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In this paper, is presented a mathematical deduction of a new improved model for heat transfer calculations during fluid flow in singlephase inside tubes. The proposal model was verified by comparison with available experimental data of 35 different fluids, including water, air, gases and organic substances. The proposal model is valid for a range of Reynolds number for singlephase from 2.4·10^{3 }to 8.2·10^{6}, Prandtl number for singlephase from 0.65 to 4.71·10^{4}, dimensionless length in the interval 2≤l⁄d≤450 and values of Petukhov’s correction in the interval 0006≤μ_{F}⁄μ_{P}≤177. In 3096 data analyzed, for R_{e}<1·10^{4}, the mean deviation found was 13.91% in the 80.32% of the experimental data, while for 1·10^{4}≤R_{e}, the mean deviation found was 13.96% in 80.94% of experimental data.
single phase, model, heat transfer coefficient , average deviation
Currently, heat transfer calculations for turbulent fluid flow within straight conduits in singlephase media are made by the DittusBoelter equation, or by the improved version of SiederTate [1]. This procedure is a requirement for the evaluation of industrial facilities or production equipment. A drawback of these equations is their high dispersion value, reaching compute errors close to ± 40 %.
At the Moscow Energy Institute, Petukhov and his collaborators constructed a model based on experimental quantity adjustments, using the Prandtl analogy as an adjustment function [2]. This equation gives results with a lower margin of error, and allows us to estimate the mean error by the dimensionless number of Prandtl.
if Pr≤200 Error<5%
if Pr≥200 Error≤10%
Although the application of the Petukhov’s Equation is more laborious, the results obtained have a minor dispersion, therefore, a smaller safety margin in the design calculations.
A major drawback is its applicability range, because this is only valid for a fully developed turbulent flow regime, 1∙$10^4$<Re, and is not valid for the flow that operate in the transition zone 2.3∙$10^3$<Re<1∙$10^4$. This problem was later solved by Gnielinsky [34], who modified the Petukhov’s Equation, adjusting it to experimental data that do take into account the transition flow zone.
In the literature can be found an important group of works that facilitate the calculation of heat transfer inside of straight tubes with turbulent flow, this is mainly associated with the changing nature of the turbulent flow, which hinders the development of analytical expressions. This element makes it necessary to resort to the experimentation and subsequent adjustment of experimental quantities through the theory of dimensional analysis.
2.1 Analogy between heat transfer and momentum in singlephase fluid flow inside pipes
The Darcy friction factor f allows determining the heat transfer coefficient α, by analogy between heat transfer and momentum. The shear stress $\tau$ in the turbulent boundary layer is composed of two terms [5]:
$\tau ={{\tau }_{Visc}}+{{\tau }_{Turb}}=\mu \frac{dV}{dx}\rho V_{X}^{*}V_{Y}^{*}$ (1)
In Equation (1) ${\tau }_{Turb}$ is the Reynolds stress; $V*_X$ is the fluctuation of the instantaneous velocity $V^M_X$ in the coordinate axis x; $V*_Y$ is the fluctuation of the instantaneous velocity $V^M_X$ in the coordinate axis y.
The instantaneous velocity $V^M_X$ and $V^M_Y$ are determined as:
$V_{X}^{M}=V_{M}^{X}\pm V_{X}^{*}=V_{M}^{X}\pm V_{agit}^{X}$ (2)
$V_{Y}^{M}=V_{M}^{Y}\pm V_{Y}^{*}=V_{M}^{Y}\pm V_{agit}^{Y}$ (3)
For turbulent heat flow, it can be considered that the total heat flow $q*$ is composed of a sum that includes the conductive component $q_{cond}$ and the turbulent component $q_{turb}$, then:
${{q}^{*}}={{q}_{cond}}+{{q}_{turb}}=\lambda \frac{dT}{dx}+\rho CpV_{Y}^{*}T_{F}^{*}$ (4)
In Equation (3) there are three temperature references, which are:
$\left\{ \begin{align} & T_{F}^{*}={{T}_{F}}\pm T_{F}^{*}\ \text{,}\ \text{ }\quad \quad \to \text{is the instantaneous temperature} \\ & {{T}_{F}}={{T}_{\infty }}\ \text{, }\ \quad \quad \quad \quad \to \text{is the average temperature of the fluid} \\ & T_{F}^{*}\ \text{, }\ \quad \quad \quad \quad \quad \quad \to \text{is the temperature due to the fluctuation } \\ \end{align} \right.$ (5)
Terms $V*_X$ and $V*_Y$ are obtained from their physical meaning from the Prandtl mixing number, which suggests that the fluctuation of velocity $V*_X$ is related with $dV/dx$ as:
$V_{X}^{*}\approx {{L}_{M}}\frac{dV}{dx}$ (6)
In Equation (6), $L_M$ is the mixture length of the thickness film ${{\delta }_{2}}$ of the momentum in boundary layer. Similarly, transverse fluctuation $V*_Y$ is admitted to be of the same order of magnitude $V*_X$ but opposite in sign, [6]:
$V_{Y}^{*}\approx {{L}_{M}}\frac{dV}{dx}$ (6.a)
Combining the Equations (6) and (6.a):
$V_{X}^{*}V_{Y}^{*}\approx {{\left( {{L}_{M}}\frac{dV}{dx} \right)}^{2}}$ (7)
Equation (7) can be transformed to:
$V_{X}^{*}V_{Y}^{*}\approx {{\varepsilon }_{M}}{dV}/{dx}\;$ (8)
In Equation (8) ${{\varepsilon }_{M}}$ is the momentum turbulent diffusivity, then:
${{\varepsilon }_{M}}\approx L_{M}^{2}{dV}/{dx}\;$ (9)
To find the relationship of the term $V_{Y}^{*}T_{F}^{*}$, with the mean local temperature gradient, a similar method is applied, in the form [6].
$T_{F}^{*}={{L}_{C}}\frac{dT}{dx}$ (10)
$V_{Y}^{*}={{L}_{C}}{dV}/{dx}\;$ (11)
In the expression (11) $L_C$ is the mixture length of the energy in the thickness ${{\delta }_{3}}$ of the thermal boundary layer, then:
$V_{Y}^{*}T_{F}^{*}=L_{C}^{2}\frac{dV}{dx}\frac{dT}{dx}={{\varepsilon }_{C}}\frac{dT}{dx}$ (12)
In the Equation (12), the term ${{\varepsilon }_{C}}=L_{C}^{2}{dV}/{dx}\;$ is the heat turbulent diffusivity. Substituting Equation (8) into Equation (1):
$\tau ={{\tau }_{Visc}}+{{\tau }_{Turb}}=\mu {dV}/{dx}\;\rho {{\varepsilon }_{M}}{dV}/{dx}\;$ (13)
Dividing by the density $\rho $ to both members of the Equation (13) and taken the derivative ${dV}/{dx}\;$ as a common factor [78], then:
$\frac{\tau }{\rho }={{\tau }_{Visc}}+{{\tau }_{Turb}}=\left( \frac{\mu }{\rho }+{{\varepsilon }_{M}} \right)\frac{dV}{dx}=\left( \nu +{{\varepsilon }_{M}} \right)\frac{dV}{dx}$ (14)
Substituting Equation (12) into Equation (4):
${{q}^{*}}={{q}_{cond}}+{{q}_{turb}}=\lambda \frac{dT}{dx}\rho Cp{{\varepsilon }_{C}}\frac{dT}{dx}$ (15)
If in Equation (15) the derivative ${dT}/{dx}\;$ is taken as a common factor, then
${{q}^{*}}={{q}_{cond}}+{{q}_{turb}}=\left( \lambda +\rho Cp{{\varepsilon }_{C}} \right){dT}/{dx}\;$ (16)
In Equation (16), both members are divided by the product of the density and specific heat at constant pressure $\rho Cp$.
$\frac{{{q}^{*}}}{\rho Cp}=\left( \frac{\lambda }{\rho Cp}+{{\varepsilon }_{C}} \right)\frac{dT}{dx}=\left( a+{{\varepsilon }_{C}} \right)\frac{dT}{dx}$ (17)
Dividing Equation (17) by the Equation (14) is obtained the basic relationships for the fluid flow inside of tubes [69]:
$\frac{\tau }{{{q}^{*}}}=\frac{\nu +{{\varepsilon }_{M}}}{Cp\left( a+{{\varepsilon }_{C}} \right)}\frac{dV}{dT}$ (18)
In Equation (18), the kinematic viscosity v and the thermal diffusivity a are properties of the fluid, while ${{\varepsilon }_{C}}$ and ${{\varepsilon }_{M}}$ are properties of the flow.
2.2 Development of one linear model for convective heat transfer calculation in singlephase inside pipes
Development of the new model for to calculate the convective heat transfer in singlephase inside pipes is a complex task. Initially is taken the criterion established by Prandtl, which considers that the flow is divided into two zones, a viscous zone and a turbulent zone. In his analysis Prandtl makes the additional assumptions that in the turbulent zone the molecular diffusivities of momentum v and of heat a, are negligible in comparison with the turbulent diffusivities $\nu <<{{\varepsilon }_{M}}$; $a<<{{\varepsilon }_{C}}$, so they do not intervene in the process. It would be very useful for such a purpose to assume that the relationship between molecular diffusivities a and v is equal to the relationship between diffusivities ${{\varepsilon }_{C}}$ and ${{\varepsilon }_{M}}$.
Since the dimensionless Prandtl number is a relation between diffusivities, then the previous assumption is fulfilled [10].
$\Pr =\frac{\nu }{a}=\frac{{{\varepsilon }_{M}}}{{{\varepsilon }_{C}}}$ (19)
Clearing v and ${{\varepsilon }_{M}}$ in the Equation (19):
$\nu =a\Pr $ (20)
${{\varepsilon }_{M}}={{\varepsilon }_{C}}\Pr $(21)
Substituting Equations (20) and (21) into Equation (18):
$\frac{\tau }{{{q}^{*}}}=\frac{a\Pr +{{\varepsilon }_{C}}\Pr }{Cp\left( a+{{\varepsilon }_{C}} \right)}\frac{dV}{dT}$ (22)
If the Prandtl number Pr is taken as a common factor in Equation (22), it is reduced to [11]:
$\frac{\tau }{{{q}^{*}}}=\frac{{{\tau }_{0}}}{q_{0}^{*}}=\frac{\left( a+{{\varepsilon }_{C}} \right)\Pr }{\left( a+{{\varepsilon }_{C}} \right)Cp}\frac{dV}{dT}=\frac{\Pr }{Cp}\frac{dV}{dT}$ (23)
Separating variables in (23) and integrating
$\int\limits_{0}^{{{V}_{M}}}{dV}=\frac{Cp{{\tau }_{0}}}{\Pr q_{0}^{*}}\int\limits_{{{T}_{P}}}^{{{T}_{F}}}{dT}$ (24)
Resolving the integrals present in Equation (24) and grouping conveniently:
${{V}_{M}}=\frac{Cp{{\tau }_{0}}}{\Pr q_{0}^{*}}\left( {{T}_{P}}{{T}_{F}} \right)$ (25)
In Equation (25) the terms ${{\tau }_{0}}$ and $q_{0}^{*}$ are taken on the surface. It is known from the fluid mechanics courses that [10]:
$\pi dL{{\tau }_{0}}=\Delta p\frac{\pi {{d}^{2}}}{4}$ (26)
Clearing $\Delta p$ in Equation (26)
$\Delta p={4L{{\tau }_{0}}}/{d}\;$ (27)
The Darcy Equation for surface is:
$\Delta p=f\frac{l}{d}\frac{{{V}^{2}}}{2}\rho $ (28)
Equaling the Equations (28) and (27)
$\frac{4L{{\tau }_{0}}}{d}=f\frac{L\rho V_{M}^{2}}{2d}$ (29)
The shear stress ${{\tau }_{0}}$ on the surface is given in the left term of the Equation (29), therefore, clearing it, can be obtaining the expression that allows determining the shear stress ${{\tau }_{0}}$ on the surface [12].
${{\tau }_{0}}={f\rho V_{M}^{2}}/{8}\;$ (30)
The mean drag coefficient is taken as a quarter of the Darcy friction factor
${{C}_{W}}={f}/{4}\;$ (31)
Then, substituting the Equation (31) into Equation (30)
${{\tau }_{0}}=\frac{{{C}_{W}}\rho V_{M}^{2}}{2}$ (32)
The quantity of heat transferred is obtained with the Newton’s law of cooling [1113].
$q_{0}^{*}=\alpha \left( {{T}_{P}}{{T}_{F}} \right)$ (33)
Substituting the Equations (33) and (32) into Equation (25):
${{V}_{M}}=\frac{Cp\,f\,\rho \,V_{M}^{2}}{8\Pr \alpha \left( {{T}_{P}}{{T}_{F}} \right)}\left( {{T}_{P}}{{T}_{F}} \right)$ (34)
Equation (34) is transformed to:
${{V}_{M}}=\frac{Cp\,f\,\rho \,V_{M}^{2}}{8\Pr \alpha }$ (35)
Clearing the mean heat transfer coefficient in Equation (35):
$\alpha =\frac{Cp\,f\,\rho \,{{V}_{M}}}{8\Pr }$ (36)
Equation (36) contains all the physical properties necessary to form the Stanton dimensionless group.
$St=\frac{Nu}{\operatorname{Re}\Pr }=\frac{\alpha }{\rho CpV}$ (37)
Substituting Equation (37) into Equation (36)
$\frac{\alpha }{Cp\rho {{V}_{M}}}=St=\frac{Nu}{\operatorname{Re}\Pr }=\frac{f}{8\Pr }$ (38)
From the Equation (31), Equation (38) is transformed to:
$\frac{\alpha }{Cp\rho {{V}_{M}}}=St=\frac{Nu}{\operatorname{Re}\Pr }=\frac{{{C}_{W}}}{2\Pr }$ (39)
Solving the average drag coefficient ${{C}_{W}}$ in the Equation (39)
$\frac{\alpha}{CpρV_M}=\frac{Nu}{RePr}=St=\frac{C_W}{2Pr}$ (39.a)
Equation (39.a) is a good approximation to the model given by Pohlhausen [1213].
${{C}_{W}}=2St\Pr $ (40)
Equation (40) agrees very well with the experimental values for $\Pr \approx 1$. The friction factor is obtained with the Equation given by Eckert [1214].
$f=0,184{{\operatorname{Re}}^{0,2}}$ (41)
Equation (41) is valid for:
${{10}^{4}}<\operatorname{Re}<{{10}^{5}}\quad \quad ;\quad \quad \frac{L}{d}=0,623\sqrt[4]{\operatorname{Re}}$ (42)
In Equation (42) L is the initial section of hydrodynamic compensation (necessary distance so that in turbulent flow the Darcy's f friction factor becomes constant). Substituting the Equations (41) into Equation (40):
$\begin{align} & Nu=St\operatorname{Re}\Pr =\frac{f}{8\sqrt[3]{\Pr }}\operatorname{Re}\Pr = \\ & =\frac{0,184}{8}{{\operatorname{Re}}^{0,8}}{{\Pr }^{{1}/{3}\;}}=0,023{{\operatorname{Re}}^{0,8}}{{\Pr }^{{1}/{3}\;}} \\ \end{align}$ (43)
Equation (43) is valid for:
${{10}^{4}}<\operatorname{Re}<{{10}^{5}}\quad ;\quad \frac{L}{d}>60\quad ;\quad 0,5<\Pr <100$ (44)
Equation (43) was later modified by DittusBoelter [15], where, the exponent 1/3 from the Prandtl number was substituted by the constant n, which takes values of 0.3 and 0.4 for cooling and heating respectively. This action broadens the area of applicability of Equation (43) to [1618]:
${{10}^{4}}<\operatorname{Re}\quad ;\quad \frac{L}{d}>60\quad ;\quad 0,5<\Pr <160$ (45)
2.3 Deduction and development of the proposal model
If it is considered that ${\nu }/{a}\;=1$, then ${{\varepsilon }_{M}}={{\varepsilon }_{C}}$ and the Equation (23) is transformed to:
$\frac{\tau }{{{q}^{*}}}=\frac{{{\tau }_{0}}}{q_{0}^{*}}=\frac{{{\varepsilon }_{M}}}{{{\varepsilon }_{C}}}\frac{dV}{CpdT}=\frac{dV}{CpdT}$ (46)
In the viscous sublayer it is satisfied that $\nu >>{{\varepsilon }_{M}}$ $a>>{{\varepsilon }_{C}}$, then, transforming the Equation (46):
$\frac{\tau }{{{q}^{*}}}=\frac{{{\tau }_{0}}}{q_{0}^{*}}=\frac{\nu }{a}\frac{dV}{CpdT}=\frac{\Pr }{Cp}\frac{dV}{dT}$ (47)
Separating variables in Equation (47), assuming that the profile of temperature distribution on the turbulent boundary layer is approximately a parabolicexponential curve, we obtain:
$dT={{\Pr }^{{2}/{3}\;}}\frac{q_{0}^{*}}{Cp{{\tau }_{0}}}dV$ (48)
Integrating in the Equation (48), the left member between the wall temperatures $T_p$ and the temperature on the edge of the viscous boundary layer $T_1$, the member on the right is integrated in the interval from zero until the edge velocity in the tube wall.
$\int\limits_{{{T}_{P}}}^{{{T}_{1}}}{dT}=\int\limits_{0}^{{{V}_{1}}}{{{\Pr }^{{2}/{3}\;}}\frac{q_{0}^{*}}{Cp{{\tau }_{0}}}dV}$ (49)
Solving the integrals present in Equation (49):
${{T}_{P}}{{T}_{1}}={{\Pr }^{{2}/{3}\;}}\frac{q_{0}^{*}}{Cp{{\tau }_{0}}}{{V}_{1}}$ (50)
Separating variables in the Equation (46):
$dT=\frac{q_{0}^{*}}{Cp{{\tau }_{0}}}dV$ (51)
Integrating the Equation (51), in the left member, between the temperature on the edge of the viscous boundary layer and the average temperature of the fluid flow. The right member is integrated in the interval between the velocity on the edge of the viscous boundary layer and the mean velocity of the fluid stream, then:
$\int\limits_{{{T}_{1}}}^{{{T}_{F}}}{dT}=\int\limits_{{{V}_{1}}}^{{{V}_{M}}}{P\frac{q_{0}^{*}}{Cp{{\tau }_{0}}}dV}$ (52)
Resolving the integrals present in Equation (52)
${{T}_{1}}{{T}_{F}}=\frac{q_{0}^{*}}{Cp{{\tau }_{0}}}\left( {{V}_{M}}{{V}_{1}} \right)$ (53)
Adding the Equations (53) and (50), we obtain [19]:
$\begin{align} & {{T}_{P}}{{T}_{1}}+{{T}_{1}}{{T}_{F}}={{\Pr }^{{2}/{3}\;}}\frac{q_{0}^{*}}{Cp{{\tau }_{0}}}{{V}_{1}}+ \\ & +\frac{q_{0}^{*}}{Cp{{\tau }_{0}}}\left( {{V}_{M}}{{V}_{1}} \right) \\ \end{align}$ (54)
Grouping terms in Equation (54) is up to:
${{T}_{P}}{{T}_{F}}=\frac{q_{0}^{*}{{V}_{M}}}{Cp{{\tau }_{0}}}\left[ 1+\frac{{{V}_{1}}}{{{V}_{M}}}\left( {{\Pr }^{{2}/{3}\;}}1 \right) \right]$ (55)
Substituting Equations (30) and (33) into Equation (55)
${{T}_{P}}{{T}_{F}}=\frac{8\alpha \left( {{T}_{P}}{{T}_{F}} \right){{V}_{M}}}{Cp\,f\,\rho \,V_{M}^{2}}\left[ 1+\frac{{{V}_{1}}}{{{V}_{M}}}\left( {{\Pr }^{{2}/{3}\;}}1 \right) \right]$(56)
Clearing the mean heat transfer coefficient $\alpha $ in the Equation (56) and grouping conveniently is finally obtained [2021]:
$\alpha =\frac{f}{8}\frac{\rho Cp{{V}_{M}}}{\left[ 1+\frac{{{V}_{1}}}{{{V}_{M}}}\left( {{\Pr }^{{2}/{3}\;}}1 \right) \right]}$ (57)
Substituting Equation (37) into Equation (57)
$\frac{\alpha }{Cp\rho {{V}_{M}}}=St=\frac{Nu}{\operatorname{Re}\Pr }=\frac{f}{8\left[ 1+\frac{{{V}_{1}}}{{{V}_{M}}}\left( {{\Pr }^{{2}/{3}\;}}1 \right) \right]}$ (58)
The velocity on the edge of the viscous boundary layer V_{1 }is determined with the aid of the law of velocities distribution for turbulent flows, applying the Schlichting Equation [22]:
$\frac{{{\tau }_{0}}}{\rho }={{\left( \frac{{{V}_{1}}}{12,7} \right)}^{2}}=\frac{fV_{M}^{2}}{8}$ (59)
Clearing the velocities of the left member in Equation (59), we obtain that:
$\frac{{{V}_{1}}}{{{V}_{M}}}=12,7\sqrt{\frac{f}{8}}$ (60)
Substituting Equation (60) into Equation (58) gives the final Stanton number.
$\frac{\alpha }{Cp\rho {{V}_{M}}}=St=\frac{Nu}{\operatorname{Re}\Pr }=\frac{\frac{f}{8}}{1+12,7\sqrt{\frac{f}{8}}\left( {{\Pr }^{{2}/{3}\;}}1 \right)}$ (61)
or:
$Nu=\frac{f\operatorname{Re}\Pr }{8+\sqrt{1290,3f}\cdot \left( {{\Pr }^{{2}/{3}\;}}1 \right)}$ (62)
Equation (62) is the starting point for the development of a new model that allows to obtain the coefficient of heat transfer in single phase. This includes a smaller margin of error with respect to the existing models and with a greater range of applicability.
To consider the effect of the variation of the fluid physical properties along of the tube, the Equation (62) is affected by the factor of correction given by Petukhov [1618]:
$Nu=\frac{f\operatorname{Re}\Pr }{8+\sqrt{1290,3f}\cdot \left( {{\Pr }^{{2}/{3}\;}}1 \right)}\cdot {{\left( \frac{{{\mu }_{F}}}{{{\mu }_{P}}} \right)}^{N}}$ (63)
In Equation (63), the coefficient N take values 0.25 y 0.11 for cooling and heating of the fluid respectively.
When an initial section of hydrodynamic compensation is not available, it is necessary to include this correction, transforming Equation (63)
$\begin{align} & Nu= \\ & =\frac{f\operatorname{Re}\Pr }{8+\sqrt{1290,3f}\cdot \left( {{\Pr }^{{2}/{3}\;}}1 \right)}\cdot \left( 1+{{\left( \frac{d}{l} \right)}^{{2}/{3}\;}} \right)\cdot {{\left( \frac{{{\mu }_{F}}}{{{\mu }_{P}}} \right)}^{N}} \\ \end{align}$ (64)
The friction factor is obtained with the application of the Equation of Filonenko [1516]:
$Nu=\frac{f\operatorname{Re}\Pr }{8+\sqrt{200f}\cdot \left( \Pr 1 \right)}\cdot \left( 1+{{\left( \frac{d}{l} \right)}^{{2}/{3}\;}} \right)\cdot {{\left( \frac{{{\mu }_{F}}}{{{\mu }_{P}}} \right)}^{N}}$ (65)
Equation (65) is conveniently transformed to:
${{f}^{2}}={{\left( \log {{\left( \operatorname{Re} \right)}^{1,82}}1,64 \right)}^{1}}$ (66)
Applying logarithm properties in the Equation (66), taking one constant equal to 3.25 as a common factor:
${{f}^{2}}={{\left( 3,25\cdot \left( \log {{\left( \operatorname{Re} \right)}^{1{,82}/{3,25}\;}}{1,64}/{3,25}\; \right) \right)}^{1}}$ (67)
Then:
${{f}^{2}}={{\left( 3,25\cdot \left( \log {{\left( \operatorname{Re} \right)}^{0,56}}0,505 \right) \right)}^{1}}$ (68)
Simplifying the Equation (68).
${{f}^{2}}={{\left[ 3,25\cdot \log \left( \frac{{{\operatorname{Re}}^{0,56}}}{3,196} \right) \right]}^{1}}$ (69)
or
$f=\frac{{{\left[ \log \left( \frac{{{\operatorname{Re}}^{0,56}}}{3,196} \right) \right]}^{2}}}{10,56}\quad \to \quad B=\log \left( \frac{{{\operatorname{Re}}^{0,56}}}{3,196} \right)$ (70)
Then Equation (70) is transformed to:
$f=\frac{{{B}^{2}}}{10,56}$ (71)
Substituting Equation (71) into Equation (64)
$\begin{align} & Nu=\frac{\operatorname{Re}\Pr }{10,56{{B}^{2}}\cdot \left( 8+\sqrt{1290,3\cdot 10,56\cdot {{B}^{2}}}\cdot \left( {{\Pr }^{{2}/{3}\;}}1 \right) \right)}\cdot \\ & \cdot \left( 1+{{\left( \frac{d}{l} \right)}^{{2}/{3}\;}} \right)\cdot {{\left( \frac{{{\mu }_{F}}}{{{\mu }_{P}}} \right)}^{N}} \\ \end{align}$ (72)
or
$\begin{align} & Nu=\frac{\operatorname{Re}\Pr }{84,5{{B}^{2}}+116,74B\cdot \left( {{\Pr }^{{2}/{3}\;}}1 \right)}\cdot \\ & \cdot \left( 1+{{\left( \frac{d}{l} \right)}^{{2}/{3}\;}} \right)\cdot {{\left( \frac{{{\mu }_{F}}}{{{\mu }_{P}}} \right)}^{N}} \\ \end{align}$ (73)
Equation (73) can be written as [15]:
$\begin{align} & Nu=\frac{\operatorname{Re}\Pr }{84,5{{B}^{2}}116,74B\cdot \left( 1{{\Pr }^{{2}/{3}\;}} \right)}\cdot \\ & \cdot \left( 1+{{\left( \frac{d}{l} \right)}^{{2}/{3}\;}} \right)\cdot {{\left( \frac{{{\mu }_{F}}}{{{\mu }_{P}}} \right)}^{N}} \\ \end{align}$ (74)
In Equation (74), $A=84.5$ and $D=116.74$
Equation (74) was developed for turbulent flow in singlephase inside pipes. For the transitional zone, in this work, the authors prefer the adjustments obtained with the Gnielinsky's correction, predetermining it as a functional logarithmic of base 10.
$Re\approx(Re10^D)$ (75)
Applying the Brezhneztov's method, can be obtained the coefficient D as one polynomial curve of second order, dependent of the functional $log(D)$ .
$D=0.027\cdot {{\left[ \log \left( \operatorname{Re} \right) \right]}^{\,2}}+0,2\log \left( \operatorname{Re} \right)+2,63$ (76)
Figure 1 shows the correlation between the Equation (76) and the experimental data [17]. For transitional zone, constants A and C in Equation (74) are determined through adjustment of experimental data. This correlation is showed in the figures 2 and 3 respectively.
For the turbulent flow regime, the constant D is deleted, while the constants A and C in Equation (74) are determined through adjustment of experimental data [22]. This correlation is showed in the figures 4 and 5 respectively. Table 1 provides a detailed description of the new proposal model for transition and turbulent flow regime.
The proposal model covers a greater range of validity. To show its effectiveness, a correlation is made of the values obtained from the use of Equation (74) and the experimental data available [2223], dividing the range of applicability into seven subintervals of validity and then the average error rate is determined. The results obtained are determined by determining the percent of average error. The results obtained are summarized in Tables 4 and 5.
Figure 1. Comparison of experimental data with the Equation (76)
Figure 2. Determination of the constant A for the Equation (74) in transition zone
Figure 3. Determination of the constant C for the Equation (74) in the transition zone
In the table 4, for the validity range 2.4·$10^3$≤Re≤$10^4$ and 0.65<Pr≤4.71∙$10^4$, the proposal model correlates with an average error of 13.91%, in 80.32% of the available experimental data, then, the obtained adjustment is considered excellent, very similar to those obtained by using the Gnielinsky Equation, which should be clarified that it cannot be used for Pr>2000. It is also observed that for values of Pr<200, the average error obtained is 6.96% for 90.42% of the available data, which brings it numerically to the 5% reported by Gnielinsky.
Figure 4. Determination of the constant A for the Equation (74) in turbulent flow regime
Figure 5. Determination of the constant C for the Equation (74) in turbulent flow regime
Table 1. Description of the new proposal model for transition and turbulent flow regime
Equation (74) 
$\begin{align} & Nu=\frac{(Re10^D)Pr}{A\cdot {{B}^{2}}C\cdot B\cdot \left( 1{{\Pr }^{{2}/{3}\;}} \right)}\cdot \\ & \cdot \left( 1+{{\left( \frac{d}{l} \right)}^{{2}/{3}\;}} \right)\cdot {{\left( \frac{{{\mu }_{F}}}{{{\mu }_{P}}} \right)}^{N}} \\ \end{align}$ 
Transition zone $2.3\cdot 10^3$<Re<$1\cdot 10^4$ 

A 
75.44 
C 
104 
D 
$0.027\cdot[log(Re)]^2+0.2log(Re)+2.63$ 
Turbulent zone $1\cdot 10^4<Re$ 

A 
91.415 
C 
116.74 
D 
0 
In the Table 5 for $10^4≤Re≤8.2\cdot 10^6$ and $0.65<Pr≤4.71\cdot10^4$, the Equation (74) correlates with an average error of 13.96%, in the 80.94% of the available experimental data, so the adjustment obtained is considered to be excellent, very similar to those obtained by using the Equations of Petukhov and Gnielinsky, which should be clarified that it cannot be used for Pr> 2000. It is also observed that for values of Pr <200, the average error obtained is 7.12 % for 88.35% of the available data, which brings it numerically to the 5% reported by Petukhov and Gnielinsky.
Table 2 provides a detailed summary of the range that shows a satisfactory fit with the correlation proposed in the present work.
Table 2. Summary of the validity range for the Equation (74)
Parameter 
Range 
Fluids 
Water, Air, Helium, Hydrogen, Nitrogen, Carbon Dioxide, Transformer oil, Glycerin, MC Oil, MK Oil, Butyl alcohol, Methanol, Ethanol, Ethylene glycol, Kerosene, Acetic Acid, Acetaldehyde, Butanol, Aniline, Carbon Disulfide, Ciclohexane, Ethyl ether, Ethylamine, Oil olive, Toluene, Turpentine, Propylene, Pentane, Benzene, Gasoline, Isobutene, Engine oil. Decane and Dodecane 
$Pr$ 
0.65 to $4.71\cdot 10^4$ 
$Re$ 
$2.4\cdot 10^3$ to $8.2\cdot 10^6$ 
$\mu_F/\mu_p$ 
$0.006≤\mu_F/\mu_p≤177$ 
$l/d$ 
$2≤l/d≤420$ 
In this work, the experimental data used in the validation of the developed model were extracted of the critical review available in the reference [22], which provides one large data base of experimental data compiled on heat transfer calculation during fluid flow in singlephase inside tubes. Table 3 provides the available experimental data used in this paper.
Table 3. Experimental data used in the validation of the Equation (74)
Source 
Number of data 
Fluid 
l/d 
$Re\cdot10^3$ 
Pr 
$\mu_F/\mu_p$ 
Deviation n percent 
I’lin (1951) 
188 
Air 
41 162 
7 6600 
0.68 0.7 
0.65 1.65 
5.3 3.5 
Volkov (1966) 
218 
Air 
48 370 
12.5 3700 
0.68 0.7 
0.65 1.65 
6,2 1,5 
Petukhov (1963) 
140 
Air 
39 100 
15 5800 
0.68 0.7 
0.65 1.65 
4,4 2,1 
Sukomiel (1962) 
44 
Helium 
20 50 
9 40 
0.71 0.72 
0.22 4.5 
7,1 2,3 
67 
Isobutene (2Methylpropane) 
2 60 
3200 7200 
0,73 0,75 
0.68 1.46 
9,7 6,4 

148 
Water 
6 64 
12 540 
0.9 9.4 
0.19 0.77 
8,2 7,9 

Eckert (1964) 
93 
Turpentine 
10 90 
13 110 
14.3 29.8 
0.41 2.43 
11,6 14,7 
Sabersky (1963) 
33 
Water 
48 61 
120 160 
1.2 5.9 
0.24 0.86 
10,2 1,1 
52 
Pentane 
46 88 
150 620 
4.5 7.1 
0.47 2.08 
13,1 9,6 

Yakolev (1960) 
39 
Water 
70 90 
19 140 
2 12 
0.21 1.15 
12,6 3,9 
Sabersky (1965) 
62 
Water 
60 180 
35 120 
1 9.44 
0.13 7.15 
13,1 9,9 
StermanPetukhov (1970) 
41 
Transformer oil 
89 125 
3.4 13.8 
34.9 1530 
0.01 115.2 
12,7 10,3 
29 
Glycerin 
89 125 
2.5 9.1 
1630 22650 
0.018 55,4 
9,2 5,4 

49 
MC Oil 
66 165 
5 10.4 
120 9800 
0,007 133,3 
14,8 17,1 

27 
MK Oil 
80 145 
5.4 8.7 
590 39000 
0.011 88.7 
15.8 12.6 

Kreith (1947) 
20 
Butyl Alcohol 
38 
42 78 
23 30 
0.08 0.45 
15,3 12,2 
Ykolev et al. (1965) 
50 
Benzene 
60 110 
2.6 21.1 
3.2 5 
0.31 3.17 
8,8 4,8 
Humbble (1993) 
113 
Gasoline 
60 190 
70 6900 
5.5 15.1 
0.22 4.4 
10,4 6,1 
181 
Hydrogen 
43 67 
12 8200 
0.65 0.73 
0.48 3.28 
2.4 8.8 

Kirilov (1967) 
125 
Nitrogen 
100 138 
6 8100 
0.68 0.75 
0.15 6.5 
9,8 1,9 
Efimok (1969) 
19 
Carbon Dioxide 
77 206 
14 660 
0.66 0.81 
0.3 3.3 
7,4 0,7 
YanLin (1999) 
91 
Water 
2 420 
4 250 
0.94 11 
0.19 0.96 
9,9 11.5 
Tarashmova (2001) 
23 
Water 
20 450 
400 2500 
0.94 11 
0.19 0.96 
13,6 8,9 
Karkalala (2012) 
44 
Water 
18 51 
1200 2800 
1.2 5.9 
0.24 0.96 
5.3 4.5 
Jung et al. (2008) 
71 
Transformer oil 
19 150 
2.8 8.1 
34.9 4800 
1.2 28 
16,2 7,5 
Carpenter (1957) 
66 
Methanol 
45 120 
2.9 1112.1 
2.2 7.7 
0.1 9.9 
4,4 2,1 
Vasserman (1962) 
112 
Kerosene 
30 280 
6.4 52.8 
1.35 2.9 
0.38 2.6 
7,1 2,3 
47 
Acetic acid 
55 135 
3.1 987.8 
8.5 14.2 
0.8 1.2 
4,7 3,7 

38 
Acetaldehyde 
65 120 
3.9 52.4 
2.85 4.4 
0.4 2.1 
8,2 7,9 

141 
Butanol 
40 160 
5.4 822.6 
22.5 3860 
0.04 24.6 
11,6 16,7 

187 
Aniline 
50 280 
4.4 1024.2 
11.5 111 
0.08 12.35 
9,7 3.5 

37 
Carbon Disulfide 
48 125 
13.8 76.9 
2.3 3.2 
0.59 1.68 
10,2 1,1 

23 
Ciclohexane 
85 220 
36.1 89.4 
11 19.9 
0.5 1.9 
2.3 1.7 

Sherwood (1967) 
113 
Ethanol 
80 125 
21.4 1513.8 
6.9 68.4 
0.049 20.5 
5.2 7.4 
71 
Ethyl ether 
70 135 
580 2560 
3.5 7.3 
0.3 3.6 
4.2 8.1 

17 
Ethylamine 
80 100 
12.1 17.8 
5.1 8.3 
0.55 1.8 
3.2 6.1 

21 
Propylene 
60 120 
125 284 
2.8 3.2 
0.27 3.66 
9.1 4.8 

36 
Dodecane 
70 150 
72 96 
10.7 28.2 
0.4 3.3 
11.1 12.4 

40 
Decane 
65 135 
16 47.2 
6.8 17.1 
0.25 4.1 
2.3 7.8 

53 
Ethylene glycol 
90 165 
6.3 12.1 
69 510 
0.12 8.1 
7.1 9.3 

Gordon (1937) 
11 
Oil olive 
85 120 
2.7 7.6 
700 810 
0.3 2.9 
9.1 11.4 
Gordon (1939) 
13 
Toluene 
70 150 
3.9 27.2 
4.7 21.1 
0.1 7.8 
11.6 9.4 
GMC (2012) 
103 
Engine Oil 
30 180 
2.4 7.2 
84 47100 
0.006 177 
14.1 19.4 
For all sources above 
3096 

2.0 450 
2.4 8200 
0.65 47100 
0.006 177 
16.2 19.4 
In Tables 4 and 5 can be appreciated that the Equation (74), is as accurate as the Equations of Petukhov and Gnielisky, allowing a wider range of application, while the results obtained are very similar. In the acknowledged literature was not found antecedent of a similar model with a wide range of validity. Therefore, the proposed model constitutes one contribution to the state of the art, on heat transfer calculation during fluid flow in singlephase inside pipes.
Table 4. Correlation adjustments with the experimental data for the first range of values available for Equation (74)
$2.4\cdot 10^3≤Re≤10^4$ 

$0.006≤\frac{\mu_F}{\mu_p}≤12.42$ 
$0.65<Pr≤10^2$ 
error<6.18% 91.32% data 
$0.006≤\frac{\mu_F}{\mu_p}≤18.35$ 
$0.65<Pr≤2\cdot10^2$ 
error<6.96% 90.42% data 
$0.006≤\frac{\mu_F}{\mu_p}≤22.2$ 
$0.65<Pr≤2\cdot10^3$ 
error<8.74% 89.14% data 
$0.006≤\frac{\mu_F}{\mu_p}≤34.16$ 
$0.65<Pr≤8.1\cdot10^3$ 
error<9.96% 88.05% data 
$0.006≤\frac{\mu_F}{\mu_p}≤62.2$ 
$0.65<Pr≤1.2\cdot10^4$ 
error<10.74% 86.42% data 
$0.006≤\frac{\mu_F}{\mu_p}≤105$ 
$0.65<Pr≤2.24\cdot10^4$ 
error<12.18% 83.18% data 
$0.006≤\frac{\mu_F}{\mu_p}≤177$ 
$0.65<Pr≤4.71\cdot10^4$ 
error<13.91% 80.32% data 
Table 5. Correlations with experimental data for the second range of values available for Equation (74)
$10^4≤Re≤8.2\cdot 10^6$ 

$0.006≤\frac{\mu_F}{\mu_p}≤12.42$ 
$0.65<Pr≤10^2$ 
error<6.24% 89.36% data 
$0.006≤\frac{\mu_F}{\mu_p}≤18.35$ 
$0.65<Pr≤2\cdot10^2$ 
error<7.12% 88.35% data 
$0.006≤\frac{\mu_F}{\mu_p}≤22.2$ 
$0.65<Pr≤2\cdot10^3$ 
error<8.31% 87.12% data 
$0.006≤\frac{\mu_F}{\mu_p}≤34.16$ 
$0.65<Pr≤8.1\cdot10^3$ 
error<10.17% 86.31% data 
$0.006≤\frac{\mu_F}{\mu_p}≤62.2$ 
$0.65<Pr≤1.2\cdot10^4$ 
error<11.23% 84.02% data 
$0.006≤\frac{\mu_F}{\mu_p}≤105$ 
$0.65<Pr≤2.24\cdot10^4$ 
error<13.37% 82.72% data 
$0.006≤\frac{\mu_F}{\mu_p}≤177$ 
$0.65<Pr≤4.71\cdot10^4$ 
error<13.96% 80.94% data 
Figure 6 shows the correlation between the proposed model and the experimental data reported by various authors.
Figure 6. Application of the model to experimental data reported by several authors
A new model has been obtained for the determination of the heat transfer coefficient in transition and turbulent regime, on a fluid flow in single phase inside straight tubes. The model obtained has a greater range of applicability, covering almost twice the permissible values for the models that were taken as reference, establishing its domain in a range not covered by any model established and known in the literature, for this reason recommends its use in the calculation of the mean coefficients of heat transfer by convection for straight tubes with turbulent flow and transition.
For validity range 2.4∙$10^3$≤Re<$10^4$ and 0.65<Pr≤4.71∙$10^4$, the proposal model correlates with an average error of 13.91%, in 80.32% of the available experimental data. For 2.4∙$10^3$≤Re<$10^4$ and Pr<200, the average error obtained is 6.96% for 90.42% of the available data. For validity range $10^4$≤Re≤8.2∙$10^6$ and 0.65<Pr≤4.71∙$10^4$, the proposal model correlates with an average error of 13.96%, in 80.94% of the available experimental data. For $10^4$≤Re≤8.2∙$10^6$ and Pr<200, the mean error obtained is 7.12 % for 88.35% of the available data.
The Doctoral program of the Universidad Central “Marta Abreu” de Las Villas, Santa Clara, Cuba, is gratefully acknowledged.
a Thermal diffusivity, m^{2}∙s^{1}
A Constant, defined in Equation (74)
B Constant, defined in Equation (74)
C Constant, defined in Equation (74)
$C_p$ Fluid specific heat, J∙kg^{1}∙K^{1}
$C_W$ Drag coefficient
d Equivalent inner tube diameter, m
D Constant, defined in Equation (74)
f Darcy friction factor
l Length of the tube, m
L Initial section of hydrodynamic compensation, m
$L_C$ Mixture length of the energy in the thickness $δ_3$, m
$L_M$ Mixture length of the thickness $δ_2$, m
N Exponent of the Petukhov correction in Equation (63)
$Nu$ Nusselt number
$Pr$ Prandtl number for singlephase
q* Total heat flux, kg∙m^{2}∙s^{3}
$q_0*$ Heat flux on the boundary layer surface, kg∙m^{2}∙s^{3}
$q_{cond}$ Conductive component of the total heat flux, kg∙m^{2}∙s^{3}
$q_{turb}$ Turbulent component of the total heat flux, kg∙m^{2}∙s^{3}
Re Reynolds number for singlephase
_{St }Stanton number
$T_F$ Mean fluid temperature, ^{°}C
$T_I$ Instantaneous temperature used in Equation (5), ^{°}C
$T_F*$ Temperature fluctuation used in Equation (5), ^{°}C
$T_P$_{ }Wall temperature, ^{°}C
$V_1$ Velocity at the edge of the viscous layer, m∙s^{1}
$V_M$ Mean fluid velocity, m∙s^{1}
$V_X*$ Fluctuation of the $V_X^M$, m∙s^{1}
$V_Y*$ Fluctuation of the $V_Y^M$, m∙s^{1}
$V_X^M$ Instantaneous velocity in the coordinate axis x used in Equation (2), m∙s^{1}
$V_Y^M$ Instantaneous velocity in the coordinate axis y used in Equation (3), m∙s^{1}
Greek symbols
$α$ Heat transfer coefficient in singlephase, kg∙m^{1}∙K^{1}∙s^{1}
$ε_C$ Heat turbulent diffusivity, m^{2}∙s^{1}
$ε_M$ Momentum turbulent diffusivity, m^{2}∙s^{1}
$µ_F$ Fluid dynamic viscosity at T_{F}, kg∙m^{1}∙s^{1}
$µ_P$ Fluid dynamic viscosity at T_{P}, kg∙m^{1}∙s^{1}
ρ Fluid density, kg∙m^{3}
λ Fluid thermal conductivity, W∙m^{1}∙K^{1}
v Liquid kinematic viscosity, m^{2}∙s^{1}
$δ_2$ Film thickness of the momentum in boundary layer, m
$δ_3$ Film thickness of the thermal boundary layer, m
τ Shear stress in the turbulent boundary layer, kg∙m∙s^{2}
$τ_0$ Shear stress on the surface of the turbulent boundary layer, kg∙m∙s^{2}
$τ_{Visc}$ Stress of the viscous forces, kg∙m∙s^{2}
$τ_{Turb}$ Stress of the turbulent strain, kg∙m∙s^{2}
ΔP Pressure drop, m
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