- Email: [email protected]
Compactness factors for rolled, stacked-screen regenerative heat exchangers
PII: S0011-2275(98)00061-7
Cryogenics 38 (1998) 959–965 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0011-2275/98/$ - see front matter
Compactness factors for rolled, stackedscreen regenerative heat exchangers Tim Murphy*† and W. Jerry Bowman‡ †Air Force Research Laboratory, Kirtland AFB, NM 87117, USA ‡Department of Mechanical Engineering, Brigham Young University, Provo, UT 84602, USA
Received 9 March 1998; revised 26 April 1998 The objective for the current research was to determine how the heat transfer and pressure drop characteristics changed in a stacked, wire-screen regenerative heat exchanger (regenerator), if the screens which were used to construct the matrix were rolled to decrease their thickness. More of the rolled screens would fit into a regenerator with a reduction in gas volume in the regenerator. During the course of the research, an improved method for determining the heat transfer characteristics of a porous medium regenerator was developed. The results show that the compactness factor ( jH/f ) was adversely affected by rolling. Flattening the screens by 15%, 30%, or 50%, did not improve the heat transfer rate to the matrix, but it did increase the total pressure drop. Rolling the screens caused a decrease in both wetted surface area and pore size. The heat transfer which is proportioned to the wetted surface area was reduced. The pressure drop, which is strongly dependent on pore size, increased. Rolling the screens also reduced the effectiveness of the regenerator. 1998 Elsevier Science Ltd. All rights reserved Keywords: regenerative heat exchanger; regenerator; porous media; compactness factor; heat transfer coefficient; friction factor
Cryocoolers are used in space-based assets to cool electronics1. Numerous efforts are under way to improve the efficiency of cryocoolers2. The measure of efficiency for cryocoolers is the coefficient of performance (COP). An integral component of many cryocooler systems is a regenerator. A figure of merit for regenerators is its compactness factor, which is the ratio of its heat transfer characteristics as measured by the Colburn j-factor, to its pressure drop characteristics as measured by the friction factor. The COP of the cryocooler is improved with higher regenerator compactness factor3. Although the compactness factor is an important parameter influencing system COP4, researchers have also shown that reducing the dead volume in the cryocooler also improves system COP5. Hence, if a way could be found to reduce the dead volume without causing the compactness factor to decrease, a better COP could be achieved. One way to reduce the dead volume is to flatten the screens which when stacked together constitute the matrix commonly used to construct the regenerator. Flattening the screens reduces the porosity of the regenerator matrix, allowing for more matrix mass in the regenerator and less void for gas. Even though flattening the screen will reduce
*To whom correspondence should be addressed.
the void fraction, its effect on compactness factor is unknown. Consequently, the heat transfer coefficient and friction factor for flattened screen regenerators need to be measured to determine the effect on the compactness factor. An improved approach for measuring the heat transfer coefficient in a porous matrix was developed. A total of 12 regenerators were tested using the new technique. Mesh sizes of 100, 180, and 250 per inch at screen thickness reductions of 15%, 30% and 50% were studied. The regenerators were test for a Reynolds number range of 50 to 600.
Experimental method The classic step-change transient technique for determining the heat transfer coefficient of a regenerator was originally described by Pucci et al.6. For their method, gas flows in one direction through a regenerative heat exchanger. The technique uses a step change in gas temperature at the regenerator inlet to cause heat transfer between the gas and the regenerator matrix. The heat transfer coefficient is determined by studying the temperature trace at the regenerator outlet. Pucci showed that with the introduction of the appropriate dimensionless variables, along with an adiabatic boundary condition and the step-change initial condition, the temperature of the gas could be calculated as a
Cryogenics 1998 Volume 38, Number 10 959
Compactness factors for regenerative heat exchangers: T. Murphy and W. J. Bowman function of time and location in the regenerator. Locke7 showed that the maximum slope of the temperature at the matrix outlet is solely a function of the heat transfer coefficient. This solution to the governing equations of the stepchange transient approach offered an elegant method for determining the heat transfer coefficient: provide a stepchange in inlet temperature and measure the maximum slope of the outlet temperature trace. From the gradient of temperature at the outlet, with respect to time, a value for Ntu (number of thermal units) can be determined. Since Ntu ⫽ (hconvA)/(m ˙ cp ), where A is the surface area for heat transfer (m2 ), m ˙ is the mass flow rate (kg/s), and cp is the specific heat of the gas (J/kg/K); the heat transfer coefficient, hconv, can be calculated. There are three major problems with this technique. First, obtaining an exact step change in inlet temperature in the laboratory is difficult. Second, it is very difficult to obtain an accurate slope from experimental data. Third, the model ignores heat transfer between the gas and the regenerator wall. The importance of energy exchange with the wall can be seen in Figure 1, where a temperature transient for an empty regenerator (no matrix material) is shown. The inlet and outlet temperatures start the test run at the same steadystate value. When the transient begins, the inlet temperature, Tg1, abruptly falls, and soon begins to converge on a new steady-state value. Similarly, the outlet temperature trace, Tg2, falls sharply, but the temperature drop lags the drop in Tg1 due to heat transfer from the regenerator wall to the gas. The wall is supplying enough energy to measurably influence the temperature of the gas for a significant time. A simple calculation shows that the thermal time constant of the wall and environment is much larger than the thermal time constant of the matrix. If the wall and matrix are made from the same material and treated in a lumped parameter sense, and if the heat transfer coefficient between the gas and the wall is considered the same as between the gas and the matrix, the ratio of time constants for heat transfer for a unit length of regenerator wall to that for a unit length of regenerator matrix is written:
冋 册 冋 册
冋册 冋册
Vc V hA wall A wall w tw ⫽ ⫽ ⫽ m Vc V 1⫺ hA matrix A matrix a
冉 冊
Figure 1 Temperature lag at empty regenerator outlet
960
Cryogenics 1998 Volume 38, Number 10
(1)
where w is the time constant for wall (s), m is the time constant for matrix (s), tw is the wall thickness (m), is the porosity of the matrix (dead volume/total volume) and a is the surface area for heat transfer per unit volume of the matrix (m−1 ). For the 12 regenerators used in the current research, this ratio ranged from 16.4 (100 mesh, no reduction in thickness), to 47.2 (250 mesh, 50% reduction in thickness). Consequently, the time constant for the regenerator matrix is much quicker than that for the wall. If this is the case, the early-time heat transfer should be studied to get the best estimate of the heat transfer coefficient for the matrix. But, the step-change transient technique concentrates on relatively late-time events (when the maximum temperature slope occurs) where the influence of the wall is present. Three modifications to the classical method described above were made in the current research. The first was to eliminate the assumption that an exact step-change in temperature occurs at the inlet. This was accomplished by measuring the true inlet temperature trace for each experimental test run. The method used the measure inlet and outlet temperature traces and an inverse method to predict the heat transfer coefficient between the gas and the regenerator matrix. The second major improvement to the model was the inclusion of the energy transfer to the regenerator wall. A lumped parameter approach was taken. The heat transfer coefficient between the gas and the wall was assumed to be the same as between the gas and the matrix, and the wall temperature was allowed to vary in the axial direction and with time. The third major improvement deals with the criteria used in the inverse method to select the most accurate heat transfer coefficient. In the earlier model, the heat transfer coefficient was selected by studying the maximum slope of outlet temperature trace. Because the wall effect is less important early in the transient, an earlier time in the transient was used for the convergence criteria. This time is discussed below. A characteristic of porous media is that it acts like a thermal sponge. The sponge effect of a regenerator can be demonstrated by comparing Figure 1 and Figure 2. For the empty regenerator in Figure 1, the outlet temperature trace changes almost immediately, in tandem with the inlet temperature trace. However, when a matrix material is present
Figure 2 Regenerator sponge effect delay time
Compactness factors for regenerative heat exchangers: T. Murphy and W. J. Bowman as in Figure 2, the outlet gas stays at the initial temperature for some time before it begins to change. The definition of the sponge effect delay time is as follows: it is the time it takes for the outlet gas temperature to change a measurable amount. For the current research, the definition of a measurable change is a deviation in outlet temperature from its running average by more than twice the accuracy of the temperature measuring devices, or about 0.4°C for this work. Using the sponge effect delay time gave a better value for the heat transfer coefficient than using the maximum outlet temperature slope. Two arguments can be made to substantiate this claim. First, the most accurate value for the heat transfer coefficient will be found by studying the early transient when heat transfer between the gas and the matrix dominates. The second argument for the improvement can be seen from Figure 3. Whereas a wide range of outlet temperature traces can be achieved depending on how much the tube affects the flow, the delay time is nearly insensitive to these extraneous factors. For the example in Figure 3, the maximum slopes differ for the two cases by approximately 150%, whereas the delay times are only different by about 5%. Hence, the sponge effect delay time is an easily measured parameter of the regenerator that can be used to calculate more accurate heat transfer coefficients.
Experimental procedure and apparatus The apparatus used is represented by Figure 4. It consisted of three main subsections. The first subsection was a source of both hot (around 35°C) and cold (10°C) helium. The second subsection consisted of a test section which contained a mounting bracket which held the regenerator (#11 in Figure 2), instrumentation (fast-response thermistors (#10) and differential-pressure transducers (#15)) to measure the temperatures and pressures at the inlet and outlet to the regenerator, a four-way valve (#9) which diverted the flow from cold to hot, producing the transient, and a venturi flowmeter (#12). The third subsection was the data acquisition system (#16). The procedure was straightforward. After steady-state flow was achieved at the appropriate pressure and mass flow rate at the cold gas temperature, the four-way valve was switched to hot flow and the transient inlet and outlet
Figure 3 Comparison of outlet temperature trace with and without a wall
Figure 4 Diagram of experimental apparatus
temperature traces were measured, as well as the pressure drop across the regenerator and the gas mass flow rate. The raw data consisted of temperature traces measured at the inlet and outlet to the test section; pressure and pressure drop readings from the three pressure stations in the test section; and an average value of the mass flow rate. The first parameters determined from the data were the actual start and end times of the transient. The raw temperature data were filtered with a 20th order finite impulse resolution filter. The start time 0 was selected to be the time when the inlet temperature changed by more than two standard deviations from the initial, steady state value. The end time was selected at a time after which the maximum slope in the outlet temperature trace had obviously occurred, between 10 and 15 regenerator matrix time constants after 0. Once the beginning and end times were identified, the sponge effect delay timing was found, 2. The time between when the transient begins, 0, and when the outlet temperature trace began to change measurably was defined above as the sponge effect delay time. The determination of 2 was done in a similar manner as for 0; it occurs when the outlet temperature trace changed by more than two standard deviations from its running average. The final step in the reduction process was collecting all the data into a read file for each experimental run. Results from the raw data file such as mass flow rate and pressures were assembled at the beginning. Next, some geometric properties of the regenerators such as mass, pore size, and length were added to the file. Then, the filtered temperature traces were appended to the file. Finally, the time parameters described above were included. All these data were used by the numerical model described below to determine a heat transfer coefficient. A numerical model was developed to model the flow. It used an explicit, finite difference method to solve the energy equations that modeled the regenerator during the transient. The model was the same as the one used in the approach of Pucci6 for the step-change transient solution except for the three modifications mentioned above. Assumptions in the model included no axial conduction, no conduction between the wall and the matrix, perfect gas relations, constant properties, low Mach numbers, onedimensional flow, incompressible flow, and a constant heat transfer coefficient. With these assumptions, the continuity
Cryogenics 1998 Volume 38, Number 10 961
Compactness factors for regenerative heat exchangers: T. Murphy and W. J. Bowman equation shows the mass flow rate, and consequently, the mass velocity, is a constant. Conservation of energy for three control volumes, one representing the gas, one representing the regenerator matrix and one representing the regenerator wall, are: Gas: ⫺ m ˙ cp
∂Tg ⌬x ⫹ hconvAm(Tm ⫺ Tg ) ∂x
⫹ HconvAw(Tw ⫺ Tg ) ⫽ cvA⌬x
(2)
∂Tg ∂t
Regenerator matrix: ⫺ hconvAm(Tm ⫺ Tg ) ⫽ mcmA(1 ⫺ )⌬X
Tube: kAwx
(3) ∂Tg ∂t
∂2Tw ∂Tw − hconvAw(Tw − Tg ) = wcwAwx⌬X ∂x2 ∂t
Figure 5 Friction and Nusselt number comparison
(4) where m ˙ is the gas mass flow rate (kg/s) cp, cv are the specific heat of the gas at constant pressure/volume (J/kg/K), hconv is the heat transfer coefficient (W/m2/K), Am is the matrix surface area for convection (m2 ), Aw is the wall surface area for convection (m2 ), Awx is the wall crosssectional area associated with axial conduction (m2 ), Tg, Tm, Tw are the gas, matrix, and wall temperatures, (K), is the gas density (kg/m3 ), A is the cross-sectional area of the regenerator (m2 ), is the porosity, m is the density of the matrix (kg/m3 ), cm is the specific heat of matrix (J/kg/K), and ⌬x is the length of the control volume. Using a first-order accurate forward difference in time, and a second-order accurate central difference in space, the equations were numerically integrated to predict the gas, matrix and wall temperatures as functions of time and space. Several sample problems with known solutions were used to validate the accuracy of the model and the numerical grid selected3. The initial condition was the same for each of the three equations. The gas, regenerator, and wall were all at the same initial, steady-state temperature. During the transient, the experimentally measured inlet gas temperature was used as a boundary condition for the model. Using the model, a numerically generated outlet temperature trace could be obtained for an approximate value of the heat transfer coefficient. The heat transfer coefficient was determined by requiring that at the sponge effect delay time the calculated outlet temperature trace match the experimental temperature trace. Root solving methods were employed to find the convection heat transfer coefficient which produced the closest match between the experimental and numerical outlet temperatures. For comparison, the theoretical heat transfer coefficient based on the approach of Pucci6 was also found. Two comparisons were made to give confidence in the method. The first can be seen in Figure 5 where results for friction factor and Nusselt number are compared with published correlations for friction factors and Nusselt numbers for unrolled screens. In the case of the friction factors, Armour and Cannons’8 study included similar mesh sizes and a Reynolds number range, 1.0 ⬍ Re ⬍ 100. The agreement is very good with a mean difference of ⫾ 2.5%. The published Nusselt number data are due to Hamaguchi9, who
962
Cryogenics 1998 Volume 38, Number 10
also used identical mesh sizes, and a Reynolds number range of 1.0 ⬍ Re ⬍ 900. Another test of confidence for the methodology is repeatability. In Figure 5, data is shown for both friction factor and Nusselt number for four sets of test runs. The test runs were done on different days but with approximately the same operating conditions. The test runs were chosen at random and cover a range of Reynolds numbers. The friction data is particularly good, with a mean difference of only 2.0% between the members of each set. The fit for the Nusselt numbers is also good, with a mean difference of 8.4% between members of the same set. Since the results can be duplicated so closely from one test run to the next, confidence in the measurements is warranted.
Experimental data Two preliminary items are addressed to facilitate using the data which follows: the first is the definition of the characteristic length used to define Reynolds and Nusselt numbers. The second is an explanation of how the specific surface area for each matrix was found. Many definitions for the characteristic length for heat transfer/fluid flow in a porous medium appear in the literature. One common length used was presented in an article by Armour and Cannon8. Their definition for the characteristic length is LA ⫽ 1/(a2 ⫻ D), where LA is the characteristic length, a is the surface area per unit volume of the matrix, and D is the pore diameter. The pore diameter was found by taking the cube root of the free volume between the wire threads that make up the screen matrix. The sides of the volume have length (1/n ⫺ dw ). The volume was assumed to have a depth of 2dw. Thus, the pore diameter is: D ⫽ [(2dw )(1/n ⫺ dw )2 ]1/3
(5)
where n is the mesh number (m−1 ) and dw is the wire diameter (m). In order to use the Armour and Cannon dimension, the surface area per unit volume (a) needed to be determined for each regenerator. Wiese10 took electron microscope photographs of the screens which he rolled to the same reduction factors as those used in the current research. The photographs show that the rolling caused the round wire surfaces to be flattened where wires crossed. When screen
Compactness factors for regenerative heat exchangers: T. Murphy and W. J. Bowman
Figure 6 Specific surface area vs. reduction factor
disks are stacked one against another, the flattened areas can contact each other, and reduce the surface area available for heat transfer from the fluid to the regenerator matrix. For this work, it was assumed that the wetted surface area of the regenerator matrix was reduced by the flattened regions on the rolled screens. The magnitude of the area of the flattened regions was estimated from the photographs taken of the screen. The results are presented in Figure 6. To use this figure, the area per unit volume for the unrolled screen (as ) is required. It was calculated by assuming the screen wires were woven cylinders and the screen thickness was equal to one wire diameter. The resulting relationship for the area to volume ratio is:
In Figure 8 Nusselt number is plotted as a function of reduction factor. It is interesting to note that there does not seem to be a strong dependence. During the first attempt to reduce the data, the surface area of the regenerator was assumed to be the same for the reduced thickness screens and for the unrolled screens of the same mesh number. When this was the case, the Nusselt number was smaller for the rolled screens. When the reduction in wetted surface area was taken into account (discussed earlier, Figure 6, by using a characteristic length which included the effects of surface area reduction) the Nusselt number was no longer dependent on the reduction factor. The plot shows that within the accuracy of the testing, changing the reduction factor had no effect on the Nusselt number. When considering the heat transfer from a gas to a regenerator matrix, a regenerator matrix made from rolled screen would experience less heat transfer. This is due to either a smaller area for heat transfer or a decrease in the heat transfer coefficient or some combination of the two effects. The friction factor is defined as follows8: f ⫽
⌬pgc 2D LU2g
(7)
This value for as is multiplied by the appropriate ratio obtained from Figure 6 to get the modified surface area per unit volume for rolled screen (a). The Nusselt number is defined as Nu ⫽ hconv ⫻ LA/k, where hconv is the convective heat transfer coefficient between the gas and the matrix, LA is the Armour and Cannon dimension, and k is the thermal conductivity of the gas. The experimental results and a least-squares curve fit for Nusselt number versus Reynolds number are shown in Figure 7.
where ⌬p is the pressure drop across the regenerator (N/m2 ), is the porosity, D is the pore diameter (m), L is the matrix length (m), is the gas density (kg/m3 ), and Ug is the gas approach velocity (m/s). The behavior of the friction factor differs from the behavior of the Nusselt number. The reduction factor plays a significant role in the value of the total pressure drop. The experimental results and a least-squares curve fit for friction factor versus Reynolds number is shown in Figure 9. The solid line is the relationship by Armour and Cannon8 for unrolled screens. The first thing to notice about the experimental data is the group of points which are in the upper left-hand corner of the figure. All of these data points are for test runs with a reduction factor ⫽ 0.5. Apparently, reducing the thickness of the screen by 50% causes a large increase in the friction coefficient. The second thing to notice in Figure 9 is that the remaining data points fall close to the line with only a 6% mean difference. To insure this was the case, the friction factor was plotted as a reduced friction factor in Figure 10. The reduced friction factor was defined as the friction factor for a specific screen unit divided by the friction factor for the unrolled screen
Figure 7 Nusselt number as a function of Reynolds number
Figure 8 Nusselt number as a function of reduction factor
as ⫽ 2n2[d 2w ⫹ (1/n)2 ]1/2
(6)
Cryogenics 1998 Volume 38, Number 10 963
Compactness factors for regenerative heat exchangers: T. Murphy and W. J. Bowman
Figure 11 Compactness factor for each screen mesh size Figure 9 Friction factor as a function of Reynolds number
unit operating at the same Reynolds number. It appears that the friction factor increases for all values of screen thickness reduction, but most pronouncedly so for reductions greater than 30%. An important result from the current research is the measurement of compactness factor. The definition of compactness factor is CF ⫽ jH/f, where jH is the Colburn jfactor, and f is the friction factor11. The Colburn j-factor is defined as jH ⫽ St ⫻ Pr1/3, where St is the Stanton number and Pr is the Prandtl number. In Figure 11, the results for the compactness factor as a function of reduction factor are shown for each mesh size. For the 100 and 180 mesh screens, the trend is clear: the compactness factor decreases with decreasing reduction factor. Hence, rolling the screens tends to reduce the compactness factor. For the 250 mesh screens, the compactness factor seems to increase at a reduction factor of 0.7. The data for this case was examined more closely, and whereas the heat transfer coefficient is about what one would expect (compared to the other data,) the friction factor is smaller than expected. The reason for this discrepancy is not apparent. Since all of the other data follow a predictable trend, it is considered an outlier. A reduced compactness factor was defined to be the ratio of the compactness factor for rolled screen to the com-
Figure 10 Reduced friction factor as a function of reduction factor
964
Cryogenics 1998 Volume 38, Number 10
pactness factor for unrolled screens. The reduced compactness factors data is plotted in Figure 12, along with a least-squares curve fit. The spread is moderate (a standard deviation of 21.5% from the best-fit curve). The trend clearly shows a reduction in the compactness factor as the screen thickness gets smaller. Another remark can be made about the use of the compactness factor in numerical models. Due to the lack of heat transfer data for regenerators, some researchers resort to measuring the friction factor and applying Reynolds analogy12. For Reynolds analogy to apply to flow in a regenerator, the compactness factor should be a constant. If the results for flow in a tube are used, Reynolds analogy states the compactness factor should be 0.125. For the unrolled screens, this is not a bad guess, being 16.7% high versus the average of the unrolled values. But the analogy breaks down at smaller reduction factors, reaching a mean value of 0.031 for a 50% reduction in screen thickness. Another parameter was used to evaluate the merit of reducing the thickness of the screens. Effectiveness is defined for regenerators in terms of the ability of the regenerator to remove the maximum amount of energy from the gas relative to an ideal case. Effectiveness, , can be defined for the case of one-directional flow as
⫽
m ˙ cp2 msucm
(8)
Figure 12 Reduced compactness factor vs. reduction factor
Compactness factors for regenerative heat exchangers: T. Murphy and W. J. Bowman
Figure 13 Reduced effectiveness vs. reduction factor
where m ˙ is the mass flow rate (kg/s), cp is the specific heat of the gas (J/kg/K), msu is the mass of matrix (kg), cm is the specific heat of the matrix (J/kg/K), and 2 is the sponge effect delay time (s)3. In Figure 13, the reduced effectiveness is shown as a function of the reduction factor. Effectiveness is the dominant parameter in cryocooler systems performance4,13. The data show a reduction in effectiveness whenever the screen thickness (reduction factor) is reduced. The effect is small at reduction factors of 0.7 and 0.85, and falls to about a 13% decrement in effectiveness at a reduction factor of 0.5. Again, these results would rule out any benefit of reducing the screen thickness by 50%. The interesting thing about the figures is that one would expect the effectiveness to increase by making pore sizes smaller. Rolling the screens does not change the material properties of the gas or matrix, hence another reason must exist for the decrement. As noted above, rolling the screens does reduce the wetted surface area of the matrix since abutting round wires have a small area of contact while flattened surfaces can have a larger contact area. Large surface-area-to-volume ratio is one of the factors responsible for the large effectiveness of wire-screen matrices. Since the wetted surface area is diminished when flattened surfaces are introduced by rolling the screens, the effectiveness is also reduced.
Conclusions The results presented in this paper give a convincing explanation for the conclusion that rolling the screens to reduce their thickness causes the ratio of heat transfer coefficient to the friction factor to decrease. The data show that the heat transfer between the gas and the matrix is reduced for regenerators made from rolled screens. This reduction is a result of reduced wetted surface area or reduced convection heat transfer coefficient. The friction factor, however,
increased marginally for small thickness reductions, but substantially for the 50% reduction case. The crux of the matter with regard to the compactness factor is how the geometry of the matrix changes due to rolling. Rolling the screens causes the round surface areas of wires to be replaced with flattened areas with abrupt edges. These flattened areas abut against one another, causing the total wetted surface area in the matrix to decrease. This reduces the heat transfer. Also, the smaller pore sizes and sharp edges cause the inertial component of the pressure drop to increase to the point where the total pressure drop is higher. The overall effect of the geometry change is to increase the compactness factor. The effect of reducing the screen thickness on cryocooler COP was not addressed. It was shown that reducing the screen thickness caused a decrease in regenerator effectiveness which will have a negative impact on COP. However, reducing the screen thickness should also decrease the void fraction in a cryocooler, which will have a positive impact on COP. The optimum screen thickness for best COP still needs to be studied.
References 1. Chan, C. K., et al., Overview of cryocooler technologies for spacebased electronics and sensors. Advances in Cryogenic Engineering, 1990, 35, 1239–1250. 2. Thomas, P. J., SDIO and Air Force cryocooler technology developments at USAF Phillips Laboratory. In Proceedings of the 7th International Cryocooler Conference, Santa Fe, NM, November 1992, pp. 3–13. 3. Murphy, T. J., How reducing the thickness of the round wire screens in a porous medium regenerator affects its compactness factor (jH/f). PhD dissertation, AFIT/DS/ENY/96-5, School of Engineering, Air Force Institute of Technology (AU), Wright-Patterson AFB, OH, June 1996. 4. Urieli, I. and Berchowitz, D. M., Stirling Cycle Engine Analysis. Adam Hilger, Bristol, UK, 1984. 5. Atria, M. D., et al., Microcomputer aided cyclic analysis of the Stirling cryocooler with different regenerator meshes. Cryogenics (September Supplement), 1990, 30, 236–240. 6. Pucci, P. F., et al., The single blow transient testing technique for compact heat exchangers surfaces. Journal of Engineering for Power, 1967, 89, 29–40. 7. Locke, G. L., Heat transfer and flow friction characteristics of porous solids. Test Report No. 10, Department of Mechanical Engineering, Stanford University, Stanford, CA, 1 June 1950. 8. Armour, J. C. and Cannon, J. N., Fluid flow through woven screens. AICHE Journal, 1968, 14(3), 415–420. 9. Hamaguchi, K. et al., Effects of regenerator size change on Stirling engine performance. In Proceedings of the 26th IECEC, Vol. 5. 1991, pp. 293–298. 10. Wiese, J. L., Experimental analysis of heat transfer characteristics and pressure drop through screen regenerative heat exchangers. Masters thesis, AFIT/GAE/ENY/93D-30, School of Engineering, Air Force Institute of Technology (AU), Wright-Patterson AFB, OH, December 1993. 11. Kays, W. M. and London, A. L., Compact Heat Exchangers, 3rd edn. McGraw-Hill, New York, 1984. 12. Radebaugh, R. and Louie, B., A simple 1st step to optimizing regenerator geometry. In Proceedings of the 3rd Cryocooler Conference, Boulder, CO, 17–18 September 1986, pp. 177–198. 13. Martini, W. R., Stirling Engine Design Manual. NASA CR135382 (1980).
Cryogenics 1998 Volume 38, Number 10 965
Cryogenics 38 (1998) 959–965 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0011-2275/98/$ - see front matter
Compactness factors for rolled, stackedscreen regenerative heat exchangers Tim Murphy*† and W. Jerry Bowman‡ †Air Force Research Laboratory, Kirtland AFB, NM 87117, USA ‡Department of Mechanical Engineering, Brigham Young University, Provo, UT 84602, USA
Received 9 March 1998; revised 26 April 1998 The objective for the current research was to determine how the heat transfer and pressure drop characteristics changed in a stacked, wire-screen regenerative heat exchanger (regenerator), if the screens which were used to construct the matrix were rolled to decrease their thickness. More of the rolled screens would fit into a regenerator with a reduction in gas volume in the regenerator. During the course of the research, an improved method for determining the heat transfer characteristics of a porous medium regenerator was developed. The results show that the compactness factor ( jH/f ) was adversely affected by rolling. Flattening the screens by 15%, 30%, or 50%, did not improve the heat transfer rate to the matrix, but it did increase the total pressure drop. Rolling the screens caused a decrease in both wetted surface area and pore size. The heat transfer which is proportioned to the wetted surface area was reduced. The pressure drop, which is strongly dependent on pore size, increased. Rolling the screens also reduced the effectiveness of the regenerator. 1998 Elsevier Science Ltd. All rights reserved Keywords: regenerative heat exchanger; regenerator; porous media; compactness factor; heat transfer coefficient; friction factor
Cryocoolers are used in space-based assets to cool electronics1. Numerous efforts are under way to improve the efficiency of cryocoolers2. The measure of efficiency for cryocoolers is the coefficient of performance (COP). An integral component of many cryocooler systems is a regenerator. A figure of merit for regenerators is its compactness factor, which is the ratio of its heat transfer characteristics as measured by the Colburn j-factor, to its pressure drop characteristics as measured by the friction factor. The COP of the cryocooler is improved with higher regenerator compactness factor3. Although the compactness factor is an important parameter influencing system COP4, researchers have also shown that reducing the dead volume in the cryocooler also improves system COP5. Hence, if a way could be found to reduce the dead volume without causing the compactness factor to decrease, a better COP could be achieved. One way to reduce the dead volume is to flatten the screens which when stacked together constitute the matrix commonly used to construct the regenerator. Flattening the screens reduces the porosity of the regenerator matrix, allowing for more matrix mass in the regenerator and less void for gas. Even though flattening the screen will reduce
*To whom correspondence should be addressed.
the void fraction, its effect on compactness factor is unknown. Consequently, the heat transfer coefficient and friction factor for flattened screen regenerators need to be measured to determine the effect on the compactness factor. An improved approach for measuring the heat transfer coefficient in a porous matrix was developed. A total of 12 regenerators were tested using the new technique. Mesh sizes of 100, 180, and 250 per inch at screen thickness reductions of 15%, 30% and 50% were studied. The regenerators were test for a Reynolds number range of 50 to 600.
Experimental method The classic step-change transient technique for determining the heat transfer coefficient of a regenerator was originally described by Pucci et al.6. For their method, gas flows in one direction through a regenerative heat exchanger. The technique uses a step change in gas temperature at the regenerator inlet to cause heat transfer between the gas and the regenerator matrix. The heat transfer coefficient is determined by studying the temperature trace at the regenerator outlet. Pucci showed that with the introduction of the appropriate dimensionless variables, along with an adiabatic boundary condition and the step-change initial condition, the temperature of the gas could be calculated as a
Cryogenics 1998 Volume 38, Number 10 959
Compactness factors for regenerative heat exchangers: T. Murphy and W. J. Bowman function of time and location in the regenerator. Locke7 showed that the maximum slope of the temperature at the matrix outlet is solely a function of the heat transfer coefficient. This solution to the governing equations of the stepchange transient approach offered an elegant method for determining the heat transfer coefficient: provide a stepchange in inlet temperature and measure the maximum slope of the outlet temperature trace. From the gradient of temperature at the outlet, with respect to time, a value for Ntu (number of thermal units) can be determined. Since Ntu ⫽ (hconvA)/(m ˙ cp ), where A is the surface area for heat transfer (m2 ), m ˙ is the mass flow rate (kg/s), and cp is the specific heat of the gas (J/kg/K); the heat transfer coefficient, hconv, can be calculated. There are three major problems with this technique. First, obtaining an exact step change in inlet temperature in the laboratory is difficult. Second, it is very difficult to obtain an accurate slope from experimental data. Third, the model ignores heat transfer between the gas and the regenerator wall. The importance of energy exchange with the wall can be seen in Figure 1, where a temperature transient for an empty regenerator (no matrix material) is shown. The inlet and outlet temperatures start the test run at the same steadystate value. When the transient begins, the inlet temperature, Tg1, abruptly falls, and soon begins to converge on a new steady-state value. Similarly, the outlet temperature trace, Tg2, falls sharply, but the temperature drop lags the drop in Tg1 due to heat transfer from the regenerator wall to the gas. The wall is supplying enough energy to measurably influence the temperature of the gas for a significant time. A simple calculation shows that the thermal time constant of the wall and environment is much larger than the thermal time constant of the matrix. If the wall and matrix are made from the same material and treated in a lumped parameter sense, and if the heat transfer coefficient between the gas and the wall is considered the same as between the gas and the matrix, the ratio of time constants for heat transfer for a unit length of regenerator wall to that for a unit length of regenerator matrix is written:
冋 册 冋 册
冋册 冋册
Vc V hA wall A wall w tw ⫽ ⫽ ⫽ m Vc V 1⫺ hA matrix A matrix a
冉 冊
Figure 1 Temperature lag at empty regenerator outlet
960
Cryogenics 1998 Volume 38, Number 10
(1)
where w is the time constant for wall (s), m is the time constant for matrix (s), tw is the wall thickness (m), is the porosity of the matrix (dead volume/total volume) and a is the surface area for heat transfer per unit volume of the matrix (m−1 ). For the 12 regenerators used in the current research, this ratio ranged from 16.4 (100 mesh, no reduction in thickness), to 47.2 (250 mesh, 50% reduction in thickness). Consequently, the time constant for the regenerator matrix is much quicker than that for the wall. If this is the case, the early-time heat transfer should be studied to get the best estimate of the heat transfer coefficient for the matrix. But, the step-change transient technique concentrates on relatively late-time events (when the maximum temperature slope occurs) where the influence of the wall is present. Three modifications to the classical method described above were made in the current research. The first was to eliminate the assumption that an exact step-change in temperature occurs at the inlet. This was accomplished by measuring the true inlet temperature trace for each experimental test run. The method used the measure inlet and outlet temperature traces and an inverse method to predict the heat transfer coefficient between the gas and the regenerator matrix. The second major improvement to the model was the inclusion of the energy transfer to the regenerator wall. A lumped parameter approach was taken. The heat transfer coefficient between the gas and the wall was assumed to be the same as between the gas and the matrix, and the wall temperature was allowed to vary in the axial direction and with time. The third major improvement deals with the criteria used in the inverse method to select the most accurate heat transfer coefficient. In the earlier model, the heat transfer coefficient was selected by studying the maximum slope of outlet temperature trace. Because the wall effect is less important early in the transient, an earlier time in the transient was used for the convergence criteria. This time is discussed below. A characteristic of porous media is that it acts like a thermal sponge. The sponge effect of a regenerator can be demonstrated by comparing Figure 1 and Figure 2. For the empty regenerator in Figure 1, the outlet temperature trace changes almost immediately, in tandem with the inlet temperature trace. However, when a matrix material is present
Figure 2 Regenerator sponge effect delay time
Compactness factors for regenerative heat exchangers: T. Murphy and W. J. Bowman as in Figure 2, the outlet gas stays at the initial temperature for some time before it begins to change. The definition of the sponge effect delay time is as follows: it is the time it takes for the outlet gas temperature to change a measurable amount. For the current research, the definition of a measurable change is a deviation in outlet temperature from its running average by more than twice the accuracy of the temperature measuring devices, or about 0.4°C for this work. Using the sponge effect delay time gave a better value for the heat transfer coefficient than using the maximum outlet temperature slope. Two arguments can be made to substantiate this claim. First, the most accurate value for the heat transfer coefficient will be found by studying the early transient when heat transfer between the gas and the matrix dominates. The second argument for the improvement can be seen from Figure 3. Whereas a wide range of outlet temperature traces can be achieved depending on how much the tube affects the flow, the delay time is nearly insensitive to these extraneous factors. For the example in Figure 3, the maximum slopes differ for the two cases by approximately 150%, whereas the delay times are only different by about 5%. Hence, the sponge effect delay time is an easily measured parameter of the regenerator that can be used to calculate more accurate heat transfer coefficients.
Experimental procedure and apparatus The apparatus used is represented by Figure 4. It consisted of three main subsections. The first subsection was a source of both hot (around 35°C) and cold (10°C) helium. The second subsection consisted of a test section which contained a mounting bracket which held the regenerator (#11 in Figure 2), instrumentation (fast-response thermistors (#10) and differential-pressure transducers (#15)) to measure the temperatures and pressures at the inlet and outlet to the regenerator, a four-way valve (#9) which diverted the flow from cold to hot, producing the transient, and a venturi flowmeter (#12). The third subsection was the data acquisition system (#16). The procedure was straightforward. After steady-state flow was achieved at the appropriate pressure and mass flow rate at the cold gas temperature, the four-way valve was switched to hot flow and the transient inlet and outlet
Figure 3 Comparison of outlet temperature trace with and without a wall
Figure 4 Diagram of experimental apparatus
temperature traces were measured, as well as the pressure drop across the regenerator and the gas mass flow rate. The raw data consisted of temperature traces measured at the inlet and outlet to the test section; pressure and pressure drop readings from the three pressure stations in the test section; and an average value of the mass flow rate. The first parameters determined from the data were the actual start and end times of the transient. The raw temperature data were filtered with a 20th order finite impulse resolution filter. The start time 0 was selected to be the time when the inlet temperature changed by more than two standard deviations from the initial, steady state value. The end time was selected at a time after which the maximum slope in the outlet temperature trace had obviously occurred, between 10 and 15 regenerator matrix time constants after 0. Once the beginning and end times were identified, the sponge effect delay timing was found, 2. The time between when the transient begins, 0, and when the outlet temperature trace began to change measurably was defined above as the sponge effect delay time. The determination of 2 was done in a similar manner as for 0; it occurs when the outlet temperature trace changed by more than two standard deviations from its running average. The final step in the reduction process was collecting all the data into a read file for each experimental run. Results from the raw data file such as mass flow rate and pressures were assembled at the beginning. Next, some geometric properties of the regenerators such as mass, pore size, and length were added to the file. Then, the filtered temperature traces were appended to the file. Finally, the time parameters described above were included. All these data were used by the numerical model described below to determine a heat transfer coefficient. A numerical model was developed to model the flow. It used an explicit, finite difference method to solve the energy equations that modeled the regenerator during the transient. The model was the same as the one used in the approach of Pucci6 for the step-change transient solution except for the three modifications mentioned above. Assumptions in the model included no axial conduction, no conduction between the wall and the matrix, perfect gas relations, constant properties, low Mach numbers, onedimensional flow, incompressible flow, and a constant heat transfer coefficient. With these assumptions, the continuity
Cryogenics 1998 Volume 38, Number 10 961
Compactness factors for regenerative heat exchangers: T. Murphy and W. J. Bowman equation shows the mass flow rate, and consequently, the mass velocity, is a constant. Conservation of energy for three control volumes, one representing the gas, one representing the regenerator matrix and one representing the regenerator wall, are: Gas: ⫺ m ˙ cp
∂Tg ⌬x ⫹ hconvAm(Tm ⫺ Tg ) ∂x
⫹ HconvAw(Tw ⫺ Tg ) ⫽ cvA⌬x
(2)
∂Tg ∂t
Regenerator matrix: ⫺ hconvAm(Tm ⫺ Tg ) ⫽ mcmA(1 ⫺ )⌬X
Tube: kAwx
(3) ∂Tg ∂t
∂2Tw ∂Tw − hconvAw(Tw − Tg ) = wcwAwx⌬X ∂x2 ∂t
Figure 5 Friction and Nusselt number comparison
(4) where m ˙ is the gas mass flow rate (kg/s) cp, cv are the specific heat of the gas at constant pressure/volume (J/kg/K), hconv is the heat transfer coefficient (W/m2/K), Am is the matrix surface area for convection (m2 ), Aw is the wall surface area for convection (m2 ), Awx is the wall crosssectional area associated with axial conduction (m2 ), Tg, Tm, Tw are the gas, matrix, and wall temperatures, (K), is the gas density (kg/m3 ), A is the cross-sectional area of the regenerator (m2 ), is the porosity, m is the density of the matrix (kg/m3 ), cm is the specific heat of matrix (J/kg/K), and ⌬x is the length of the control volume. Using a first-order accurate forward difference in time, and a second-order accurate central difference in space, the equations were numerically integrated to predict the gas, matrix and wall temperatures as functions of time and space. Several sample problems with known solutions were used to validate the accuracy of the model and the numerical grid selected3. The initial condition was the same for each of the three equations. The gas, regenerator, and wall were all at the same initial, steady-state temperature. During the transient, the experimentally measured inlet gas temperature was used as a boundary condition for the model. Using the model, a numerically generated outlet temperature trace could be obtained for an approximate value of the heat transfer coefficient. The heat transfer coefficient was determined by requiring that at the sponge effect delay time the calculated outlet temperature trace match the experimental temperature trace. Root solving methods were employed to find the convection heat transfer coefficient which produced the closest match between the experimental and numerical outlet temperatures. For comparison, the theoretical heat transfer coefficient based on the approach of Pucci6 was also found. Two comparisons were made to give confidence in the method. The first can be seen in Figure 5 where results for friction factor and Nusselt number are compared with published correlations for friction factors and Nusselt numbers for unrolled screens. In the case of the friction factors, Armour and Cannons’8 study included similar mesh sizes and a Reynolds number range, 1.0 ⬍ Re ⬍ 100. The agreement is very good with a mean difference of ⫾ 2.5%. The published Nusselt number data are due to Hamaguchi9, who
962
Cryogenics 1998 Volume 38, Number 10
also used identical mesh sizes, and a Reynolds number range of 1.0 ⬍ Re ⬍ 900. Another test of confidence for the methodology is repeatability. In Figure 5, data is shown for both friction factor and Nusselt number for four sets of test runs. The test runs were done on different days but with approximately the same operating conditions. The test runs were chosen at random and cover a range of Reynolds numbers. The friction data is particularly good, with a mean difference of only 2.0% between the members of each set. The fit for the Nusselt numbers is also good, with a mean difference of 8.4% between members of the same set. Since the results can be duplicated so closely from one test run to the next, confidence in the measurements is warranted.
Experimental data Two preliminary items are addressed to facilitate using the data which follows: the first is the definition of the characteristic length used to define Reynolds and Nusselt numbers. The second is an explanation of how the specific surface area for each matrix was found. Many definitions for the characteristic length for heat transfer/fluid flow in a porous medium appear in the literature. One common length used was presented in an article by Armour and Cannon8. Their definition for the characteristic length is LA ⫽ 1/(a2 ⫻ D), where LA is the characteristic length, a is the surface area per unit volume of the matrix, and D is the pore diameter. The pore diameter was found by taking the cube root of the free volume between the wire threads that make up the screen matrix. The sides of the volume have length (1/n ⫺ dw ). The volume was assumed to have a depth of 2dw. Thus, the pore diameter is: D ⫽ [(2dw )(1/n ⫺ dw )2 ]1/3
(5)
where n is the mesh number (m−1 ) and dw is the wire diameter (m). In order to use the Armour and Cannon dimension, the surface area per unit volume (a) needed to be determined for each regenerator. Wiese10 took electron microscope photographs of the screens which he rolled to the same reduction factors as those used in the current research. The photographs show that the rolling caused the round wire surfaces to be flattened where wires crossed. When screen
Compactness factors for regenerative heat exchangers: T. Murphy and W. J. Bowman
Figure 6 Specific surface area vs. reduction factor
disks are stacked one against another, the flattened areas can contact each other, and reduce the surface area available for heat transfer from the fluid to the regenerator matrix. For this work, it was assumed that the wetted surface area of the regenerator matrix was reduced by the flattened regions on the rolled screens. The magnitude of the area of the flattened regions was estimated from the photographs taken of the screen. The results are presented in Figure 6. To use this figure, the area per unit volume for the unrolled screen (as ) is required. It was calculated by assuming the screen wires were woven cylinders and the screen thickness was equal to one wire diameter. The resulting relationship for the area to volume ratio is:
In Figure 8 Nusselt number is plotted as a function of reduction factor. It is interesting to note that there does not seem to be a strong dependence. During the first attempt to reduce the data, the surface area of the regenerator was assumed to be the same for the reduced thickness screens and for the unrolled screens of the same mesh number. When this was the case, the Nusselt number was smaller for the rolled screens. When the reduction in wetted surface area was taken into account (discussed earlier, Figure 6, by using a characteristic length which included the effects of surface area reduction) the Nusselt number was no longer dependent on the reduction factor. The plot shows that within the accuracy of the testing, changing the reduction factor had no effect on the Nusselt number. When considering the heat transfer from a gas to a regenerator matrix, a regenerator matrix made from rolled screen would experience less heat transfer. This is due to either a smaller area for heat transfer or a decrease in the heat transfer coefficient or some combination of the two effects. The friction factor is defined as follows8: f ⫽
⌬pgc 2D LU2g
(7)
This value for as is multiplied by the appropriate ratio obtained from Figure 6 to get the modified surface area per unit volume for rolled screen (a). The Nusselt number is defined as Nu ⫽ hconv ⫻ LA/k, where hconv is the convective heat transfer coefficient between the gas and the matrix, LA is the Armour and Cannon dimension, and k is the thermal conductivity of the gas. The experimental results and a least-squares curve fit for Nusselt number versus Reynolds number are shown in Figure 7.
where ⌬p is the pressure drop across the regenerator (N/m2 ), is the porosity, D is the pore diameter (m), L is the matrix length (m), is the gas density (kg/m3 ), and Ug is the gas approach velocity (m/s). The behavior of the friction factor differs from the behavior of the Nusselt number. The reduction factor plays a significant role in the value of the total pressure drop. The experimental results and a least-squares curve fit for friction factor versus Reynolds number is shown in Figure 9. The solid line is the relationship by Armour and Cannon8 for unrolled screens. The first thing to notice about the experimental data is the group of points which are in the upper left-hand corner of the figure. All of these data points are for test runs with a reduction factor ⫽ 0.5. Apparently, reducing the thickness of the screen by 50% causes a large increase in the friction coefficient. The second thing to notice in Figure 9 is that the remaining data points fall close to the line with only a 6% mean difference. To insure this was the case, the friction factor was plotted as a reduced friction factor in Figure 10. The reduced friction factor was defined as the friction factor for a specific screen unit divided by the friction factor for the unrolled screen
Figure 7 Nusselt number as a function of Reynolds number
Figure 8 Nusselt number as a function of reduction factor
as ⫽ 2n2[d 2w ⫹ (1/n)2 ]1/2
(6)
Cryogenics 1998 Volume 38, Number 10 963
Compactness factors for regenerative heat exchangers: T. Murphy and W. J. Bowman
Figure 11 Compactness factor for each screen mesh size Figure 9 Friction factor as a function of Reynolds number
unit operating at the same Reynolds number. It appears that the friction factor increases for all values of screen thickness reduction, but most pronouncedly so for reductions greater than 30%. An important result from the current research is the measurement of compactness factor. The definition of compactness factor is CF ⫽ jH/f, where jH is the Colburn jfactor, and f is the friction factor11. The Colburn j-factor is defined as jH ⫽ St ⫻ Pr1/3, where St is the Stanton number and Pr is the Prandtl number. In Figure 11, the results for the compactness factor as a function of reduction factor are shown for each mesh size. For the 100 and 180 mesh screens, the trend is clear: the compactness factor decreases with decreasing reduction factor. Hence, rolling the screens tends to reduce the compactness factor. For the 250 mesh screens, the compactness factor seems to increase at a reduction factor of 0.7. The data for this case was examined more closely, and whereas the heat transfer coefficient is about what one would expect (compared to the other data,) the friction factor is smaller than expected. The reason for this discrepancy is not apparent. Since all of the other data follow a predictable trend, it is considered an outlier. A reduced compactness factor was defined to be the ratio of the compactness factor for rolled screen to the com-
Figure 10 Reduced friction factor as a function of reduction factor
964
Cryogenics 1998 Volume 38, Number 10
pactness factor for unrolled screens. The reduced compactness factors data is plotted in Figure 12, along with a least-squares curve fit. The spread is moderate (a standard deviation of 21.5% from the best-fit curve). The trend clearly shows a reduction in the compactness factor as the screen thickness gets smaller. Another remark can be made about the use of the compactness factor in numerical models. Due to the lack of heat transfer data for regenerators, some researchers resort to measuring the friction factor and applying Reynolds analogy12. For Reynolds analogy to apply to flow in a regenerator, the compactness factor should be a constant. If the results for flow in a tube are used, Reynolds analogy states the compactness factor should be 0.125. For the unrolled screens, this is not a bad guess, being 16.7% high versus the average of the unrolled values. But the analogy breaks down at smaller reduction factors, reaching a mean value of 0.031 for a 50% reduction in screen thickness. Another parameter was used to evaluate the merit of reducing the thickness of the screens. Effectiveness is defined for regenerators in terms of the ability of the regenerator to remove the maximum amount of energy from the gas relative to an ideal case. Effectiveness, , can be defined for the case of one-directional flow as
⫽
m ˙ cp2 msucm
(8)
Figure 12 Reduced compactness factor vs. reduction factor
Compactness factors for regenerative heat exchangers: T. Murphy and W. J. Bowman
Figure 13 Reduced effectiveness vs. reduction factor
where m ˙ is the mass flow rate (kg/s), cp is the specific heat of the gas (J/kg/K), msu is the mass of matrix (kg), cm is the specific heat of the matrix (J/kg/K), and 2 is the sponge effect delay time (s)3. In Figure 13, the reduced effectiveness is shown as a function of the reduction factor. Effectiveness is the dominant parameter in cryocooler systems performance4,13. The data show a reduction in effectiveness whenever the screen thickness (reduction factor) is reduced. The effect is small at reduction factors of 0.7 and 0.85, and falls to about a 13% decrement in effectiveness at a reduction factor of 0.5. Again, these results would rule out any benefit of reducing the screen thickness by 50%. The interesting thing about the figures is that one would expect the effectiveness to increase by making pore sizes smaller. Rolling the screens does not change the material properties of the gas or matrix, hence another reason must exist for the decrement. As noted above, rolling the screens does reduce the wetted surface area of the matrix since abutting round wires have a small area of contact while flattened surfaces can have a larger contact area. Large surface-area-to-volume ratio is one of the factors responsible for the large effectiveness of wire-screen matrices. Since the wetted surface area is diminished when flattened surfaces are introduced by rolling the screens, the effectiveness is also reduced.
Conclusions The results presented in this paper give a convincing explanation for the conclusion that rolling the screens to reduce their thickness causes the ratio of heat transfer coefficient to the friction factor to decrease. The data show that the heat transfer between the gas and the matrix is reduced for regenerators made from rolled screens. This reduction is a result of reduced wetted surface area or reduced convection heat transfer coefficient. The friction factor, however,
increased marginally for small thickness reductions, but substantially for the 50% reduction case. The crux of the matter with regard to the compactness factor is how the geometry of the matrix changes due to rolling. Rolling the screens causes the round surface areas of wires to be replaced with flattened areas with abrupt edges. These flattened areas abut against one another, causing the total wetted surface area in the matrix to decrease. This reduces the heat transfer. Also, the smaller pore sizes and sharp edges cause the inertial component of the pressure drop to increase to the point where the total pressure drop is higher. The overall effect of the geometry change is to increase the compactness factor. The effect of reducing the screen thickness on cryocooler COP was not addressed. It was shown that reducing the screen thickness caused a decrease in regenerator effectiveness which will have a negative impact on COP. However, reducing the screen thickness should also decrease the void fraction in a cryocooler, which will have a positive impact on COP. The optimum screen thickness for best COP still needs to be studied.
References 1. Chan, C. K., et al., Overview of cryocooler technologies for spacebased electronics and sensors. Advances in Cryogenic Engineering, 1990, 35, 1239–1250. 2. Thomas, P. J., SDIO and Air Force cryocooler technology developments at USAF Phillips Laboratory. In Proceedings of the 7th International Cryocooler Conference, Santa Fe, NM, November 1992, pp. 3–13. 3. Murphy, T. J., How reducing the thickness of the round wire screens in a porous medium regenerator affects its compactness factor (jH/f). PhD dissertation, AFIT/DS/ENY/96-5, School of Engineering, Air Force Institute of Technology (AU), Wright-Patterson AFB, OH, June 1996. 4. Urieli, I. and Berchowitz, D. M., Stirling Cycle Engine Analysis. Adam Hilger, Bristol, UK, 1984. 5. Atria, M. D., et al., Microcomputer aided cyclic analysis of the Stirling cryocooler with different regenerator meshes. Cryogenics (September Supplement), 1990, 30, 236–240. 6. Pucci, P. F., et al., The single blow transient testing technique for compact heat exchangers surfaces. Journal of Engineering for Power, 1967, 89, 29–40. 7. Locke, G. L., Heat transfer and flow friction characteristics of porous solids. Test Report No. 10, Department of Mechanical Engineering, Stanford University, Stanford, CA, 1 June 1950. 8. Armour, J. C. and Cannon, J. N., Fluid flow through woven screens. AICHE Journal, 1968, 14(3), 415–420. 9. Hamaguchi, K. et al., Effects of regenerator size change on Stirling engine performance. In Proceedings of the 26th IECEC, Vol. 5. 1991, pp. 293–298. 10. Wiese, J. L., Experimental analysis of heat transfer characteristics and pressure drop through screen regenerative heat exchangers. Masters thesis, AFIT/GAE/ENY/93D-30, School of Engineering, Air Force Institute of Technology (AU), Wright-Patterson AFB, OH, December 1993. 11. Kays, W. M. and London, A. L., Compact Heat Exchangers, 3rd edn. McGraw-Hill, New York, 1984. 12. Radebaugh, R. and Louie, B., A simple 1st step to optimizing regenerator geometry. In Proceedings of the 3rd Cryocooler Conference, Boulder, CO, 17–18 September 1986, pp. 177–198. 13. Martini, W. R., Stirling Engine Design Manual. NASA CR135382 (1980).
Cryogenics 1998 Volume 38, Number 10 965