Theoretical Foundations of Chemical Engineering, Vol. 39, No. 6, 2005, pp. 658–662. Translated from Teoreticheskie Osnovy Khimicheskoi Tekhnologii, Vol. 39, No. 6, 2005, pp. 698–702. Original Russian Text Copyright © 2005 by G. Mikhailov, V. Mikhailov, Kondakova, Reva.
SHORT COMMUNICATIONS
Prediction of the Convective Heat Transfer Coefficient for the Transient and Turbulent Flows in a Tube G. M. Mikhailov, V. G. Mikhailov, L. A. Kondakova, and L. S. Reva Volgograd State Technical University, pr. Lenina 28, Volgograd, 400131 Russia e-mail:
[email protected] Received May 25, 2004
To date, the most reliable equations for estimating the heat transfer coefficient for the flow of Newtonian media in a tube are the following: for the laminar flow (Re < 2320), this is the Nusselt equation
or Nu -------------------------------0.25 0.43 ⎛ Pr ⎞ -------Pr ⎝ Pr w⎠
λ α = 3.66 --- , d
3.66 2/3 = -------------------------------- + 0.0855 ( Re – 2320 ) , 0.25 0.43 ⎛ Pr ⎞ -------Pr ⎝ Pr w⎠
or, in terms of dimensionless numbers, (1)
Nu = 3.66,
and for the turbulent region,
and for the turbulent flow (Re > 10000), this is the Mikheev equation [1] Nu 0.8 -------------------------------- = 0.021Re . 0.25 0.43 Pr Pr ⎛ --------⎞ ⎝ Pr w⎠
(2)
For the transient flow (2320 < Re < 10000), reliable equations for Newtonian media are absent and it is conventional to use a graphic representation, which is approximately described by the general equation (see, e.g., [2]) Nu -------------------------------- = f ( Re ). 0.25 0.43 ⎛ Pr ⎞ -------Pr ⎝ Pr w⎠
(3)
The results of predicting the convective heat transfer coefficient for the flow of Newtonian media in a tube using Eqs. (1)–(3) are presented in Fig. 1a. The observed mismatch of the curves at the boundary between the laminar and the transient regions perplexes and forces a formalistic estimator (such as a computer) to make fundamentally wrong decisions, e.g., that it is more correct to terminate the calculation process at Re = 2300 rather than at Re = 2400. In this work, we propose the following equations: for the transient region, Nu = 3.66 + 0.08555 ( Re – 2320 )
2/3
Pr
Pr 0.25 --------⎞ ,(4) ⎝ Pr w⎠
0.43 ⎛
(5)
0.8
Nu = 3.66 + 0.021Re Pr
Pr 0.25 --------⎞ , ⎝ Pr w⎠
0.43 ⎛
(6)
or 3.66 Nu 0.8 -------------------------------- = -------------------------------- + 0.021Re . 0.25 0.25 0.43 Pr 0.43 Pr Pr ⎛ --------⎞ Pr ⎛ --------⎞ ⎝ Pr w⎠ ⎝ Pr w⎠
(7)
The results of predicting the heat transfer coefficient for the flow of Newtonian media using the equations proposed are given in Fig. 1b. The curves at the boundaries of the regions of different flows are seen to match well. In Eqs. (4)–(7), the first terms describe the contribution of purely molecular heat conduction and the second terms characterize the contribution of forced convection. If the contribution of purely molecular heat conduction of the medium is ignored, Eq. (7) appears as Mikheev equation (2) and Eq. (5) takes the form Nu 2/3 -------------------------------- = 0.08555 ( Re – 2320 ) , 0.25 0.43 ⎛ Pr ⎞ -------Pr ⎝ Pr w⎠
(8)
which describes Eq. (3) with high accuracy. However, in the turbulent flow, we also should not unconditionally ignore the contribution of purely molecular heat conduction of the medium, i.e., the first terms of Eqs. (4) and (7). It is another matter that, at
0040-5795/05/3906-0658 © 2005 MAIK “Nauka /Interperiodica”
PREDICTION OF THE CONVECTIVE HEAT TRANSFER COEFFICIENT Nu
log -------------------------------------0.43 Pr 0.25 Pr ⎛⎝ ----------⎞⎠ Pr
Nu l o g -------------------------------------0.43 ⎛ Pr ⎞ 0.25 Pr ⎝ ----------⎠ Pr w
(a)
w
659
(b)
3
2 2 3
1
3
‡ b c d e f
2 1
1
2 1
g
0
0 3
‡ b c d e f
4
5
3
logRe
4
5
logRe
Pr 0.25 Fig. 1. Heat transfer coefficient in a round tube as determined from the following equations: (a) (1) Eq. (1) at Pr0.43 ⎛ --------⎞ = (a) ⎝ Pr w⎠ 0.1, (b) 0.5, (c) 0, (d) 1.0, (e) 2.0, and (f) 3.0; (2) Eq. (3); and (3) Eq. (2) and (b) (1) the Nusselt equation; (2) Eq. (5); and (3) Eq. Pr 0.25 (7) at Pr0.043 ⎛ --------⎞ = (a) 0.1, (b) 0.2, (c) 0.5, (d) 1.0, (e) 2.0, ( f ) 3.0, and (g) ∞. ⎝ Pr w⎠
Pr 0.25 large values of Pr0.43 ⎛ --------⎞ , the first terms of these ⎝ Pr w⎠ equations automatically become negligibly small. Equations (2) and (3), in which the contribution of purely molecular heat conduction is completely ignored, in some cases can be not only significantly inaccurate but also fundamentally absurd. For example, when mercury cools as it flows in a tube at t = 800°ë and tw = 100°ë, we have Pr = 0.0074 and Prw = 0.01796 [3]. In this case, at Re = 10000, from the Mikheev equation, we obtain Nu = 0.021 ( 10 000 )
0.8
× 0.0074
0.0074 0.25 -------------------⎞ ⎝ 0.01796⎠
0.48 ⎛
= 3.23 < 3.66, which is known to be theoretically impossible. In the case Re = 5000, from Eq. (3), we have Nu = 16.3 × 0.0074
0.0074 0.25 -------------------⎞ = 1.584 < 3.66, ⎝ 0.01796⎠
0.43 ⎛
which is all the more impossible.
Let us give one more example. In the case of cooling Pr of monoatomic gases, we have Pr = 0.67 and -------- = 1. Pr w At Re = 2500, from Eq. (3), we obtain Nu = 4 × 0.67
0.43
×1
0.25
= 3.367 < 3.66.
In the context of the above, it is obvious that Eqs. (2) and (3) are conventionally used to characterize the flow of Newtonian liquids not because they accurately describe reality but only because they predict guaranteed values of the heat transfer coefficient; i.e., the heat transfer coefficient cannot be lower than that predicted by Eqs. (2) and (3), but, in some cases, it can be somewhat higher. The use of these equations leads to a certain, sometimes unreasonably large, overestimation of the heat-exchange surface area. The table compares the data obtained using Eq. (4) with the experimental data of [4] on the heat transfer for the transient flow of water and benzene in a tube. The following notation is accepted in the table: tin and tout are the temperatures of the medium at the inlet and the outlet, °C, respectively; tw is the wall temperature, °C; ∆tm is the logarithmic mean temperature difference, °C;
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Comparison of the data obtained using Eq. (4) with the experimental data of [4] tin , °C
tout , °C
tw , °C
66 59.5 67.5 30 30 59.5 32 61 34 32.5 Benzene 16 12 30 15 12.5
84.5 81 84.5 64.5 66.5 80 61.5 81.5 68 62 33 39.5 42 30 25
95 95 95 89.5 93.5 95 94 94 92.5 91.5 59 87.5 61 55.5 56
Liquid Water
α λ, Prw ∆tm , t = tw – ∆tm , W/(m K) Nuexp Pr [2] Re °C °C 2 2 [2] kcal/(m h °C) W/(m K) [2]
Nucal δ, %
18.20 23.11 17.66 39.79 42.68 23.80 45.67 21.12 39.06 42.56 33.79 60.72 24.51 32.42 36.90
38.63 42.56 44.14 46.41 48.83 53.80 54.43 54.97 57.08 63.81 38.40 40.67 61.30 68.74 74.23
76.79 71.89 77.34 49.71 50.83 71.20 48.33 72.88 53.43 48.54 25.21 26.78 36.49 23.08 19.10
6740 7230 7870 6250 6600 9320 7160 9770 8100 8710 4340 4570 6920 7530 8180
1620 2000 2000 2200 2300 2400 2380 2700 2700 2750 422 380 580 645 750
t = (tw – ∆tm) is the mean wall temperature, °ë; Nuexp is the experimental Nusselt number; Nucal is the Nusselt number calculated from Eq. (4); and δ is the relative deviation of the calculated Nusselt number from the experimental one, %. The table shows that the deviations of the data calculated using Eq. (4) from the experimental results do not exceed ±10%. This correlation of the experimental data of [4] should be considered quite satisfactory, taking into account that the experimental data tables give too rounded values of both Re numbers and heat transfer coefficients. The proposed equations are confirmed by the following facts. First, these equations match very well Nusselt equation (1) at the boundary between the laminar and the transient regions and Mikheev equation (2) at the boundary between the transient and the developed turbulent regions. Second, Eq. (5) in a particular case at 3.66 -------------------------------0.25 0.43 ⎛ Pr ⎞ -------Pr ⎝ Pr w⎠
0
agrees very well with conventional Eq. (3) and Fig. 1b. Third, Eq. (5) is very convincingly supported by the statement [2, p. 154] that “approximate (overestimated) calculation can be carried out using the plot” of Eq. (3). This is equivalent to the statement that the heat transfer coefficient calculated from Eq. (3) is somewhat underestimated in comparison with the mean values for vari-
1886 2327 2327 2561 2677 2793 2770 3142 3142 3201 491.2 442.3 675.1 750.7 872.9
0.666 0.663 0.667 0.643 0.644 0.663 0.641 0.664 0.646 0.642 0.1449 0.1445 0.1416 0.1454 0.1467
35.40 43.87 43.61 49.79 51.96 52.66 54.02 59.15 60.80 62.32 42.36 38.27 59.62 64.56 74.36
2.309 2.476 2.290 3.562 3.493 2.499 3.670 2.442 3.344 3.623 7.82 7.45 7.35 7.82 7.85
1.835 1.835 1.835 1.953 1.866 1.835 1.856 1.856 1.887 1.908 6.45 5.85 6.45 6.55 6.55
9.12 –2.98 1.21 –6.78 –6.03 2.17 0.76 –7.07 –6.11 2.39 –9.35 6.27 2.82 6.47 –0.17
ous liquids. It is this conclusion that follows from Fig. 1b constructed using Eq. (5). Finally, Eqs. (5) and (7) are confirmed by the fact that it is in the region in Fig. 1b where the plots of these equations, as well as Nusselt equation (1), lie that the plots of numerous equations proposed by various researchers in corresponding coordinates [5] also lie. This indicates that these equations are obtained using quite reliable experimental data but these data were processed incorrectly: the molecular heat conduction was ignored and, in some cases, the effect of Pr and Prw was either neglected or taken into account improperly. We consider it appropriate to dwell here briefly on the problem of the effect of natural convection on the heat transfer within a tube since neither Nusselt equation (1) nor Eqs. (4)–(7) include the Grashof number. If a medium flows in small-diameter tubes, which are used in heat exchangers, any intense natural convection is unlikely since the Grashof number is directly proportional to the cube of the tube diameter. Meanwhile, at small tube diameters and high viscosities of the medium, the Grashof number is too small for any intense natural convection to occur, whereas, at large tube diameters and low viscosities of the medium, the Reynolds numbers are already large and the contribution of natural convection is negligibly small in comparison with the contribution of forced convection. Therefore, it is not accidental that the Grashof number is absent from recently published equations of heat transfer in a forced flow in a tube even for the laminar region. Even Mikheev, who originally introduced the Grashof number into the equation for the laminar
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PREDICTION OF THE CONVECTIVE HEAT TRANSFER COEFFICIENT
region, excluded the Grashof number for the flow within a tube even for the laminar region in the last works, e.g., [6]. It seems likely that the problem of the effect of natural convection on the heat transfer in the flow of Newtonian media within a tube is virtually nonexistent. The question of whether or not the equations proposed are applicable to molten metals, which are characterized by very small Prandtl numbers (Pr = 0.005– 0.05), remains open. However, this circumstance does not diminish the significance of the equations proposed since the prediction of the heat transfer coefficient from molten metals is of no practical importance because the thermal resistance 1/αmet to heat transfer from molten metals is always negligibly small in comparison with other thermal resistances that determine the heat transfer coefficient. However, from a purely theoretical standpoint, it is of interest to discuss previously published equations for predicting the heat transfer coefficient from molten metals. The following equations of heat transfer in the flow of molten metals are known [7, 8]: 0.8
(9)
0.8
(10)
Nu = 3.3 + 0.014Pe , Nu = 4.36 + 0.025Pe .
Since Pe = Re · Pr, these equations are equivalent to the equations 0.8
0.8
Nu = 3.3 + 0.014Re Pr , 0.8
0.8
Nu = 4.36 + 0.0025Re Pr . The considerable quantitative discrepancy of these equations is an alarming fact. It is unclear why the Pr effect of the ratio -------- does not manifest itself since the Pr w Prandtl number Pr for molten metals depends strongly on temperature and the values of this ratio should differ significantly from unity. Much becomes clear in the context of the remark of the author of Eq. (9) [6] that the experiments were carried out in “oxidized tubes” and, moreover, that a layer of oxides, which intensely formed as the molten metal contacted air, deposited on the heat-exchange surface. Consequently, in the experiments, not the true heat transfer coefficient from the molten metal was determined but the apparent heat transfer coefficient αapp, which equals 1 α app = ---------------------. r ox + 1/α Here, rox is the total thermal resistance of the oxide film and the layer of oxides of the molten metal that are deposited on the heat-exchange surface.
661
The thermal resistance of the fouled surface with respect to purified water is supposed to be at least [2] 1 –4 2 r = ------------ ≅ 1.7 × 10 ( m K )/W. 5800 Even if the total thermal resistance of the oxide film and the layer of oxides that are deposited on the surface is taken to be the same as in the literature [7, 8], then, instead of, e.g., the true heat transfer coefficients α1 = 40000 W/(m2 K) and α2 = 20000 W/(m2 K), incommensurably lower and virtually identical values of the apparent heat transfer coefficient are obtained: 1 2 α app 1 = --------------------------------------------------= 5128 W/m K; –4 1.7 × 10 + 1 /40 000 1 2 α app 2 = --------------------------------------------------= 4550 W/m K. –4 1.7 × 10 + 1/20 000 The examples given confirm the above doubt regarding the expedience of investigations of the heat transfer from molten metals. It would be much more important to investigate the additional thermal resistances that are formed on the heat-exchange surface in contact with molten metals. In determining the heat transfer from molten metal to a medium through a wall, the thermal resistance to heat transfer from the molten metal can be taken to be negligibly low. Thus, Eqs. (4) and (6) allow one to predict the convective heat transfer coefficient for the flow of Newtonian media in a tube somewhat more accurately than the known graphic representation for the transient region and the Mikheev equation for the turbulent region. In conclusion, we note that a relatively recent source [9] presented an equation for the turbulent region that is accepted in foreign practice. This equation virtually coincides with the Mikheev equation. The absence of specific equations for the transient region in that work [9] is an eloquent recognition that there are still no reliable equations for the transient region and is an indirect confirmation of the topicality of this paper. NOTATION a—thermal diffusivity, m2/s; d—inner tube diameter, m; r—thermal resistance, (m2 K)/W; t—temperature, °C; w—velocity of a medium in a tube, m/s; α—heat transfer coefficient, W/(m2 K); λ—thermal conductivity, W/(m K); ν—kinematic viscosity, m2/s; αd Nu = ------- —Nusselt number; λ Pe = Re · Pr—Peclet number;
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a Pr = --- —Prandtl number at the mean temperature of ν a medium; Prw—Prandtl number at the wall temperature; wd Re = ------- —Reynolds number. ν SUBSCRIPTS AND SUPERSCRIPTS cal—calculated; exp—experimental; in—inlet; m—mean; out—outlet; w—wall. REFERENCES 1. Mikheev, M.A., Osnovy teploperedachi (Principles of Heat Transfer), Moscow: Gosenergoizdat, 1956. 2. Pavlov, K.F., Romankov, P.G., and Noskov, A.A., Primery i zadachi po kursu protsessov i apparatov khimicheskoi tekhnologii (Examples and Problems for Course in Unit Operations and Equipment of Chemical Engineering), Leningrad: Khimiya, 1987.
3. Vargaftik, N.B., Spravochnik po teplofizicheskim svoistvam zhidkostei i gazov (Handbook of Thermophysical Properties of Liquids and Gases), Moscow: Nauka, 1972. 4. Balajka, B. and Sykora, K., Vymena tepla v zarizenich chemickeho prumyslu (Heat Exchange in Chemical Engineering Equipment), Praha: Statni Nakladatelstvi Technicke Literatury, 1959. Translated under the title Protsessy teploobmena v apparatakh khimicheskoi promyshlennosti, Moscow: Mashgiz, 1962. 5. Tarasov, F.M., Teoriya i raschet protochnykh teploobmennikov (Theory and Calculation of Flow Heat Exchangers), Leningrad: Len. Gos. Univ., 1975. 6. Mikheev, M.A. and Mikheeva, I.M., Osnovy teploperedachi (Principles of Heat Transfer), Moscow: Energiya, 1977. 7. Mikheev, M.A., Fedynskii, O.S., Deryugin, V.M., and Petrov, V.I., Heat Transfer in Metals in Tubes, in Teploperedacha i teplovoe modelirovanie (Heat Transfer and Thermal Modeling), Moscow: Acad. Nauk SSSR, 1959. 8. Subbotin, V.I., Ushakov, P.A., Gabrilovich, B.N.. et al., Heat Transfer in the Flow of Liquid Metals in Tubes, Inzh.–Fiz. Zh., 1963, no. 4, p. 16. 9. Obshchii kurs protsessov i apparatov khimicheskoi tekhnologii (General Course in Unit Operations and Equipment of Chemical Engineering), Ainshtein, V.G., Ed., Moscow: Logos, 2002.
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