Experimental and numerical investigation of forced convection of subsonic gas flows in microtubes

International Journal of Heat and Mass Transfer 78 (2014) 732–740

Contents lists available at ScienceDirect

International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt

Experimental and numerical investigation of forced convection of subsonic gas flows in microtubes Yahui Yang a, Chungpyo Hong b, Gian Luca Morini a,⇑, Yutaka Asako c a

DIN, Alma Mater Studiorum, Università di Bologna, Bologna, Italy Dept. of Mechanical Engineering, Kagoshima University, Kagoshima, Japan c Dept. of Mechanical Engineering, Tokyo University of Science, Tokyo, Japan b

a r t i c l e

i n f o

Article history: Received 20 January 2014 Received in revised form 8 July 2014 Accepted 8 July 2014

Keywords: Micro convection Nusselt number Flow compressibility Kinetic energy Mach number

a b s t r a c t The aim of this paper is to investigate the impact of gas compressibility on forced convection through commercial stainless steel microtubes with an inner diameter of 750 lm, 510 lm and 170 lm by combining experimental data with numerical simulations. The analysis covers both transitional and turbulent flow regimes (3000 < Re < 12,000). The results have evidenced that compressibility effects can significantly enhance convective heat transfer when the gas flow is heated by the walls (H boundary condition). This enhancement turns out to be more remarkable for microtubes with smaller inner diameter (lower than 200 lm). In order to explore in-depth the heat transfer mechanism along the microtube in presence of non negligible compressibility effects, the experimental data have been integrated with the numerical results obtained by modeling the fluid flow through the microtube with the adoption of the Arbitrary– Lagrangian–Eulerian (ALE) method and the Lam–Bremhorst Low–Reynolds number turbulence model in order to evaluate eddy viscosity coefficient and turbulence energy within the gas flow. The results presented in this work put in evidence that the integration of the experimental data with the numerical results is strongly beneficial in order to obtain a deep investigation of the physics of micro convection for compressible flows. The experimental values of the Nusselt numbers obtained for three different microtubes have been compared with both classical correlations validated for conventional pipes and specific correlations proposed for microtubes. This comparison highlights that the conventional correlations still holds for gas flow convection through microtubes when the compressibility effect is not significant. On the contrary, when compressibility is no longer negligible, the conventional correlations tend to underestimate the value of the Nusselt number. It is also demonstrated that the specific correlations proposed for the prediction of the Nusselt number in microtubes fail in presence of strong compressibility effects. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction During the last decades a rapid progress there has been in the technology of miniaturization, with devices and systems being scaled down from macro-metric sizes to micro-metric dimensions. This trend was not only driven by extensive engineering and industrial applications with promising market potential but also stimulated by multidisciplinary research intersecting chemistry, physics, biology, life science, pharmaceutics and engineering, etc. The ⇑ Corresponding author. Address: Department of Industrial Engineering (DIN), School of Engineering and Architecture, Alma Mater Studiorum, University of Bologna, viale del Risorgimento 2, 40136 Bologna, Italy. Tel.: +39 051 2093381; fax: +39 051 2093296. E-mail address: [email protected] (G.L. Morini). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2014.07.017 0017-9310/Ó 2014 Elsevier Ltd. All rights reserved.

analysis of flow and heat transfer mechanisms at microscale level is an interesting topic not completely investigated up to now as remarked in a recent review by Kandlikar et al. [1]. One of the main questions remaining unanswered is in which conditions the macroscale rules for single-phase flow heat transfer are still valid for microscale phenomena. For this reason, in the last years a large amount of experimental analyses has been addressed to the analysis of fluid-dynamics and heat transfer characteristics of singleand two-phase flows in microchannels [1]; however, the results are not always univocal as evidenced by Morini [2] and Hetsroni et al. [3]. As underlined in [1], forced convection of single-phase liquid flows in microchannels has been extensively investigated in the past and now it is possible to conclude that the conventional theory developed at macroscale is able to predict the convective

Y. Yang et al. / International Journal of Heat and Mass Transfer 78 (2014) 732–740

733

Nomenclature Br cp d D dh h k L Lh Lth _ m M Ma Nu P Pr q R r Re T

Brinkman number (–) specific heat at constant pressure (J/kg K) inner diameter (m) external diameter (m) hydraulic diameter (m) heat transfer coefficient (W/m2 K) thermal conductivity (W/m K) microtube total length (m) microtube heated length (m) microtube thermal entrance length (m) mass flow rate (kg/s) wall axial conduction parameter (–) Mach number (–) Nusselt number (–) heating power (W) Prandtl number (–) heat flux (W/m2) specific gas constant (J/kg K) radius (m) Reynolds number (–) temperature (K)

heat transfer coefficient for liquid flows in microchannels, with minor deviations caused by experimental errors [4] and/or by the presence of non negligible scaling effects [5–7]. On the contrary, in the case of gas micro convection very few experiments support the theoretical models and a significant effort is still needed in this direction [1]; this fact has been also stressed by Colin [8] in a recent review focused on gas micro convection. The first experimental investigation devoted to the analysis of gas micro convection can be traced back to the work by Wu and Little [9] in the 1980s, who explored the flow and heat transfer characteristics of N2, H2 and Ar through miniaturized rectangular channels with hydraulic diameter from 40 to 80 lm. The comparison of their experimental Nusselt numbers with the predictions of the classical correlations (i.e. Sieder and Tate [10], Hausen [11] and Dittus and Boelter [12]) evidenced a strong disagreement in laminar, transitional and turbulent regimes. The experimental data published by Choi et al. [13] successively, obtained with nitrogen flow, confirmed that the Nusselt number in turbulent regime was larger than the prediction of the Dittus–Boelter correlation [12] but their data were not in agreement with the correlation proposed previously by Wu and Little [9]. In laminar regime Choi et al. [13] obtained very low values of Nusselt number compared to the predictions of the conventional correlations. The authors gave no justification to this trend, both in laminar and turbulent regime, and proposed two new correlations for the prediction of the Nusselt number in microtubes by fitting their own experimental data. Yu et al. [14] investigated the convective heat transfer of nitrogen flow in turbulent regime through microtubes with inner diameters between 19 lm and 102 lm. Also in this case, the Nusselt numbers obtained in turbulent regime were larger than those predicted by means of conventional correlations but the authors avoided any physical explanation of their results and proposed a new correlation for the prediction of the Nusselt number in microtubes, not in agreement with the previous correlations of Wu and Little [9] and Choi et al. [13]. Hara et al. [15] experimentally investigated the convective heat transfer of air flows through square minichannels with hydraulic diameters between 0.3 mm and 2 mm. Unlike the other researchers, they found that the deviation of Nusselt number from

u

velocity (m/s)

Greek symbols c specific heat ratio (cp/cv) (–) e absolute roughness (m) l dynamic viscosity (kg/ms) q density (kg/m3) Subscript ad b e f i in m out T w w6

adiabatic bulk value external surface fluid internal surface inlet value mean value outlet value total wall wall value measured by the thermocouple close to the outlet

conventional theory may depend on the hydraulic diameter and length of the tested minichannels. A possible physical explanation of the large Nusselt numbers evidenced by the experimental runs in [9,13–15] is linked to the significant role played by gas compressibility in microtubes. When the inner dimensions of a tube are reduced, for a fixed mass flow rate, the fluid velocity increases and the local Mach number can reach large values (larger than 0.3) even for Reynolds numbers less than 5000–10,000, especially close to the tube exit. In this case the gas compressibility cannot be ignored inducing conversion from thermal energy to kinetic energy when the flow accelerates along the microtube and this can justify an augmentation of the Nusselt number. However, this effect has been ignored in [9,13–15] and for this reason in the proposed correlations the Mach number is never involved. The beneficial effect of gas compressibility on the Nusselt number has been also demonstrated numerically for very large values of the Mach number (up to Ma > 1) by Hong et al. [16] and Lijo et al. [17]. More recently, Chen et al. [18] and Yang et al. [19] conducted experimental research on forced convection of air and carbon dioxide through microtubes with an inner diameter from 86 lm to 920 lm. The trend of the Nusselt number in both laminar and turbulent regime was found in agreement with the classical correlation proposed by Gnielinski [20], which was validated for incompressible flow at macro scale. These experimental results seem to indicate that the heat transfer coefficient is not influenced by the flow compressibility effect even at very large Reynolds numbers (Re  20 000), if the Mach number at the outlet is lower than 0.3. It is possible to conclude that the experimental results obtained recently by Chen et al. [18] and Yang et al. [19] and the numerical results of Hong et al. [16] and Lijo et al. [17] are in disagreement with the results of the previous works due to Wu and Little [9] and Choi et al. [13] Yu et al. [14] and Hara et al. [15] and these last works are not in agreement each to other. For these reasons, the main objective of the present work is to investigate the limits of validity of the conventional correlations for the prediction of Nusselt numbers in microtubes having an inner diameter down to 170 lm by using nitrogen as working gas. The experimental investigation is made with the aim to quantify

734

Y. Yang et al. / International Journal of Heat and Mass Transfer 78 (2014) 732–740

the impact of the gas compressibility on the convective heat transfer coefficient when the inner diameter of the microtube is reduced. The analysis is conducted by varying the inner diameter of the microtubes from 750 lm down to 170 lm and the Reynolds number between 3000 up to 12,000 in order to investigate the behavior of the gas flow both in transitional and turbulent regime.

Table 1 Main geometrical characteristics of the tested stainless steel microtubes.

2. Experimental test rig

have been selected for the experiment. The main geometrical characteristics of the tested microtubes are summarized in Table 1, where d is the average inner diameter of microtube, D the external diameter, e the average roughness of the wall surface, L the length of microtube and Lh the heated length of the tube. It is important to bear in mind that the real inner diameter of a microtube may deviate greatly from the nominal values provided by the manufacturer [4]. From the experience of the authors, this is especially true for microtubes with inner diameters smaller than 200 lm. The use of an inaccurate inner diameter will easily lead to wrong experimental results. Therefore in this work the inner diameters of the three microtubes were carefully checked before to start the experimental runs. Firstly, the configurations of the microtube end sections were imaged with SEM (Scanning Electron Microscope), and the diameter information were obtained by fitting the images. Then, the average hydraulic diameter was further checked by means of experimental tests for the determination of the friction factor in laminar regime at low Reynolds numbers (Re < 300); in fact, the experimental results accumulated in the last years confirm that the Poiseuille law (f = 64/Re) is still valid for gas flow in microtubes when very low values of the Reynolds number are considered. In this case the gas can be considered as incompressible and the Poiseuille law can be used in order to calculate the average inner diameter of the microtube. Since friction factor of incompressible laminar flow is inversely proportional to the fifth power of hydraulic diameter, its value is extremely sensible to the hydraulic diameter of the tube and can be used in order to verify the inner diameter determined by SEM. This enables to tune very finely the microtube hydraulic diameter until the trend of experimental friction factor at low Reynolds number agrees well with the Poiseuille law. This method can be very accurate as long as the experiment is carefully carried out and it provides the values of hydraulic diameter averaged over the whole length of microtube, taking into account the possible deformation of local cross section along the microtube. The hydraulic diameters of the tested microtubes determined by using this method are equal to 750 lm, 510 lm and 170 lm respectively while the values declared by the manufacturer are 750 lm, 500 lm and 175 lm, respectively.

The lay-out of the test rig used for the experimental runs of this work is shown in Fig. 1. Nitrogen is used as working fluid; it is stored in a high pressure flask (200 bar) (1) and it is expanded using a valve (2) to approximately 10 bar at ambient temperature. Then, the gas flow enters a 7 lm particle filter (3, Hamlet) which is used to prevent possible impurities from clogging the microtubes or the mass flow controllers. A three-way valve directs the flow to the proper flow sensor, as the facility is equipped with three Bronkhorst EL-Flow E7000 operating in the 0–5000 nml/min (4a), 0–500 nml/min (4b) and 0–50 nml/min (4c) ranges respectively. The mass flow controllers can impose an expected mass flow rate by means of a computer-steered valve which makes an indirect regulation of the pressure at the inlet of the microtube. Microtubes (5) of various lengths (from 1 to 100 cm) and variable inner diameters, providing they have the same external diameter, namely 1/1600 can be tested. After exiting the microtube, the gas is vented to the atmosphere. The total pressure drop between the inlet and the outlet of the microtube is measured by a differential pressure transducer (6, Validyne DP15). To measure the temperature distribution along the external surface of the microtube, five calibrated K-type thermocouples are glued to the outer surface of the microtube using an electrically non-conductive epoxy resin. In addition, at the inlet and outlet of the channel two additional K-type thermocouples are carefully inserted into the manifolds. The microtubes are heated by Joule effect by means of a programmable DC power supply (8, HP6032A) in order to impose a uniform heat flux along the walls of the tube (H boundary condition). The microtube is thermally insulated at the external surface (ArmaflexÒ, kin = 0.035 W/m K) and enclosed in an insulated box in order to decouple the test section from the outside thermal conditions. An infrared thermo-camera is also used (7, AVIO TVS 200EX) to check all the possible thermal bridges of the test section. An absolute pressure sensor (9, Validyne AP42) is used to monitor the ambient pressure at the exit of the test rig. Three commercial stainless steel microtubes (Upchurch) having different inner diameters and the same outlet diameter (1/1600 )

Microtube

d (lm)

D (lm)

e (lm)

L (mm)

Lh (mm)

e/d (%)

L/d

#1 #2 #3

750 510 170

1588 1588 1588

4 3 3

500 200 100

470 180 75

0.53 0.6 1.76

667 392 588

Tin

Tout T1

T2

T3

T4

Δ

Fig. 1. Schematic lay-out of the test rig.

T5

T6

Y. Yang et al. / International Journal of Heat and Mass Transfer 78 (2014) 732–740

3. Data reduction In order to determine the axial trend of the local convective heat transfer coefficient the value of the bulk fluid temperature at each location along the microtube is needed. However, this quantity is not easy to measure inside a microtube. On the contrary, it is comparatively easier to deduce the mean value of the convective heat transfer coefficient along the whole microtube. This mean value can be expressed as a function of the average fluid bulk temperature and of the mean wall temperature. Under a H boundary condition (uniform wall heat flux) the mean Nusselt number can be calculated from the experimental parameters by using the following equation:

Num ¼

d qw;i hd ¼ kf kf ðT w;i  T b Þ

ð1Þ

where T w;i is the average value of the inner wall temperature along the tube, T b is the gas mean static bulk temperature averaged between the inlet and outlet, kf is the fluid thermal conductivity calculated at the fluid average bulk temperature, d is the inner hydraulic diameter of the microtube and qw,i is the heat flux on the inner wall of the microtube. The maximum temperature difference between the external and internal surfaces of the microtube wall is calculated to be of the order of 0.15 K for stainless steel (kw = 15 W/m K) under the heat fluxes used in these tests. This temperature difference is smaller than the typical uncertainty of the thermocouples. Therefore, it is possible to replace the wall internal surface temperature by the external surface temperature (Tw,e) with negligible influence on the Nusselt number. The average temperature of the wall external surface is directly determined by the measurement of a set of 5 K-type thermocouples attached to the microtubes. The local bulk static temperature averaged on the cross section is defined as follows:

R qcp ruTdr Tb ¼ R qcp rudr

ð2Þ

where r is the radius of microtube and u is velocity. In addition, for compressible flows the gas total temperature can be introduced:

R TT ¼

R

qcp ruTdrR þ qruðu2 =2ÞTdr qcp rudr

ð3Þ

in which the contribution of the kinetic energy of the flow is taken into account. When the compressibility effects are negligible the heat flux can be calculated by means of an energy balance between the inlet and the outlet of the microtube:

qw;i ¼

_ pf ðT b;out  T b;in Þ mc p d Lh

ð4Þ

_ is the mass flow rate, cpf is the gas specific heat at the averwhere m age bulk temperature, Tb,in and Tb,out are the gas temperature measured by K-type thermocouples inserted at the inlet and outlet plena of the test section and Lh is the heated length of the microtube (see Table 1). The evaluation of the heat flux on the inner wall of the microtube by means of Eq. (4) allows to avoid an analytical determination of the heat losses of the test rig; in fact, by using the difference between the bulk temperature of the gas at the inlet and the outlet of the microtube only the heat transferred to the gas flow is taken into account in the evaluation of the Nusselt number. On the contrary, when the compressibility effects are no longer negligible the energy balance (Eq. (4)) must be corrected in order

735

to take into account that the acceleration of the gas flow due to the pressure drop determines an increase of the kinetic energy and a decrease of the internal energy of the gas flow. With the presence of an energy conversion between thermal energy and kinetic energy, Eq. (4) is still valid only if one uses the gas total temperature at the inlet and at the outlet of the microtube. In a similar way, the temperature difference between bulk and wall needs to be re-considered. The convective heat transfer is driven by the local temperature gradient. Therefore, even if the gas local total temperature is higher than the wall temperature, the heat flux remains from wall to fluid as long as the gas bulk temperature is lower than the local wall temperature. As a result, the temperature difference in the calculation of the local convective heat transfer coefficient h, or Nusselt number, is the difference between wall temperature and gas bulk temperature:

Nu ¼

_ pf T T;out  T T;in mc

pkf L T w  T b

ð5Þ

The local value of the gas static bulk temperature and of the stagnation temperature are linked by means of the local Mach number:

  c1 2 TT ¼ Tb 1 þ Ma 2

ð6Þ

where c is the gas specific heat ratio. By means of local measurements of gas pressure, temperature and mass flow rate, the Mach number can be obtained directly. However, the fluid temperature measured in practice is neither exactly the total temperature nor the bulk temperature, due to incomplete conversion from kinetic energy to thermal energy at the measurement points. For this reason the measured temperature can be linked to the bulk temperature and to the local Mach number by means of the following equation:

  c1 2 T meas ¼ T b 1 þ c Ma 2

ð7Þ

where c is the recovery coefficient which indicates the incompleteness of energy conversion and it assumes values between 0 and 1. Although some recommended values for the recovery coefficient can be found in literature, the exact value of this coefficient is more sensitive and related to the specific features of the test rig, flow conditions, position and placement of thermocouples and so on. In this work the estimation of the recovery coefficients have been made by following the procedure suggested by Hong and Asako in [21] for each tested microtube mounted on the aforementioned test rig under adiabatic conditions. Since the value of the recovery coefficient does not depend on the heating conditions of the flow, the results obtained in the adiabatic case can be assumed equal to those in the heated case by fixing the other conditions. To be sure that this assumption is verified, in practical operations during experiments, greatest caution should be paid to the gas temperature measurement. After the test section is connected to the microtube and the thermal insulation has been placed, the test section should be carefully preserved from any mechanical action, otherwise the value of the recovery coefficient may change from one test to another one. Under adiabatic conditions, the difference between the wall temperature and the gas static bulk temperature is generally very small and the wall temperature can be considered as indicative of the gas static temperature; in this way the recovery coefficient can be determined by using the following expression:

c¼

  T meas;ad 2 1 1 T w;ad c  1 Ma2

where the subscript ad indicates the adiabatic condition.

ð8Þ

736

Y. Yang et al. / International Journal of Heat and Mass Transfer 78 (2014) 732–740

0.30

29

Tout

28

Tw6

d=750 µm, L=0.5 m

recovery coefficient

Temperature ( )

30

27 26 25 24 23 22 21

d=750 µm L=0.5 m P= 0 W

20 3000

5000

7000

9000

1.0

Mach number at outlet

0.15 0.10 0.05

4000

6000

8000

10000

12000

Re Fig. 4. Recovery coefficients as a function of the Reynolds number.

As an example, the measured gas temperature at the outlet of microtube #1 (Tout) and wall temperature measured by the thermocouple close to the outlet (Tw6) are shown in Fig. 2 when Joule heating on the microtube surface is switched off (adiabatic test). Nitrogen gas enters in the microtube at ambient temperature. The wall temperature at the inlet coincides with the ambient one. Due to the large pressure drop, compressibility effects occur and the fluid accelerates along the tube. This causes a reduction in the gas internal energy, which decreases the fluid static temperature. As a result, the local temperature of the wall inner surface, which is in direct contact with the fluid, is reduced. At the outlet plenum where the fluid outlet temperature is measured, the gas thermal energy is partially recovered from kinetic energy due to flow deceleration in the larger flow area compared with the microtube. Therefore, the fluid temperature measured at the outlet is surprisingly higher than the wall temperature at the outlet, the latter being measured by the last thermocouple attached along the microtube wall. As the Reynolds number is increased, the compressibility effects become more significant. Consequently, the microtube wall at the outlet is further cooled by the gas flow, and the difference between wall temperature and measured fluid temperature at the outlet becomes larger. Similar trend has been experimentally evidenced for the other microtubes over a large range of Reynolds numbers.

0.9 0.8 0.7 0.6 0.5 0.4 0.3

d=750 µm, L=0.5 m

0.2

d=510 µm. L=0.2 m

0.1

d=170 µm, L=0.1 m 6000

d=170 µm, L=0.1 m

11000

Fig. 2. Gas temperature (Tout) measured at the outlet and wall temperature (Tw6) measured by the thermocouple close to the outlet under adiabatic condition (microtube #1).

4000

d=510 µm. L=0.2 m

0.20

0.00 2000

Re

0.0 2000

0.25

8000

10000

12000

Re Fig. 3. Experimental Mach number at the outlet as a function of the Reynolds number for the tested microtubes.

The outlet Mach number evaluated experimentally as a function of Reynolds number for all the microtubes tested in this work is plotted in Fig. 3. For the same Reynolds number, the outlet Mach number increases dramatically for lower microtube inner diameters: for microtube #3, the outlet Mach number approaches to 0.85 when the Reynolds number reaches 10,000. The recovery coefficients c obtained by means of Eq. (8) are shown in Fig. 4 as a function of the Reynolds number for the three microtubes. It is possible to note how the value of the recovery coefficients decreases considerably with the reduction of the microtube inner diameter, which means that it is more difficult to put in evidence experimentally the conversion from kinetic energy into internal energy in smaller tubes due to the geometry of the exit region and to the right position of the thermocouple tip at the centerline of the bulk flow close to the outlet section as observed by Yang et al. [22]. In addition, it can be noticed that the recovery coefficients for microtube #1 and #2 are nearly constant over the whole range of Reynolds numbers tested (c = 0.17 for microtube #1 and c = 0.1 for microtube #2). For the smaller microtube (#3) the value of the recovery coefficient is very low and its value decreases gradually with the increase of the Reynolds number. This trend can be explained by considering the smaller percentage of internal energy recovered from kinetic energy at the outlet when the flow velocity is extremely large (Mach number larger than 0.7). The trends of data in Fig. 4 suggest that, for moderately compressible flows (0.3 < Mach < 0.7) the recovery coefficient is independent of the flow velocity or Mach number. On the contrary, when the flows are highly compressible (Mach > 0.7), attention must be paid in order to use a constant value of the recovery coefficient at different Reynolds numbers. The values of the recovery coefficients obtained with the adiabatic tests have been used for the estimation of the Nusselt number in the next sections. To assess the accuracy of the experimental Nusselt numbers presented in this paper, the uncertainty associated to each measurement device used in the test rig is reported in Table 2. The uncertainty on the inner diameter evaluated through SEM imaging and by means of the friction factor comparison is estimated to be of the order of ±2%. The uncertainty on the microtube length is of the order of ±0.3%. By applying the conventional theory on the propagation of errors and considering the typical uncertainty of each measured parameter, the estimated uncertainty of the experimental Nusselt numbers presented in this paper is in the range from 8% to 37%. 4. Numerical model The experimental evaluation of the convective heat transfer has been coupled to the numerical analysis of the heat transfer within microtubes in presence of compressibility effects.

737

Y. Yang et al. / International Journal of Heat and Mass Transfer 78 (2014) 732–740 Table 2 Typical operative range and uncertainty of instrumentation. Instrument

Range (0-FS)

Uncertainty

Flow meter (Bronkhorst EL-Flow E7000) Flow meter (Bronkhorst EL-Flow E7000) Flow meter (Bronkhorst EL-Flow E7000) Differential pressure transducer (Validyne DP15)

0–5000 (nml/min) 0–500 (nml/min) 0–50 (nml/min) 0–35 (kPa) 0–86 (kPa) 0–220 (kPa) 0–860 (kPa) 0–1440 (kPa) 0–200 (°C)

±0.6% ±0.5% ±0.5% ±0.5%

Thermocouple (K type)

FS FS FS FS

±0.25% FS

F.S. = Full scale.

contribution of viscous dissipation and fluid expansion on temperature change can be considered negligible. This result has been further evidenced by Lin and Kandlikar [26] who use a turbulence model in order to quantify viscous heating in high speed gas flows and found that the influence of viscous heating on heat transfer can be neglected. However, for sake of completeness, in the energy equation of the present model the flow work term and the contribution of viscous dissipation have been taken into account. The computations were performed for 3 cases (Re = 3736, 6218 and 9878) for d = 510 lm by imposing axial-symmetric conditions. The computational domain is divided into quadrilateral cells: 200 in the axial direction (x-direction) and 20 in the radial direction (r-direction). The choice of the grid size is the result of an accurate analysis on the sensitivity of the numerical results on the grid dimensions described by Hong et al. in [21,25]. No-slip boundary conditions are assumed for the gas velocity at the walls because the gas rarefaction can be considered negligible in these operative conditions characterized by a Knudsen number lower than 0.001. For the same reason, no temperature jump between the solid walls and the gas close to the wall is considered. Additional details on the numerical procedure followed for the resolution of the governing equations can be found in [24]. Fig. 6 shows the axial trend of the local Mach number calculated numerically for microtube #2 at different Reynolds numbers. It is evident that the flow enters microtube #2 at a relatively low Mach number, which increases gradually as the gas flows downstream. However, near the outlet an abrupt increase of the Mach number is observed. A similar trend has been found by Hong et al. [24] and by Yang et al. [27]. Therefore, high speed gas flows through

1.0

Constant heat flux wall

pin pstg

pout

r x

Tstg

TT,in

pin(kPa) 204.07 297.83 446.52

0.8

Re 3736 6218 9878

0.6

Ma

The microtube has been modeled by neglecting the conjugate effects between the solid wall and fluid; in this way the solid region of the tube wall is not taken into account in the model. In fact, as observed by Morini and Yang [4], for a fixed inner diameter and Reynolds number, the reduction of Nusselt number due to wall axial conduction is more important in the case of microtube with smaller inner diameter and it can play an important role when the inner diameter of the microtube is lower than 200 lm especially for Re < 1000. However, an increase of the Reynolds number tends to weaken the influence of wall axial conduction on convective heat transfer. Under the experimental conditions adopted in the present work (Re > 3500), by using the correlation proposed by Lin and Kandlikar [23] the influence of wall axial conduction on the Nusselt number is evaluated to be smaller than 1% and for this reason this aspect can be neglected in the numerical model of the tube. A numerical computation based on the Arbitrary–Lagrangian– Eulerian (ALE) method to solve the two-dimensional compressible momentum and energy equations has been conducted. The Lam– Bremhorst’s Low–Reynolds number turbulence model has been employed to calculate the eddy viscosity coefficient and turbulence energy. The problem is modeled by considering a microtube subject to a constant and uniform heat flux at the walls as shown in Fig. 5 (H boundary condition). As evidenced in Fig. 5, the abrupt variation of the flow area due to the presence of the outlet and inlet manifolds is not considered in the numerical domain. On the contrary, a chamber at the stagnation temperature, Tstg, and pressure pstg is considered as smoothly connected to its upstream section in order to fix the inlet conditions. The momentum and energy governing equations for compressible flows are the same used by Hong et al. in their previous work [24] and for this reason, the complete set of equations is not detailed here. When heat transfer between high speed gas flows and microtube wall surface is activated viscous dissipation, flow expansion and convective heat transfer are simultaneously contributing to the change in fluid temperature. The numerical work of Hong et al. [25] has shown that under such condition the fluid temperature change is mainly caused by convective heat transfer, while the combined

0.4 0.2

d 0

L Reservoir Fig. 5. Computational domain and inlet reservoir.

0

0.05

0.10

0.15

0.20

x (m) Fig. 6. Axial distribution of the local Mach number through microtube #2 determined numerically at different Reynolds numbers.

Y. Yang et al. / International Journal of Heat and Mass Transfer 78 (2014) 732–740

microtubes can hardly become fully developed either hydrodynamically or thermally. By comparing the numerical values of the outlet Mach number quoted in Fig. 6 for microtube #2 with the corresponding experimental values shown in Fig. 3, it is evident that the experimental values are slightly lower than those obtained numerically and this difference increases for large Reynolds numbers. This difference is mainly due to the significant minor losses at the outlet which reduce the value of the outlet Mach number in the experiment. These minor losses were not considered in the numerical model and they are the main reason of this deviation. However, the indication of the numerical data is precious in order to analyze what happens within the microtube where sensors cannot be arranged.

350

330 320 310 300 290

5. Results and discussion

0

0.05

0.10

0.15

0.20

x (m) Fig. 8. Axial distribution of wall temperature, static bulk temperature and total temperature determined numerically at Re = 9878 for microtube #2.

70

T [ ]

Fig. 7 shows the numerical 2D temperature field in microtube #2 subject to an uniform heat flux at the wall for Reynolds numbers of 3736 and 9878. The static bulk temperature increases as the gas proceeds downstream due to the convective heat transfer from the wall. It is very interesting to observe from Fig. 7 that the bulk temperature along the tube center line begins to decrease when the flow approaches the microtube outlet due to the occurrence of energy conversion from internal energy to kinetic energy linked to the flow acceleration. The axial trend of the local bulk static temperature obtained by means of Eq. (2) from numerical simulation is shown in Fig. 8 for Reynolds number at 9878 (microtube #2), where the fluid total temperature obtained by Eq. (3) and wall temperature are also plotted. The temperature difference between the wall and the fluid is constant along large part of the microtube length, except for the outlet section where the static bulk temperature sharply drops at the outlet of the microtube (see Fig. 8). This also brings the wall temperature down leading to non-linear distribution of the wall temperature. In addition, it can be noted from Fig. 8 that the total temperature tends to exceed the wall temperature close to the outlet. Even in such case there is still potential to transfer thermal energy from the wall to the fluid, as long as the fluid static temperature is lower than the local wall temperature. Therefore, the flow compressibility effects have a favorable influence on the convective heat transfer if the wall surface is subject to a positive heat flux.

pstg=446.52 kPa, Re=9878 Tb TT Tw

340

T (K)

738

65

Tout

60

Tw6

55 50 45 40 35 2000

d=510 µm L=0.2 m P=2 W 4000

6000

8000

10000

Fig. 9. Gas temperature (Tout) measured at the outlet and wall temperature (Tw6) measured by the thermocouple close to the outlet as a function of the Reynolds number (microtube #2, heated).

r (m)

0.00025

0

0

0.05

0.1

0.15

0.2

x (m)

(a)

r (m)

0.00025

0

0.05

0.1

T (K) 340 335 330 325 320 315 310 305 300 295 290

Re=3736

0

12000

Re

0.15

0.2

x (m)

Re=9878

(b) Fig. 7. 2D local temperature distribution through microtube #2 determined numerically at Reynolds numbers equal to 3736 (a) and 9878 (b).

739

Y. Yang et al. / International Journal of Heat and Mass Transfer 78 (2014) 732–740

1000

Nu

100

Gnielinski Wu and Little Choi et al. Yu et al. Nu experiment

10

d=750 µm L=0.5 m 1 2000

Re

10000

Fig. 10. Nusselt number as a function of the Reynolds number for microtube #1 and comparison with the existing correlations.

1000

Nu

100

Gnielinski Wu and Little Choi et al. Yu et al. Nu experiment

10 d=510 µm L=0.2 m 1 2000

Re

10000

Fig. 11. Nusselt number as a function of the Reynolds number for microtube #2 and comparison with the existing correlations.

1000

100

Gnielinski Wu and Little Choi et al. Yu et al. Nu experiment

Nu

The wall temperature (Tw6, measured by the thermocouple close to the microtube exit) and the measured outlet gas temperature (Tout) are shown in Fig. 9 for Reynolds numbers from 2780 up to 10,840 for microtube #2. The data were obtained under a constant heating power fixed at 2 W. It can be noted that under heated conditions the measured bulk temperature is lower than the local wall temperature at low Reynolds numbers. However, as the Reynolds number increases, the measured outlet gas temperature overcomes the local wall temperature due to the internal energy recovered from kinetic energy at the outlet plenum where the temperature is measured. The Nusselt number calculated by Eq. (5) is shown in Fig. 10 for microtube #1, which was heated by a constant power of 0.41 W. For Reynolds numbers lower than 6000, the Mach number at the outlet is lower of 0.3, as shown in Fig. 3, and compressibility can be considered negligible; the Nusselt number follows the trend of the Gnielinski correlation [20]. On the contrary, as the outlet Mach number increases the compressibility effects gain in significance and the experimental Nusselt number deviates from the Gnielinski correlation. However, the experimental trend of the Nusselt number is in disagreement with the correlations proposed specifically for microtubes by Wu and Little [9], Choi et al. [13] and Yu et al. [14]. A similar trend was evidenced for microtube #2, as shown in Fig. 11 under a heating power of 2 W. For Reynolds numbers lower than 4000 the experimental results can be well predicted by using the Gnielinski correlation [20] because the Mach number at the outlet is lower than 0.3 (see Fig. 3) and hence the compressibility effects are negligible. When the Reynolds number increases, the compressibility becomes more and more significant and the Nusselt numbers becomes larger than the values predicted by the Gnielinski correlation in which the flow is assumed incompressible. These results are qualitatively in agreement with the numerical results by Croce and D’Agaro [28], who found that the gas convective heat transfer in microtubes are dominated by the compressibility effects which increases the Nusselt number at high Mach numbers for heated flows. Fig. 12 shows the Nusselt number as a function of the Reynolds number for microtube #3 under a heating power fixed at 4 W. The Nusselt number is surprisingly high compared with that of the other two microtubes. This is related to the larger outlet Mach numbers even at low Reynolds numbers, as highlighted by Fig. 3. In fact, at Reynolds number equal to 3894 the outlet Mach number reaches 0.69 for microtube #3, while for the same Reynolds number, the outlet Mach numbers of flow through microtube #1 and #2 are smaller than 0.35.

10 d=170 µm L=0.1m 1 2000

Re

10000

Fig. 12. Nusselt number as a function of Reynolds number for microtube #3 and comparison with the existing correlations.

The significant increase of the Nusselt number evidenced in Fig. 12 can be explained by observing that, in the case of fast flow, the dimensionless wall and bulk temperature difference decreases approaching to the outlet. Since the thermal conductivity of the gas is low, the fall of the wall temperature is smaller than that of the bulk temperature: for this reason the temperature difference between wall and fluid becomes larger due to flow acceleration, which tend to reduce the fluid static temperature. On the other hand, the total temperature difference between the inlet and the outlet increases due to flow acceleration and this increase is larger than the increase of the temperature difference between wall and fluid: for this reason the Nusselt number increases when the compressibility effects become more important. The correlations proposed for microtubes by Wu and Little [9], Choi et al. [13] and Yu et al. [14] are in agreement to highlight that for gas micro convection higher values of the Nusselt number than that predicted by the Gnielinski correlation [20] can be experienced, but these correlations avoid to link the Nusselt number to the Mach number, as this increase is mainly due to the compressibility effect. It is possible to note that the experimental data obtained for microtube #3 seem to be in agreement with the correlation proposed by Choi et al. [13] if one takes into account the typical error bar. However, this agreement can be considered as a coincidence because the correlation of Choi et al. [13] fails to predict the trends of the Nusselt numbers obtained using the other two microtubes. This fact confirms that the correlations proposed by Wu and Little [9], Choi et al. [13] and Yu et al. [14] are correlations not physically-based.

740

Y. Yang et al. / International Journal of Heat and Mass Transfer 78 (2014) 732–740

If a correlation has to be suggested for the prediction of the Nusselt number for gas flows in microtubes this correlation is expected to have the following functional expression:

L Tw Nu ¼ Nu Re;Pr; Maout ; Br; ; d Tb

! ð9Þ

which is different from the expression utilized by Wu and Little [9], Choi et al. [13] and Yu et al. [14] in which only Reynolds number and Prandtl number are involved. For sake of completeness, in Eq. (9) the Brinkman number is also recalled; this parameter can be important especially when large Mach numbers (close to 1) are considered, which is not the case of the present results. More experimental and numerical data are needed in order to propose this kind of correlation for the prediction of the Nusselt number for gas flows in microtubes and this point will be the subject of a further work.

6. Conclusions Convective heat transfer of compressible nitrogen flows in transitional and turbulent regimes through three microtubes was investigated experimentally and numerically. The main conclusions of this work can be summarized as follows: 1. The conventional correlations for the prediction of the Nusselt numbers validated for incompressible flows are not able to work correctly in microtubes having an inner diameter smaller than 200 lm for Reynolds numbers larger than 3000–6000 because the compressibility effects tend to become significant and increase the average Nusselt number. 2. It has been demonstrated that in microtubes having an inner diameter lower than 200 lm the local Mach number assumes very high values (>0.3) also in transitional regime (Reynolds between 3000 and 10,000). In these cases, the acceleration of the gas flow due to the strong pressure drop determines an increase of the gas kinetic energy to the detriment of the gas internal energy. This conversion is very strong especially close to the microtube outlet. 3. Gnielinski’s correlation for the prediction of Nusselt number, rather than those proposed specifically for micro convection, still holds for gas flow convection through microtubes until the gas outlet Mach number is lower than 0.3. When compressibility is no longer negligible, the value of the Nusselt number overcomes the values predicted by Gnielinski’s correlation. 4. Specific correlations for the calculation of the Nusselt number in microtubes must involve the Mach number in order to take into account the non negligible effect due to gas compressibility.

Conflict of interest None declared.

Acknowledgements This research has received funding from the European Community’s Seventh Framework Programme 2007–2012 under Grant agreement No. 215504 and Italian PRIN 2009.

References [1] S.G. Kandlikar, S. Colin, Y. Peles, S. Garimella, R.F. Pease, J.J. Brandner, D.B. Tuckerman, ‘‘Heat transfer in microchannels – 2012 status and research needs’’, ASME J. Heat Transfer 25 (9) (2013) 091017. [2] G.L. Morini, Single-phase convective heat transfer in microchannels: a review of experimental results, Int. J. Therm. Sci. 43 (7) (2004) 631–651. [3] G. Hetsroni, A. Mosyak, E. Pogrebnyak, L. Yarin, Heat transfer in microchannels: comparison of experiments with theory and numerical results, Int. J. Heat Mass Transf. 48 (25–26) (2005) 5580–5601. [4] G.L. Morini, Y. Yang, Guidelines for the determination of single-phase forced convection coefficients in microchannels, ASME J. Heat Transfer 135 (2013) 101004. [5] T.J. Lin, S.G. Kandlikar, An experimental investigation of structured roughness effect on heat transfer during single-phase liquid flow at microscale, ASME J. Hear Transfer 134 (2012) 101701. [6] G.L. Morini, Scaling effects for liquid flows in micro-channels, Heat Transfer Eng. 27 (4) (2006) 64–73. [7] G.L. Morini, M. Lorenzini, S. Salvigni, G.P. Celata, Experimental analysis of the microconvective heat transfer in the laminar and transition regions, Exp. Heat Transfer 23 (2010) 73–93. [8] S. Colin, Gas microflows in the slip flow regime: a critical review on convective heat transfer, ASME J. Heat Transfer 134 (2) (2012) 020908. [9] P. Wu, W.A. Little, Measurement of the heat transfer characteristics of gas flow in fine channel heat exchangers used for microminiature refrigerators, Cryogenics 24 (8) (1984) 415–420. [10] E.N. Sieder, G.E. Tate, Heat transfer and pressure drop of liquids in tubes, Ind. Eng. Chem. 28 (12) (1936) 1429–1435. [11] H. Hausen, Neue gleichungen fuer die waermeuebertragung bei freier und erzwungener stroemung, Allg. Waermetechnik 9 (1959) 75–79. [12] F.W. Dittus, L.M.K. Boelter, ‘‘Heat transfer in automobile radiators of the tubular type’’, University of California (Berkeley) Publication in Engineering 2 (1930) 443. [13] S.B. Choi, R. Barron, R. Warrington, Fluid flow and heat transfer in microtubes, Am. Soc. Mech. Eng.: Dyn. Syst. Control Div. (Publication) DSC 32 (1991) 123– 134. [14] D. Yu, R. Warrington, R. Barron, T. Ameel, An experimental and theoretical investigation of fluid flow and heat transfer in microtubes, in Proceedings of ASME/JSME Thermal Engineering Joint Conf., Maui, HI, 1995. [15] K. Hara, M. Inoue, M. Furukawa, Heat transfer in minichannel gaseous cooling, J. Environ. Eng. 2 (2007) 525–534. [16] C. Hong, Y. Asako, I. Ueno, M. Motosuke, Total temperature measurement of laminar gas flow at microtube outlet: cooled from the wall, Heat Transfer Eng. 35 (1) (2014) 142–149. [17] V. Lijo, H.D. Kim, T. Setoguchi, Effects of choking on flow and heat transfer in micro-channels, Int. J. Heat Mass Transf. 55 (2012) 701–709. [18] C.W. Chen, T.Y. Lin, C.Y. Yang, S.G. Kandlikar, An experimental investigation on heat transfer characteristics or air and co2 in microtubes, in: Proceedings of 9th ASME International Conference on Nanochannels, Microchannels, and Minichannels, June 19–22, Edmonton, Canada, 2011. [19] C.Y. Yang, C.W. Chen, T.Y. Lin, S.G. Kandlikar, Heat transfer and friction characteristics of air flow in microtubes, Exp. Thermal Fluid Sci. 37 (3) (2012) 12–18. [20] V. Gnielinski, Ein neues berechnungsverfahren fuer die waermeuebertragung im uebergangsbereich zwischen laminarer und turbulenter roehrstromung, Forsch. Ingenieurwes. 61 (9) (1995) 240–248. [21] C. Hong, Y Asako, Heat transfer characteristics of gaseous flows in a microchannel and a microtube with constant wall temperature, Numer. Heat Transfer A: Appl. 52 (3) (2007) 219–238. [22] Y. Yang, G.L. Morini, H. Chalabi, M. Lorenzini, The effect on the Nusselt number of the non-linear axial temperature distribution of gas flows through commercial microtubes, Heat Transfer Eng. 35 (2) (2014) 159–170. [23] T.Y. Lin, S.G. Kandlikar, A theoretical model for axial heat conduction effects during single-phase flow in microchannels, ASME J. Heat Transfer 134 (2012) 020902. [24] C. Hong, Y. Asako, I. Ueno, Flow and heat transfer characteristics of turbulent gas flow in microtube with constant wall heat flux, J. Phys: Conf. Ser. 362 (1) (2012) 012022. [25] C. Hong, Y. Asako, J.-H. Lee, Heat transfer characteristics of gaseous flows in micro-channel with constant heat flux, Int. J. Therm. Sci. 46 (11) (2007) 1153– 1162. [26] T.Y. Lin, S.G. Kandlikar, Heat transfer investigation of air flow in microtube – Part I: effects of heat loss, viscous heating, and axial conduction, ASME J. Heat Transfer 135 (2014) 031703. [27] Y. Yang, T. Schakenbos, H. Chalabi, M. Lorenzini, G.L. Morini, Compressible gas flow through heated commercial microtubes, in: Proceedings of the 4th International Conference on Heat Transfer and Fluid Flow in Microscale, September 4–9, Fukuoka, Japan, 2011. [28] G. Croce, P. D’Agaro, Compressibility and rarefaction effect on heat transfer in rough microchannels, Int. J. Therm. Sci. 48 (2) (2009) 252–260.