International Journal of Heat and Technology

Vol. 37, No. 1, March, 2019, pp. 257-266

Journal homepage: http://iieta.org/Journals/IJHT

Suggested Model for Heat Transfer Calculation During Fluid Flow in Single Phase Inside Pipes (II)

Yanán Camaraza-Medina1*, Ken Mortensen-Carlson2, Pratijay Guha3, Ángel M. Rubio-Gonzales1, Oscar M. Cruz-Fonticiella1,
Osvaldo F. García-Morales4
1
Center of Energy Studies and Environmental Technology, Universidad Central “Marta Abreu” de Las Villas 54440, Cuba
2
Department of Chemical Engineering, University of California, Santa Bárbara CA 93106, USA
3
Department of Mechanical Engineering, Birla Institute of Technology and Sciences, Pilani Hyderabad 333031, India
4
Technical Sciences Faculty, Universidad de Matanzas, Matanzas 44440, Cuba

Corresponding Author Email: ycamaraza1980@yahoo.com

https://doi.org/10.18280/ijht.370131 ABSTRACT

Received: 11 December 2018 In this paper, is presented a mathematical deduction of a new improved model for heat transfer
Accepted: 5 March 2019 calculations during fluid flow in single-phase inside tubes. The proposal model was verified
by comparison with available experimental data of 35 different fluids, including water, air,
Keywords: gases and organic substances. The proposal model is valid for a range of Reynolds number for
single phase, model, heat transfer single-phase from 2.4 ∙ 103 to 8.2 ∙ 106 , Prandtl number for single-phase from 0.65 to 4.71 ∙
coefficient , average deviation 104 , dimensionless length in the interval 2 ≤ 𝑙 ⁄𝑑 ≤ 450 and values of Petukhov’s correction
in the interval 0.006 ≤ 𝜇𝐹 ⁄𝜇𝑃 ≤ 177. In 3096 data analyzed, for 𝑅𝑒 < 1 ∙ 104 , the mean
deviation found was 13.91% in the 80.32% of the experimental data, while for 1 ∙ 104 ≤
𝑅𝑒, the mean deviation found was 13.96% in 80.94% of experimental data.

1. INTRODUCTION necessary to resort to the experimentation and subsequent

adjustment of experimental quantities through the theory of
Currently, heat transfer calculations for turbulent fluid flow dimensional analysis.
within straight conduits in single-phase media are made by the
Dittus-Boelter equation, or by the improved version of Sieder-
Tate [1]. This procedure is a requirement for the evaluation of 2. METHODS AND VALIDATION
industrial facilities or production equipment. A drawback of
these equations is their high dispersion value, reaching 2.1 Analogy between heat transfer and momentum in
compute errors close to ±40 %. single-phase fluid flow inside pipes
At the Moscow Energy Institute, Petukhov and his
collaborators constructed a model based on experimental The Darcy friction factor 𝑓 allows determining the heat
quantity adjustments, using the Prandtl analogy as an transfer coefficient 𝛼, by analogy between heat transfer and
adjustment function [2]. This equation gives results with a momentum. The shear stress 𝜏 in the turbulent boundary layer
lower margin of error, and allows us to estimate the mean error is composed of two terms [5]:
by the dimensionless number of Prandtl.
dV
𝑖𝑓 𝑃𝑟 ≤ 200 𝐸𝑟𝑟𝑜𝑟 < 5%  =  Visc +  Turb =  − VX*VY* (1)
dx
𝑖𝑓 𝑃𝑟 ≥ 200 𝐸𝑟𝑟𝑜𝑟 ≤ 10%
In Equation (1) 𝜏 𝑇𝑢𝑟𝑏 is the Reynolds stress; 𝑉𝑋∗ is the
Although the application of the Petukhov’s Equation is fluctuation of the instantaneous velocity 𝑉𝑋𝑀 in the coordinate
more laborious, the results obtained have a minor dispersion, axis x; 𝑉𝑌∗ is the fluctuation of the instantaneous velocity 𝑉𝑌𝑀 in
therefore, a smaller safety margin in the design calculations. the coordinate axis y.
A major drawback is its applicability range, because this is
The instantaneous velocity 𝑉𝑋𝑀 and 𝑉𝑌𝑀 are determined as:
only valid for a fully developed turbulent flow regime, 1 ∙
104 < 𝑅𝑒, and is not valid for the flow that operate in the
transition zone 2.3 ∙ 103 < 𝑅𝑒 < 1 ∙ 104 . This problem was VXM = VMX  VX* = VMX  Vagit
X
(2)
later solved by Gnielinsky [3-4], who modified the Petukhov’s
Equation, adjusting it to experimental data that do take into VYM = VMY  VY* = VMY  Vagit
Y
(3)
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 For turbulent heat flow, it can be considered that the total
tubes with turbulent flow, this is mainly associated with the heat flow q∗ is composed of a sum that includes the conductive
changing nature of the turbulent flow, which hinders the component qcond and the turbulent component qturb , then:
development of analytical expressions. This element makes it

257
 dT    dV
= ( +  M )
q * = qcond + qturb = − + CpVY*TF* dV
(4) =  Visc +  Turb =  +  M  (14)
dx    dx dx
In Equation (3) there are three temperature references, Substituting Equation (12) into Equation (4):
which are:
dT dT
𝑇𝐼 = 𝑇𝐹 ± 𝑇𝐹∗ 𝑖𝑠 𝑡ℎ𝑒 𝑖𝑛𝑠𝑡𝑎𝑛𝑡𝑎𝑛𝑒𝑜𝑢𝑠 𝑡𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒 q* = qcond + qturb = − − Cp C (15)
𝑇𝐹 = 𝑇∞ 𝑖𝑠 𝑡ℎ𝑒 𝑚𝑒𝑎𝑛 𝑡𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑓𝑙𝑢𝑖𝑑 (5) dx dx
𝑇𝐹∗ 𝑖𝑠 𝑡ℎ𝑒 𝑡𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒 𝑑𝑢𝑒 𝑡𝑜 𝑡ℎ𝑒 𝑓𝑙𝑢𝑐𝑡𝑢𝑎𝑡𝑖𝑜𝑛
If in Equation (15) the derivative 𝑑𝑇⁄𝑑𝑥 is taken as a
Terms 𝑉𝑋∗ and 𝑉𝑌∗ are obtained from their physical meaning common factor, then
from the Prandtl mixing number, which suggests that the
fluctuation of velocity 𝑉𝑋∗ is related with dV dx as: q * = qcond + qturb = −( + Cp  C ) dT dx (16)

VX*  LM dV dx (6) In Equation (16), both members are divided by the product
of the density and specific heat at constant pressure 𝜌𝐶𝑝.
In Equation (6), 𝐿𝑀 is the mixture length of the thickness
film 𝛿2 of the momentum in boundary layer. Similarly, q*    dT
= −(a +  C )
dT
transverse fluctuation 𝑉𝑌∗ is admitted to be of the same order of = − +  C  (17)
 Cp   Cp  dx dx
magnitude 𝑉𝑋∗ but opposite in sign, [6]:

VY*  − LM dV dx (6.a) Dividing Equation (17) by the Equation (14) is obtained the
basic relationships for the fluid flow inside of tubes [6-9]:
Combining the Equations (6) and (6.a):
  +  M dV
=− (18)
V V  −(LM dV dx )
* *
X Y
2
(7) q *
Cp(a +  C ) dT

Equation (7) can be transformed to: In Equation (18), the kinematic viscosity 𝜈 and the thermal
diffusivity 𝑎 are properties of the fluid, while 𝜀𝐶 and 𝜀𝑀 are
VX*VY*   M dV dx (8) properties of the flow.

2.2 Development of one linear model for convective heat

In Equation (8) 𝜀𝑀 is the momentum turbulent diffusivity, transfer calculation in single-phase inside pipes
then:
Development of the new model for to calculate the
𝜀𝑀 ≈ 𝐿2𝑀 𝑑𝑉 ⁄𝑑𝑥 (9) convective heat transfer in single-phase inside pipes is a
complex task. Initially is taken the criterion established by
To find the relationship of the term 𝑉𝑌∗ 𝑇𝐹∗ , with the mean Prandtl, which considers that the flow is divided into two
local temperature gradient, a similar method is applied, in the zones, a viscous zone and a turbulent zone. In his analysis
form [6]. Prandtl makes the additional assumptions that in the turbulent
zone the molecular diffusivities of momentum 𝜈 and of heat 𝑎,
𝑇𝐹∗ ≈ −𝐿𝐶 𝑑𝑇 ⁄𝑑𝑥 (10) are negligible in comparison with the turbulent diffusivities,
𝜈 ≫ 𝜀𝑀 and 𝑎 ≫ 𝜀𝐶 , so they do not intervene in the process.
VY* = LC dV dx (11) It would be very useful for such a purpose to assume that the
relationship between molecular diffusivities 𝑎and𝜈 is equal to
In the expression (11) 𝐿𝐶 is the mixture length of the energy the relationship between diffusivities 𝜀𝐶 and 𝜀𝑀 .
in the thickness 𝛿3 of the thermal boundary layer, then: Since the dimensionless Prandtl number is a relation
between diffusivities, then the previous assumption is fulfilled
dV dT dT [10].
VY*TF* = − L2C = − C (12)
dx dx dx
Pr =  a =  M  C (19)
In the Equation (12), the term 𝜀𝐶 = −𝐿2𝐶𝑑𝑉 ⁄𝑑𝑥 is the heat
turbulent diffusivity. Substituting Equation (8) into Equation Clearing 𝜈 and 𝜀𝑀 in the Equation (19):
(1):
𝜈 = 𝑎𝑃𝑟 (20)
 =  Visc +  Turb =  dV dx −  M dV dx (13)
𝜀𝑀 = 𝜀𝐶 𝑃𝑟 (21)
Dividing by the density 𝜌 to both members of the Equation
Substituting Equations (20) and (21) into Equation (18):
(13) and taken the derivative 𝑑𝑉 ⁄𝑑𝑥 as a common factor [7-
8], then:
 a Pr +  C Pr dV (22)
=−
q* Cp(a +  C ) dT

258
 If the Prandtl number Pr is taken as a common factor in Cp f  VM2
VM = (TP − TF ) (34)
Equation (22), it is reduced to [11]: 8 Pr  (TP − TF )


=
0
=−
(a +  C ) Pr dV =−
Pr dV (23) Equation (34) is transformed to:
q *
q*
0 (a +  C )Cp dT Cp dT
Cp f  VM2 (35)
Separating variables in (23) and integrating VM =
8 Pr 
Cp 0 F
VM T
(24) Clearing the mean heat transfer coefficient  in Equation
 dV = −
0
Pr q0* TP
dT
(35):

Resolving the integrals present in Equation (24) and Cp f  VM

= (36)
grouping conveniently: 8 Pr
Cp 0
VM = (TP − TF ) (25) Equation (36) contains all the physical properties necessary
Pr q0* to form the Stanton dimensionless group.

In Equation (25) the terms 𝜏0 and 𝑞0∗ are taken on the Nu  (37)
St = =
surface. It is known from the fluid mechanics courses that [10]: Re Pr  CpVM

dL 0 = p d 2 4 (26) Substituting Equation (37) into Equation (36)

Clearing Δ𝑃 in Equation (26)  Nu f (38)

= = St =
Cp VM Re Pr 8 Pr
Δ𝑃 = 4𝐿 𝜏0 ⁄𝑑 (27)
From the Equation (31), Equation (38) is transformed to:
The Darcy Equation for surface is:
 Nu C (39)
= = St = W
f L VM2  (28) Cp VM Re Pr 2 Pr
P =
2d
Solving the average drag coefficient 𝐶𝑊 in the Equation (39)
Equaling the Equations (28) and (27)
2 CW = 2St Pr (39.a)
𝐿𝑉𝑀 𝜌
4𝐿 𝜏0 ⁄𝑑 = 𝑓 (29)
2𝑑
Equation (39.a) is a good approximation to the model given
The shear stress 𝜏0 on the surface is given in the left term of by Pohlhausen [12-13].
the Equation (29), therefore, clearing it, can be obtaining the
expression that allows determining the shear stress 𝜏0 on the CW = 2St Pr 2 3 (40)
surface [12].
Equation (40) agrees very well with the experimental values
f V2 (30) for Pr  1 . The friction factor is obtained with the Equation
0 = M
2 given by Eckert [12-14].

The mean drag coefficient is taken as a quarter of the Darcy f = 0.184 Re−0.2 (41)
friction factor
Equation (41) is valid for:
CW = f 4 (31)
10 4  Re  10 5 ; L d = 0.623 4 Re (42)
Then, substituting the Equation (31) into Equation (30)
In Equation (42) L is the initial section of hydrodynamic
C V 2 (32) compensation (necessary distance so that in turbulent flow the
0 = W M
2 Darcy's friction factor 𝑓 becomes constant). Substituting the
Equations (41) into Equation (40):
The quantity of heat transferred is obtained with the
Newton’s law of cooling [11-13]. f
Nu = St Re Pr = Re Pr 2 3 =
8  Pr
3
q0* =  (TP − TF ) (33) (43)
0,184 0.8 1 3
= Re Pr = 0,023 Re 0.8 Pr1 3
Substituting the Equations (33) and (32) into Equation (25): 8

259
 Equation (43) is valid for: the viscous boundary layer and the mean velocity of the fluid
stream, then:
L (44)
104  Re  105 ;  60 ; 0.5  Pr  100 TF VM
d q0* (52)
 dT =
T1

V1
−P
Cp 0
dV
Equation (43) was later modified by Dittus-Boelter [15],
where, the exponent 1/3 from the Prandtl number was
Resolving the integrals present in Equation (52)
substituted by the constant n, which takes values of 0.3 and 0.4
for cooling and heating respectively. This action broadens the
q0*
area of applicability of Equation (43) to [16-18]: T1 − TF = (VM − V1 ) (53)
Cp 0
L
104  Re ;  60 ; 0.5  Pr  160 (45)
d Adding the Equations (53) and (50), we obtain [19]:

2.3 Deduction and development of the proposal model q0*

TP − T1 + T1 − TF = − Pr 2 3 V1 + (54)
Cp 0
If it is considered that 𝜈 ⁄𝑎 = 1 , then 𝜀𝑀 = 𝜀𝐶 and the
Equation (23) is transformed to: + (q0* Cp 0 ) (VM − V1 )

 0  dV dV Grouping terms in Equation (54) is up to:

= =− M =− (46)
q* q0*  C CpdT CpdT
q0*VM  
1 + 1 (Pr 2 3 − 1)
V (55)
TP − TF = 
In the viscous sublayer it is satisfied that 𝜈 ≫ 𝜀𝑀 and 𝑎 ≫ Cp 0  VM 
𝜀𝐶 , then, transforming the Equation (46):
Substituting Equations (30) and (33) into Equation (55)
 0  dV Pr dV
= =− =− (47)
q* q0* 8 (TP − TF )VM  V1 
a CpdT Cp dT
TP − TF =
Cp f  VM2
1 + ( )
Pr 2 3 − 1  (56)
 V M 
Separating variables in Equation (47), assuming that the
profile of temperature distribution on the turbulent boundary
Clearing the mean heat transfer coefficient 𝛼 in the
layer is approximately a parabolic-exponential curve, we
Equation (56) and grouping conveniently is finally obtained
obtain:
[20-21]:

q0* f  CpVM (57)

dT = − Pr 2 3 dV (48) =
Cp 0 8  V1 
1 + (
Pr 2 3 − 1  )
 VM 
Integrating in the Equation (48), the left member between
the wall temperatures 𝑇𝑃 and the temperature on the edge of
Substituting Equation (37) into Equation (57)
the viscous boundary layer 𝑇1 , the member on the right is
integrated in the interval from zero until the edge velocity in
 Nu f (58)
the tube wall. = St = =
CpVM  
Re Pr V
( )
8 1 + 1 Pr 2 3 − 1 
T1 V1
q *
(49)  V 
T dT = 0 − Pr Cp 0 dV
M
23 0

P
The velocity on the edge of the viscous boundary layer V1 is
determined with the aid of the law of velocities distribution for
Solving the integrals present in Equation (49): turbulent flows, applying the Schlichting Equation [22]:

q0*  0  V1 
2
f VM2
TP − T1 = − Pr 2 3 V1 (50) =  = (59)
Cp 0   12.7  8

Separating variables in the Equation (46): Clearing the velocities of the left member in Equation (59),
we obtain that:
q0* (51)
dT = − dV
Cp 0 V1
= 12.7
f (60)
VM 8
Integrating the Equation (51), in the left member, between
the temperature on the edge of the viscous boundary layer and Substituting Equation (60) into Equation (58) gives the final
the average temperature of the fluid flow. The right member is Stanton number.
integrated in the interval between the velocity on the edge of

260
 f Then Equation (70) is transformed to:
 Nu 8 (61)
= St = =
CpVM B −2
Re Pr
1 + 12.7
f
8
(
Pr 2 3 − 1 ) f = (71)
10.563
or:
Substituting Equation (71) into Equation (64)
f Re Pr (62)
Nu =
8 + 1290,3 f  (Pr 2 3 − 1) Re Pr
Nu =
(
10.563B  8 + 1290.3  10.563  B 2  (Pr 2 3 − 1)
2
) (72)
Equation (62) is the starting point for the development of a
 d    F
N

23
new model that allows to obtain the coefficient of heat transfer  1 +    
 l   P
in single phase. This includes a smaller margin of error with    
respect to the existing models and with a greater range of
applicability. or
To consider the effect of the variation of the fluid physical
properties along of the tube, the Equation (62) is affected by Re Pr
Nu = 
the factor of correction given by Petukhov [16-18]: 84.5 B + 116.74 B  Pr 2 3 − 1
2
( ) (73)
 d    F
N

N 23
f Re Pr   (63)  1 +    
Nu =   F   l    P
(
8 + 1290.3 f  Pr 2 3 − 1 )  P    

In Equation (63), the coefficient N take values 0.25 and Equation (73) can be written as [15]:
0.11 for cooling and heating of the fluid respectively.
When an initial section of hydrodynamic compensation is Re Pr
Nu = 
not available, it is necessary to include this correction, A  B 2 − C  B  (1 − Pr 2 3 ) (74)
transforming Equation (63)
 d    F
N

23

 1 +    
 l   P
𝑁𝑢 =    
𝑓𝑅𝑒𝑃𝑟
= 2 (64)
2 𝑁
𝑑 3 𝜇
8+√1290.3𝑓⋅(𝑃𝑟 3 −1)(1+( ) )( 𝐹 )
𝑙 𝜇𝑃
In Equation (74), 𝐴 = 84.5 and 𝐶 = 116.74

The friction factor is obtained with the application of the 3. EXPERIMENTAL VALIDATION OF THE PROPOSED
Equation of Filonenko [15-16]: MODEL

f = (1.82 log (Re ) − 1.64 ) Equation (74) was developed for turbulent flow in single-
−2
(65)
phase inside pipes. For the transitional zone, in this work, the
Equation (65) is conveniently transformed to: authors prefer the adjustments obtained with the Gnielinsky's
correction, predetermining it as a functional logarithmic of
𝑓 2 = [𝑙𝑜𝑔(𝑅𝑒)1.82 − 1.64]−1 (66) base 10.

Applying logarithm properties in the Equation (66), taking Re  (Re− 10 D ) (75)

one constant equal to 3.25 as a common factor:
Applying the Brezhneztov’s method, can be obtained the
2
( (
f = 3.25  log(Re )
1.82 3.25
− 1.64 3.25 )) −1
(67) coefficient D as one polynomial curve of second order,
dependent of the functional 𝑙𝑜𝑔(𝐷).
Then:
D = −0.027  log(Re ) + 0.2 log(Re ) + 2.63
2
(76)
( (
f 2 = 3.25  log (Re )
0.56
− 0.505 ))−1
(68)
Figure 1 shows the correlation between the Equation (76)
and the experimental data [17]. For transitional zone, constants
Simplifying the Equation (68).
A and C in Equation (74) are determined through adjustment
−1 of experimental data. This correlation is showed in the figures
𝑅𝑒 0.56
𝑓 2 = [3.25 ⋅ 𝑙𝑜𝑔 ( )] (69) 2 and 3 respectively.
3.196
For the turbulent flow regime, the constant D is deleted,
or while the constants A and C in Equation (74) are determined
through adjustment of experimental data [22]. This correlation
−2 is showed in the figures 4 and 5 respectively. Table 1 provides
  Re 0.56  a detailed description of the new proposal model for transition
log  
  3.196   Re 0.56  (70) and turbulent flow regime.
f = → B = log 
10.563  3.196  The proposal model covers a greater range of validity. To
show its effectiveness, a correlation is made of the values

261
obtained from the use of Equation (74) and the experimental experimental data, then, the obtained adjustment is considered
data available [22-23], dividing the range of applicability into excellent, very similar to those obtained by using the
seven subintervals of validity and then the average error rate Gnielinsky Equation, which should be clarified that it cannot
is determined. The results obtained are determined by be used for Pr> 2000. It is also observed that for values of Pr
determining the percent of average error. The results obtained <200, the average error obtained is 6.96% for 90.42% of the
are summarized in Tables 4 and 5. available data, which brings it numerically to the 5% reported
by Gnielinsky.

Figure 1. Comparison of experimental data with the Figure 4. Determination of the constant A for the Equation
Equation (76) (74) in turbulent flow regime

Figure 2. Determination of the constant A for the Equation

Figure 5. Determination of the constant C for the Equation
(74) in transition zone
(74) in turbulent flow regime

Table 1. Description of the new proposal model for transition

and turbulent flow regime

Nu =
(Re− 10 ) Pr
D

A  B − C  B  (1 − Pr )
2 23

Equation (74)
 d    F 
23 N

 1 +    
 l     P 
 
Transition zone 2.3 ∙ 103 < 𝑅𝑒 < 1 ∙ 104
A 75.44
C 104
− 0.027  log (Re ) + 0.2 log (Re ) + 2.63
2
D
Turbulent zone 1 ∙ 104 < 𝑅𝑒
A 91.415
Figure 3. Determination of the constant C for the Equation C 116.74
(74) in the transition zone D 0

In the table 4, for the validity range 2.4 ∙ 103 ≤ 𝑅𝑒 < 104 In the Table 5 for 104 ≤ 𝑅𝑒 ≤ 8.2 ∙ 106 and 0.65 < Pr ≤
and 0.65 < Pr ≤ 4.71 ∙ 104 , the proposal model correlates 4.71 ∙ 104 , the Equation (74) correlates with an average error
with an average error of 13.91%, in 80.32% of the available of 13.96%, in the 80.94% of the available experimental data,

262
so the adjustment obtained is considered to be excellent, very Table 4. Correlation adjustments with the experimental data
similar to those obtained by using the Equations of Petukhov for the first range of values available for Equation (74)
and Gnielinsky, which should be clarified that it cannot be
used for Pr> 2000. It is also observed that for values of Pr <200, 2.4 ∙ 103 ≤ 𝑅𝑒 < 104
the average error obtained is 7.12 % for 88.35% of the 𝜇𝐹 𝑒𝑟𝑟𝑜𝑟 < 6.18%
available data, which brings it numerically to the 5% reported 0.006 < ≤ 12.42 0.65 < Pr ≤ 102
𝜇𝑃 91.32% data
by Petukhov and Gnielinsky. 𝜇𝐹 𝑒𝑟𝑟𝑜𝑟 < 6.96%
0.006 < ≤ 18.35 0.65 < Pr ≤ 2 ∙ 102
Table 2 provides a detailed summary of the range that shows 𝜇𝑃 90.42% data
a satisfactory fit with the correlation proposed in the present 𝜇𝐹 𝑒𝑟𝑟𝑜𝑟 < 8.74%
0.006 < ≤ 22.2 0.65 < Pr ≤ 2 ∙ 103
work. 𝜇𝑃 89.14% data
𝜇𝐹 𝑒𝑟𝑟𝑜𝑟 < 9.96%
0.006 < ≤ 34.16 0.65 < Pr ≤ 8.1 ∙ 103
𝜇𝑃 88.05% data
Table 2. Summary of the validity range for the Equation (74) 𝜇𝐹 𝑒𝑟𝑟𝑜𝑟 < 10.74%
0.006 < ≤ 62.2 0.65 < Pr ≤ 1.2 ∙ 104
𝜇𝑃 86.42% data
Parameter Range 𝜇𝐹 𝑒𝑟𝑟𝑜𝑟 < 12.18%
Water, Air, Helium, Hydrogen, Nitrogen, Carbon 0.006 < ≤ 105 0.65 < Pr ≤ 2.24 ∙ 104
𝜇𝑃 83.18% data
Dioxide, Transformer oil, Glycerin, MC Oil, MK 𝜇𝐹 𝑒𝑟𝑟𝑜𝑟 < 13.91%
Oil, Butyl alcohol, Methanol, Ethanol, Ethylene 0.006 < ≤ 177 0.65 < Pr ≤ 4.71 ∙ 104
𝜇𝑃 80.32% data
glycol, Kerosene, Acetic Acid, Acetaldehyde,
Fluids
Butanol, Aniline, Carbon Disulfide, Ciclohexane,
Ethyl ether, Ethylamine, Oil olive, Toluene, Table 5. Correlations with experimental data for the second
Turpentine, Propylene, Pentane, Benzene, Gasoline, range of values available for Equation (74)
Isobutene, Engine oil. Decane and Dodecane
𝑃𝑟 0.65 to 4.71 ∙ 104 104 ≤ 𝑅𝑒 ≤ 8.2 ∙ 106
𝑅𝑒 2.4 ∙ 103 to 8.2 ∙ 106 𝜇𝐹 𝑒𝑟𝑟𝑜𝑟 < 6.24%
𝜇𝐹 ⁄𝜇𝑃 0.006 ≤ 𝜇𝐹 ⁄𝜇𝑃 ≤ 177 0.006 < ≤ 12.42 0.65 < Pr ≤ 102
𝜇𝑃 89.36% data
𝑙 ⁄𝑑 2 ≤ 𝑙 ⁄𝑑 ≤ 420 𝜇𝐹 𝑒𝑟𝑟𝑜𝑟 < 7.12%
0.006 < ≤ 18.35 0.65 < Pr ≤ 2 ∙ 102
𝜇𝑃 88.35% data
In this work, the experimental data used in the validation of 𝜇𝐹 𝑒𝑟𝑟𝑜𝑟 < 8.31%
the developed model were extracted of the critical review 0.006 < ≤ 22.2 0.65 < Pr ≤ 2 ∙ 103
𝜇𝑃 87.12% data
available in the reference [22], which provides one large data 𝜇𝐹 𝑒𝑟𝑟𝑜𝑟 < 10.17%
base of experimental data compiled on heat transfer 0.006 < ≤ 34.16 0.65 < Pr ≤ 8.1 ∙ 103
𝜇𝑃 86.31% data
calculation during fluid flow in single-phase inside tubes. 𝜇𝐹 𝑒𝑟𝑟𝑜𝑟 < 11.23%
0.006 < ≤ 62.2 0.65 < Pr ≤ 1.2 ∙ 104
Table 3 provides the available experimental data used in this 𝜇𝑃 84.02% data
𝜇𝐹
paper. 0.006 < ≤ 105 0.65 < Pr ≤ 2.24 ∙ 104
𝑒𝑟𝑟𝑜𝑟 < 13.37%
𝜇𝑃 82.72% data
In Tables 4 and 5 can be appreciated that the Equation (74),
𝜇𝐹 𝑒𝑟𝑟𝑜𝑟 < 13.96%
is as accurate as the Equations of Petukhov and Gnielisky, 0.006 < ≤ 177 0.65 < Pr ≤ 4.71 ∙ 104
𝜇𝑃 80.94% data
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 Figure 6 shows the correlation between the proposed model
validity. Therefore, the proposed model constitutes one and the experimental data reported by various authors.
contribution to the state of the art, on heat transfer calculation
during fluid flow in single-phase inside pipes.

Table 3. Experimental data used in the validation of the Equation (74)

Deviatio
Source Number of data Fluid l/d 𝑅𝑒 ∙ 103 Pr 𝜇𝐹 ⁄𝜇𝑃
n percent
41 7 0.68 0.65 5.3
I’lin (1951) 188 Air
162 6600 0.7 1.65 3.5
48 12.5 0.68 0.65 6,2
Volkov (1966) 218 Air
370 3700 0.7 1.65 1,5
39 15 0.68 0.65 4,4
Petukhov (1963) 140 Air
100 5800 0.7 1.65 2,1
20 9 0.71 0.22 7,1
44 Helium
50 40 0.72 4.5 -2,3
Isobutene (2- 2 3200 0,73 0.68 9,7
Sukomiel (1962) 67
Methylpropane) 60 7200 0,75 1.46 -6,4
6 12 0.9 0.19 8,2
148 Water
64 540 9.4 0.77 -7,9
10 13 14.3 0.41 11,6
Eckert (1964) 93 Turpentine
90 110 29.8 2.43 -14,7
48 120 1.2 0.24 10,2
33 Water
61 160 5.9 0.86 1,1
Sabersky (1963)
46 150 4.5 0.47 13,1
52 Pentane
88 620 7.1 2.08 -9,6
70 19 2 0.21 12,6
Yakolev (1960) 39 Water
90 140 12 1.15 -3,9

263
 60 35 1 0.13 13,1
Sabersky (1965) 62 Water
180 120 9.44 7.15 9,9
89 3.4 34.9 0.01 12,7
41 Transformer oil
125 13.8 1530 115.2 -10,3
89 2.5 1630 0.018 9,2
29 Glycerin
Sterman-Petukhov 125 9.1 22650 55,4 -5,4
(1970) 66 5 120 0,007 14,8
49 MC Oil
165 10.4 9800 133,3 -17,1
80 5.4 590 0.011 15.8
27 MK Oil
145 8.7 39000 88.7 -12.6
42 23 0.08 15,3
Kreith (1947) 20 Butyl Alcohol 38
78 30 0.45 -12,2
60 2.6 3.2 0.31 8,8
Ykolev et al. (1965) 50 Benzene
110 21.1 5 3.17 -4,8
60 70 5.5 0.22 10,4
113 Gasoline
190 6900 15.1 4.4 -6,1
Humbble (1993)
43 12 0.65 0.48 -2.4
181 Hydrogen
67 8200 0.73 3.28 -8.8
100 6 0.68 0.15 9,8
Kirilov (1967) 125 Nitrogen
138 8100 0.75 6.5 1,9
77 14 0.66 0.3 7,4
Efimok (1969) 19 Carbon Dioxide
206 660 0.81 3.3 0,7
2 4 0.94 0.19 9,9
Yan-Lin (1999) 91 Water
420 250 11 0.96 -11.5
20 400 0.94 0.19 13,6
Tarashmova (2001) 23 Water
450 2500 11 0.96 -8,9
18 1200 1.2 0.24 5.3
Karkalala (2012) 44 Water
51 2800 5.9 0.96 4.5
19 2.8 34.9 1.2 16,2
Jung et al. (2008) 71 Transformer oil
150 8.1 4800 28 -7,5
45 2.9 2.2 0.1 4,4
Carpenter (1957) 66 Methanol
120 1112.1 7.7 9.9 2,1
30 6.4 1.35 0.38 7,1
112 Kerosene
280 52.8 2.9 2.6 -2,3
55 3.1 8.5 0.8 4,7
47 Acetic acid
135 987.8 14.2 1.2 -3,7
65 3.9 2.85 0.4 8,2
38 Acetaldehyde
120 52.4 4.4 2.1 -7,9
40 5.4 22.5 0.04 11,6
Vasserman (1962) 141 Butanol
160 822.6 3860 24.6 -16,7
50 4.4 11.5 0.08 9,7
187 Aniline
280 1024.2 111 12.35 -3.5
48 13.8 2.3 0.59 10,2
37 Carbon Disulfide
125 76.9 3.2 1.68 -1,1
85 36.1 11 0.5 2.3
23 Ciclohexane
220 89.4 19.9 1.9 -1.7
80 21.4 6.9 0.049 5.2
113 Ethanol
125 1513.8 68.4 20.5 7.4
70 580 3.5 0.3 4.2
71 Ethyl ether
135 2560 7.3 3.6 8.1
80 12.1 5.1 0.55 3.2
17 Ethylamine
100 17.8 8.3 1.8 -6.1
60 125 2.8 0.27 9.1
Sherwood (1967) 21 Propylene
120 284 3.2 3.66 -4.8
70 72 10.7 0.4 11.1
36 Dodecane
150 96 28.2 3.3 -12.4
65 16 6.8 0.25 2.3
40 Decane
135 47.2 17.1 4.1 -7.8
90 6.3 69 0.12 7.1
53 Ethylene glycol
165 12.1 510 8.1 -9.3
85 2.7 700 0.3 9.1
Gordon (1937) 11 Oil olive
120 7.6 810 2.9 -11.4
70 3.9 4.7 0.1 11.6
Gordon (1939) 13 Toluene
150 27.2 21.1 7.8 -9.4
30 2.4 84 0.006 14.1
GMC (2012) 103 Engine Oil
180 7.2 47100 177 -19.4
2.0 2.4 0.65 0.006 16.2
For all sources above 3096
450 8200 47100 177 -19.4

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[7] Zhang W, Du X, Yang L, Yang Y. (2016). Research on
A new model has been obtained for the determination of the performance of finned tube bundles of indirect air-cooled.
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Laminar forced convective heat transfer in helical pipe
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𝜌 Fluid density, kg∙m-3
𝜆 Fluid thermal conductivity, W∙m-1∙K-1
NOMENCLATURE 𝑣 Liquid kinematic viscosity, m2∙s-1
𝛿2 Film thickness of the momentum in boundary layer, m
𝑎 Thermal diffusivity, m2∙s-1 𝛿3 Film thickness of the thermal boundary layer, m
𝐴 Constant, defined in Equation (74) 𝜏 Shear stress in the turbulent boundary layer, kg∙m∙s-2
𝐵 Constant, defined in Equation (74) 𝜏0 Shear stress on the surface of the turbulent boundary
𝐶 Constant, defined in Equation (74) layer, kg∙m∙s-2
𝐶𝑝 Fluid specific heat, J∙kg-1∙K-1 𝜏𝑉𝑖𝑠𝑐 Stress of the viscous forces, kg∙m∙s-2
𝐶𝑊 Drag coefficient 𝜏 𝑇𝑢𝑟𝑏 Stress of the turbulent strain, kg∙m∙s-2
Δ𝑃 Pressure drop, m

266