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Heat transfer coefficient of gas flowing in a circular tube under rarefied condition
International Journal of Thermal Sciences 49 (2010) 1994e1999
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Heat transfer coefficient of gas flowing in a circular tube under rarefied condition Anwar Demsis, Bhaskar Verma, S.V. Prabhu, Amit Agrawal* Department of Mechanical Engineering, Indian Institute of Technology e Bombay, Powai, Mumbai 400076, India
a r t i c l e i n f o
a b s t r a c t
Article history: Received 17 November 2009 Received in revised form 8 March 2010 Accepted 25 May 2010 Available online 22 June 2010
The purpose of this paper is to present heat transfer measurements of gas in a tube under rarefied condition. The measurements are made in a circular tube of inner diameter 25 mm for approximately constant wall temperature boundary conditions, with nitrogen, oxygen, argon, and helium as the working fluids. The range of Knudsen and Reynolds numbers covered in this study are 0.0022e0.032 and 0.13e14.7, respectively. Whereas the continuum values are correctly reproduced in our setup, the measured values for Nusselt numbers are very small in the slip regime. The measured values are two-five orders of magnitude smaller than the corresponding values in the continuum regime, and suggest that the Nusselt number is a strong function of Reynolds, Knudsen and Brinkmann numbers in the slip flow regime. These are among the first heat transfer measurements in the slip flow regime and the current theoretical and simulation models are inadequate to explain such low values of Nusselt number. Ó 2010 Elsevier Masson SAS. All rights reserved.
Keywords: Slip flow Nusselt number Knudsen number LMTD method
1. Introduction
ug uw ¼
The subject of this study is measurement of the Nusselt number in a circular tube with flow in the slip regime. Although, the Nusselt number in the continuum regime is well documented, the corresponding numbers in the slip regime (103 < Kn < 101) are available only from theoretical analysis and no experimental data verifying the predictions have been reported. A new nondimensional number, the Knudsen number (Kn) becomes relevant in this regime. A relatively large Knudsen number can be achieved by either reducing the dimensions of the tube or by increasing the mean free path of the gas (i.e., by reducing the pressure in the pipe). We have chosen the latter approach because of difficulty in making measurements at small scales required in the former case. However, the results are applicable to both micro and macro domains under similar values of the non-dimensional parameters. Several theoretical attempts have been made. Sparrow and Lin [1], Ameel et al. [2] and Larrode et al. [3] solved the simplified energy equation:
a v vT vT r u ¼ r vr vr vx
(1)
with velocity slip and temperature jump at the walls of the tube, given by the following equations:
* Corresponding author. Tel.: þ91 22 2576 7516; fax: þ91 22 2572 6875. E-mail address: [email protected] (A. Agrawal). 1290-0729/$ e see front matter Ó 2010 Elsevier Masson SAS. All rights reserved. doi:10.1016/j.ijthermalsci.2010.05.018
Tg Tw ¼
2 sv
sv
2 sT
sT
vu Kn vn w
(2)
2g Kn vT 1 þ g Pr vn w
(3)
Here u represents velocity in streamwise direction, T represents temperature and a represents thermal diffusivity. Radial and streamwise coordinates are represented by r and x respectively. In Eqs. (2) and (3), sv refers to tangential momentum accommodation coefficient, sT to thermal accommodation coefficient, Pr is the Prandtl number, g is the ratio of specific heat, and subscripts ‘g’ and ‘w’ refer to gas and wall, respectively. They [1,2] found a substantial reduction in Nusselt number (Nu) with an increase in Knudsen number (Kn). Larrode et al. [3] considered a circular tube with constant wall temperature at the wall, hydodynamically fully developed flow, constant properties fluid, high Peclet number, negligible viscous dissipation and obtained the Nusselt number as a series solution. The authors concluded that with increasing rarefaction, the Nusselt number may increase, decrease or remain unchanged depending on the values of velocity slip and temperature jump coefficients in Eqs. (2) and (3). The reason offered for the higher heat transfer coefficient with rarefaction as compared to continuum regime, was an increased velocity at the wall (because of velocity slip) leading to an increase in the convective heat transfer coefficient. A larger than continuum value of Nusselt number has however not been reported by other theoretical studies. Tunc and Bayazitoglu [4], Aydin and Avci [5], Hooman [6] and Zhang et al. [7] included the viscous dissipation term in the
A. Demsis et al. / International Journal of Thermal Sciences 49 (2010) 1994e1999
simplified energy equation (Eq. (1) given above). In particular, Tunc and Bayazitoglu [4] solved the two-dimensional energy equation with viscous dissipation using the integral transform technique, for both uniform temperature and uniform heat flux boundary conditions. The viscous term was non-dimensionalized in terms of the Brinkman number and the effect of Brinkman number on Nusselt number was studied. Under fully developed condition, the Nusselt number was found to increase with Brinkman number for the constant wall temperature case, while the opposite was reported for the constant heat flux case. Further, a decrease in Prandtl number was found to increase the Nusselt number because of a corresponding decrease in the temperature jump. However, no explanation for the difference in behaviour of Brinkman number with boundary condition was offered. The values of Nusselt number obtained from these studies are summarized in Table 1. Note that most of the analyses assume full thermal and momentum accommodation. In a recent review of the available experimental values of tangential momentum accommodation coefficient, Agrawal and Prabhu [8] found that sv ¼ 0.93 can be assigned to monatomic gases under normal conditions. However, sv for non-monatomic gases shows a dependence on Knudsen number. Zhu and Liao [9] and Satyapathy [10] included the axial conduction term in the simplified energy equation given above with constant wall temperature in the slip and transition regimes. They found a decrease in Nusselt number with an increase in Knudsen number. Jeong and Jeong [11] solved the simplified energy equation with both axial conduction and viscous dissipation terms included:
vT v2 T v2 T rCp u ¼ kf þ vx vx2 vy2
!
2 vu þm vy
(4)
where, r is density, Cp is specific heat, kf is thermal conductivity of fluid, m is viscosity, and x and y denote the streamwise and lateral coordinate, respectively. The equation was solved for both uniform heat flux and constant wall temperature boundary conditions, and the effect of velocity slip and temperature jump was considered in the solution. The Nusselt number was found to decrease with Knudsen and Peclet numbers, and with an increase in the Brinkman number. Vasudevaiah and Balamurugan [12] solved the energy equation for an ideal Newtonian gas for both plane and corrugated channels. Zhu and Liao [9] and Hooman [13] solved the more general problem of heat transfer in microchannels of arbitrary cross-sections. Some three-dimensional simulation results on gas flow and heat transfer in microchannels are also available [14,15]. The above theoretical studies have simplified the original governing equation in several different ways. The dominant terms to be retained in the equation have however not been clearly agreed upon. An order of magnitude analysis of various terms can be obtained by comparison with experimental data, which can serve as a guide for further theoretical analysis. There are however only a few experimental studies. Choi et al. [18] measured the friction factor and heat transfer coefficient for microtubes with
Table 1 Nusselt numbers from theoretical analysis and present experiments for circular tube configuration, at Kn ¼ 0.01 and constant wall temperature case. Most of the analyses assume full thermal and momentum accommodation. Source
Nu
Ameel et al. [2] Larrode et al. [3] Tunc and Bayazitoglu [4] Aydin and Avci [5] Hooman [6] Present (Experimental, nitrogen)
3.75 3.6 3.55 3.55 8.58 0.00166
1995
inside diameter of 3 and 81 mm, with nitrogen as the working fluid. However, due to the large inlet pressures (5.7e10 MPa), the Knudsen number is relatively small in their measurements and the results were not presented in terms of Knudsen number. Interestingly, the Nusselt number was found to depend on Reynolds number (Re), with the exponent of Re being 1.17 and 1.96 in the laminar and turbulent regimes respectively. Yan and Farouk [19] used simulations (DSMC) to compute fluid flow and heat transfer for a mixture of noble gases. These authors proposed a correlation for Nusselt number in terms of Knudsen and Peclet numbers. We therefore recognize that not many experimental studies are available and in view of numerous theoretical attempts, experimental data to validate them is urgently needed. The present study has been undertaken to partly fill this gap in the literature. Demsis et al. [20] have provided the first dataset in the slip flow regime; we have extended their work to include other gases. A correlation for Nusselt number is also proposed. 2. Experimental setup and procedure The experimental setup, data reduction and validation of the experimental setup have already been provided elsewhere [20]. The following is a brief summary of the essential points. Fig. 1 shows the schematic of experimental set-up used for the measurement of heat transfer coefficient in a tube with gas flowing in the slip regime. The experimental facility consists of a vacuum pump connected to a tube-in-tube heat exchanger. The flow of the gas in the inner tube of the heat exchanger is driven by the vacuum pump. A mass flow controller (MKS 1179A) is used to control and meter the mass of the gas flowing in the tube. The range of the mass flow controller is 0e200 sccm (standard cubic centimeter per minute). The pressures at the inlet and outlet of the inner tube are measured by an absolute pressure transducer (MKS 626A, range of 0.01e100 mbar), so that the corresponding Knudsen numbers can be calculated. The upstream tap is kept at a distance greater than the development length for laminar flow in a tube [21]. The setup is tested for leakage by stopping both the pump and the flow of gas, and noting the increase of pressure in the test section with time owing to leakage. From such data, the leakage into the setup is calculated to be less than 0.1% of the flow rate in the tube, i.e. the maximum amount of ambient air that leaks into the test section is less than 0.1% of the minimum main flow. The tube-in-tube heat exchanger is made of stainless steel. The inner tube is circular with a length of 0.96 m and an inner diameter of 24.8 mm (wall thickness of 2.5 mm). The surface roughness of the commercial stainless steel pipe chosen in the present study is
2
3
4
5
To vacuum pump
6 1
7
Fig. 1. Layout of the experimental set-up (1. The gas cylinder 2. Pressure regulator 3. Particle filter 4. Mass flow controller 5. Counter flow tube-in-tube heat exchanger 6. Hot water tank 7. Water pump).
1996
A. Demsis et al. / International Journal of Thermal Sciences 49 (2010) 1994e1999
around 0.05 mm. This heat exchanger is used for the measurement of heat transfer coefficient with the gas at low pressure (<150 Pa) flowing in the inner tube and hot water in the annulus. The hot water heats up the gas. The mass flow rate for water is maintained constant at about 44 g/s. The inlet and outlet temperatures of both the gas and water are measured using thermocouples situated at the inlet and outlet of each tube. The water is heated in a tank to a temperature (maintained constant) of around 60 C to 80 C. The entire heat exchanger unit is insulated with thermowool. The temperature of water and gas at the inlet and outlet of the heat exchanger, pressure at the inlet and outlet of the tube on the gas side are recorded after the system reaches steady state. The overall heat transfer coefficient for the heat exchanger (UA) is obtained from the energy balance, using
_ p ðTco Tci Þ ðUAÞTLMTD ¼ mC
(5)
where UA is also given by
1 1 lnðro =ri Þ 1 þ þ ¼ UA hi Ai 2pkw L ho Ao
(6)
where ri is inner radius of the inner tube, ro is outer radius of the inner tube, kw is the thermal conductivity of the tube material, A is surface area, L is length of the heat exchanger, h is the heat transfer coefficient, and subscripts i and o refer to the inner (gas), outer (water) side of the tube. Tci and Tco refer to the inlet and outlet _ is mass flow rate, TLMTD is logtemperatures of the cold fluid, m mean temperature difference. It turns out that for this case, the second and third terms on the right hand side of the above equation are very small as compared to (hiAi)1; therefore, Eq. (6) reduces to:
UA ¼ hi Ai
(7)
and hi can be readily calculated. The assumption of neglecting the other two terms in Eq. (6) was verified for all measurements [20]. The maximum uncertainty in the various parameters measured and derived is summarized in Table 2. The friction factor under rarefied condition and measurement of the heat transfer coefficient under conventional (continuum) conditions are considered as validation parameters for the setup. The friction factor is calculated using the definition given by Wu and Little [22]. The friction factor (f) times Reynolds number as a function of Knudsen number matches with the earlier results of Sreekanth [21] and other researchers [23,20]. The set up is also validated for heat transfer coefficient in the continuum regime. Table 3 shows that the Nusselt number from the present experiment compare with the DittuseBoelter correlation for both values of Reynolds Number investigated. The overall agreement with available results establishes confidence in the experimental setup and measurement technique.
Table 3 Comparison of experimental value of Nusselt number against the DittuseBoelter correlation for turbulent flow in tube in the continuum regime. Re
Nu from DittuseBoelter Correlation
Nu from the present experiment
Percentage difference
10,870 8420
34.3 28.0
38.3 25.0
12 11
increase in Knudsen number is expected from physical considerations e as the gas becomes increasing rarefied, its ability to transfer heat from a hot surface reduces. This result is therefore inline with the theoretical and simulation results available in the literature and is analogous to the reduction of friction factor with Knudsen number. However, while the friction factor can reduce by 50% [23], the Nusselt number shows a substantially larger reduction with an increase in Knudsen number. Notice the unusually low values of Nusselt number (0.000062e0.028). Fig. 2 suggests a large (about two orders of magnitude) reduction in Nu over a one order of magnitude change in Kn. Some effect of the gas on the result is also noted from the figure. The data for Nusselt number for a lower value of Knudsen number (104 < Kn < 103) for one gas (nitrogen) is presented in Fig. 3 (taken from Demsis et al. [20]). It is clear from the figure that Nusselt number tends to its continuum value (of 3.65) as Kn tends to zero. It is not possible to go to still lower values of Knudsen number (i.e. cover a bigger range of Knudsen number) with the present facility. However, if one extrapolates this curve, the value of Nu ¼ 3.65 is obtained at Kn ¼ 0.56 104. The experiments were repeated several times and pointed out that the numbers obtained were quite stable; it is observed in Fig. 4 that the scatter with nitrogen is within 3%. Note that the wall temperature is approximately constant in these experiments e this can be seen from the fact that almost no heat is picked up by the cold fluid and therefore little heat is lost by the hot fluid. Therefore, the hot fluid in the annulus of the heat exchanger is nearly isothermal. Theoretically, the heat gained by nitrogen is exactly balanced by that lost by water. It is however not possible to check for such energy balance in the present setup because extremely small amount of power is involved in these measurements. Only a few milli-watt of power is required to heat up the gas and therefore the temperature drop in water is very low. In practice, it is 0.1 Nitrogen Oxygen
0.01
Argon Helium
Nu
3. Results 0.001
Fig. 2 shows that the Nusselt number reduces with an increase in Knudsen number. A reduction in Nusselt number with an Table 2 Uncertainty for different measured and derived parameters at the smallest flow rate. Parameter
Maximum uncertainty
Mass flow rate Pressure Temperature Length Reynolds Number Knudsen Number Nusselt Number
2 sccm 0.15% of reading 0.5 C 0.1% 10% 0.5% 12%
0.0001
0.00001 0.001
0.01
0.1
Kn Fig. 2. Variation of Nusselt number as a function of Knudsen numbers, for different gases.
A. Demsis et al. / International Journal of Thermal Sciences 49 (2010) 1994e1999
1.2
The Nusselt number seems to be strongly dependent on the Knudsen, Reynolds and Brinkmann numbers (Br), and the following correlation is proposed:
1.1 1
Nu ¼ 0:001837 Re0:5 Kn0:72 Br 0:47
0.9
0.7
Nu
(8)
where
0.8
Br ¼
0.6
mu2 : kf DT
The Brinkmann number is a measure of the ratio of viscous energy dissipation to heat conduction. Note that the above correlation applies to all four gases covered in this study and suggests that the Nusselt number increases with Reynolds and Brinkman numbers and decreases with Knudsen number. The range of Knudsen, Reynolds and Brinkmann numbers covered are 0.0022e0.032, 0.13e14.7 and 0.000023e0.002, respectively. The above correlation is based on 47 data points and predicts all but 6 points within 30% to þ20% error bands.
0.5 0.4 0.3 0.2 0.1 0 10-4
1997
10-3
Kn
10-2
10-1
Fig. 3. Variation of Nusselt number as a function of Knudsen number with nitrogen as the working fluid.
not possible to have perfect insulation and the temperature drop in water basically reflects the heat lost to the surrounding. We are not aware of any experimental results in the open literature against which these measurements could be compared. The theoretical (Table 1) and simulation results available in the literature [2e6,24,25] are unable to predict such low values of Nusselt number. The smallest value of Nusselt number for nitrogen that has been reported in the literature is Nu ¼ 0.112 at Re ¼ 0.067 and Kn ¼ 1.619 [19]. This result has been obtained in a microchannel through the direct simulation Monte Carlo technique. The plot shows that the values of Nusselt number corresponding to a certain Knudsen number in the slip regime are considerably very small as compared to theoretically suggested values in the literature.
Fig. 4. Variation of Nusselt number as a function of Knudsen number for four different runs, showing repeatability in the measurements with nitrogen as the working fluid.
4. Discussion The accuracy of the measurements hinges on Eq. (5), which implies that the temperatures and mass flow rates are correct. Measurement of gas temperature can become inaccurate under rarefied condition; with mean pressure in the tube of 0.1e1 mbar the gas is however not extremely rarefied in these measurements. Nonetheless care has been taken while measuring the temperatures. For example, thermocouple was inserted in the tube in the following two ways: radially and with the uninsulated part of the thermocouple lying along an isocline. As documented in Demsis et al. [20], the difference in the value of heat transfer coefficient between the two cases is within the experimental uncertainty. Similarly, the measurements were cross-checked with the help of a resistance temperature detector (RTD) and similar conclusion holds [20]. The mass flow is indirectly verified (besides using stateof-the-art and calibrated instrument) by comparing the pressure drop values against theoretical and other experimental data, as discussed in Ref. [23]. Other checks like validation and repeatability were systematically performed; the latter has already been demonstrated through Fig. 4. Relatively small values of Nusselt number can be encountered in the free convection regime. The value of ðGr=Re2 Þ (where Gr is the Grashof number) is calculated to check this and indicates that the flow is indeed in the forced convection regime [20]. The large reduction in the value of Nusselt number as compared to theoretical and simulation values appears rather paradoxical. The vast disagreement between the present experimental results and existing theoretical analysis needs to be resolved; perhaps the earlier theoretical works need to be re-examined in light of these results. One possible reason is presence of a large temperature jump at the wall; this can happen for a low value of sT in Eq. (3). The reported values of sT, for example, with stainless steel-nitrogen pair is between 0.75e0.80 [26], which implies that the temperature jump is less than 0.5 C for these measurements and the corresponding change in the Nusselt number is less than 4%. The Prandtl number for different gases tested is slightly different, which may lead (through Eq. (3)) to a slight change in the Nusselt number values, as noted here. Note that the Brinkman number is also different for the various gases tested and affects the Nusselt number through Eq. (8). The effect of variation in property with temperature on Nusselt number has recently been examined by Hooman et al. [27], Ramon et al. [28], Gulhane and Mahulikar [29]
1998
A. Demsis et al. / International Journal of Thermal Sciences 49 (2010) 1994e1999
and Hooman and Ejlali [30]. Property variation was found to have relatively small effect and cannot adequately explain the present observation. An order of magnitude analysis suggests that both the advection terms vvT=vr (where v is the radial velocity which is non-zero in microchannels [16,17]) and uvT=vx are of the same order, it is therefore important to add the vvT=vr in Eq. (1) (and Eq. (4)) [20]. It can further be argued that because vvT=vr leads to a convective flux which is opposite to the conduction flux near the wall, this term should lead to a reduction in the Nusselt number [31,32]. This convective term has however been retained in the simulations of Hong and Asako [33e35]. No slip boundary condition was however employed in these simulations and only limited computations involving slip and temperature jump at the wall were presented. If one examines the simulations of Kavehpour et al. [36], Hong and Asako [33e35] carefully, there seems to a significant influence of the viscous dissipation and compressibility effect terms. In particular, Hong and Asako have shown that the Nusselt number reduces drastically when the compressibility effect (pressure work term in the energy equation) is included, at a Knudsen number of approximately 0.0014. Note that the compressibility effect terms have not been included in any of the theoretical analysis. Our experimental results also do not show a change in temperature across the tube without heating, suggesting that compressibility effects may not be important in the present experiments. The presence of a temperature jump at the walls of the tube, property variation, along with the vvT=vr term and compressibility effects can lead to a reduction in Nusselt number. However, we suspect that incorporating these effects may still not be adequate to explain the rather large change in the values of Nusselt numbers observed here. Perhaps the governing equations and/or boundary conditions need to be re-examined in light of these experimental observations. 5. Conclusions This study is motivated by gas flow and heat transfer in microchannels and flow of rarefied gases in tubes, where the phenomenon of slip plays an important role. There is a paucity of experimental data in slip regime in the literature. A standard tubein-tube heat exchanger was constructed for the purpose of these measurements. Hot water is the fluid passing in the outer side of the counter-flow heat exchanger while the gas passes in the inner tube at low pressure. Investigations are made for four gases e nitrogen, oxygen, argon and helium. The range of Knudsen number covered under this study is 0.0022e0.032. Measurements of mass flow rates, temperatures, and pressures are taken to obtain the value of the heat transfer coefficient, using the LMTD approach. The values obtained for Nusselt number are between 0.000062e0.028 which is 2e5 orders of magnitude smaller than the corresponding values in the continuum regime. The available theoretical and simulation results are unable to predict such a drastic reduction in the value of Nusselt number. Such low values of Nusselt number appear inexplicable for the moment. The governing equations and/or boundary conditions for the problem should be carefully examined in light of these experimental observations. Acknowledgement We are grateful to Professor M.D. Atrey for making helium and argon gases available to us. The financial support of IIT Bombay and ISRO is acknowledged.
Nomenclature
A Br Cp D f Gr h kw kf Kn L _ m Nu Pr r R Re T v u U x y
cross-sectional area, m2 Brinkmann number specific heat of nitrogen, J/kg K diameter of a circular tube, m friction factor Grashof number ðrg bðTs TN ÞD3 =m2 Þ heat transfer coefficient, W/m2 K thermal conductivity of steel, W/m K thermal conductivity of nitrogen, W/m K Knudsen number, (l/D) total length of the tube, m mass flow rate, kg/s Nusselt number, (hD/kf) Prandtl number ðmCp =kf Þ radius; radial coordinate, m specific gas constant, J/kg K _ pDmÞ Reynolds number ð4m= temperature, K radial velocity, m/s mean streamwise velocity, m/s overall heat transfer coefficient, W/m2 K axial coordinate, m lateral coordinate, m
Greek letters a thermal diffusivity, m2/s r density of nitrogen, kg/m3 m dynamic viscosity of nitrogen, Pa s l mean free path, m aT thermal accommodation coefficient av tangential momentum accommodation coefficient g ratio of specific heat b coefficient of volume expansion, K1 Subscripts ci cold nitrogen inlet co cold nitrogen outlet g surface close to wall hi hot water inlet ho hot water outlet i inlet m mean o outlet w wall l at a distance of mean free path LMTD log mean temperature difference References [1] E.M. Sparrow, S.H. Lin, Laminar heat transfer in tubes under slip flow conditions. Journal of Heat Transfer 84 (1962) 362e369. [2] T.A. Ameel, X. Wang, R.F. Barron, R.O. Warrington, Laminar forced convection in a circular tube with constant heat flux and slip flow. Microscale Thermophysics Engineering 1 (1997) 303e320. [3] F.E. Larrode, C. Housiadas, Y. Drossinos, Slip-flow heat transfer in circular tubes. International Journal of Heat and Mass Transfer 43 (2000) 2669e2680. [4] G. Tunc, Y. Bayazitoglu, Heat transfer in microtubes with viscous dissipation. International Journal of Heat and Mass Transfer 44 (2001) 2395e2403. [5] O. Aydin, M. Avci, Heat and fluid flow characteristics of gases in micropipes. International Journal of Heat and Mass Transfer 49 (2006) 1723e1730. [6] K. Hooman, Entropy generation for microscale forced convection: effects of different thermal boundary conditions, velocity slip, temperature jump, viscous dissipation, and duct geometry. International Communications in Heat and Mass Transfer 34 (2007) 945e957.
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Heat transfer coefficient of gas flowing in a circular tube under rarefied condition Anwar Demsis, Bhaskar Verma, S.V. Prabhu, Amit Agrawal* Department of Mechanical Engineering, Indian Institute of Technology e Bombay, Powai, Mumbai 400076, India
a r t i c l e i n f o
a b s t r a c t
Article history: Received 17 November 2009 Received in revised form 8 March 2010 Accepted 25 May 2010 Available online 22 June 2010
The purpose of this paper is to present heat transfer measurements of gas in a tube under rarefied condition. The measurements are made in a circular tube of inner diameter 25 mm for approximately constant wall temperature boundary conditions, with nitrogen, oxygen, argon, and helium as the working fluids. The range of Knudsen and Reynolds numbers covered in this study are 0.0022e0.032 and 0.13e14.7, respectively. Whereas the continuum values are correctly reproduced in our setup, the measured values for Nusselt numbers are very small in the slip regime. The measured values are two-five orders of magnitude smaller than the corresponding values in the continuum regime, and suggest that the Nusselt number is a strong function of Reynolds, Knudsen and Brinkmann numbers in the slip flow regime. These are among the first heat transfer measurements in the slip flow regime and the current theoretical and simulation models are inadequate to explain such low values of Nusselt number. Ó 2010 Elsevier Masson SAS. All rights reserved.
Keywords: Slip flow Nusselt number Knudsen number LMTD method
1. Introduction
ug uw ¼
The subject of this study is measurement of the Nusselt number in a circular tube with flow in the slip regime. Although, the Nusselt number in the continuum regime is well documented, the corresponding numbers in the slip regime (103 < Kn < 101) are available only from theoretical analysis and no experimental data verifying the predictions have been reported. A new nondimensional number, the Knudsen number (Kn) becomes relevant in this regime. A relatively large Knudsen number can be achieved by either reducing the dimensions of the tube or by increasing the mean free path of the gas (i.e., by reducing the pressure in the pipe). We have chosen the latter approach because of difficulty in making measurements at small scales required in the former case. However, the results are applicable to both micro and macro domains under similar values of the non-dimensional parameters. Several theoretical attempts have been made. Sparrow and Lin [1], Ameel et al. [2] and Larrode et al. [3] solved the simplified energy equation:
a v vT vT r u ¼ r vr vr vx
(1)
with velocity slip and temperature jump at the walls of the tube, given by the following equations:
* Corresponding author. Tel.: þ91 22 2576 7516; fax: þ91 22 2572 6875. E-mail address: [email protected] (A. Agrawal). 1290-0729/$ e see front matter Ó 2010 Elsevier Masson SAS. All rights reserved. doi:10.1016/j.ijthermalsci.2010.05.018
Tg Tw ¼
2 sv
sv
2 sT
sT
vu Kn vn w
(2)
2g Kn vT 1 þ g Pr vn w
(3)
Here u represents velocity in streamwise direction, T represents temperature and a represents thermal diffusivity. Radial and streamwise coordinates are represented by r and x respectively. In Eqs. (2) and (3), sv refers to tangential momentum accommodation coefficient, sT to thermal accommodation coefficient, Pr is the Prandtl number, g is the ratio of specific heat, and subscripts ‘g’ and ‘w’ refer to gas and wall, respectively. They [1,2] found a substantial reduction in Nusselt number (Nu) with an increase in Knudsen number (Kn). Larrode et al. [3] considered a circular tube with constant wall temperature at the wall, hydodynamically fully developed flow, constant properties fluid, high Peclet number, negligible viscous dissipation and obtained the Nusselt number as a series solution. The authors concluded that with increasing rarefaction, the Nusselt number may increase, decrease or remain unchanged depending on the values of velocity slip and temperature jump coefficients in Eqs. (2) and (3). The reason offered for the higher heat transfer coefficient with rarefaction as compared to continuum regime, was an increased velocity at the wall (because of velocity slip) leading to an increase in the convective heat transfer coefficient. A larger than continuum value of Nusselt number has however not been reported by other theoretical studies. Tunc and Bayazitoglu [4], Aydin and Avci [5], Hooman [6] and Zhang et al. [7] included the viscous dissipation term in the
A. Demsis et al. / International Journal of Thermal Sciences 49 (2010) 1994e1999
simplified energy equation (Eq. (1) given above). In particular, Tunc and Bayazitoglu [4] solved the two-dimensional energy equation with viscous dissipation using the integral transform technique, for both uniform temperature and uniform heat flux boundary conditions. The viscous term was non-dimensionalized in terms of the Brinkman number and the effect of Brinkman number on Nusselt number was studied. Under fully developed condition, the Nusselt number was found to increase with Brinkman number for the constant wall temperature case, while the opposite was reported for the constant heat flux case. Further, a decrease in Prandtl number was found to increase the Nusselt number because of a corresponding decrease in the temperature jump. However, no explanation for the difference in behaviour of Brinkman number with boundary condition was offered. The values of Nusselt number obtained from these studies are summarized in Table 1. Note that most of the analyses assume full thermal and momentum accommodation. In a recent review of the available experimental values of tangential momentum accommodation coefficient, Agrawal and Prabhu [8] found that sv ¼ 0.93 can be assigned to monatomic gases under normal conditions. However, sv for non-monatomic gases shows a dependence on Knudsen number. Zhu and Liao [9] and Satyapathy [10] included the axial conduction term in the simplified energy equation given above with constant wall temperature in the slip and transition regimes. They found a decrease in Nusselt number with an increase in Knudsen number. Jeong and Jeong [11] solved the simplified energy equation with both axial conduction and viscous dissipation terms included:
vT v2 T v2 T rCp u ¼ kf þ vx vx2 vy2
!
2 vu þm vy
(4)
where, r is density, Cp is specific heat, kf is thermal conductivity of fluid, m is viscosity, and x and y denote the streamwise and lateral coordinate, respectively. The equation was solved for both uniform heat flux and constant wall temperature boundary conditions, and the effect of velocity slip and temperature jump was considered in the solution. The Nusselt number was found to decrease with Knudsen and Peclet numbers, and with an increase in the Brinkman number. Vasudevaiah and Balamurugan [12] solved the energy equation for an ideal Newtonian gas for both plane and corrugated channels. Zhu and Liao [9] and Hooman [13] solved the more general problem of heat transfer in microchannels of arbitrary cross-sections. Some three-dimensional simulation results on gas flow and heat transfer in microchannels are also available [14,15]. The above theoretical studies have simplified the original governing equation in several different ways. The dominant terms to be retained in the equation have however not been clearly agreed upon. An order of magnitude analysis of various terms can be obtained by comparison with experimental data, which can serve as a guide for further theoretical analysis. There are however only a few experimental studies. Choi et al. [18] measured the friction factor and heat transfer coefficient for microtubes with
Table 1 Nusselt numbers from theoretical analysis and present experiments for circular tube configuration, at Kn ¼ 0.01 and constant wall temperature case. Most of the analyses assume full thermal and momentum accommodation. Source
Nu
Ameel et al. [2] Larrode et al. [3] Tunc and Bayazitoglu [4] Aydin and Avci [5] Hooman [6] Present (Experimental, nitrogen)
3.75 3.6 3.55 3.55 8.58 0.00166
1995
inside diameter of 3 and 81 mm, with nitrogen as the working fluid. However, due to the large inlet pressures (5.7e10 MPa), the Knudsen number is relatively small in their measurements and the results were not presented in terms of Knudsen number. Interestingly, the Nusselt number was found to depend on Reynolds number (Re), with the exponent of Re being 1.17 and 1.96 in the laminar and turbulent regimes respectively. Yan and Farouk [19] used simulations (DSMC) to compute fluid flow and heat transfer for a mixture of noble gases. These authors proposed a correlation for Nusselt number in terms of Knudsen and Peclet numbers. We therefore recognize that not many experimental studies are available and in view of numerous theoretical attempts, experimental data to validate them is urgently needed. The present study has been undertaken to partly fill this gap in the literature. Demsis et al. [20] have provided the first dataset in the slip flow regime; we have extended their work to include other gases. A correlation for Nusselt number is also proposed. 2. Experimental setup and procedure The experimental setup, data reduction and validation of the experimental setup have already been provided elsewhere [20]. The following is a brief summary of the essential points. Fig. 1 shows the schematic of experimental set-up used for the measurement of heat transfer coefficient in a tube with gas flowing in the slip regime. The experimental facility consists of a vacuum pump connected to a tube-in-tube heat exchanger. The flow of the gas in the inner tube of the heat exchanger is driven by the vacuum pump. A mass flow controller (MKS 1179A) is used to control and meter the mass of the gas flowing in the tube. The range of the mass flow controller is 0e200 sccm (standard cubic centimeter per minute). The pressures at the inlet and outlet of the inner tube are measured by an absolute pressure transducer (MKS 626A, range of 0.01e100 mbar), so that the corresponding Knudsen numbers can be calculated. The upstream tap is kept at a distance greater than the development length for laminar flow in a tube [21]. The setup is tested for leakage by stopping both the pump and the flow of gas, and noting the increase of pressure in the test section with time owing to leakage. From such data, the leakage into the setup is calculated to be less than 0.1% of the flow rate in the tube, i.e. the maximum amount of ambient air that leaks into the test section is less than 0.1% of the minimum main flow. The tube-in-tube heat exchanger is made of stainless steel. The inner tube is circular with a length of 0.96 m and an inner diameter of 24.8 mm (wall thickness of 2.5 mm). The surface roughness of the commercial stainless steel pipe chosen in the present study is
2
3
4
5
To vacuum pump
6 1
7
Fig. 1. Layout of the experimental set-up (1. The gas cylinder 2. Pressure regulator 3. Particle filter 4. Mass flow controller 5. Counter flow tube-in-tube heat exchanger 6. Hot water tank 7. Water pump).
1996
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around 0.05 mm. This heat exchanger is used for the measurement of heat transfer coefficient with the gas at low pressure (<150 Pa) flowing in the inner tube and hot water in the annulus. The hot water heats up the gas. The mass flow rate for water is maintained constant at about 44 g/s. The inlet and outlet temperatures of both the gas and water are measured using thermocouples situated at the inlet and outlet of each tube. The water is heated in a tank to a temperature (maintained constant) of around 60 C to 80 C. The entire heat exchanger unit is insulated with thermowool. The temperature of water and gas at the inlet and outlet of the heat exchanger, pressure at the inlet and outlet of the tube on the gas side are recorded after the system reaches steady state. The overall heat transfer coefficient for the heat exchanger (UA) is obtained from the energy balance, using
_ p ðTco Tci Þ ðUAÞTLMTD ¼ mC
(5)
where UA is also given by
1 1 lnðro =ri Þ 1 þ þ ¼ UA hi Ai 2pkw L ho Ao
(6)
where ri is inner radius of the inner tube, ro is outer radius of the inner tube, kw is the thermal conductivity of the tube material, A is surface area, L is length of the heat exchanger, h is the heat transfer coefficient, and subscripts i and o refer to the inner (gas), outer (water) side of the tube. Tci and Tco refer to the inlet and outlet _ is mass flow rate, TLMTD is logtemperatures of the cold fluid, m mean temperature difference. It turns out that for this case, the second and third terms on the right hand side of the above equation are very small as compared to (hiAi)1; therefore, Eq. (6) reduces to:
UA ¼ hi Ai
(7)
and hi can be readily calculated. The assumption of neglecting the other two terms in Eq. (6) was verified for all measurements [20]. The maximum uncertainty in the various parameters measured and derived is summarized in Table 2. The friction factor under rarefied condition and measurement of the heat transfer coefficient under conventional (continuum) conditions are considered as validation parameters for the setup. The friction factor is calculated using the definition given by Wu and Little [22]. The friction factor (f) times Reynolds number as a function of Knudsen number matches with the earlier results of Sreekanth [21] and other researchers [23,20]. The set up is also validated for heat transfer coefficient in the continuum regime. Table 3 shows that the Nusselt number from the present experiment compare with the DittuseBoelter correlation for both values of Reynolds Number investigated. The overall agreement with available results establishes confidence in the experimental setup and measurement technique.
Table 3 Comparison of experimental value of Nusselt number against the DittuseBoelter correlation for turbulent flow in tube in the continuum regime. Re
Nu from DittuseBoelter Correlation
Nu from the present experiment
Percentage difference
10,870 8420
34.3 28.0
38.3 25.0
12 11
increase in Knudsen number is expected from physical considerations e as the gas becomes increasing rarefied, its ability to transfer heat from a hot surface reduces. This result is therefore inline with the theoretical and simulation results available in the literature and is analogous to the reduction of friction factor with Knudsen number. However, while the friction factor can reduce by 50% [23], the Nusselt number shows a substantially larger reduction with an increase in Knudsen number. Notice the unusually low values of Nusselt number (0.000062e0.028). Fig. 2 suggests a large (about two orders of magnitude) reduction in Nu over a one order of magnitude change in Kn. Some effect of the gas on the result is also noted from the figure. The data for Nusselt number for a lower value of Knudsen number (104 < Kn < 103) for one gas (nitrogen) is presented in Fig. 3 (taken from Demsis et al. [20]). It is clear from the figure that Nusselt number tends to its continuum value (of 3.65) as Kn tends to zero. It is not possible to go to still lower values of Knudsen number (i.e. cover a bigger range of Knudsen number) with the present facility. However, if one extrapolates this curve, the value of Nu ¼ 3.65 is obtained at Kn ¼ 0.56 104. The experiments were repeated several times and pointed out that the numbers obtained were quite stable; it is observed in Fig. 4 that the scatter with nitrogen is within 3%. Note that the wall temperature is approximately constant in these experiments e this can be seen from the fact that almost no heat is picked up by the cold fluid and therefore little heat is lost by the hot fluid. Therefore, the hot fluid in the annulus of the heat exchanger is nearly isothermal. Theoretically, the heat gained by nitrogen is exactly balanced by that lost by water. It is however not possible to check for such energy balance in the present setup because extremely small amount of power is involved in these measurements. Only a few milli-watt of power is required to heat up the gas and therefore the temperature drop in water is very low. In practice, it is 0.1 Nitrogen Oxygen
0.01
Argon Helium
Nu
3. Results 0.001
Fig. 2 shows that the Nusselt number reduces with an increase in Knudsen number. A reduction in Nusselt number with an Table 2 Uncertainty for different measured and derived parameters at the smallest flow rate. Parameter
Maximum uncertainty
Mass flow rate Pressure Temperature Length Reynolds Number Knudsen Number Nusselt Number
2 sccm 0.15% of reading 0.5 C 0.1% 10% 0.5% 12%
0.0001
0.00001 0.001
0.01
0.1
Kn Fig. 2. Variation of Nusselt number as a function of Knudsen numbers, for different gases.
A. Demsis et al. / International Journal of Thermal Sciences 49 (2010) 1994e1999
1.2
The Nusselt number seems to be strongly dependent on the Knudsen, Reynolds and Brinkmann numbers (Br), and the following correlation is proposed:
1.1 1
Nu ¼ 0:001837 Re0:5 Kn0:72 Br 0:47
0.9
0.7
Nu
(8)
where
0.8
Br ¼
0.6
mu2 : kf DT
The Brinkmann number is a measure of the ratio of viscous energy dissipation to heat conduction. Note that the above correlation applies to all four gases covered in this study and suggests that the Nusselt number increases with Reynolds and Brinkman numbers and decreases with Knudsen number. The range of Knudsen, Reynolds and Brinkmann numbers covered are 0.0022e0.032, 0.13e14.7 and 0.000023e0.002, respectively. The above correlation is based on 47 data points and predicts all but 6 points within 30% to þ20% error bands.
0.5 0.4 0.3 0.2 0.1 0 10-4
1997
10-3
Kn
10-2
10-1
Fig. 3. Variation of Nusselt number as a function of Knudsen number with nitrogen as the working fluid.
not possible to have perfect insulation and the temperature drop in water basically reflects the heat lost to the surrounding. We are not aware of any experimental results in the open literature against which these measurements could be compared. The theoretical (Table 1) and simulation results available in the literature [2e6,24,25] are unable to predict such low values of Nusselt number. The smallest value of Nusselt number for nitrogen that has been reported in the literature is Nu ¼ 0.112 at Re ¼ 0.067 and Kn ¼ 1.619 [19]. This result has been obtained in a microchannel through the direct simulation Monte Carlo technique. The plot shows that the values of Nusselt number corresponding to a certain Knudsen number in the slip regime are considerably very small as compared to theoretically suggested values in the literature.
Fig. 4. Variation of Nusselt number as a function of Knudsen number for four different runs, showing repeatability in the measurements with nitrogen as the working fluid.
4. Discussion The accuracy of the measurements hinges on Eq. (5), which implies that the temperatures and mass flow rates are correct. Measurement of gas temperature can become inaccurate under rarefied condition; with mean pressure in the tube of 0.1e1 mbar the gas is however not extremely rarefied in these measurements. Nonetheless care has been taken while measuring the temperatures. For example, thermocouple was inserted in the tube in the following two ways: radially and with the uninsulated part of the thermocouple lying along an isocline. As documented in Demsis et al. [20], the difference in the value of heat transfer coefficient between the two cases is within the experimental uncertainty. Similarly, the measurements were cross-checked with the help of a resistance temperature detector (RTD) and similar conclusion holds [20]. The mass flow is indirectly verified (besides using stateof-the-art and calibrated instrument) by comparing the pressure drop values against theoretical and other experimental data, as discussed in Ref. [23]. Other checks like validation and repeatability were systematically performed; the latter has already been demonstrated through Fig. 4. Relatively small values of Nusselt number can be encountered in the free convection regime. The value of ðGr=Re2 Þ (where Gr is the Grashof number) is calculated to check this and indicates that the flow is indeed in the forced convection regime [20]. The large reduction in the value of Nusselt number as compared to theoretical and simulation values appears rather paradoxical. The vast disagreement between the present experimental results and existing theoretical analysis needs to be resolved; perhaps the earlier theoretical works need to be re-examined in light of these results. One possible reason is presence of a large temperature jump at the wall; this can happen for a low value of sT in Eq. (3). The reported values of sT, for example, with stainless steel-nitrogen pair is between 0.75e0.80 [26], which implies that the temperature jump is less than 0.5 C for these measurements and the corresponding change in the Nusselt number is less than 4%. The Prandtl number for different gases tested is slightly different, which may lead (through Eq. (3)) to a slight change in the Nusselt number values, as noted here. Note that the Brinkman number is also different for the various gases tested and affects the Nusselt number through Eq. (8). The effect of variation in property with temperature on Nusselt number has recently been examined by Hooman et al. [27], Ramon et al. [28], Gulhane and Mahulikar [29]
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A. Demsis et al. / International Journal of Thermal Sciences 49 (2010) 1994e1999
and Hooman and Ejlali [30]. Property variation was found to have relatively small effect and cannot adequately explain the present observation. An order of magnitude analysis suggests that both the advection terms vvT=vr (where v is the radial velocity which is non-zero in microchannels [16,17]) and uvT=vx are of the same order, it is therefore important to add the vvT=vr in Eq. (1) (and Eq. (4)) [20]. It can further be argued that because vvT=vr leads to a convective flux which is opposite to the conduction flux near the wall, this term should lead to a reduction in the Nusselt number [31,32]. This convective term has however been retained in the simulations of Hong and Asako [33e35]. No slip boundary condition was however employed in these simulations and only limited computations involving slip and temperature jump at the wall were presented. If one examines the simulations of Kavehpour et al. [36], Hong and Asako [33e35] carefully, there seems to a significant influence of the viscous dissipation and compressibility effect terms. In particular, Hong and Asako have shown that the Nusselt number reduces drastically when the compressibility effect (pressure work term in the energy equation) is included, at a Knudsen number of approximately 0.0014. Note that the compressibility effect terms have not been included in any of the theoretical analysis. Our experimental results also do not show a change in temperature across the tube without heating, suggesting that compressibility effects may not be important in the present experiments. The presence of a temperature jump at the walls of the tube, property variation, along with the vvT=vr term and compressibility effects can lead to a reduction in Nusselt number. However, we suspect that incorporating these effects may still not be adequate to explain the rather large change in the values of Nusselt numbers observed here. Perhaps the governing equations and/or boundary conditions need to be re-examined in light of these experimental observations. 5. Conclusions This study is motivated by gas flow and heat transfer in microchannels and flow of rarefied gases in tubes, where the phenomenon of slip plays an important role. There is a paucity of experimental data in slip regime in the literature. A standard tubein-tube heat exchanger was constructed for the purpose of these measurements. Hot water is the fluid passing in the outer side of the counter-flow heat exchanger while the gas passes in the inner tube at low pressure. Investigations are made for four gases e nitrogen, oxygen, argon and helium. The range of Knudsen number covered under this study is 0.0022e0.032. Measurements of mass flow rates, temperatures, and pressures are taken to obtain the value of the heat transfer coefficient, using the LMTD approach. The values obtained for Nusselt number are between 0.000062e0.028 which is 2e5 orders of magnitude smaller than the corresponding values in the continuum regime. The available theoretical and simulation results are unable to predict such a drastic reduction in the value of Nusselt number. Such low values of Nusselt number appear inexplicable for the moment. The governing equations and/or boundary conditions for the problem should be carefully examined in light of these experimental observations. Acknowledgement We are grateful to Professor M.D. Atrey for making helium and argon gases available to us. The financial support of IIT Bombay and ISRO is acknowledged.
Nomenclature
A Br Cp D f Gr h kw kf Kn L _ m Nu Pr r R Re T v u U x y
cross-sectional area, m2 Brinkmann number specific heat of nitrogen, J/kg K diameter of a circular tube, m friction factor Grashof number ðrg bðTs TN ÞD3 =m2 Þ heat transfer coefficient, W/m2 K thermal conductivity of steel, W/m K thermal conductivity of nitrogen, W/m K Knudsen number, (l/D) total length of the tube, m mass flow rate, kg/s Nusselt number, (hD/kf) Prandtl number ðmCp =kf Þ radius; radial coordinate, m specific gas constant, J/kg K _ pDmÞ Reynolds number ð4m= temperature, K radial velocity, m/s mean streamwise velocity, m/s overall heat transfer coefficient, W/m2 K axial coordinate, m lateral coordinate, m
Greek letters a thermal diffusivity, m2/s r density of nitrogen, kg/m3 m dynamic viscosity of nitrogen, Pa s l mean free path, m aT thermal accommodation coefficient av tangential momentum accommodation coefficient g ratio of specific heat b coefficient of volume expansion, K1 Subscripts ci cold nitrogen inlet co cold nitrogen outlet g surface close to wall hi hot water inlet ho hot water outlet i inlet m mean o outlet w wall l at a distance of mean free path LMTD log mean temperature difference References [1] E.M. Sparrow, S.H. Lin, Laminar heat transfer in tubes under slip flow conditions. Journal of Heat Transfer 84 (1962) 362e369. [2] T.A. Ameel, X. Wang, R.F. Barron, R.O. Warrington, Laminar forced convection in a circular tube with constant heat flux and slip flow. Microscale Thermophysics Engineering 1 (1997) 303e320. [3] F.E. Larrode, C. Housiadas, Y. Drossinos, Slip-flow heat transfer in circular tubes. International Journal of Heat and Mass Transfer 43 (2000) 2669e2680. [4] G. Tunc, Y. Bayazitoglu, Heat transfer in microtubes with viscous dissipation. International Journal of Heat and Mass Transfer 44 (2001) 2395e2403. [5] O. Aydin, M. Avci, Heat and fluid flow characteristics of gases in micropipes. International Journal of Heat and Mass Transfer 49 (2006) 1723e1730. [6] K. Hooman, Entropy generation for microscale forced convection: effects of different thermal boundary conditions, velocity slip, temperature jump, viscous dissipation, and duct geometry. International Communications in Heat and Mass Transfer 34 (2007) 945e957.
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