Mathematical Modelling and Design Software for Cryogenic Regenerator

DOI : 10.17577/IJERTCONV6IS16012

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Mathematical Modelling and Design Software for Cryogenic Regenerator

Mathematical Modelling and Design Software for Cryogenic Regenerator

Debashis Panda, Manoj Kumar

Ph.D. Scholar, Cryogenics Engineering Laboratory Department of Mechanical Engineering

National Institute of Technology, Rourkela ,Odisha, India, 769008

  1. K. Satapathy, R. K. Sahoo

    Professor, Cryogenics Engineering Laboratory, Department of Mechanical Engineering National Institute of Technology,

    Rourkela, Odisha, India,769008

    Sunil K. Sarangi

    Emeritus, Process Equipment and Design Laboratory, Cryogenic Engineering Centre Indian Institute of Technology, Kharagpur Professorial fellow,

    IIT Bhubaneswar

    AbstractClosed cycle regenerative cryocooler provides a wide variety of technical merits, among them are ease of design and construction, highly reliable operation, minimal maintenance requirements and moderate cost. Stirling and pulse tube cryocoolers offer the simplicity of operation and vibration free operation. It also covers a wide range of cold end temperature and cooling capacity. The space required by Stirling and pulse tube cryocoolers are very small, so they are mostly used in space-based applications. In all regenerative cryocoolers, the regenerator is the chief component of these machines. Over past three decades, the mechanism of refrigeration has been understood and it is obvious that the performance of these machines directly depends upon the effectiveness of the regenerator. However, the design of regenerator may be complicated and tedious task to the engineers and scientists, pursing in this field due to the complex fluid flow and heat transfer process occurring inside it. To overcome these difficulties, design tools and software becomes very much essential to design regenerators to achieve required output. This paper is directed towards the creation of a general purpose simulation software to design regenerators. Both primary and secondary losses are taken into consideration. The effect of longitudinal conduction and wall conduction are also taken into consideration. The software is written in MATLAB, which has a decent graphical user interface. It assists the designer, not only in feeding input information but also in generating reports, those would support the user in future. The design is implemented through a process of successive simulation. The results are compared with published results and a good agreement is observed. A regenerator is simulated by using this software and various thermo-hydrodynamic phenomena are validated with published experimental results. The effect of various operating parameters on the performance of a typical cryocooler is also presented with interactive charts.

    KeywordsRegenerator,Softwrare, MATLAB, IR Sensors


      Regenerator is the crucial component of all regenerative type machines including Stirling, GM, VM, pulse tube cryocoolers and Stirling engines of alpha, beta and gamma configurations. Stirling and pulse tube cryocooler are used in IR sensors because of its compact size. The design of such miniature cryocoolers is a challenging problem to the

      scientists and engineers working in this field. The performance of cryocoolers and heat engines is an active function of the effectiveness of the regenerator. Regenerator is thermal energy storage device, consists of a hollow cylinder filled with meshes (woven meshes, random fiber meshes etc.) or lead balls, which stores heat from the fluid during heating period to cool the working fluid. To provide the required refrigerating effect, it releases heat to the fluid during the cooling period. Because of periodic and unsteady flow of fluid inside the matrix of the regenerator, its design may be a little bit complicated. The use of regenerator in hot air engine was reported by Stirling [1]. However, early mathematical modelling of regenerator was found in a German publication [2], in which Nusselt did the mathematical analysis of regenerator assuming infinite matrix heat capacity. The comparison of the regenerative heat exchanger with recuperative heat exchanger was carried out by Hausen [2]. IIife [3] performed the first order numerical analysis considering a transient variation of matrix temperature.

      A lot of numerical and experimental work on regenerator was carried out by London et al. [4] for gas turbine plants. Wilmot [5-7] expanded the reduced length and reduced period method proposed by Hausen et al. by considering three-dimensional effects to examine multidimensional effects of a regenerator. Radebaugh [8, 9] gave an approximate design procedure for designing of the regenerator. REGEN series software developed by Radebaugh et al. is used for designing of regenerators. Several investigators used REGEN for carrying out parametric studies of regenerator by assuming various materials [10-15]. Kapitza [2] first reported the application of regenerators in cryogenic refrigerators. Regenerators are classified into two different types, depending upon the fact that regenerator matrix would be stationary or rotating. In stationary regenerator, the matrix is rest whereas in rotating type the matrix is rotating. In cryocoolers and heat engines, mostly stationary regenerators are used. While the REGEN software is mainly used for fixed matrix regenerators, whereas, the present software can be used for the design of fixed matrix regenerator of various meshes (woven mesh,

      random fiber mesh, spherical balls) and also rotary regenerators. In the present article, modelling procedure and results for fixed matrix regenerator are presented.

      The significant losses that affect the performance of regenerator involve ineffective loss, conduction loss of matrix, temperature swing loss and wall friction loss and pressure drop loss. This article is directed towards the solution of such complex equations to develop general purpose software with an advanced interface that will help the designers not only to design regenerators but also to predict its performance from thermal aspects. The results of this software are tested with the results available in open

      Eq. (1) represents the unsteady term or accumulation of heat in regenerator matrix, whereas the term on the right-hand side signifies the convection heat transfer between fluid and matrix. Similarly, in Eq. (2) left-hand side represent the accumulation of heat in fluid, and right-hand side represents convection heat transfer between fluid and matrix.

      Longitudinal conduction effect

      In regenerator the matrix absorbs heat alternatively from hot fluid and cold fluid so there is conduction loss along the matrix of regenerator. Matrix thermal equation is

      T T

      literature. A good agreement is obtained between them.


      m V

      • h A T T

      c V





      m z

      m s m p m t

      The mathematical model presented here is based upon numerical solutions of thermal energy equations of working fluid and matrix of regenerator. Detailed description of this can be found elsewhere [2, 16]. The mathematical model is based on the assumption that considers one-dimensional longitudinal heat flow.

      Fluid thermal equation is


      h As (T Tm ) (cpV ) f



      Ideal Model

      Thermal energy equation of matrix

      Longitudinal conduction and wall effects.

      Matrix thermal equation is

      T aK




      (c V )

      m h A (T T )

      m m

      K m A z hA

      (T )


      m s m

      p m t s m


      Tm z

      z2 z


      Thermal energy equation of working fluid



      p )m ]z



      p f

      (c V )

      s m

      h A (T T )



      Wall thermal equation is

      2 2

      Nt bK T

      wl wl K

      Twl A

      z hA

      (T T )

      T z

      wl z2

      wl wl z wl



      p )wl ]z




      Fluid thermal equation is



      2 hAs z

      (T Tm ) hAwl z

      (T Twl ) (mcp ) f t



      1 2 3 Nz

      FIG: 1 Discretization of regenerator geometry section.

      where i is number of spatial nodes and j is number of time step given as follows

      i 1, 2,3,…………………..Nz

      j 1, 2,3,………………….Nt

      The governing equations are converted to algebraic equations according to the procedure proposed by Ackermann et al. [2]. Then, the algebraic equations are solved by means of an iterative method. The temperature on the left hand side is assumed to be Th and cold end is Tc. Linear variation of temperature is assumed as initial condition. The effect of various losses are affect the performance of regenerator are presented by Panda et al. [16]. Pressure drop has been taken into consideration as reported by Kays et al. [4]. All other parameters (e.g. inefficiency) are calculated as reported by Ackermann et al. [2].

      Where, Nz

      the total number of spatial nodes and Nt

      is the

      total number of time step. The term on the left-hand side of



      Screen diameter

      19.05 mm

      Wire diameter

      0.069 mm

      Regenerator thickness

      0.152 mm

      Number of screens


      Mesh size

      #150 phosphorous bronze

      Flow rate


      Cold end temperature

      80 K

      Hot end temperature

      300 K


      This section explains the detailed design and explanation of the Graphical User Interface of CRESP-REGEN package, which has been written in MATLAB [17]. Fig. 2 shows the input interface of woven matrix configuration that accepts various input parameters from the user including geometrical parameters and operating parameters. Also, it allows the user to choose the solid and fluid materials. After entering the values of parameters, the user has to choose the model ideal, the


      model including longitudinal effect and longitudinal effect and wall effect, then hitting run it will solve the equations and produce results, which will be displayed after user will

      Parameter Ackerman et al.

      Present result


      ve error (%)

      click results. The detailed description of various menu and toolbar icons as well as other configurations for e.g. Random fiber meshes, spherical balls, parallel plates and rotary regenerators are beyond the scope of this paper. The GUI is converted into Microsoft Visual Studio 2012 to develop the executable package that runs in Windows operating system

      Length(cm) 10.16 10.14 -0.19

      Heat transfer area(m2) 0.44 0.48 8.33

      Reynolds number 53 53.77 1.43

      NTU 192 210 8.57

      Stanton number – 0.16 –

      Mass of regenerator 0.1 0.14 28.5

      Ineffectiveness (%)

      Ideal 0.62 0.68 8.82

      without MATLAB [18]. In an ideal model, least time is required to get results, on the other hand non-ideal case (Longitudinal effect and longitudinal conduction) more computational time is needed.

      Longitudinal conduction

      Longitudinal conduction + wall effect

      .71 0.79 10.12

      0.99 1.02 2.94

      FIG: 2 input GUI of CRESP-REGEN woven matrix configuration.


      First, the numerical programme is validated with the previously published results by Ackerman et al. [2]. The input parameters for validation are shown in Table 1. Since the solution procedure is based upon discretization approach, the results will vary with the change in time intervals and space coordinates. If the number of space coordinates and time intervals are small, then accuracy will be better but the computational time will be more. Hence, there must be a comfortable balance between accuracy and computational time. The results of present software with results published by Ackerman et al.[2] is shown in Table. 2.




      A parametric study is conducted to examine the effect of various geometrical and operating parameters upon the performance of a regenerator for cryocooler based applications. The input parameters chosen for parametric investigation is given in Table 3.



      Regenerator Screen Diameter(mm) 10

      Thickness of wall(mm) 0.5

      Wire diameter(mm) 0.04

      Mesh size # 250SS

      Number of Mesh 670

      Cold end temperature(K) 80

      Hot End temperature(K) 300

      Frequency(Hz) 28

      Number of spatial Nodes 580

      Number of time interval 85

      Maximum period 120


      Mesh size

      Mesh sizes are available in various ranges in markets (50 50 to 635 635) for both academic and industrial-based applications. 50 50 mesh means 50 numbers of openings per

      inches. As the mesh size increases, then number of mesh opening per inch increases, results in decreasing porosity. Also, thermal penetration depth increases that lead to an increase in heat transfer area. It is clearly observed from Fig. 3, with an increase in mesh size inefficiency decreases. Also, the pressure drop increases with increase in mesh size.

      Wire diameter

      Wire diameter is also another significant geometrical parameter related to the screen mesh that affects the performance of regenerator. Increase in wire diameter decreases the porosity of mesh. Since porosity is inversely proportional to the mass of regenerator, a decrease in porosity increases the mass of regenerator and also heat transfer area. From Fig. 4, it is evident that there is an inefficiency of regenerator with increase in wire diameter of regenerator.

      FIG: 3 Eeffect of mesh size on regenerator inefficiency

      Wall thickness

      Figure 5 indicates that, with increase in regenerator wall thickness, inefficiency increases due to the increase in conduction heat loss through the wall. However, this has negligible effect as compared to mesh size and wire diameter. In case of low temperature applications (e.g. 4 K) the conduction loss is a significant parameter to decrease its performance.

      FIG: 4 Effect of wire diameter on regenerator inefficiency.

      FIG: 5 Effect of wall thickness on regenerator inefficiency.

      FIG: 6 Effect of volumetric flow rate on regenerator inefficiency.

      Volumetric flow rate variation

      As volumetric flow rate increases, the effectiveness of regenerator decreases due to increase in inefficiency and reduction in NTU as indicated in Fig 6.

      FIG: 7 Effect of operating frequency on Regenerator inefficiency.

      Operating Frequency

      The operating frequency of Stirling cryocoooler and Stirling type pulse tube cryocooler is higher (near about 30-45 Hz), on the other hand, frequency of GM cryocooler and GM type pulse tube cyocooler is about 2-5 Hz. Thermoacoustic driven pulse tube cryocoolers operates at a frequency higher than 300

      Hz to generate thermoacoustic waves. Also, the frequency of typical pulse tube cryocoolers can be increased to 120-150 Hz. With increase in operating frequency results in decrease in the size of the system, so that it will be suited for space based applications. With increase in operating frequency of the regenerator, heat flow loss starts decreasing, so the inefficiency decreases.These results are in aggrement with those of found in the literaature [14]. But in special types of Stirling type pulse tube refrigerators (e.g., Inertance type pulse tube refrigerartor), increase in operating frequency generates the inertance effect inside the inertance tube that causes an additional phase shift between pressure wave and mass flow rate, thus resulting in increase in cooling power and COP.

      FIG: 8 Effect of length of regenerator on Regenerator inefficiency.

      FIG: 9 Effect of NTU of regenerator on Regenerator inefficiency.

      Length of regenerator

      There is an increase in axial conduction loss and viscous loss with increase in length of regenerator, as a result of this inefficiency of regenerator decreases, as shown in Fig. 8.

      NTU of regenerator

      NTU (Number of transfer unit) is related to the size of regenerator. With increase in NTU of regenerator, its inefficiency decreases.


A software has been developed by solving the governing equations of working fluid and matrix of the regenerator. Both ideal and non-ideal cases (effects including longitudinal conduction and both longitudinal conduction and wall effect) are taken into consideration. The present package is validated

with the available results in the literature and a good agreement is observed between them. Parametric studies have been performed to examine the effect of important geometrical and operating parameters on the performance of a typical regenerator for cryogenic based applications. It is observed that parameters related to screen mesh of regenerator, such as mesh size and wire diameter, have a significant effect on the inefficiency of the regenerator, as compaired to thickness of the regenerator wall. Also, volumetric flow rate and operating frequency have a considerable impact on the performance over other operating parameters. The software will be extended in future by adding more materials and its temperature dependent properties. Also, the numerical model presented here will be extended in to higher order models considering the effect of additional losses. These works will be published in future papers.


[1] Walker G., 1973. Stirling-cycle machines, 1st ed., Clarendon Press, Oxford, England.

[2] Ackermann R.A., 2013. Cryogenic regenerative heat exchangers. Springer.

[3] Iliffe, C., 1948. Thermal analysis of the contra-flow regenerative heat exchanger. Proc. of the Institution of Mechanical Engineers, Vol.159 (1), pp. 363-372.

[4] Kays, W.M. and London A.L., 1984. Compact heat exchangers.

[5] Hill, A. and A.J. Willmott, 1989. Accurate and rapid thermal regenerator calculations. Int. J. of Heat and mass transfer, Vol. 32(3), pp. 465-476.

[6] Willmott, A. and R. Thomas, 1974. Analysis of the long contra-flow regenerative heat exchanger. IMA J. of Applied Mathematics, Vol. 14(3), pp. 267-280

[7] Willmott, A., 1964. Digital computer simulation of a thermal regenerator. Int. J. of Heat and Mass Transfer. Vol. 7(11), pp. 1291- 1302.

[8] Daney, D. and R. Radebaugh, 1984. Non-ideal regenerator performancethe effect of void volume fluid heat capacity, Cryogenics, Vol. 24(9), pp. 499-501.

[9] Radebaugh, R. and B. Louie. 1985. A simple, first step to the optimization of regenerator geometry. Proc. of the 3rd Cryocooler Conf, pp. 177-198.

[10] Lewis, M., et al., 1998.Effects of regenerator geometry on pulse tube refrigerator performance. Adv. in cryogenic engineering, Springer, pp. 1999-2005.

[11] Garaway, I., et al., 2009. Development of a miniature 150 Hz pulse tube cryocooler. Cryocoolers, 15: pp. 105-113.

[12] Pfotenhauer, J., J. Shi, and G. Nellis, 2005. A parametric optimization of a single stage regenerator using REGEN 3.2, Cryocoolers 13, Springer, pp. 463-470.

[13] Pfotenhauer, J., et al., 2003. Regenerator Loss Measurements at Low Temperatures and High Frequencies.Cryocoolers 12, Springer, pp. 523-530.

[14] Qiu, L., et al., 2012. Operating characteristics of a three-stage Stirling pulse tube cryocooler operating around 5K. Cryogenics, Vol. 52(7), pp. 382-388.

[15] Shi, J., et al., 2007, Dimensionless analysis for regenerator design, Cryocoolers, Vol.14, pp. 419-4270.

[16] Panda D., 2017. Mathematical modelling and Design software for pulse tube cryocoolers. MS Thesis, National Institute of Technology,

Rourkela, India, January. See also URL [17]



A Area of crossections, m2

a Constant depends upon material property

Cp Specific heat at constant pressure, J/kg-K

D Hydraulic diameter, m

L Length of the regenerator, m

K Thermal conductivity of working fluid, W/m-K

Greek Symbol

Kinematic viscosity, m2/s

Density, kg/m3

Length of control volume,m

Dynamic viscosity, kg/m-s


h Instantaneous local heat transfer coefficient, W/m2-K f

p Static pressure, Pa m


Matrix of regenerator

Re Reynolds number

Nu Average Nusselt number

x Cartesian x-coordinate, m

wl Wall of regenerator

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