Thermal Analysis of Shell-and-Tube Thermoacoustic Heat Exchangers

Transcription

1 entropy Article Thermal Analysis of Shell-and-Tube Thermoacoustic Heat Exchangers Mohammad Gholamrezaei * and Kaveh Ghorbanian Department of Aerospace Engineering, Sharif University of Technology, Tehran , Iran; ghorbanian@sharifedu * Correspondence: gholamrezaei@aesharifedu; Tel: Academic Editor: Kevin H Knuth Received: 7 June 2016; Accepted: 8 August 2016; Published: 16 August 2016 Abstract: Heat exchangers are of key importance in overall performance and commercialization of rmoacoustic devices The main goal in designing efficient rmoacoustic heat exchangers (TAHXs) is achievement of required heat transfer rate in conjunction with low acoustic energy dissipation A numerical investigation is performed to examine effects of geometry on both viscous and rmal-relaxation losses of shell-and-tube TAHXs Furr, impact of drive ratio as well as temperature difference between oscillating gas and TAHX tube wall on acoustic energy dissipation are explored While viscous losses decrease with d i /δ κ, rmal-relaxation losses increase; however, rmal relaxation effects mainly determine acoustic power dissipated in TAHXs The results indicate existence of an optimal configuration for which acoustic energy dissipation minimizes depending on both TAHX metal temperature and drive ratio Keywords: heat exchangers; rmoacoustics; shell-and-tube; acoustic energy; dissipation 1 Introduction Today, rmoacoustic engines, coolers, and heat pumps are receiving more industrial interest While rmoacoustic engines utilize heat to generate acoustic power, rmoacoustic refrigerators consume acoustic power to transfer heat from cold reservoir In general, re are two basic types of rmoacoustic systems: standing-wave [1] and travelling-wave [2] systems Thermoacoustic systems have three distinct advantages: first, y have no moving parts and are simple in structure, have low manufacturing costs, and are highly reliable Furr, y are environmentally friendly due to usage of inert gases as working fluid Additionally, heat-driven rmoacoustic devices can be driven by low quality energy sources such as waste heat and solar energy The main components of a rmoacoustic device are stack/regenerators and two heat exchangers (HXs) that are placed at both ends of stack/regenerators in an acoustic network Although steady flow heat exchangers are a mature technology, ir design is of key importance in commercialization of rmoacoustic devices where flow is oscillatory All engines and refrigerators must reject waste heat to ambient temperature where ambient heat sink is often available as a flowing water stream Similarly, engines must also accept heat from a heat source at a higher temperature (ie, burner, waste heat, solar energy, etc) The efficiency of rmoacoustic engines is defined as ratio of produced acoustic power to heat received by system [3] However, irreversible dissipation of acoustic power in different components of rmoacoustic engines due to viscous and rmal relaxation losses is main source for performance degradation Two highly dissipative components of rmoacoustic engines are regenerators/stacks and heat exchangers These components have recently been subject of many studies for system performance optimization Wu et al [4] presented a generalized heat transfer model using a complex heat transfer exponent to optimize performance of a rmoacoustic engine Entropy 2016, 18, 301; doi:103390/e wwwmdpicom/journal/entropy

2 Entropy 2016, 18, of 15 Zhibin and Jaworski [5] studied role of configuration and geometrical dimensions of a regenerator on acoustic power dissipation The irreversibility of porous rmoacoustic stacks in terms of entropy generation is analyzed by Tasnim et al [6] Chaitou and Nika [7] considered dependence of acoustic energy on parameters such as stack s hydraulic radius and stack position As pointed out, an optimum design of TAHXs is a critical and challenging task for commercialization of rmoacoustic devices The relationship between acoustic energy dissipation, heat exchanger geometry as well as operating conditions is of significant importance In general, while different parallel plate, finned-tube, and shell-and-tube HXs configurations are commonly used, present knowledge of acoustic energy dissipation in TAHXs is relatively limited It should be emphasized that designing efficient TAHXs in terms of required heat transfer rate, in addition to low acoustic energy dissipation, remains a challenge Ishikawa and Hobson [8] derived an analytical expression of time averaged entropy generation in parallel plate HXs Piccolo [9,10] developed a computational model for studying entropy generation characteristics of TAHXs with plane fins of a standing-wave rmoacoustic device Previous studies focused mostly on parallel plate and finned-tube HXs It should be pointed out that, in case of high-capacity refrigerators and engines, rmal conductivity of solids is insufficient to carry required heats without significant temperature differences; hence, advanced and likely complex heat exchangers have to be used to interweave process fluid and working gas, bringing m into intimate rmal contact A shell-and-tube heat exchanger system is classical approach [11] The present paper is focused on shell-and-tube heat exchanger design of rmoacoustic systems and will investigate relationship between acoustic energy dissipation, viscous loss, rmal relaxation loss, and heat exchanger geometry Much of motivation behind current work is driven by interest as to wher re is a oretical optimum of se parameters and/or optimum of ir combination for a specific operating point Second, and more generally, current research attempts to develop a simple but constructive methodology for comparison and evaluation of related heat exchangers at different operating conditions In this work, heat exchangers are considered in travelling-wave mode with ideal gases based on linear rmoacoustic ory In this regard, heat exchangers with different geometries are investigated numerically and compared to each or The dependence of acoustic power dissipation on heat exchanger metal temperature and drive ratio for a fixed value of heat transfer rate is examined 2 Theoretical Analysis 21 Thermal Modeling of Heat Exchangers For heat transfer analysis, a local heat transfer coefficient is defined as ratio of heat transfer per unit area to temperature difference between surface and bulk fluid adjacent to surface However, in practice, a global heat transfer coefficient derived from a lumped analysis of heat exchanger (UA-LMTD or effectiveness-ntu methods) is utilized [12] The surface area, A s, of heat exchanger is expressed as A s = Q T lm U i (1) where Q is heat transfer rate, T lm denotes log-mean temperature and U i is overall heat transfer coefficient (W m 2 K 1 ) In above equation, log-mean temperature difference model is used for rmal modeling of shell-and-tube heat exchanger operating in unsteady flow The log-mean temperature difference for rmoacoustic shell-and-tube heat exchangers is formulated as follows:

3 Entropy 2016, 18, of 15 T lm = T w,o T ( w,i ) Tm (2) T ln w,i T m T w,o where T m is gas mean temperature, T w,i and T w,o are water inlet and outlet temperature, respectively In general, a shell-and-tube heat exchanger is considered with gas flowing through tubes and water flowing over tubes The heat transfer in heat exchanger involves rmal resistance consisting of three parts: resistance for heat transfer from gas to tube wall, rmal resistance of tube, and resistance for heat transfer from tube to water Therefore, overall heat transfer coefficient is estimated based on tube inner diameter according to U i = [ ( ) ] 1 di N t d h i (ln (d o /d i ) /2ε) + (3) i d o h o where d o and d i are outside and inside tube diameters; h i and h o are tube and shell-side heat transfer coefficient, respectively Furr, ε is rmal conductivity of tube The shell-side heat transfer coefficient for water, h o, may be recalled from classical heat exchanger textbooks, such as [12] It should be pointed out that, for most rmoacoustic-related investigations, heat transfer from oscillatory gas to heat exchanger inner surface area is mainly considered In or words, heat transfer across tube wall and n furr to water is usually embedded Thus, Equation (1) might be rewritten for heat transfer from oscillatory gas to tube wall as follows A s = Q (T m T s ) h i (4) where T s is temperature of tube wall at gas side This temperature is noted as HX metal temperature in DeltaEC [13] and also as solid temperature in some TA studies [8,10] Focusing on heat transfer in oscillatory gas and employing T s instead of T w aids to eliminate shell side parameters like tube thickness, tube length and tube pitch which have a negligible influence on acoustic energy dissipation The surface area, A s, of a shell-and-tube heat exchanger is given as A s = N t (πd i L t ) (5) where N t is number of tubes and L t is length of tubes A s is calculated according to heat transfer Equation (4) Thermoacoustic Considerations It is important to recognize that problem of heat transfer from an acoustically oscillating gas medium to a solid surface is fundamentally different from that addressed in most discussions of compact heat exchangers that are based on steady unidirectional flow of fluid through tubes of heat exchanger Indeed, TAHXs are distinguished by an oscillatory flow with zero mean velocity and this circumstance has two important implications [14] The most significant difference between classical HXs and TAHXs is acoustically oscillating gas parcels which only move a limited distance before reversing ir direction of flow The consequence of ir periodic flow reversal is that one cannot arbitrarily increase length of heat exchange surfaces (tubes) in direction of flow to increase effective surface area available for heat transfer Thus, optimum length, L TAHX, of heat exchanger should be of order of peak-to-peak displacement of working gas acoustic displacement amplitude Anor important aspect of TAHXs is that conventional steady flow heat transfer correlations cannot be directly applied for estimation of convective heat transfer coefficient, h i, between gas and solid wall of TAHXs Different approaches are currently proposed to overcome this limitation

4 Entropy 2016, 18, of 15 Swift [3,15] and Garrett [14] suggest an approximate estimation of heat transfer coefficient on basis of a simple boundary layer conduction heat transfer model given as h i = k y e f f (6) and y e f f = min [δ k, r h ] (7) r h = d i /4 (8) for shell-and-tube heat exchangers The rmal conductivity of gas is denoted by k and ( ) δ k = 2k/ω is rmal penetration depth that is, distance through which heat will diffuse in an acoustic cycle Mozurkewich [16], Peak et al [17], Nsofar et al [18] and Kamsanam et al [19] also developed different numerical and experimental models of h i for mostly parallel plate and finned-tube TAHXs However, for shell-and-tube TAHXs, correlations of heat transfer coefficient are still limited In this regard, Swift s model [15] appears to be a sound approximation of h i for oscillatory flow in shell-and-tube HXs especially when flow in tubes is laminar The tube side acoustic Reynolds number can be evaluated as follows Re d = ρ m u 1 d h µ (9) where d h (= 4r h ) is hydraulic diameter of tube 22 Acoustic Energy in TAHXs According to linear rmoacoustic ory [1,3], calculated using du 1 dx = iωa g ρ m a 2 dp 1 dx = wave propagation in TAHXs is ( 1 + (γ 1) f ) κ p 1 + ɛ 1 (10) s iωρ m (1 f v ) A g U 1 (11) where U 1 and p 1 are oscillating volume velocity and pressure, respectively A g is cross-sectional area of gas channels in heat exchanger f v and f k are spatially averaged rmoviscous functions and are given in [1,3] in detail It should be pointed out that, in this paper, rmoviscous functions for cylindrical geometry is considered for shell-and-tube TAHXs γ, a, ɛ s and ρ m are ratio of specific heat capacities, sound speed, correction factor for finite solid heat capacity, and mean density of working gas, respectively Furrmore, A g is expressed as A g = N t πd i 2 /4 (12) The time averaged acoustic power d E 2 produced in a length dx of channel is given in complex notation in general form as d E 2 dx = 1 [ ] 2 Re dp Ũ 1 1 dx + p du 1 1 dx Here, tilde, ~, indicates a complex conjugation Subscript 2 indicates second order quantity and Re denotes real part of complex number Substituting Equations (10) and (11) into Equation (13), one obtain as [3]: (13) d E 2 dx = r v 2 U r κ p Re [g p 1U 1 ] (14)

5 Entropy 2016, 18, of 15 In above equation, viscous resistance per unit length of channel, r v, rmal relaxation conductance per unit length of channel, 1/r k, and complex gain/attenuation constant for volume flow rate, g, are defined as follows: and g = r v = ωρ m A g Im [ f v ] 1 f v 2 (15) 1 = γ 1 ωa g Im [ f κ ] (16) r κ γ (1 + ɛ s ) p m ( f κ f v ) 1 dt m (1 f κ ) (1 σ) (1 + ɛ s ) T m dx Here, σ, p m and T m are Prandtl number, mean pressure, and mean temperature of working gas, respectively On right hand side of Equation (14), regardless of temperature gradient along length of channel, first two terms represent viscous and rmal-relaxation dissipation, which always consume acoustic power The third term denotes acoustic power produced (or consumed) by channel due to axial temperature gradient depending on magnitude and direction of axial temperature gradient In TAHXs, it is mostly assumed that length of heat exchanger is short; refore, T m remains constant and re is no axial temperature gradient (dt m /dx = 0), so third term vanishes Therefore, it will be more convenient to refer to dė 2/dx as a net time averaged acoustic power dissipated per unit length of heat exchanger due to viscous and rmal relaxation loss ( first two terms of RHS of Equation (14)) To calculate acoustic power dissipated in heat exchanger, dė 2/dx is integrated over length of heat exchanger Thus, Ė diss is written as LTAHX E diss = 0 (17) d E 2 dx dx = Ė diss,v + Ė diss,κ (18) where Ė diss,v and Ė diss,k represent viscous and rmal dissipation of acoustic power in heat exchanger, respectively LTAHX ωρ m Im [ f v ] E diss,v = 0 A g 1 f v 2 U 1 2 dx (19) LTAHX E diss,κ = γ 1 ωa g Im [ f κ ] p 0 2γ (1 + ɛ s ) p 1 2 dx m (20) For shell-and-tube heat exchangers, replacing A g with A s in Equations (18) and (19), one may rewrite Ė diss,v and Ė diss,k as follows 3 Results and Discussion In this paper, LTAHX 4ωρ m L E diss,v = hx Im [ f v ] 0 d i A s 1 f v 2 U 1 2 dx (21) LTAHX E diss,κ = γ 1 ωd i A s Im [ f κ ] p 0 2γ (1 + ɛ s ) 4p m L 1 2 dx (22) hx E diss is used as an evaluation index for TAHXs However, it is also important to understand both Ė diss,v and Ė diss,κ which provide an estimation of total dissipation if integrated toger Therefore, for shell-and-tube heat exchangers, both Ė diss,v and Ė diss,κ, are numerically

6 Entropy 2016, 18, of 15 determined The normalized tube diameter, d i /δ κ, is used for analysis All calculations are performed for helium as working fluid (σ = 2/3, γ = 5/3) According to Equations (1) and (21), knowledge of heat transfer rate ( Q), gas mean temperature (T m ), TAHX metal temperature (T s ), and L TAHX (which is equal to L t in this paper) are required for numerical analysis The gas mean temperature is determined in design process of rmoacoustic systems For example, in Backhaus and Swift s travelling-wave engine [2], heat load of ambient heat exchanger is 1614 W, L TAHX of heat exchanger is 20 mm, and gas mean temperature is 319 K at middle of HX The operating frequency is 84 Hz with helium at a mean pressure of 31 MPa as working gas In addition, HX material is stainless steel Parameters of TAHX and working conditions are presented in Table 1 In this paper, for purpose of comparison of proposed model, experimental results of shell-and-tube HX of Backhaus and Swift s rmoacoustic Stirling heat engine (TASHE) [2] is considered Table 1 Parameters of shell-and-tube heat exchanger and working fluid Parameter Symbol Value Units Mean pressure p m 31 MPa Mean gas temperature T m 319 K Gas sound speed a m/s Gas polytropic coefficient γ Gas Prandtl number σ Gas rmal conductivity k 0159 W/(m K) Gas specific heat c p 5193 J/(kg K) Gas kinematic viscosity µ kg/(s m) Q 1614 W HX heat load HX length L TAHX 20 mm HX rmal conductivity k s W/(m K) HX specific heat c s J/(kg K) HX material density ρ s kg/m 3 Operating frequency f 84 Hz Gas rmal penetration depth δ κ 0157 mm In following sections, analysis of acoustic energy in shell-and-tube HXs is presented The TAHXs are considered with a fixed value of heat load Furr, impact of geometry configuration (d i /δ κ ), TAHX metal temperature (T s ), and drive ratio (DR) on acoustic energy dissipation is examined The optimum value of (d i /δ κ ) opt as a function of drive ratio at various T s is determined Changes on acoustic energy dissipation due to geometry is investigated for different values of T s The effect of T s on heat exchanger geometry as well as acoustic energy dissipation is assessed Moreover, impact of drive ratio as an important design parameter is discussed The validation of model is performed by comparing numerical predictions of present model with experimental results of TASHE s shell-and-tube HX [2] The results are shown in Table 2 As illustrated, regarding design conditions presented in Table 1, proposed model can design a shell-and-tube HX and predict acoustic power dissipated in HX with good accuracy compared to shell-and-tube HX of TASHE [2] Table 2 Comparison of present model with experimental results of TAHX of TASHE [2] at DR = 01 and T s = 287 K Parameter Present Model TAHX of TASHE [2] Error (%) Number of tubes (N t ) % Tubes diameter (d t ) 24 mm 25 mm 4% Acoustic energy dissipation W 25%

7 Entropy 2016, 18, of Variation of Acoustic Energy Dissipation (Ė diss) due to Geometry Configuration Entropy 2016, 18, of 15 The31 impact Variation of of Acoustic geometry Energy Dissipation acoustic energy due to dissipation, Geometry Configuration both viscous and rmal losses, is examined The Itimpact should of be geometry pointedon out acoustic that at energy any dissipation, T s of interest, both viscous heat exchangers and rmal losses, with different is configurations examined (different It should tube diameters pointed out and that tube at any numbers) of interest, are designed heat exchangers with maintaining with different surface area as constant configurations Moreover, (different each tube configuration diameters and can tube transfer numbers) are specified designed heat with load, maintaining at designed T s with different surface value area as of constant dissipation Moreover, of acoustic each configuration energy can transfer specified heat load, at designed Figure 1 shows with Ė different value of dissipation of acoustic energy diss,v for designed shell-and-tube heat exchanger as a function of number of Figure 1 shows, for designed shell-and-tube heat exchanger as a function of number tubes atof various tubes at Tvarious s, ranging, ranging from 270 from K 270 to 305 K to K, 305 for K, normalized for normalized tube tube diameters (d ( i /δ κ ) ) It It can can be be seen by inspection seen by that, inspection for a fixed that, for T s, a viscous fixed dissipation, viscous dissipation decreases decreases with increasing with increasing d i /δ κ and and requiring less number requiring of tubes less number of tubes (a) (b) Figure 1 (a), of HXs with different and at DR = 01; (b), of HXs with Figure 1 (a) different Ė diss,v of HXs with different N t and d i at DR = 01; (b) and (Zoom for Nt < 500) Ė diss,v of HXs with different N t and d i (Zoom for N t < 500) One may state that at any, growth of viscous dissipation for heat exchangers with more One narrow may tubes stateis that associated at anywith T s, increase growthof of viscous cross-sectional dissipation area of for heat gas channels exchangers in heat with more exchanger ( ) while heat transfer area ( ) is kept constant for specified heat load Indeed, narrow tubes is associated with increase of cross-sectional area of gas channels in heat when tube diameter decreases in heat exchanger, number of tubes has to be increased exchanger (A to keep g ) while heat transfer area (A surface area unchanged However, s ) is kept constant for specified heat load Indeed, this triggers a decrease of which furr leads when to tube an increase diameter of decreases viscous resistance in heat This exchanger, is in agreement number with Equation of (21) tubes has to be increased to keep surface A simulation area unchanged on, analogous However, to this that triggers of a decrease, in Figure1 ofis performed A g which The furr results leads are to an increasedemonstrated of viscous in Figure resistance 2 It is This evident is in that agreement, has with a different Equation behavior (21) than, At constant A simulation on Ė diss,κ analogous to that of Ė diss,v in Figure 1 is performed The results are demonstrated in Figure 2 It is evident that Ė diss,κ has a different behavior than Ė diss,v At constant T s, an increase of normalized tube diameter (d i /δ κ ) has an increasing impact on rmal dissipation of acoustic power

8 Entropy 2016, 18, of 15 Entropy 2016, 18, of 15, an increase of normalized tube diameter ( ) has an increasing impact on rmal dissipation of acoustic power Entropy 2016, 18, of 15 Table 4 Parameters of optimal shell-and-tube heat exchangers at various at DR = 01 (, ) ( ) ( ) (K) (m 2 ) (-) (W) (-,mm) (-) (-) (203,25) Figure 2, of different and at DR = Figure 2 E46450 diss,κ of HXs895 with (225,25) N t and d i at DR = Considering different behaviors of 999, and,, (317,2) while viscous dissipation 317 decreases 127 Considering 285 with increasing different, rmal behaviors relaxation of Ė dissipation diss,v 1130 and Ė decreases diss,κ, (485,15) while A furr viscous investigation dissipation 238 made decreases 95 on with increasing 290 d i impact /δ κ, rmal of geometric relaxation parameters dissipation of 1303 HXs decreases on total (567,15) acoustic A furr energy investigation dissipation 238 Figure is made 953 on illustrates total acoustic energy dissipation that is, combined, and, as a impact of 295 geometric parameters of HXs on1525 total acoustic (1025,1) energy dissipation 159 Figure 363 illustrates function of number of tubes at various for normalized tube diameter ( ) total 300 acoustic A close energy inspection dissipation of results Ė 1837 diss that presented in is, Figure combined (1986,0065) 3b indicates Ė diss,v existence and 103 Ė of diss,κ as a configuration a 41 function of (2680,0065) number of for tubes which at various total Tdissipation, s for normalized, becomes tube diameter a minimum, (d i /δ κ ) The corresponding normalized tube diameters ( ) is referred to ( ) In fact, strong dependence of acoustic energy dissipation on both number and diameter of tubes suggests that minimization in acoustic energy dissipation can constitute an effective design criteria to choose optimal configuration of TAHXs for given conditions The optimal TAHXs configuration in correspondence with different operating conditions is presented in Tables 3 and 4 In fact, Tables 3 and 4 are illustrating shell-and-tube HXs with minimum acoustic energy dissipation at different values for drive ratios DR = 006 and DR = 01, respectively Furrmore, it should be mentioned that weight of viscous and rmal relaxation losses on total acoustic energy dissipation is different; rmal relaxation plays dominant role especially at higher values A close inspection of Figure 3b reveals that at high values rmal dissipation of acoustic power is much more than viscous dissipation; thus, an optimum configuration for which becomes a minimum is not available Table 3 Parameters of optimal shell-and-tube heat exchangers at various at DR = 006 (, ) ( ) ( ) (K) (m 2 ) (-) (W) (-,mm) (-) (-) (a) (92,55) (125,45) (141,45) (208,35) (341,25) (513,2) (861,15) (1742,1) (b) Figure 3 of HXs with different and (a) DR 006; (b) DR = 01 Figure 3 E diss of HXs with different N t and d i (a) DR = 006; (b) DR = 01 Moreover, considering that is related to total time averaged acoustic power dissipation of heat exchanger, one may state that both HX metal temperature as well as drive ratio significantly influence acoustic power dissipation of shell-and-tube HXs in travelling-wave engines The dependence of acoustic energy dissipation on HX metal temperature and drive ratio are discussed in following sections

9 Entropy 2016, 18, of 15 A close inspection of results presented in Figure 3b indicates existence of a configuration for which total dissipation, Ė diss, becomes a minimum, (Ė ) diss The corresponding normalized min tube diameters (d i /δ κ ) is referred to (d i /δ κ ) opt In fact, strong dependence of acoustic energy dissipation on both number and diameter of tubes suggests that minimization in acoustic energy dissipation can constitute an effective design criteria to choose optimal configuration of TAHXs for given conditions The optimal TAHXs configuration in correspondence with different operating conditions is presented in Tables 3 and 4 In fact, Tables 3 and 4 are illustrating shell-and-tube HXs with minimum acoustic energy dissipation (Ė ) diss min at different T s values for drive ratios DR = 006 and DR = 01, respectively Table 3 Parameters of optimal shell-and-tube heat exchangers at various T s at DR = 006 ( ) T s A s Re d Ediss (N min t, d i ) opt (d i /δ κ ) opt (r h /δ κ ) opt (K) (m 2 ) (-) (W) (-,mm) (-) (-) (92,55) (125,45) (141,45) (208,35) (341,25) (513,2) (861,15) (1742,1) Table 4 Parameters of optimal shell-and-tube heat exchangers at various T s at DR = 01 ( ) T s A s Re d Ediss (N min t, d i ) opt (d i /δ κ ) opt (r h /δ κ ) opt (K) (m 2 ) (-) (W) (-,mm) (-) (-) (203,25) (225,25) (317,2) (485,15) (567,15) (1025,1) (1986,0065) (2680,0065) Furrmore, it should be mentioned that weight of viscous and rmal relaxation losses on total acoustic energy dissipation is different; rmal relaxation plays dominant role especially at higher T s values A close inspection of Figure 3b reveals that at high T s values rmal dissipation of acoustic power is much more than viscous dissipation; thus, an optimum configuration for which Ė diss becomes a minimum is not available Moreover, considering that Ė diss is related to total time averaged acoustic power dissipation of heat exchanger, one may state that both HX metal temperature as well as drive ratio significantly influence acoustic power dissipation of shell-and-tube HXs in travelling-wave engines The dependence of acoustic energy dissipation on HX metal temperature and drive ratio are discussed in following sections 32 The Influence of T s on Acoustic Energy Dissipation (Ė ) diss The influence of T s on acoustic energy dissipation is furr examined While viscous dissipation decreases with T s, rmal relaxation dissipation increases as shown in Figures 1 and 2, respectively As it can be seen in Figure 3a,b, strong dependence of total acoustic energy dissipation on T s is caused essentially by rmal relaxation term The results show that by increasing T s,

10 32 The Influence of 32 The Influence of on Acoustic Energy Dissipation on Acoustic Energy Dissipation The influence of The influence of on acoustic energy dissipation is furr examined While viscous on acoustic energy dissipation is furr examined While viscous dissipation decreases with dissipation decreases with, rmal relaxation dissipation increases as shown in Figures 1 and 2, rmal relaxation dissipation increases as shown in Figures and 2, Entropy respectively 2016, 18, As 301 it can be seen in Figure 3a,b, strong dependence of total acoustic energy respectively As it can be seen in Figure 3a,b, strong dependence of total acoustic energy 10 of 15 dissipation on dissipation on is caused essentially by rmal relaxation term The results show that by is caused essentially by rmal relaxation term The results show that by increasing increasing, larger surface area is needed for heat transfer and consequently rmal larger surface area larger surface area is needed for heat transfer and consequently rmal dissipation of acoustic is needed power for increases heat transfer and consequently rmal dissipation of acoustic dissipation of acoustic power increases power Furr, increases it can be seen that both viscous and rmal relaxation dissipations do not vary linearly Furr, it can be seen that both viscous and rmal relaxation dissipations do not vary linearly proportional Furr, it to can be seen that both viscous and rmal relaxation dissipations do not vary linearly proportional to As temperature difference between gas and metal decreases, proportional to T s As temperature difference between gas and metal decreases, acoustic energy dissipation As temperature increases progressively difference between gas and metal decreases, acoustic acoustic energy dissipation increases progressively energy The dissipation design strategy, increases developed progressively in previous section and which is based on acoustic energy The design strategy, developed in previous section and which is based on acoustic energy dissipation The design minimization, strategy, developed is applied in to previous model section TAHX and under which study is based at different on acoustic operating energy dissipation minimization, is applied to model TAHX under study at different operating conditions dissipation minimization, At various drive is applied ratios and to model TAHX under study at different operating conditions conditions At various drive ratios and, optimal configuration and its corresponding At various drive ratios and T s, optimal configuration and its corresponding (Ė optimal configuration and its corresponding ) is determined, Figure 4 It can be seen that an increase of diss is determined, min is determined, Figure 4 It can be seen that an increase of has a minor Figure 4 It can be seen that an increase of T s has a minor effect on (Ė effect on has minor effect ) on at lower diss at lower at lower drive ratios; however, it has a major effect at larger drive ratios To be exact, drive ratios; drive ratios; however, it has major effect at larger drive ratios To be exact, min for a TAHX metal for TAHX metal temperature however, it has of a major effect at larger drive ratios To be exact, for a TAHX metal temperature of temperature of = 305 K, an increase of drive ratio from 004 to 01 will increase six-fold T s = 305 K, an increase 305 K, an increase of drive ratio from 004 to 01 will increase six-fold total dissipation of drive ratio from 004 to 01 will increase six-fold total dissipation total dissipation Figure 4 Figure 4 (Ė ) diss min at at any any T s The variation of ( ( ) with at different drive ratios is shown in Figure 5 It is evident The variation of (d i /δ κ ) opt with T s at at different drive ratios is is shown in in Figure 5 5 It It is evident that ( that (d( i /δ κ ) ) slightly decreases as opt slightly slightly decreases decreases T s increases increases This decrease in ( increases This decrease This decrease in (d i /δ κ ) in opt has ( a ) lower has a lower magnitude has lower at magnitude at larger drive ratios One may recall that as TAHX metal temperature increases, larger magnitude drive ratios larger One drive may ratios recallone thatmay as recall TAHX that metal as temperature TAHX metal increases, temperature corresponding increases, corresponding ( (d corresponding i /δ κ ) opt decreases that ( ) decreases that is, optimum diameter of tube decreases is, decreases that optimum diameter is, of optimum tube decreases diameter Theoretically, of tube it is decreases possible Theoretically, it is possible to obtain an optimum combination between to Theoretically, obtain an optimum it is possible combination to obtain an between optimum combination d i of tubebetween and TAHX metal of tube and TAHX of temperature tube and TAHX for a metal temperature for a minimum acoustic power dissipation minimum metal temperature acoustic power for minimum dissipation acoustic power dissipation Figure 5 ( (d Figure 5 ( i /δ κ ) opt at at various T s at various The variation of heat exchanger surface area A s versus T s is plotted in Figure 6 It is evident that A s is proportional to T s and hence acoustic power dissipated in HX and so (Ė ) diss is nearly in min proportion to T s This result is in good agreement with analysis in Section 21 that is, rmal

11 Entropy 2016, 18, of 15 Entropy 2016, 18, of 15 The variation of heat exchanger surface area versus is plotted in Figure 6 It is evident The variation of heat exchanger surface area versus is plotted in Figure 6 It is evident that is proportional to and hence acoustic power dissipated in HX and so Entropy that 2016, is proportional 18, 301 to and hence acoustic power dissipated in HX and so 11 is of 15 is nearly in proportion to This result is in good agreement with analysis in Section 21 that is, nearly in proportion to This result is in good agreement with analysis in Section 21 that is, rmal relaxation dissipation, which has biggest impact in total dissipation in HX, is rmal relaxation dissipation, which has biggest impact in total dissipation in HX, is relaxation proportional dissipation, to which has biggest impact in total dissipation in HX, is proportional proportional to to A s Figure 6 TAHX surface area variation with Figure 6 TAHX surface area variation with T s 33 The Influence of Drive Ratio (DR) on Acoustic Energy Dissipation 33 The Influence of Drive Ratio (DR) on Acoustic Energy Dissipation (Ė ) diss Furr Furr investigation investigation is is made made on on impact impact of of drive drive ratio ratio on on total total acoustic acoustic energy energy dissipation, dissipation, Figure Figure 3a,b 3a,b A A comparison comparison between between between se se se figures figures figures reveals reveals reveals strong strong strong dependency dependency dependency of of of total total total acoustic acoustic acoustic energy energy energy dissipation dissipation dissipation on on on drive drive drive ratio ratio ratio Furr, Furr, considering considering drive drive ratio ratio as as an an operating condition, analysis of influence of drive ratio on optimal TAHX configuration operating condition, analysis of influence of drive ratio on optimal TAHX configuration indicates increase of minimum energy dissipation indicates increase of minimum energy dissipation (Ė ) diss with drive ratio This fact is with min drive ratio This fact is clearly clearly supported supported by by rearrangement rearrangement of of previous previous results results illustrated illustrated in in in Figure Figure Figure7 7 (Ė ) diss Figure 7 variation with DR min variation with DR Finally, variation of Reynolds number with TAHX metal temperatures T Finally, variation of Reynolds number with TAHX metal temperatures s is provided in is provided in Figure Figure 8 8 As As mentioned mentioned previously, previously, one one may may specify specify values values for for Q,, T m, and and T s to to to determine determine A s, surface surface area area area of of of HX HX HX However, However, However, different different different HX configurations HX configurations HX configurations (N t,d (, ) might be designed to i ) ( might, be might designed be designed to provide to provide provide same A s same same Moreover, Moreover, Moreover, since since volume velocity of gas entering HX is constant for since volume volume velocity velocity of gas of entering gas entering HX is constant HX is constant for all for possible all possible configurations, volume velocity in each tube as well as changes Neverless, all possible configurations, configurations, volume volume velocity velocity in each in tube each as tube well as well as A g as changes changes Neverless, Neverless, Re d remains Re remains constant at each and decreasing temperature difference between oscillating Re remains constant constant at each at T each s and decreasing and decreasing temperature temperature difference difference between between oscillating oscillating gas and gas and heat exchanger will result in a larger heat transfer area as shown in Figure 6 Furr, gas heat and exchanger heat exchanger will result will in a result larger in heat larger transfer heat area transfer as shown area in as Figure shown 6 in Furr, Figure 6 Furr, increment of heat transfer surface area is provided by increasing number of tubes, consequently leading to a decrease of Re d in correspondence with T s

12 Entropy 2016, 18, of 15 increment of heat transfer surface area is provided by increasing number of tubes, consequently leading to a decrease of Re in correspondence with Entropy 2016, 18, of 15 Figure 8 Re d variation with T s 4 4 Conclusions Today, rmoacoustic engines, coolers, and heat pumps are gaining more interest for for commercialization While rmoacoustic engines utilize heat to generate acoustic power, rmoacoustic refrigerators consume acoustic acousticpower powerto totransfer transfer heat heat from from cold cold reservoir reservoir A Akey keyparameter in inmaking makingrmoacoustic rmoacousticdevices devicesmore competitivewith classical products is optimum design of of heat exchangers Hence, a deeper knowledge of of heat exchangers is is required so so that gas gas mean temperature does not not change along flow flow oscillation and and dissipate acoustic energy in in form of of viscous and and rmal relaxation while while transferring specified heat heat load load This research is is motivated by by developing a simple but constructive methodology for comparison and evaluation of ofrelated relatedheat heatexchangers exchangers with with respect respect to acoustic to acoustic energy energy dissipation, dissipation, viscous viscous loss, loss, rmal rmal relaxation relaxation loss, tube loss, dimension, tube dimension, and drive andratio drivein ratio this paper, In this paper, performance performance of shell-and- of shell-and-tube TAHXs in TAHXs travelling-wave travelling-wave mode with mode ideal with gases ideal is gases investigated is investigated through through a numerical a numerical model model based on based on classical classical linear rmoacoustic linear rmoacoustic ory ory The The results indicate that, that, at at equal equal operating conditions, viscous dissipation and and rmal relaxation dissipation have opposing behaviors While, decreases with relaxation dissipation have opposing behaviors While,, increases Consequently, an optimum value of exists, Ė diss,v decreases with d i /δ k, denoted as ( ), at Ė diss,κ increases Consequently, an optimum value of d which total i /δ k exists, denoted as (d i /δ k ) opt, at which total acoustic acoustic dissipation becomes a minimum, The impact of heat exchanger metal dissipation becomes a minimum, (Ė ) diss The impact of heat exchanger metal temperature on temperature on is also investigated min It is determined that, in general, increases with E Furr, diss is also investigated ( ) It is determined that, in general, are obtained to be decreasing with metal Ė diss increases with T temperature In s Furr, fact, heat (d transfer i /δ k ) opt are obtained to be decreasing with metal temperature T with minimum acoustic energy dissipation at lower s In fact, heat transfer with minimum acoustic temperature defects leads to For energy dissipation at lower temperature defects leads to d example, at a drive ratio of 006 and metal temperature i δ of 285, k For example, at a drive ratio of 006 minimum dissipation occurs and metal temperature around ( ) of 285, minimum dissipation occurs around (d 16; whereas at = 300 and higher, minimum i /δ dissipation k ) opt 16; whereas at appears at T relatively s = 300 and higher, minimum dissipation appears at relatively lower d lower The relation between acoustic energy dissipation i /δ k The relation between and drive ratio for acoustic energy dissipation and drive ratio for various T various is also presented It is shown that acoustic energy s is also presented It is shown that dissipation increases with drive acoustic energy dissipation increases with drive ratio The growth rate, however, was lower for ratio The growth rate, however, was lower for smaller values of smaller values of T The proposed s model is able to put in evidence strong dependence of acoustic energy The proposed model is able to put in evidence strong dependence of acoustic dissipation on both diameter and number of tubes and existence of minima in energy dissipation on both diameter and number of tubes and existence of minima correspondence with tube diameter normalized by rmal penetration depth The different weight in correspondence with tube diameter normalized by rmal penetration depth The different of viscous and rmal losses affecting HXs behavior is also highlighted with second one weight of viscous and rmal losses affecting HXs behavior is also highlighted with dominating Also important for design purposes is that acoustic energy dissipation minimization second one dominating Also important for design purposes is that acoustic energy dissipation criterion can be effectively employed for optimization of configuration of minimization criterion can be effectively employed for optimization of configuration of shell-and-tube TAHXs shell-and-tube TAHXs Acknowledgments: The present work has been supported in part by office of research at Sharif University of Technology

13 Entropy 2016, 18, of 15 Author Contributions: Mohammad Gholamrezaei developed algorithm, provided results and wrote draft Kaveh Ghorbanian was in charge of technical examination and language proficiency Both authors have read and approved final manuscript Conflicts of Interest: The authors declare no conflict of interest Nomenclature English letters: a sound speed (m s 1 ) A s surface area (m 2 ) A g cross-sectional area of gas channels (m 2 ) d o tube outside diameters (m) d i tube inside diameters (m) DR drive ratio (= p 1 /p m ) - E acoustic power dissipated (W) E diss acoustic power dissipated (W) E diss,v viscous dissipation of acoustic power (W) E diss,κ rmal dissipation of acoustic power (W) E 2 time averaged acoustic power (W) f spatially averaged diffusion function - f k spatially averaged rmal diffusion function - f v spatially averaged viscous diffusion function - g complex gain/attenuation constant for volume flow rate - h o shell-side heat transfer coefficient (W m 2 K 1 ) h i tube-side heat transfer coefficient (W m 2 K 1 ) k gas rmal conductivity (W m 1 K 1 ) L t length of tubes (m) L TAHX optimum length of rmoacoustic heat exchanger (m) N t number of tube - p Acoustic pressure (Pa) p m mean pressure (Pa) Q heat transfer rate (W) r h hydraulic diameter (m) r v viscous resistance per unit length (Pa m 4 s) r k rmal resistance per unit length (Pa m 4 s) r acoustic resistance per unit length (Pa m 4 s) Re d tube side acoustic Reynolds number - T lm log-mean temperature (K) T gas temperature (K) T w,o water outlet temperature (K) T w,i water inlet temperature (K) T 0 ambient temperature (K) T m mean temperature (K) U oscillating volume flow rate (m 3 s) U 0 overall heat transfer coefficient (W m 2 K 1 ) U i overall heat transfer coefficient (W m 2 K 1 ) u 1 oscillating velocity (m s 1 )

14 Entropy 2016, 18, of 15 Greek symbols: δ k rmal penetration depth (m) γ ratio of isobaric to isochoric specific heats - ε rmal conductivity (W m 1 K 1 ) ɛ s correction factor for finite solid heat capacity - µ dynamic viscosity (kg m 1 s 1 ) ρ density of working gas (kg m 3 ) σ Prandtl number - ω angular frequency (s 1 ) ( ) E diss minimum value of min diss (W) (d i /δ κ ) opt optimum value of d i /δ κ - Subscripts: diss dissipation m mean min minimum opt optimum κ rmal ν viscous 0 environment or ambient 1 first order 2 second order References 1 Swift, GW Thermoacoustic engines J Acoust Soc Am 1988, 84, [CrossRef] 2 Backhaus, S; Swift, GW A rmoacoustic Stirling heat engine: Detailed study J Acoust Soc Am 2000, 107, [CrossRef] [PubMed] 3 Swift, GW Thermoacoustics: A Unifying Perspective for Some Engines and Refrigerators; Acoustical Society of America: Melville, NY, USA, Wu, F; Wu, C; Guo, F; Li, Q; Chen, L Optimization of a Thermoacoustic Engine with a Complex Heat Transfer Exponent Entropy 2003, 5, [CrossRef] 5 Zhibin, Y; Jaworski, AJ Impact of acoustic impedance and flow resistance on power output capacity of regenerators in travelling-wave rmoacoustic engines Energy Conv Manag 2010, 51, Tasnim, SH; Mahmud, S; Fraser, RA Second law analysis of porous rmoacoustic stack systems Appl Therm Eng 2011, 31, [CrossRef] 7 Chaitou, H; Nika, P Exergetic optimization of a rmoacoustic engine using particle swarm optimization method Energy Conv Manag 2012, 55, [CrossRef] 8 Ishikawa, H; Hobson, PA Optimization of heat exchanger design in a rmoacoustic engine using a second law analysis Int Commun Heat Mass Transf 1996, 23, [CrossRef] 9 Piccolo, A Optimization of rmoacoustic refrigerators using second law analysis Appl Energy 2013, 103, [CrossRef] 10 Piccolo, A Numerical Study of Entropy Generation Within Thermoacoustic Heat Exchangers with Plane Fins Entropy 2015, 17, [CrossRef] 11 Swift, GW; Backhaus, S A resonant, self-pumped, circulating rmoacoustic heat exchanger J Acoust Soc Am 2004, 116, [CrossRef] 12 Incropera, FP; DeWitt, DP; Bergman, TL; Lavine, AS Introduction to Heat Transfer; John Wiley & Sons: Hoboken, NJ, USA, 2006

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