International Journal of Heat and Mass Transfer 121 (2018) 187–195

Contents lists available at ScienceDirect

International Journal of Heat and Mass Transfer

journal homepage: www.elsevier .com/locate / i jhmt

Flow characteristics of gaseous flow through a microtube dischargedinto the atmosphere

https://doi.org/10.1016/j.ijheatmasstransfer.2017.12.1040017-9310/� 2018 Elsevier Ltd. All rights reserved.

⇑ Corresponding author.E-mail addresses: hong@mech.kagoshima-u.ac.jp (C. Hong), k3944613@kadai.jp

(G. Tanaka), y.asako@utm.my (Y. Asako), katanoda@mech.kagoshima-u.ac.jp(H. Katanoda).

Chungpyo Hong a,⇑, Goku Tanaka a, Yutaka Asako b, Hiroshi Katanoda aaDepartment of Mechanical Engineering, Kagoshima University, 1-21-40 Korimoto, Kagoshima 890-0065, JapanbDepartment of Mechanical Precision Engineering, Malaysia-Japan International Institute of Technology, University Technology Malaysia, Jalan Sultan Yahya Petra, 54100Kuala Lumpur, Malaysia

a r t i c l e i n f o

Article history:Received 21 October 2016Received in revised form 25 August 2017Accepted 21 December 2017Available online 5 January 2018

Keywords:Friction factorTurbulentCompressible gaseous flowMicrotubeChoked flow

a b s t r a c t

Flow characteristics for a wide range of Reynolds number up to turbulent gas flow regime, including flowchoking were numerically investigated with a microtube discharged into the atmosphere. The numericalmethodology is based on the Arbitrary-Lagrangian-Eulerian (ALE) method. The LB1 turbulence model wasused in the turbulent flow case. Axis-symmetric compressible momentum and energy equations of anideal gas are solved to obtain the flow characteristics. In order to calculate the underexpanded (choked)flow at the microtube outlet, the computational domain is extended to the downstream region of thehemisphere from the microtube outlet. The back pressure was given to the outside of the downstreamregion. The computations were performed for adiabatic microtubes whose diameter ranges from 10 to500 lm and whose aspect ratio is 100 or 200. The stagnation pressure range is chosen in such a way thatthe flow becomes a fully underexpanded flow at the microtube outlet. The results in the wide range ofReynolds number and Mach number were obtained including the choked flow. With increasing the stag-nation pressure, the flow at the microtube outlet is underexpanded and choked. Although the velocity islimited, the mass flow rate (Reynolds number) increases. In order to further validate the present numer-ical model, an experiment was also performed for nitrogen gas through a glass microtube with 397 lm indiameter and 120 mm in length. Three pressure tap holes were drilled on the glass microtube wall. Thelocal pressures were measured to determine local values of Mach numbers and friction factors. Local fric-tion factors were numerically and experimentally obtained and were compared with empirical correla-tions in the literature on Moody’s chart. The numerical results are also in excellent agreement withthe experimental ones.

� 2018 Elsevier Ltd. All rights reserved.

1. Introduction

Advanced development to the design technology of MEMS(micro electro mechanical system) have increased the need foran understanding of fluid flow and heat transfer of micro flowdevices such as micro-heat exchangers, micro-reactors and manyother micro-fluid systems. Therefore numerous experimental andnumerical studies have been performed in an effort to betterunderstand flow characteristics in microchannels.

In the case of gaseous flow in microchannels, it is well knownthat the rarefaction, the surface roughness, and the compressibilitysignificantly affect the flow characteristics separately or simulta-neously [1]. For the microchannels with 10 lm or more in hydrau-

lic diameter, the effect of compressibility is more dominant on flowcharacteristics than that of surface roughness and rarefaction. Thecompressibility effect leads that the flow accelerates along thelength and the pressure steeply falls near the outlet due to gasexpansion. Therefore to obtain the local value of friction factor isimportant for an understanding of flow phenomenon of gaseousflow in microchannels. The compressibility effect on laminar gasflow in microchannels have been numerically investigated bymany researchers, e.g. Prud’homme et al. [2], Berg et al. [3], Kaveh-pour et al. [4], Guo et al. [5], Sun and Faghri [6]. Recently, Asakoet al. [7,8] and Hong et al. [9–11] conducted numerical investiga-tions of gas flow in microchannels. They obtained f�Re correlationsas functions of Mach number and Knudsen number. The f�Re corre-lation obtained for rectangular microchannels are in excellentagreement with the experimental values of f�Re obtained by Honget al. [12] who measured the local pressure along the channellength, to determine the local values of Mach number and frictionfactor for the range of 58 � Re � 7965 for nitrogen.

http://crossmark.crossref.org/dialog/?doi=10.1016/j.ijheatmasstransfer.2017.12.104&domain=pdfhttps://doi.org/10.1016/j.ijheatmasstransfer.2017.12.104mailto:hong@mech.kagoshima-u.ac.jpmailto:k3944613@kadai.jpmailto:y.asako@utm.mymailto:katanoda@mech.kagoshima-u.ac.jp https://doi.org/10.1016/j.ijheatmasstransfer.2017.12.104http://www.sciencedirect.com/science/journal/00179310http://www.elsevier.com/locate/ijhmt

10D

L

x

r

D

PstgTstg

flow

Microtube(adiabatic wall)

Reservoir

10D

L

x

r

D

PstgTstg

flow

Microtube(adiabatic wall)

Reservoir

Fig. 1. Schematic diagram of a problem.

Nomenclature

a speed of sound [m/s]D microtube diameter [m]i specific internal energy [J/kg]k turbulence energy [m2/s2]L micro-tube length [m]Ma Mach number [–]n pressure port numberp static pressure [Pa]p� modified pressure, ¼ pþ 23qk [Pa]r, x coordinates [m]R gas constant [J/(kg�K)]Re Reynolds number [–]T static temperature [K]tu turbulent intensity [–]u, v velocity components [m/s]yþ dimensionless wall distance [–]d� displacement thickness based on mass flow [m]e turbulence dissipation rate [m2/s3]c specific heat ratio [–]k thermal conductivity [W/(m�K)]keff effective thermal conductivity, ¼ kþ kt [W/(m�K)]kt turbulent thermal conductivity [W/(m�K)]

l viscosity [Pa�s]leff effective viscosity, ¼ lþ lt [Pa�s]lk diffusion coefficient for k equation, ¼ lþ lt=rk [Pa�s]lt turbulent viscosity [Pa�s]le diffusion coefficient for e equation, ¼ lþ lt=re [Pa�s]q density [kg/m3]r turbulent Prandtl number [–]s shear stress [Pa]sw shear stress on wall [Pa]/ dissipation function [1/s2]

Subscriptave cross sectional average valuein inletout outlet of micro-tubestg stagnation value

Superscript� Reynolds-averaged value~ Favre-averaged value

188 C. Hong et al. / International Journal of Heat and Mass Transfer 121 (2018) 187–195

In the case of turbulent gas flow in microchannels, Chen et al.[13] performed numerical procedures to solve compressible, tur-bulent boundary-layer equations by using the Baldwin-Lomaxtwo-layer turbulence model. The numerically calculated f�Re1/4 val-ues are higher than those predicted by the Fanno line flow.Recently, Murakami and Asako [14] investigated numerically toobtain the effect of compressibility on the local pipe friction factorsof laminar and turbulent gas flow in microtubes. They reportedthat the ratio of the Fanning friction factor to Blasius formula forturbulent flow is equal to unity but the ratio of the Darcy frictionfactor to Blasius formula is still a function of the Mach number.However, the Fanning friction factor of the turbulent flow in a PEEKmicrotube measured by Kawashima and Asako [15] is 12–20%higher than the value predicted from Blasius formula.

Attention will now be focused on choked flow in microchan-nels; the choked flow has been extensively investigated over theyears under the conditions that the inlet pressure is preserved ata specific (atmospheric) pressure and the back pressure is decom-pressed. Lijo et al. [16] numerically investigated the effect of chok-ing on flow and heat transfer in a microchannel whose hydraulicdiameter is 300 lm. They considered the flow to be choked whenthe mass flow rate does not change with the conditions of a specificinlet pressure and a further decrease in the back pressure. Theyreported that for a higher-pressure ratio the Mach number nearthe exit of the channel is well above 1.0 since thinning boundarylayer near the exit. On the other hand, in the case of atmosphericback pressure and the further increase in inlet pressure, the gasvelocity becomes limited and the mass flow rate (Reynolds num-ber) is increased. In this situation the outlet pressure of the channelis higher than the back pressure and the flow becomes underex-panded. Kawashima et al. [17] investigated numerically the Machnumber and pressure at outlet plane of a straight microtube forboth laminar and turbulent flow cases. They found that the Machnumber at the outlet plane of the choked flow depends on the tubediameter and ranges from 1.16 to 1.25. However, details of choked(underexpanded) flow in a micro-tube are still open problemsbecause of measurement limitation. There also seems to be noparametric study to investigate flow characteristics of non-choking and choking turbulent gas flows through a microtube dis-

charged into atmosphere with an experimental validation. This isthe motivation of the present numerical study with microtubeswhose diameters range from 10 to 500 lm and whose aspect ratiosare 100 and 200. In order to further validate the present numericalmodel, an experiment was also conducted with a glass microtubewith 397 lm in diameter and 120 mm in length.

2. Description of the problem

The schematic diagram of gaseous flow in a microtube fornumerical calculation is shown in Fig. 1. The numerical calcula-tions were performed under the assumption that the flow is steadyand axisymmetric and laminar or turbulent. Compressible fluid in areservoir at the stagnation pressure, pstg and the stagnation tem-perature, Tstg, passes through an adiabatic microtube into theatmosphere at the pressure, pb (105 Pa). The calculational domainis extended to the downstream region of hemisphere to calculatethe underexpanded flow as shown in Fig. 1. The physical quantitiesare the time mean values and the physical properties such as theviscosity and thermal conductivity of the fluid are assumed to beconstant. For a microtube, the following governing equations areused [17]:

@qeu@x

þ 1r@qrev@r

¼ 0 ð1Þ

@qeueu@x

þ 1r@qrev eu

@r¼ � @p

�

@xþ @sxx

@xþ 1

r@rsrx@r

ð2Þ

C. Hong et al. / International Journal of Heat and Mass Transfer 121 (2018) 187–195 189

@qeuev@x

þ 1r@qrev ev

@r¼ � @p

�

@rþ @sxr

@xþ 1

r@rsrr@r

� shhr

ð3Þ

@qeuei@x

þ 1r@qrevei@r

¼ �p� @eu@x

þ 1r@rev@r

� �þ leff/þ

@qx@x

þ 1r@rqr@r

ð4Þ

where the viscous stresses are expressed as

sxx ¼ leff 2@eu@x

� 23

@eu@x

þ 1r@rev@r

� �� �ð5Þ

srr ¼ leff 2@ev@r

� 23

@eu@x

þ 1r@rev@r

� �� �ð6Þ

shh ¼ leff 2evr� 23

@eu@x

þ 1r@rev@r

� �� �ð7Þ

sxr ¼ srx ¼ leff@ev@x

þ @eu@r

� �ð8Þ

and the heat fluxes caused by thermal conduction are

qx ¼ keff@eT@x

; qr ¼ keff@eT@r

ð9Þ

/ is the viscous dissipation function expressed as

/ ¼ 2 @eu@x

� �2þ ev

r

� �2þ @ev

@r

� �2( )� 23

@eu@x

þ 1r@rev@r

� �2

þ @eu@r

þ @ev@x

� �2ð10Þ

Also, the equation of state for the ideal gas can be expressed as

ei ¼ 1c� 1

pq¼ Rc� 1

eT ð11Þ

2.1. Turbulence model

The k-e low Reynolds number model is used. In this model, theturbulence kinetic energy k and the turbulence dissipation rate eare solved to determine the coefficient of turbulent viscosity lt.The various turbulence models have been proposed. In this paper,the Lam-Bremhorst low Reynolds number (LB1) model [18,19] ischosen since LB1 model is widely used and very stable.

@qkeu@x

þ 1r@qrkev@r

¼ Pk þ @@x

lk@k@x

� �þ 1

r@

@rrlk

@k@r

� �� qe ð12Þ

@qeeu@x

þ 1r@qreev@r

¼ Ce1f 1ekPk þ @

@xle

@e@x

� �þ 1

r@

@rrle

@e@r

� �� Ce2f 2q

e2

kð13Þ

Pk ¼ lt/�23qk

@eu@z

þ 1r@rev@r

� �ð14Þ

lt ¼ Clf lqk2

e; kt ¼ cpltrT ð15Þ

Constants and functions are

Cl ¼ 0:09; rk ¼ 1:0; re ¼ 1:3; rT ¼ 0:9;Ce1 ¼ 1:44; Ce2 ¼ 1:92;

f 1 ¼ 1þ0:05f l

!3; f 2 ¼ 1� e�R

2T ; f l ¼ ð1� e�0:0165Ry Þ

21�20:5

RT

� �;

RT ¼ qk2

el; Ry ¼ q

ffiffiffik

pyw

lð16Þ

These values are standard values of the model and yw repre-sents the minimum distance from the tube wall.

2.2. Boundary conditions

The effect of rarefaction is negligible because the tube diameteris much larger than the molecular mean free path. Therefore no-slip condition at the wall is used for velocity. The thermal bound-ary condition on the wall is adiabatic (not like isentropic flow, flowon an adiabatic tube wall will be called adiabatic flow hereinafter).It is also assumed that the velocity, pressure, temperature and den-sity profiles at the inlet are uniform. From these assumptions, theboundary conditions are expressed as

on the tube wall : u ¼ v ¼ k ¼ 0; @T=@r ¼ @e=@r ¼ 0on the wall at the outlet plane :u ¼ v ¼ k ¼ 0; @T=@r ¼ @e=@r ¼ 0

on the symmetric axis :v ¼ 0; @u=@r ¼ 0; @T=@r ¼ @k=@r ¼ @e=@r ¼ 0

at the inlet : v ¼ 0; u ¼ uin; p ¼ pin; q ¼ qin; T ¼ T in;k ¼ kin; e ¼ ein

at the hemispheric outlet : p ¼ pb

ð17Þ

The values of velocity, pressure, temperature and density at theinlet are evaluated by the stagnation treatment proposed by Karki[20].

pinpstg

¼ qinqstg

!c¼ T in

Tstg

� �c=ðc�1Þ; uin

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2cRTstgc� 1 1�

pinpstg

!ðc�1Þ=c8

190 C. Hong et al. / International Journal of Heat and Mass Transfer 121 (2018) 187–195

Note that the Re is constant along the tube but the Maave variesin the flow direction.

The product of friction factor and Reynolds number is calledPoiseuille number for laminar flow regime. The friction factorsbased on the Darcy’s and Fanning definitions will be introduced.The Darcy friction factor is defined as

f d ¼�2D

qaveu2avedpavedx

� �ð22Þ

The modified Fanning friction factors (four times of Fanningfriction factor) is based on the wall shear stress and is defined as

f f ¼4sw

ð1=2Þqaveu2ave¼ 4lð@u=@rÞr¼0:5Dð1=2Þqaveu2ave

ð23Þ

2.4. Numerical simulation

The simulation code used is SALE [21]. The methodology of thecode is based on the ALE (Arbitary-Langrangian-Eulerian) method.The detailed description of the ALE method is documented in theliterature by Amsden et al. [21] and will not be presented here.In SALE, the computational domain is divided into quadrilateralcells. The velocity components are assigned at the vertices of thecell and the other values such as pressure, specific internal energy(temperature), and density are assigned at the cell centers. The cellsize for all diameter gradually increased as the power of the spac-ing number [22] in the x-direction to x/L = 0.8, and it graduallydecreased to the outlet of the tube. The number of the cells in x-direction in the tube section is 200 and the index of the power-law spacing is 1.8 in all case. The cell size gradually increased againin the radial direction of the hemispheric downstream section. Thenumber of the cell in the downstream section in the radial direc-tion is 300 for all diameters. The Low-Reynolds number turbulencemode requires sufficiently small cell near the tube wall. The cellsize in r-direction gradually increased as the power of the spacingnumber form the tube wall to the center of the tube. The cell cen-ters adjacent to the tube wall are placed in such a way that thedimensionless wall distance of the grid point y+, defined as

yþ ¼ ywql

ffiffiffiffiffiffiswq

rð24Þ

is less than 2 and least two cell centers are allocated in the viscoussub layer. The number of the grids in r-direction is 40 for all diam-eters. The index of the power-law spacing is 1.2 for D � 100 lm andfor 1.4 for D � 250 lm, respectively. The effects of cell size, down-stream size and thermal boundary condition of tube wall on flowcharacteristics are well documented in Ref. [17].

3. Results and discussion

Numerical computations were conducted for microtubes of D =10, 20, 50, 100, 250 and 500 lm whose aspect ratio is 100 and 200.All the computations were performed as laminar flow or turbulentflow. The tube diameter, the tube length, the stagnation pressureand the corresponding Reynolds number are listed in Table 1.

Table 1Tube diameter, length, pstg and Re.

D (lm) L (m) pstg (kPa) Re

10 0.001 500–4000 117–48480.002 300–9000 22–4957

20 0.002 500–3000 384–40730.004 500–5500 234–6182

50 0.005 200–900 330–28420.01 200–1800 193–4930

3.1. Mach number, temperature and pressure

The contour plots of Mach numbers and temperatures in thetube and its downstream region of D = 100 lm and L = 0.02 m arerepresented in Figs. 2 and 3, respectively. These are typical contourplots of Mach numbers and temperatures for fast flow (Ma > 0.3).As seen in Fig. 2 the flow is accelerated and the Mach numberincreases near the outlet due to gas expansion caused by the pres-sure drop. Therefore, the temperature decreases can be seen nearthe outlet due to thermal energy conversion into kinetic energyas shown in Fig. 3. Since the contour plots of Mach number andtemperature for pstg = 1200 and 1600 kPa are almost the same asshown in Figs. 2(b) and (c) and 3(b) and (c), the flow is chokedand underexpanded at the outlet. Fig. 4 shows the pressure varia-tions for all cases of D = 100 lm and L = 0.02 m. The pressure in thefigure is the average value at a cross-section. The pressure curvebecomes increasingly nonlinear as the stagnation pressureincreases. The pressure falls steeply near the outlet due to the flowacceleration and gas expansion. The pressure at the outlet (pstg �500 kPa) is higher than back pressure (100 kPa) since the flow atthe outlet is choked (underexpanded). The Mach number is repre-sented in Fig. 5. It is the average value at a cross-section. The Machnumber increases along the tube length due to the flow accelera-tion. It also increases with increasing stagnation pressure. And itis very slightly increases at pstg � 500 kPa since the flow at the out-let becomes underexpanded and choked. The Mach number at theoutlet for some cases exceeds the speed of sound.

The outlet Mach numbers obtained for all computations areplotted as a function of Reynolds number in Fig. 6. They are theaverage values at the outlet cross-section. The outlet Mach numberincreases with increasing Reynolds number and levels off at theReynolds number where the flow is choked. Then, the outlet Machnumbers for all diameters are more than unity. The outlet Machnumber of the smaller diameter reaches a higher maximum valueat a smaller Reynolds number. The microtubes with the samediameter and different aspect ratio have almost the same outletMach number at an arbitrary Reynolds number. In the case of D= 10 and 20 lm, the flow is choked in the laminar flow regime.And the outlet Mach number leveled off, then decreases aroundRe = 2300 and levels off since the velocity profile at a cross sectionbecomes flatter due to the flow transition from laminar to turbu-lent. As a result of that, the outlet Mach number with flow chokingis more than unity and depends on the diameter. The Mach num-ber on the outlet plane of a microtube was discussed in detail inour previous study [17].

If a tube flow of the fluid with very high thermal conductivitycan be considered, the fluid flow becomes isothermal, which thefluid inside the tube has almost the same temperature.

Therefore, in order to compare the outlet Mach numberobtained by adiabatic flow (the present study) with that obtainedby isothermal flow, supplement runs were performed with isother-mal flow condition of T = 300 K at tubes of D = 20, 100 and 500 lm.The obtained outlet Mach numbers for isothermal flow are alsoplotted in Fig. 6 by the black colored symbols. They increase withincreasing Reynolds number and level off with similar trends ofthose of adiabatic flow. However, the values obtained by isother-

D (lm) L (m) pstg (kPa) Re

100 0.01 200–1000 994–70730.02 300–1800 669–10,657

250 0.025 200–1000 3095–19,0790.05 125–1200 809–18,403

500 0.05 200–500 6676–19,0810.1 105–800 174–24,926

0.3 0.36 0.42 0.48 0.54 0.6 0.66 0.72 0.78 0.84 0.9 0.96 1.02 1.08 1.14 1.2

Ma

0.30 0.84 1.38 1.92 2.46 3.00

(a) 0 0.005 0.01 0.015 0.020

5E-05

x (m)

x (m)

x (m)

x (m)

x (m)

x (m)

0.02 0.0201 0.0202

(b) 0 0.005 0.01 0.015 0.020

5E-05

0.02 0.0201 0.0202

(c) 0 0.005 0.01 0.015 0.020

5E-05

r (m

)r

(m)

r (m

)

0.02 0.0201 0.0202

Fig. 2. Contour plots of Mach number for D = 100 lm and L = 0.02 m: (a) pstg = 300 kPa (Re = 1414 andMaave, out = 0.587), (b) pstg = 1200 kPa (Re = 6817 andMaave, out = 1.230),(c) pstg = 1600 kPa (Re = 9322 and Maave, out = 1.238).

(a) 0 0.005 0.01 0.015 0.020

5E-05

240 246 252 258 264 270 276 282 288 294 300

T (K)

r (m

)r

(m)

r (m

)

0.02 0.0201 0.0202

130 165 200 235 270 305

(b) 0 0.005 0.01 0.015 0.020

5E-05

0.02 0.0201 0.0202

(c) 0 0.005 0.01 0.015 0.020

5E-05

x (m)

x (m)

x (m) x (m)

x (m)

x (m)

0.02 0.0201 0.0202

Fig. 3. Contour plots of temperature for D = 100 lm and L = 0.02 m: (a) pstg = 300 kPa (Re = 1414 and Maave, out = 0.587), (b) pstg = 1200 kPa (Re = 6817 and Maave, out = 1.230),(c) pstg = 1600 kPa (Re = 9322 and Maave, out = 1.238).

0 0.005 0.010 0.015 0.020

500

1000

1500

2000

300, 1414500, 2540

800, 4323

1000, 55711200, 68171400, 80631600, 9322

pstg

=1800kPa, Re=10657 D=100µm, L=0.02m

p ave

(kP

a)

x (m)

Fig. 4. Pressure as a function of x for D = 100 lm and L = 0.02 m.

0 0.005 0.010 0.015 0.020

0.2

0.4

0.6

0.8

1.0

1.2

1.4

200, 669

300, 1414

pstg

=400-1800kPa, Re=2194-10657

D=100µm, L=0.02m

Ma a

ve

x (m)

Fig. 5. Mach number as a function of x for D = 100 lm and L = 0.02 m.

C. Hong et al. / International Journal of Heat and Mass Transfer 121 (2018) 187–195 191

0 4000 8000 12000 16000 200000

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Ma a

ve, o

ut

Re

Adiabatic flowL/D=200 100 D(µm)

10 20 50 100 250 500

Isothermal flowL/D=200 D(µm)

20 100 500

Fig. 6. Outlet Mach number as a function of Re.

0.2

0.4

0.6

0.8

1.0

1.2

x/L

Ma a

ve

D=50µmL(m) 0.01 0.005p

stg(kPa) 700 600

Re 1817 1833Ma

ave, out 1.1328 1.1253

Maave

(a) D = 50 µm

0 0.2 0.4 0.6 0.8 1.0

0.8

1.0

1.2

1.4

ave

D=250µmL(m) 0.05 0.025p

stg(kPa) 1000 900

Re 15152 15078Ma

ave, out 1.1994 1.2001

192 C. Hong et al. / International Journal of Heat and Mass Transfer 121 (2018) 187–195

mal flow are lower than those obtained by adiabatic flow (the pre-sent study) since the speed of sound does not include the effect oftemperature decrease as seen in Fig. 3. However the valuesobtained by isothermal flow leveled off are slightly higher thanunity. Fig. 7 represents curves of the average Mach number, veloc-ity and speed of sound at a cross section for D = 100 lm and L =0.02 m obtained under both the adiabatic and isotherm flows.The curve of the average velocity for isothermal flow coincideswith that for the adiabatic flow. However, the values of speed ofsound for isothermal flow are constant along the length and thosefor the adiabatic flow decrease near the outlet with decreasingtemperature as seen in Fig. 3. As a result of that, Mach numberof isothermal flow is lower than that of the adiabatic flow. If thethermal conductivity of the gas is extremely high, the flow isalmost close to isothermal flow. However in actual situations, theflow seems to be closer to the adiabatic flow without heat inputfrom the wall.

The average Mach number curves at a cross section obtained formicrotubes with the same diameter and different length are plot-

0.2

0.4

0.6

0.8

1.0

1.2

1.4

100

200

300

400

0 0.005 0.010 0.015 0.020250

300

350

400

Maave, out, adiabatic

=1.2378

Maave, out, Iso

=1.067

Ma a

ve

Vel .ave, out, Iso

=359.6 m/s

Vel.ave, out, adiabatic

=377.0 m/s

Vel

. ave

(m

/s)

D=100µm, L=0.02m, pstg

=1600kPaRe Ma

ave, Vel.

ave, a

ave

Adiabatic 9322 Isothermal 9072

aave, out, adiabatic

=304.6 m/s

aave, out, Iso

=351.7 m/s

a ave

(m

/s)

x (m)

Fig. 7. Mach number, velocity and speed of sound as a function of x.

ted in Fig. 8 as a function of x/L. One case is for the laminar flowregime (Fig. 8(a)) and the other case is for turbulent one (Fig. 8(b)). The Reynolds number for each case is almost the same, butthe stagnation pressure of the longer microtube is lower than thatof the shorter one. As seen in both figures Mach number curves risein parallel until they reach x/L � 0.9, then they become the samesince at the outlet Mach number is a function of mass flow rate(Reynolds number) and tube diameter due to the outlet faced tothe atmosphere. As mentioned above, the outlet Mach numberon the outlet plane of the choked flow is only a function of tubediameter [17].

3.2. Friction factor

Fanning friction factor, ff for all computations except D = 10 lmwas obtained by Eq. (23). The values of the product of Fanning fric-tion factor and Reynolds number, ff�Re at 0.8L are plotted as a func-tion of the Mach number at a cross sectional average in Fig. 9. Inthe case of laminar flow regime, the values increased with increas-ing Mach number since compressibility effect is dominant due toflow acceleration. Hong et al. [10] numerically obtained the ff�Re

x/L0 0.2 0.4 0.6 0.8 1.0

0.2

0.4

0.6Ma Maave

(b) D = 250 µm

Fig. 8. Mach number as a function of x/L.

0.1 0.2 0.3 0.4 0.5 0.60

100

200

300

400

500

2170

Laminar flow

Turbulent flow

30783172

Re=2060

f f R

e

Maave

ff Re=64+1.39Ma+49.07Ma2+33.51Ma3

ff Re at x=0.8L

D=20 50 100 250 500 (µm)

Fig. 9. ff�Re as a function of Re.

C. Hong et al. / International Journal of Heat and Mass Transfer 121 (2018) 187–195 193

correlation for a circular microtube as a quadratic function of Machnumber. The solid line in the figure represents the ff�Re correlation.The present results for all computation coincide with the ff�Re cor-relation. In the case of turbulent regime, the values of ff�Re increasewith a large slope in deviation from the solid line. And theyincrease steeply because the flow is choked. This means that thegas velocity at each location remains nearly constant althoughthe Reynolds number increase as shown in Fig. 6.

The values of Fanning and Darcy friction factor, ff and fd, at 0.4L,0.6L, 0.8L and 0.9L obtained for D = 250 lm and L = 0.05 m are plot-ted onMoody’s chart in Fig. 10. The corresponding Mach number ateach location is also plotted in the figure. The solid line and dottedline in the figure represent the values obtained from f = 64/Re and f= 0.3164Re�0.25 (Blasius formula) for incompressible flow, respec-

1000 100000.02

0.10

0.2

0.4

0.6

0.8

1.0

0.08

0.06

200003000

0.04

5000

D=250µm, L=50mm x Maave fd ff0.4L0.6L0.8L0.9L

f

Re

choked flow

Ma a

ve

Fig. 10. Mach number and f on Moody chart.

(a)

(c)

Fig. 11. Pictures of a glass microtube: (a) cross section, (b) c

tively. As shown in the laminar flow regime, in the case of Ma <0.2 (slow flow), both values of ff and fd are slightly higher thanthose of incompressible flow theory (f = 64/Re). The differencebetween ff and fd for the range from x = 0.4L to 0.9L is small becausethe Mach numbers in the range are almost the same. However inthe case of Ma > 0.2, they deviate more and more from that of anincompressible flow and the difference between ff and fd is largerthan that of Ma < 0.2 because the Mach number from x = 0.4L to0.9L increase due to the compressibility effect.

As seen in the turbulent flow regime, the values of ff closelycoincide with the Blasius formula and might be only a functionof Reynolds number. The compressibility (Mach number) effectincluding the choked flow on ff is quite small. However the valuesof fd deviate from those of Blasius formula with increasing Machnumber. And they are parallel to those of Blasius formula in therange of Re > 6000 because of flow choking. Qualitatively similarresults for the microtube of D = 10, 20, 50, 100 and 500 lm areobtained.

3.3. Validation with experimental data

In order to further validate the present numerical model, anexperimental investigation whose setup was analogous to that ofKawashima et al. [15] was performed for nitrogen gas through aglass microtube with 397 lm in diameter and 12 cm in length.The picture of the cross section of the microtube is shown inFig. 11(a). In order to measure the roughness of inner surface ofthe tube, a part of the microtube is cut and the picture of a cut tubeis shown in Fig. 11(b). The microscopic image of its roughness fea-ture (surface morphology) is also shown in Fig. 11(c), where theimage is typical of all test sections used in the present study. Thearithmetic mean height of the surface (Sa) of the glass microtubewas measured with a 3D laser scanning confocal microscope forprofilometry (Keyence VK-X260, Display resolution: 1 nm). The

(b)

ut tube, (c) microscopic image of its roughness feature.

0

2.0

4.0

6.0

8.0

100 200 300 400 500 600 7000.2

0.3

0.4

0.5

0.6

0.7

Glass microtube D=397µm, L=12cm Exp. Num.

[ x10-5 ]

m (

kg/s

)M

a ave

pstg (kPa)

x (cm) Exp. Num. 9 10 11

Fig. 14. Mass flow rates and Mach numbers as a function of stagnation pressure.

3000 10000

0.03

0.05

16000

Blasius eq.x(cm) Exp. Num.9~10

Glass microtube D=397µm, L=12cm

0.04

5000

f f

Re

Fig. 15. ff on Moody chart.

194 C. Hong et al. / International Journal of Heat and Mass Transfer 121 (2018) 187–195

arithmetic mean height of the tube (Sa) was around 0.073 lm, andrelative roughness which is the ratio of the Sa and the diameter wasabout 0.0002%. Therefore the glass microtube is seemed to besmooth. Three pressure tap holes near the outlet were drilled asshown in Fig. 12. The local pressures were measured to determinethe local values of Mach number and friction factor. Supplemen-tary runs were also conducted for 6 cases whose tube dimensionscoincide with those of the experiment. Since the thermal boundarycondition of the computation is adiabatic, the glass microtubetested in the present study was covered with a foamed polystyreneto avoid heat gain and loss from the surroundings. The measuredpressures at the pressure tap holes are plotted Fig. 13 as a functionof x. The stagnation pressure is the curve parameter and the curvesof the pressure are plotted for every 100 kPa. The pressures at theinlet that are obtained by solving Eq. (18) are also plotted in thefigure. Numerically obtained pressures are also plotted in the fig-ure by solid lines. Note that the pressure at the outlet of the tube(x = 0.12 m) is higher that the back pressure (pb = 100 kPa) in thecase of pstg � 400 kPa. Both experimentally measured pressuresand numerically obtained ones almost agree well. Experimentallyand numerically obtained mass flow rates and Mach numbers atx = 0.09, 0.1 and 0.11 m and are plotted as a function of the stagna-tion pressure in Fig. 14. The numerical Mach numbers wereobtained by Eq. (20) and the experimental ones were obtainedwith the local static gas temperature determined by the measuredlocal pressure. The white symbols represent numerical results andthe black symbols represent experimental ones. Mach numberincreases with increasing stagnation pressure and levels off inthe range pstg � 400 kPa. This is the reason why the flow is chokedin that range. Both experimental and numerical mass flow rate andlocal Mach number are in excellent agreement.

The quasi-local Fanning friction factor between pressure tapholes for Fanno flow can be expressed with Eq. (23) as [15]

f f ¼4sw

ð1=2Þqaveu2ave

¼ 2Dxnþ1 � xn

p2ave;n � p2ave;nþ1RðTave;n þ Tave;nþ1Þ � ln

pave;npave;nþ1

!� ln Tave;n

Tave;nþ1

� �" #ð25Þ

Both experimental and numerical values of the quasi-local Fan-ning friction factors were obtained from Eq. (25) between x = 9 cm

Fig. 12. Picture of pressure tap holder.

0 0.02 0.04 0.06 0.08 0.10 0.12100

200

300

400

500

600

700

200

300

400

500

600

pstg

=700kPa

p ave

(kP

a)

x (m)

Glass microtube D=397µm, L=12cm Num. Exp. estimated by Eq. (18)

Fig. 13. Pressure distributions as a function of x.

and 10 cm. They are also plotted in Fig. 15 as a function of Reynoldsnumber on Moody chart. The solid line represents Blasius formula.Both of them nearly coincide with Blasius formula. As mentionedabove, the Fanning friction factor in the turbulent flow regimecoincides with Blasius formula and it is not a function of Machnumber since the compressibility effect including the choked flowon the Fanning friction factor is quite small. It is noteworthy thatboth experimentally and numerically obtained quasi local Fanningfriction factors of a flow in microtubes with a smooth inner surfacedischarged into the atmosphere are in good agreement with Bla-sius formula in the turbulent flow regime including the chokedflow, showing the validity to the present numerical model.

4. Conclusion

Numerical investigations to obtain flow characteristics for awide range of Reynolds number (22 � Re � 24,926) up to the tur-bulent gas flow regime including the choked flow were performedfor microtubes whose inner diameters are 10, 20, 50, 100, 200 and500 lm and whose aspect ratio is 100 or 200 in the case of atmo-spheric back pressure and the case of further increases in inletpressure with an experimental validation. The following conclu-sions were reached.

(1) The Fanning friction factor, flow rate and pressure drop ofthe turbulent flow in a microtube with a smooth surfacecan be estimated from Blasius formula. Measured quasi-local Fanning friction factors are in good agreement withBlasius formula in the range of 3167 � Re � 14,581 includingthe choked flow. The values of mass flow rates, local pres-sures, local Mach numbers and quasi-local Fanning frictionfactors obtained experimentally for a glass microtube of D

C. Hong et al. / International Journal of Heat and Mass Transfer 121 (2018) 187–195 195

= 397 lm and L = 120 mm with a smooth inner surfacenearly coincide with those obtained numerically under sameconditions.

(2) The correlation between ff�Re and Mach number can bephysically explained. This is, the values of ff�Re in the turbu-lent flow regime increase with a large slope in deviationfrom the ff�Re correlation obtained for the laminar flowregime as Mach number increases. And they increase steeplydue to flow choking where the Mach number stays constantbut Reynolds number increases.

(3) In the case of the unchoked flow, the outlet Mach numbersof the flow in microtubes with the same diameter and differ-ent lengths are identical when Reynolds numbers of the floware identical. And in the case of the choked flow, they haveidentical outlet Mach numbers depending on the microtubediameters.

Conflict of Interest

Authors declare that there is no conflict of interest.

References

[1] C. Hong, Y. Asako, J.-H. Lee, Poiseuille number correlation for high speed micro-flows, J. Phys. D Appl. Phys. 41 (2008), p. 105111(1-10).

[2] R.K. Prud’homme, T.W. Chapman, J.R. Bowen, Laminar compressible flow in atube, Appl. Sci. Res. 43 (1986) 67–74.

[3] H.R. Berg, C.A. Seldam, P.S. Gulik, Compressible laminar flow in a capillary, J.Fluid Mech. 246 (1993) 1–20.

[4] H.P. Kavehpour, M. Faghri, Y. Asako, Effects of compressibility and rarefactionon gaseous flows in microchannels, Numer. Heat Transfer, Part A 32 (1997)677–696.

[5] Z.Y. Guo, X.B. Wu, Compressibility effect on the gas flow and heat transfer in amicro tube, Int. J. Heat Mass Transfer 40 (13) (1997) 3251–3254.

[6] H. Sun, M. Faghri, Effect of rarefaction and compressibility of gaseous flow inmicro channel using DSMC, Numer. Heat Transfer, Part A 38 (1999) 153–158.

[7] Y. Asako, K. Nakayama, T. Shinozuka, Effect of compressibility on gaseous flowsin a micro-tube, Int. J. Heat Mass Transfer 48 (2005) 4985–4994.

[8] Y. Asako, T. Pi, S.E. Turner, M. Faghri, Effect of compressibility on gaseous flowsin micro-channels, Int. J. Heat Mass Transfer 46 (2003) 3041–3050.

[9] C. Hong, Y. Asako, S.E. Turner, M. Faghri, Friction factor correlations for gas flowin slip flow regime, J. Fluids Eng. 129 (2007) 1268–1276.

[10] C. Hong, Y. Asako, M. Faghri, J.-H. Lee, Poiseuille number correlations for gasslip flow in micro-tubes, Numer. Heat Transfer, Part A 56 (2009) 785–806.

[11] C. Hong, Y. Asako, K. Suzuki, M. Faghri, Frction factor correlations forcompressible gaseous flow in a concentric micro annular tube, Numer. HeatTransfer, Part A 61 (2012) 163–179.

[12] C. Hong, T. Yamada, Y. Asako, M. Faghri, Experimental investigation of laminar,transitional and turbulent gas flow in microchannels, Int. J. Heat Mass Transfer55 (2012) 4397–4403.

[13] C.S. Chen, W.J. Kuo, Numerical study of compressible turbulent flow inmicrotubes, Numer. Heat Transfer, Part A 45 (2004) 85–99.

[14] S. Murakami, Y. Asako, Local pipe friction factor of compressible laminar orturbulent flow in micro-tubes, 9th International Conference on Nanochannels,Microchannels, and Minichannels, ICNMM2011-58036, 2011.

[15] D. Kawashiam, Y. Asako, Measurement of quasi-local friction factor of gas flowin a micro-tube, J. Mech. Eng. Sci. 230 (5) (2016) 782–792.

[16] V. Lijo, H.D. Kim, T. Setoguchi, Effects of choking on flow and heat transfer inmicro-channels, Int. J. Heat Mass Transfer 55 (4) (2012) 701–709.

[17] D. Kawashima, T. Yamada, C. Hong, Y. Asako, Mach number at outlet plane of astraight micro-tube, J. Mech. Eng. Sci. 230 (19) (2016) 3420–3430.

[18] C.K.G. Lam, K. Bremhorst, A modified form of the k-emodel for predicting wallturbulence, ASME J. Fluid Eng 103 (1981) 456–460.

[19] V.C. Patel, W. Rodi, G. Scheuerer, Turbulence models for near-wall and lowReynolds number flows: a review, AIAA J. 23–9 (1984) 1308–1319.

[20] K.C. Karki, A Calculation Procedure for Viscous Flows at All Speeds in ComplexGeometries PhD Thesis, University of Minnesota, 1986.

[21] A.A. Amsden, H.M. Rupell, C.W. Hire, SALE a Simplified ALE Computer Programof Fluid Flow at All Speeds, Los Alamos Scientific Lab Rep., LA-8095, 1980.

[22] S.V. Patankar, Computation of Conduction and Duct Flow Heat Transfer,Innovative Research, Inc., Minnesota, 1991.

http://refhub.elsevier.com/S0017-9310(16)33336-1/h0005http://refhub.elsevier.com/S0017-9310(16)33336-1/h0005http://refhub.elsevier.com/S0017-9310(16)33336-1/h0010http://refhub.elsevier.com/S0017-9310(16)33336-1/h0010http://refhub.elsevier.com/S0017-9310(16)33336-1/h0015http://refhub.elsevier.com/S0017-9310(16)33336-1/h0015http://refhub.elsevier.com/S0017-9310(16)33336-1/h0020http://refhub.elsevier.com/S0017-9310(16)33336-1/h0020http://refhub.elsevier.com/S0017-9310(16)33336-1/h0020http://refhub.elsevier.com/S0017-9310(16)33336-1/h0025http://refhub.elsevier.com/S0017-9310(16)33336-1/h0025http://refhub.elsevier.com/S0017-9310(16)33336-1/h0030http://refhub.elsevier.com/S0017-9310(16)33336-1/h0030http://refhub.elsevier.com/S0017-9310(16)33336-1/h0035http://refhub.elsevier.com/S0017-9310(16)33336-1/h0035http://refhub.elsevier.com/S0017-9310(16)33336-1/h0040http://refhub.elsevier.com/S0017-9310(16)33336-1/h0040http://refhub.elsevier.com/S0017-9310(16)33336-1/h0045http://refhub.elsevier.com/S0017-9310(16)33336-1/h0045http://refhub.elsevier.com/S0017-9310(16)33336-1/h0050http://refhub.elsevier.com/S0017-9310(16)33336-1/h0050http://refhub.elsevier.com/S0017-9310(16)33336-1/h0055http://refhub.elsevier.com/S0017-9310(16)33336-1/h0055http://refhub.elsevier.com/S0017-9310(16)33336-1/h0055http://refhub.elsevier.com/S0017-9310(16)33336-1/h0060http://refhub.elsevier.com/S0017-9310(16)33336-1/h0060http://refhub.elsevier.com/S0017-9310(16)33336-1/h0060http://refhub.elsevier.com/S0017-9310(16)33336-1/h0065http://refhub.elsevier.com/S0017-9310(16)33336-1/h0065http://refhub.elsevier.com/S0017-9310(16)33336-1/h0070http://refhub.elsevier.com/S0017-9310(16)33336-1/h0070http://refhub.elsevier.com/S0017-9310(16)33336-1/h0070http://refhub.elsevier.com/S0017-9310(16)33336-1/h0070http://refhub.elsevier.com/S0017-9310(16)33336-1/h0075http://refhub.elsevier.com/S0017-9310(16)33336-1/h0075http://refhub.elsevier.com/S0017-9310(16)33336-1/h0080http://refhub.elsevier.com/S0017-9310(16)33336-1/h0080http://refhub.elsevier.com/S0017-9310(16)33336-1/h0085http://refhub.elsevier.com/S0017-9310(16)33336-1/h0085http://refhub.elsevier.com/S0017-9310(16)33336-1/h0090http://refhub.elsevier.com/S0017-9310(16)33336-1/h0090http://refhub.elsevier.com/S0017-9310(16)33336-1/h0095http://refhub.elsevier.com/S0017-9310(16)33336-1/h0095http://refhub.elsevier.com/S0017-9310(16)33336-1/h0100http://refhub.elsevier.com/S0017-9310(16)33336-1/h0100http://refhub.elsevier.com/S0017-9310(16)33336-1/h0100http://refhub.elsevier.com/S0017-9310(16)33336-1/h0110http://refhub.elsevier.com/S0017-9310(16)33336-1/h0110http://refhub.elsevier.com/S0017-9310(16)33336-1/h0110

Flow characteristics of gaseous flow through a microtube discharged into the atmosphere1 Introduction2 Description of the problem2.1 Turbulence model2.2 Boundary conditions2.3 Dimensionless valuables2.4 Numerical simulation

3 Results and discussion3.1 Mach number, temperature and pressure3.2 Friction factor3.3 Validation with experimental data

4 ConclusionConflict of InterestReferences