Simple heat transfer correlations for turbulent tube flow

The paper presents three power-type correlations of a simple form, which are valid for Reynolds numbers range from 3·10 3 ≤ Re ≤ 10 6 , and for three different ranges of Prandtl number: 0.1 ≤ Pr ≤ 1.0, 1.0 3 . Heat transfer correlations developed in the paper were compared with experimental results available in the literature. The comparisons performed in the paper confirm the good accuracy of the proposed correlations. They are also much simpler compared with the relationship of Gnielinski, which is also widely used in the heat transfer calculations.


Introduction
Heat transfer correlations for turbulent fluid flow in the tubes are commonly used in the design and performance calculations of heat exchangers [1][2][3][4]. The value of heat transfer coefficient significantly affects the value of the thermal stress [5]. Time changes in optimum fluid temperature determined from the condition of not exceeding the stress at a point lying on the inner surface of the pressure component, very strong depend on the heat transfer coefficient [6][7][8][9]. Therefore, continuous experimental research is carried out to find a straightforward and accurate heat transfer correlations. The empirical correlation of Dittus-Boelter [10][11][12] has gained widespread acceptance for prediction of the Nusselt number with turbulent flow in the smoothsurface tubes The exponent of the Prandtl number is n = 0.4 for heating of the fluid and n = 0.3 if the fluid is being cooled. The similar power-type relationship of Colburn is based on the Chilton-Colburn analogy for heat and momentum transfer [13]   The constant n is equal zero for gases. The relationship (4) of Gnielinski is very widely used, as well approximates the experimental data [17,[23][24][25].
In recent years, however, a new rise in popularity of power-type correlation to an approximation of the measurement data is observed [24][25][26].
The thermal-hydraulic performance of a dimpled enhanced tube was studied experimentally and numerically by Li et al. [24]. The power type correlation based on the experimental results was proposed. The Wilson method was used to adjust unknown constants in the searched relationship. The Prandtl number changed from 5.2 to 30.7 and the Reynolds number from 500 to 8,000.
Simple power correlations are also proposed to fit experimental data for turbulent flows of nanofluids in tubes. Li and Xuan [25] and Azmi and al. [26] also used electrically heat tubes to study hydrodynamically fully developed flow. Before starting the actual experiments with nanofluids, tests were conducted in which the working fluid was water [25] or a mixture of water and ethylene glycol [26]. The Reynolds number varied in the study of Azmi et al. from 3,000 to 25,000. The Nusselt numbers determined experimentally showed good agreement with the Nusselt numbers obtained from the Dittus-Boelter equation.
Li and Xuan [25] also found a heat transfer correlation for internal flow of Cu-water nanofluid that for water reduces to Previous studies indicate that power-type correlations are still used successfully to approximate the experimental results when the Reynolds and Prandtl numbers vary in a narrow range.
Unknown constants occurring in the power-type relationships can be readily determined by standard [27] or modified [28] Wilson method while maintaining reasonable effort.
Easy determination of searched constants based on the experimental results is also the reason for the use of power type functions to find heat transfer correlations for the turbulent flow of nanofluids or molten salts in tubes.

Nusselt numbers for turbulent tube flow
Energy conservation equation for turbulent tube flow averaged by Reynolds has the following form [23,29] ( ) where: q -heat flux density, x -Cartesian coordinate, r -radius, u -time averaged velocity.
The heat flux q consists of the molecular m q and where Pr , Pr where w q is the wall heat flux. The mass averaged The time averaged velocity profile ( ) u r in tube crosssection is necessary to determine the temperature distribution in the fluid. The eddy diffusivity for momentum transfer ετ and time averaged fluid velocity ( ) u r were calculated using Reichardt's [30][31] empirical relationships, which are straightforward and accurate. Equation (8) with the boundary conditions (11)-(13) was solved using the finite difference method [29]. The Nusselt number was determined based on the temperature distribution determined numerically using the following formula The Nusselt number was evaluated for various Reynolds and Prandtl numbers, and the results are listed in Table 1 [29]. The Nusselt numbers shown in Table 1 with Prandtl numbers ranging from 0.1 to 1,000 and Reynolds numbers ranging from 3,000 to 10 6 were approximated by a product of two power functions of the Reynolds and Prandtl numbers  Table 1 were taken. The values of Nu c i j were evaluated using Eq. (17).
Substituting the constants 1 2 , c c , and 3 c determined from the condition (17)

Nu
Re Pr 1 The heat transfer correlations (18) - (20) were compared with the experimental data available in the literature taking into account the correction factor ( )

correlations with experimental data
The proposed correlation (18) was compared with the experimental results of turbulent air flow in the pipes and the correlation (20) with experimental data for water, which are available in the literature. Comparison of the results of experimental studies of doctoral dissertations [32][33][34][35] for the air flowing inside the heating pipe is shown in Fig. 1. Abraham [36] was the first who presented the experimental data for smooth tubes with uniform wall heat flux reported in these theses and used by him to verify the CFD results. Equation (18) gives lower values of Nusselt number as compared to experimental results, whereas the correlation of Dittus-Boelter (1) overpredicts them. The proposed relationship (18) was also compared in Fig. 2 with the experimental data of Eiamsa-ard and Promvonge [37] for turbulent air flow in a tube with constant wall heat flux. Measurements were performed in an electrically heated tube. The outer and inner diameter of the copper tube were 47 mm and 50 mm, respectively. The tube was heated uniformly over the length of 1,250 mm. An unheated calming section preceded heated pipe section.   (18) proposed in the present paper and the correlation of Dittus-Boelter [10] and Gnielinski [19][20] with the experimental data of Eiamsa-ard| and Promvonge [37] obtained for turbulent air flow in a tube with constant wall heat flux.
The hydrodynamically developed turbulent flow of water enters a uniformly heated tube. The Nusselt number determined experimentally was 10 to 20% higher than values predicted by the Dittus-Boelter equation. The discrepancy between the results of measurements and values of Nusselt number obtained from the Dittus-Boelter formula is greater for higher Reynolds numbers.
Then, Eq. (20) was compared with experimental results available of Allen and Eckert [16]. The smoothtube heat transfer results reported by Allen and Eckert [16] were obtained for developed turbulent flow of water under the uniform wall heat flux boundary condition at Pr=7 and Pr=8, and 1.3·10 4 ≤ Re ≤ 1.11·10 5 [16]. Steel pipe with an inner diameter of 19.05 mm and a wall thickness of 1.5875 mm was heated electrically so that the heat flux at the internal surface of the tube was constant.
The correlation (20) found in the paper closely approximates the experimental equation (23) of Allen and Eckert for Reynolds numbers greater than 20,000 as compared to the Dittus-Boelter equation (1).
Comparisons of the correlations (8) and (20) to experimental data obtained by different researchers confirm quite well accuracy of the proposed relationships, especially when taking into account a variety of test conditions and different ways of experimental data processing.

Conclusions
The paper presents three power-type correlations of a simple form, which are valid for Reynolds numbers range from 3·10 3 ≤ Re ≤ 10 6 , and for three different ranges of Prandtl number: 0.1 ≤ Pr ≤ 1.0, 1.0 < Pr ≤ 3.0, , and 3.0 < Pr ≤ 10 3 .
Formulas proposed in the paper have good theoretical basis, as they have been obtained by approximating Nusselt numbers obtained from the solution of the energy conservation equation for turbulent flow in a pipe by the method of least squares. Heat transfer correlations developed in the paper were compared with experimental results available in the literature. The performed comparisons confirm the good accuracy of the proposed correlations, better than the accuracy of the Dittus-Boelter correlation. Heat transfer correlations proposed in the paper can be used in a broader ranges of Reynolds and Prandtl numbers compared with a widely used correlation of Dittus-Boelter. They are also much simpler in comparison to the relationship of Gnielinski, which is also widely used in the heat transfer calculations.