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Heat transfer in a small diameter tube at high Reynolds numbers
International Journal of Heat and Mass Transfer 109 (2017) 997–1003
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Heat transfer in a small diameter tube at high Reynolds numbers Oleg V. Vitovsky, Maksim S. Makarov ⇑, Vladimir E. Nakoryakov, Viktor S. Naumkin Kutateladze Institute of Thermophysics, Siberian Branch, Russian Academy of Sciences, Ac. Lavrentiev ave. 1, Novosibirsk 630090, Russia
a r t i c l e
i n f o
Article history: Received 26 September 2016 Received in revised form 3 January 2017 Accepted 16 February 2017
Keywords: Heat transfer Low Prandtl number Compressible flow Internal turbulent flow Accelerated flow Recovery factor Stagnation parameters Mass-average recovery temperature
a b s t r a c t Investigation results on heat transfer in low-Prandtl helium-xenon gas mixture and air, flowing in a small diameter tube, are presented. New experimental data on the heat transfer coefficient in the flow of helium-xenon mixture are obtained; results of numerical simulation are compared with the experimental data and known empirical correlations. Based on the simulation data it is shown that in a heated tube an increase in the Reynolds number due to an increase in the flow rate intensifies the heat transfer, and it may be due to the flow acceleration. It is shown that the high flow velocity and significant acceleration have a considerable effect on heat transfer in a tube, and the use of mean-mass stagnation temperatures as the determining one for generalization of data on heat transfer is insufficient. For the studied conditions, the known correlations give a significant error in determination of the heat transfer coefficient: the lower the Prandtl number and gas density, the higher the error. Ó 2017 Elsevier Ltd. All rights reserved.
1. Introduction Determination of heat transfer intensity on the tube wall is a classical problem of the heat transfer theory. The engineering methods for determining the heat transfer coefficient in the fully-established turbulent flow, based on correlations of Dittus– Boelter, Mikheev, Petukhov–Popov, and others are widely known. The basis for the construction of such correlations is the data of numerous experiments and numerical simulations [1–5]. Applying these correlations for the analysis of thermal processes in the tubular heat exchangers of traditional configurations for the flow of air, steam or water is well-founded. The transition to the compact heat exchangers makes it necessary to reduce the flow cross-section of the tubes and, as a result, to increase the speed of coolant pumping. High pumping speed at simultaneous use vapor or gas mixture as a coolant leads to a significant increase in the effects of compressibility, Prandtl number and flow acceleration on heat transfer intensity, which was not taken into account at derivation of the above correlations. The relevance of the effect of these factors on heat transfer is also confirmed by the fact that the mixed gas coolants are increasingly used at various power plants [6]. The projects of compact nuclear reactors, providing energy for the research equipment on the surface of Mars, are known [7]. The equipment for ⇑ Corresponding author. E-mail addresses: [email protected] (O.V. Vitovsky), [email protected] (M.S. Makarov), [email protected] (V.E. Nakoryakov), [email protected] (V.S. Naumkin). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2017.02.041 0017-9310/Ó 2017 Elsevier Ltd. All rights reserved.
flameless heating and cooling of gas pipelines are being developed [8]. By some characteristics (for example, efficiency of compressor turbine, heat exchanger mass) the helium-xenon mixture with helium mass concentration of 5–10% is more effective coolant than pure gases or mixtures of different composition [9]. The properties of helium-xenon mixture have been studied in detail [10,11]. The main feature of helium-xenon mixture of this composition is the low Prandtl number (0.23). In literature there are few experimental data and data of numerical simulation on heat transfer in such mixtures [12–15]. The working media in most other papers, dealt with heat transfer in the substances with low-Prandtl number, are liquid metals [16,17]. Gas heat-transfer agents, in contrast to liquid metals, are the compressible media. Gas heating causes it expansion, which in turn leads to a significant acceleration of the flow in tubes with a small cross-section and, as a consequence, to a change in heat transfer conditions. The modern numerical models and methods of heat transfer investigation allow sufficiently accurate calculation of the velocity and temperature fields in a compressible flow and determination of the local and length-average heat transfer coefficient on the tube wall. The numerical experiment allows us to extend the range of tested parameters and complete the full-scale experiment. For the mixed gas heat-transfer agents with low molecular Prandtl number Pr, it is crucial to choose the model for the turbulent Prandtl number Prt. According to the data of direct numerical simulation (DNS) at Pr ¼ 0:1, Prt can be an order of magnitude higher and have substantial non-uniformity in the cross-section of the
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Nomenclature cp d G K L M Nu p Pr Prt Q q r rf Red S T ux ; ur
specific heat at constant pressure [J/kgK] tube diameter [m] mass flow rate [kg/s] mass concentration tube length [m] Mach number Nusselt number pressure [Pa] Prandtl number turbulent Prandtl number heat load [W] heat flux [W/m2] tube radius [m] recovery factor Reynolds number cross-sectional area [m2] temperature [K] axial and radial velocity [m/s]
wall boundary layer [18,19]. Heat transfer in the flow in the wide range of molecular Prandtl numbers was studied in [20–24]. The dependence of the turbulent Prandtl number on the Reynolds number and molecular Prandtl number was derived in [20]. It is shown that for low Reynolds and Prandtl numbers, the value of Prt can be higher than one. By increasing Red and Pr, the value of Prt decreases and tends to the constant value of 0.7. This paper presents the results of numerical studies of heat transfer in the flow of helium-xenon mixture and air in a small diameter tube in the wide range of Reynolds numbers, covering the areas of gas compressibility influence on the flow dynamics and heat transfer. The used mathematical model and method for solving the heat transfer problems take into account the changes in the turbulent Prandtl number over the tube cross-section and they are verified by the data of the experiment under the conditions similar to the considered ones. 2. Problem statement and main features of the flow Heat transfer was studied in a circular tube with diameter d = 5.5 mm and length L from 0.75 to 1 m. The flow scheme is shown in Fig. 1a. The helium-xenon mixture with helium mass concentration KHe = 5% (Pr = 0.23) or air (Pr = 0.71) was used as a coolant. The properties of helium-xenon mixture were calculated by dependences suggested in [10]. The properties of air were determined by the data of [25]. In all calculations, excluding the test cases, specific heat flux qw of 2895 and 5790 W/m2 was fed to the wall along the entire tube, and for the tube of 1-m length it corresponded to heat power Q = 50 and 100 W. The stagnation temperature of gas at the channel inlet was T in ¼ 293 K. At numerical simulation, mass flow rate G was determined by the gas parameters at the tube inlet, at that, total pressure pin was varied from 0.5 to 2 atm, and Mach number Min was varied from 0.003 to 0.31. Total pressure pout , stagnation temperature of the flow T out , and Mach number Mout at the tube outlet were determined through calculations. For the flow velocity at the tube outlet close to the sonic one, the maximal tube length corresponded to zero pressure at the outlet. During the full-scale experiment, the mass flow of gas through the tube of a constant length was regulated; the static pressure and temperature were measured at the tube inlet and outlet by the pressure gauges and thermocouple converters. To verify the computational model, the tube geometry and gas param-
x
a k
l q Indexes ax in out r t w ⁄
axial coordinate [m] heat transfer coefficient [W/m2K] thermal conductivity [W/mK] dynamic viscosity [Pas] density [kg/m3]
at the axis at the tube inlet at the tube outlet recovery turbulent at the wall mass-average value stagnation value
eters at the inlet and outlet were chosen as close as possible to the experimental conditions. For the flows at a high Reynolds number, the drop of static pressure along the tube and change in the gas temperature lead to a substantial change in the flow velocity even in the area of fullyestablished flow, as can be seen from the characteristic velocity profiles in Fig. 1b. The gas flow velocity at the outlet is substantially higher than the flow velocity at the inlet and can reach the local sound velocity, and the flow in the tube is accelerated. The solid lines represent the changes in the boundary layer thickness d, thickness of displacement d and momentum thickness d along the tube. The line marked with letter d corresponds to the distance from the tube axis, where the longitudinal flow velocity is 0.995 of its value on the axis. For the initial section, this parameter corresponds to the thickness of the wall boundary layer. It is evident that in the initial section, the boundary layer increases according to the law for the laminar flow, followed by the transition to the turbulent flow and boundary layer closing. At low Reynolds numbers, an incompressible flow downstream is established with the power velocity distribution. As it can be seen, at high Reynolds numbers, a compressible flow can be called established only conditionally. The near-axial section with almost uniform velocity distribution increases downstream together with substantial increase in the longitudinal velocity. We can speak about flow acceleration due to the changes in displacement thickness, momentum thickness and distributions of transverse velocities along the tube radius in different cross-sections (see Fig. 1c). In the accelerated flow, d and d values decrease and transverse velocity changes the sign [26]. Increasing the momentum thickness at the beginning of the tube and its reduction at the end leads to a change in aerodynamic curvature of the tube wall and an expansion of the gas flow at the tube outlet at high Reynolds numbers. 3. Mathematical model and method of solution To describe all features of the gas mixture flow with low Prandtl number in a small diameter tube at high Reynolds numbers and their effect on heat transfer, the following mathematical model was used. The problem was considered in 2D axisymmetrical statement. Fluid dynamics and heat transfer are described by the following equation set of the boundary layer type [27]: Continuity equation
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Fig. 1. The scheme of the flow (a), characteristic distributions of longitudinal (b), and radial (c) flow velocity and integral parameters of the boundary layer.
@ðr qux Þ @ðr qur Þ þ ¼ 0; @x @r
grid points in the radial direction is 200. The integration step in the longitudinal direction does not exceed 103 m.
Momentum equation
@ðqu2x Þ @ðqrux ur Þ @ @ux dp þ ¼ r leff ; @x r@r r@r dx @r and energy equation
cp
2 @ðqux TÞ @ðqrur TÞ @ @T @ux @p þ ux þ ¼ rkeff þ leff ; @x r@r r@r @r @x @r
where ux ; ur are the axial and radial velocity components; T is temperature; p is static pressure; q is density; leff ¼ l þ lt is effective coefficient of dynamic viscosity; keff ¼ k þ lt cp =Prt is effective heat conductivity; cp is specific heat capacity at constant pressure; x is axial coordinate; r is radial coordinate. Turbulent viscosity lt is calculated by the q x model of turbulence [28]. The scale and turbulence level at the tube inlet are set 1 m and 2%, respectively. To determine turbulence Prandtl number Prt, the extended KaysCrawford model is used [29]:
" !# 1 1 c 1 2 ; ¼ þ c 1 exp Prt 2Prt0 Pr0:5 cPr0:5 t0 t0 where c ¼ 0:3Prlt =l, Prt0 ¼ 0:85 is turbulent Prandtl number in the undisturbed flow, 0.3 is empirical constant. The problem is solved by numerical integration of equations in the physical coordinates by means of the implicit scheme on the non-uniform rectangular grid with compression near the high velocity gradients. The nonlinear character of differential equations is eliminated by the method of simple iterations at each step of integration with the accuracy of 105. The number of calculation
4. Experimental setup To verify the mathematical model and numerical method, a series of experiments on helium-xenon circuit, described in detail in [13], was carried out. The working sections were made of nickel and nichrome thin-wall smooth tubes with the outer diameters of 5 mm and 6 mm; the wall thickness was 0.15 and 0.3 mm. The length of the heated part of the tubes was changed from 450 to 800 mm. The tube was heated by AC. The working sections were electrically isolated from the other parts of set-up. The junctions of copper-constantan thermocouples were welded on the tube surface to measure the local values of wall temperatures. The number of thermocouples along the tube was from 6 to 9. The chromelalumel thermocouples with isolated junction were located in the flow at the inlet and outlet of the working section and they registered the inlet and outlet gas temperature, respectively. Distributions of the flow core temperature along the tube length were measured by a thermocouple probe along the axial line of the tube with a step of 5 mm. The diameter of the thermocouple probe bead was 100 lm. The chromel-alumel and copper-constantan thermocouples had individual calibration characteristics. To regulate the mass flow rate of gas through the working section in the desired range of Reynolds numbers, the Bronkhorst EL-FLOW controller was used. 5. Verification of mathematical model The local temperatures on the tube wall and on its axis in the flow of helium-xenon mixture (KHe = 7%, Pr = 0.24) were measured
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with different flow rates and different thermal loads. It should be noted that due to the small area of the tube cross-section, the thermocouple probes, used in the experiment, were the opened thermocouple beads without screens. Such probe design brings minimal distortions to the flow, and this is important for the high gas velocities, but the temperature measurement error becomes higher. In particular, the open thermocouples at high flow velocity allow the measurement of the recovery temperature, which, as it is known, is higher than the thermodynamic flow temperature, but it is below than the stagnation temperature and depends on the gas Prandtl number. For the turbulent flows at low Prandtl numbers, this relationship is not strictly defined, which introduces additional uncertainty to the results of experiment. The experimental data obtained for Reynolds numbers 14,800, 40,200 and 83,300 are shown in Fig. 2 by points, while the specific heat fluxes on the tube wall are 3600, 9600 and 20,400 W/m2, respectively. The Reynolds numbers are determined by gas viscosity at the tube inlet, and the heat fluxes are determined on the inner surface of the tube wall. Simulation results obtained with the parameters close to the experimental conditions are shown in Fig. 2 by the lines. The solid lines correspond to the stagnation temperature; the dashed lines correspond to thermodynamic temperature of the flow. Due to set-up construction, the tube had an initial unheated section with the length of 0.175 m, and this was taken into account at simulation. In the initial unheated section,
the wall temperature remains almost constant and equal to the gas temperature at the tube inlet. Leakages of heat from the heated tube section to the unheated one are negligible due to the small wall thickness and low coefficient of heat conductivity of the used materials. Immediately after the initial unheated section the wall temperature increases sharply, followed by the section of thermal stabilization of the flow with a nonlinear increase in temperature and section of established heat transfer. The section of stabilization increases with increasing Reynolds number and in these experiments it reached 30% of the length of the heated section of the tube. In the area of established heat transfer, the temperature at the tube axis increases linearly in proportion to the wall temperature. As it can be seen from Fig. 2, simulation data are in a good agreement with experimental results at low Reynolds numbers. Under these conditions, the influence of gas compressibility is small. With an increase in the Reynolds number, the difference between the stagnation temperature and thermodynamic temperature increases as well as the uncertainty in experimental temperatures. Experimental temperature at the tube axis determined for Red = 83,300 is higher than the thermodynamic one and lower than the stagnation temperature. Although at high Reynolds numbers, verification of the mathematical model is difficult, in general it can be said that the obtained simulation results are consistent with the experimental data and known ideas about the measured temperature of the high-velocity flow. 6. Heat transfer in a small diameter tube In the generalized form, heat transfer intensity in tubes is determined by Nusselt number Nu ¼ ad=k, where d is tube diameter; k is coefficient of heat conductivity of the medium, determined by characteristic temperature T 0 ; a ¼ qw =ðT w T 0 Þ is heat transfer coefficient. Determination of the characteristic flow temperature is a key moment for calculation of the Nusselt number. In the initial section of the tube, temperature at the tube axis T ax is a convenient characteristic temperature because this value does not change along the length, and the heat transfer coefficient depends linearly on its value. The mass-average gas temperature in a local R tube cross-section T ¼ quTdS=G is taken as the characteristic temperature for the developed low-velocity flow because under the constant heat flux, this temperature changes linearly along the tube in proportion to the wall temperature; at that, the heat transfer coefficient remains constant. According to the known transformations of the high-velocity (supersonic) boundary layer at Pr ¼ 1 stagnation temperature of the flow T ¼ T ax þ u2ax =2cp should be taken as the characteristic temperature, and at Pr – 1, this should be recovery temperature T r ¼ r f T þ ð1 rf ÞT ax , where rf ¼ f ðPr;Prt Þ is the recovery factor. As it is shown above, the considered flow includes all the mentioned situations: extended initial section (up to 30% of the tube length), flow velocity (Mach number) along the tube length can be changed by the factor of 3; low Prandtl number. It can be assumed that under the considered conditions, the mass-average stagnation temperature: R T ¼ quT dS=G, where S is the area of tube cross-section and G is mass flow rate through the tube, is the convenient characteristic temperature. At low velocities, T in the limit tends to T. In the developed high-velocity flow with a constant heat flux, T changes linearly. It is easy to determined T in experiment at the known
Fig. 2. Distribution of the tube wall temperature (a) and temperature of heliumxenon mixture on the tube axis (b) along its length for Reynolds numbers: 1 – 14,800, 2 – 40,200, and 3 – 83,300 at specific heat fluxes: 1 – 3600, 2 – 9600, and 3 – 20,400 W/m2.
heat losses: T ¼ T in þ gQ =Gcp , where Tin is gas temperature at the tube inlet, g is coefficient of isolation efficiency; as usual it is from 0.95 to 0.99. Distributions of local Nusselt number along the tube for the helium-xenon mixture are presented in Fig. 3 at different Reynolds numbers calculated by mass-average temperature T (Fig. 3a) and
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the situations, when the wall temperature becomes lower than the mass-average stagnation temperature due to reduction in the static temperature in the flow core (see Fig. 4). In these cases, the function of local Nusselt number, determined by T , is discontinuous at some distance from the tube inlet, and further downstream it becomes negative (upper line 4 in Fig. 3b is presented as an absolute value of the Nusselt number). Obviously, generalization of calculated and experimental data through the mass-average stagnation temperature becomes impossible. Dependences Nu ¼ f ðxÞ with a singularity in distribution at different Reynolds numbers are shown in Fig. 5. Here, the Reynolds number changes due to a change in the gas pressure at the tube inlet pin from 0.5 to 2 atm. As it can be seen from Figs. 4 and 5, the value of Mach number is critical. A singularity in determination of the heat transfer coefficient occurs at the Mach number, derived by massaverage flow velocity M from 0.35 to 0.55. It can be noted that at a constant heat load, the pressure leads to a decrease in M and increase in Red, corresponding to the specified singularity. Partly, the approach adopted for the external supersonic flows allows us to solve the problem of data generalization on heat transfer for the high-velocity flows. The analogue of recovery temperature: mass-average recovery temperature T r ¼ r f T þ ð1 rf ÞT, where rf ¼ 0:9Pr0:1 , can be taken as a characteristic temperature (lines 4 in Fig. 4.). According to data of [30,31], the above formula for rf is valid for the turbulent flow in the range of molecular Prandtl numbers from 0.1 to 1. The resultant distribution of the Nusselt number along the tube for Red = 76,200 and pin = 2 atm is
Fig. 3. Distribution of Nusselt number along the tube, calculated by mass-average static temperature (a) and mass-average stagnation temperature (b) of the heliumxenon mixture for inlet Reynolds numbers (the inlet Mach number is shown in brackets): 1 – 762 (0.007), 2 – 7800 (0.03), 3 – 38,900 (0.15), and 4 – 76,200 (0.29); dashed line – calculation by Dittus-Boelter formula.
shown in Figs. 3 and 4 by the line marked T 0 ¼ T r . As it can be seen, the data on heat transfer in this form have no singularity and correlate much better with the Dittus-Boelter formula. An increase in heat transfer at the tube outlet is associated with flow acceleration, mentioned above. Dependence of rf on acceleration rate for gases with a low Prandtl number is not known. However, the proposed approach allows us to extend the application range for the Dittus-Boelter formula under the considered conditions up to the Reynolds numbers of 60,000. Generalized data on heat transfer in a small diameter tube are shown in Fig. 6 depending on the Reynolds number for all considered versions of calculations and experiments. All parameters are
mass-average stagnation temperature T (Fig. 3b). Heat load is 50 W. Data are presented in comparison with the Dittus-Boelter formula for the developed turbulent flow in a tube: 0:4 Nu ¼ 0:021Re0:8 . This correlation was used as a simplest case. d Pr In the studied conditions the correlations of Gnielinski, Mikheev, Petukhov-Popov are equal to the Dittus-Boelter formula. At Reynolds numbers less than 10,000 (lines 1 and 2 in Fig. 3), the Dittus-Boelter formula allows us to determine accurately the heat transfer intensity. We do not observe the differences in determina-
tion of Nusselt number by T and T , evidently, due to the low flow velocity (M < 0.1). With an increase in the Reynolds number (by increasing the flow rate), the effect of compressibility becomes significant, and even at 38,900 (M = 0.15, lines 3 in Fig. 3), the Nusselt number determined by T is 1.5 times lower than that determined by T and Dittus-Boelter formula. When Red = 76,200 (M = 0.29, lines 4 in Fig. 3), heat transfer decreases almost by an order. Generalization of data through the mass-average stagnation temperature allows the extension of Dittus-Boelter formula applicability by the Reynolds number up to 50,000. A further increase in the Reynolds number leads to strong acceleration and cooling of the flow core along the tube. In the studied range of heat loads, there can be
Fig. 4. Dependence of static flow temperature (1), wall temperature (2), massaverage stagnation temperature (3) and mass-average recovery temperature (4) on the Mach number at heat load of 50 W and different pressures at the tube inlet.
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limited applicability range by the Reynolds number, whose upper limit is determined by the zone of significant flow acceleration in the tube (a vertical line in Fig. 6). It should be noted that in the studied range of parameters, an increase in heat transfer due to acceleration of air in the flow is observed only at the flow velocity close to the sound velocity and it depends weak on the pressure. In the flow of helium-xenon mixture, the effect of acceleration appears at lower Reynolds numbers, and typical velocities, at which acceleration leads to an increase in heat transfer, are substantially less than the sound velocity. A low Prandtl number for the helium-xenon mixture leads to better flow core heating as compared to air that, in turn, leads to greater expansion and acceleration of gas in the tube. Application of gases with low Prandtl number as the coolants at low pressures and high flow rates is the basis for adjustment of heat transfer coefficients obtained by the Dittus-Boelter formula and its equivalents in the engineering practice.
Fig. 5. Distributions of Nusselt numbers, calculated by mass-average stagnation temperature (solid lines) and Mach number (dashed lines) along the tube at different pressures and Reynolds numbers (in brackets) at the tube inlet: 1–2 atm (76,000), 2–1 atm (57,000), and 3–0.5 atm (38,000).
Fig. 6. Generalization of data on heat transfer in the small diameter tube depending on Reynolds number.
7. Conclusion The effect of gas compressibility on heat transfer in a small diameter tube at Prandtl numbers of 0.71 (air) and 0.23 (heliumxenon mixture) was studied. It is shown that the known methods of generalization of experimental data and numerical simulations do not provide a power heat transfer law at high Reynolds numbers for gases with different Prandtl numbers. The method for determining the mass-average recovery temperature as a characteristic flow temperature, allowing extension for the applicability range of the Dittus-Boelter formula by the Reynolds number, was suggested. However, the upper limit of applicability of Dittus-Boelter formula at low Prandtl number turned out to be dependent on the gas density. A decrease in density leads to an increase in the pressure gradient along the tube, the flow becomes significantly accelerated, and this apparently leads to an increase in heat transfer as compared with the established incompressible flow. Heat transfer enhancement in the studied air flow is observed only at the flow velocities close to the sound velocity, and in the flow of helium-xenon mixture, it is observed at velocities substantially lower than the sound velocity. The use of gas coolants with a low Prandtl number in engineering practice at low pressures and high flow rates requires adjustment of the heat transfer coefficients obtained by the known correlations. Acknowledgements
determined at the tube outlet. Light points indicate data for air, dark points indicate data for the helium-xenon mixture. Different shapes of points correspond to different pressures at the tube inlet. The Nusselt and Reynolds numbers are determined by formulas:
Nu ¼
qw d 4G ; Red ¼ ; T w T r k pdl
where k and l are the coefficients of heat conductivity and dynamic viscosity at the tube outlet. In processing the experimental data T r
was taken equal to T , which was calculated according to heat balance. The inlet pressure in the experiments was more than 3.8 atm, and therefore the significant effect of gas compressibility on heat transfer is not obtained. Despite consideration of the effect of gas compressibility on the heat transfer coefficient in the numerical modeling data, we failed to establish a general correlation of the Nusselt number on the Reynolds and Prandtl numbers. With a decrease in the Reynolds number due to the gas density (a change in pressure at the tube inlet) the pressure gradient along the tube increases, the flow becomes significantly accelerated, and this leads to an increase in heat transfer. The Dittus-Boelter formula has the
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[20] V.P. Bobkov, Anisotropic turbulent heat transfer in channels of nuclear power reactors, Sov. Atom. Energy 59 (5) (1986) 886–892. [21] Y.H. Dong, X.Y. Lu, L.X. Zhuang, An investigation of the Prandtl number effect on turbulent heat transfer in channel flows by large eddy simulation, Acta Mech. 159 (1–4) (2002) 39–51. [22] B.M. Mitrovic, P.M. Le, D.V. Papavassiliou, On the Prandtl or Schmidt number dependence of the turbulent heat or mass transfer coefficient, Chem. Eng. Sci. 59 (3) (2004) 543–555. [23] Lei. Wang, Lu. Xi-Yun, Large eddy simulation of stably stratified turbulent open channel flows with low- to high-Prandtl number, Int. J. Heat Mass Transfer 48 (2005) 1883–1897. [24] M. Piller, E. Nobile, T.J. Hanratty, DNS study of turbulent transport at low Prandtl numbers in a channel flow, J. Fluid Mech. 458 (2002) 419–441. [25] S. Gordon, B.J. McBride Computer Program for Calculation of Complex Chemical Equilibrium Compositions and Applications. I. Analysis, Washington, NASA RP1311, vol. 1, 1994, 58p. [26] E.P. Volchkov, M.S. Makarov, A. Yu, Sakhnov Boundary layer with asymptotic favourable pressure gradient, Int. J. Heat Mass Transfer 53 (13–14) (2010) 2837–2843. [27] R.H. Pletcher, J.C. Tannehill, D.A. Anderson, Computational Fluid Mechanics and Heat Transfer, third ed., CRC Press, Hoboken, 2012, p. 763. [28] T.J. Coakley, P.G. Huang, Turbulence modeling for high speed flows, in: 30th Aerospace Sciences Meeting and Exhibit, Aerospace Sciences Meetings, January 6-9, 1992, Reno, NV, USA, AIAA Paper, No. 92-0436, 15p. [29] B. Weigand, J.R. Ferguson, M.E. Crawford, An extended Kays and Crawford turbulent Prandtl number model, Int. J. Heat Mass Transfer 40 (17) (1997) 4191–4196. [30] V.E. Nakoryakov, M.S. Makarov, Yu.I. Petukhov, O.V. Vitovsky, S.L. Elistratov Teplovye processy v potokah gasovyh smesey s malym chislom Prandtlya, Akademizdat, Novosibirsk, 2015, 283p. (in Russian). [31] V.G. Lushchik, M.S. Makarova, Numerical investigation of the effect of the Prandtl number on the temperature recovery and the Reynolds analogy factors in the boundary layer on a plate, High Temp. 54 (3) (2016) 377–382.
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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Heat transfer in a small diameter tube at high Reynolds numbers Oleg V. Vitovsky, Maksim S. Makarov ⇑, Vladimir E. Nakoryakov, Viktor S. Naumkin Kutateladze Institute of Thermophysics, Siberian Branch, Russian Academy of Sciences, Ac. Lavrentiev ave. 1, Novosibirsk 630090, Russia
a r t i c l e
i n f o
Article history: Received 26 September 2016 Received in revised form 3 January 2017 Accepted 16 February 2017
Keywords: Heat transfer Low Prandtl number Compressible flow Internal turbulent flow Accelerated flow Recovery factor Stagnation parameters Mass-average recovery temperature
a b s t r a c t Investigation results on heat transfer in low-Prandtl helium-xenon gas mixture and air, flowing in a small diameter tube, are presented. New experimental data on the heat transfer coefficient in the flow of helium-xenon mixture are obtained; results of numerical simulation are compared with the experimental data and known empirical correlations. Based on the simulation data it is shown that in a heated tube an increase in the Reynolds number due to an increase in the flow rate intensifies the heat transfer, and it may be due to the flow acceleration. It is shown that the high flow velocity and significant acceleration have a considerable effect on heat transfer in a tube, and the use of mean-mass stagnation temperatures as the determining one for generalization of data on heat transfer is insufficient. For the studied conditions, the known correlations give a significant error in determination of the heat transfer coefficient: the lower the Prandtl number and gas density, the higher the error. Ó 2017 Elsevier Ltd. All rights reserved.
1. Introduction Determination of heat transfer intensity on the tube wall is a classical problem of the heat transfer theory. The engineering methods for determining the heat transfer coefficient in the fully-established turbulent flow, based on correlations of Dittus– Boelter, Mikheev, Petukhov–Popov, and others are widely known. The basis for the construction of such correlations is the data of numerous experiments and numerical simulations [1–5]. Applying these correlations for the analysis of thermal processes in the tubular heat exchangers of traditional configurations for the flow of air, steam or water is well-founded. The transition to the compact heat exchangers makes it necessary to reduce the flow cross-section of the tubes and, as a result, to increase the speed of coolant pumping. High pumping speed at simultaneous use vapor or gas mixture as a coolant leads to a significant increase in the effects of compressibility, Prandtl number and flow acceleration on heat transfer intensity, which was not taken into account at derivation of the above correlations. The relevance of the effect of these factors on heat transfer is also confirmed by the fact that the mixed gas coolants are increasingly used at various power plants [6]. The projects of compact nuclear reactors, providing energy for the research equipment on the surface of Mars, are known [7]. The equipment for ⇑ Corresponding author. E-mail addresses: [email protected] (O.V. Vitovsky), [email protected] (M.S. Makarov), [email protected] (V.E. Nakoryakov), [email protected] (V.S. Naumkin). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2017.02.041 0017-9310/Ó 2017 Elsevier Ltd. All rights reserved.
flameless heating and cooling of gas pipelines are being developed [8]. By some characteristics (for example, efficiency of compressor turbine, heat exchanger mass) the helium-xenon mixture with helium mass concentration of 5–10% is more effective coolant than pure gases or mixtures of different composition [9]. The properties of helium-xenon mixture have been studied in detail [10,11]. The main feature of helium-xenon mixture of this composition is the low Prandtl number (0.23). In literature there are few experimental data and data of numerical simulation on heat transfer in such mixtures [12–15]. The working media in most other papers, dealt with heat transfer in the substances with low-Prandtl number, are liquid metals [16,17]. Gas heat-transfer agents, in contrast to liquid metals, are the compressible media. Gas heating causes it expansion, which in turn leads to a significant acceleration of the flow in tubes with a small cross-section and, as a consequence, to a change in heat transfer conditions. The modern numerical models and methods of heat transfer investigation allow sufficiently accurate calculation of the velocity and temperature fields in a compressible flow and determination of the local and length-average heat transfer coefficient on the tube wall. The numerical experiment allows us to extend the range of tested parameters and complete the full-scale experiment. For the mixed gas heat-transfer agents with low molecular Prandtl number Pr, it is crucial to choose the model for the turbulent Prandtl number Prt. According to the data of direct numerical simulation (DNS) at Pr ¼ 0:1, Prt can be an order of magnitude higher and have substantial non-uniformity in the cross-section of the
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Nomenclature cp d G K L M Nu p Pr Prt Q q r rf Red S T ux ; ur
specific heat at constant pressure [J/kgK] tube diameter [m] mass flow rate [kg/s] mass concentration tube length [m] Mach number Nusselt number pressure [Pa] Prandtl number turbulent Prandtl number heat load [W] heat flux [W/m2] tube radius [m] recovery factor Reynolds number cross-sectional area [m2] temperature [K] axial and radial velocity [m/s]
wall boundary layer [18,19]. Heat transfer in the flow in the wide range of molecular Prandtl numbers was studied in [20–24]. The dependence of the turbulent Prandtl number on the Reynolds number and molecular Prandtl number was derived in [20]. It is shown that for low Reynolds and Prandtl numbers, the value of Prt can be higher than one. By increasing Red and Pr, the value of Prt decreases and tends to the constant value of 0.7. This paper presents the results of numerical studies of heat transfer in the flow of helium-xenon mixture and air in a small diameter tube in the wide range of Reynolds numbers, covering the areas of gas compressibility influence on the flow dynamics and heat transfer. The used mathematical model and method for solving the heat transfer problems take into account the changes in the turbulent Prandtl number over the tube cross-section and they are verified by the data of the experiment under the conditions similar to the considered ones. 2. Problem statement and main features of the flow Heat transfer was studied in a circular tube with diameter d = 5.5 mm and length L from 0.75 to 1 m. The flow scheme is shown in Fig. 1a. The helium-xenon mixture with helium mass concentration KHe = 5% (Pr = 0.23) or air (Pr = 0.71) was used as a coolant. The properties of helium-xenon mixture were calculated by dependences suggested in [10]. The properties of air were determined by the data of [25]. In all calculations, excluding the test cases, specific heat flux qw of 2895 and 5790 W/m2 was fed to the wall along the entire tube, and for the tube of 1-m length it corresponded to heat power Q = 50 and 100 W. The stagnation temperature of gas at the channel inlet was T in ¼ 293 K. At numerical simulation, mass flow rate G was determined by the gas parameters at the tube inlet, at that, total pressure pin was varied from 0.5 to 2 atm, and Mach number Min was varied from 0.003 to 0.31. Total pressure pout , stagnation temperature of the flow T out , and Mach number Mout at the tube outlet were determined through calculations. For the flow velocity at the tube outlet close to the sonic one, the maximal tube length corresponded to zero pressure at the outlet. During the full-scale experiment, the mass flow of gas through the tube of a constant length was regulated; the static pressure and temperature were measured at the tube inlet and outlet by the pressure gauges and thermocouple converters. To verify the computational model, the tube geometry and gas param-
x
a k
l q Indexes ax in out r t w ⁄
axial coordinate [m] heat transfer coefficient [W/m2K] thermal conductivity [W/mK] dynamic viscosity [Pas] density [kg/m3]
at the axis at the tube inlet at the tube outlet recovery turbulent at the wall mass-average value stagnation value
eters at the inlet and outlet were chosen as close as possible to the experimental conditions. For the flows at a high Reynolds number, the drop of static pressure along the tube and change in the gas temperature lead to a substantial change in the flow velocity even in the area of fullyestablished flow, as can be seen from the characteristic velocity profiles in Fig. 1b. The gas flow velocity at the outlet is substantially higher than the flow velocity at the inlet and can reach the local sound velocity, and the flow in the tube is accelerated. The solid lines represent the changes in the boundary layer thickness d, thickness of displacement d and momentum thickness d along the tube. The line marked with letter d corresponds to the distance from the tube axis, where the longitudinal flow velocity is 0.995 of its value on the axis. For the initial section, this parameter corresponds to the thickness of the wall boundary layer. It is evident that in the initial section, the boundary layer increases according to the law for the laminar flow, followed by the transition to the turbulent flow and boundary layer closing. At low Reynolds numbers, an incompressible flow downstream is established with the power velocity distribution. As it can be seen, at high Reynolds numbers, a compressible flow can be called established only conditionally. The near-axial section with almost uniform velocity distribution increases downstream together with substantial increase in the longitudinal velocity. We can speak about flow acceleration due to the changes in displacement thickness, momentum thickness and distributions of transverse velocities along the tube radius in different cross-sections (see Fig. 1c). In the accelerated flow, d and d values decrease and transverse velocity changes the sign [26]. Increasing the momentum thickness at the beginning of the tube and its reduction at the end leads to a change in aerodynamic curvature of the tube wall and an expansion of the gas flow at the tube outlet at high Reynolds numbers. 3. Mathematical model and method of solution To describe all features of the gas mixture flow with low Prandtl number in a small diameter tube at high Reynolds numbers and their effect on heat transfer, the following mathematical model was used. The problem was considered in 2D axisymmetrical statement. Fluid dynamics and heat transfer are described by the following equation set of the boundary layer type [27]: Continuity equation
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Fig. 1. The scheme of the flow (a), characteristic distributions of longitudinal (b), and radial (c) flow velocity and integral parameters of the boundary layer.
@ðr qux Þ @ðr qur Þ þ ¼ 0; @x @r
grid points in the radial direction is 200. The integration step in the longitudinal direction does not exceed 103 m.
Momentum equation
@ðqu2x Þ @ðqrux ur Þ @ @ux dp þ ¼ r leff ; @x r@r r@r dx @r and energy equation
cp
2 @ðqux TÞ @ðqrur TÞ @ @T @ux @p þ ux þ ¼ rkeff þ leff ; @x r@r r@r @r @x @r
where ux ; ur are the axial and radial velocity components; T is temperature; p is static pressure; q is density; leff ¼ l þ lt is effective coefficient of dynamic viscosity; keff ¼ k þ lt cp =Prt is effective heat conductivity; cp is specific heat capacity at constant pressure; x is axial coordinate; r is radial coordinate. Turbulent viscosity lt is calculated by the q x model of turbulence [28]. The scale and turbulence level at the tube inlet are set 1 m and 2%, respectively. To determine turbulence Prandtl number Prt, the extended KaysCrawford model is used [29]:
" !# 1 1 c 1 2 ; ¼ þ c 1 exp Prt 2Prt0 Pr0:5 cPr0:5 t0 t0 where c ¼ 0:3Prlt =l, Prt0 ¼ 0:85 is turbulent Prandtl number in the undisturbed flow, 0.3 is empirical constant. The problem is solved by numerical integration of equations in the physical coordinates by means of the implicit scheme on the non-uniform rectangular grid with compression near the high velocity gradients. The nonlinear character of differential equations is eliminated by the method of simple iterations at each step of integration with the accuracy of 105. The number of calculation
4. Experimental setup To verify the mathematical model and numerical method, a series of experiments on helium-xenon circuit, described in detail in [13], was carried out. The working sections were made of nickel and nichrome thin-wall smooth tubes with the outer diameters of 5 mm and 6 mm; the wall thickness was 0.15 and 0.3 mm. The length of the heated part of the tubes was changed from 450 to 800 mm. The tube was heated by AC. The working sections were electrically isolated from the other parts of set-up. The junctions of copper-constantan thermocouples were welded on the tube surface to measure the local values of wall temperatures. The number of thermocouples along the tube was from 6 to 9. The chromelalumel thermocouples with isolated junction were located in the flow at the inlet and outlet of the working section and they registered the inlet and outlet gas temperature, respectively. Distributions of the flow core temperature along the tube length were measured by a thermocouple probe along the axial line of the tube with a step of 5 mm. The diameter of the thermocouple probe bead was 100 lm. The chromel-alumel and copper-constantan thermocouples had individual calibration characteristics. To regulate the mass flow rate of gas through the working section in the desired range of Reynolds numbers, the Bronkhorst EL-FLOW controller was used. 5. Verification of mathematical model The local temperatures on the tube wall and on its axis in the flow of helium-xenon mixture (KHe = 7%, Pr = 0.24) were measured
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with different flow rates and different thermal loads. It should be noted that due to the small area of the tube cross-section, the thermocouple probes, used in the experiment, were the opened thermocouple beads without screens. Such probe design brings minimal distortions to the flow, and this is important for the high gas velocities, but the temperature measurement error becomes higher. In particular, the open thermocouples at high flow velocity allow the measurement of the recovery temperature, which, as it is known, is higher than the thermodynamic flow temperature, but it is below than the stagnation temperature and depends on the gas Prandtl number. For the turbulent flows at low Prandtl numbers, this relationship is not strictly defined, which introduces additional uncertainty to the results of experiment. The experimental data obtained for Reynolds numbers 14,800, 40,200 and 83,300 are shown in Fig. 2 by points, while the specific heat fluxes on the tube wall are 3600, 9600 and 20,400 W/m2, respectively. The Reynolds numbers are determined by gas viscosity at the tube inlet, and the heat fluxes are determined on the inner surface of the tube wall. Simulation results obtained with the parameters close to the experimental conditions are shown in Fig. 2 by the lines. The solid lines correspond to the stagnation temperature; the dashed lines correspond to thermodynamic temperature of the flow. Due to set-up construction, the tube had an initial unheated section with the length of 0.175 m, and this was taken into account at simulation. In the initial unheated section,
the wall temperature remains almost constant and equal to the gas temperature at the tube inlet. Leakages of heat from the heated tube section to the unheated one are negligible due to the small wall thickness and low coefficient of heat conductivity of the used materials. Immediately after the initial unheated section the wall temperature increases sharply, followed by the section of thermal stabilization of the flow with a nonlinear increase in temperature and section of established heat transfer. The section of stabilization increases with increasing Reynolds number and in these experiments it reached 30% of the length of the heated section of the tube. In the area of established heat transfer, the temperature at the tube axis increases linearly in proportion to the wall temperature. As it can be seen from Fig. 2, simulation data are in a good agreement with experimental results at low Reynolds numbers. Under these conditions, the influence of gas compressibility is small. With an increase in the Reynolds number, the difference between the stagnation temperature and thermodynamic temperature increases as well as the uncertainty in experimental temperatures. Experimental temperature at the tube axis determined for Red = 83,300 is higher than the thermodynamic one and lower than the stagnation temperature. Although at high Reynolds numbers, verification of the mathematical model is difficult, in general it can be said that the obtained simulation results are consistent with the experimental data and known ideas about the measured temperature of the high-velocity flow. 6. Heat transfer in a small diameter tube In the generalized form, heat transfer intensity in tubes is determined by Nusselt number Nu ¼ ad=k, where d is tube diameter; k is coefficient of heat conductivity of the medium, determined by characteristic temperature T 0 ; a ¼ qw =ðT w T 0 Þ is heat transfer coefficient. Determination of the characteristic flow temperature is a key moment for calculation of the Nusselt number. In the initial section of the tube, temperature at the tube axis T ax is a convenient characteristic temperature because this value does not change along the length, and the heat transfer coefficient depends linearly on its value. The mass-average gas temperature in a local R tube cross-section T ¼ quTdS=G is taken as the characteristic temperature for the developed low-velocity flow because under the constant heat flux, this temperature changes linearly along the tube in proportion to the wall temperature; at that, the heat transfer coefficient remains constant. According to the known transformations of the high-velocity (supersonic) boundary layer at Pr ¼ 1 stagnation temperature of the flow T ¼ T ax þ u2ax =2cp should be taken as the characteristic temperature, and at Pr – 1, this should be recovery temperature T r ¼ r f T þ ð1 rf ÞT ax , where rf ¼ f ðPr;Prt Þ is the recovery factor. As it is shown above, the considered flow includes all the mentioned situations: extended initial section (up to 30% of the tube length), flow velocity (Mach number) along the tube length can be changed by the factor of 3; low Prandtl number. It can be assumed that under the considered conditions, the mass-average stagnation temperature: R T ¼ quT dS=G, where S is the area of tube cross-section and G is mass flow rate through the tube, is the convenient characteristic temperature. At low velocities, T in the limit tends to T. In the developed high-velocity flow with a constant heat flux, T changes linearly. It is easy to determined T in experiment at the known
Fig. 2. Distribution of the tube wall temperature (a) and temperature of heliumxenon mixture on the tube axis (b) along its length for Reynolds numbers: 1 – 14,800, 2 – 40,200, and 3 – 83,300 at specific heat fluxes: 1 – 3600, 2 – 9600, and 3 – 20,400 W/m2.
heat losses: T ¼ T in þ gQ =Gcp , where Tin is gas temperature at the tube inlet, g is coefficient of isolation efficiency; as usual it is from 0.95 to 0.99. Distributions of local Nusselt number along the tube for the helium-xenon mixture are presented in Fig. 3 at different Reynolds numbers calculated by mass-average temperature T (Fig. 3a) and
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the situations, when the wall temperature becomes lower than the mass-average stagnation temperature due to reduction in the static temperature in the flow core (see Fig. 4). In these cases, the function of local Nusselt number, determined by T , is discontinuous at some distance from the tube inlet, and further downstream it becomes negative (upper line 4 in Fig. 3b is presented as an absolute value of the Nusselt number). Obviously, generalization of calculated and experimental data through the mass-average stagnation temperature becomes impossible. Dependences Nu ¼ f ðxÞ with a singularity in distribution at different Reynolds numbers are shown in Fig. 5. Here, the Reynolds number changes due to a change in the gas pressure at the tube inlet pin from 0.5 to 2 atm. As it can be seen from Figs. 4 and 5, the value of Mach number is critical. A singularity in determination of the heat transfer coefficient occurs at the Mach number, derived by massaverage flow velocity M from 0.35 to 0.55. It can be noted that at a constant heat load, the pressure leads to a decrease in M and increase in Red, corresponding to the specified singularity. Partly, the approach adopted for the external supersonic flows allows us to solve the problem of data generalization on heat transfer for the high-velocity flows. The analogue of recovery temperature: mass-average recovery temperature T r ¼ r f T þ ð1 rf ÞT, where rf ¼ 0:9Pr0:1 , can be taken as a characteristic temperature (lines 4 in Fig. 4.). According to data of [30,31], the above formula for rf is valid for the turbulent flow in the range of molecular Prandtl numbers from 0.1 to 1. The resultant distribution of the Nusselt number along the tube for Red = 76,200 and pin = 2 atm is
Fig. 3. Distribution of Nusselt number along the tube, calculated by mass-average static temperature (a) and mass-average stagnation temperature (b) of the heliumxenon mixture for inlet Reynolds numbers (the inlet Mach number is shown in brackets): 1 – 762 (0.007), 2 – 7800 (0.03), 3 – 38,900 (0.15), and 4 – 76,200 (0.29); dashed line – calculation by Dittus-Boelter formula.
shown in Figs. 3 and 4 by the line marked T 0 ¼ T r . As it can be seen, the data on heat transfer in this form have no singularity and correlate much better with the Dittus-Boelter formula. An increase in heat transfer at the tube outlet is associated with flow acceleration, mentioned above. Dependence of rf on acceleration rate for gases with a low Prandtl number is not known. However, the proposed approach allows us to extend the application range for the Dittus-Boelter formula under the considered conditions up to the Reynolds numbers of 60,000. Generalized data on heat transfer in a small diameter tube are shown in Fig. 6 depending on the Reynolds number for all considered versions of calculations and experiments. All parameters are
mass-average stagnation temperature T (Fig. 3b). Heat load is 50 W. Data are presented in comparison with the Dittus-Boelter formula for the developed turbulent flow in a tube: 0:4 Nu ¼ 0:021Re0:8 . This correlation was used as a simplest case. d Pr In the studied conditions the correlations of Gnielinski, Mikheev, Petukhov-Popov are equal to the Dittus-Boelter formula. At Reynolds numbers less than 10,000 (lines 1 and 2 in Fig. 3), the Dittus-Boelter formula allows us to determine accurately the heat transfer intensity. We do not observe the differences in determina-
tion of Nusselt number by T and T , evidently, due to the low flow velocity (M < 0.1). With an increase in the Reynolds number (by increasing the flow rate), the effect of compressibility becomes significant, and even at 38,900 (M = 0.15, lines 3 in Fig. 3), the Nusselt number determined by T is 1.5 times lower than that determined by T and Dittus-Boelter formula. When Red = 76,200 (M = 0.29, lines 4 in Fig. 3), heat transfer decreases almost by an order. Generalization of data through the mass-average stagnation temperature allows the extension of Dittus-Boelter formula applicability by the Reynolds number up to 50,000. A further increase in the Reynolds number leads to strong acceleration and cooling of the flow core along the tube. In the studied range of heat loads, there can be
Fig. 4. Dependence of static flow temperature (1), wall temperature (2), massaverage stagnation temperature (3) and mass-average recovery temperature (4) on the Mach number at heat load of 50 W and different pressures at the tube inlet.
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limited applicability range by the Reynolds number, whose upper limit is determined by the zone of significant flow acceleration in the tube (a vertical line in Fig. 6). It should be noted that in the studied range of parameters, an increase in heat transfer due to acceleration of air in the flow is observed only at the flow velocity close to the sound velocity and it depends weak on the pressure. In the flow of helium-xenon mixture, the effect of acceleration appears at lower Reynolds numbers, and typical velocities, at which acceleration leads to an increase in heat transfer, are substantially less than the sound velocity. A low Prandtl number for the helium-xenon mixture leads to better flow core heating as compared to air that, in turn, leads to greater expansion and acceleration of gas in the tube. Application of gases with low Prandtl number as the coolants at low pressures and high flow rates is the basis for adjustment of heat transfer coefficients obtained by the Dittus-Boelter formula and its equivalents in the engineering practice.
Fig. 5. Distributions of Nusselt numbers, calculated by mass-average stagnation temperature (solid lines) and Mach number (dashed lines) along the tube at different pressures and Reynolds numbers (in brackets) at the tube inlet: 1–2 atm (76,000), 2–1 atm (57,000), and 3–0.5 atm (38,000).
Fig. 6. Generalization of data on heat transfer in the small diameter tube depending on Reynolds number.
7. Conclusion The effect of gas compressibility on heat transfer in a small diameter tube at Prandtl numbers of 0.71 (air) and 0.23 (heliumxenon mixture) was studied. It is shown that the known methods of generalization of experimental data and numerical simulations do not provide a power heat transfer law at high Reynolds numbers for gases with different Prandtl numbers. The method for determining the mass-average recovery temperature as a characteristic flow temperature, allowing extension for the applicability range of the Dittus-Boelter formula by the Reynolds number, was suggested. However, the upper limit of applicability of Dittus-Boelter formula at low Prandtl number turned out to be dependent on the gas density. A decrease in density leads to an increase in the pressure gradient along the tube, the flow becomes significantly accelerated, and this apparently leads to an increase in heat transfer as compared with the established incompressible flow. Heat transfer enhancement in the studied air flow is observed only at the flow velocities close to the sound velocity, and in the flow of helium-xenon mixture, it is observed at velocities substantially lower than the sound velocity. The use of gas coolants with a low Prandtl number in engineering practice at low pressures and high flow rates requires adjustment of the heat transfer coefficients obtained by the known correlations. Acknowledgements
determined at the tube outlet. Light points indicate data for air, dark points indicate data for the helium-xenon mixture. Different shapes of points correspond to different pressures at the tube inlet. The Nusselt and Reynolds numbers are determined by formulas:
Nu ¼
qw d 4G ; Red ¼ ; T w T r k pdl
where k and l are the coefficients of heat conductivity and dynamic viscosity at the tube outlet. In processing the experimental data T r
was taken equal to T , which was calculated according to heat balance. The inlet pressure in the experiments was more than 3.8 atm, and therefore the significant effect of gas compressibility on heat transfer is not obtained. Despite consideration of the effect of gas compressibility on the heat transfer coefficient in the numerical modeling data, we failed to establish a general correlation of the Nusselt number on the Reynolds and Prandtl numbers. With a decrease in the Reynolds number due to the gas density (a change in pressure at the tube inlet) the pressure gradient along the tube increases, the flow becomes significantly accelerated, and this leads to an increase in heat transfer. The Dittus-Boelter formula has the
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