CHAPTER 1

INTRODUCTION

1.1 DEFINITION OF CRYOCOOLER

Cryocooler is a device or ensemble of equipment for producing refrigeration at

temperatures less than 120K.

(i) The quality or worth of a unit of such refrigeration effect depends on the

temperature at which the refrigeration is available

(ii) Theoretical ideal work required to generate a unit of refrigeration as a function of

temperature

Temperature (K) Work(W)

0.5 600

4 74

10 29

1.1.1 Classification of cryogenic cooling systems

Cryogenic cooling systems are classified based on the cooling capacity and refrigeration temperature

Designation Cooling capacity

  1K 4K 20K 80K 120K

Micro miniature <0.25w <1w <1.5w

Miniature <0.5w <2w <8w <12w

Small <1w <10w <100w <0.8kw <1.2kw

Intermediate <25w <100w <1kw <15kw <25kw

Large >25w >100w >1kw >15kw >25kw

The bench mark temperature levels of 1K,4K,20K,80K,and120K are the approximately

liquefaction temperatures at normal pressure of Helium(4K), Hydrogen (20K), Nitrogen

(80K) and Methane (120K)

1

1.1.2 Miniature Stirling Cryocooler

Originally Stirling machines were all driven kinematically, that is, by way of crank shafts

and connecting rods as is used in most positive displacement machinery. The kinematic

configurations have lead to a number of problems peculiar to the Stirling The free-piston

Stirling employs the internal gas pressures and a linear motor to move the reciprocating

components in the proper fashion

Fig.1.1: Miniature Stirling Cryocooler

1.1.3 Benefits of Free Piston Stirling Cryocooler

Simplicity of construction. The basic machine has only two moving parts

and no valves.

A linear motor for supplying power to the piston is easily placed within the pressure vessel making it possible to hermetically seal the unit which avoids the working gas leakage problem

2

Fig.1.2: Free Piston Stirling Cryocooler

1.2 REGENERATOR

The regenerator acts as a large thermal capacity that exchanges heat with the gas it takes

up heat when the gas moves from the hot to the cold side, and it gives off heat when the

gas moves back from the cold to the hot side

Fig.1.3: Regenerator

3

1.2.1 Requirements of Regenerators

• A maximum ratio of the regenerator heat capacity to the heat capacity of the gas.

• A maximum heat transfer between the gas and the regenerator. This can be achieved by using a long and fine-meshed matrix with a large contact area

• A minimum pressure drop over the regenerator. This can be achieved by using a short, highly porous matrix

• Minimum heat conduction from the hot to the cold side of the regenerator.

• Complete penetration of the heat in the regenerator material when it is heated or cooled .This can be achieved by using finely divided regenerator material with a small characteristic dimension

1.2.2 Types of Regenerators

The applications described in above chapter all use what is commonly referred to as a

static regenerator .in general, regenerators are classified as either dynamic of static

regenerator depending on the whether the matrix material is moving or stationary

(i) Rotary type regenerators

(ii) Static type regenerators

1.2.3 Rotary type regenerators

Again in rotary type regenerator they are two types

Axial flow rotary regenerators

Radial flow rotary regenerators

Fig.1.4: Axial flow rotary regenerator Fig.1.5: Radial flow rotary regenerator

4

In rotary type dynamic regenerators, the fluid flows through the matrix material. At a

constant rate and the matrix rotates slowly through the two fluids .Thus the matrix

material is periodically heated and cooled as it passes through the warm and cold

fluids seals between the matrix and housing separate the two fluids entering the

regenerator, and sealing surfaces (partitions) within the matrix keep the flows from

mixing inside the matrix material .leakage past he seal and fluid left in the matrix as it

passes from one stream to other present difficulty is design problems for dynamic

regenerators are found primarily in magnetic refrigeration where the matrix is made

up of a magneto caloric material which is periodically heated and cooled by passing

it through a magnetic field

As the matrix rotates through the magnetic field a fluid also passes through the matrix

to remove heat generated during the magnetization process and transfer the

refrigeration produce during the demagnetization process to the thermal load

1.2.4 Static Type Regenerators

Fig.1.6: Valve-type static regenerator

For the static and rotary regenerators theoretically, the only difference between he two

occurs in the calculation of the matrix heating and cooling durations .for the static

regenerators the flow is periodically switched to the heat and cool the matrix

material ,and the duration is defined by the time interval from when the fluid starts to

flow through the regenerator to the time reversal and the flow starts in the opposite

direction for the rotary regenerator ,the flow is continuous in both directions and the

5

heating and cooling of the matrix occurs by the rotation of the matrix through the flow

streams

For the rotary regenerator ,the duration of heating or a cooling period is defined as the

time interval that begins when the matrix enter one flow streams and ends when it leaves

and enter the opposite stream ,specified as some fraction o rotation of speed of the matrix

Thus the heating and cooling period of rotary regenerator is given by

The successful application of rotary regenerators in cryogenic equipment is rare because

of the need for low-temperature seals to prevent leakage past the matrix and between the

flow streams in above fig the axial seal is required to prevent the leakage past through the

matrix and the annular space between the matrix and housing ,and flow separation seal is

required to prevent the leakage between the in let and exhaust flow streams at either end

of the regenerator .The design and selection of materials for use at low temperatures

always been a particularly difficult task because of thermal contractions and the rigidity

of elastomers in cryogenic temperatures

1.2.5 Classification of wire meshes in Regenerators

The classification of weave style wire meshes are two types

Twill mesh Plain mesh

Fig.1.7: Magnified picture of twill mesh (500X) Fig.1.8: Magnified picture of plain mesh (500X)

Weave density is #200, #250, #300.so.on

6

1.2.6 Specification of Regenerators

Random wire regenerator 1Wire dia. : 30 mm, Material: stainless steelPorosity: 80 %, Mass of the matrix: 25 gHeat transfer area: 0.4237 m2

Fig.1.9(a): Random wire regenerator 1

Random wire regenerator 2Wire dia. : 12 mm, Material : stainless steelPorosity: 90 %, Mass of the matrix: 14.5 gHeat transfer area: 0.5296 m2

Fig.1.9 (b): Random wire regenerator 2

Non-metallic regenerator Wire dia. : 12 mm, Material : Aramid Porosity: 80 %, Mass of the matrix: 4.4 g Heat transfer area: 1.0592 m2

Fig.1.10: Non-metallic regenerator

7

CHAPTER 2

LITERATURE REVIEW

M.D. Atrey, et. al [2] investigated the numerical model that attempts to simulate the

actual conditions existing in the regenerator of Stirling cycle Cryocooler. The model

presented is solved by finite difference techniques and found the performance behaviors

of the regenerators under different operating conditions has studied the present analysis

considers the sinusoidal variation of mass and pressure fluctuations with a phase

difference between them. The analysis shows that the effectiveness of a regenerator is a

function of the porosity of the mesh used and is maximum when the porosity is at a

minimum for a particular mesh number. The effectiveness of the regenerator varies with

mass flow rate and shows a maximum for a particular mass flow rate. Mesh material and

the mean pressure of the system do not affect the effectiveness significantly. For a

particular speed and mass flow rate, an increase in the blow period increases the

effectiveness, which passes through a minimum when the blow period is around 1/4 of

the cycle period.

U. V. Joshi and L. N. Patel et. al [3].Hybrid regenerator with combination of three

different-sized mesh matrix is investigated analytically and attempt is made to minimize

the heat loss. Three-mesh hybrid regenerator is analyzed and results are compared with

homogeneous regenerator. Suitably hybridized regenerator can give much better

performance and increased effectiveness.The performance of three-mesh hybrid

regenerator is much better as compared to two-mesh hybrid regenerator and a simple

homogeneous regenerator due to more uniformly decreasing hydraulic diameter.

8

Shaowei Zhu, Yoichi Matsubara et. al [5]. A numerical method for regenerators has

introduced in this paper. It is not only suitable for the regenerators in the Cryocooler and

Stirling engines, but also suitable for the pulse tubes in pulse tube refrigerators. The

numerical model is one dimensional periodic unsteady flow model. The numerical

method is based on the control volume concept with the implicitly solve method. The

iteration acceleration method, which considers the one-dimensional periodic unsteady

problem as the steady two-dimensional problem, is used for decreasing the calculation

time. By this method, the regenerator in an inertance tube pulse tube refrigerator was

simulated. The result is useful for understanding how the inefficiency of the regenerator

changes with the inertance effect.

Xiaoqin Yang, J.N. Chung et. al [6]. Investigated the Size effects on the performance of

Stirling cycle Cryocooler by examining each individual loss associated with the

regenerator and combining these effects. For the fixed cycle parameters and given

regenerator length scale, it was found that only for a specific range of the hydrodynamic

diameter the system can produce net refrigeration and there is an optimum hydraulic

diameter at which the maximum net refrigeration is achieved. When the hydraulic

diameter is less than the optimum value, the regenerator performance is controlled by the

pressure drop loss; when the hydraulic diameter is greater than the optimum value, the

system performance is controlled by the thermal losses. It was also found that there exists

an optimum ratio between the hydraulic diameter and the length of the regenerator that

offers the maximum net refrigeration. As the regenerator length is decreased, the

optimum hydraulic diameter-to-length ratio increases and the system performance is

increased that is controlled by the pressure drop loss and heat conduction loss. Choosing

appropriate regenerator characteristic sizes in small-scale systems are more critical than

in large-scale ones

Burns and A. J. Willmott, et. al [8] has found the Instantaneous response by a thermal

regenerator to step changes in operation is prevented by the thermal inertia of the system.

The transient response of periodic flow regenerators to simultaneous step changes in inlet

gas temperature and gas flow rate are examined. It is shown how the transient response to

9

a single step change in one of these two input parameters can be superimposed upon that

of the other in order to obtain the response of simultaneous step changes.

G.Venkatarathnam and Sunil Sarangi, et.al [9] has examined the necessity of high

effectiveness in a small volume has led to the development of perforated plate matrix heat

exchangers for cryogenic applications. Although the basic principles have remained the

same, the techniques of fabrication and bonding have changed considerably with the

introduction of all metal construction these exchangers are finding increasing use in

cryogenic refrigerators.The mechanism of heat transfer in a matrix heat exchanger is

complex convection in three different surfaces and conduction in two different directions

are coupled together in determining the temperature profiles. While early analyses were

based on simple empirical correlations and approximate analytical solutions, they have

given way to accurate numerical models. This paper traces the chronological

development of the MHE and different methods of fabrication, heat transfer and fluid

flow characteristics and design and simulation procedures.

J.P. Harvey, P.V. Desai et.al [12], has reported various models for predicting the flow

and heat transfer in a porous Cryocooler regenerator have been proposed in the literature,

such model utilizes a semi-implicit set of equations after making some simplifying

assumptions, resulting in a momentum equation that is decoupled from the energy and

continuity equations. This work addresses concerns with a semi-implicit model an

important result is that the pressure gradient term in the energy equation, which has been

neglected in the semi-implicit model, is leading order.

10

CHAPTER 3

OBJECTIVE AND METHODOLOGY

3.1 OBJECTIVE

(i) To evaluate the performance of homogeneous regenerator by solving the

continuity, gas energy and matrix energy equations for different mesh sizes

with appropriate length and diameter of the regenerator

(ii) Performance investigation of 3-zone Hybrid –Regenerator used in miniature

Cryocooler and compared with homogeneous regenerator

(iii) To finding the optimum combination of regenerator matrix in a 3-zone

Hybrid regenerator

3.2 METHODOLOGY

(i) For the analysis of regenerator from the basic gas energy equation, matrix

energy equation and gas continuity equations, the finite difference technique

have adopted. Gas continuity equation and gas energy equations are solved by

explicit method from the data of ‘t’ time level

(ii) Gauss-eliminations techniques are used in hybrid-regenerator performance by

varying the length of warm end sub-regenerator

11

CHAPTER 4

REGENERATOR HEAT EXCHANGER THEORY

Regenerator theory deals with the physical equations defining the thermal and fluid flow

fields that exist in a regenerator. These equations describes the temperature distribution in

the matrix and the fluid as functions of both space and time and lead to a complex set of

differential equations for-which no closed form of solutions exist .in the following

sections we explore the different types of regenerator designs commonly used in

cryogenic devices and develop both thermal and fluid dynamic equations that defines

their performance

4.1 THE IDEAL REGENERATOR

The regenerator thermal equations are sufficiently complex that has no closed form of

solutions and finite element analysis in combination with high–speed digital computers

are required to obtain the analytical solutions, however, to gain some insight and intuitive

feel for the operation of a regenerator, we will examine several assumptions commonly

employed to reduce the problem to manageable form

Fig.4.1: Static Regenerator Layout

12

The Regenerator we are considering is a static type in many reciprocating Cryocooler is a

Fine phosphor bronze wire mesh screens in a cylindrical housing .The flow is

periodically reversed with warm fluid entering from the right and heating the matrix for

the half of the cycle, referred to as the heating period and cold gas entering to the left and

cooling the matrix for the second half of the cycle, referred to as the cooling period.

The concept of the ideal regenerator is that the warm fluid enters the regenerator at

constant temperature, gives up its heat to the matrix on its way through the regenerator,

and leaves with a lower variable temperature at the cold end .The warm fluid supply is

then turned off, and when, all of the warm fluid is exhausted from the regenerators the

flow is reversed with cold gas entering the cold end at constant temperature .the cold gas

cools the matrix and leaves the regenerator with a variable warmer temperature at the

warmed end .after several reversal of the flow ,a steady state condition is reached where

the matrix temperature distribution will vary periodically with time and the gas and

matrix temperature at any location within the regenerator will repeat them selves from

one cycle to the next .

4.1.1 The assumptions for ideal regenerators

(i) The heat stored in the fluid is small compared with the heat stored in the

matrix material

(ii) The flow is one dimensional or both the radial ,wr, and circumferential wө

flows are zero ,and longitudinal flows wz are finite

(iii) Thermal conductivity of matrix is zero in the longitudinal direction and

infinite in the radial and circumferential directions

13

(iv) The fluid and matrix properties are constant, with temperature and therefore

they do not vary over the length of the regenerator.

(v) The heat transfer co-efficient between fluid and matrix is constant throughout

the regenerator

(vi) The fluid pass in counter flow direction

(vii) No mixing of the fluids occurs during the reversal from hot to cold

(viii) Entering fluid temperature are uniform over the flow cross section and

constant with time

(ix) Regular periodic conditions are established for al matrix elements

With these assumptions the continuity equations

=constant

4.1.2 The matrix thermal equation

The matrix thermal equation will becomes

Where,

Mm is the mass of matrix material in the regenerator,

As is total heat transfer area of matrix material in the regenerator

L is the length of regenerator

4.1.3 The fluid thermal equation

The fluid thermal equation will becomes

The boundary conditions for the ideal regenerator are as follows

14

(i) For the interval of warm flow Tf,in=constant (Tw) at z=0

(ii) For the interval of cold flow Tf,in=constant (Tc) at z=L

4.2 REGENERATOR THERMAL DESIGN PARAMETERS

4.2.1 Exchanger heat transfer effectiveness

The effectiveness defines how well a real heat exchanger is performing relative to an

ideal exchanger operating across the same temperature differences “Efectiveness”and

“efficiency” are used interchangeability, and are defined as

Where,

Q is the actual heat exchange between the fluids

Qideal is the ideal amount that could be exchanged if no temperature difference

between the inlet and outlet streams

For a balanced flow Ch/Cc=1

Regenerator performance is given in terms of inefficiency

15

4.2.2 Number of Exchanger Heat Transfer Units

The number of heat transfer units is a non dimensional expression that is related to a

heat exchangers heat transfer size when the NTU is small the exchanger effectiveness

is low, and when NTU is large the effectiveness approaches a limit physically

imposed by flow and thermodynamic considerations

4.2.3 Fluid capacity Ratio

The fluid capacity ratio measures the thermal imbalance of the flow streams

Fluid capacity ratio=

Where Cmin and Cmax are the smaller and larger of the two magnitudes Ch and Cc

4.2.4 Matrix capacity ratio

Matrix capacity ratio measures the thermal capacity of the matrix relative to the

minimum flow stream capacity

Matrix capacity =

The larger the matrix capacity ratio, the smaller is the matrix temperature swing and,

In general, the more efficient the regenerator

4.2.5 Nusselt Number

The Nusselt number is relates the convective heat transfer coefficient to the thermal

conductivity of the fluid and a characteristic flow dimension l;

16

The physical interpretation of the Nusselt number is that in convective heat transfer,

between a solid and fluid flowing over the solid occurs in a thin boundary layer

region of the fluid near the surface of the solid when the fluid velocity and turbulence

are small, the transfer of heat between the fluid and wall occurs mostly by conduction

and is not aided materially by mixing currents on a microscopic scale. The

temperature difference between the bulk fluid and solid surface occurs across the thin

boundary layer region and the region and heat transfer is given by

Qh=hAs(Tm-Tf)

4.2.6 Pressure drop

Two commonly used expressions for calculating the pressure drop in heat exchanger

(i) fanning pressure drop equation

(ii) kays and London (1964).this equation express the pressure drop for flow

through porous material that is based on the flow acceleration and core

friction

(Flow acceleration core friction)

4.3 REGENERATOR FLOW DESIGN PARAMETERS

The flow parameters most commonly used to define heat exchanger performance are

the hydraulic radius and the Reynolds number

4.3.1 Hydraulic Radius

The hydraulic radius defines the critical flow dimension as

rh= flow cross sectional area /wetted perimeter =Dh/4

Dh, is the hydraulic diameter for a porous matrix

Wetted perimeter is defined as the heat transfer area (As/L)

17

4.3.2 Reynolds Number

In regenerator theory ,the Reynolds number used to characterize nature and similarity

of the flow fields in various porous matrix pickings .Thus ,for a given fluid ,it relates

the mass flow rate to the heat transfer and frictional pressure drop coefficients over a

wide range of operating conditions .We define the Reynolds number in terms of the

average mass flow rate per unit as defined by ,the hydraulic diameter ,and the

dynamic viscosity

4.4 REGENERATOR MATERIAL SELECTION AND DESIGN

In cryogenic applications ,regenerator design deals with the selection of materials and

geometries that optimize performance over the desired temperature range .the selection of

materials is important because materials that perform well at relatively warm

temperatures are generally inappropriate for temperatures below 50K.also geometries that

provide good performance characteristics, such as low pressure drop. may be suitable for

relatively warm temperatures but most likely will perform poorly at colder temperatures

because they contain too much void volume .thus a through understanding of critical

regenerator properties such as specific heat, fluid viscosity,porosity,and heat transfer area

are all critical to a good design

4.4.1 Regenerator packing Geometries

The regenerator matrix should have the following characteristics:

(i) Maximum heat transfer area

(ii) Minimum axial conduction

(iii) Minimum pressure drop losses

(iv) High heat capacity

(v) Minimum dead volume

18

4.4.2 Common regenerator matrix geometries

Fig.4.2: Annular gap regenerator Fig.4.3: Wire mesh screen regenerator

Fig.4.4 Random packed sphere matrix Fig.4.5: Dimpled ribbon

4.4.3 Annular gap regenerators

The annular gap regenerators uses the cylindrical space between two closely fitted

cylinders to provide the heat transfer surface area .This configuration was first employed

in early stirling cycle engines because of the simplicity of the geometry and low pressure

drop it produces. It also has been used with limited success with cryogenic refrigerators,

19

where low pressure drop and minimum dead volume are essential to the overall

performance of the refrigerator

Disadvantage: Limited amount of heat transfer area that can be achieved and the

minimum surface capacity (thermal penetration depths) of the wall to store the heat The

geometrical parameter used to define the flow passage in an annular gap regenerator is

the hydraulic diameter, Dh

Where, Dh=4rh;

Where,

Length of the regenerator matrix, L

free flow area

Total heat transfer area,

4.4.4 Wire mesh screens

The woven wire mesh screen regenerators is commonly used regenerators material its

advantages are that it provides a high heat transfer area with minimum pressure drop it is

readily available in useful mesh sizes from 50mesh(50x50 openings /inch) to over 250

mesh , it is available in many different materials ,it is relatively in expensive to use and

the small diameter and high thermal conductivity of the wire used to weave the screens

provides full utilization of the thermal capacity of the material. Woven bronze screen

regenerators are used in the first stage of all commercial regenerative cryogenic

refrigerators to provide cooling down to 30K.Below 30K,t he loss in specific heat of the

commercially available materials, such as bronze and stainless steel ,limits the effective

ness of the screen pickings.

20

The geometrical parameters used in the description of screen regenerators are the porosity

and area density they are defined as

α = (total volume of connected void spaces /total volume of matrix) = porosity

β= (total surface area of connected voids/total volume of matrix) = area density

From the porosity and area density, the important relationship for the hydraulic radius for

a screen packing is given by

Volume of matrix flow passage =porosity times the volume of regenerator

Where As, is total heat transfer area

4.4.4.1 Porosity

Experimentally the porosity and area density can be found from the dimensions and

weight of the screens.

Where, Wp is the weight of packed matrix material,

is the density of packing matrix

Vr volume of regenerator

4.4.4.2 Area density

Geometry of woven screen

21

Total regenerator heat transfer area

4.4.5 Spherical Packing

Spherical packings such as lead spheres, are used primarily in very low-temperature

applications where commercially available screen materials lose their specific heat and

become ineffective for thermal storage

The characteristics of spherical particle matrix are:

1. For a perfectly packed matrix of uniform particle size, δd = 0, the porosity is

independent of the particle size and approaches a limit of 30%.

%

area density is given by

2. The use of spherical packing leads to lower porosities and higher pressure drop

and, therefore, becomes most effective at temperature below 25K. Where the

viscosity of the working fluid in a cryogenic refrigerator is low.

3. The lower porosity of spherical packing results in larger heat transfer surface area

than obtained with screen matrices.

4. The low temperature, higher heat capacity materials such as lead are generally

only available in spherical form. The materials most commonly used below 25K

or lead, lead antimony, and for very low temperature applications below 10K, a

22

new family of regenerator materials consisting of magnetic intermetallic

compounds, which have been used to achieve refrigeration below 4.2K.

4.4.6 Ribbon Regenerator

In theory, the ribbon generator should be superior to either the woven screen or

spherical particle regenerator because it has a much higher ratio of heat transfer area to

pressure drop. The concept approximates a gap regenerator with multiple flow channels.

The channels are formed by either dimpling or embossing the ribbon and then winding it

on a mandrel. A flow header at either end of the regenerator distributes the flow to the

channels to provide a uniform radial flow distribution across the generator.

4.4.7 Regenerator materials

Regenerator materials and geometries generally fall into three groups, based on

the temperature range over which they are most commonly used in the first group are the

woven screen materials- such as stainless steel, bronze, and copper-which are easy to

weave into the screen geometry. These materials are used over the temperature range

from 30 to 300K, where they provide the following advantages:

Low pressure drop

High heat transfer area

Low axial conduction

High heat capacity

23

CHAPTER 5

NUMERICAL METHODS FOR EVALUATING REGENERATOPERFORMANCE

In this chapter we will review the techniques used to obtain the solutions in the open

form. The analytical models are based on finite difference techniques, in which the

differential equations are replaced with difference equation and solutions are obtained by

a step wise iteration procedure Open –form models are divided into several levels of

complexity distinguished by the assumptions employed the first order model represents

the independent performance of the regenerator without integration into a cryogenic

refrigerator

5.1 ASSUMPTIONS OF THE MODEL

(i) The mass flow through the regenerator and the fluid pressure are constant and

their magnitude s equal during the both flow periods.

(ii) The fluid inlet temperature is constant during the heating and cooling periods

(iii) The void volume in the regenerator is zero, and thus the fluid stored energy is

zero.

(iv) No longitudinal thermal conduction

5.2 NUMERICAL SOLUTION FOR THE IDEAL REGENERATOR THERMAL

EQUATIONS

24

Fig.5.1: Regenerator finite element

5.2.1 Matrix thermal equation

5.2.2 Fluid thermal equation

First the regenerator is sub divided into series of spatial nodes and as in the i=1,

2,3,4…………Nz (number of spatial nodes)second the heating and the cooling periods are

sub divided into a series of small time intervals j=1,2,3,4………….Nt the first order

derivatives are replaced by the finite difference equation

Where the average temperature between the fluid and matrix over the time interval is

25

Fluid and the matrix temperatures at the beginning and end of the each interval, the

temperature change can be expressed as

(i) The ideal matrix thermal equation

(ii) The ideal fluid thermal equation

Substitute the equation (2) into Equation (1) .

By solving he above equations we willl get the Two algebraic equations, those will define

the outlet temperature for the fluid and matrix in terms of known inlet temperature of

fluid and matrix

Final equations:

5.2.3 Regenerator finite element scheme

26

Fig.5.2: Regenerator finite element Schematic

5.2.4 Boundary conditions

(i) Initial matrix temperature profile:

(ii) Heating period boundary condition:

(iii) Cooling period boundary condition

27

5.2.5 Computational procedure

(i) Assume the initial matrix temperature profile between Tw and Tc

(ii) Calculate the outlet temperatures for each node from known boundary

temperature values

(iii) The computational process is repeated for each node (1, 2, 3,4,5….Nz) by

using the calculated outlet temperatures as the inlet temperature for the next

nodal calculations

(iv) Stepwise procedure is repeated for every spatial node over each of the time

interval until a matrix of (Nz*Nt) equations is solved for both the heating and

cooling periods

(v) the final matrix temperatures is used as the initial conditions for the start of

the cooling period

(Reversal condition)

(vi) This procedure is repeated for each period until steady –state behavior of the

temperature distribution is achieved .steady state temperature occurs when the

matrix temperature distribution becomes cyclic and reversal condition

consisting the similar fluid and matrix temperature distributions

5.2.6 Criteria for Convergence

The main use of convergence criteria is to choose the maximum size of the spatial nodes

from these two convergent criteria’s we found that large number of spatial nodes and

time steps will leads convergent to open- form solutions

28

Fig.5.3: Finite difference convergence criteria

For Heating and cooling period the criteria for convergence are:

(i) (Tm)o> (Tf)o heating mode

(ii) (Tf)o >(Tm)o cooling mode

These two conditions define convergence because a crossing of out let temperatures

results in a temperature reversal of inlet temperatures for next node and a switch occurs

in the computation from the heating of the matrix to the cooling of matrix.

5.3 REGENERATOR OPTIMIZATION

Regeneration optimization is the process of choosing the regenerator design parameters

that maximize the system performance. for cryogenic refrigerators, optimization

generally refers to maximizing the available refrigeration by systematically selecting the

regenerator performance that the geometry type matrix design, and matrix material that

achieve this goal .The critical parameters affecting the thermal performance

(effectiveness) of a regenerator are the number of heat transfer units, the fluid heat

capacity ratio, the matrix heat capacity ratio, and thermal losses such as the Longitudinal

conduction

29

These parameters establish the thermal performance of a regenerator because they

determine the temperature difference between the fluid and the matrix, the temperature

swing of the matrix material ,and any other irreversible heat transfer process that

contribute to a degradation in regenerator performance .

To maximize the regenerator effectiveness both the parameters NTU and matrix capacity

ratio must be made as large as possible. However in designing a regenerator for an actual

cryogenic refrigerator the major obstacle limiting the magnitude of these parameters is

the additional requirement to keep the pressure drop and regenerator void volume small

these conflicting requirements that lead to the regenerator optimization in cryogenic

refrigerator and thorough understanding of the interaction of all key parameters.

5.3.1 Conflicting requirements

The optimization problem for the designer as the task of satisfying the following

(i) the temperature swing of the matrix must be minimized ,thus the matrix heat

capacity ratio must be a maximum this can be achieved by a large ,solid

matrix

(ii) The pressure drop across the regenerator must be small it will achieved by the

highly porous matrix

(iii) For the maximum refrigeration the pressure ration must be large or the void

volume small .this can be achieved by a small, dense matrix

5.3.2 Optimization Analysis

To illustrate the conflicting requirements that occur in the optimization of a

regenerator for a cryogenic refrigerator ,we shall consider a procedure for maximizing

the available refrigeration selected to illustrate the problem is a stirling cycle

refrigerator operating between the an ambient temperature of 300K and a

30

refrigeration temperature of 80 K in this case ,the objective of the optimization is to

maximize the available refrigeration by determining the values of the key regenerator

parameters such as length ,cross sectional area matrix material and porosity required

to maximize performance of the regenerator .

The optimization is performed given the following selected operating conditions:

(i) temperature difference across the regenerator is 300K to 80K

(ii) the frequency of operation of the Cryocooler is fixed

(iii) the mean operating pressure of the refrigerator is fixed

(iv) the piston and displacer motions are sinusoidal

CHAPTER 6

MATHEMATICAL MODEL

6.1 MATHEMATICAL MODEL FOR HOMOGENEOUS REGENERATOR

The mathematical model presented is solved by finite difference techniques for different

mesh size of the regenerator with appropriate length and diameter. The analysis is further

extended to study the performance behavior of the regenerators under different operating

conditions

6.2 REGENERATOR CONTROL VOLUME

The Regenerator in Fig 6.1 is assumed to be connected to some restricted volume, thus

the mass of gas, temperature of the gas and matrix are vary with both regenerator length

and time. The time is equal to blow period

31

TC

Fig.6.1 Regenerator domain

6.3 BASIC DIFFERENTIAL EQUATIONS

The governing equations are derived by making energy and mass balances for the gas and

for the matrix in an elemental control volume, dx, at position x within the regenerator

6.3.1 Gas energy equation

Energy flux entering the control volume.= (energy flux leaving the control volume) +

(rate of change of energy within the control volume)

Substituting h=ho+CpTg, where

For a perfect gas u=h-PV and ρg = p/RTg

the final equation is

6.3.2 Matrix Energy Equation

Energy flux entering the control volume = (energy flux leaving the control volume) +

(rate of change of energy within the control volume

32

TE

The final equation is reduced to

6.3.3 Gas continuity Equation

Mass flow entering the control volume = (mass flow leaving the control volume) + (rate

of storage of mass flow within the control volume)

Variation of mass flow: W= WaSinwt +Wm

6.4 NON-DIMENSIONALIZED PARAMETERS

1) W * W/Wa

2) P* P/Pa

3) x* x/Lr

4) t* wt

5) E kgAoWaCpLr

6) B AoPowLr/WaR (Tmc-TE)

7) NTU HTAtLr/waCp

8) C KmA/ MmCmLr2w

9) D HTAt/MmCmw

6.5 COMPUTATIONAL SOLUTION DOMAIN

33

TC

Fig.6.2: Computational domain

6.6 NON-DIMENSIONALIZED EQUATIONS

6.7 DIMENSIONAL BOUNDARY CONDITIONS

At x=0,

At x=Lr ,

At t=0,

6.8 NON-DIMENSIONAL BOUNDARY CONDITIONS

34

TE

At x*=0,

At x*=1,

At t*=0,

6.9 DESCRITIZED EQUATIONS

6.10 FLOW-CHART

Flowchart is used for finding the Mass flow rate, Temperatures of fluid and Matrix

35

Fig.6.3: Sequential steps in Flow-chart

6.11 MATHEMATICAL MODEL FOR 3-ZONE HYBRID REGENERATOR

Find the Tg*,Tm*From W* values(i=2 to Nz-1) nodesFor j=1 (1st iteration)

Run with initial values of W*

Convergence(y/No)

Find Tg*,Tm* at Nz(Back ward difference formula)

stop

Go to next time iterations

Enter the input of regeneratorMeshsize, length diameter, geometric properties

start

36

The hybrid regenerator is build by combining different size mesh matrix, so that porous

matrix is at warm end and dense matrix at cold end. Hybrid regenerator with combination

of three different-sized mesh matrix is investigated analytically and attempt is made to

minimize the heat loss .Attention is given to mesh matrix size of regenerator. Three-mesh

hybrid regenerator is analyzed and results are compared with homogeneous regenerator.

Suitably hybridized regenerator can give much better performance and increased

effectiveness. The performance of three-mesh hybrid regenerator is much better as

compared to two-mesh hybrid regenerator and a simple homogeneous regenerator due to

more uniformly decreasing hydraulic diameter.

The selection of optimum proportion of each mesh matrix is the key problem in designing

three-mesh hybrid regenerator .the optimum combination serves the purpose of minimum

heat loss due to pressure drop and improved effectiveness. It is difficult to guess the

optimum proportion by random selection of matrix proportion in three-mesh hybrid

regenerator

Fig.6.4: Simple Homogeneous Regenerator

Fig.6.5: 3-zone Hybrid Regenerator

6.12 REGENERATOR CONSTRAINTS

The constraints in the selection of matrix are:

(i) The total length of each sub-regenerator must be equal to length of regenerator

L1+L2 +L3= Lr

(ii) The sum of mass of each sub regenerator must be limited to

the mass of

Homogeneous regenerator

M1+M2+ M3= Mm

37

6.13 METHODOLOGY

Solving (1) and (2) through the gauss-elimination technique for a selected value of warm-

end sub regenerator, a computer code has developed to solve the simultaneous equations

and obtained the optimum combination

6.14 ASSUMPTIONS INVOLVED

Constant temperature of fluid at the inlet of each matrix.

Linear axial temperature distribution through regenerator matrix each sub

regenerator is analyzed separately and the results for hybrid regenerator is

calculated as

Similarly we have to calculated the refrigeration loss due to inefficiency, pressure

drop and axial conduction can be calculated

Where,

Qnet = Net Refrigeration capacity, watts.

Qideal = Ideal Refrigeration capacity calculated from

Thermodynamic analysis for design data [3] = 5.83 watts.

QI,H = Total refrigeration loss due to regenerator ineffectiveness, watts.

QΔP,H = Total refrigeration loss due to pressure drop due to flow resistance, watts.

QA,H = Total refrigeration loss due to axial heat conduction, watts

38

6.15 FLOW-CHART FOR HYBRID REGENERATOR

Constraints(i) Mass of matrix(ii) Length of Reg

No of punches ,n

Net cooling effect

Hydraulic dia.Massflow ,velocity,

Re,Nu,h

Fixed parametersFrequency

Tc.Te ,Blow time

start

stop

Chose mesh sizeReglen (Lr)

Chose L1

Gauss-eliminationLength of sub reg

L2,L3

Physical properties at mean temperature(i) Working fluid(ii) Matrix material

Performance parametersCapacity ratio ,NTUEffectiveness,pressure drop

Regenerator losses due to(i) axial conduction(ii) flow resistance

(iii) Ineffectiveness of reg

Fig.6.6:Flow-chart for Hybrid Regenerator

6.16 REGENERATOR HEAT LOSS PARAMETERS

(i) loss due to ineffectiveness:

(ii) loss due to axial conduction

(iii) Pressure drop loss

39

‘

CHAPTER 7

RESULTS AND DISCUSSION

7.1 REGENERATOR PERFORMANCE RESULTS

The present analysis is applied to miniature Regenerator of fixed length and

diameter .The quantities of various parameters are calculated from the available data for

different mesh sizes this analysis is carried out with Helium as a cryogenic fluid and

phosphor Bronze is the matrix material

(a) 200mesh (b) 250mesh (c) 300 mesh

Fig.7.0 Regenerator Mesh configurations

7.1.1 Input Data

InputLength of regenerator 52mmDiameter of regenerator 8.6mmTemperature of compression side 320KTemperature of expansion side 80KMatrix material Phosphor bronzeDensity 8860kg/m3

Specific heat 376j/kg-Kcryogenic fluid HeliumDensity 3.86kg/m3

Specific heat 376j/kg-KThermal conductivity 0.1052w/m-kKinematic viscosity 0.000003965m2/sType of mesh Mesh wire screenPorosity 0.668X200,0.665X250,0.609X300wire diameter(mm) 0.05X200,0.04X250,0.03X300

40

7.1.2 Temperature distribution along the length of Regenerator

0

50

100

150

200

250

300

350

0 0.01 0.02 0.03 0.04 0.05 0.06

RegeneratorLength(m)

Tem

per

atu

re o

f g

as(K

)

200mesh

250mesh

300mesh

Fig 7.1: Temperature distribution of fluid along the length of Regenerator

7.1.3Temperature of matrix along the length of Regenerator

0

50

100

150

200

250

300

350

0 0.02 0.04 0.06

Regenerator length(m)

Te

mp

era

ture

of

ma

trix

(K)

200mesh

250mesh

300mesh

Fig. 7.2: Temperature of matrix along the length of Regenerator

41

7.1.4 Ratio of mass flow variations along the length of Regenerator

3.495

3.53.505

3.513.515

3.52

3.5253.53

3.5353.54

3.545

0 0.02 0.04 0.06

Regenerator Length(m)

Rat

io o

f m

ass

flo

w f

luct

uat

ion

s

200mesh

250mesh

300mesh

Fig .7.3: Ratio of mass flow variations along the length of Regenerator

The above three graphs figs 7.1,7.2,7.3,will shows the typical performance of fluid

temperature, matrix temperature and ratio of mass flow variation along the length of a

regenerator

The temperature of fluid along the length is decreasing and the temperature of matrix

along the regenerator is increasing because we have considered the heating blow in this

blow the matrix has been getting heated and the fluid losses the heat, the fine mesh

having less porosity the slope value in the graph is less as compared to the coarse mesh

size because of fine meshes having high free flow area so that they have good heat

transfer

42

7.2 HOMOGENEOUS REGENERATOR (FIXED LENGTH, DIAMETER)

7.2.1 Regenerator effectiveness

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4 5

Mass flow rate(g/s)

Eff

ec

tiv

en

es

s

150

200

250

300

Fig.7.4: Effectiveness v/s Mass flow rate

0

0.2

0.4

0.6

0.8

1

1.2

0 0.02 0.04 0.06 0.08 0.1 0.12

Blowtime(S)

Eff

ecti

ven

es 150

200

250

300

Fig .7.5: Effectiveness v/s blow time

43

The variation of effectiveness versus mass flow rates are represented in fig 7.4, for

different mesh sizes, The mesh size influence the effectiveness of the regenerator

significantly, due to a change in the porosity of the regenerator it can be observed from

the graph maximum effectiveness is obtained at the given mesh of 300-49, where 300

indicates number of wires per square inch and 49 indicates the standard wire gauge

From the above graph fig 7.4, it is evident that at lower mass flow rates the conduction

effect is more dominant than the loss due to imperfect heat transfer and at higher mass

flow rates the loss due to heat transfer is more than that of conduction

The variation of effectiveness versus Blow time (fraction of cycle period) represented in

fig 7.5 for different mesh. The blow period may be defined as the time taken for the total

quantity of fluid to pass any point in the regenerator. The blow period is varied from 0.01

to 0.1 S this graph shows as the blow period is increase ,the effectiveness is decreasing

for fine mesh and increasing for coarse meshes as the blow period is increasing the Mass

flow fluctuations are increasing so, fine meshes having high effectiveness

44

7.2.2 Heat transfer co-efficient

Heat transfer co-efficient

0

2

4

6

8

10

12

0 1 2 3 4 5

Mass flowrate(g/s)

He

at

tra

ns

fer

co

eff

icie

nt(

kW

/m-K

)

150

200

250

300

Fig .7.6: Heat transfer co-efficient v/s Mass flow rate

Heat transfer co-efficient

0

2

4

6

8

10

12

0 0.02 0.04 0.06 0.08 0.1 0.12

Blow time(S)

He

at

tra

ns

fer

co

-eff

icie

nt(

kW

/m-k

)

150

200

250

300

Fig .7.7: Heat transfer co-efficient v/s Blow-time

45

7.2.3 Nusselt Number

0

1

2

3

4

5

6

7

8

9

0 1 2 3 4 5

Mass flow rate(g/s)

Nu

ss

elt

nu

mb

er

150

200

250

300

Fig. 7.8: Nusselt number vs. Mass flow rate

0

1

2

3

4

5

6

7

8

9

0 0.02 0.04 0.06 0.08 0.1 0.12

Blowtime(S)

Nu

ss

elt

nu

mb

er

150

200

250

300

Fig .7.9: Nusselt Number v/s blow time

46

7.3 HOMOGENEOUS REGENERATOR FIXED MESH (VARIABLE LENGTH)

7.3 Input data

Regenerator data Matrix dataDiameter =8.6mm mesh 200, porosity 0.668Number of screens 440, 520, 635, 1060

7.3.1Regenerator Inefficiency

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 0.2 0.4 0.6 0.8 1

Mass flow rate(g/s)

Ineff

icie

ncy(%

) 44.mm

63.5mm

52mm

82mm

106mm

Fig .7.10: Regenerator Inefficiency v/s Mass flow rate

7.3.2 Heat loss in regenerator

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 0.02 0.04 0.06 0.08 0.1 0.12

Blowtime(S)

He

at

los

s i

n R

eg

en

era

tor(

W)

44.mm

63.5mm

52mm

82mm

106mm

Fig .7.11: Heat loss in the regenerator v/s Blow time

47

From the Fig7.6 it is conclude that as go on increasing the mass flow rate the value of

heat transfer co-efficient increasing, that is more for porous mesh because of high heat

transfer area more number of wire screens, and the Nusselt number is more for denser

mesh (coarse) because the hydraulic diameter for coarse mesh is more as compared to

fine mesh.

The Fig 7.7 explains the behavior of the heat transfer co-efficient versus the blow time as

go on increasing the blow period the heat transfer co-efficient is decreasing because of

the frequency of operation is decreasing i.e number of cycle per sec, so it will leads to

decrease the flow velocity, Reynolds number based on geometric properties 300 mesh

having high heat transfer co-efficient

The Fig 7.9, explains the behavior of the Nusselt number versus the blow time as go on

increasing the blow period the Nusselt number is decreasing because of the frequency of

operation is decreasing i.e number of cycle per sec, so it will leads to decrease the flow

velocity, Reynolds number the value of Nusselt number values are more for coarse mesh

because of the hydraulic diameter is more as compared to the fine mesh

The Fig 7.10, explains the behavior of the percentage of in efficiency versus the mass

flow for different length of regenerators, the inefficiency is less for long regenerator

because of as the length of matrix increases the average leaving outlet fluid temperature

approaches the cold end temperature of regenerator so the percentage of in efficiency is

decreasing

The Fig 7.11explains the behavior of heat loss in the regenerator versus the blow time as

increasssing the blow time the mass flow increases, inefficiency is decreasing means

regenerator losses due to inefficiency are decreasing (heat losses in the regenerator)

As incressing the matrix length with fixed mesh.

48

7.4 RESULTS FOR 3-ZONE HYBRID REGENERATOR

Fig 4.12: Simple homogeneous Regenerator

Fig.7.13: Three-mesh hybrid Regenerator

Different combinations of three-mesh hybrid regenerator are developed by assuming the

length of warm-end sub-regenerator. Results for the selected combinations are obtained

using the computer programme and compared with 200-mesh matrix that represents the

homogeneous regenerator.

The results are plotted for different selected values of warm-end regenerator, the fig

shows the effect of matrix size and mesh fraction on the performance of regenerator with

different combinations

49

7.4.1 Input data for Hybrid regenerator

(i) Temperature of compression space, Tc = 320 K.

Temperature of expansion space, Te=80 K.

Regenerator Length, L = 52 mm.

Mesh Punch diameter, D = 8.6 mm

Working fluid = Helium.

Blow time, t = 0.01 sec.

Mesh material = Phosphor-bronze

(ii) Regenerator / Displacer

Regenerator tube material = Stainless steel

Tube size = 52 mm length,

Inner diameter =8.6mm

7.4.2 Mesh matrix characteristics

MeshSize No.

WireSize No.

Pitch,P(mm)

Mesh Distance(mm)

Opening area ratio

porosity

150 46 0.061 0.108 0.408 0.699

200 47 0.050 0.077 0.368 0.668

250 48 0.040 0.061 0.365 0.665

300 49 0.030 0.054 0.409 0.609

50

7.4.3 Regenerator Effectiveness

regenerator effectiveness

0.984

0.986

0.988

0.99

0.992

0.994

0.996

0.998

0 0.01 0.02 0.03 0.04 0.05

LengthL1(m)

Eff

ecti

ven

ess

200-250-300

150-200-300

150-250-300

200

Fig.7.14: Effect of matrix size and mesh fraction on effectiveness of three-mesh hybrid regenerator

7.4.4 Pressure drop

pressure drop

0

50

100

150

200

250

0 0.02 0.04 0.06

LengthL1(m)

Pre

ss

ure

dro

p(m

pa

)

150-200-300

150-250-300

200

Fig.7.15: Effect of matrix size and mesh fraction on Pressure drop of three-mesh hybrid regenerator

51

7.4.5 NTU

NTU

0

50

100

150

200

250

0 0.02 0.04 0.06

LengthL1(m)

NT

U

200-250-300

150-200-300

150-250-300

200

Fig.7.16: Effect of matrix size and mesh fraction on Number of Transfer units of three-mesh hybrid regenerator

Fig.7.17: Effectiveness versus NTU for three-mesh hybrid regenerator

52

0.984

0.986

0.988

0.99

0.992

0.994

0.996

0.998

0 50 100 150 200 250

NTU

Effe

ctiv

enes

s

200-250-300 150-200-300 150-250-300

The above two graphs Fig 7.14, 7.15, explains the behavior of the effectiveness and

pressure drop for a hybrid regenerator with different selected values of length of warm

end sub regenerator, with (150-250-300) combination, for the same effectiveness as that

of homogeneous regenerator, much improved performance can be obtained wit reduction

in pressure drop

(200-250-300) combination has largest density thus it’s having high effectiveness, and

net refrigeration capacity also more. because of uniform decreasing in the hydraulic

diameter

The above graphs Fig 7.16,7.17 explain the behavior of NTU, effectiveness for different

selected values of length of warm-end sub regenerator the length of warm –end sub

regenerator should be small and that should be a porous mesh then we will get the high

NTU values, Because of (200-250-300) mesh having the high density and uniform

decreasing the hydraulic diameter

For 220-250-300 mesh corresponding NTU the value of effectiveness is more, it means

The regenerator thermal losses are minimizing which leads the net refrigeration capacity

is more.

53

7.5 Optimized parameters for three-mesh hybrid regenerators

For a fixed length and diameter of regenerator, the optimized length and mesh in three

zones are calculated in such a way that to get the maximum effectiveness the tabular

column should fulfils the requirements for design of regenerator

Fig.7.18: 3-mesh hybrid regenerator

Fig.7.19: physical dimensions of regeneratorLr=52mm

dr=86mm

Housing diameter=0.1mm,Housing material stainless steel

Matrix material=phosphor Bronze (wire screen meshes)

Cryogenic fluid =Helium gas

Warm end side temperature Tw=320K

Cold end side Temperature=80K

Zone Length of sub regenerator (mm)

Mesh size Number of screens

Type of matrix

Type of mesh

porosity

1st zone 5 200X200dw=0.05mm

50 Phosphor bronze

Wire screen

0.668

2nd zone 19.423250X250dw=0.04mm

242 Phosphor bronze

Wire screen

0.665

54

3rd zone 27.57300x300dw=0.03mm

460 Phosphor bronze

Wire screen

0.609

7.20: Optimized length, mesh size for a fixed diameter, length of regenerator 7.6 Comparision of ideal regenerator with Ackermann Model

150X150 Screen meshDiameter of Regenerator 19.05mmlength 7.62mmscreens 610

7.6.1. Percentage of Inefficiency of Regenerator

0

0.2

0.4

0.6

0.8

1

1.2

0 0.05 0.1 0.15 0.2 0.25

blow period(S)

%o

f In

eff

icie

ncy(%

)

ideal regeneratormodel

ackermann results

Fig.7.21: Percentage of Inefficiency v/s Blow time

7.6.2 Percentage of Inefficiency of Regenerator

55

0

0.2

0.4

0.6

0.8

1

1.2

0 5 10 15

Mass flow rate (x10^3cc/s)

% o

f In

eff

icie

ncy(%

)

ideal regenerator model

ackermann results

Fig.7.22: Percentage of Inefficiency v/s mass flow rate

CHAPTER 8CONCLUSION

Regenerator is the heart of miniature Stirling Cryocooler that strongly influences the

performance of Stirling machines. The present analysis considers the sinusoidal variation

of mass flow.

The analysis shows that the effectiveness of a regenerator is a function of porosity

of mesh used and is maximum when the porosity is at a minimum for a particular

mesh number

The hybrid regenerator is build by combining different size mesh matrix, so that porous

matrix is at warm end and dense matrix at cold end. Hybrid regenerator with combination

of three different-sized mesh matrix is investigated analytically and attempt is made to

minimize the heat loss

The hybrid regenerator analysis shows that for particular length ,mass of matrix

the performance of 3-mesh hybrid regenerator much better as compared to

homogeneous regenerator, due to hybrid regenerator having the greater density of

mesh

56

Under the same operating conditions The Hybrid regenerator is more effective

than than simple Homogeneous regenerators the Concept of hybrid regenerator

combinations will have implemented in a miniature Cryocooler Refrigeration

systems, to compare this results with experimental results same work is to be

tested experimentally

REFERENCES

[1] A. Ackermann, 1997, Cryogenic Regenerative Heat Exchangers, Plenum Press, New

York.

[2] M.D. Atrey, S.L. Bapat and K.G. Narayankhedkar “Theoretical analysis and

performance investigation of Stirling cycle regenerators” journal of cryogenics –

vol.31(1991)December

[3] U. V. Joshi and L. N. Patel, “Performance Investigation of Two-Mesh Hybrid

regenerator for Miniature Stirling Cryocooler,” Proc. of National Conference on

Emerging Trends in Mechanical Engineering, pp. 297, Sept. 10-12,2004.

[4] U. V. Joshi and L. N. Patel, Analysis, “Design and Performance Investigation of

Three-Mesh Hybrid Regenerator for Miniature Stirling Cryocooler”,

[5] Shaowei Zhu, Yoichi Matsubara, “A numerical method of regenerator” journal of

cryogenics - vol.44 (2004), pp.131-140

[6] Xiaoqin Yang, J.N. Chung., “Size effects on miniature Stirling cycle cryocoolers”

Journal of cryogenics - vol.45 (2005), pp.537-545

[7] W. M. Kays, A. L. London, Compact Heat Exchanger, McGraw-Hill Company

(1976).

[8] A. Burns and A. J. Willmott, “Transient performance of Periodic flow Regenerators

“International journal of heat and mass transfer-vol.23, pp623-627

57

[9] G. Venkatarathnam and Sunil Sarangi, “Matrix heat exchangers and their application

in cryogenic systems”, journal of cryogenics-vol.30 (1990), pp.907-918

[10] Walker.G, cryocoolers, part1, fundamentals and part 2; applications, plenum press,

New York

[11] S. Sarangi and H.S. Baral“Effects of axial conduction in the fluid on cryogenic

regenerator performance” journal of cryogenics-vol.27 (1987), pp 505-509

[12] J.P. Harvey, P.V. Desai, and C.S. Kirkconnel, “A Comparative Evaluation of

Numerical Models for Cryocooler Regenerators”, Cryocoolers-12, Kluwer Academic

Plenum Publishers, 2003

58