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1 Research Archive Citation or published version: Lijing Zhai, et al, Numerical analysis o the axial heat conduction with variable luid properties in a orced laminar low tube, International Journal o Heat and Mass ranser, Vol. 114: , November DOI: Document Version: his is the Accepted Manuscript version. he version in the University o Hertordshire Research Archive may dier rom the inal published version. Copyright and Reuse: 2017 Elsevier Ltd. his manuscript version is made available under the terms o the Creative Commons Attribution-NonCommercial- NoDerivatives License CC BY NC-ND 4.0 ( ), which permits non-commercial re-use, distribution, and reproduction in any medium, provided the original work is properly cited, and is not altered, transormed, or built upon in any way. Enquiries I you believe this document inringes copyright, please contact the Research & Scholarly Communications eam at rsc@herts.ac.uk

2 Numerical analysis o the axial heat conduction with variable luid properties in a orced laminar low tube Lijing Zhai a, Guoqiang Xu a, Yongkai Quan a,, Gu Song a, Bensi Dong a, Hongwei Wu b, a National Key Laboratory o Science and echnology on Aero-Engine Aero-thermodynamics, School o Energy and Power Engineering, Beihang University, Beijing , China b School o Engineering and echnology, University o Hertordshire, Hatield, AL10 9AB, United Kingdom Corresponding author. quanyongkai@buaa.edu.cn el. +86(010) Corresponding author. h.wu6@herts.ac.uk el. +44(0) ; Fax. +44(0) /41

3 Abstract In this article, a theoretical model is developed to investigate the eects o the axial heat conduction on the laminar orced convection in a circular tube with uniorm internal heat generation in the solid wall. In the current work, three dierent luids, i.e. water, n-decane and air, are selected on purpose since their thermophysical properties show dierent behavior with temperature. he eects o the axial heat conduction with varying dynamic viscosity and/or varying thermal conductivity are investigated in a systematic manner. Results indicate that the variable-property eects could alleviate the reduction in Nusselt number (Nu) due to the axial heat conduction. For the case o Peclet number (Pe) equal to 100, wall thickness to inner diameter ratio o 1 and solid wall to luid thermal conductivity ratio o 100, the maximum Nu deviation between constant and variable properties are up to 7.33% at the entrance part or water in the temperature range o 50, and 4.45% at the entrance part or n-decane in the temperature range o 120, as well as 2.20% at the ending part or air in the temperature range o 475, respectively. In addition, the average Nu deviation or water, n-decane and air are 3.24%, 1.94% and 1.74%, respectively. Besides, Nu decreases drastically with decreasing Pe when Pe 500 and with increasing solid wall to luid thermal conductivity ratio ( k s ) when ks 100. It is also ound that variable properties have more obvious eects on the velocity proile at the upstream part while more obvious eects on the temperature proile at the downstream part. Keywords Conjugate heat transer; laminar low; axial heat conduction; variable luid properties; Nusselt number. 2/41

4 Nomenclature u in uniorm inlet velocity (m/s) k s solid wall to luid thermal conductivity ratio t in uniorm inlet temperature (K) Nu z local Nusselt number u m average axial velocity (m/s) Re Reynolds number u max maximum axial velocity (m/s) Pr Prandtl number t temperature (K) Pe Peclet number dimensionless temperature dimensionless luid bulk temperature p pressure (Pa) w dimensionless interacial wall temperature P dimensionless pressure q dimensionless interacial heat lux u axial velocity (m/s) q w,in actual interacial heat lux (W/m 2 ) U dimensionless axial velocity q w,in ideal interacial heat lux (W/m 2 ) v radial velocity (m/s) z dimensionless axial distance V dimensionless radial velocity Nu Nusselt number deviation (%) r radial coordinate (m) R dimensionless radial coordinate Greek symbols z axial coordinate (m) ρ density (kg/m 3 ) Z dimensionless axial coordinate internal heat generation (W/m 3 ) u(r) axial velocity proile in radial cross-section δ wall thickness to inner diameter ratio t(r) temperature proile in radial cross-section μ dynamic viscosity (Pa s) L tube length (m) wall thickness (m) A cross-sectional area o the tube (m 2 ) m mass low rate (kg/s) Subscripts r i tube inner radius (m) luid r o tube outer radius (m) s solid d i tube inner diameter (m) w wall d o tube outer diameter (m) cp constant luid properties c p speciic heat capacity (J/kg K) vp variable luid properties k thermal conductivity (W/m K) wall considering the axial heat conduction L h L t theoretical hydrodynamic length (m) theoretical thermal entrance length (m) 3/41

5 1. Introduction Conjugate heat transer problems are not new in concept and have been still extensively studied over the past decades because o its signiicance in a large variety o engineering applications, such as pipeline petroleum transport, micro-devices as well as aerospace technology. he axial heat conduction has strong eects on heat patterns o the internal low in channels with thick walls, which are ar rom the idealized boundary conditions, such as constant outside wall temperature and uniorm outside heat lux, on which the available standard heat transer correlations are based. he discrepancies between experimental results and numerical predictions based on the conventional correlations which neglect the axial heat conduction are requently reported in scientiic literature [1, 2]. Faghri and Sparrow [3] reported that the inluence o the axial heat conduction was relatively greater or laminar low than that or turbulent low. heir results also showed that the wall axial conduction could easily overwhelm the luid axial conduction. hereore, more attention should be paid to the axial heat conduction in the solid region at low Reynolds number [4, 5]. Many analytical, numerical and experimental works have been done in order to reveal the eects o the axial heat conduction on the laminar low. Zarieh et al. [6] and Bilir [7] used a inite-dierence method to investigate the combined eects o the wall and luid axial conduction. hese studies were based on the assumption o one-dimensional conduction in the solid wall. Quintero and Vera [8] theoretically and numerically studied the multilayered, counterlow, parallel-plate heat exchangers, considering both axial and transverse wall conduction eects. Davis and Gill [9] employed analytical and experimental methods to investigate the Couette low between parallel plates, studying the parameters that determined the relative importance o the axial wall heat conduction. Adelaja et al. [10] used the separation o 4/41

6 the variables in order to investigate the conjugate heat transer in tube laminar low. Axtmann et al. [11] ocused on the convective heat transer in the thermal entrance region o the concentric annuli and proposed Nusselt number correlations o laminar and turbulent internal lows. Furthermore, the axial heat conduction is more prominent in microchannels since the wall thickness is signiicant compared to the tube s hydraulic diameter, and thus almost all o the microchannels should be considered as a thick-wall type. Detailed research activities were perormed in microchannels over the past decades, such as Nonino et al. [12], S. X. Zhang et al. [13], oh et al. [14] and Gamrat et al. [15]. With respect to the evaluation criteria o the axial heat conduction, Faghri and Sparrow [3], Maranzana et al. [16], Rahimi and Mehryar [17] and Lin et al. [18] deined their own dimensionless wall conduction number M, which represented the ratio o the axial heat conduction in the solid wall to the heat convection in the luid rom dierent angles. Maranzana et al. [16] concluded that the axial conduction could be neglected when M was lower than 0.01 or most cases. Lin et al. [18] took the temperature gradient o the solid wall and luid into consideration to deine a modiied axial conduction number. According to the conclusions obtained rom above studies, whether the eects o the axial heat conduction can be neglected or not is highly situation-dependent and the axial conduction number is not the only criterion. On the other hand, the assumption o the constant luid properties is inaccurate, since the luid low and heat transer characteristics in channels with variable luid properties reveal signiicant deviations rom that with constant properties. Many previous studies paid attention to the inluence o the variable properties in macro- or micro-convection within the continuum regime during the last ew decades. Kumar and Mahulikar [19] investigated the eects o the temperature-dependent viscosity 5/41

7 variation on the ully developed low through a microchannel and observed our dierent low regions in the low regime. Ho et al. [20] studied the eects o the temperature-dependent thermophysical properties on the laminar orced convection eectiveness o the Al 2 O 3 -water nanoluids in a circular tube imposed to a constant heat lux. It was ound that the eects o the thermophysical properties on the Nusselt number were more notable with an increase in nanoluid concentration. Ghosh et al. [21] developed a simulation algorithm o the multistream plate in heat exchangers incorporating the eects o the axial conduction in the heat exchanger core, heat leakage and variable luid properties. Lelea [22] investigated the conjugate heat transer o the variable-property water low inside the microtube and analyzed the inluence o the heating position, tube material, wall thickness and Re upon the thermal parameters. Nonino et al. [23] investigated the laminar orced convection at the entrance region o the straight ducts with variable viscosity according to an exponential relation by a inite element procedure, considering dierent cross-sectional geometries. Aterwards, they adopted the same procedure to study the eects o the viscous dissipation and temperature dependent viscosity in thermally and simultaneously developing laminar lows [24]. A series o work by Mahulikar and Herwig were related to the variable-property eects in laminar convection o the incompressible liquid and compressible gas. Mahulikar and Herwig [25] investigated the convection rom macro-to-microscale by re-examing the dimensionless governing equations including the dynamic viscosity and thermal conductivity variation. hey concluded that the eects o the property variations became highly signiicant rom macro-to-microscale convection, and that the eects o the property variations along the low became more signiicant relative to the property variations over the cross section. Similar conclusions were also drawn rom their research [26]. Later on, they 6/41

8 made a urther analysis o the laminar micro-convection eects due to the variation o the dynamic viscosity and thermal conductivity o the incompressible liquid [27]. Ater that they paid attention to the laminar micro-convection due to variation o the gas properties [28], incorporating the physical eects o the variation o the gas density with pressure and temperature as well as gas viscosity and thermal conductivity with temperature. Gulhane and Mahulikar [29, 30] explored the orced laminar gas micro-convection due to density, speciic heat capacity, viscosity and thermal conductivity variations with entrance eects. Ramiar and Ranjbar [31] studied the eects o the axial conduction and variable properties on two-dimensional conjugate heat transer o Al 2 O 3 -EG/water mixture nanoluid in microchannel, concluding that considering variable properties caused higher Nu and lower shear stresses compared with constant properties. he main conclusion rom these researches is that taking variable-property into consideration is more accurate and more realistic, especially at low Re [32, 33]. It appears rom the aorementioned investigations that the variable-property eects play a signiicant role on the luid low and heat transer characteristics. o the best knowledge o the authors, however, the comprehension o the low and heat transer mechanism o the axial heat conduction with varying luid properties has been ar rom complete and there is still much room to be enhanced in this area. From the mechanism research s point o view, the main objective o the present work is to propose a theoretical model capable o investigating the conjugate heat transer characteristics with varying luid properties such as dynamic viscosity and thermal conductivity with an emphasis on the axial heat conduction in a orced laminar low tube. hereore, in this work three dierent luids, i.e. water, n-decane (C10) and air, are selected in order to achieve a proound comprehension on the heat transer 7/41

9 mechanism o conjugate heat transer problems with varying luid properties. 2. Mathematical model and numerical analysis 2.1. Mathematical model Fig. 1 illustrates the schematic diagram o the mathematical model. Fluid with uniorm velocity u in and temperature t in enters a circular tube with inner radius r i, wall thickness, and length L. In the current study, the tube length is ive times o the thermal entrance length in order to incorporate both developing and ully developed regions. he heat source is associated with a uniorm internal heat generation applied in the solid wall. he two ends and outside wall o the tube are set as adiabatic conditions. he current work investigates both developing and developed laminar low, which occurs when velocity and temperature ields develop simultaneously as the heat transer begins at the duct inlet. Fig. 1. Schematic diagram o the mathematical model and coordinate system. he mathematical descriptions are derived rom continuum-based equations o mass, momentum and energy. he ollowing major assumptions are employed in the derivations o the governing equations: (1) Incompressible Newtonian luid and steady laminar low. (2) hermal conductivity and dynamic viscosity are the only physical properties 8/41

10 varying as a unction o temperature while other luid properties are kept unchanged. (3) Constant thermal properties o the solid wall. (4) Negligible eects o electromagnetic orces, gravity and other body orces. (5) Negligible radiation heat transer. (6) Negligible heat generation due to viscous dissipation. (7) No slip low and no temperature jump. Other simpliications are described in due course in the rest o the paper. Based on the above assumptions, in the present study, two-dimensional steady low and heat transer o the incompressible luids will be taken into account. he continuity equation (1), momentum equations (2) and (3) in axial and radial directions, as well as energy equations (4) and (5) in the luid and solid regions, respectively, are as ollows. Continuity equation: u 1 rv z r r 0 (1) Momentum equations: uu 1 r vu p u 1 u r z r r z z z r r r uv 1 r vv p v 1 v v r 2 z r r r z z r r r r (2) (3) Energy equations: Liquid Solid,, cp ut 1 r cp vt t 1 t k rk z r r z z r r r ts 1 ts ks rk s 0 z z r r r (4) (5) where, p, c, k and are the luid density, speciic heat capacity, thermal 9/41

11 conductivity and dynamic viscosity, respectively; is the internal heat generation in the solid wall, W/m 3. According to the physical problem described in Fig.1, the boundary conditions can be mathematically summarized as ollows: z 0, 0 r ri, u uin, v 0, t tin ts z 0, ri r ro, u v 0, 0 z ts z L, ri r ro, u v 0, 0 z u t r 0, 0 z L, 0, v 0, 0 r r t r r, 0 z L, u 0, v 0, t t, k k r i s s ts r (6) In order to generalize results, the non-dimensional governing equations and boundary conditions are derived in the ollowing: Continuity equation: U 1 RV Z R R 0 (7) Momentum equations: UU 1 RUV P 2 U 1 2 U R Z R R Z Z Re Z R R Re R UV 1 RVV P 2 V 1 2 V 2 V R Z R R R Z Re Z R R Re R Re R 2 (8) (9) Energy equations: U 1 RV R Z R R Z Pe Z R R Pe R Liquid (10) Solid 2 s R Z R R R k s (11) he non-dimensionl parameters in Eqs. (7)-(11) are deined as below: 10/41

12 di i i = k,, r r z r u v p, Z, R, U, V, P, q r r d r r u u u ' w, in o i i i i in in in k t t ud c ud c, k, Re, Pr =, Pe Re Pr. in s i p i p ' s qw, in ri k k k where k s stands or solid wall to luid thermal conductivity ratio, deined as k k / k s s ; stands or wall thickness to inner diameter ratio, deined as / di ; is the wall thickness. hese equations are solved under the ollowing boundary conditions: Z 0, 0 R 1.0, U 1.0, V 0, 0 ro Z 0, 1.0 R, U V 0, 0 ri Z L ro s Z, 1.0 R, U V 0, 0 (12) ri ri Z L U R 0, 0 Z, 0, V 0, 0 ri R R L s R 1.0, 0 Z, U 0, V 0, s, k ks ri R R he dimensionless governing equations above suggest that the axial heat conduction mainly depends on our parameters, such as Pr, Re, the dimensionless parameter k s and. For a given luid, axial heat conduction depends on only three parameters: Pe, k s and since Pe equals to the product o Pr and Re. he local Nusselt number or the circular cross-section tube is deined in Eq. (13). Nu z t di r q d rr w, in i i tw t k tw t (13) where t is the luid bulk temperature, which is the average luid temperature weighted by the mass low rate; t w is the wall temperature at the solid-luid interace; q w, in is the actual heat lux density with the unit o W/m 3 calculated at the inside 11/41

13 surace o the tube wall which depends on axial distance z; q is the ideal heat lux ' w, in density calculated according to as ollows: q ' w, in 2 2 ro ri (14) 2r i In the ollowing sections the dimensionless interacial heat lux q, temperature and Re are deined in Eq.(15-17) respectively: q q (15) q w, in ' w, in in ' w, in t t k q r i (16) 2ru i m Re (17) where u m is the average velocity o the luid, deined as um m / A. According to [34], the theoretical hydrodynamic and thermal entrance lengths are given in Eq. (18a) and (18b). L h 0.05d Re (18a) i L t 0.05d RePr (18b) i 2.2. Numerical method and validation In the current work, the commercially available computational luid dynamics (CFD) sotware, ANSYS CFX 15.0, was used to solve the governing equations with imposed boundary conditions mentioned above. A mass-low-inlet and temperature-inlet were set as the inlet boundary conditions, and a pressure-outlet was set as the outlet boundary condition. he element-based inite volume method (FVM) was used to discretize the governing equations and imposed boundary conditions [35]. 12/41

14 he second order high resolution was used or the discretization o the convective term. Furthermore, the SIMPLE algorithm [36] was adopted to deal with the pressure-velocity coupling. he internal wall was set to satisy no slip and no penetration conditions. Since the governing equations are coupled and thus there was no need to speciy the interacial boundary condition between the luid and solid domain. he iterations were continued until a converged solution was obtained with root mean square residuals less than or all the variables. he laminar low o water with constant properties in a circular tube without solid region and with uniorm heat lux over the entire wall surace was considered as a reerence. Under these conditions, the accurate analytical value o Nu or ully developed low is reported 48/11(appropriate 4.364) [37]. he grids in axial direction were uniorm while the grids in radial direction were nonuniorm with mesh reinement near the inner wall surace shown in Fig. 2. Fig. 2. Computational channel mesh. he grid independence study was conducted to identiy an appropriate grid density or the aimed calculations. he computational domain included both the luid and solid wall regions, which were discretized using ive dierent grid arrangements o , , , and he numerical results o 13/41

15 the local Nusselt number ( Nu ) versus dimensionless axial distance z z r Pe z / i are presented in Fig. 3. Besides, table 1 illustrates the relative dierences o Nu z in the ully developed region between simulated and analytical values or dierent grid arrangements. Fig. 3. Grid independent study. (water, Re=14.3, Pr=7, / d i 0) able 1 Relative dierences between simulated values and analytical values. Grid arrangement Simulation error (%) RadialAxial (Nu-4.364)/ In Fig. 3, it is apparent that the simulated ully developed Nu z generally agrees with the analytical value with a relative error lower than 0.425% at various grid arrangements. he vertical line o z =0.1 stands or the dimensionless thermal entrance length predicted by Eq. (18b), which is only an experimental correlation not 14/41

16 a precise expression, and hence a deviation can be observed. he axial velocity proile and temperature proile in cross-section in ully developed laminar low have analytical solutions as ollows [37]. 2 2 r r m max ri ri u 2u 1 u 1 (19) 4 q w 3 2 r 2 t tw ri r 2 k ri 4 4ri (20) Velocity/temperature proiles According to the above equations, or grid arrangement o , the analytical velocity/temperature proiles and simulated velocity/temperature proiles at z =0.25 are shown in able 2, in which the adjust R-square o simulations is 1. able 2 Velocity and temperature proiles o analytical solution and simulated solution. Analytical solution Simulated solution u r r r r tr r r r r r r From able 2, it can be seen that at the grid arrangement o , both the temperature and velocity proiles in the ully developed region show good agreements with analytical solutions. Furthermore, as shown in able 1, rom grid arrangement o to , the grid number increases by 37.12% while simulation error decreases by just 0.01%. hereore, the grid arrangement o or the computational domain has satisactory grid-independence and is suicient to resolve the conjugate heat transer problem in orced laminar low. 3. Results and discussion 3.1. Axial heat conduction with constant luid properties For the purpose o comparison, the eects o the axial heat conduction with 15/41

17 constant properties are studied irstly in order to gain a deep understanding o the axial conduction eects with variable luid properties. As previously stated, or a given luid, the degree o the axial heat conduction is only determined by Pe, k s and. he inluence o these parameters on the axial heat conduction will be discussed in detail in this section. In the present study, water with constant properties is chosen as the benchmark case. Four dierent values o Pe i.e., 30, 100, 500 and 1000, together with three dierent k s values o 1, 100 and 500, as well as three dierent values o Δ, namely, 0.5d i, 1d i and 2d i are considered. he k s values representing the thermal conductivity ratio o solid wall to luid are selected according to [13] in practical applications. able 3 is a summary o the value arrangements or numerical simulations. A total o 36 simulations are carried out to allow or all the possible combinations. he eects o Pe, Nu z, the dimensionless interacial heat lux k s and on the local Nusselt number q, dimensionless luid bulk temperature and dimensionless interacial wall temperature w, are shown in Fig.4, Fig.5 and Fig.6, respectively, with reerence to representative cases. here are three straight dot lines in Fig.4a, Fig.5a and Fig.6a. he horizontal lines o Nu=4.364 and Nu=3.66 stand or the analytical values o ully developed Nu under constant heat lux boundary condition and constant temperature boundary condition, respectively. he vertical line o z =0.1 stands or the thermal entrance length predicted by Eq. (18b). able 3 Value arrangements or numerical simulations. Re Pe ks / k / di L/ d i /41

18 Inluence o Pe, k s and on q, and It is recognized that the axial wall heat conduction can result in more heat conduction toward the entrance where temperature o the wall is lower due to the higher convective heat transer coeicient, as shown in Fig. 4d, Fig. 5d and Fig. 6d. hus, more heat enters the luid at the entrance whereas less heat enters the luid at the exit according to the conservation o the total heat. he eects o the axial heat w conduction on q, and w can be summarized as ollows: (1) axial heat conduction can lead to the luid bulk temperature increase, delecting rom linear distribution or those without axial heat conduction, as demonstrated in Fig. 4c, Fig. 5c and Fig. 6c; (2) axial heat conduction orces the interacial wall temperature to increase in the upstream part while to decrease in the downstream part, approaching uniorm wall temperature boundary condition, as shown in Fig. 4d, Fig. 5d and Fig. 6d; (3) axial heat conduction can aect the interacial heat lux higher in the upstream part while lower in the downstream part, departing rom uniorm heat lux boundary condition, as illustrated in Fig. 4b, Fig. 5b and Fig. 6b; (4) consequently, axial heat conduction lowers Nu z along the tube relative to those under constant heat lux boundary condition at the solid-luid interace, as shown in Fig. 4a, Fig. 5a and Fig. 6a. 17/41

19 Fig. 4. Eects o Pe on the axial distribution o a: Nu z ; b: q ; c: ; d: w. Fig.5. Eects o k s on the axial distribution o a: Nu z ; b: q ; c: ; d: w. 18/41

20 Fig. 6. Eects o δ on the axial distribution o a: Nu z ; b: q ; c: ; d: w. Fig. 4 shows that the axial heat conduction becomes more dominant with the increase o Pe. Increasing Pe can decrease the thermal resistance across the low and urther decrease the eects o axial heat conduction. As shown in Fig. 5, the axial heat conduction becomes more signiicant as k s increases. Since k s is the ratio o the thermal resistance in the low to that in the wall, with increasing k s axial heat conduction in the wall increases due to the wall thermal resistance reduction. It can be observed rom Fig. 6 that the larger implies the more signiicant axial heat conduction. With respect to, compared to that in the axial direction, the higher the ratio, the larger the thermal resistance in the radial direction. hus, it is more likely to conduct more heat to the upstream o the tube. It can be generally observed rom above igures that the axial heat conduction becomes more dominant with increasing k s and as well as with decreasing Pe. 19/41

21 Eects o the axial heat conduction on Nusselt number he ollowing noticeable eatures o Nu can be observed rom Fig. 4a, Fig. 5a and Fig. 6a: (1) It is noted that the axial heat conduction has the generalized eects o reducing Nu z with respect to the no wall case. As the eects o the axial heat conduction become more signiicant, Nu z in the intermediate part gradually decreases rom 4.36 to aking the axial heat conduction into account, the circular tube may experience a boundary condition transormation: rom the uniorm outside heat lux boundary condition to the constant outside wall temperature boundary condition. (2) he results show that the axial heat conduction causes Nu z to decrease sharply at both the entrance and exit o the tube, which can lead to a reduction in average Nu. Similar behavior is also ound but will not be repeated here. he sharp drop o Nu z at the entrance can be explained by the act o the entrance eects [32] while the sharp reduction in Nu z at the exit can be explained by the inluence o the axial heat conduction, which causes more heat enters the luid at the entrance whereas less heat enters the luid at the exit on account o the conservation o the total heat. hus, it appears to be a sharp drop o Nu z at the exit. (3) It is recognized that the axial heat conduction causes Nu z to decrease aster at the entrance, which contributes to the laminar low towards a thermal ully developed region. his will have a negative inluence on the enhancement o the heat transer. Unlike the results presented by Rahimi and Mehryar [17], Fig. 4a, Fig. 5a and Fig. 6a present that Nu z in the ully developed region is not always a constant value. Furthermore, in some cases Nu z does not monotonically decrease rom the inlet to the outlet, but demonstrates a local minimum near the entrance. Such minima are also reported in some numerical [12, 22, 31] and experimental [39] researches. 20/41

22 From the point view o Nu, in the entrance and ending regions, Nu varies signiicantly while in the ully developed region Nu varies less. Speciically, the average Nu deviations o all the 36 cases or the entrance region, or the ully development region and or the ending region are up to %, 1.829% and 4.415%, respectively. Here, the ully developed region is dierent rom the no wall case, in which Nu remains constant. In the ully developed region discussed here, the thermal boundary layer converges at the central streamline though the temperature proile changes along the channel as shown in Fig. 15a. Besides, Fig. 15b indicates that the velocity proile does not change in the ully developed region even considering axial heat conduction. Fig. 7 shows the Nusselt number deviation between the case with axial heat conduction and that without axial heat conduction as a unction o Pe, k s and. Erro_Nu is calculated by Eq. (21). In Fig. 7, Nu decreases drastically with decreasing Pe at low Pe (Pe 500) and with increasing k s at low k s ( k s 100). As axial heat conduction becomes dominant, Pe has more eects than k s and. Nu Nu (21) Nu cp, wall cp Erro _ Nu 100 cp 21/41

23 22/41

24 Fig.7. Eects o Pe, k s and δ on Erro_Nu Axial heat conduction with variable luid properties As mentioned earlier, in a real case the solid wall with strong axial heat conduction conducts a portion o heat rom the downstream part back to the upstream part o the channel, leading to the redistribution o q, and w, which inally contributes to the change o Nu z. But most o the open published literatures take the luid properties as constant when investigating the axial heat conduction. his section will urther study the eects o the axial heat conduction allowing or variations in luid properties. When the low is incompressible, property variations may occur due to the temperature or pressure dependent viscosity, thermal conductivity, speciic heat capacity and other luid properties. Among all the thermophysical properties, the present work selected dynamic viscosity and thermal conductivity as the most prominent and the most important luid properties while other properties were kept 23/41

25 constant. Prandtl number ( Pr c / k ) is the characteristic number o a luid. In the p temperature ranges examined in this study, dynamic viscosity and thermal conductivity have the greatest change among the three kinds o thermophysical properties (, c p, k ). Fluid-property change rate or water, C10 and air rom inlet to outlet is listed in able 4. In addition, the inluence o the pressure on the luid properties was not considered. his assumption, though somehow unrealistic or certain luid especially or gas, can clearly reveal the role played by the most prominent luid properties rom a research point o view. In this section, three dierent working luids, i.e., water, C10 and air are selected since the behavior o both and k o the three luids are dierent, which can represent the change o and k with temperature increase or most luids. k used in the calculations o Nu z by Eq. (13) was obtained according to the luid bulk temperature. For water and C10, the numerical simulations were conducted within the temperature range o the liquid phase at atmospheric pressure. he air considered in this study was not ideal gas since its density remained constant. For the three kinds o luids, the temperature-dependent and k were given by cubic polynomial itting results o data rom luid property database REFPROP Version 9.0 developed by the National Institute o Standards and echnology [40] in the orm o 2 3 / k a a a a, where a0, a1, a2, a 3 were constant coeicients he ollowing work was conducted or the case with Pe=100, ks 100 and 1 since variations in luid properties were the ocus o this section. While or water, the present work studied cases under dierent parameters related to the axial heat conduction. 24/41

26 Fluid able 4 Fluid-property change rate or water, C10 and air (rom inlet to outlet). emperature range( ) Dynamic viscosity change rate (%) Water C Air hermal conductivity change rate (%) Eects o both separately and simultaneously variable and k on Nu z, q, and w Fig. 8. Comparisons o the local values o relevant parameters between constant and variable properties o water. (a: Nu z ; b: q ; c: ; d: w ) Comparisons o the local values o relevant parameters (Nu z, q, and between constant and variable properties or water are shown in Fig. 8. It is clear that Nu z with variable is much higher than that with constant properties. It is generally believed that a lower viscosity contributes to a higher velocity and a smaller boundary layer thickness, which is avorable or promoting convective heat transer in the channel. Fig. 8 presents that Nu z with variable k is a little higher than that w ) 25/41

27 with constant properties. he thermal boundary layer thickness indicates the scale o temperature gradient across the low and thus directly determines the local heat transer perormance. A higher thermal conductivity enhances the heat transer perormance. hereore, the increase in thermal conductivity account or the higher Nu z o water as temperature rises. Fig. 8 shows that Nu z with both variable and k is much higher than that with constant properties. Since water has a lower level and a higher k level as temperature rises, the heat transer in the heated tube is enhanced with variable and k or water, both separately and simultaneously, within the temperature range considered here. It can also be seen rom Fig. 8 that variations in luid properties have little eects on q, and w or water due to its small change rate o luid properties. Fig. 9. Comparisons o the local values o relevant parameters between constant and variable properties o C10. (a: Nu z ; b: q ; c: ; d: w ) In Fig. 9 the comparisons o the local values o relevant parameters (Nu z, q, and w ) between constant and variable properties or C10 are shown. Since C10 26/41

28 has a lower level as temperature rises, which contributes to promoting convective heat transer in the channel, Nu z o C10 with variable is much higher than that with constant properties. Nu z o C10 with variable k is lower than that with constant properties due to its lower k level as temperature rises. In addition, Nu z o C10 with both variable and k is higher than that with constant properties but lower than that with only variable. It appears that dynamic-viscosity variation dominates or C10. Fig. 9 also presents that variations in properties have little eects on q and or C10 but cause w to be slightly higher than that o the constant properties. Fig. 10. Comparisons o the local values o relevant parameters between constant and variable properties o air. (a: Nu z ; b: q ; c: ; d: w ) In Fig. 10 our igures are presented or the comparisons o the local values o relevant parameters (Nu z, q, and w ) between constant and variable properties 27/41

29 or air. he higher and k levels o air as temperature rises can explain the characteristics o Nu z with constant and variable properties shown in Fig.10: (1) Nu z with variable is lower than that with constant properties; (2) Nu z with variable k is higher than that with constant properties; (3) Nu z with both variable and k is higher than that with constant properties but lower than that with only variable k ; (4) Variations in properties have little eects on q and or air, but causes w to be lower than that o the constant properties. It is intuitive to predict a steep Nu jump near the ront part o the tube due to the thin thermal boundary layer, regardless o variable- or constant-property low, as hinted in Figs hough the variations o q and are small, their combined eects together with the variations o w can cause Nu z to deviate rom that with constant properties. Variable-property eects cannot change the variation trend o the distribution o Nu z, but variable dynamic viscosity and thermal conductivity aect more the upstream part o the channel or water and C10, while or air their eects are more notable at the downstream part o the channel. his so-called variable-property eects are evaluated by the Nu deviation, deined as ollows: Nu Nu vp, wall cp, wall (22) Nu Nu cp, wall 100 he maximum Nu o water, C10 and air are up to 7.33% at the entrance part, 4.45% at the entrance part, 2.20% at the ending part, respectively, and the average Nu o water, C10 and air are about 3.24%, 1.94%, 1.74%, respectively. It is also worth noting that, Nu or water and C10 due to variable exceeds that due 28/41

30 to variable k, while or air it is the opposite. his might be explained by the dierence o luid-property change rate considered in this work, as shown in able 4. From water, C10 to air, the temperature range becomes larger and larger. hese dierences o dynamic-viscosity variation and thermal-conductivity variation are the reason why the dynamic-viscosity variation dominates or water and C10, but or air thermal-conductivity variation is more dominant. In any case, it seems that the eects o the simultaneous variations o and k can be estimated by combining their separate eects qualitatively but not quantitatively. 29/41

31 Fig. 11. Comparisons o Nu z between constant and variable properties o water. Fig. 11 illustrates the axial distribution o Nu z with constant and variable properties with water as the working luid in 14 cases. It is noted that as axial heat conduction becomes more dominant, variable-property eects become more notable at the entrance and weaken gradually as a result o axial heat conduction. hereore, or water variable-property eects should be taken seriously at the entrance when axial heat conduction is dominant. 30/41

32 Eects o property variations on velocity and temperature proiles In order to have a deep understanding o why the changes o relevant parameters (Nu z, q, and w ) happen considering variable-property eects, this could be identiied by analyzing the velocity and temperature proile characteristics. he velocity and temperature proiles o variable- and constant-property lows or water, C10 and air are depicted in Fig , respectively, where black curves stand or constant-property low while the other curves stand or variable-property low. he axial velocity and temperature proiles in cross-section are drawn at two typical streamwise locations, i.e., z =0.025 at the entrance region and z =0.25 at the ully developed region. Fig. 12. Variations o axial velocity and temperature proiles in cross-section. (water, a, b: z =0.025; c, d: z =0.25) 31/41

33 Fig. 13. Variations o axial velocity and temperature proiles in cross-section. (C10, a, b: z =0.025; c, d: z =0.25) 32/41

34 Fig.14. Variations o axial velocity and temperature proiles in cross-section. (air, a, b: z =0.025; c, d: z =0.25) Above igures show that the variable luid properties have more notable eects on the velocity proile at the entrance region while more notable eects on temperature proile at the ully developed region. Since the air has the biggest changes o thermophysical properties compared with water and C10, variations in properties have the greatest eects on its velocity and temperature proiles. It is also noted that in Fig. 12a and c, Fig.13a and c, Fig.14a and c curves o constant k and variable k overlap very well with each other, which indicates that variable 33/41 k has little eects on velocity proile. But variable k can aect the temperature proile as shown in Fig. 13b and d, Fig. 14b and d. Likewise, variations in have greater eects on velocity proile than variations in k. On the one hand, or water and C10, velocity proile with both variable and k are lattened compared to that with constant properties while or air it is the opposite. Decrease in due to higher temperature reduces axial velocity at the center and thus causes the lattening eect. his lattening eect leads to more mass low rate near the wall, thereby enhancing heat convection. On the other hand, or water, the temperature proile o the variable properties changes a little. But or C10, the temperature proile

35 with variable and k are steeper compared to that with constant properties while or air it is also the opposite. Decrease in k due to higher temperature reduces temperature gradient near the wall and thus weakens the heat convection. All these dierences can be explained by the disparity o luid-property changes as temperature rises. Fig. 15. Variations o axial velocity and temperature proiles in cross-section along the tube between constant and variable properties. (water, Pe=100, k = 100, / = 1) Fig. 15 plots the velocity and temperature proiles in cross-section along the tube with constant- and variable-property or water. For axial heat conduction with constant properties, the temperature proile changes along the channel while the velocity proile remains unchanged. As can be seen in Fig. 15c and d, there is a s d i 34/41

36 change in the velocity and temperature proiles with varying and k along the heated region. he velocity proile, however, changes slightly along the low, not as much as the temperature proile. he velocity proile becomes steeper along the low direction as a result o decreasing o water. Likewise, as long as k varies with increasing luid bulk temperature, the temperature proile changes unceasingly along the tube and never achieves an unchanged status. 4. Conclusions A comprehensive numerical study o the eects o the axial heat conduction on tube laminar low and heat transer with uniorm internal heat generation has been conducted considering temperature-dependent and k. he main conclusions can be described as ollows: (1) Axial heat conduction has generalized eects o reducing the local Nusselt number with respect to the no wall case. Nu is still changed even at the ully developed region, though the thermal boundary layer is converged at the central streamline. Axial heat conduction can make the entrance length decrease and sometimes cause a minimum in Nu z distribution. Axial heat conduction becomes more dominant with increasing k s and, as well as with decreasing Pe. Nu decreases drastically with decreasing Pe when Pe 500 and with increasing k s when ks 100. As the axial heat conduction becomes more dominant, Pe has more eects than k s and. (2) Variable-property eects alleviate the reduction in Nu z due to the axial heat conduction, which enhances heat transer characteristics o the channel. For the case o Pe=100, =1 and k s =100, the maximum Nu between constant and variable 35/41

37 properties are up to 7.33% at the entrance part or water in the temperature range o 50, and 4.45% at the entrance part or n-decane in the temperature range o 120, as well as 2.20% at the ending part or air in the temperature range o 475, respectively. In addition, the average Nu or water, n-decane and air are 3.24%, 1.94% and 1.74%, respectively. Namely, or water and C10 variable-property eects aect more the upstream part o the channel, while or air their eects are more notable at the downstream part o the channel. (3) For the axial heat conduction with constant properties, the temperature proile changes along the channel while the velocity proile remains unchanged. On the other hand, or the axial heat conduction with variable properties, both temperature and velocity proiles keep changing along the channel and the change o temperature proile is larger. Besides, variable properties have more obvious eects on the velocity proile at the upstream part while more obvious eects on the temperature proile at the downstream part. Reerences [1] H. Herwig, O. Hausner, Critical view on new results in micro-luid mechanics : an example. International Journal o Heat and Mass ranser, 2003, 46(5): [2] G. Hetsroni, A. Mosyak, E. Pogrebnyak, L.P. Yarin, Heat transer in micro-channels: Comparison o experiments with theory and numerical results. International Journal o Heat and Mass ranser, 2005, 48(25): [3] M. Faghri, E. M.Sparrow, Simultaneous wall and luid axial conduction in laminar pipe-low heat transer. Journal o Heat ranser, 1980, 102(1): [4] C. J. Kroeker, H. M. Soliman, S. J. Ormiston, hree-dimensional thermal analysis o heat sinks with circular cooling micro-channels. International Journal o Heat and 36/41

38 Mass ranser, 2004, 47(22): [5] J. Li, G. P. Peterson, P. Cheng, hree-dimensional analysis o heat transer in a micro-heat sink with single phase low. International Journal o Heat and Mass ranser, 2004, 47(19-20): [6] E. K. Zarieh, H. M. Soliman, A. C. rupp, Combined eects o wall and luid axial conduction on laminar heat transer in circular tubes. Begel House Inc, 1982: [7] S. Bilir, Laminar low heat transer in pipes including two-dimensional wall and luid axial conduction. International Journal o Heat and Mass ranser, 1995, 38(9): [8] A. E. Quintero, M. Vera, Laminar counterlow parallel-plate heat exchangers: An exact solution including axial and transverse wall conduction eects. International Journal o Heat and Mass ranser, 2017, 104: [9] E. J. Davis, W. N. Gill, he eects o axial conduction in the wall on heat transer with laminar low. International Journal o Heat and Mass ranser, 1970, 13(3): [10] A. O. Adelaja, J. Dirker, J. P. Meyer, Eects o the thick walled pipes with convective boundaries on laminar low heat transer. Applied Energy, 2014, 130(5): [11] M. Axtmann, M. Heier, W. Hilali, B. Weigand, Axial heat conduction eects in the thermal entrance region or lows in concentric annular ducts: Correlations or the local bulk-temperature and the Nusselt number at the outer wall. International Journal o Heat and Mass ranser, 2016, 103: [12] C. Nonino, S. Savino, S. D. Giudice, L. Mansutti, Conjugate orced convection and heat conduction in circular microchannels. International Journal o Heat and Fluid 37/41

39 Flow, 2009, 30(5): [13] S. X. Zhang, Y. L. He, G. Lauriat, W. Q. ao, Numerical studies o simultaneously developing laminar low and heat transer in microtubes with thick wall and constant outside wall temperature. International Journal o Heat and Mass ranser, 2010, 53(19): [14] K. C. oh, X. Y. Chen, J. C. Chai, Numerical computation o luid low and heat transer in microchannels. International Journal o Advanced Manuacturing echnology, 2005, 26(5-6): [15] G. Gamrat, M. Favre-Marinet, D. Asendrych, Conduction and entrance eects on laminar liquid low and heat transer in rectangular microchannels. International Journal o Heat and Mass ranser, 2005, 48(14): [16] G. Maranzana, I. Perry, D. Maillet, Mini- and micro-channels: inluence o axial conduction in the walls. International Journal o Heat and Mass ranser, 2004, 47(17): [17] M. Rahimi, R. Mehryar, Numerical study o axial heat conduction eects on the local Nusselt number at the entrance and ending regions o a circular microchannel. International Journal o hermal Sciences, 2012, 59(2): [18] M. Lin, Q. W. Wang, Z. Guo, Investigation on evaluation criteria o axial wall heat conduction under two classical thermal boundary conditions. Applied Energy, 2016, 162: [19] R. Kumar, S. P. Mahulikar, Eect o temperature-dependent viscosity variation on ully developed laminar microconvective low. International Journal o hermal Sciences, 2015, 98: [20] C. J. Ho, C. Y. Chang, C. Y. Cheng, S. J. Cheng, Y. W. Guo, S.. Hsu, W. M. Yan, Laminar orced convection eectiveness o Al 2 O 3 water nanoluid low in a circular 38/41

40 tube at various operation temperatures: Eects o temperature-dependent properties]. International Journal o Heat and Mass ranser, 2016, 100: [21] I. Ghosh, S. K. Sarangi, P. K. Das, Simulation algorithm or multistream plate in heat exchangers including axial conduction, heat leakage, and variable luid property. Journal o Heat ranser, 2007, 129(7): [22] D. Lelea, he conjugate heat transer o the partially heated microchannels. Heat and Mass ranser, 2007, 44(1): [23] C. Nonino, S. D. Giudice, S. Savino, emperature dependent viscosity eects on laminar orced convection in the entrance region o straight ducts. International Journal o Heat and Mass ranser, 2006, 49(23-24): [24] S. D. Giudice, C. Nonino, S. Savino, Eects o viscous dissipation and temperature dependent viscosity in thermally and simultaneously developing laminar lows in microchannels. International Journal o Heat and Fluid Flow, 2007, 28(1): [25] S. P. Mahulikar, H. Herwig, heoretical investigation o scaling eects rom macro-to-microscale convection due to variations in incompressible luid properties. Applied Physics Letters, 2005, 86(1): [26] H. Herwig, S. P. Mahulikar, Variable property eects in single-phase incompressible lows through microchannels. International Journal o hermal Sciences, 2006, 45(10): [27] S. P. Mahulikar, H. Herwig, Physical eects in laminar microconvection due to variations in incompressible luid properties. Physics o Fluids, 2006, 18(7): [28] S. P. Mahulikar, H. Herwig, Physical eects in pure continuum-based laminar micro-convection due to variation o gas properties. Journal o Physics D: Applied 39/41

41 Physics, 2006, 39(18): [29] H. P. Gulhane, S. P. Mahulikar, Variations in gas properties in laminar micro-convection with entrance eect. International Journal o Heat and Mass ranser, 2009, 52(7-8): [30] H. P. Gulhane, S. P. Mahulikar, Numerical study o compressible convective heat transer with variations in all luid properties. International Journal o hermal Sciences, 2010, 49(5): [31] A. Ramiar, A. A. Ranjbar, S. F. Hosseinizadeh, Eect o axial conduction and variable properties on two-dimensional conjugate heat transer o Al2O3-EG/water mixture nanoluid in microchannel. Journal o Applied Fluid Mechanics, 2012, 5(3): [32] G. L. Morini, Scaling eects or liquid lows in microchannels. Heat ranser Engineering, 2006, 27(27): [33] Z. G. Li, X. L. Huai, Y. J. ao, H. Z. Chen, Eects o thermal property variations on the liquid low and heat transer in microchannel heat sinks. Applied hermal Engineering, 2007,27(17-18): [34] L. C. Burmerister, Convective Heat ranser, John Weily & Sons, [35] ANSYS CFX-Solver heory Guide. ANSYS Inc., [36] S. V. Patankar, D. B. Spalding, A calculation procedure or heat, mass and momentum transer in three-dimensional parabolic lows. International Journal o Heat and Mass ranser, 1972, 15(10): [37] W. M. Kays, M. E. Craword, B. Weigand, Convective Heat and Mass ranser, ourth ed., McGraw-Hill, New York,1980. [38] R. K. Shah, A. L. London, Laminar Flow Forced Convection in Ducts, Academic Press, New York, /41