal process. An ideal process is also called a reversible process, which is defined as a process that has taken place, can be reversed without any change in the system or surroundings. A reversible process has an efficiency of 100%.
From the point of view of a vortex tube it can be viewed as a heat pump and as a refrigerator. The aim of a heat pump is to transfer heat from a low temperature source to a high temperature source. The aim of a refrigerator is to transfer heat from a low temperature source to a high temperature source. The efficiency of a heat pump or refrigerator is defined by its Coefficient of Performance (COP).
The COP of a heat pump can be expressed as
COP_HP=Q ??_H/W ??_net
QH = Rate of heat absorbed from the low temperature source
Wnet = Rate of net-work done by the compressor
The COP of a refrigerator can be expressed as
COP_R=Q ??_L/W ??_net
QL = Rate of heat rejected from the high temperature source
Wnet = Rate of net-work done by the compressor
As shown, the rate of net-work is required to calculate the COP. (Mulryan, n.d.)
Mathematically work is expressed as force time’s distance. In the compressor, the work done is the amount of force applied to move a piston a certain distance in the cylinder (Figure 3 3). (Borgnakke & Sonntag, 2008)
Figure 3 3: Force Applied to a Piston (Dmic001, n.d.)
Using the sign convention for mechanical work, the work done by the compressor is negative. But, as the vortex tube is being analysed and not the compressor, the work done is taken to be positive as the mechanical work is done on the vortex tube. Therefore the work done is expressed as
W=F.s
Knowing that the area times the distance is equal to the volume,
W=P.V
P = Pressure applied to the piston.
V = Volume displaced by the piston in the cylinder.
This can also be written as
w=P.v
w = specific work done
v = specific volume
Obeying the first law of thermodynamics, the total internal energy (E) is a working substance is the sum of all energies which exist in the substance. The total internal energy can be written as
E=E_k+E_p+U
Ek = kinetic energy of the working substance
Ep = potential energy of the working substance
U = internal energy of the working substance. (Mulryan, n.d.)
In a closed system, there is no change in Ek and Ep. Therefore
E = U
The first law of thermodynamics states that in a closed system
‘Q+’W=’E
Or
‘q+’w=’e
With q = 0
‘w=’u=C_p.’T
Cp = isochoric specific heat capacity = ‘h/’T = ??R/((??-1) )
??R/((??-1)).’T=(??.R.T_2-??.R.T_1)/(??-1)
For an ideal gas
Pv=RT
Therefore
w=(‘??.P’_2.v_2-??.P_1 ‘.v’_1)/((??-1))=(??.P_1.v_1)/((??-1)) [(P_2.v_2)/(P_1.v_1 )-1]
But
v_2/v_1 =(‘P_1/P_2 )’^(1/??) =v_2/v_1 =(‘P_1/P_2 )’^((-1)/??)
From knowing P_1.v_1=R.T_1
Therefore
w=(??.R.T_O)/((??-1) ).[‘P_2/P_1 ‘^((??-1)/??)-1]
(Christy, 2005)
The gas constant R can also be written as
R=Cp((??-1)/??)
Subbing in the gas constant, the equation for specific work becomes
w=Cp.T_1 (‘r_c’^((??-1)/??)-1)
Therefore
The C.O.P of an isentropic vortex tube acting as a refrigerator can be expressed as
‘C.O.P’_R=Q ??_C/W ??_net
‘C.O.P’_R=(m ??_C.Cp.(T_O-T_C))/(m ??_O.Cp.T_O (‘r_c’^((??-1)/??)-1) )
‘C.O.P’_R=??_C.(1-??_C ).(‘r_c’^((??-1)/??)-1)^(-1)
The C.O.P of an isentropic vortex tube acting as a heat pump can be expressed as
‘C.O.P’_HP=Q ??_H/W ??_net
‘C.O.P’_HP=(m ??_H.Cp.(T_H-T_O))/(m ??_O.Cp.T_O (‘r_c’^((??-1)/??)-1) )
‘C.O.P’_HP=??_H.(??_H-1).’ (‘r_c’^((??-1)/??)-1)’^(-1)
The Second Law of Thermodynamics
A process cannot take place unless it satisfies both the first and second laws of thermodynamics. If a cup of hot tea is left in a cooler room, it eventually cools off. This process obeys the first law as the energy lost by the tea is equal to the amount of energy gained by the room. If this process was reversed with the hot tea getting hotter in the cool room, this process would not happen as it violates the first law.
A process occurs in a certain direction and not in the reverse direction. The second law of thermodynamics does not only identify the direction of the process but also states that energy has quality as well as quantity. The second law of thermodynamics is used when finding theoretical limits for the performance of engineering systems. The second law defines perfection for thermodynamic processes and eliminates imperfections in a system. (Cengel & Boles, 2010)
Kelvin Planck Statement
The second law of thermodynamics is an expression of the fact that no heat engine or heat pump has efficiency greater than 100%. The Kelvin Planck Statement states that that it is impossible to create a thermodynamic system which operates in a cycle to take heat from a high temperature source and does an equal amount of work on the surrounding. Heat will flow naturally from a high temperature source to a low temperature source to gain work from a heat engine. The heat engine operates between two temperature sources with a difference in temperature. (Mulryan, n.d.)
The Clausius Statement
The Clausius Statement is another statement relating to the second law of thermodynamics. It states that it is impossible to create a device which operates in a cycle and creates no effects other than the movement of heat from a cool source to a hot a source. The statement is related to the operation of a heat pump or refrigerator, meaning that a heat pump or refrigerator cannot operate without the input of work.
From investigating the two statements, three observations should be made about them. The first observation is that the second law of thermodynamics depends on experimental evidence. Every experiment that has been carried out, relating to the second law, has never contradicted the second law. Therefore the second law is based on experimental evidence.
The second observation is that the two statements are true to one another. This means that if the Kelvin Planck Statement violates the second law, then the Clausius Statement must also violate the second law, and vice-versa.
The third observation is from the second law, stating that it is impossible to create a perpetual-motion machine of the second kind. A perpetual-motion machine of the second kind is a device that would take heat from a source and convert the entire amount of heat into another form of energy; thus violating the second law. (Borgnakke & Sonntag, 2008)
Entropy Balance
Entropy (S) is a property that must be taken into consideration when carrying out the second law analysis of engineering devices. Entropy can be viewed as a measure of molecular disorder. When a system becomes distorted, the molecules become more predictable and the entropy increases. During a process, the change in entropy of a system is greater than the entropy transfer by an amount equal to the entropy generated during the process.
The entropy balance equation for control volumes can be expressed as
‘Q/T+’m_i s_i-‘m_e s_e+S_gen=??S_cv
Or in rate form as
‘Q ??/T+’m ??_i s_i-‘m ??_e s_e+S ??_gen=??S ??_cv
The entropy balance equation states that the rate of change in entropy within a control volume is equal to the sum of the rate of entropy transfer by heat transfer through a control volume, the net rate of entropy transfer by mass flow through a control volume and the rate of entropy generation within the control volume due to irreversibility’s.
As the vortex tube operates in a steady-flow process, there is no change in entropy. Therefore the entropy balance for a steady-flow process can be obtained by letting ??S ??_cv= 0, to give
S ??_gen=’m ??_e s_e-‘m ??_i s_i-‘Q ??/T
For the case of an isentropic vortex tube where the process is adiabatic, there is no entropy generated during the process. The entropy balance equation can be written as
0=(m ??_H s_H+m ??_C s_C )-(m ??_C+m ??_H)s_O
0=m ??_H s_H+m ??_H s_O+m ??_C s_C-m ??_C s_O
0 = m ??_H (s_H-s_O )+m ??_C (s_C-s_O)
For an ideal gas, the change in entropy can be obtained.
s_H-s_O=C_p ln'(T_H/T_O )-Rln(P_H/P_O )
s_C-s_O=C_p ln'(T_C/T_O )-Rln(P_C/P_O )
The entropy balance equation can be rearranged.
0=m ??_H [C_p ln'(T_H/T_O )-Rln(P_H/P_O )]+m ??_C [C_p ln'(T_C/T_O )-R.ln(P_C/P_O )]
0=??_H [C_p ln'(??_H )-Rln(P_H/P_O )]+??_C [C_p ln'(??_C )-R.ln(P_C/P_O )]
0=’1-‘??_C [C_p ln'(??_H )-Rln(P_H/P_O )]+??_C [C_p ln'(??_C )-R.ln(P_C/P_O )]
By multiplying out the equation.
0=C_p.ln??_H-R.ln P_H/P_O -??_C.C_p.ln??_H+??_C.R.ln'(P_H/P_O )+??_C ‘.C’_p.ln??_C-??_C.R.ln(P_C/P_O )
Because the air from the hot and cold outlets are released at atmospheric pressure.
P_H/P_O =P_C/P_O =1/r_C
r_C=P_O/P_atm
Therefore
C_p.ln??_H-R.ln'(r’_C)-??_C.C_p.ln'(‘??_H)+??_C ‘.C’_p.ln'(‘??_C)=0
ln??_H (C_p-??_C.C_p )+??_C ‘.C’_p ln.??_C=-R.ln(r_C)
C_p (1-??_C )ln'(??_H)+??_C.C_p ln'(‘??_C)=-(C_p-C_v )ln'(r_C)
C_p.??_H.ln'(??_H)+??_C ‘.C’_p ln.??_C=-(C_p-C_v )ln'(r_c)
??_H.ln'(‘??_H)+??_C.ln'(??_C)=-((C_p-C_v)/C_p )ln'(r_c)
??_H.ln'(??_H)+??_C.ln'(‘??_C)=((C_v-C_p)/C_p )ln'(r_c)
??_H.ln'(‘??_H)+??_C.ln'(‘??_C)=(1/??-1)ln'(r_c)
r_C=P_O/P_C =P_O/P_H
For air, ?? = 1.4
??_H.ln'(??_H)+??_C.ln'(??_C)=(-0.4)/1.4 ln'(r_C )
ln'(‘??_H^(??_H ))+ln'(‘??_C^(??_C ))=ln'(r_C^(-2/7) )
ln'(??_H^(??_H ).??_C^(??_C ) )=ln”(1/r_c )^(2/7) ‘
”??_H^(??_H ) ‘.??_C^(??_C ) ‘='(1/r_c )’^(2/7)
For a fixed volume of rc, and knowing the value of ??H or ??c, ??H and ??c can be found for an isentropic vortex tube.
Analysis of limits of operation and performance of an isentropic process and analysis of commercial RHVT
In real life there is no such thing as an isentropic device, since the process is ideal. However, when analysing devices such as a vortex tube an isentropic model can be used. Below in Figure 3 4 is an image of an isentropic process on a T-s diagram and an adiabatic process on an h-s diagram for a steady flow device.
Figure 3 4: Isentropic & Adiabatic Process (Amirault, n.d.)
From using the previous equation, for air or diatomic gas, it can be manipulated to solve for the hot and cold temperatures fractions with the aid of MATLAB. Solving for ??c vs. ??c and ??H vs. ??H the equation was manipulated to become the following form
??_c^(??_c ) [(1-??_c ??_c)/(1-??_c )]^(1-??_c )=(1/r_c )^(2/7)
??_H^(??_H ) [(1-??_H ??_H)/(1-??_H )]^(1-??_H )=(1/r_c )^(2/7)
Using the Newton Raphson method for a range of values for the cold mass fraction between 0.15 and 0.85, the hot and cold temperature fractions ??H and ??c were calculated for an isentropic vortex tube. Below in Figure 3 5 and Figure 3 6 is an image of the fx function file generated in MATLAB.
Figure 3 5: MATLAB Fx Function File for ??c
Figure 3 6: MATLAB Fx Function File for ??H
Below in Figure 3 7 and Figure 3 8 is an image of the Newton function files generated in MATLAB to solve for these.
Figure 3 7: MATLAB Newton Function File Generated for ??c
Figure 3 8: MATLAB Newton Function File Generated for ??H
Nex Flow ‘ Air Products Corp. manufactures compressed air-operated products made to improve plant efficiency, improve energy efficiency, improve quality, increase product life and enhance the environment. Their range of products include air blow-off products that reduce noise levels and improve the use of compressed air, and vortex tubes for spot cooling, control panel cooling, tool cooling and other applications. Nex Flow’ has only been set up approximately 10 years but has become a recognised name with sales in large number of countries around the world. (NexFlow, n.d.)
Nex Flow’ provides vortex tubes in a wide range of sizes to meet the needs of many process and spot cooling applications. Their vortex tubes take compressed air and convert into two air streams, one stream is hot air and the other stream is cold air. The main advantage of their vortex tube is that it has no moving parts, which means no maintenance. The cold air can be adjusted down to -28??C, and the hot side can be adjusted up to a temperature of 139??C. (NexFlow, n.d.)
Using the data provided from Nex Flow’, it was possible to analyse a commercial RHVT. The tests were conducted with a 15H generator in a medium sized vortex tube (Model 50015H). There are different types of generators for compressed air capacity. There are also two basic types of generators; one to produce the maximum cold temperature out called the C generator and one to produce the maximum amount of cooling, called the H generator. In the figure below are the temperature drops and rises from the inlet air temperature produced by a vortex tube set at various cold fractions, assuming constant inlet pressure and temperature. (NexFlow, n.d.)
Figure 3 10: Data of a Commercial RHVT by Nex Flow (NexFlow, n.d.)
In Figure 3 10, when the vortex tube is operating at a gauge pressure of 100 psi (6.9 bar) the data for the temperature drop and temperature rise were put into MS Excel, to produce a table and graph. Assuming constant inlet temperature of 20??C.
Cold Mass Fraction (Xc) 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Temperature Drop 68 65 61 55 48 39 30
Temperature Rise 14 25 37 50 66 84 106
Outlet Cold Temperature (??C) -48 -45 -41 -35 -28 -19 -10
Outlet Hot Temperature (??C) 34 45 57 70 86 104 126
Inlet Temperature (K) 293 293 293 293 293 293 293
Outlet Cold Temperature (K) 225 228 232 238 245 254 263
Outlet Hot Temperature (K) 307 318 330 343 359 377 399
Cold Temperature Fraction (??c) 0.768 0.778 0.792 0.812 0.836 0.867 0.898
Hot Temperature Fraction (??h) 1.048 1.085 1.126 1.171 1.225 1.287 1.362
Table 3 1: Commercial RHVT Data Generated using MS Excel
Table 3 1 was created using MS Excel. From this the data was plotted on a graph showing the relationship of XC vs. ??C and XH vs. ??H.
Results of analysis between cold temperature ratio and cold mass fraction
Results of analysis between hot temperature ratio and hot mass fraction
Results of performance analysis between COP and cold mass fraction
Future Work
Upon the completion of the literature review in chapters 3 and 4, a listing of the future work involved in the project is as follows:
Design a device viable for in-house operational conditions, such as, compression ratio, compressor power output and flow rate.
Manufacture a device once the design is successful.
Analyse the performance of the manufactured device and record the results.
Plot the recorded results and compare the performance of the device against the commercial device.
Identify opportunities to optimise the performance of the manufactured device through the investigation of the devices geometry.
Conclusion
References
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