Entropy Generation Minimization analysis of oscillating-flow regenerators

International Journal of Heat and Mass Transfer 87 (2015) 347–358

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Entropy Generation Minimization analysis of oscillating-flow regenerators Paulo V. Trevizoli, Jader R. Barbosa Jr. ⇑ POLO – Research Laboratories for Emerging Technologies in Cooling and Thermophysics, Department of Mechanical Engineering, Federal University of Santa Catarina, Florianopolis, SC, Brazil

a r t i c l e

i n f o

Article history: Received 16 December 2014 Received in revised form 14 March 2015 Accepted 25 March 2015

Keywords: Passive regenerator Effectiveness Entropy Generation Minimization Performance evaluation criteria

a b s t r a c t The thermodynamic efficiencies of regenerative cooling cycles are directly linked to the heat transfer effectiveness and thermal losses in the regenerator. This paper proposes a performance analysis for regenerators based on the Entropy Generation Minimization (EGM) theory. The mathematical model consists of the one-dimensional Brinkman–Forchheimer equation to describe the fluid flow in the porous matrix and coupled energy equations to determine the temperatures in the fluid and solid phases. The cycle-average entropy generation contributions due to axial heat conduction, fluid friction and interstitial heat transfer are calculated. The influences of parameters such as the mass flow rate, operating frequency, regenerator cross sectional area, housing aspect ratio, utilization factor and particle diameter are evaluated according to the variable geometry (VG) and fixed face (cross-section) area (FA) performance evaluation criteria (PEC). Optimal regenerator configurations are found for each PEC for flow rates between 40 and 300 kg/h (0.01 and 0.083 kg/s) and frequencies between 1 and 4 Hz with constraints of regenerator effectiveness equal to 95% and temperature span of 40 K. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction Oscillating-flow regenerators are storage-type heat exchangers in which hot and cold fluid streams flow in alternating directions through a porous matrix. Intermittent heat transfer takes place between the solid and the fluid so that during a hot blow the high-temperature fluid warms up the solid matrix that accumulates thermal energy. In the cold blow, the matrix releases the stored energy as heat, and warms up the fluid [1–3]. Regenerators are widely employed in power and cooling gas cycles such as the Stirling, pulse-tube, thermoacoustic, Gifford–McMahon and Vuillemier cycles. Regenerators that use liquids as thermal fluids are encountered in some magnetic cooling cycles. In the Brayton magnetic cooling cycle, for example, the regenerator can be classified as active because the solid matrix is made of a magnetocaloric material that is heated up or cooled down (with respect to the ambient temperature) when the regenerator is magnetized or demagnetized adiabatically [4–6]. The active regenerator concept can be extended to other energy conversion mechanisms, such as the electrocaloric and mechanocaloric effects [7–9]. Although working prototypes exploring the latter effects have not yet reached the ⇑ Corresponding author. E-mail address: [email protected] (J.R. Barbosa Jr.). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2015.03.079 0017-9310/Ó 2015 Elsevier Ltd. All rights reserved.

development stage of their magnetocaloric counterparts, it is likely that liquids will also be the heat transfer fluid of choice for near room temperature applications [10]. In active regenerators, the structure and the geometry of the solid matrix have to be optimized to reduce the thermal, viscous and other losses and achieve the desired operating conditions of temperature span, cooling capacity and cycle efficiency. An ideal regenerative matrix geometry is one with a large thermal mass, large surface area and high thermal conductance, but negligible viscous and axial conduction losses. Due to the sometimes prohibitive manufacturing and processing costs of magnetocaloric, electrocaloric and mechanocaloric materials, it is not always possible to make systematic experiments with different solid matrix geometries. Nevertheless, experiments with passive solid matrices may help to quantify the influence of the porous medium geometry on the thermal–hydraulic performance of the regenerator independently of magnetic related (or similar types of) losses in the matrix [11,12]. In a heat exchanger, there must be good thermal contact between the fluid and solid phases, but the latter should offer a small resistance to the fluid flow. These conflicting requirements are often equalized using thermal optimization. A regenerator can be designed for optimal performance according to the Entropy Generation Minimization (EGM) method [13]. A recent

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Nomenclature Roman Ac As At c cp cE Dk Dh;h Dp f  h k K L _ m NTU Nu p Pe Pr Re Sg S_ 000 g

t T TC TH u

cross-section area [m2] interstitial heat transfer area [m2] amplitude of the pressure waveform [m/s2] specific heat capacity [J/kg K] specific heat capacity at constant pressure [J/kg K] Ergun constant [–] longitudinal thermal dispersion coefficient [m2/s] regenerator housing hydraulic diameter [m] particle diameter [m] cycle frequency [Hz] convective heat transfer coefficient [W/m2 K] thermal conductivity [W/m K] permeability of the porous medium [m2] regenerator length [m] mass flow rate [kg/h] Number of heat transfer units [–] Nusselt number [–] pressure [kPa] Péclet number [–] Prandtl number [–] Reynolds number [–] cycle average entropy generation [J/K] entropy generation rate per unit volume [W/K m3] time [s] temperature [K] cold source temperature [K] hot source temperature [K] superficial (Darcy) velocity [m/s]

review of applications of the method in the context of heat exchangers and storage systems was presented by Awad and Muzychka [14]. Krane [15] evaluated the performance of regenerators using gases as working fluids and concluded that the storage and removal processes need to be analyzed together to determine the optimum characteristics of these devices, which were observed to be quite inefficient (i.e., 70–90% of the available exergy is destroyed by the end of a cycle). Das and Sahoo [16] used the EGM method in the thermodynamic optimization of regenerators under single blow operation. Their model disregarded the axial heat conduction and was valid only for low values of NTU. An optimum operating condition was identified in terms of the cycle time and NTU. In a subsequent work, Das and Sahoo [17] included the time dependence and the axial conduction in the EGM analysis, thus extending the validity of their model to more densely packed regenerators operating at higher values of NTU. de Waele et al. [18] and Steijaert [19] applied the EGM method to pulse-tube cryocoolers, taking into consideration the entropy production in every component (orifice, heat exchangers, regenerator, switching valves). The model was used to evaluate the thermodynamic performance of a cryocooler prototype. Based on the work of de Waele et al. [18], Nam and Jeong [20] employed the EGM method in the analysis of parallel-wire (segmented and unsegmented) mesh regenerators. They observed a better performance of the unsegmented parallel-wire configuration (in comparison to a screen mesh matrix) as a result of lower values of porosity and friction factor. However, axial heat conduction was identified as the main source of irreversibility in the parallel-wire case. To overcome this loss, a segmented parallel-wire geometry was used to decrease the axial conduction irreversibility and improve the thermodynamic performance of the parallel-wire regenerator. The present work proposes a calculation procedure based on the EGM method to design optimal passive oscillating-flow

~ v z

velocity vector [m/s] axial coordinate [m]

Greek

a b

 e /

l m x q s f

thermal diffusivity [m2/s] surface area density [m2/m3] effectiveness [–] porosity [–] utilization factor [–] dynamic viscosity [Pa s] kinematic viscosity [m2/s] angular frequency [rad/s] density [kg/m3] cycle period [s] regenerator housing aspect ratio [–]

Subscripts and superscripts Dp particle diameter f fluid phase FAC fluid axial conduction HT heat transfer min minimum s solid phase SAC solid axial conduction VD viscous dissipation eff effective x average value (overbar)

regenerators. The mathematical model is composed of the one-dimensional Brinkman–Forchheimer equation for momentum transfer in porous media coupled with energy balance equations for the fluid and solid phases. The local instantaneous velocity and temperature fields are used in the calculation of the local rates of entropy generation per unit volume due to fluid friction, axial heat conduction in both media and interstitial heat transfer with a finite temperature difference between the phases. Since the ultimate application of the present method involves the optimization of active magnetic regenerators, the thermal fluid has been treated as water. As performed by Pussoli et al. [21] in the optimization of peripheral-finned tube recuperators, the total entropy generation, Sg , was used as the objective function in a calculation procedure to identify optimal regenerator configurations making use of the performance evaluation criteria (PEC) of Webb and Kim [22]. In these PEC, the heat exchange device can be optimized according to variable geometry (VG), fixed face area (FA) or fixed geometry (FG) constraints, which may be useful in the context of regenerator design for both passive and active magnetic applications. For instance, while the FA PEC can be used to evaluate the most suitable operating conditions of existing devices (i.e., those whose physical dimensions cannot be changed), the VG PEC is applicable in the earlier stages of system design in order to determine the optimal aspect ratio of the regenerator for a given volume of solid material. It is worth pointing out that the so-called ‘‘entropy generation paradox’’ [23] is not applicable in the present analysis because the heat transfer rate (heat duty) is a fixed constraint in all calculations and the rate of entropy generation due to fluid friction can be of the same order of magnitude (i.e., non-negligible) as those associated with heat transfer. As shown in [24] for recuperators, there is an optimum NTU associated with the minimum entropy generation rate. Additionally, the heat exchanger

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effectiveness was proven not to be a suitable performance parameter to evaluate the heat exchanger performance because it increases monotonically with the NTU and does not take into account the power required to pump the fluid through the heat exchanger.

2.2. Energy balances 2.2.1. Fluid phase The energy equation for the one-dimensional, laminar, incompressible fluid flow in the porous medium is given by,

2. Mathematical modeling

qf cp;f e



The mathematical model for the thermal-hydraulic analysis of passive regenerators is based on a similar model for active magnetic regenerators described elsewhere [25]. The model comprises the momentum equation to calculate the fluid flow through the porous matrix and conjugate energy equations for the fluid and solid phases. The basic geometry of the model is presented in Fig. 1. The simplifying assumptions are one-dimensional, laminar and incompressible fluid flow, low-porosity porous medium (e < 0.6) and absence of body forces. There are no thermal losses to the surroundings and the temperatures of the cold and hot thermal reservoirs, T C and T H , are constant and uniform. The volume of fluid between each thermal reservoir and the nearest regenerator end is negligible (zero void volume). 2.1. Momentum balance The Brinkman–Frochheimer equation for momentum transfer in porous media is given by [26,27]:





lf 2 l qf @~ c q v ~ ~ fþ þ v  rv ¼ rp þ qf~ r~ v f~ v  E1=2f j~ v j~ v e @t eqf K K

ð1Þ

where the term on the left is the macroscopic inertial force and those on the right are the pore pressure gradient, body force, macroscopic viscous shear stress (Brinkman viscous term), microscopic shear stress (Darcy term) and microscopic inertial force (Ergun inertial term), respectively. With the simplifying assumptions one has:





qf @u cE qf @p lf ¼   u  1=2 juju @z K e @t K

ð2Þ

where u is the Darcian (superficial) velocity, t is time, p is the pressure, qf is the fluid density, lf is the fluid kinematic viscosity, e is the porosity, z is the axial distance and Dp is the particle diameter. K ¼ e3 Dp =150ð1  eÞ2 is the permeability of the porous medium and 0:5

cE ¼ 1:75=ð150e3 Þ is the Ergun constant [26–28]. In the fluid flow model, the fluid properties were treated as constants obtained at an average fluid temperature. The empirical constants in the expressions for K and cE were validated experimentally through experiments using packed beds of spheres, which will be reported at a later stage. To describe the oscillatory flow in the regenerator, the pressure gradient was approximated by a time-dependent function, which contains a characteristic waveform as follows [29–31]:



@p ¼ qf At sin ð2pftÞ @z

ð3Þ

where At is the amplitude and f is the cycle frequency.

@T f @T f þu @t @z



  @ 2 T  @p eff f ¼ hbðT f  T s Þ þ e kf þ qf cp;f Dk þ u  @z2  @z  ð4Þ

where the terms on the left are due to inertial (thermal capacity) effects and longitudinal advection, and those on the right are the transversal heat transfer term calculated using a convective heat transfer coefficient, the axial conduction and the viscous dissipation terms, respectively. cp;f is the fluid specific heat capacity, T f is the eff

fluid temperature and T s is the solid temperature. kf ¼ ekf is the effective thermal conductivity of the fluid phase.

Dk

af

¼ 0:75 Pe is

the longitudinal thermal dispersion in a bed of spheres, for Re

Pe  1. Pe ¼ 2Dp Pr is the Peclet number, ReDp is the Reynolds number based on the particle diameter and Pr is the Prandtl number [26,32]. The interstitial convective heat transfer coefficient for a packed h, was calculated using the Nusselt number corbed of spheres,  relation of [33]. This correlation was preferred to more traditional ones [34,35] because it is valid for porosities between 0.2 and 0.9 and for a wide range of Prandtl numbers [36]. The results discussed in this paper are such that the particle Reynolds numbers lie between 1 and 500, which for water (Pr  6) correspond to 2

1=3 ðRe0:6 Þ between 2 and 6 103 . However, the majority of the Dp Pr data in the present simulations are for ReDp  100, or 2

1=3 ðRe0:6 Þ  800, which is exactly the range where the Wakao Dp Pr and Kaguei [34] correlation exhibits the largest discrepancy with respect to experimental data [33]. When comparing against our own experimental data for heat transfer in oscillating flow in packed spheres regenerators, the Pallares and Grau [33] correlation also 1=3 exhibited the smallest deviations for the range of ðRe0:6 Þ Dp Pr

between 7 and 2.5 103 . These data will be published in a future paper. 2.2.2. Solid phase After the simplifying assumptions, the energy equation for the solid phase is given by:

qs cs ð1  eÞ

2 @T s eff @ T s ¼ hbðT s  T f Þ þ ð1  eÞks @t @z2

ð5Þ

where the term on the left accounts for thermal inertia in the solid, and those on the right are due to interstitial heat convection and axial heat conduction, respectively. qs is the solid density, cs is the eff

solid specific heat capacity and ks is the solid effective thermal conductivity calculated using the Hadley correlation [26,37]. 2.3. Entropy generation In the regenerator, entropy is generated due to heat transfer and viscous dissipation. The local rate of entropy generation per unit volume is given by [19]:

   2 eff hbðT s  T f Þ2 ðkf þ qf cp;f Dk Þ dTf 2 keff dTs s S_ 000 ¼ þ þ g TsTf dz T 2s dz T 2f    1 dp  þ u  Tf dz  Fig. 1. Problem geometry.

2

ð6Þ

where the first term on the right is the entropy generation rate per unit volume due to interphase heat transfer with a finite

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temperature difference. The second and third terms are the entropy generation rates due to axial conduction in the fluid and solid matrix, and the fourth term is the entropy generation rate per unit volume due to viscous friction. The cycle-average entropy generation in the regenerator, Sg , is defined as:

Sg ¼ A c

Z 0

L

Z s 0

S_ 000 g dt dz

ð7Þ

Sg will be used as the objective function to be minimized in the regenerator optimization described next.

3. Performance evaluation criteria The simulations for the Entropy Generation Minimization were carried out based on the PEC of Webb and Kim [22]. The first criterion is of variable geometry (VG), for which the regenerator housing cross sectional area, or the housing diameter, Dh;h , and the length, L, are allowed to vary, keeping a constant housing volume. The second criterion is of fixed face area (FA), where the regenerator housing cross sectional area is kept constant and the regenerator length can vary, thus changing also the housing volume. For reasons that will be discussed later, the fixed geometry (FG) criterion was not evaluated in this paper. The baseline (reference) housing geometry has the following geometric characteristics: Dh;h ¼ 25 mm and f ¼ 2, where f ¼ L=Dh;h is the aspect ratio of the regenerator housing. Thus, the baseline regenerator housing volume is 24.544 cm3. The ranges of the variables explored in the analysis are presented in Table 1 and a flow chart of the various scenarios is shown in Fig. 2. The operating conditions (i.e., frequency and mass flow rates) and ranges of geometric parameters evaluated in this paper were chosen so as to be consistent with figures currently encountered in magnetic regenerator systems [38,39]. As can be seen from Table 1, all of the possible Dh;h and f combinations in the VG criterion result in a fixed housing volume of 24.544 cm3. The particle diameter is varied from 0.2 to 2 mm, for each possible combination. In the FA criterion, Dh;h is fixed at

Table 1 Ranges of the geometric variables for each PEC. PEC

Dh;h [mm]

f

Housing volume [cm3]

Dp [mm]

VG FA

12.5–75 25

16–0.074 1–8

24.544 12.272–98.175

0.2–2 0.2–2

25 mm, while f is changed from 1 to 8. The particle diameter is also varied from 0.2 to 2 mm for each possible combination. For each combination of Dh;h ; f and Dp in each PEC, simulations were carried out considering two different scenarios. In the first scenario, the frequency was kept constant at 1 Hz and the mass flow rate was varied from 40 to 100 kg/h (steps of 10 kg/h) for the VG PEC and from 100 to 300 kg/h (steps of 25 kg/h) for the FA PEC, as presented in Table 2. The mass flow rate is set by changing the amplitude of the pressure gradient term, At , in Eq. (3). Eq. (2) is then solved iteratively until the instantaneous velocity, _ are such that their uðtÞ, and the instantaneous mass flow rate, mðtÞ, integration over a blow period corresponds to the prescribed mass flow rate. In the second scenario, the frequency was varied from 1 to 4 Hz (steps of 0.5 Hz) and the mass flow rate was kept constant at 60 kg/h for the VG PEC and at 250 kg/h for the FA PEC, as presented in Table 3. As seen from Tables 2 and 3, the utilization factor, /, can be constant or variable depending on the evaluation criterion (VG or FA). The utilization factor is defined as the ratio of the thermal masses of the fluid and solid phases:

/¼

_ cp;f mf cp;f m ¼ cp;s ms fcs qs ð1  eÞAc L

ð8Þ

The utilization factor is an important parameter in regenerator analysis, since high regeneration effectiveness is generally associated with / < 1, i.e., a large thermal mass of the regenerative matrix [12,40]. Keeping in mind that the solid regenerative material may be one of the most expensive items of the thermal system, the lowest possible value of utilization factor that maximizes the thermal performance should be always sought in regenerator design.

Table 2 Constraints of the regenerator analysis for the VG and FA PEC for a fixed frequency and a variable mass flow rate. Variable

Value or range

Units

e

0.36 280 320 1 _ ¼ 10) 40–100 (Dm 0.40–0.99 _ ¼ 25) 100–300 (Dm Variable (0.25–5.95)

– K K Hz kg/h – kg/h –

TH TC f _ VG – m VG – / _ FA – m FA – /

Fig. 2. Flow chart of the various scenarios evaluated in the simulations.

P.V. Trevizoli, J.R. Barbosa Jr. / International Journal of Heat and Mass Transfer 87 (2015) 347–358 Table 3 Constraints of the regenerator analysis for the VG and FA PEC for a fixed mass flow rate and a variable frequency. Variable

Value or range

Units

e

0.36 280 320 1–4 (Df ¼ 0:5) 60 0.60–0.15 250 Variable (4.96–0.15)

– K K Hz kg/h – kg/h –

TH TC f _ VG – m VG – / _ FA – m FA – /

In the VG cases, if the frequency is fixed the utilization factor is only a function of the mass flow rate because the housing volume and porosity do not change. Similarly, if the mass flow rate is fixed, the utilization factor is only a function of frequency. On the other hand, for the FA PEC, the utilization factor decreases with f because the mass of the regenerative matrix increases with L. Thus, for the FA PEC, / varies with the mass flow rate (for a fixed frequency), or with the frequency (for a fixed mass flow rate) and with the housing aspect ratio. For this reason, the mass flow rate and the frequency were chosen as independent variables instead of a normalized parameter, i.e., the utilization. Because the utilization and the aspect ratio are not independent of each other in the FA PEC, it is not possible to know the individual roles of the mass flow rate or the frequency in the regenerator entropy minimization if the utilization and the aspect ratio are used as independent input variables. The regenerator effectiveness is a measure of the rate at which heat is transferred between the fluid and solid phases in a given blow [1]. The hot-blow regenerator effectiveness is defined as:

¼

_ pf ðT HB  T C Þ mc Q_ HB ðT HB  T C Þ ¼1 ¼1 _ pf ðT H  T C Þ ðT H  T C Þ mc Q_ max

ð9Þ

_ is the cycle-average mass flow rate, T HB is the cycle-averwhere m age temperature of the fluid exiting the regenerator at the cold end (after the hot blow). In an ideal regenerator [1], the large volumetric heat capacity of the solid material and the perfect heat contact between the fluid and the matrix result in T HB ¼ T C . In the VG and FA PEC, a fixed regenerator effectiveness is equivalent to a fixed heat transfer rate when the temperatures of the reservoirs and the frequency or the mass flow rate are fixed (Tables 2 and 3). In the present analyses, a target value of  was set at 95% (performance constraint). In the VG cases, for fixed values of frequency, mass flow rate and utilization, the regenerator effectiveness will change as a result of changes in Dh;h and f (which are reciprocal because of the constant housing volume constraint) or Dp . Changes in crosssection area affect the fluid superficial velocity, the particle Reynolds number (which is also affected by Dp ) and the magnitude of the axial heat conduction. The surface area per unit volume is directly affected by the particle diameter. In the FA cases, the interstitial heat transfer coefficient is only a function of Dp , since the superficial velocity is constant for a given mass flow rate. On the other hand, the interstitial area changes with both Dp and f. The utilization factor decreases with increasing f, which contributes to achieving higher values of regenerator effectiveness due to the larger thermal mass of the solid phase. It should be noted that the fixed geometry (FG) PEC [22], i.e., that in which both the regenerator length and cross-section area are kept fixed, were not evaluated in the present paper because fixing a value of  in this case does not necessarily correspond to a fixed cycle-average heat transfer rate.

351

4. Numerical implementation Each operating condition is defined by a set of constraints specified according to Table 2 (variable mass flow rate) or Table 3 (variable frequency). For a specific operating condition, the ranges of the geometric variables (e.g., housing and particle diameters and aspect ratio) are chosen according to each PEC (VG or FA), as shown in Table 1. For each point in the range associated with a given PEC, the momentum and energy equations are solved numerically and the cycle-average entropy generation is calculated. Dimensionless forms of the momentum and energy equations for the fluid and solid phases were solved using the finite volume method [41,42]. The energy equations were implemented using a fully implicit scheme (in both the spatial and temporal terms) so that a coupled solution of these equations is performed at a given time. Since the momentum equation is not position-dependent, a fully explicit discretization scheme was adopted. Nevertheless, the Ergun inertial term contains a strong non-linearity, which requires an iterative solution [41,42]. The Weighted Upstream Differencing Scheme (WUDS) was used in the fluid energy equation and the Central Difference Scheme (CDS) was applied in the solid. The solver was based on a line-by-line method (Thomas Algorithm). In regenerator modeling, the solution of the energy equations can be strongly affected by variations in the solid and fluid thermophysical properties, especially the solid specific heat capacity [12]. Thus, the dependence of the fluid and solid thermophysical properties on the temperature has been incorporated in the energy equations solver. For simplicity, however, this dependence has been omitted from the solution of the momentum equation. This did not have an important influence in the heat transfer results and significantly reduced the computational cost. To accelerate the numerical convergence, the fluid and solid temperatures are initialized as linear profiles between T C at z ¼ 0 and T H at z ¼ L. The initial condition for the momentum equation is u ¼ 0 at t ¼ 0. Symmetry boundary conditions are assumed at z ¼ 0 and z ¼ L for the solid temperature. The boundary conditions for the fluid phase depend of the direction of the fluid flow. For convenience, if u is positive, then at z ¼ 0 an inlet boundary condition is used with T f ¼ T C , and at z ¼ L an outflow boundary condition (symmetry) is applied. On the other hand, if u is negative, an inlet boundary condition is applied at z ¼ L with T f ¼ T H and an outflow boundary condition is set at z ¼ 0. A numerical mesh consisting of 360 dimensionless time steps and 200 dimensionless control volumes has been used in all simulations. Based on a mesh study performed with several frequencies, mass flow rates, housing diameters, aspect ratios and particle diameters, the proposed mesh size was proven to be satisfactory. A comparison of results obtained with the 200/360 volume/time mesh with a 400/720 volume/time mesh gave maximum differences of 0.22% for the hot-blow effectiveness with up to 6 times lower computing time. For each scenario of the VG PEC (i.e., variable frequency or variable mass flow rate), discrete data points were selected in the housing diameter and particle diameter ranges shown in Table 1 (incremental steps of 5 mm for Dh;h and 0.1 mm for Dp ), resulting in 266 independent cases. Similarly, for the FA PEC, the aspect ratio range was divided in incremental steps of 0.5 (plus the 0.1-mm steps for Dp ), resulting in 342 cases for the variable frequency and variable mass flow rate scenarios. In total, 9196 different simulations were performed for the VG and FA PEC. The searches for the points of minimum entropy generation were refined further by means of 4th-order (or less) polynomial interpolations (R2 > 0:9999), which guaranteed the stability of the numerical solutions and a better resolution (finer than 0.5 mm for Dh;h , 0.05

0.1

15 0.1

0.1 0 25.9 5

0.1

1

0.175

7

0.115 0.11

8

for f and 0.01 mm for Dp ) for the minimum Sg value at a reasonable computational cost.

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0.25

352

4

0.2

0.9

2 0.

0.1

5

5

0.85

75 0.1

3

0.9

0. 17 5 0. 12

5

0. 15

0.125

0.15

1.955 0.10

0.175

5

0.9

0.2

0.85

0.3

5

5 0.3

2 0.

2

0.15

0.4

5 17 0.

As numerical convergence is obtained for the momentum and energy equations for a specific case (defined by values of frequency, mass flow rate and geometric parameters associated with each PEC), the regenerator configurations that yield the desired value of hot-blow effectiveness can be identified. To illustrate this, _ ¼ 60 kg/h and freFig. 3 presents the results for the VG PEC with m quencies of 1 and 3 Hz. Fig. 4, in turn, presents the results for the _ ¼ 250 kg/h and frequencies of 1 and 3 Hz. Lines of FA PEC with m constant Sg (red lines) are plotted together with lines of constant effectiveness (blue lines), as a function of the particle diameter

Aspect Ratio [-]

6

5.1. Entropy generation-effectiveness contour maps

2 0.

0.2

0.3

5. Results and discussions

0.2

.11

1 0.2

0.4

0.6

0.8

0.3 5

0.3 5

1

1.2

1.4

1.6

1.8

2

Particle Diameter [mm]

(a)

75

5 0. 04

25

0.0

02 0.

0.0 15

20

0. 02

0.00.0 252

15

0.2

0.4

0.6

0.8

5

1

0.0

5

1.2

1.4

2

4

1.6

1.8

2

0. 03

0.06

4

0. 02

0. 02

0. 9 5

0.0

55 50

0.9

0.0

01 0.

0.0

25

0.03

2

5

35 0.0

30 25

0.0

1

0.0 0.0

20

05

15 0

75

0.02

15

95 0.

0. 00

0.0 .020.010.01075 5

0.2

0.4

0.6

0.8

1

0. 05

0.0

4

0.035

0.04

35

0. 95

0.0

0. 05

04 0.

0.85

0.4

0.0

75

0.0

6

0.8

1

1.2

0.1

0.9

0.85

0.12

0.10.85

0.9

0.6

75

1.4

5

1.6

0.15

1.8

2

0.04

5 0.9

5

00 0.

40

0. 01

05 0.

0.0

3

75

45

75

Fig. 4. Lines of constant effectiveness (blue lines) and Sg (red lines) as a function of _ ¼ 250 kg/h and frequencies of (a) 1 Hz and (b) 3 Hz. Dp and f for the FA PEC with m (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

0.05

60

0.0

(b)

0.9 5

00 0.

Housing Diameter [mm]

65

0.0

1 0.2

06 0.

Particle Diameter [mm]

(a) 70

95 0.

0.9

Particle Diameter [mm]

75

4 3

5

0.0

3 0.9

5

0.0

30

6

4

0.1

0.0

0.0

9 0.

85 0.

75

0.9

03 0. 5

02 0.

35

0.0

25

35

0.1

0.05

6

0.035

02 0.

0. 05

5

0.075 0.06

85 0.

0.1

04 0.

45 40

7

5 0.1

50

8

25

9 0.

55

0. 07

0. 95

75

0.04

0.1

60

05 0.

Housing Diameter [mm]

65

0.1

5

0. 95

0.1

0. 1

Aspect Ratio [-]

70

0.2

1.2

1.4

1.6

1.8

2

Particle Diameter [mm]

(b) Fig. 3. Lines of constant effectiveness (blue lines) and Sg (red lines) as a function of _ ¼ 60 kg/h and frequencies of (a) 1 Hz and (b) Dp and Dh;h for the VG PEC with m 3 Hz. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

and housing diameter for the VG PEC (Fig. 3) and particle diameter and housing aspect ratio for the FA PEC (Fig. 4). As can be seen from Figs. 3 and 4, the target value of  ¼ 95% (blue solid line) can be achieved with different combinations of _ being held constant for each PEC. Dp and Dh;h or f, with f and m _ ¼ 60 kg/h and f = 1 Hz, Fig. 3(a), According to the VG PEC with m  ¼ 95% can be achieved with small values of Dp in the 0.2– 1.3 mm range and small housing diameters (12.5–58 mm), yielding values of f between 0.16 and 16. Following the line of constant hotblow effectiveness of 95%, different values of Sg can be verified with a minimum at around Dp = 1.07 mm and Dh;h = 17.23 mm. As the operating frequency is increased, Fig. 3(b), the minimum Sg for  ¼ 95% is shifted to larger values of Dp , yielding other combinations of Dh;h and f. _ ¼ 250 kg/h and f = 1 Hz, Fig. 4(a), the For the FA PEC with m minimum Sg for  ¼ 95% was identified in the vicinity of Dp ¼ 0.59 mm, with an aspect ratio of around 5.12. As expected, for other frequencies the minimum Sg range changes and  ¼ 95%

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0.1

is achieved with other combinations of Dp ; Dh;h and f, as presented in Fig. 4(b).

Heat transfer Solid axial conduction Fluid axial conduction

0.09

Viscous dissipation

0.08

5.2. Individual contributions to the total entropy generation

Total

Housing diameter = 12.5mm Aspect ratio = 16

0.06 0.05 0.04 0.03 0.02 0.01 0

0.2

0.4

0.8

0.6

1

1.2

1.4

1.6

1.8

2

Particle diameter [mm]

(a) 0.25

Housing diameter = 75mm Aspect ratio = 0.074

Heat transfer Solid axial conduction Fluid axial conduction

0.225

Viscous dissipation

0.2

Total

0.175

Sg [J/K]

This section illustrates the behavior of the three sources of entropy generation (interstitial heat transfer, axial conduction and viscous dissipation) for two regenerators with the same housing volume, but very distinct values of housing diameter and aspect ratio, as seen in Table 4. In these two cases, the particle diameter is varied between 0.2 and 2 mm. The flow rate was 60 kg/h and the frequency was 1 Hz in both cases. However, it must be noted that the hot-blow effectiveness is not the same for the two simulated cases, as illustrated in Fig. 5. Fig. 6 shows the total entropy generation and the individual contributions as a function of the particle diameter for Cases 1 and 2 of Table 4. Case 1, Fig. 6(a), is a thin (small housing diameter) and long regenerator. As expected, the axial heat conduction contribution is very small and the total entropy generation is basically a combination of viscous dissipation (because of the long matrix length and the high superficial velocity) and interstitial heat transfer. For small particle diameters, viscous dissipation is the main source of entropy because of the large values of pressure drop. As the particle diameter increases, the pressure drop decreases but the interstitial heat transfer becomes less effective. As a result, the entropy generated due to a finite temperature difference between the solid and the fluid becomes more important. A local minimum Sg at a particle diameter of around 0.8 mm can be verified. However, at this point of minimum entropy generation, the effectiveness is slightly higher than the desired performance constraint of 95% effectiveness (see Fig. 5). Case 2, Fig. 6(b), illustrates an opposite situation, i.e., a largediameter short regenerator, for which the contributions of axial heat conduction are expected to be more important because of

Sg [J/K]

0.07

0.15 0.125 0.1 0.075 0.05 0.025

Table 4 Parameters of the case study on the contributions to the total entropy generation. Case

Dh;h [mm]

f

Dp [mm]

Case 1 Case 2

12.5 75

16 0.074

0.2–2 0.2–2

1

0.9

Effectiveness [-]

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Particle diameter [mm]

(b) Fig. 6. Entropy generation due to each contribution. (a) Case 1: Dh ¼ 12:5 mm and f ¼ 16; (b) Case 2: Dh ¼ 75 mm and f ¼ 0:074.

the low superficial velocity and the short regenerator length. The total entropy generation in Case 2 is a combination of the entropy production by axial heat conduction and interstitial heat transfer. As shown in Fig. 6(b), for small particle diameters, axial heat conduction in the solid is the main source of entropy generation. As the particle diameter increases, interstitial heat transfer is poorer and the dispersion axial conduction in the fluid phase becomes more important. In this case, the target effectiveness of 95% is not achieved. It can be concluded, therefore, that this is a poor regenerator matrix, with low effectiveness and larger Sg values when compared with Case 1.

0.95

0.85 0.8 0.75 0.7

5.3. Variable geometry (VG) evaluation criteria

0.65 0.6

0

Housing diameter = 12.5mm, Aspect ratio = 16 Housing diameter = 75mm, Aspect ratio = 0.074

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Particle diameter [mm] Fig. 5. Effectiveness as a function of the particle diameter for Cases 1 and 2 at 60 kg/ h and 1 Hz.

5.3.1. Fixed frequency In this section, the results for the VG PEC at a fixed frequency and variable mass flow rate are presented as a function of changes in the geometric parameters. To exemplify the analysis, Fig. 7 shows the variation of Sg as a function of Dp ; Dh;h and f, for a mass flow rate of 60 kg/h and a frequency of 1 Hz. These results are for a constant effectiveness of 95%, as demonstrated in Fig. 3(a).

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0.04

60 57.5 55 14 52.5 13 50 12 47.5 11 45 42.5 10 40 9 37.5 8 35 7 32.5 6 30 27.5 5 25 4 22.5 3 20 2 17.5 1 15 0 12.5 1.1 1.2 1.3 1.4 16

Sg [J/K] Aspect Ratio Housing diameter [mm]

0.038 0.037

Aspect Ratio [-]

0.036

Sg [J/K]

0.035 0.034 0.033 0.032 0.031 0.03 0.029 0.028 0.027 0.026

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

As can be seen in Fig. 7, a minimum Sg is clearly achieved. The dashed lines indicate the optimal values of Dp ; Dh;h and f — 1.07 mm, 17.23 mm and 6.11, respectively — that result in an effectiveness of 95% with a minimum entropy production. Repeating the analysis for the other mass flow rates (see Table 2) gives the optimal combinations of Dp ; Dh;h and f that guarantee a hot-blow effectiveness of 95%. The results for the VG PEC for a fixed frequency and variable mass flow rates are further illustrated in Fig. 8. A summary of the conditions associated with the points of minimum Sg is presented in Table 5. In addition to the values of the utilization factor, /, the particle Reynolds number, _ p;f are  As =mc ReDp ¼ uDp =mf , the number of transfer units, NTU ¼ h presented. The individual contributions to the total entropy at the minimum, Sg;min , due to interstitial heat transfer with a finite temperature difference, Sg;HT , axial conduction in the solid, Sg;SAC , axial conduction in the fluid, Sg;FAC , and viscous dissipation, Sg;VD , are also presented. A more in-depth evaluation of the results in Fig. 8 and Table 5 _ = 40 kg/h the optimal region involves bigger parreveals that for m ticle diameters, with larger values of ReDp . As the interstitial heat transfer is less effective (lower NTU and higher Sg;HT ) and the

15

Housing Diameter [mm]

0.039

Particle diameter [mm] Fig. 7. Entropy generation in the VG PEC, for a mass flow rates of 60 kg/h and f = 1 Hz, at a constant effectiveness of 95%.

0.055

100 kg/h

0.055

90 kg/h

100 kg/h

0.05

0.05

0.045

0.045

80 kg/h

90 kg/h

80 kg/h

0.04

Sg [J/K]

Sg [J/K]

0.04 70 kg/h

0.035 0.03

70 kg/h 60 kg/h

0.035

50 kg/h

0.03

60 kg/h

40 kg/h

0.025

0.025

50 kg/h

0.02 0.015 0.2

0.02

40 kg/h

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0.015

2

15

20

25

30

35

40

45

50

55

60

Particle Diameter [mm]

Housing Diameter [mm]

(a)

(b)

0.055

100 kg/h

65

70

75

90 kg/h

0.05 0.045

80 kg/h

Sg [J/K]

0.04 70 kg/h

0.035 0.03

60 kg/h

0.025

50 kg/h

0.02

40 kg/h

0.015

0

1

2

3

4

5

6

7

8

9 10 11 12 13 14 15 16

Aspect Ratio [-]

(c) Fig. 8. Minimum entropy analysis for the VG PEC for function of Dh;h ; Sg as a function of f.

 ¼ 95%, frequency of 1 Hz and variable flow rates in the range of 40–100 kg/h: (a) Sg

as a function of Dp ; (b) Sg as a

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P.V. Trevizoli, J.R. Barbosa Jr. / International Journal of Heat and Mass Transfer 87 (2015) 347–358 Table 5 Minimum Sg parameters for the VG PEC for

 ¼ 95 %, at a fixed frequency of 1 Hz and flow rates between 40 and 100 kg/h.

_ [kg/h] m

/ [–]

ReDp [–]

NTU

Dp [mm]

Dh;h [mm]

f [–]

Sg;min [J/K]

Sg;HT [J/K] (%)

Sg;SAC [J/K] (%)

Sg;FAC [J/K] (%)

Sg;VD [J/K] (%)

40 50 60 70 80 90 100

0.397 0.496 0.595 0.695 0.794 0.893 0.992

148.3 115.4 89.5 75.6 61.0 43.8 31.1

20.0 23.0 26.6 31.2 37.9 48.4 66.2

1.66 1.32 1.07 0.89 0.73 0.58 0.45

13.610 15.386 17.227 18.460 19.900 22.206 24.468

12.396 8.579 6.113 4.968 3.965 2.854 2.133

0.01971 0.02360 0.02704 0.02997 0.03210 0.03343 0.03377

0.01569 0.01829 0.02029 0.02171 0.02215 0.02146 0.01948

0.00007 0.00011 0.00019 0.00027 0.00039 0.00069 0.00107

0.00304 0.00404 0.00519 0.00615 0.00721 0.00873 0.01009

0.00091 0.00115 0.00137 0.00184 0.00235 0.00259 0.00313

0.04

0.04

0.035

0.035

(0.3) (0.5) (0.7) (0.9) (1.2) (2.0) (3.2)

(15.4) (17.1) (19.2) (20.5) (22.5) (26.1) (29.9)

(4.6) (4.9) (5.0) (6.1) (7.3) (7.7) (9.3)

1.0 Hz

0.03

1.0 Hz

0.03

0.025

Sg [J/K]

Sg [J/K]

(79.6) (77.5) (75.0) (72.4) (69.0) (64.2) (57.7)

1.5 Hz

0.02

1.5 Hz

0.025 2.0 Hz

0.02

2.5 Hz

2.0 Hz

0.015

3.0 Hz

0.015

2.5 Hz

3.5 Hz 4.0 Hz

3.0 Hz

0.01

0.01

3.5 Hz 4.0 Hz

0.005 0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0.005

2

15

20

25

30

35

40

45

50

55

60

Particle Diameter [mm]

Housing Diameter [mm]

(a)

(b)

65

70

75

0.04 0.035 1.0 Hz

Sg [J/K]

0.03 0.025

1.5 Hz

0.02 2.0 Hz

0.015

2.5 Hz 3.0 Hz 3.5 Hz

0.01 4.0 Hz

0.005

0

2

4

6

8

10

12

14

16

Aspect Ratio [-]

(c) Fig. 9. Minimum entropy analysis for the VG PEC for  ¼ 95%, fixed mass flow rate of 60 kg/h and variable frequencies in the range of 1–4 Hz: (a) Sg as a function of Dp ; (b) Sg as a function of Dh;h ; Sg as a function of f.

Table 6 Minimum Sg parameters for the VG PEC for

 ¼ 95 %, at a fixed mass flow rate of 60 kg/h and frequencies between 1 and 4 Hz.

f [Hz]

/ [–]

ReDp [–]

NTU [–]

Dp [mm]

Dh;h [mm]

f [–]

Sg;min [J/K]

Sg;HT [J/K] (%)

Sg;SAC [J/K] (%)

Sg;FAC [J/K] (%)

Sg;VD [J/K] (%)

1.0 1.5 2.0 2.5 3.0 3.5 4.0

0.595 0.397 0.298 0.238 0.198 0.170 –

89.5 121.3 139.4 154.5 175.2 193.1 –

26.6 19.8 17.8 16.1 14.6 12.8 –

1.07 1.31 1.42 1.52 1.64 1.78 –

17.227 16.373 15.903 15.630 15.246 15.127 –

6.113 7.120 7.770 8.185 8.818 9.029 –

0.02704 0.02032 0.01565 0.01277 0.01079 0.00933 –

0.02029 0.01586 0.01229 0.01005 0.00847 0.00726 –

0.00019 0.00009 0.00006 0.00004 0.00003 0.00003 –

0.00519 0.00347 0.00258 0.00210 0.00178 0.00163 –

0.00137 0.00091 0.00072 0.00058 0.00051 0.00041 –

(75.0) (78.0) (78.6) (78.7) (78.5) (77.8)

(0.7) (0.5) (0.4) (0.3) (0.3) (0.3)

(19.2) (17.1) (16.5) (16.4) (16.5) (17.5)

(5.0) (4.5) (4.6) (4.5) (4.7) (4.4)

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viscous losses are smaller for bigger particle diameters, long regenerative matrices with smaller housing diameters are required in order to achieve the 95% regenerator effectiveness, giving rise to the lowest contributions of axial conduction in the fluid and solid. _ = 100 kg/h the optimal region involves On the other hand, for m small particle diameters, larger housing diameters and smaller values of aspect ratio. Since at high mass flow rates and small particle

0.25 0.225 0.2

Sg [J/K]

0.175

300 kg/h

0.15

275 kg/h 250 kg/h

0.125

225 kg/h

0.1

200 kg/h 175 kg/h 150 kg/h 125 kg/h

0.075

100 kg/h

0.05 0.025 0 0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Particle Diameter [mm]

(a) 0.25 0.225

Sg [J/K]

0.2 0.175

300 kg/h

0.15

275 kg/h 250 kg/h

0.125

225 kg/h

0.1

200 kg/h

0.075

175 kg/h 150 kg/h 125 kg/h

0.05

100 kg/h

0.025 0

diameters the viscous losses become more important (see Sg;VD column), a big housing diameter is required to decrease the superficial velocity, which results in short matrices. Nevertheless, smaller particle diameters are necessary to enhance the interstitial heat transfer and guarantee an optimal performance of the regenerator, increasing NTU and decreasing Sg;HT . As a result, the axial conduction contributions become more important since the spheres size becomes smaller and the superficial velocity decreases (see Sg;SAC and Sg;FAC columns). In summary, Sg;HT was found to be the main contribution (in %) to the total entropy generation for the VG PEC at a fixed frequency (although its importance decreases with flow rate as the NTU increases). The second largest contribution is Sg;FAC , which increases with the flow rate since the matrices becomes shorter and the velocity decreases with increasing housing diameters. The Sg;SAC and Sg;VD contributions are less important to Sg;min , with Sg;VD increasing with flow rate as a result of the decrease in particle diameter.

1

2

3

4

5

6

7

8

Aspect Ratio [-]

(b) Fig. 10. Minimum entropy analysis for the FA PEC for  ¼ 95%, fixed frequency of 1 Hz and variable mass flow rates in the range of 100–300 kg/h: (a) Sg as a function of Dp ; (b) Sg as a function of f.

Table 7 Minimum Sg parameters for the FA PEC for

5.3.2. Fixed mass flow rate The results for the VG PEC at a fixed mass flow rate and variable frequency (see Table 3) are presented as a function of changes in Dh;h ; f and Dp for a constant effectiveness of 95%. The discussion involving Fig. 7 enabled the construction of diagrams to quantify the effect of frequency and geometric parameters on the cycleaverage entropy generation for  ¼ 95%, as illustrated in Fig. 9. In general, the regions of minimum Sg with respect to Dp ; Dh;h and f become wider as the frequency increases. A summary of the conditions that lead to the minimum Sg results is presented in Table 6. An analysis of the results in Fig. 9 and Table 6 reveals that for f = 1 Hz the optimal region involves smaller particle diameters, which guarantee values of NTU that are sufficiently large to achieve 95% effectiveness. In turn, larger housing diameters and shorter matrices are needed to decrease the superficial velocity and reduce the viscous losses, which leads to significant axial conduction contributions to Sg;min . On the other hand, for f = 3.5 Hz the optimal region involves bigger particle diameters, smaller housing diameters and larger values of the aspect ratio. Since the utilization factor decreases with increasing frequency, bigger particle diameters, i.e., larger solid thermal mass, are sufficient to guarantee 95% effectiveness, even at lower values of the NTU. In general, at the highest frequencies, the effectiveness is higher than 95% due to the small values of the utilization factor. Thus, longer regenerators are required to produce sufficiently large heat transfer areas and decrease the axial conduction losses. As a consequence, the housing diameter decreases. It should be noted in Table 6 that for f = 4.0 Hz, a minimum Sg was not found within the ranges of the simulated parameters. A comparison between the contributions of each term (in %) for the variable frequency (Table 6) and variable mass flow rate (Table 5) cases shows that these are much more sensitive to variations in flow rate for a fixed frequency.

 ¼ 95%, at a fixed frequency of 1 Hz and variable flow rates between 100 and 300 kg/h.

_ [kg/h] m

/ [–]

ReDp [–]

NTU [–]

Dp [mm]

Dh;h [mm]

f [–]

Sg;min [J/K]

Sg;HT [J/K] (%)

Sg;SAC [J/K] (%)

Sg;FAC [J/K] (%)

Sg;VD [J/K] (%)

100 125 150 175 200 225 250 275 300

1.186 1.178 1.149 1.114 1.070 1.027 0.969 0.910 0.850

13.9 20.7 29.8 41.7 56.9 74.5 97.7 125.6 158.9

214.1 164.4 127.3 99.5 79.3 66.6 55.5 47.3 41.2

0.21 0.25 0.30 0.36 0.43 0.50 0.59 0.69 0.80

25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0

1.674 2.106 2.591 3.119 3.709 4.350 5.120 6.000 7.003

0.02615 0.03616 0.04787 0.06167 0.07752 0.09547 0.11562 0.13825 0.16363

0.00797 0.01254 0.01839 0.02582 0.03465 0.04382 0.05462 0.06615 0.07828

0.00180 0.00141 0.00109 0.00086 0.00069 0.00055 0.00044 0.00036 0.00029

0.00761 0.00879 0.00983 0.01086 0.01177 0.01246 0.01307 0.01355 0.01394

0.00877 0.01342 0.01856 0.02412 0.03042 0.03863 0.04749 0.05819 0.07112

(30.5) (34.7) (38.4) (41.9) (44.7) (45.9) (47.2) (47.8) (47.8)

(6.9) (3.9) (2.3) (1.4) (0.9) (0.6) (0.4) (0.3) (0.2)

(29.1) (24.3) (20.5) (17.6) (15.2) (13.1) (11.3) (9.8) (8.5)

(33.5) (37.1) (38.8) (39.1) (39.2) (40.5) (41.1) (42.1) (43.5)

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5.4. Fixed face area (FA) evaluation criteria 5.4.1. Fixed frequency This section presents the results for the FA PEC for a fixed frequency and variable mass flow rate as a function of f and Dp . As mentioned above, Dh;h is constant and kept fixed at 25 mm (see Table 2). Again, the results are illustrated for a target effectiveness of 95%. The FA PEC is further explored in Fig. 10, which shows Sg as

0.25 0.225 0.2

Sg [J/K]

0.175 0.15 1.0 Hz

0.125 0.1

1.5 Hz

0.075

2.0 Hz 2.5 Hz

0.05

3.0 Hz

3.5 Hz 4.0 Hz

0.025 0 0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Particle Diameter [mm]

(a) 0.25 0.225 0.2

Sg [J/K]

0.175 0.15 1.0 Hz

0.125 0.1

1.5 Hz

0.075

2.0 Hz

0.05

2.5 Hz 3.0 Hz

0.025 0

a function of Dp and f for mass flow rates ranging from 100 to 300 kg/h. Table 7 summarizes the parameters associated with the optimal conditions for each mass flow rate. _ = 100 kg/h the minimum As can be seen from the results, for m Sg was associated with a small particle diameter (Dp = 0.21 mm) _ = 300 kg/h the minimum and a short matrix (f = 1.674), while for m Sg was for a bigger particle diameter (Dp = 0.80 mm) and a longer matrix (f = 7.003). As the mass flow rate is increased (increasing the superficial velocity and ReDp ), the advection term becomes more important in the energy equation and the value of NTU is decreased. Therefore, a longer regenerator (i.e., a larger interstitial area and a lower utilization factor) is required to keep the effectiveness at 95%. However, when f increases, a larger particle diameter is needed to compensate for the entropy generated due to the viscous dissipation. For the FA PEC at a fixed frequency, Sg;HT and Sg;VD are the main contributions (in %) to the total entropy generation. As the mass flow rate increases, NTU decreases and f increases, making the heat transfer less effective (higher Sg;HT ) and, even for increasing particle diameters, the viscous dissipation becomes more important. Also, the Sg;FAC and Sg;SAC contributions were found to be significant at lower mass flow rates, i.e., lower superficial velocities and shorter matrices.

4.0 Hz

1

2

3

4

5

6

7

3.5 Hz

8

Aspect Ratio [-]

5.4.2. Fixed mass flow rate This section presents the results for the FA PEC for a fixed mass flow rate of 250 kg/h and variable frequency between 1 and 4 Hz. Again, changes in f and Dp are evaluated along the line of 95% effectiveness (see Table 3). Fig. 11 shows the behavior of Sg as a function of Dp and of f and Table 8 summarizes the optimal parameters for each frequency. As seen from Table 8, for f = 1 Hz the minimum Sg was found at a relatively large particle diameter (Dp = 0.59 mm) and a large aspect ratio (f = 5.120), while for f = 4 Hz the minimum Sg was found at a smaller particle size (Dp = 0.27 mm) and a shorter matrix length (f = 1.337). A different trend is observed when compared with the variable mass flow rate cases. As the frequency is increased, the utilization factor decreases due to the shorter blow period. Nevertheless, the matrix length also decreases, reducing the thermal mass of the solid phase and the total interstitial area. As a result, the utilization factor and NTU did not present significant variations with frequency, which resulted in a nearly constant contribution (in %) of Sg;HT to Sg;min . At small frequencies Sg;VD was higher even for bigger particle diameters due to the large value of the matrix length. On the other hand, higher frequencies need shorter matrix length and smaller particle diameters, which increases the penalty associated with axial heat conduction (Sg;FAC and Sg;SAC ).

(b) 6. Conclusions Fig. 11. Minimum entropy analysis for the FA PEC for  ¼ 95%, fixed mass flow rate of 250 kg/h and variable frequencies in the range of 1–4 Hz: (a) Sg as a function of Dp ; (b) Sg as a function of f.

Table 8 Minimum Sg parameters for the FA PEC for

This paper presented a performance analysis of passive regenerators based on the Entropy Generation Minimization

 ¼ 95%, at a fixed mass flow rate of 250 kg/h and variable frequency between 1 and 4 Hz.

f [Hz]

/ [–]

ReDp [–]

NTU [–]

Dp [mm]

Dh;h [mm]

f [–]

Sg;min [J/K]

Sg;HT [J/K] (%)

Sg;SAC [J/K] (%)

Sg;FAC [J/K] (%)

Sg;VD [J/K] (%)

1.0 1.5 2.0 2.5 3.0 3.5 4.0

0.969 0.988 0.977 0.970 0.959 0.945 0.928

97.7 74.5 62.9 54.6 49.7 46.3 44.7

55.5 56.9 58.0 58.7 58.2 57.0 54.1

0.59 0.45 0.37 0.33 0.30 0.28 0.27

25.0 25.0 25.0 25.0 25.0 25.0 25.0

5.120 3.349 2.539 2.047 1.724 1.500 1.337

0.11562 0.07723 0.05752 0.04630 0.03874 0.03336 0.02945

0.05462 0.03623 0.02648 0.02105 0.01761 0.01525 0.01383

0.00044 0.00046 0.00045 0.00044 0.00043 0.00042 0.00041

0.01307 0.01031 0.00841 0.00736 0.00657 0.00596 0.00554

0.04749 0.03022 0.02218 0.01744 0.01414 0.01172 0.00967

(47.2) (46.9) (46.0) (45.5) (45.4) (45.7) (47.0)

(0.4) (0.6) (0.8) (1.0) (1.1) (1.3) (1.4)

(11.3) (13.4) (14.6) (15.9) (17.0) (17.9) (18.8)

(41.1) (39.1) (38.6) (37.7) (36.5) (35.1) (32.8)

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(EGM) method [13] combined with performance evaluation criteria (PEC) of variable geometry (VG) and fixed face area (FA) [22] to determine optimal regenerator configurations subjected to constant effectiveness constraints. A one-dimensional model was developed to solve the fluid flow and coupled heat transfer in the porous regenerative matrix. Entropy generation contributions due to axial heat conduction, fluid friction and interstitial heat transfer were taken into account in the mathematical model. The main conclusions arising from this work are as follows: 1. For the fixed frequency/variable flow rate VG PEC, the optimal configuration for low flow rates involves a long regenerator with a small housing diameter and large particles to achieve higher particle Reynolds numbers and increase the interstitial heat transfer coefficient. Conversely, for high flow rates, the optimal regenerator configuration is one with a small particle diameter, a large housing diameter and a small aspect ratio to compensate for the larger viscous losses. 2. For the fixed flow rate/variable frequency VG PEC, the optimal regenerator configuration for small frequencies has a small particle diameter, a short length and a big housing diameter, which contributes to reducing the superficial velocity and hence the viscous losses. At higher frequencies, the utilization factor is reduced and longer regenerators with big particles are sufficient to guarantee a 95% effectiveness. 3. For the fixed frequency/variable flow rate FA PEC, the optimal configuration for low flow rates is associated with a short regenerator matrix and a small particle size, which is required to guarantee a sufficiently large NTU and hence the desired 95% effectiveness. As the flow rate increases, the minimum Sg shifts to larger particle sizes to decrease the viscous losses. Longer matrices are required to compensate for the reduction in the interstitial area. 4. For the fixed flow rate/variable frequency FA PEC, the optimal configuration for low the lowest frequency is one with a comparatively large regenerator aspect ratio and a large particle diameter. As the frequency increases and the utilization factor becomes lower, the use of smaller particles is required to guarantee an effective interstitial heat transfer. Conflict of interest None declared. Acknowledgements The authors thank Embraco and the CNPq – Brazil (Grant No. 573581/2008-8 – National Institute of Science and Technology in Cooling and Thermophysics) for financial support. Technical assistance from Mr. Diego Alcalde (undergraduate student) in the numerical implementation of the model is also acknowledged. References [1] F.W. Schmidt, A.J. Willmott, Thermal Energy Storage and Regeneration, Hemisphere Publishing Co., 1981. [2] R.A. Ackermann, Cryogenic Regenerative Heat Exchangers, Plenum Press, New York, 1997. [3] R.K. Shah, D.P. Sekulic, Fundamentals of Heat Exchanger Design, Wiley, 2003. [4] G.V. Brown, Magnetic heat pumping near room temperature, J. Appl. Phys. 47 (1976) 3673–3680. [5] V.K. Pecharsky, K.A. Gschneidner Jr., Magnetocaloric effect and magnetic refrigeration, J. Magn. Magn. Mater. 200 (1999) 44–56. [6] A. Kitanovski, P.W. Egolf, Thermodynamics of magnetic refrigeration, Int. J. Refrig. 29 (2006) 3–21. [7] X. Moya, S. Kar-Narayan, N.D. Mathur, Caloric materials near ferroic phase transitions, Nat. Mater. 13 (2014) 439–450.

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