- Email: [email protected]
Normalized charging exergy performance of stratified sensible thermal stores
Solar Energy 136 (2016) 487–498
Contents lists available at ScienceDirect
Solar Energy journal homepage: www.elsevier.com/locate/solener
Normalized charging exergy performance of stratified sensible thermal stores H.O. Njoku ⇑, O.V. Ekechukwu 1, S.O. Onyegegbu Department of Mechanical Engineering, University of Nigeria, Nsukka 410001, Nigeria
a r t i c l e
i n f o
Article history: Received 25 May 2015 Received in revised form 28 December 2015 Accepted 18 July 2016
Keywords: Stratified thermal storage CFD analysis Exergy performance analysis Normalized exergy efficiency
a b s t r a c t This paper presents the performance assessment of stratified sensible thermal energy stores (SSTES) on the basis of the normalized exergy efficiency, gx . Assessments based on gx provide comparisons with performances of both the perfectly stratified and the fully mixed stores, which offer the best and worst performances, respectively. This is in contrast with energy and exergy efficiencies, which compare SSTES with only the perfectly stratified store. A dimensionless unsteady axisymmetric model of vertical cylindrical SSTES was implemented using a finite volume numerical scheme. The effect of some significant parameters on SSTES performance were considered by performing computations for aspect ratios (AR) between 1 and 4, Peclet number (PeD ) varying from 5 103 to 100 103 , Richardson number (Ri) varying from 10 to 104, and overall heat loss coefficients (U) varying from 0 to 100 W m2 K1. gx increases with PeD , Ri and AR, with the most significant increases occurring at low values of these parameters, and appreciable increases are no longer obtained beyond PeD 30 103 , Ri 103 and AR 3. gx also falls monotonically as the U values increase. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction During the charging and discharging of a stratified thermal energy store, different regions are at different temperatures due to the action of buoyancy forces. When charging a stratified hot water storage tank, for example, hot water flows into the tank as the water in the tank is simultaneously withdrawn for heating. The difference in the densities of the incoming hot water and the cooler water already in the tank causes the hot water to rise to the top and the cooler water to fall to the bottom of the storage tank, creating a transition layer, known as the thermocline, which separates the hot upper zone from the cold lower zone. The same situation obtains, albeit in the reverse, during the discharging of the tank, when hot water is withdrawn from the tank to service a load while cooler water is simultaneously let into the tank from the mains. An illustration of the typical temperature profile existing in a stratified thermal storage tank is presented in Fig. 1. Compared to a situation in which the liquid storage media (water, in this case) is fully mixed during the charging process, the storage tank’s performance is improved by the presence of ⇑ Corresponding author. E-mail addresses: [email protected] (H.O. Njoku), [email protected] (O.V. Ekechukwu), [email protected] (S.O. Onyegegbu). 1 Formerly at the National Universities Commission, 26 Aguiyi-Ironsi Street, Maitama District, Abuja FCT, Nigeria. http://dx.doi.org/10.1016/j.solener.2016.07.032 0038-092X/Ó 2016 Elsevier Ltd. All rights reserved.
temperature stratification in several ways. If the storage tank is connected in closed circuit to a heat source and stratification maintained during charging, a shorter charging period will be required to extract an equal amount of heat in comparison with a fully mixed tank. This is so because the return storage fluid stream from the stratified tank is at a quasi-constant low temperature as it passes through the heat source, thus creating a higher heat transfer gradient at the heat source. For a solar collector heat source, for example, this will translate to higher mean collector heat output and thus higher efficiencies (Abu-Hamdan et al., 1992; Cristofari et al., 2003; Hollands and Lightstone, 1989). During discharge also, more heat will be available for supply to the load because of the quasi-constant high temperature at which the storage fluid is withdrawn from the storage tank to service the load.
2. Background In order to quantify the extent of improvements obtainable in sensible thermal energy stores (STES) due to the presence of stratification, objective and rational performance assessment criteria are needed. Several of such criteria have been proposed in the literature, including familiar dimensionless numbers, efficiency measures, and other miscellaneous parameters. Some are solely based on first law (energy) considerations, others on second law (entropy) considerations, while the rest result from a combination
488
H.O. Njoku et al. / Solar Energy 136 (2016) 487–498
Nomenclature Ain AR cp D DE g H m _ m P PeD r ReD Ri t T T T U U1 v V Vd z
inlet cross-sectional area (m2) aspect ratio (=H/D) constant pressure specific heat (J kg1 K1) tank diameter (m) energy accumulation (J) gravitational acceleration (m s2) tank height (m) mass of tank content (kg) inlet mass flow rate (m) pressure (N m2) Peclet number (¼ U 1 D=a) radial distance (m) Reynolds number (¼ U 1 D=m) Richardson number (¼ gbHðT T ini Þ=U 21 ) time (s) temperature (K) equivalent energy temperature (K) equivalent exergy temperature (K) overall heat loss coefficient (W m2 K1 ) _ qAin m s1) free stream velocity (¼ m= velocity (m s1) tank volume (m3) discharge volume fraction axial distance (m)
DN gx ðtÞ
gx;max u m q h h h #
s
N
exergy accumulation (J) normalized exergy efficiency maximum normalized exergy efficiency _ volume change fraction (¼ mt=m) kinematic viscosity (m2 s1) density (kg m3) dimensionless temperature (¼ ðT T ini Þ=ðT h T ini Þ) dimensionless equivalent energy temperature dimensionless equivalent exergy temperature dimensionless temperature ratio (¼ T=ðT h T ini Þ) dimensionless time (¼ U 1 t=D) exergy (J)
Subscripts and superscripts 0 reference state h hot in inlet ini initial m fully mixed max maximum min minimum r radial direction s perfectly stratified z axial direction dimensionless quantities
Greek symbols a thermal diffusivity (m2 s1) b compressibility coefficient (K1)
of first and second law considerations (exergy analysis). Reviews of these assessment criteria are well discussed in the literature (Zurigat and Ghajar, 2007; Haller et al., 2009; Castell et al., 2010; Njoku et al., 2014). The incorporation of second law considerations, (e.g., in the form of exergy analysis,) into the assessment of stratified sensible thermal energy stores (SSTES) has however been shown to be more rational and illuminating than assessments based only on energy analysis (Rosen, 2001; Rosen and Dincer, 2003; Rosen et al., 2004). Therefore, in addition to energy efficiencies, exergy efficiencies, which incorporate second law considerations, have been applied to the assessment of SSTES (e.g., Rosen, 2001; Solé et al., 2008). The analysis of SSTES is best undertaken using computer aided multi-dimensional numerical models (otherwise known as computational fluid dynamics (CFD)), as these better account for the inherently multi-dimensional temperature and flow regimes developed within the SSTES during its operation. Also, the number of simplifying assumptions made are limited, unlike what obtains with analytical models. The use of multi-dimensional numerical models in the study of SSTES, however, has traditionally focused on the simulation of temporal evolution of temperature and velocity profiles within the storage units (Eames and Norton, 1998; Yee and Lai, 2001; Zachar et al., 2003; Shin et al., 2004), the determination thereby, of thermal energy accumulation in the units, and the computation of energy based efficiencies (Abdoly and Rapp, 1982; Ismail et al., 1997; Hahne and Chen, 1998; Shah et al., 2005). In such studies therefore, the influence of such factors as inlet flow rate and temperature, tank aspect ratio, and the presence of various stratifier mechanisms, on the efficiency of energy accumulation in SSTES have been investigated. Ismail et al. (1997) developed a 2D CFD model which was used to simulate temperature profiles within a SSTES during charging and discharging and obtained temperature profiles which showed close agreement with
experimental results. Using a 3D CFD model, Shah et al. (2005) investigated the effectiveness of a thermal stratifier within a SSTES. Their model results compared well with experimental PIV and temperature measurements. They computed stratifier efficiencies, (ratios of actual energy supplied to the tank to the maximum possible energy supplied to the tank in the absence of mixing,) and indicated that a range of optimal flow rates – 5–8 l/min existed for the SSTES configurations they studied. Results of multi-dimensional numerical simulation studies have also led to better insights into the processes that lead to both the enhancement of and the breakdown of stratification in SSTES. Such studies have shown that placing obstacles close to the entrances or exits invariably enhances temperature stratification in SSTES (Zachar et al., 2003; Altuntop et al., 2005). 2D CFD simulation studies of Van Berkel (1996) and Van Berkel et al. (1999) have shown that destratification caused by mixing in SSTES thermocline regions can be approximated by a two stage process of fluid withdrawal from the thermocline by drag, and subsequent mixing due to the stretching and folding of fluid particles. A limited number of studies in which exergy methods and multi-dimensional numerical models were simultaneous used, have been reported. Time varying temperature profiles in a horizontal SSTES tank were obtained by Cónsul et al. (2004) using 3D numerical simulations. The performances of the tank when assessed using a ‘‘non-dimensional thermocline thickness”, the ‘‘MIX number” and the ‘‘non-dimensional exergy” were determined and compared, leading to the conclusion that the nondimensional exergy gave the best assessment of the tank’s performance. Farmahini-Farahani (2012) performed a 2D CFD study of the effect of tank aspect ratios (AR), inlet/outlet geometries (diameter, vertical position and inclination) on stratification in SSTES tanks. Stratification was measured using a dimensionless exergy parameter and was found to improve with higher AR, smaller
H.O. Njoku et al. / Solar Energy 136 (2016) 487–498
489
Fig. 2. Simplified axisymmetric representation of stratified sensible thermal storage tank.
Fig. 1. Typical temperature profile in a stratified thermal storage tank.
velocities in the radial and axial directions, respectively, where D, the tank diameter, is chosen as the characteristic length scale. _ qAin , the bulk inlet fluid velocity is the characteristic velocU 1 ¼ m= _ is the inlet mass flow rate, q, the fluid density and ity scale, where m Ain , the inlet cross-sectional area.
inlet/outlet diameters, and by inclining the inlets/outlets away from the top/bottom of the tanks. Shah and Furbo (2003) also used CFD simulations to study the behavior of SSTES when three different inlet designs were used. From these, they obtained time varying temperature profiles in the store with the different inlet designs. Based on limited experiments, they also computed exergy and entropy efficiencies, showing how these were influenced by Richardson number, draw-offs and initial conditions in the store. In the present work, by a non-dimensional presentation of the governing conservation equations, the major dimensionless parameters that influence the thermal and fluid flow regimes in the SSTES are identified. With the time-varying temperatures in the SSTES obtained by the CFD solutions of these equations, the exergy-based performance of the SSTES is assessed by computing its normalized exergy efficiency. This is done for a wide range of the dimensionless parameters to evaluate the parametric dependence of SSTES’ exergy performance on these parameters.
In Eq. (2), h ¼ ðT T ini Þ=ðT h T ini Þ is the time-dependent dimensionless temperature distribution, where Tðr; z; tÞ is the timedependent temperature distribution in the tank, and T h T ini , the temperature difference between inlet fluid and the initial fluid in the tank, is the characteristic temperature scale. PeD ð¼ U 1 D=aÞ is the Peclet number, where a is the fluid thermal diffusivity, while sð¼ U1 t=DÞ is the dimensionless time.
3. Analysis
3.1.3. Momentum conservation equations
3.1. SSTES tank mathematical model A schematic of the SSTES tank under consideration is shown in Fig. 2. The fluid temperature and flow fields in the SSTES tank are governed by the mass, energy and momentum conservation equations. Dimensionless forms of these conservation equations in cylindrical co-ordinates are presented in Eqs. (1)–(4), highlighting the dimensionless groups which influence the performance of the SSTES unit. 3.1.1. Continuity equation
1 @ r v ðr Þ @r
r
þ
v
@ z @z
¼0
ð1Þ
In Eq. (1), r ¼ r=D and z ¼ z=D are the dimensionless radial and axial distances, v r ¼ v r =U 1 and v z ¼ v z =U 1 are the dimensionless
3.1.2. Energy conservation equation
" # @h 1 1 @ @2h @h @h @h þ 2 þ vr þ vz ¼ r @s @r @z PeD r @r @r @z
" # @ v r 1 @ 1 @r v r @ 2 v r @P @v r @v r þ þ vr þ vz ¼ ReD @r r @r @s @r @z @r @z2
ð2Þ
ð3Þ
" # @ v z @v @v 1 1 @ @ 2 v z @P Ri @v z þ þ þ v r z þ v z z ¼ r 2 ReD r @r AR @s @r @z @r @z @z ð4Þ Momentum conservation in the radial and axial directions are expressed by Eqs. (3) and (4), respectively, where ReD ð¼ U 1 D=mÞ is the Reynolds’ number based on the tank diameter D; P ð¼ P=ðqU 21 ÞÞ is the dimensionless pressure, Rið¼ gbHðT T ini Þ=U 21 Þ is the Richardson’s number, and ARð¼ H=DÞ is the tank aspect ratio. m and b are the fluid kinematic viscosity and compressibility coefficient, respectively. In the development of the STESS model above, it is assumed that flow in the tank is axisymmetric, there are no internal sources of
490
H.O. Njoku et al. / Solar Energy 136 (2016) 487–498
heat generation, and only gravitational body forces, parallel to the z axis, act on the system. All fluid properties are determined at the mean of the inlet and initial temperatures. 3.2. Initial and boundary conditions
gx ðtÞ ¼
Initially, the SSTES tank contents are uniformly at rest at the initial temperature, i.e.,
h ¼ v r ¼ v z ¼ 0 at
s ¼ 0:
ð5Þ
During the operation of the tank, hot fluid at fixed uniform temperature and velocity flows in at the inlet boundary, viz.,
9 h¼1 > = v r ¼ 0 for > ; vz ¼ 1
s > 0; r < rin ; z ¼ 0;
ð6Þ
while downstream to the outlet, fully developed flow is assumed:
@h @ v r @ v z ¼ ¼ ¼ 0 for r < rin ; z AR @z @z @z
ð7Þ
The no-slip condition is specified on all the tank’s walls, i.e.,
v r ¼ v z
8 > < r > r in ; z ¼ 0 ðtopÞ r > r in ; z ¼ AR ðbottomÞ ¼ 0 for > : r ¼ 0:5; 0 6 z 6 AR ðsideÞ
ð8Þ
The assumption of axisymmetry implies that along the tank axis,
@h @ v r @ v z ¼ ¼ ¼ 0 for r ¼ 0: @r @r @r
A time-dependent normalized exergy efficiency, gx ðtÞ, which compares the exergy accumulated in an actual operating SSTES with the exergy accumulated if the tanks operated according to the limiting modes described above, is defined as
ð9Þ
3.3. Exergy analysis The state of stratification in an SSTES is strongly timedependent. Fig. 3(a) shows realistic temperature profiles along the axis of the tank at different time instants. As stated earlier, the effect of the time-varying degrees of stratification in the SSTES on its performance is only quantifiable by an exergy analysis. However, two modes of operation define limiting bounds for the exergy performance of SSTES – the perfectly stratified mode in which exergy accumulation is maximum and the fully mixed mode in which exergy accumulation is minimum for equal inlet mass flow rates and temperatures. The exergy performance of a SSTES may therefore be measured on a scale ranging from zero to one, with the extremes on the scale corresponding to the fully mixed and perfectly stratified modes of operation, respectively.
DNa ðtÞ DNm ðtÞ DNs ðtÞ DNm ðtÞ
ð10Þ
where after any period of charging t; DNa ðtÞ is the exergy accumulated in an actual SSTES, DNm ðtÞ is the exergy accumulated in the SSTES if operating as fully mixed, and DNs ðtÞ is the exergy accumulated in the SSTES if operating as perfectly stratified. The normalized exergy efficiency, gx , is similar to the non-dimensional exergy, n, used by Cónsul et al. (2004) and Farmahini-Farahani (2012). However, the definition of gx , which assigns a zero value to the fully mixed store (with the least exergy accumulation) and a unity value to the perfectly stratified store (with the most exergy accumulation), is the reverse of the assignments made by n, and seemingly logically so. Consider the SSTES of volume V, initially filled with storage fluid of mass m, specific heat capacity, cp , density, q, and at uniform temperature T ini . During charging (i.e., t > 0), hot water at a constant temperature T h flows into top of the tank at a mass flow rate _ displacing cold fluid from the bottom of the tank simultanem, ously at the same rate. At any time t after the initiation of charging, the temperature distribution within the tank is Tðr; z; tÞ and the exergy accumulated in the tank, DNa ðtÞ, above that at the start, is given by
DNa ðtÞ ¼ Na ðtÞ Nð0Þ
ð11Þ
Na ðtÞ is the exergy content of the fluid in the tank at time t, given by N X T i ðtÞ Vi Na ðtÞ ¼ q cp ðT i ðtÞ T o Þ T o ln ð12Þ To i¼1 where T i ðtÞ is the temperature of the ith cell of volume V i at time t; T o is the reference temperature, and the summation is carried out over all N cells into which the store volume has been discretized. Nð0Þ is the exergy of the fluid initially in the tank given by
Nð0Þ ¼ m cp ðT ini T o Þ T o ln
T ini To
ð13Þ
Substituting Eqs. (12) and (13) into Eq. (11) results in
T ðtÞ DNa ðtÞ ¼ m cp ðTðtÞ T ini Þ T o ln T ini
ð14Þ
where
Fig. 3. Temperature profiles at different charging times, 0 ¼ t 0 < t 1 < t2 < t3 < t4 < < t end1 < tend , along the axis of (a) an actual operating SSTES tank, (b) perfectly stratified SSTES tank, and (c) fully mixed SSTES tank.
491
H.O. Njoku et al. / Solar Energy 136 (2016) 487–498
TðtÞ ¼
N 1 X T i ðtÞ V i V i¼1
ð15Þ
DNm ðtÞ ¼ Nm ðtÞ Nð0Þ
and
" T ðtÞ ¼ exp
N 1 X ln T i ðtÞV i V i¼1
DNs ðtÞ ¼ Ns ðtÞ Nð0Þ
ð16Þ
ð17Þ
ð18Þ
where u is the fraction of the tank volume that has been replaced at the time of consideration, given by
u¼
_ mt m
1
_ 6m for mt _ >m for mt
ð19Þ
i.e., beyond u ¼ 1, additional inflow of hot water into the perfectly stratified store will only displace an equal amount of water from the store, with the temperature of the store contents remain fixed at T h . With Eqs. (13), (18) and (19), the expression for DNs ðtÞ in Eq. (17) may be written as
Th DNs ðtÞ ¼ umcp ðT h T ini Þ T o ln T ini
by
_ p t ðT m ðtÞ T o Þ T o ln Nm ðtÞ ¼ mc
Ns ðtÞ is the exergy content of the perfectly stratified tank at time t, which is the sum of the exergies of the hot and cold fluid zones given by T Ns ðtÞ ¼ umcp ðT h T o Þ T o ln h To T ini þ ð1 uÞmcp ðT ini T o Þ T o ln To
ð21Þ
Nm ðtÞ is the exergy content of the fully mixed tank at time t, given
#
Physically, TðtÞ is the temperature that will exist in the SSTES tank if its contents were fully mixed at time t, while T ðtÞ is the storage fluid temperature that will exist in a fully mixed store having the same exergy content as the SSTES at time t (Cónsul et al., 2004; Rosen, 2001). When operating as perfectly stratified, only two uniform temperature zones exist in the tank, a hot zone whose temperature is equal to the charging fluid temperature, T h , and a cold zone at the initial tank temperature, T ini . The zones are separated by an insulating thermocline of infinitesimal thickness. Fig. 3(b) shows temperature profiles at different time instants along the axis of a perfectly stratified tank. In an actual storage operation, this scenario may be approximated by the insertion of a diaphragm to physically separate the incoming fluid stream from the outgoing one. The perfectly stratified operation will result in the highest exergy accumulation in the tank since entropy generation due to fluid mixing and heat exchange across fluid layers is practically nil. The exergy accumulated in the SSTES at time t, if operating as perfectly stratified, DNs ðtÞ, is given by
(
exergy accumulated in the fully mixed SSTES tank at time t; DNm ðtÞ, is given by
ð20Þ
Thus exergy accumulation in the perfectly stratified tank is a maximum at u ¼ 1. When operating as fully mixed, there is instantaneous temperature redistribution in the tank as the incoming fluid mixes completely with the resident fluid, resulting in a uniform temperature in the tank. Fig. 3(c) shows temperature profiles at different time instants along the axis of the fully mixed storage tank. In actual storage tank operations, this is approximated by very low PeD number scenarios for which there is sufficient time for diffusion mechanisms to result in temperature redistributions, and by high ReD flows accompanied by vigorous mixing due to turbulence. The fully mixed operation will result in the least exergy accumulation in the tank since the level of mixing present in the tank is accompanied by in the maximum entropy generation. The
T m ðtÞ To
ð22Þ
where T m ðtÞ is the temperature of the fully mixed tank at time t, given by
_ m T m ðtÞ ¼ T h ðT h T ini Þ exp t m
ð23Þ
With Eqs. (13), (22) and (23), the expression for DNm ðtÞ in Eq. (21) may be written as
T m ðtÞ DNm ðtÞ ¼ mcp T m ðtÞ T ini T o ln T ini
ð24Þ
Substituting Eqs. (14), (20) and (24) into Eq. (10), the normalized exergy efficiency may be expressed as
gx ðtÞ ¼
TðtÞ T m ðtÞ T o lnðT ðtÞ=T m ðtÞÞ
u1 uðT h T ini Þ ðT m ðtÞ T ini Þ T o lnðT uh =T m ðtÞT ini Þ
ð25Þ
In terms of dimensionless quantities,
h i h ðsÞþ#min hðsÞ 1 þ expðuÞ #0 ln #max expðuÞ u ð1uÞ gx ðsÞ ¼ # #min u 1 þ expðuÞ #0 ln #maxmaxexpð uÞ
ð26Þ
where, hðsÞ ¼ ðTðtÞ T ini Þ=ðT h T ini Þ is the dimensionless temperature in a fully mixed store having the same energy content as the SSTES at dimensionless time s; h ðsÞ ¼ ðT ðtÞ T ini Þ=ðT h T ini Þ is the dimensionless temperature in a fully mixed store having the same exergy content as the SSTES at dimensionless time s; #min ¼ T ini =ðT h T ini Þ and #max ¼ T h =ðT h T ini Þ are the minimum and maximum dimensionless temperature ratios, respectively, while #0 ¼ T 0 =ðT h T ini Þ is the reference dimensionless temperature ratio. 4. Results and discussion SSTES of 0.4 m diameter, inlet to tank diameter ratio, din =D ¼ 0:1, and containing water initially at 25 °C are considered. Parametric effects are studied by setting up different cases with different inlet temperatures and charging velocities, to obtain different Richardson number and Peclet number conditions, respectively, as well as different tank aspect ratios and tank wall heat loss coefficients. The ranges of the parameters considered in this study are summarized in Table 1. Inlet flow rates in the cases investigated were limited to the laminar range in order to avoid turbulence in the stores, which is undesirable when the aim is to maintain stratification.
Table 1 Range of parameters considered. Parameter
Range
Aspect ratio, AR Overall heat loss coefficient, U Peclet number, PeD
1–4 0–100 W m2 K1
Reynolds number, ReD Richardson number, Ri Max. dimensionless temperature ratio, #max Min. dimensionless temperature ratio, #min
5 103 –100 103 107–2885 10–105 5.58–12.92 4.58–11.92
492
H.O. Njoku et al. / Solar Energy 136 (2016) 487–498
4.1. Numerical solution of the STESS model The finite volume numerical solution of the STESS model was implemented using the OpenFOAM CFD C++ toolbox. A wedge shaped, conformal block-structured mesh was used, consisting of three blocks – a 6 10 cell block for the tank inlet pipe, a 120 750 cell block for the storage tank, and a 6 40 cell block for the tank outlet pipe, i.e., a total of 90,300 cells. The discretization of diffusive and convective terms were performed using second-order linear and upwind schemes, respectively, while temporal discretization was second-order Euler implicit with time steps automatically adjusted subject to Courant number limits of between 0.1 and 0.25. The ensuing algebraic equation sets were solved using the PCG (preconditioned conjugate gradient) solver in the PIMPLE (merged PISO-SIMPLE) algorithm (OpenFOAM, 2014). The convergence criteria set for the solution were a tolerance,
kþ1 ¼ jwkþ1 wk j 6 108
and a relative tolerance, kþ1 k 6 0, where w is any dependent variable. The discretization schemes described here were implemented using the buoyantPimpleFoam and buoyantBoussinesqPimpleFoam OpenFOAM utilities.
reversed flow rises to the top of the store, establishing a fluid circulation zone, which is stable at the top of the store for the entire duration of the charging. Flow within the thermocline is parallel to the store axis, whereas within the cold region of the store, the flow converges smoothly to the store’s exit. Evidently, the state of stratification within the store is continuously evolving with the charging duration, necessitating a performance measure which adequately accounts for these temporal distinctions in the states of stratification within the SSTES. Dimensionless temperature and flow profiles after a charging duration corresponding to u ¼ 0:42, for SSTES having different different combinations of the dimensionless parameters Ri, PeD and AR, are presented in Fig. 5(a)–(d). Some qualitative deductions may be made from the plots, viz., as PeD increases (comparing Fig. 5(a) and (b)) multiple fluid circulation zones appear in the hot region of the store; with decreasing Ri, (comparing Fig. 5(a) and (c)) a radial temperature gradient within the hot region becomes more evident; whereas as AR decreases, (comparing Fig. 5(a) and (d)) the thermocline occupies a greater proportion of the store height. These observed variations in SSTES operational characteristics also need to be adequately accounted for by the measures used in assessing SSTES performance.
4.2. Transient evolution of temperature and flow profiles 4.3. Comparison of energy vs exergy accumulations in STES The temperature and flow fields within a representative SSTES (with AR = 3, Ri = 103 and PeD ¼ 20 103 ) are presented in Fig. 4 (a)–(d) as dimensionless temperature profiles and flow streamlines at four instants during the charging of the SSTES. Two distinct temperature regions in the store are clearly visible after the commencement of charging, together with the separating thermocline region which travels down the store as the charging progresses. The streamlines show that the flow of the incoming charge is reversed just before it comes in contact with the thermocline. This is because the buoyancy force resulting from the temperature difference between the incoming charge and the resident fluid is greater than the inertial force of the charge entering the store. Hence, the flow of the incoming charge is unable to penetrate the thermocline and is reversed. Subsequently, the
When the total energy accumulation in the SSTES is considered, there is no difference between the energy accumulated in a real store and that in a perfectly stratified store for most of the charging duration of the stores. This shown in Fig. 6 for a SSTES operating under
the
conditions,
AR = 3,
PeD ¼ 20 103 ,
Ri = 103
and
1
U ¼ 0 W m K . This is because, until the thermocline arrives at the exit of the SSTES, all the thermal energy accompanying the charging stream into the store is retained within it, just as obtains in a perfectly stratified store for 0 6 u 6 1. Thereafter, the outgoing stream will be accompanied by some of the input energy. In contrast, some of the thermal energy charged into a fully mixed store is carried out by the outgoing stream from the onset of charging, hence the lower value of accumulated energy in a fully mixed 2
Fig. 4. Temperature profiles and velocity streamlines in a typical stratified thermal store (AR = 3, Ri = 102, PeD ¼ 20 103 ) after tank volume change fractions u ¼ (a) 0.20, (b) 0.42, (c) 0.65, and (d) 0.87.
H.O. Njoku et al. / Solar Energy 136 (2016) 487–498
493
Fig. 5. Temperature profiles and velocity streamlines after tank volume change fraction, u ¼ 0:42 in stratified thermal stores having (a) Ri = 103 , PeD ¼ 20 103 , AR = 3; (b) Ri = 103 , PeD ¼ 60 103 , AR = 3; (c) Ri = 102 , PeD ¼ 20 103 , AR = 3; (d) Ri = 103 , PeD ¼ 20 103 , AR = 1.
Fig. 6. The accumulation of energy in a perfectly stratified store, a fully mixed store and the actual SSTES. (AR = 3, PeD ¼ 20 103 , Ri = 103 ; U ¼ 0 W m2 K1 ).
store shown in Fig. 6. An attempt to assess the SSTES using an energy efficiency, which essentially compares the energy accumulated in the store to that in perfectly stratified store, posses a problem because for most of its charging duration this will result in a 100% efficiency. The energy efficiency measure thus seems to suggest that no loss is incurred during the charging process and that charging duration has no influence on overall performance. It will therefore be impossible to measure how performance progresses with charging duration on this basis. Again, since all SSTES configurations produce this result, it will be impossible to make distinctions in the performances of differently configured SSTES (different AR, PeD , Ri, etc.) on this basis.
The plots of exergy accumulations for the same stores are shown in Fig. 7. Clearly, the exergy accumulation in the SSTES differs from both of those in the perfectly stratified store and the fully mixed store, for the entire duration of charging. This is in keeping earlier conclusions that an exergy-based measure of assessment is more desirable. One such measure is the exergy efficiency, which, similar to the energy efficiency, compares the exergy accumulated in the SSTES with that in a perfectly stratified store. While this measure recognizes the existence of an upper bound on exergy accumulation in the store, which is achieved in the perfectly stratified store, and thus measures ‘‘how nearly the operation of [the] system approaches the . . . theoretical upper limit” (Rosen, 2001),
494
H.O. Njoku et al. / Solar Energy 136 (2016) 487–498
Fig. 7. The accumulation of exergy in a perfectly stratified store, a fully mixed store and the actual SSTES. (AR = 3, PeD ¼ 20 103 , Ri = 103 ; U ¼ 0 W m2 K1 ).
the existence of a lower bound (corresponding to the situation in a fully mixed store) is overlooked. In a real store, the exit temperature is raised by fluid mixing in the store, resulting in a lower temperature difference at the heat source and thus a longer charging period. The normalized exergy efficiency, gx , defined in Eq. (10), in addition to measuring the deviation from the upper limit, also evaluates the deviation of the system’s performance from the theoretical lower limit, which is of equal importance. It takes both bounds into consideration and is used in succeeding discussions. 4.4. Normalized exergy efficiency, gx under different charging rate regimes The plots of the time-varying normalized exergy efficiencies,
gx ðsÞ, for different Peclet numbers are shown for Ri ¼ 102 , in Fig. 8(a) and for Ri ¼ 103 in Fig. 8(b). The shapes of the plots are essentially the same, with gx ðsÞ rising gradually from the start of charging (u ¼ 0) till it peaks at u 0:90, and drops sharply thereafter. At any u; gðsÞ increases sharply with PeD at low PeD values 3
(< 30 10 ) but thereafter remains fairly constant as PeD continues to increase. A range of PeD values for peak SSTES tank performance was also reported by Ji and Homan (2007). They found that when storage performance was assessed using a discharge volume fraction, V d , computed based on the simplified Uniform Diffusivity SSTES tank model of Cole and Bellinger (1982) and the NonUniform Diffusivity SSTES tank model of Zurigat et al. (1991), V d was found to be maximum at Peclet numbers between 1 103 and 3 103 . (A Peclet number based on tank height, Pe ¼ U 1 H=a, was used by Ji and Homan (2007). For PeD based on tank diameter, as used in this study, and with AR = 3, this optimal PeD range will be equivalent to 33:3 103 –100 103 , which is consistent with the current observation.) Bearing in mind that gx ðsÞ compares processes in an actual SSTES with those in a perfectly stratified unit (the best case scenario) and those in a fully mixed unit (the worst case scenario), we observe that for most of the charging period, gx ðsÞ is greater than 0.5, implying that the behavior of the actual SSTES does approach that of a perfectly stratified unit than that of a fully mixed unit. Since heat loss through store walls have been ignored, the
departure from the perfectly stratified behavior is due to entropy generations resulting from fluid mixing and heat transfer between the incoming fluid stream and the fluid ab initio in the tank which is mainly confined to the thermocline region in the actual tank. Whereas these processes occurs everywhere in a fully mixed tank, they are completely absent in a perfectly stratified tank. As the charging of a SSTES progresses, a time is reached when some of the exergy accumulated in the tank starts being conveyed out by the out-going fluid stream – this is the situation in a fully mixed unit from the very onset of charging, but never occurs in a perfectly stratified tank until u is unity. An increasing fraction of accumulated exergy will be carried out of the SSTES in this way as u approaches and goes beyond unity. Hence, we observe from these plots peaks in gx ðsÞ; gx;max , occurring at u 0:90, beyond which charging is increasingly dominated by the fully mixed behavior. 4.5. Normalized exergy efficiency, gx for different PeD and Ri When gx ðsÞ for different Ri cases are considered, as shown in Fig. 9, it is observed that gx ðsÞ generally increases as Ri increases. This is because stabilizing buoyancy forces in the SSTES increase when the characteristic temperature difference in the store, ðT h T ini Þ, and hence Ri in the store, increases. Thus, a store’s behavior approaches that of the perfectly stratified store as Ri increases. At small Ri values, huge gains in gx ðsÞ are obtained as Ri increases, however, beyond Ri = 103, the changes in gx ðsÞ with increasing Ri are no longer significant. Peak normalized exergy efficiencies, gx;max , for the different Ri are observed at u 0:90 as were observed also in the previous section. The timing of the occurrence of gmax seems therefore to be independent of PeD and Ri. gx;max may be utilized as a criteria to determine when to cease charging a SSTESS if the system design objective is the optimization of the exergy stream from the heat source. This will be justified if a system were to depend on expensive (fossil fuel based) energy sources and simultaneously operate as an open system in which the out-going fluid stream is not returned to the heat source. For most practical systems, which are closed, and systems based on solar thermal, geothermal, waste heat, or any other such inexpensive heat sources, the design criteria will usually be to get the store contents to a uniform temperature equal or close to the charging
H.O. Njoku et al. / Solar Energy 136 (2016) 487–498
495
Fig. 8. Evolution of normalized exergy efficiencies, gx , over the charging duration, u, for cases with different PeD . (AR = 3, U ¼ 0 W m2 K1 ).
temperature, i.e., hðsÞ ¼ h ðsÞ ! 1. Whatever the objective is, gx;max may be used for comparing the performances of different SSTES, since for differently configured systems, gx;max occurs after seemingly identical charging durations. Whereas the u corresponding to gmax seems to be invariant with PeD and Ri, gx;max varies with both PeD and Ri as shown by the surface plot of their dependencies presented in Fig. 10. gx;max increases with both PeD and Ri. There is a sharp increase in gx;max with increasing PeD and Ri at low values of both. But we observe that the rate of increase with Ri is steeper than that with Ri at low values of both. Beyond Ri = 102, the change in gx;max with increasing Ri is not appreciable, suggesting that there is a limit to which improvements in gx;max may be obtained by increasing Ri. gx;max increases continuously with PeD within the range of PeD values considered and thus has a more general influence on gx;max than Ri.
4.6. Effect of aspect ratio on normalized exergy efficiency The results presented thus far have been for SSTES with AR = 3. Now we consider the influence of AR on gx;max . The destruction of exergy in the SSTES is due to the mixing of the incoming hot fluid with the resident cold fluid, which is accompanied by the degradation of thermal gradients and these occur primarily in the thermocline. As the AR increases, the proportion of the store’s volume occupied by the thermocline reduces and thus there is a reduction in the proportion of accumulated exergy that is destroyed. Expectedly therefore, from the surface plot of Fig. 11, we observe gx;max increasing monotonously as AR increases within the range of ARs considered. Whereas this may suggest that AR may be increased indefinitely in order to achieve higher performance, the choice of appropriate AR, as noted by Lavan and Thompson (1977), will be
496
H.O. Njoku et al. / Solar Energy 136 (2016) 487–498
Fig. 9. Evolution of normalized exergy efficiencies, gx , over the charging duration, u, for different cases with different Ri. (AR = 3, PeD ¼ 20 103 ; U ¼ 0 W m2 K1 ).
Fig. 10. Effect of PeD and Ri on maximum normalized exergy efficiencies, gx;max (AR = 3, U ¼ 0 W m2 K1 ).
Fig. 11. The effects of Peclet number, PeD , and aspect ratio, AR on the maximum normalized exergy efficiency, gx;max . (U ¼ 0 W m2 K1 and Ri = 103, i.e., #max ¼ 7:62; #min ¼ 6:62).
H.O. Njoku et al. / Solar Energy 136 (2016) 487–498
497
Fig. 12. Evolution of normalized exergy efficiencies, gx , over the SSTES charging duration, u, for cases with different PeD and levels of insulation. (AR = 3 and Ri = 103 , i.e., #max ¼ 7:62; #min ¼ 6:62).
a trade-off between performance and cost, with an AR between 3 and 4 seeming reasonable. Furthermore, the increases in gx;max with AR are more significant at low AR and PeD values, and the gx;max profile flattens out at higher values of both. 4.7. Effect of heat losses The influence of the presence of heat losses across the store walls on the gx , was studied by relaxing the assumption of perfect store wall insulation. Uniform heat loss coefficients, U, were imposed on the store walls, varying from U ¼ 0 W m2 K1 to 100 W m2 K1 , with PeD fixed at 40 103 and Ri = 103. Fig. 12 shows the variation of gx over the charging duration for different U values. Generally, gx rises rapidly initially as charging commences. Due to the exergy accompanying heat losses across the walls and the additional entropy generation due to the heat losses, the cases with finite heat losses present peaks in gx , a short while after the commencement of charging. Reductions in gx are observed beyond these peaks, which are as a result of increasing proportions of the input exergy being lost through the walls and the accompanying entropy generation due to the heat losses. A second dip in gx is also observed at 0:95 K u K 1:0. As described earlier, this coincides with the exit of the thermocline from the stores. In addition, as U increases, gx falls
distinctions for periods of charging within 0 6 u 6 1. By using gx , SSTES are compared with both the perfectly stratified and the fully mixed stores, and thus SSTES performances are placed in the proper context with reference to the best possible and the worst possible STES performances, respectively. This analysis, which incorporates second law considerations, is superior to those that account for only first law effects. The influence of AR, PeD and Ri on gx was also investigated. As AR increased four folds, gx ;max increased by 4.6–7.7% suggesting that slender tanks are to be preferred to large diameter tanks. Beyond aspect ratios of between 3 and 4, however, the improvements in gx ;max with increasing AR are no longer significant. Also, within the range of conditions considered in this study, a twenty fold increase in PeD resulted in an increase in gx ;max of 7.2–16.8%, while a thousand fold increase in Ri led to a 4.3–13.6% increase in gx ;max . The maximum possible temperature difference needs to be maintained between the incoming charge and the resident fluid in the store for improved stratification. High Richardson numbers correspond with laminar conditions in the SSTES since Ri is related
monotonically, such that cases with U ¼ 0 W m2 K1 represent upper limits of actual SSTES performance. Though the drop in gx isn’t appreciable at low U values, there is need nonetheless for proper insulation since the effects considered here only apply to the charging stage and not the storage phase when heat losses would occur over extended resident times.
to the Reynolds number by Ri = Gr/Re2D , where Gr is the Grashoff number. Under these conditions in which stratification does not breakdown in the store, higher PeD (hence faster charging rates) are desirable in order to minimize the contact time for conduction across the thermocline. Such conduction leads to a thickening of the thermocline and hence a deterioration of stratification. The use of inlet modifier is beneficial in this regard, in order to accommodate faster charging rates in SSTES. The steepest increases in gx ;max due to increases in the dimensionless parameters occurred at the lower values of the parameters.
5. Conclusion
Acknowledgements
The extent and state of thermal stratification in sensible thermal energy stores during charging, and hence the stores’ performances, vary depending on the conditions at store inlets, the stores’ geometries, and the extent of charging. Clear distinctions between the performances of SSTES under these different conditions are not very clear on the basis of energy considerations alone. The normalized exergy efficiency, gx , has been utilized in this study, which is shown to be capable of providing the required
Computing facilities of the University of Nigeria-UNESCO-HP Grid Computing Brain Drain-to-Brain Gain Initiative and of the Abdus-Salam International Centre Theoretical Physics (ICTP), Trieste, Italy were employed in this study. We are grateful to Dr. Collins Udanor of the Computer Science Department, University of Nigeria, and to Prof. Joe Niemela, Dr. Clement Onime and Ivan Girotto of the ICTP, for facilitating access granted the first author to these facilities.
498
H.O. Njoku et al. / Solar Energy 136 (2016) 487–498
References Abdoly, M.A., Rapp, D., 1982. Theoretical and experimental studies of stratified thermocline storage of hot water. Energy Convers. Manage. 22, 275–285. Abu-Hamdan, M.G., Zurigat, Y.H., Ghajar, A.J., 1992. An experimental study of a stratified thermal storage under variable inlet temperature for different inlet designs. Int. J. Heat Mass Transf. 35 (8), 1927–1934. Altuntop, N., Arslan, M., Ozceyhan, V., Kanoglu, M., 2005. Effect of obstacles on thermal stratification in hot water storage tanks. Appl. Therm. Eng. 25, 2285– 2298. Castell, A., Medrano, M., Sol, C., Cabeza, L.F., 2010. Dimensionless numbers used to characterize stratification in water tanks for discharging at low flow rates. Renew. Energy 35, 2192–2199. Cole, R.L., Bellinger, F.O., 1982. Natural Thermal Stratification in Tanks. Phase 1 Final Report, Report No. ANL-82-5. Cónsul, R., Rodríguez, I., Pérez-Segarra, C.D., Soria, M., 2004. Virtual prototyping of storage tanks by means of three-dimensional CFD and heat transfer numerical simulations. Sol. Energy 77, 179–191. Cristofari, C., Notton, G., Poggi, P., Louche, A., 2003. Influence of the flow rate and the tank stratification degree on the performances of a solar flat-plate collector. Int. J. Therm. Sci. 42, 455–469. Eames, P.C., Norton, B., 1998. The effect of tank geometry on thermally stratified sensible heat storage subject to low reynolds number flows. Int. J. Heat Mass Transf. 41 (14), 2131–2142. Farmahini-Farahani, M., 2012. Investigation of four geometrical parameters on thermal stratification of cold water tanks by exergy analysis. Int. J. Exergy 10 (3), 332–345. Hahne, E., Chen, Y., 1998. Numerical study of flow and heat transfer characteristics in hot water stores. Sol. Energy 64 (1–3), 9–18. Haller, M.Y., Cruickshank, C.A., Streicher, W., Harrison, S.J., Andersen, E., Furbo, S., 2009. Methods to determine stratification efficiency of thermal energy storage processes – review and theoretical comparison. Sol. Energy 83, 1847– 1860. Hollands, K.G.T., Lightstone, M.F., 1989. A review of low-flow, stratified-tank solar water heating systems. Sol. Energy 43, 97–105. Ismail, K.A.R., Leal, J.F.B., Zanardi, M.A., 1997. Models of liquid storage tanks. Energy 22 (8), 805–815.
Ji, Y., Homan, K.O., 2007. On simplified models for the rate- and time-dependent performance of stratified thermal storage. J. Energy Resour. Technol. 129 (4), 214–222, September, Transactions of the ASME. Lavan, Z., Thompson, J., 1977. Experimental study of thermally stratified hot water storage tanks. Sol. Energy 19, 519–524. Njoku, H.O., Ekechukwu, O.V., Onyegegbu, S.O., 2014. Analysis of stratified thermal storage systems: an overview. Heat Mass Transf. 50 (7), 1017–1030. http://dx. doi.org/10.1007/s00231-014-1302-8. OpenFOAM, 2014.
Contents lists available at ScienceDirect
Solar Energy journal homepage: www.elsevier.com/locate/solener
Normalized charging exergy performance of stratified sensible thermal stores H.O. Njoku ⇑, O.V. Ekechukwu 1, S.O. Onyegegbu Department of Mechanical Engineering, University of Nigeria, Nsukka 410001, Nigeria
a r t i c l e
i n f o
Article history: Received 25 May 2015 Received in revised form 28 December 2015 Accepted 18 July 2016
Keywords: Stratified thermal storage CFD analysis Exergy performance analysis Normalized exergy efficiency
a b s t r a c t This paper presents the performance assessment of stratified sensible thermal energy stores (SSTES) on the basis of the normalized exergy efficiency, gx . Assessments based on gx provide comparisons with performances of both the perfectly stratified and the fully mixed stores, which offer the best and worst performances, respectively. This is in contrast with energy and exergy efficiencies, which compare SSTES with only the perfectly stratified store. A dimensionless unsteady axisymmetric model of vertical cylindrical SSTES was implemented using a finite volume numerical scheme. The effect of some significant parameters on SSTES performance were considered by performing computations for aspect ratios (AR) between 1 and 4, Peclet number (PeD ) varying from 5 103 to 100 103 , Richardson number (Ri) varying from 10 to 104, and overall heat loss coefficients (U) varying from 0 to 100 W m2 K1. gx increases with PeD , Ri and AR, with the most significant increases occurring at low values of these parameters, and appreciable increases are no longer obtained beyond PeD 30 103 , Ri 103 and AR 3. gx also falls monotonically as the U values increase. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction During the charging and discharging of a stratified thermal energy store, different regions are at different temperatures due to the action of buoyancy forces. When charging a stratified hot water storage tank, for example, hot water flows into the tank as the water in the tank is simultaneously withdrawn for heating. The difference in the densities of the incoming hot water and the cooler water already in the tank causes the hot water to rise to the top and the cooler water to fall to the bottom of the storage tank, creating a transition layer, known as the thermocline, which separates the hot upper zone from the cold lower zone. The same situation obtains, albeit in the reverse, during the discharging of the tank, when hot water is withdrawn from the tank to service a load while cooler water is simultaneously let into the tank from the mains. An illustration of the typical temperature profile existing in a stratified thermal storage tank is presented in Fig. 1. Compared to a situation in which the liquid storage media (water, in this case) is fully mixed during the charging process, the storage tank’s performance is improved by the presence of ⇑ Corresponding author. E-mail addresses: [email protected] (H.O. Njoku), [email protected] (O.V. Ekechukwu), [email protected] (S.O. Onyegegbu). 1 Formerly at the National Universities Commission, 26 Aguiyi-Ironsi Street, Maitama District, Abuja FCT, Nigeria. http://dx.doi.org/10.1016/j.solener.2016.07.032 0038-092X/Ó 2016 Elsevier Ltd. All rights reserved.
temperature stratification in several ways. If the storage tank is connected in closed circuit to a heat source and stratification maintained during charging, a shorter charging period will be required to extract an equal amount of heat in comparison with a fully mixed tank. This is so because the return storage fluid stream from the stratified tank is at a quasi-constant low temperature as it passes through the heat source, thus creating a higher heat transfer gradient at the heat source. For a solar collector heat source, for example, this will translate to higher mean collector heat output and thus higher efficiencies (Abu-Hamdan et al., 1992; Cristofari et al., 2003; Hollands and Lightstone, 1989). During discharge also, more heat will be available for supply to the load because of the quasi-constant high temperature at which the storage fluid is withdrawn from the storage tank to service the load.
2. Background In order to quantify the extent of improvements obtainable in sensible thermal energy stores (STES) due to the presence of stratification, objective and rational performance assessment criteria are needed. Several of such criteria have been proposed in the literature, including familiar dimensionless numbers, efficiency measures, and other miscellaneous parameters. Some are solely based on first law (energy) considerations, others on second law (entropy) considerations, while the rest result from a combination
488
H.O. Njoku et al. / Solar Energy 136 (2016) 487–498
Nomenclature Ain AR cp D DE g H m _ m P PeD r ReD Ri t T T T U U1 v V Vd z
inlet cross-sectional area (m2) aspect ratio (=H/D) constant pressure specific heat (J kg1 K1) tank diameter (m) energy accumulation (J) gravitational acceleration (m s2) tank height (m) mass of tank content (kg) inlet mass flow rate (m) pressure (N m2) Peclet number (¼ U 1 D=a) radial distance (m) Reynolds number (¼ U 1 D=m) Richardson number (¼ gbHðT T ini Þ=U 21 ) time (s) temperature (K) equivalent energy temperature (K) equivalent exergy temperature (K) overall heat loss coefficient (W m2 K1 ) _ qAin m s1) free stream velocity (¼ m= velocity (m s1) tank volume (m3) discharge volume fraction axial distance (m)
DN gx ðtÞ
gx;max u m q h h h #
s
N
exergy accumulation (J) normalized exergy efficiency maximum normalized exergy efficiency _ volume change fraction (¼ mt=m) kinematic viscosity (m2 s1) density (kg m3) dimensionless temperature (¼ ðT T ini Þ=ðT h T ini Þ) dimensionless equivalent energy temperature dimensionless equivalent exergy temperature dimensionless temperature ratio (¼ T=ðT h T ini Þ) dimensionless time (¼ U 1 t=D) exergy (J)
Subscripts and superscripts 0 reference state h hot in inlet ini initial m fully mixed max maximum min minimum r radial direction s perfectly stratified z axial direction dimensionless quantities
Greek symbols a thermal diffusivity (m2 s1) b compressibility coefficient (K1)
of first and second law considerations (exergy analysis). Reviews of these assessment criteria are well discussed in the literature (Zurigat and Ghajar, 2007; Haller et al., 2009; Castell et al., 2010; Njoku et al., 2014). The incorporation of second law considerations, (e.g., in the form of exergy analysis,) into the assessment of stratified sensible thermal energy stores (SSTES) has however been shown to be more rational and illuminating than assessments based only on energy analysis (Rosen, 2001; Rosen and Dincer, 2003; Rosen et al., 2004). Therefore, in addition to energy efficiencies, exergy efficiencies, which incorporate second law considerations, have been applied to the assessment of SSTES (e.g., Rosen, 2001; Solé et al., 2008). The analysis of SSTES is best undertaken using computer aided multi-dimensional numerical models (otherwise known as computational fluid dynamics (CFD)), as these better account for the inherently multi-dimensional temperature and flow regimes developed within the SSTES during its operation. Also, the number of simplifying assumptions made are limited, unlike what obtains with analytical models. The use of multi-dimensional numerical models in the study of SSTES, however, has traditionally focused on the simulation of temporal evolution of temperature and velocity profiles within the storage units (Eames and Norton, 1998; Yee and Lai, 2001; Zachar et al., 2003; Shin et al., 2004), the determination thereby, of thermal energy accumulation in the units, and the computation of energy based efficiencies (Abdoly and Rapp, 1982; Ismail et al., 1997; Hahne and Chen, 1998; Shah et al., 2005). In such studies therefore, the influence of such factors as inlet flow rate and temperature, tank aspect ratio, and the presence of various stratifier mechanisms, on the efficiency of energy accumulation in SSTES have been investigated. Ismail et al. (1997) developed a 2D CFD model which was used to simulate temperature profiles within a SSTES during charging and discharging and obtained temperature profiles which showed close agreement with
experimental results. Using a 3D CFD model, Shah et al. (2005) investigated the effectiveness of a thermal stratifier within a SSTES. Their model results compared well with experimental PIV and temperature measurements. They computed stratifier efficiencies, (ratios of actual energy supplied to the tank to the maximum possible energy supplied to the tank in the absence of mixing,) and indicated that a range of optimal flow rates – 5–8 l/min existed for the SSTES configurations they studied. Results of multi-dimensional numerical simulation studies have also led to better insights into the processes that lead to both the enhancement of and the breakdown of stratification in SSTES. Such studies have shown that placing obstacles close to the entrances or exits invariably enhances temperature stratification in SSTES (Zachar et al., 2003; Altuntop et al., 2005). 2D CFD simulation studies of Van Berkel (1996) and Van Berkel et al. (1999) have shown that destratification caused by mixing in SSTES thermocline regions can be approximated by a two stage process of fluid withdrawal from the thermocline by drag, and subsequent mixing due to the stretching and folding of fluid particles. A limited number of studies in which exergy methods and multi-dimensional numerical models were simultaneous used, have been reported. Time varying temperature profiles in a horizontal SSTES tank were obtained by Cónsul et al. (2004) using 3D numerical simulations. The performances of the tank when assessed using a ‘‘non-dimensional thermocline thickness”, the ‘‘MIX number” and the ‘‘non-dimensional exergy” were determined and compared, leading to the conclusion that the nondimensional exergy gave the best assessment of the tank’s performance. Farmahini-Farahani (2012) performed a 2D CFD study of the effect of tank aspect ratios (AR), inlet/outlet geometries (diameter, vertical position and inclination) on stratification in SSTES tanks. Stratification was measured using a dimensionless exergy parameter and was found to improve with higher AR, smaller
H.O. Njoku et al. / Solar Energy 136 (2016) 487–498
489
Fig. 2. Simplified axisymmetric representation of stratified sensible thermal storage tank.
Fig. 1. Typical temperature profile in a stratified thermal storage tank.
velocities in the radial and axial directions, respectively, where D, the tank diameter, is chosen as the characteristic length scale. _ qAin , the bulk inlet fluid velocity is the characteristic velocU 1 ¼ m= _ is the inlet mass flow rate, q, the fluid density and ity scale, where m Ain , the inlet cross-sectional area.
inlet/outlet diameters, and by inclining the inlets/outlets away from the top/bottom of the tanks. Shah and Furbo (2003) also used CFD simulations to study the behavior of SSTES when three different inlet designs were used. From these, they obtained time varying temperature profiles in the store with the different inlet designs. Based on limited experiments, they also computed exergy and entropy efficiencies, showing how these were influenced by Richardson number, draw-offs and initial conditions in the store. In the present work, by a non-dimensional presentation of the governing conservation equations, the major dimensionless parameters that influence the thermal and fluid flow regimes in the SSTES are identified. With the time-varying temperatures in the SSTES obtained by the CFD solutions of these equations, the exergy-based performance of the SSTES is assessed by computing its normalized exergy efficiency. This is done for a wide range of the dimensionless parameters to evaluate the parametric dependence of SSTES’ exergy performance on these parameters.
In Eq. (2), h ¼ ðT T ini Þ=ðT h T ini Þ is the time-dependent dimensionless temperature distribution, where Tðr; z; tÞ is the timedependent temperature distribution in the tank, and T h T ini , the temperature difference between inlet fluid and the initial fluid in the tank, is the characteristic temperature scale. PeD ð¼ U 1 D=aÞ is the Peclet number, where a is the fluid thermal diffusivity, while sð¼ U1 t=DÞ is the dimensionless time.
3. Analysis
3.1.3. Momentum conservation equations
3.1. SSTES tank mathematical model A schematic of the SSTES tank under consideration is shown in Fig. 2. The fluid temperature and flow fields in the SSTES tank are governed by the mass, energy and momentum conservation equations. Dimensionless forms of these conservation equations in cylindrical co-ordinates are presented in Eqs. (1)–(4), highlighting the dimensionless groups which influence the performance of the SSTES unit. 3.1.1. Continuity equation
1 @ r v ðr Þ @r
r
þ
v
@ z @z
¼0
ð1Þ
In Eq. (1), r ¼ r=D and z ¼ z=D are the dimensionless radial and axial distances, v r ¼ v r =U 1 and v z ¼ v z =U 1 are the dimensionless
3.1.2. Energy conservation equation
" # @h 1 1 @ @2h @h @h @h þ 2 þ vr þ vz ¼ r @s @r @z PeD r @r @r @z
" # @ v r 1 @ 1 @r v r @ 2 v r @P @v r @v r þ þ vr þ vz ¼ ReD @r r @r @s @r @z @r @z2
ð2Þ
ð3Þ
" # @ v z @v @v 1 1 @ @ 2 v z @P Ri @v z þ þ þ v r z þ v z z ¼ r 2 ReD r @r AR @s @r @z @r @z @z ð4Þ Momentum conservation in the radial and axial directions are expressed by Eqs. (3) and (4), respectively, where ReD ð¼ U 1 D=mÞ is the Reynolds’ number based on the tank diameter D; P ð¼ P=ðqU 21 ÞÞ is the dimensionless pressure, Rið¼ gbHðT T ini Þ=U 21 Þ is the Richardson’s number, and ARð¼ H=DÞ is the tank aspect ratio. m and b are the fluid kinematic viscosity and compressibility coefficient, respectively. In the development of the STESS model above, it is assumed that flow in the tank is axisymmetric, there are no internal sources of
490
H.O. Njoku et al. / Solar Energy 136 (2016) 487–498
heat generation, and only gravitational body forces, parallel to the z axis, act on the system. All fluid properties are determined at the mean of the inlet and initial temperatures. 3.2. Initial and boundary conditions
gx ðtÞ ¼
Initially, the SSTES tank contents are uniformly at rest at the initial temperature, i.e.,
h ¼ v r ¼ v z ¼ 0 at
s ¼ 0:
ð5Þ
During the operation of the tank, hot fluid at fixed uniform temperature and velocity flows in at the inlet boundary, viz.,
9 h¼1 > = v r ¼ 0 for > ; vz ¼ 1
s > 0; r < rin ; z ¼ 0;
ð6Þ
while downstream to the outlet, fully developed flow is assumed:
@h @ v r @ v z ¼ ¼ ¼ 0 for r < rin ; z AR @z @z @z
ð7Þ
The no-slip condition is specified on all the tank’s walls, i.e.,
v r ¼ v z
8 > < r > r in ; z ¼ 0 ðtopÞ r > r in ; z ¼ AR ðbottomÞ ¼ 0 for > : r ¼ 0:5; 0 6 z 6 AR ðsideÞ
ð8Þ
The assumption of axisymmetry implies that along the tank axis,
@h @ v r @ v z ¼ ¼ ¼ 0 for r ¼ 0: @r @r @r
A time-dependent normalized exergy efficiency, gx ðtÞ, which compares the exergy accumulated in an actual operating SSTES with the exergy accumulated if the tanks operated according to the limiting modes described above, is defined as
ð9Þ
3.3. Exergy analysis The state of stratification in an SSTES is strongly timedependent. Fig. 3(a) shows realistic temperature profiles along the axis of the tank at different time instants. As stated earlier, the effect of the time-varying degrees of stratification in the SSTES on its performance is only quantifiable by an exergy analysis. However, two modes of operation define limiting bounds for the exergy performance of SSTES – the perfectly stratified mode in which exergy accumulation is maximum and the fully mixed mode in which exergy accumulation is minimum for equal inlet mass flow rates and temperatures. The exergy performance of a SSTES may therefore be measured on a scale ranging from zero to one, with the extremes on the scale corresponding to the fully mixed and perfectly stratified modes of operation, respectively.
DNa ðtÞ DNm ðtÞ DNs ðtÞ DNm ðtÞ
ð10Þ
where after any period of charging t; DNa ðtÞ is the exergy accumulated in an actual SSTES, DNm ðtÞ is the exergy accumulated in the SSTES if operating as fully mixed, and DNs ðtÞ is the exergy accumulated in the SSTES if operating as perfectly stratified. The normalized exergy efficiency, gx , is similar to the non-dimensional exergy, n, used by Cónsul et al. (2004) and Farmahini-Farahani (2012). However, the definition of gx , which assigns a zero value to the fully mixed store (with the least exergy accumulation) and a unity value to the perfectly stratified store (with the most exergy accumulation), is the reverse of the assignments made by n, and seemingly logically so. Consider the SSTES of volume V, initially filled with storage fluid of mass m, specific heat capacity, cp , density, q, and at uniform temperature T ini . During charging (i.e., t > 0), hot water at a constant temperature T h flows into top of the tank at a mass flow rate _ displacing cold fluid from the bottom of the tank simultanem, ously at the same rate. At any time t after the initiation of charging, the temperature distribution within the tank is Tðr; z; tÞ and the exergy accumulated in the tank, DNa ðtÞ, above that at the start, is given by
DNa ðtÞ ¼ Na ðtÞ Nð0Þ
ð11Þ
Na ðtÞ is the exergy content of the fluid in the tank at time t, given by N X T i ðtÞ Vi Na ðtÞ ¼ q cp ðT i ðtÞ T o Þ T o ln ð12Þ To i¼1 where T i ðtÞ is the temperature of the ith cell of volume V i at time t; T o is the reference temperature, and the summation is carried out over all N cells into which the store volume has been discretized. Nð0Þ is the exergy of the fluid initially in the tank given by
Nð0Þ ¼ m cp ðT ini T o Þ T o ln
T ini To
ð13Þ
Substituting Eqs. (12) and (13) into Eq. (11) results in
T ðtÞ DNa ðtÞ ¼ m cp ðTðtÞ T ini Þ T o ln T ini
ð14Þ
where
Fig. 3. Temperature profiles at different charging times, 0 ¼ t 0 < t 1 < t2 < t3 < t4 < < t end1 < tend , along the axis of (a) an actual operating SSTES tank, (b) perfectly stratified SSTES tank, and (c) fully mixed SSTES tank.
491
H.O. Njoku et al. / Solar Energy 136 (2016) 487–498
TðtÞ ¼
N 1 X T i ðtÞ V i V i¼1
ð15Þ
DNm ðtÞ ¼ Nm ðtÞ Nð0Þ
and
" T ðtÞ ¼ exp
N 1 X ln T i ðtÞV i V i¼1
DNs ðtÞ ¼ Ns ðtÞ Nð0Þ
ð16Þ
ð17Þ
ð18Þ
where u is the fraction of the tank volume that has been replaced at the time of consideration, given by
u¼
_ mt m
1
_ 6m for mt _ >m for mt
ð19Þ
i.e., beyond u ¼ 1, additional inflow of hot water into the perfectly stratified store will only displace an equal amount of water from the store, with the temperature of the store contents remain fixed at T h . With Eqs. (13), (18) and (19), the expression for DNs ðtÞ in Eq. (17) may be written as
Th DNs ðtÞ ¼ umcp ðT h T ini Þ T o ln T ini
by
_ p t ðT m ðtÞ T o Þ T o ln Nm ðtÞ ¼ mc
Ns ðtÞ is the exergy content of the perfectly stratified tank at time t, which is the sum of the exergies of the hot and cold fluid zones given by T Ns ðtÞ ¼ umcp ðT h T o Þ T o ln h To T ini þ ð1 uÞmcp ðT ini T o Þ T o ln To
ð21Þ
Nm ðtÞ is the exergy content of the fully mixed tank at time t, given
#
Physically, TðtÞ is the temperature that will exist in the SSTES tank if its contents were fully mixed at time t, while T ðtÞ is the storage fluid temperature that will exist in a fully mixed store having the same exergy content as the SSTES at time t (Cónsul et al., 2004; Rosen, 2001). When operating as perfectly stratified, only two uniform temperature zones exist in the tank, a hot zone whose temperature is equal to the charging fluid temperature, T h , and a cold zone at the initial tank temperature, T ini . The zones are separated by an insulating thermocline of infinitesimal thickness. Fig. 3(b) shows temperature profiles at different time instants along the axis of a perfectly stratified tank. In an actual storage operation, this scenario may be approximated by the insertion of a diaphragm to physically separate the incoming fluid stream from the outgoing one. The perfectly stratified operation will result in the highest exergy accumulation in the tank since entropy generation due to fluid mixing and heat exchange across fluid layers is practically nil. The exergy accumulated in the SSTES at time t, if operating as perfectly stratified, DNs ðtÞ, is given by
(
exergy accumulated in the fully mixed SSTES tank at time t; DNm ðtÞ, is given by
ð20Þ
Thus exergy accumulation in the perfectly stratified tank is a maximum at u ¼ 1. When operating as fully mixed, there is instantaneous temperature redistribution in the tank as the incoming fluid mixes completely with the resident fluid, resulting in a uniform temperature in the tank. Fig. 3(c) shows temperature profiles at different time instants along the axis of the fully mixed storage tank. In actual storage tank operations, this is approximated by very low PeD number scenarios for which there is sufficient time for diffusion mechanisms to result in temperature redistributions, and by high ReD flows accompanied by vigorous mixing due to turbulence. The fully mixed operation will result in the least exergy accumulation in the tank since the level of mixing present in the tank is accompanied by in the maximum entropy generation. The
T m ðtÞ To
ð22Þ
where T m ðtÞ is the temperature of the fully mixed tank at time t, given by
_ m T m ðtÞ ¼ T h ðT h T ini Þ exp t m
ð23Þ
With Eqs. (13), (22) and (23), the expression for DNm ðtÞ in Eq. (21) may be written as
T m ðtÞ DNm ðtÞ ¼ mcp T m ðtÞ T ini T o ln T ini
ð24Þ
Substituting Eqs. (14), (20) and (24) into Eq. (10), the normalized exergy efficiency may be expressed as
gx ðtÞ ¼
TðtÞ T m ðtÞ T o lnðT ðtÞ=T m ðtÞÞ
u1 uðT h T ini Þ ðT m ðtÞ T ini Þ T o lnðT uh =T m ðtÞT ini Þ
ð25Þ
In terms of dimensionless quantities,
h i h ðsÞþ#min hðsÞ 1 þ expðuÞ #0 ln #max expðuÞ u ð1uÞ gx ðsÞ ¼ # #min u 1 þ expðuÞ #0 ln #maxmaxexpð uÞ
ð26Þ
where, hðsÞ ¼ ðTðtÞ T ini Þ=ðT h T ini Þ is the dimensionless temperature in a fully mixed store having the same energy content as the SSTES at dimensionless time s; h ðsÞ ¼ ðT ðtÞ T ini Þ=ðT h T ini Þ is the dimensionless temperature in a fully mixed store having the same exergy content as the SSTES at dimensionless time s; #min ¼ T ini =ðT h T ini Þ and #max ¼ T h =ðT h T ini Þ are the minimum and maximum dimensionless temperature ratios, respectively, while #0 ¼ T 0 =ðT h T ini Þ is the reference dimensionless temperature ratio. 4. Results and discussion SSTES of 0.4 m diameter, inlet to tank diameter ratio, din =D ¼ 0:1, and containing water initially at 25 °C are considered. Parametric effects are studied by setting up different cases with different inlet temperatures and charging velocities, to obtain different Richardson number and Peclet number conditions, respectively, as well as different tank aspect ratios and tank wall heat loss coefficients. The ranges of the parameters considered in this study are summarized in Table 1. Inlet flow rates in the cases investigated were limited to the laminar range in order to avoid turbulence in the stores, which is undesirable when the aim is to maintain stratification.
Table 1 Range of parameters considered. Parameter
Range
Aspect ratio, AR Overall heat loss coefficient, U Peclet number, PeD
1–4 0–100 W m2 K1
Reynolds number, ReD Richardson number, Ri Max. dimensionless temperature ratio, #max Min. dimensionless temperature ratio, #min
5 103 –100 103 107–2885 10–105 5.58–12.92 4.58–11.92
492
H.O. Njoku et al. / Solar Energy 136 (2016) 487–498
4.1. Numerical solution of the STESS model The finite volume numerical solution of the STESS model was implemented using the OpenFOAM CFD C++ toolbox. A wedge shaped, conformal block-structured mesh was used, consisting of three blocks – a 6 10 cell block for the tank inlet pipe, a 120 750 cell block for the storage tank, and a 6 40 cell block for the tank outlet pipe, i.e., a total of 90,300 cells. The discretization of diffusive and convective terms were performed using second-order linear and upwind schemes, respectively, while temporal discretization was second-order Euler implicit with time steps automatically adjusted subject to Courant number limits of between 0.1 and 0.25. The ensuing algebraic equation sets were solved using the PCG (preconditioned conjugate gradient) solver in the PIMPLE (merged PISO-SIMPLE) algorithm (OpenFOAM, 2014). The convergence criteria set for the solution were a tolerance,
kþ1 ¼ jwkþ1 wk j 6 108
and a relative tolerance, kþ1 k 6 0, where w is any dependent variable. The discretization schemes described here were implemented using the buoyantPimpleFoam and buoyantBoussinesqPimpleFoam OpenFOAM utilities.
reversed flow rises to the top of the store, establishing a fluid circulation zone, which is stable at the top of the store for the entire duration of the charging. Flow within the thermocline is parallel to the store axis, whereas within the cold region of the store, the flow converges smoothly to the store’s exit. Evidently, the state of stratification within the store is continuously evolving with the charging duration, necessitating a performance measure which adequately accounts for these temporal distinctions in the states of stratification within the SSTES. Dimensionless temperature and flow profiles after a charging duration corresponding to u ¼ 0:42, for SSTES having different different combinations of the dimensionless parameters Ri, PeD and AR, are presented in Fig. 5(a)–(d). Some qualitative deductions may be made from the plots, viz., as PeD increases (comparing Fig. 5(a) and (b)) multiple fluid circulation zones appear in the hot region of the store; with decreasing Ri, (comparing Fig. 5(a) and (c)) a radial temperature gradient within the hot region becomes more evident; whereas as AR decreases, (comparing Fig. 5(a) and (d)) the thermocline occupies a greater proportion of the store height. These observed variations in SSTES operational characteristics also need to be adequately accounted for by the measures used in assessing SSTES performance.
4.2. Transient evolution of temperature and flow profiles 4.3. Comparison of energy vs exergy accumulations in STES The temperature and flow fields within a representative SSTES (with AR = 3, Ri = 103 and PeD ¼ 20 103 ) are presented in Fig. 4 (a)–(d) as dimensionless temperature profiles and flow streamlines at four instants during the charging of the SSTES. Two distinct temperature regions in the store are clearly visible after the commencement of charging, together with the separating thermocline region which travels down the store as the charging progresses. The streamlines show that the flow of the incoming charge is reversed just before it comes in contact with the thermocline. This is because the buoyancy force resulting from the temperature difference between the incoming charge and the resident fluid is greater than the inertial force of the charge entering the store. Hence, the flow of the incoming charge is unable to penetrate the thermocline and is reversed. Subsequently, the
When the total energy accumulation in the SSTES is considered, there is no difference between the energy accumulated in a real store and that in a perfectly stratified store for most of the charging duration of the stores. This shown in Fig. 6 for a SSTES operating under
the
conditions,
AR = 3,
PeD ¼ 20 103 ,
Ri = 103
and
1
U ¼ 0 W m K . This is because, until the thermocline arrives at the exit of the SSTES, all the thermal energy accompanying the charging stream into the store is retained within it, just as obtains in a perfectly stratified store for 0 6 u 6 1. Thereafter, the outgoing stream will be accompanied by some of the input energy. In contrast, some of the thermal energy charged into a fully mixed store is carried out by the outgoing stream from the onset of charging, hence the lower value of accumulated energy in a fully mixed 2
Fig. 4. Temperature profiles and velocity streamlines in a typical stratified thermal store (AR = 3, Ri = 102, PeD ¼ 20 103 ) after tank volume change fractions u ¼ (a) 0.20, (b) 0.42, (c) 0.65, and (d) 0.87.
H.O. Njoku et al. / Solar Energy 136 (2016) 487–498
493
Fig. 5. Temperature profiles and velocity streamlines after tank volume change fraction, u ¼ 0:42 in stratified thermal stores having (a) Ri = 103 , PeD ¼ 20 103 , AR = 3; (b) Ri = 103 , PeD ¼ 60 103 , AR = 3; (c) Ri = 102 , PeD ¼ 20 103 , AR = 3; (d) Ri = 103 , PeD ¼ 20 103 , AR = 1.
Fig. 6. The accumulation of energy in a perfectly stratified store, a fully mixed store and the actual SSTES. (AR = 3, PeD ¼ 20 103 , Ri = 103 ; U ¼ 0 W m2 K1 ).
store shown in Fig. 6. An attempt to assess the SSTES using an energy efficiency, which essentially compares the energy accumulated in the store to that in perfectly stratified store, posses a problem because for most of its charging duration this will result in a 100% efficiency. The energy efficiency measure thus seems to suggest that no loss is incurred during the charging process and that charging duration has no influence on overall performance. It will therefore be impossible to measure how performance progresses with charging duration on this basis. Again, since all SSTES configurations produce this result, it will be impossible to make distinctions in the performances of differently configured SSTES (different AR, PeD , Ri, etc.) on this basis.
The plots of exergy accumulations for the same stores are shown in Fig. 7. Clearly, the exergy accumulation in the SSTES differs from both of those in the perfectly stratified store and the fully mixed store, for the entire duration of charging. This is in keeping earlier conclusions that an exergy-based measure of assessment is more desirable. One such measure is the exergy efficiency, which, similar to the energy efficiency, compares the exergy accumulated in the SSTES with that in a perfectly stratified store. While this measure recognizes the existence of an upper bound on exergy accumulation in the store, which is achieved in the perfectly stratified store, and thus measures ‘‘how nearly the operation of [the] system approaches the . . . theoretical upper limit” (Rosen, 2001),
494
H.O. Njoku et al. / Solar Energy 136 (2016) 487–498
Fig. 7. The accumulation of exergy in a perfectly stratified store, a fully mixed store and the actual SSTES. (AR = 3, PeD ¼ 20 103 , Ri = 103 ; U ¼ 0 W m2 K1 ).
the existence of a lower bound (corresponding to the situation in a fully mixed store) is overlooked. In a real store, the exit temperature is raised by fluid mixing in the store, resulting in a lower temperature difference at the heat source and thus a longer charging period. The normalized exergy efficiency, gx , defined in Eq. (10), in addition to measuring the deviation from the upper limit, also evaluates the deviation of the system’s performance from the theoretical lower limit, which is of equal importance. It takes both bounds into consideration and is used in succeeding discussions. 4.4. Normalized exergy efficiency, gx under different charging rate regimes The plots of the time-varying normalized exergy efficiencies,
gx ðsÞ, for different Peclet numbers are shown for Ri ¼ 102 , in Fig. 8(a) and for Ri ¼ 103 in Fig. 8(b). The shapes of the plots are essentially the same, with gx ðsÞ rising gradually from the start of charging (u ¼ 0) till it peaks at u 0:90, and drops sharply thereafter. At any u; gðsÞ increases sharply with PeD at low PeD values 3
(< 30 10 ) but thereafter remains fairly constant as PeD continues to increase. A range of PeD values for peak SSTES tank performance was also reported by Ji and Homan (2007). They found that when storage performance was assessed using a discharge volume fraction, V d , computed based on the simplified Uniform Diffusivity SSTES tank model of Cole and Bellinger (1982) and the NonUniform Diffusivity SSTES tank model of Zurigat et al. (1991), V d was found to be maximum at Peclet numbers between 1 103 and 3 103 . (A Peclet number based on tank height, Pe ¼ U 1 H=a, was used by Ji and Homan (2007). For PeD based on tank diameter, as used in this study, and with AR = 3, this optimal PeD range will be equivalent to 33:3 103 –100 103 , which is consistent with the current observation.) Bearing in mind that gx ðsÞ compares processes in an actual SSTES with those in a perfectly stratified unit (the best case scenario) and those in a fully mixed unit (the worst case scenario), we observe that for most of the charging period, gx ðsÞ is greater than 0.5, implying that the behavior of the actual SSTES does approach that of a perfectly stratified unit than that of a fully mixed unit. Since heat loss through store walls have been ignored, the
departure from the perfectly stratified behavior is due to entropy generations resulting from fluid mixing and heat transfer between the incoming fluid stream and the fluid ab initio in the tank which is mainly confined to the thermocline region in the actual tank. Whereas these processes occurs everywhere in a fully mixed tank, they are completely absent in a perfectly stratified tank. As the charging of a SSTES progresses, a time is reached when some of the exergy accumulated in the tank starts being conveyed out by the out-going fluid stream – this is the situation in a fully mixed unit from the very onset of charging, but never occurs in a perfectly stratified tank until u is unity. An increasing fraction of accumulated exergy will be carried out of the SSTES in this way as u approaches and goes beyond unity. Hence, we observe from these plots peaks in gx ðsÞ; gx;max , occurring at u 0:90, beyond which charging is increasingly dominated by the fully mixed behavior. 4.5. Normalized exergy efficiency, gx for different PeD and Ri When gx ðsÞ for different Ri cases are considered, as shown in Fig. 9, it is observed that gx ðsÞ generally increases as Ri increases. This is because stabilizing buoyancy forces in the SSTES increase when the characteristic temperature difference in the store, ðT h T ini Þ, and hence Ri in the store, increases. Thus, a store’s behavior approaches that of the perfectly stratified store as Ri increases. At small Ri values, huge gains in gx ðsÞ are obtained as Ri increases, however, beyond Ri = 103, the changes in gx ðsÞ with increasing Ri are no longer significant. Peak normalized exergy efficiencies, gx;max , for the different Ri are observed at u 0:90 as were observed also in the previous section. The timing of the occurrence of gmax seems therefore to be independent of PeD and Ri. gx;max may be utilized as a criteria to determine when to cease charging a SSTESS if the system design objective is the optimization of the exergy stream from the heat source. This will be justified if a system were to depend on expensive (fossil fuel based) energy sources and simultaneously operate as an open system in which the out-going fluid stream is not returned to the heat source. For most practical systems, which are closed, and systems based on solar thermal, geothermal, waste heat, or any other such inexpensive heat sources, the design criteria will usually be to get the store contents to a uniform temperature equal or close to the charging
H.O. Njoku et al. / Solar Energy 136 (2016) 487–498
495
Fig. 8. Evolution of normalized exergy efficiencies, gx , over the charging duration, u, for cases with different PeD . (AR = 3, U ¼ 0 W m2 K1 ).
temperature, i.e., hðsÞ ¼ h ðsÞ ! 1. Whatever the objective is, gx;max may be used for comparing the performances of different SSTES, since for differently configured systems, gx;max occurs after seemingly identical charging durations. Whereas the u corresponding to gmax seems to be invariant with PeD and Ri, gx;max varies with both PeD and Ri as shown by the surface plot of their dependencies presented in Fig. 10. gx;max increases with both PeD and Ri. There is a sharp increase in gx;max with increasing PeD and Ri at low values of both. But we observe that the rate of increase with Ri is steeper than that with Ri at low values of both. Beyond Ri = 102, the change in gx;max with increasing Ri is not appreciable, suggesting that there is a limit to which improvements in gx;max may be obtained by increasing Ri. gx;max increases continuously with PeD within the range of PeD values considered and thus has a more general influence on gx;max than Ri.
4.6. Effect of aspect ratio on normalized exergy efficiency The results presented thus far have been for SSTES with AR = 3. Now we consider the influence of AR on gx;max . The destruction of exergy in the SSTES is due to the mixing of the incoming hot fluid with the resident cold fluid, which is accompanied by the degradation of thermal gradients and these occur primarily in the thermocline. As the AR increases, the proportion of the store’s volume occupied by the thermocline reduces and thus there is a reduction in the proportion of accumulated exergy that is destroyed. Expectedly therefore, from the surface plot of Fig. 11, we observe gx;max increasing monotonously as AR increases within the range of ARs considered. Whereas this may suggest that AR may be increased indefinitely in order to achieve higher performance, the choice of appropriate AR, as noted by Lavan and Thompson (1977), will be
496
H.O. Njoku et al. / Solar Energy 136 (2016) 487–498
Fig. 9. Evolution of normalized exergy efficiencies, gx , over the charging duration, u, for different cases with different Ri. (AR = 3, PeD ¼ 20 103 ; U ¼ 0 W m2 K1 ).
Fig. 10. Effect of PeD and Ri on maximum normalized exergy efficiencies, gx;max (AR = 3, U ¼ 0 W m2 K1 ).
Fig. 11. The effects of Peclet number, PeD , and aspect ratio, AR on the maximum normalized exergy efficiency, gx;max . (U ¼ 0 W m2 K1 and Ri = 103, i.e., #max ¼ 7:62; #min ¼ 6:62).
H.O. Njoku et al. / Solar Energy 136 (2016) 487–498
497
Fig. 12. Evolution of normalized exergy efficiencies, gx , over the SSTES charging duration, u, for cases with different PeD and levels of insulation. (AR = 3 and Ri = 103 , i.e., #max ¼ 7:62; #min ¼ 6:62).
a trade-off between performance and cost, with an AR between 3 and 4 seeming reasonable. Furthermore, the increases in gx;max with AR are more significant at low AR and PeD values, and the gx;max profile flattens out at higher values of both. 4.7. Effect of heat losses The influence of the presence of heat losses across the store walls on the gx , was studied by relaxing the assumption of perfect store wall insulation. Uniform heat loss coefficients, U, were imposed on the store walls, varying from U ¼ 0 W m2 K1 to 100 W m2 K1 , with PeD fixed at 40 103 and Ri = 103. Fig. 12 shows the variation of gx over the charging duration for different U values. Generally, gx rises rapidly initially as charging commences. Due to the exergy accompanying heat losses across the walls and the additional entropy generation due to the heat losses, the cases with finite heat losses present peaks in gx , a short while after the commencement of charging. Reductions in gx are observed beyond these peaks, which are as a result of increasing proportions of the input exergy being lost through the walls and the accompanying entropy generation due to the heat losses. A second dip in gx is also observed at 0:95 K u K 1:0. As described earlier, this coincides with the exit of the thermocline from the stores. In addition, as U increases, gx falls
distinctions for periods of charging within 0 6 u 6 1. By using gx , SSTES are compared with both the perfectly stratified and the fully mixed stores, and thus SSTES performances are placed in the proper context with reference to the best possible and the worst possible STES performances, respectively. This analysis, which incorporates second law considerations, is superior to those that account for only first law effects. The influence of AR, PeD and Ri on gx was also investigated. As AR increased four folds, gx ;max increased by 4.6–7.7% suggesting that slender tanks are to be preferred to large diameter tanks. Beyond aspect ratios of between 3 and 4, however, the improvements in gx ;max with increasing AR are no longer significant. Also, within the range of conditions considered in this study, a twenty fold increase in PeD resulted in an increase in gx ;max of 7.2–16.8%, while a thousand fold increase in Ri led to a 4.3–13.6% increase in gx ;max . The maximum possible temperature difference needs to be maintained between the incoming charge and the resident fluid in the store for improved stratification. High Richardson numbers correspond with laminar conditions in the SSTES since Ri is related
monotonically, such that cases with U ¼ 0 W m2 K1 represent upper limits of actual SSTES performance. Though the drop in gx isn’t appreciable at low U values, there is need nonetheless for proper insulation since the effects considered here only apply to the charging stage and not the storage phase when heat losses would occur over extended resident times.
to the Reynolds number by Ri = Gr/Re2D , where Gr is the Grashoff number. Under these conditions in which stratification does not breakdown in the store, higher PeD (hence faster charging rates) are desirable in order to minimize the contact time for conduction across the thermocline. Such conduction leads to a thickening of the thermocline and hence a deterioration of stratification. The use of inlet modifier is beneficial in this regard, in order to accommodate faster charging rates in SSTES. The steepest increases in gx ;max due to increases in the dimensionless parameters occurred at the lower values of the parameters.
5. Conclusion
Acknowledgements
The extent and state of thermal stratification in sensible thermal energy stores during charging, and hence the stores’ performances, vary depending on the conditions at store inlets, the stores’ geometries, and the extent of charging. Clear distinctions between the performances of SSTES under these different conditions are not very clear on the basis of energy considerations alone. The normalized exergy efficiency, gx , has been utilized in this study, which is shown to be capable of providing the required
Computing facilities of the University of Nigeria-UNESCO-HP Grid Computing Brain Drain-to-Brain Gain Initiative and of the Abdus-Salam International Centre Theoretical Physics (ICTP), Trieste, Italy were employed in this study. We are grateful to Dr. Collins Udanor of the Computer Science Department, University of Nigeria, and to Prof. Joe Niemela, Dr. Clement Onime and Ivan Girotto of the ICTP, for facilitating access granted the first author to these facilities.
498
H.O. Njoku et al. / Solar Energy 136 (2016) 487–498
References Abdoly, M.A., Rapp, D., 1982. Theoretical and experimental studies of stratified thermocline storage of hot water. Energy Convers. Manage. 22, 275–285. Abu-Hamdan, M.G., Zurigat, Y.H., Ghajar, A.J., 1992. An experimental study of a stratified thermal storage under variable inlet temperature for different inlet designs. Int. J. Heat Mass Transf. 35 (8), 1927–1934. Altuntop, N., Arslan, M., Ozceyhan, V., Kanoglu, M., 2005. Effect of obstacles on thermal stratification in hot water storage tanks. Appl. Therm. Eng. 25, 2285– 2298. Castell, A., Medrano, M., Sol, C., Cabeza, L.F., 2010. Dimensionless numbers used to characterize stratification in water tanks for discharging at low flow rates. Renew. Energy 35, 2192–2199. Cole, R.L., Bellinger, F.O., 1982. Natural Thermal Stratification in Tanks. Phase 1 Final Report, Report No. ANL-82-5. Cónsul, R., Rodríguez, I., Pérez-Segarra, C.D., Soria, M., 2004. Virtual prototyping of storage tanks by means of three-dimensional CFD and heat transfer numerical simulations. Sol. Energy 77, 179–191. Cristofari, C., Notton, G., Poggi, P., Louche, A., 2003. Influence of the flow rate and the tank stratification degree on the performances of a solar flat-plate collector. Int. J. Therm. Sci. 42, 455–469. Eames, P.C., Norton, B., 1998. The effect of tank geometry on thermally stratified sensible heat storage subject to low reynolds number flows. Int. J. Heat Mass Transf. 41 (14), 2131–2142. Farmahini-Farahani, M., 2012. Investigation of four geometrical parameters on thermal stratification of cold water tanks by exergy analysis. Int. J. Exergy 10 (3), 332–345. Hahne, E., Chen, Y., 1998. Numerical study of flow and heat transfer characteristics in hot water stores. Sol. Energy 64 (1–3), 9–18. Haller, M.Y., Cruickshank, C.A., Streicher, W., Harrison, S.J., Andersen, E., Furbo, S., 2009. Methods to determine stratification efficiency of thermal energy storage processes – review and theoretical comparison. Sol. Energy 83, 1847– 1860. Hollands, K.G.T., Lightstone, M.F., 1989. A review of low-flow, stratified-tank solar water heating systems. Sol. Energy 43, 97–105. Ismail, K.A.R., Leal, J.F.B., Zanardi, M.A., 1997. Models of liquid storage tanks. Energy 22 (8), 805–815.
Ji, Y., Homan, K.O., 2007. On simplified models for the rate- and time-dependent performance of stratified thermal storage. J. Energy Resour. Technol. 129 (4), 214–222, September, Transactions of the ASME. Lavan, Z., Thompson, J., 1977. Experimental study of thermally stratified hot water storage tanks. Sol. Energy 19, 519–524. Njoku, H.O., Ekechukwu, O.V., Onyegegbu, S.O., 2014. Analysis of stratified thermal storage systems: an overview. Heat Mass Transf. 50 (7), 1017–1030. http://dx. doi.org/10.1007/s00231-014-1302-8. OpenFOAM, 2014.