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New friction factor equations developed for turbulent flows in rough helical tubes
International Journal of Heat and Mass Transfer 95 (2016) 525–534
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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
New friction factor equations developed for turbulent flows in rough helical tubes Houjian Zhao, Xiaowei Li ⇑, Xinxin Wu Key Laboratory of Advanced Reactor Engineering and Safety of Ministry of Education, Collaborative Innovation Center of Advanced Nuclear Energy Technology, Institute of Nuclear and New Energy Technology, Tsinghua University, Beijing 100084, China
a r t i c l e
i n f o
Article history: Received 17 July 2015 Received in revised form 7 December 2015 Accepted 16 December 2015
Keywords: Helical tube Curvature ratio Friction factor Relative roughness Turbulent flow
a b s t r a c t Helical tubes are widely used and there lacks friction factor equations for turbulent flows in rough helical tubes. Turbulent flows in rough helical tubes were investigated theoretically in this paper. Friction factor equations for transitionally and fully rough regime turbulent flows in rough helical tubes were derived based on the logarithmic velocity distribution law. The parameters of the equations were obtained by regression analysis of experimental data of Clancy (1949) and verified by experimental data of McElligott (1948). The characteristics of the friction factors in rough helical tubes were discussed based on the equations. Friction factors of helical tubes are influenced by Reynolds numbers, relative roughnesses and curvature ratios. The friction factors of rough helical tubes can also be divided into a transitionally rough regime and a fully rough regime according to the roughness height and the Reynolds number like that for straight tubes. Roughnesses have greater effects on the transition from the transitionally rough regime to the fully rough regime than that of curvature ratios. Roughness effects on friction factors increase with the increasing of the curvature ratio and the Reynolds number. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction Helical tubes are widely used in chemical, petroleum and nuclear industries due to the advantages of compact structures, high heat transfer coefficients and good thermal expansion performances. In nuclear industries, helical tubes are usually used in steam generators [3–5], especially for gas cooled reactors. Flows in helical tubes are more complex than those in straight tubes. Secondary flows occur in helical tubes in the cross section due to the centrifugal force, which causes more friction losses. Since the classical works by Dean [6,7], flows in smooth curved tubes have been extensively studied. Dean obtained an approximate solution through perturbation over the Poiseuille flow in straight tubes and introduced the Dean number. Gnielinski [8] presented equations for laminar flows and turbulent flows to calculate the friction factors and heat transfer coefficients in helical tubes. Results were compared with experimental data from literature and the deviations were less than 15%. Srinivasan [9] measured the pressure drops of water and oil in helical tubes and developed equations to predict friction factors for laminar, transition and turbulent flow regions. Ju et al. [10] evaluated the hydraulic
⇑ Corresponding author. Tel.: +86 10 62784825; fax: +86 10 62797136. E-mail address: [email protected] (X. Li). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2015.12.035 0017-9310/Ó 2015 Elsevier Ltd. All rights reserved.
characteristics in small bending radius helical tubes. Friction factor correlations were obtained by regression analysis of the experimental data and the results showed that the critical Reynolds number for laminar flow to turbulent flow transition in helical tubes was much greater than that in straight tubes. Yamamoto et al. [11] investigated flows in helical tubes with large pitches numerically and experimentally. Simulations agreed well with experimental data and showed that two vortices transformed into one single vortex as the pitches increase. Hart et al. [12] presented a friction factor chart for flows in curved tubes which covered the laminar flow region and turbulent flow region and derived an equation to calculate friction factors. Grundman [13] developed a practical friction factor diagram for helical tubes which accounted for curvature ratio effects. The diagram also offered a graphic view of the flow conditions and other parameters. Mishra and Gupta [14] investigated the pitch effects on pressure drops in helical tubes. Pressure drops were measured and equations for friction factors were derived using the modified Dean number. They concluded that the pitch effects could be eliminated by using the modified curvature diameter. Ito [15] measured the frictional pressure drop of turbulent flows in smooth curved tubes with zero pitch and derived an equation using the 1/7th power velocity distribution law and an equation using the logarithmic velocity distribution law, which have been used widely for engineers [16].
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Nomenclature Notations Cur curvature ratio (dimensionless) fc friction factors for curved tubes (dimensionless) Ks average wall surface roughness (m) M mass flow rates (kg s1) P static pressure (Pa) r0 tube inner radius (m) R helical radius (m) Rc modified helical radius (m) V velocity magnitude in curved tubes (m s1) velocity magnitude in the r direction (m s1) Vr Vh velocity magnitude in the h direction (m s1) Vu velocity magnitude in the u direction (m s1) V⁄ ðsw =qÞ1=2 , friction velocity (m s1) mean axial velocity (m s1) Vm y distance normal to the wall (m) y0 distance normal to the wall where the velocity magnitude is zero (m). sw wall shear stress (N m2) sh wall shear stress in the h direction (N m2) su wall shear stress in the u direction (N m2)
Flows in helical tubes also have been investigated numerically. Austin and Seader [17] solved the Navier–Stokes equation in the stream-function form using the finite difference method to simulate steady, incompressible and fully developed flows in helical tubes. They presented a correlation for pressure drops in terms of the Dean number. Soeberg [18] simulated laminar flow in helical tubes based on the symmetry of the secondary flow and investigated the velocity profile changes which were influenced by the Dean number. Liu and Masliyah [19] and Yamamoto et al. [20] simulated the flows in helical tubes with finite pitches and found the friction factors would increase as the curvature ratios increase. However, the factors would decrease as the pitches increase. Yanase et al. [21] used the Forier–Chebyshev spectral method to analyze the stability of the two-vortex and the four-vortex solutions for flows in slightly curved tubes and found that the twovortex solution was stable in response to any small disturbances, while the four-vortex solution was unstable to asymmetric disturbances. Lai et al. [22] simulated turbulent flows in curved tubes and found that there were three vortex pairs in the cross section. One was the Dean-type vortex pair. Another existed in the tube core and was caused by the pressure imbalance. The third was near the outer wall and was the turbulence driven secondary flow. Manlapaz and Churchill [23] simulated laminar flows in coiled tubes with a finite pitch and developed a friction factor equation for the laminar flow region. Ivan and Michele [24] simulated turbulent flows using k e, SST k—x and RSM—x models. Pressure drops were in excellent agreement with experimental data when using SST k—x and RSM—x models, but results were unsatisfactory when using k e model with wall functions. To the authors’ knowledge, most previous studies did not consider the roughness effects on friction factors in helical tubes. There are no reported equations for calculating the friction factors in rough helical tubes. However, most helical tubes used for industries are commercial tubes with different height of roughness. Many investigations have been done on flows in rough straight tubes which can be taken as the bases for investigation on rough helical tubes. Nikuradse [25] experimentally investigated turbulent flows in rough straight tubes by roughening the tubes with uniform sand grains. At high Reynolds numbers, the results agreed
Greek letters a; b; c constants in the friction factor equation d thickness of boundary layer (m) e relative roughness, e = Ks/r0 (dimensionless) h angels in the cross section l dynamic viscosity (kg s1 m1) q density (kg m3) u angles in the axial direction Subscripts 1 at the edge of the boundary layer c curved rough tubes c, r helical rough tubes c, s helical smooth tubes rough fully rough regime str, r straight rough tubes trans transitionally rough regime w near the wall
well with others’ experimental data for commercial tubes. However, the results deviated from the performances of commercial tubes at moderate Reynolds number. Colebrook and White [26] investigated turbulent flows in roughened straight tubes experimentally and found Nikuradse’s [25] deviations from the commercial tubes were due to the uniform roughness in tubes. Then Colebrook [27] presented an implicit equation to calculated friction factors in rough straight tubes. The equation was widely used to calculate friction factors for turbulent flow in rough straight tubes. Many researchers [28–32] obtained explicit equations to calculate the friction factors in rough straight tubes with specific limits. Moody [33] presented a composite friction factor chart of all flow regions for rough straight tubes which was widely used in engineering. For rough helical tubes, Clancy [1] measured the pressure drops in rough helical tubes and straight tubes and found friction factors were influenced by Reynolds numbers, helical diameters and roughness heights. McElligott [2] compared friction factors in rough helical tubes and friction factors in rough straight tubes with comparable roughness. The results showed that friction factors in helical tubes depended more on roughnesses than on curvature ratios. This paper describes a theoretical investigation on turbulent flows in rough helical tubes and derived friction factor equations considering the roughness effects, curvature ratios and Reynolds numbers. The equations can predict the friction factors accurately compared with literature data. The comprehensive effects of roughnesses and curvature ratios on the friction factor are also discussed based on the equations.
2. Equation derivation 2.1. Physical model and mathematical description Fig. 1 shows a helical tube schematic with the main parameters and a secondary flow illustration. The helical radius is defined as R, the tube inner radius is defined as r 0 and the helical pitch is defined as b. When the helical pitch is small compared with the helical diameter, torsion effects can be neglected at moderate Reynolds
H. Zhao et al. / International Journal of Heat and Mass Transfer 95 (2016) 525–534
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Fig. 1. Schematic of a helical tube and secondary flow in the cross section.
Fig. 2. The curved tube and velocity vectors in the orthogonal coordinates.
number [19]. Moreover, the effects of pitch on friction factors can be eliminated if the helical radius is modified by Eq. (1) [14].
"
Rc ¼ R 1 þ
b
2pR
2 #
ð1Þ
For this analysis, the helical tube shown in Fig. 1 is simplified to be a helical tube with zero pitch as shown in Fig. 2(a). The toroidal orthogonal coordinate system shown in Fig. 2(b) is adopted, where r denotes the distance from the center of the tube in the cross
section, h denotes the angle between r and the symmetry of the cross section, and u denotes the angular distance from a fixed cross section. V r , V h and V u are taken as the velocity magnitudes in directions of r, h and u. The continuity and momentum equations, Eqs. (2)–(5), are obtained based on the following assumptions: (1) The helical tubes are simplified to be curved tubes with zero pitch. (2) The flows are incompressible and steady.
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(3) The flows are fully developed, so, Vr, Vh and Vu are independent of u. (4) The helical radius is much larger than the tube radius, i.e. R r0.
@V r V r 1 @V h þ þ ¼0 r @h @r r
ð2Þ
Z
@ ðV 2h Þdr þ ½V h V r rr00 d þ @h r0 d r @V h 0 m @r r0 d r0
Z
r0
V 2u sin h
r 0 d
R
Z dr ¼
r0 r0 d
1 @P
qr0 @h
dr
ð8Þ
According to Ito [36],
@P r0 ¼ q V 2u1 sin h @h R
2 @V r V h @V r V 2h V u cos h Vr þ @r r @h r R 1 @P m @ @V h V h 1 @V r ¼ þ r @h q @r r @h @r r
V r ¼ V h ¼ 0 r ¼ r0 ð3Þ
V h1 ¼
2 @V h V h @V h V h V r V u sin h þ þ þ Vr @r r @h r R 1 @P @ @V h V h 1 @V r þm þ ¼ @r @r r @h qr @h r
ð4Þ
2
2
@V u V h @V u @ V u 1 @V u 1 @ V u 1 @P Vr þm þ þ ¼ þ 2 r @r r @ u2 @r r @h qR @ u @r 2
! ð5Þ
Vr O(d), @=@r Oðd1 Þ, and d r0. In the boundary layer, the value of r is approximate to r0. Then Eqs. (2) and (4) are reduced to,
@V r 1 @V h þ ¼0 r 0 @h @r
ð6Þ 2
@V h V h @V h V u sin h 1 @P @2V h m 2 þ þ ¼ @r r 0 @h qr0 @h @r R
ð7Þ
The pressure gradient in h direction in Eq. (7) is set as independent of r and equal to the pressure gradient at the edge of the boundary layer. Eq. (7) is integrated by using Eq. (6).
@V h @r
¼ 0 r ¼ r0 d 1
where V u1 and V h1 are the velocity magnitudes at the edge of the boundary layer in u direction and h direction, respectively. Then Eq. (8) is reduced to,
q
According to Adler [34], the boundary layer will become thinner when the Dean number increases. The friction factor equation is derived based on the boundary layer approximation method. This method divides the cross section of curved tubes into two regions, the central core in the central region and the thin boundary layer near the wall region. The detailed information of the boundary layer is shown in Fig. 3. In the boundary layer, the viscous effect is significant. The thickness of the boundary layer is denoted as d. The axial velocities, V u , will gradually fall to zero when approaching the wall because of the no slip wall conditions. Then V h and V u are comparable in the boundary layer. Order-of-magnitude analysis was done to reduce Eqs. (2)–(5). Similar to the boundary layer theory for plates [35], we know that in the boundary layer, Vh O(1), Vu O(1),
Vr
1 r0
Z
@ @h
0
d
V 2h dy ¼ sh r 0 þ q
dr 0 2 r0 sin h q V sin h R u1 R
Z 0
d
V 2u dy
ð9Þ
where y is the distance normal to the wall near the wall region. As the velocities in r direction in the boundary layer are neglected. The velocities in the curved tubes are composed by velocities in axial direction, V u , and velocities in circumferential direction, V h . Then the velocities are assumed to be:
Vu ¼
V ð1 þ K 2 Þ
1=2
Vh ¼
KV ð1 þ K 2 Þ
1=2
ð10Þ
where K is the ratio of V h and V u . V is the velocity magnitude in curved tubes. According to the relationship between velocities and wall stresses,
su ¼ l
@V u @r w
sh ¼ l
@V h @r w
ð11Þ
Then wall stresses in u direction and h direction can be written as,
su ¼
sw 2 1=2
ð1 þ K Þ
sh ¼
K sw ð1 þ K 2 Þ
1=2
ð12Þ
Y is defined to have the same order-of-magnitude as V m =V .
Y
Vm V
ð13Þ
where V m is the mean axial velocity magnitude and V is the friction velocity magnitude, which is defined by,
Fig. 3. Illustration of boundary layer in the cross section.
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rffiffiffiffiffiffi
sw q
V ¼
ð14Þ
In fully rough regimes, y0 ¼ jK s , where j is constant [35]. y0 in Eq. (24) is replaced to give,
As K is the ratio of V h and V u , we can get,
Vu Vm
V h kV m
ð15Þ
Eq. (13) is combined with Eqs. (12) and (14) to get,
su
qV 2m Y2
sh
K qV 2m Y2
ð16Þ
Z p 0
su dh
The friction factor of a helical tube, f c is defined by,
@P 1 R ¼ f c qV 2m @u 2 2r 0
ð19Þ
ð20Þ
A Y2
divided by qV 2m r 0 .
R
R
K 2 eY K s r0
eY K s R
K Y
2
eY K s R
K ðr 0 =RÞ1=2
ð21Þ
ð28Þ
e ¼ K s =r0 as the relative roughness. Comparing K=Y 2
Define e, 2 Y
with K e K s =r0 , we can get that,
1 Ke
ð29Þ
The relationships in Eqs. (28) and (29) can be rewritten as,
where A is constant.
r 1=2 0 K¼D R
2.2. Logarithmic law for turbulent flow near walls
Y 2 eY ¼
The logarithmic velocity distribution law used in this analysis is Prandtl’s velocity-distribution law which was derived from the mixing length theory [35]. It is widely accepted and used for wall turbulence, usually it is called law of the wall. The law can be written as,
where D and E are constants. Eq. (31) can be solved to get Y,
Y ¼ ln
ð30Þ
E Ke
E K eY 2
ð31Þ
¼ ln K eY 2 þ ln E
ð32Þ
Eq. (32) can be rewritten by using Eqs. (21) and (30).
V ¼ aV ln y þ b where V is the velocity magnitude, a and b are constants, V is the friction velocity and y is the distance from the wall. Then for the boundary layer in the helical tube, we can get,
V 1 V ¼ aV ln ðd=yÞ
ð27Þ
K 2 eY K s =r0 with eY K s =R gives,
Y 2 eY
Define that,
fc ¼
ð26Þ
Each term must have the same order-of-magnitude. Comparing
Substitute Eq. (19) into Eq. (18) to get the relationship between Y and f c .
f c Y2
where B and C are constants.Eq. (25) is then combined with Eqs. (13) and (15), we can obtain the relationship between Y and d=K s .
d q @h@ 0d V 2h dy sh r0 q drR0 V 2u1 sin h q rR0 sin h 0 V 2u dy ¼ þ qV 2m r0 qV 2m r0 qV 2m r0 qV 2m r0
ð18Þ
ð25Þ
The order-of-magnitude of each term in Eq. (9) is obtained after
ð17Þ
Eq. (16) is combined with Eq. (17) to get,
@P RqV 2m @u r0 Y 2
V u1 ¼ B ln ðd=K s Þ þ C V
d eY K s
The pressure gradient in u direction is,
@P 2R ¼ @u pr0
2.3. Friction factor equation in the fully rough regime
ð22Þ
1 1=2
fc
¼ F ln
Ge ðr 0 =RÞ1=2 fc
2.4. Friction factor in the transitionally rough regime
y0 ¼ HK s þ Il=qV
V 1 ¼ aV ln ðd=y0 Þ
V u1 ¼ AV ln
According to Colebrook [27], in the transitionally rough regime,
d HK s þ Il=qV
Eq. (35) is combined with Eqs. (13) and (15),
V u1 ¼ aV ln ðd=y0 Þ
d eY ðHK s þ Il=qV Þ
The relationships between y0 and the mean roughness, K s , in the fully rough regime and the transitionally rough regime are different. The following analysis will be divided into two sections, friction factors in the fully rough regime and friction factors in the transitionally rough regime.
ð34Þ
where H and I are constants. Eq. (24) can be rewritten as,
At the edge of the boundary layer, V h1 ¼ 0 and V 1 ¼ V u1 . Eq. (23) can be rewritten as,
ð24Þ
ð33Þ
where F and G are constants. r 0 =R is defined as the curvature ratio.
where V 1 is the velocity magnitude at the edge of the boundary layer in the cross section. For rough helical tubes, the velocity magnitude is zero at a point, just like that for the straight rough tubes near the wall. The distance normal to the wall at this point is set to be y0 [35]. V 1 can be expressed as,
ð23Þ
ð35Þ
ð36Þ
Each side of Eq. (36) is divided by r 0 ,
d LeY Y JeY e þ r0 Re
ð37Þ
Re ¼ 2qV m r0 =l; is the Reynolds number. J and L are constants.
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The order-of-magnitude of each term in Eq. (9) is obtained after divided by qV 2m r0 ,
R
R
d q @h@ 0d V 2h dy sh r0 q drR0 V 2u1 sin h q rR0 sin h 0 V 2u dy ¼ þ qV 2m r0 qV 2m r0 qV 2m r0 qV 2m r0
d 2 K r0
K Y2
d R
ð38Þ
d R
Comparing d=R and dK 2 =r0 , we can get the relationship shown in Eq. (39), which is the same as Eq. (28).
K ðr0 =RÞ1=2
ð39Þ 2
Comparing dK =r 0 with K=Y , we can get,
ð40Þ f ¼
Then substitute Eq. (37) into Eq. (40), we can get,
JeY e þ
LeY Y 1 Re KY 2
ð41Þ
Eq. (41) can be rewritten as,
ð42Þ
where M is constant. The friction factor equation is obtained by solving Eq. (42), using Eqs. (21) and (30),
1=2
fc
"
be r0 1=2 c r0 1=2 ¼ a ln þ 3=2 Re fc R R fc
#
ð43Þ
The thickness of the viscous sub-layer decreases as the Reynolds number increases. The roughness will dominates the wall stresses when the roughnesses are all out of the viscous sub-layer. Turbulent flows in the transitionally rough regime will transform to the fully rough regime. With very high Reynolds numbers, Eq. (43) can be written as Eq. (44) by eliminating the term including the Reynolds number. 1=2
fc
be r 0 1=2 fc R
ð46Þ
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 1 2 5 1 2 2 2 2 ðDDÞ þ ðDðDPÞÞ þ ðDM Þ þ ðDLÞ M D L DP
The experimental errors for friction factors are below 5%, which is calculated by Eq. (47). 3.2. Determination of the equation parameters As Eq. (43) is a multiple parameter equation and the friction factor is in implicit expression. Eq. (43) is transformed to Eq. (48) for regression analysis to get the parameters.
c
2.5. Equation compatibility for different flow regimes
¼ a ln
Df ¼ f
," 1=2 # be eð1=a=f c Þ Re ¼ 3=2 þ fc ðr 0 =RÞ1=2 fc
where a, b and c are constants.
1
DP D 2DP D5 p 2 ¼ q 4 0:5qV 2m L M2 L
ð45Þ
ð47Þ
LeY Y M ¼ Je e þ Re KY 2 Y
1
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Xn @ ln u2 ¼ ð Dxi Þ 2 i¼1 u @xi
Du
2
d 1 r 0 KY 2
decreasing when the Reynolds number increased, the flows were in transitionally regime. The parameters in Eq. (43) can be obtained by the regression analysis of experimental data. The differential pressures were measured by water manometer at low flow rates and by mercury manometer at high flow rates. Flow rates were obtained by weighting a mass of water during a specific time interval. The experimental errors were not presented in reference [1]. The relative errors for pressure drops are set as 2% based on the minimum calibration of 1 mm. The maximum relative errors for mass flow rates and length are set as 2% and 0.1%, respectively, based on the significant digits of the experimental data. The relative errors are calculated by the following equations.
In Clancy’s experiments [1], pressure drops are measured in 12 helical tubes for different curvature ratio and different roughness. The Reynolds number varied from 1 103 to 1 105 . The experimental data for each tube are defined as one group of data and is marked by the tube number. As the equation is derived based on the logarithmic velocity distribution law, the friction factors in the laminar flow region should not be used. The critical Reynolds number is calculated by Eq. (49) [15].
ð44Þ
Eq. (44) is the same as Eq. (31), which shows that the derived friction factor equations for transitionally and fully rough regimes are compatible. 3. Equation parameter determination Friction factor equations for transitionally and fully rough regime turbulent flows have been derived. However, the constants in Eqs. (33) and (43) need to be determined using experimental data.
Recri ¼ 2 104
bers varied from 1 103 to 1 105 . As the friction factors kept
r 0:32 0 R
ð49Þ
Several groups of data are used for the regression analysis. The data with Reynolds number below 1 104 are not used. Then about 5 points in each group are left. The total points used are about 30. The curvature ratios and the relative roughnesses are set as constant in Eq. (48) for each group. The results of the regression analysis are shown in Table 1 and the confidence level is 95%. The values of a, b and c shown in Table 1 are averaged, then gives,
a = 0.923033
3.1. The experiments of Clancy The experimental data from Clancy [1] are used to approximate the derived equations. Fig. 4 shows a schematic diagram of the test apparatus of Clancy [1]. The pressure drops in helical tubes were measured with the curvature ratio varied from 0.056 to 0.063, mean roughness varied from about 0.6–4 lm and Reynolds num-
ð48Þ
b = 0.104455
c = 1.142495 Then Eq. (43) can be expressed as,
1 1=2
fc
¼ 0:923 ln
" # 0:104e r0 1=2 1:142 r 0 1=2 þ 3=2 fc R f c Re R
ð50Þ
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Fig. 4. Schematic diagram of the test apparatus of Clancy [1].
Table 1 Regression analysis results. Tube number
2
a
0.92589 b 0.10158 c 1.19671 Relative roughness 0.00016 Curvature ratio 0.06214 Adj. R-square 0.99877
Table 2 Equation verification. 3
4
5
9
10
0.92879 0.09756 1.18591 0.00018 0.06101 0.99915
0.91636 0.09101 0.98235 0.00013 0.06115 0.99411
0.92058 0.12272 1.19694 0.00010 0.06186 0.99840
0.92567 0.09693 1.11030 0.00013 0.06101 0.99858
0.92091 0.11693 1.18276 0.00011 0.06158 0.99750
Re 1 10
Relative roughness 5
0:315 10
3
Experiment
Eq. (50)
Relative error (%)
0.0257
0.025916
0.8
1 105
0:2849103
0.0247
0.024705
0.01
2 104
0:315 103
0.035
0.034533
1.3
2 104
0:302 103
0.033
0.032989
0.03
4
0:284 103
0.0324
0.032957
1.7
2 10
According to the discussion in Section 2.5, when Re is high enough, the turbulent flows in the transitionally rough regime will convert to the fully rough regime. Influence of Re will diminishes, left only the relative roughness’s influence. We set c as zero in Eq. (50). The equation for the fully rough regime is obtained.
1 1=2
fc
¼ 0:923 ln
0:104e r 0 1=2 ð Þ fc R
ð51Þ
The coefficients in Eq. (50) are obtained by regression analysis of experimental data with random roughnesses. But the uniform roughness effects and random roughness effects on friction factors are different [26,27]. Then coefficients of the equations to predict friction factors in uniform roughened helical tubes [37–40] in the transitionally rough regime and fully rough regime will be different. 3.3. Equation verification The equations are verified by the experimental data from McElligott [2]. In Ref. [2], the Reynolds numbers are 2 104 and 1 105. The relative roughnesses are from 0.284 103 to 0.315 103 which are different from those of Ref. [1]. Friction factors measured in the experiments, factors calculated by Eq. (50) and the relative errors are shown in Table 2 and Fig. 5, which shows a good prediction by using Eq. (50). 4. Discussion A good friction factor equation for rough helical tubes should be compatible with that for smooth helical tubes, which means when the roughness diminishes, the equation for rough tubes should
Fig. 5. Equation verification.
become equivalent to that for smooth tubes. We will discuss this in Section 4.1. Eq. (50) shows that the friction factors of rough helical tubes are influenced by the combined effects of the Reynolds number, the curvature ratio and the relative roughness. We will discuss these effects in Sections 4.2, 4.3 and 4.4. 4.1. Compatibility with Ito’s equations Equation (43) is derived on the bases of Ito’s theoretically investigation on turbulent flows in smooth curved tubes. Ito [15] investigated turbulent flow pressure drops in smooth curved tubes experimentally and theoretically. Two equations are derived based on the 1/7th power velocity distribution law and the logarithmic
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velocity distribution law, respectively, which are shown in Eqs. (52) and (53). The equations for smooth tubes have been widely used.
fc
1=2 R 0:316 ¼ h i1=5 2 r0 Re rR0 "
1 1=2
fc
¼
3=2 2:11log10 Ref c
1=2 # R 0:23 r0
ð52Þ
ð53Þ
We set the relative roughness as zero in Eq. (50) to calculate the friction factors in smooth helical tubes to see whether the equation is consistent with Ito’s equations. Friction factors are also calculated by Eqs. (52) and (53) with the same curvature ratio and the same Reynolds number. The results are shown in Fig. 6. The relative errors are less than 4%. The curvature ratios are 0.1 and 0.01. Cur means the curvature ratio.
Fig. 7. Comparison of friction factors in helical tubes with different roughnesses and the same curvature ratios.
4.2. Roughness effects on the friction factors Friction factors are calculated using Eq. (50) for the same curvature ratio but different relative roughnesses. The Reynolds numbers varies from 1 104 to 6 105. The relative roughness varies from 4 104 to 8 103 . The curvature ratio is 0.1, where the helical radius is 50 mm and the tube inner radius is 5 mm. The results are shown in Fig. 7. Fig. 7 shows that the friction factors decrease as the Reynolds numbers increase in a range of Re with relatively low values. The range is different for tubes with different roughness, like the transitionally rough regime in Moody chart. In the transitionally rough regime, the factors depend both on the roughness and the Reynolds number. When Re is high enough, friction factor keeps unchanged as Reynolds numbers increase, like the fully rough regime in Moody chart. 4.3. Roughness and curvature ratio effects on the transition Friction factors in the transitionally rough regime, ftrans, are calculated by Eq. (50) and friction factors in fully rough regime, f rough , are calculated by Eq. (51). The curvature ratios are 0.1 and 0.05. The relative roughnesses are 0.001, 0.002 and 0.004 for each curvature ratio. Re varies from 1 104 to 6 105 . Ratios, f trans =f rough are calculated and the results are shown in Fig. 8. When the flows in the transitionally rough regime transfer to the fully rough regime, the ratio approaches to unit. The trends of the ratio as the Reynolds number increases reflect the curvature ratio effects and roughness
Fig. 6. Comparison of friction factors calculated by Eqs. (50), (52) and (53).
effects on the transition progress. Fig. 8 shows that the curvature ratio has little effect on the transition. Roughness dominates the transition process, like that in straight tubes. 4.4. Combined effects of roughness and curvature ratio on friction factors Eq. (50) shows that the roughness effects on friction factors are influenced by Reynolds numbers and curvature ratios. The factors, f c;s , in smooth helical tubes are calculated by Eq. (50) with the relative roughness set as zero. The friction factors, f c;r , in rough helical tubes are calculated by Eq. (50) with the relative roughness set as 0.004 and the Reynolds number varies from 1 105 to 6 105. The ratio, f c;r =f c;s reflects the roughness effects on the friction factors in helical tubes. f c;r =f c;s is calculated with different curvature ratio and the results are shown in Fig. 9. The results in Fig. 9 shows that f c;r =f c;s increases as the curvature ratio increases or the Reynolds number increases. That means roughness effects on friction factors increase with increasing curvature ratios and Reynolds numbers. When the curvature ratios increase or the Reynolds numbers increase, the centrifugal force increases and the secondary flow intensities increase, which will cause more pressure drops in the helical tube. Eq. (50) also shows that the curvature ratio effects on friction factors are influenced by relative roughness and the Reynolds
Fig. 8. Comparison of f trans =f rough with different curvature ratios and different roughnesses.
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rough helical tubes are theoretically derived in this paper. The parameters in the equations are determined by regression analysis of experimental data of Clancy [1] and verified by experimental data of McElligott [2]. The influences of curvature ratio, relative roughness and the Reynolds number on friction factors in rough helical tubes are analyzed. The main conclusions are, (1) Two equations are theoretically derived for calculating the friction factors for turbulent flows in rough helical tubes. One is for the transitionally rough regime and the other is for the fully rough regime. The friction factors are influenced by the Reynolds number, curvature ratio and the relative roughness. (2) The parameters of the equations are obtained by regression analysis of experimental data. (3) The roughness has similar effects on friction factors for turbulent flow in rough helical tubes as that in rough straight tubes. When the roughness and the Reynolds number are high enough, the friction factors become independent of the Reynolds number. (4) For turbulent flow in rough helical tubes, the roughness effects on friction factors increase as the curvature ratio increases. The roughness effects and curvature ratio effects increase as the Reynolds number increase.
Fig. 9. Comparison of f c;r =f c;s with different curvature ratios.
Acknowledgments This work was financially supported by the National Natural Science Foundation of China (51576103) and the National S&T Major Project (Grant No. ZX06901). References
Fig. 10. Comparisons of f c;r =f str; r with different Reynolds numbers.
number. Friction factors in helical tubes, f c;r , are calculated by Eq. (50) with curvature ratio of 0.1, and the relative roughness varies from 0.0008 to 0.0166. Friction factors in rough straight tubes with the same relative roughness, f str;r , are calculated by Eq. (54) presented by Churchill [30].
( f str; r ¼
2log10
"
e
7:4
þ
0:9 #)2 7 Re
ð54Þ
The ratio, f c; r =f str; r , reflects the curvature ratio effects on frication factors. f c; r =f str; r is calculated with different Reynolds number and the results are shown in Fig. 10. The results in Fig. 10 shows that the ratio, f c;r =f str; r , increases with the increasing roughness height, which means the curvature ratio effects on friction factors in rough helical tubes increase with the increasing roughness.
5. Conclusions Helical tubes are extensively used as heat exchangers in chemical, petroleum and nuclear industries. Turbulent flows in helical tubes are more complex than those in straight tubes. However, previous investigations on friction factors in helical tubes are all for smooth helical tubes. All the commercially used helical tubes are rough tubes. Two equations for calculating friction factors in
[1] A.H. Clancy, Experimental Investigation of Pressure Drop in Helical Coils (AE thesis), California Institute of Technology, Pasadena, California, 1949. [2] R.H. McElligott, Investigation of Pressure Drop Through Helical Coils (AE thesis), California Institute of Technology, Pasadena, California, 1948. [3] Z.Y. Zhang, Z.X. Wu, D.Z. Wang, Y.H. Xu, Y.L. Sun, F. Li, Y.J. Dong, Current status and technical description of Chinese 2 times 250 MWth HTR-PM demonstration plant, Nucl. Eng. Des. 239 (7) (2009) 1212–1219. [4] H. Nariai, M. Kobayashi, T. Matsuoka, Friction pressure-drop and heat-transfer coefficient of 2-phase flow in helically coiled tube once-through steamgenerator for integrated type marine water-reactor, J. Nucl. Sci. Technol. 19 (11) (1982) 936–947. [5] S.H. Yang, S.H. Kim, Y.J. Chung, H.S. Park, K.K. Kim, Experimental validation of the helical steam generator model in the TASS/SMR code, Ann. Nucl. Energy 35 (1) (2008) 49–59. [6] W.R. Dean, Note on the motion of fluid in a curved pipe, London, Edinb. Dubl. Philos. Mag. J. Sci. 4 (20) (1927) 208–223. [7] W.R. Dean, The stream-line motion of fluid in a curved pipe (second paper), London, Edinb. Dubl. Philos. Mag. J. Sci. 5 (30) (1928) 673–695. [8] V. Gnielinski, Heat transfer and pressure drop in helically coiled tubes, in: Proceedings 8th International Heat Transfer Conference, San Francisco, Hemisphere, Washington DC, 1986, pp. 2847–2854. [9] P. Srinivasan, S. Nandapurkar, F. Holland, Friction factors for coils, Trans. Instrum. Chem. Eng 48 (1970) 156–161. [10] H.M. Ju, Z.Y. Huang, Y.H. Xu, B. Duan, Y. Yu, Hydraulic performance of small bending radius helical coil-pipe, J. Nucl. Sci. Technol. 38 (10) (2001) 826–831. [11] K. Yamamoto, A. Aribowo, Y. Hayamizu, T. Hirose, K. Kawahara, Visualization of the flow in a helical pipe, Fluid Dyn. Res. 30 (4) (2002) 251–267. [12] J. Hart, J. Ellenberger, P.J. Hamersma, Single-phase and 2-phase flow through helically coiled tubes, Chem. Eng. Sci. 43 (4) (1988) 775–783. [13] R. Grundmann, Friction diagram of the helically coiled tube, Chem. Eng. Process. Process Intensif. 19 (2) (1985) 113–115. [14] P. Mishra, S.N. Gupta, Momentum-transfer in curved pipes. 1. Newtonian fluids, Ind. Eng. Chem. Process Des. Dev. 18 (1) (1979) 130–137. [15] H. Ito, Friction factors for turbulent flow in curved pipes, J. Basic Eng. 81 (2) (1959) 123–134. [16] S. Vashisth, V. Kumar, K.D. Nigam, A review on the potential applications of curved geometries in process industry, Ind. Eng. Chem. Res. 47 (10) (2008) 3291–3337. [17] L.R. Austin, J. Seader, Fully developed viscous flow in coiled circular pipes, AIChE J. 19 (1) (1973) 85–94. [18] H. Soeberg, Viscous flow in curved tubes—I. Velocity profiles, Chem. Eng. Sci. 43 (4) (1988) 855–862.
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[19] S.J. Liu, J.H. Masliyah, Axially invariant laminar-flow in helical pipes with a finite pitch, J. Fluid Mech. 251 (1993) 315–353. [20] K. Yamamoto, S. Yanase, T. Yoshida, Torsion effect on the flow in a helical pipe, Fluid Dyn. Res. 14 (5) (1994) 259–273. [21] S. Yanase, N. Goto, K. Yamamoto, Dual solutions of the flow through a curved tube, Fluid Dyn. Res. 5 (3) (1989) 191. [22] Y. Lai, R. So, H. Zhang, Turbulence-driven secondary flows in a curved pipe, Theor. Comput. Fluid Dyn. 3 (3) (1991) 163–180. [23] R.L. Manlapaz, S.W. Churchill, Fully developed laminar flow in a helically coiled tube of finite pitch, Chem. Eng. Commun. 7 (1–3) (1980) 57–78. [24] D.P. Ivan, C. Michele, Numerical prediction of turbulent flow and heat transfer in helically coiled pipes, Int. J. Therm. Sci. 49 (4) (2010) 653–663. [25] J. Nikuradse, Strömungswiderstand in rauhen Rohren, ZAMM J. Appl. Math. Mech./Z. für Angew. Math. Mech. 11 (6) (1931) 409–411. [26] C.F. Colebrook, C. White, Experiments with fluid friction in roughened pipes, Proc. R. Soc. London Ser. A Math. Phys. Sci. (1937) 367–381. [27] C.F. Colebrook, Turbulent flow in pipes, with particular reference to the transition region between the smooth and rough pipe laws, J. ICE 11 (4) (1939) 133–156. [28] L.F. Moody, An approximate formula for pipe friction factors, Trans. ASME 69 (12) (1947) 1005–1011. [29] D.J. Wood, An explicit friction factor relationship, Civil Eng. 36 (12) (1966) 60– 61. [30] S.W. Churchill, Empirical expressions for the shear stress in turbulent flow in commercial pipe, AIChE J. 19 (2) (1973) 375–376.
[31] P.K. Swamee, A.K. Jain, Explicit equations for pipe-flow problems, J. Hydraul. Div. 102 (5) (1976) 657–664. [32] N.H. Chen, An explicit equation for friction factor in pipe, Ind. Eng. Chem. Fundam. 18 (3) (1979) 296–297. [33] L.F. Moody, Friction factors for pipe flow, Trans. ASME 66 (8) (1944) 671–684. [34] M. Adler, Currents in curved pipes, ZAMM J. Appl. Math. Mech./Z. für Angew. Math. Mech. 14 (1934) 257–275. [35] H. Schlichting, Boundary Layer Theory, McGraw-Hill, New York, 1979. pp. 128–131, 587–591, 615–623. [36] H. Ito, Laminar flow in curved pipes, ZAMM J. Appl. Math. Mech. 49 (11) (1969) 653–663. [37] Y.X. Li, J.H. Wu, H. Wang, L.P. Kou, X.H. Tian, Fluid flow and heat transfer characteristics in helical tubes cooperating with spiral corrugation, Energy Procedia 17 (2012) 791–800. [38] A. Zachár, Analysis of coiled-tube heat exchangers to improve heat transfer rate with spirally corrugated wall, Int. J. Heat Mass Transfer 53 (19) (2010) 3928–3939. [39] S. Rainieri, F. Bozzoli, L. Cattani, G. Pagliarini, Compound convective heat transfer enhancement in helically coiled wall corrugated tubes, Int. J. Heat Mass Transfer 59 (2013) 353–362. [40] S. Rainieri, F. Bozzoli, G. Pagliarini, Experimental investigation on the convective heat transfer in straight and coiled corrugated tubes for highly viscous fluids: Preliminary results, Int. J. Heat Mass Transfer 55 (1) (2012) 498–504.
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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
New friction factor equations developed for turbulent flows in rough helical tubes Houjian Zhao, Xiaowei Li ⇑, Xinxin Wu Key Laboratory of Advanced Reactor Engineering and Safety of Ministry of Education, Collaborative Innovation Center of Advanced Nuclear Energy Technology, Institute of Nuclear and New Energy Technology, Tsinghua University, Beijing 100084, China
a r t i c l e
i n f o
Article history: Received 17 July 2015 Received in revised form 7 December 2015 Accepted 16 December 2015
Keywords: Helical tube Curvature ratio Friction factor Relative roughness Turbulent flow
a b s t r a c t Helical tubes are widely used and there lacks friction factor equations for turbulent flows in rough helical tubes. Turbulent flows in rough helical tubes were investigated theoretically in this paper. Friction factor equations for transitionally and fully rough regime turbulent flows in rough helical tubes were derived based on the logarithmic velocity distribution law. The parameters of the equations were obtained by regression analysis of experimental data of Clancy (1949) and verified by experimental data of McElligott (1948). The characteristics of the friction factors in rough helical tubes were discussed based on the equations. Friction factors of helical tubes are influenced by Reynolds numbers, relative roughnesses and curvature ratios. The friction factors of rough helical tubes can also be divided into a transitionally rough regime and a fully rough regime according to the roughness height and the Reynolds number like that for straight tubes. Roughnesses have greater effects on the transition from the transitionally rough regime to the fully rough regime than that of curvature ratios. Roughness effects on friction factors increase with the increasing of the curvature ratio and the Reynolds number. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction Helical tubes are widely used in chemical, petroleum and nuclear industries due to the advantages of compact structures, high heat transfer coefficients and good thermal expansion performances. In nuclear industries, helical tubes are usually used in steam generators [3–5], especially for gas cooled reactors. Flows in helical tubes are more complex than those in straight tubes. Secondary flows occur in helical tubes in the cross section due to the centrifugal force, which causes more friction losses. Since the classical works by Dean [6,7], flows in smooth curved tubes have been extensively studied. Dean obtained an approximate solution through perturbation over the Poiseuille flow in straight tubes and introduced the Dean number. Gnielinski [8] presented equations for laminar flows and turbulent flows to calculate the friction factors and heat transfer coefficients in helical tubes. Results were compared with experimental data from literature and the deviations were less than 15%. Srinivasan [9] measured the pressure drops of water and oil in helical tubes and developed equations to predict friction factors for laminar, transition and turbulent flow regions. Ju et al. [10] evaluated the hydraulic
⇑ Corresponding author. Tel.: +86 10 62784825; fax: +86 10 62797136. E-mail address: [email protected] (X. Li). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2015.12.035 0017-9310/Ó 2015 Elsevier Ltd. All rights reserved.
characteristics in small bending radius helical tubes. Friction factor correlations were obtained by regression analysis of the experimental data and the results showed that the critical Reynolds number for laminar flow to turbulent flow transition in helical tubes was much greater than that in straight tubes. Yamamoto et al. [11] investigated flows in helical tubes with large pitches numerically and experimentally. Simulations agreed well with experimental data and showed that two vortices transformed into one single vortex as the pitches increase. Hart et al. [12] presented a friction factor chart for flows in curved tubes which covered the laminar flow region and turbulent flow region and derived an equation to calculate friction factors. Grundman [13] developed a practical friction factor diagram for helical tubes which accounted for curvature ratio effects. The diagram also offered a graphic view of the flow conditions and other parameters. Mishra and Gupta [14] investigated the pitch effects on pressure drops in helical tubes. Pressure drops were measured and equations for friction factors were derived using the modified Dean number. They concluded that the pitch effects could be eliminated by using the modified curvature diameter. Ito [15] measured the frictional pressure drop of turbulent flows in smooth curved tubes with zero pitch and derived an equation using the 1/7th power velocity distribution law and an equation using the logarithmic velocity distribution law, which have been used widely for engineers [16].
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Nomenclature Notations Cur curvature ratio (dimensionless) fc friction factors for curved tubes (dimensionless) Ks average wall surface roughness (m) M mass flow rates (kg s1) P static pressure (Pa) r0 tube inner radius (m) R helical radius (m) Rc modified helical radius (m) V velocity magnitude in curved tubes (m s1) velocity magnitude in the r direction (m s1) Vr Vh velocity magnitude in the h direction (m s1) Vu velocity magnitude in the u direction (m s1) V⁄ ðsw =qÞ1=2 , friction velocity (m s1) mean axial velocity (m s1) Vm y distance normal to the wall (m) y0 distance normal to the wall where the velocity magnitude is zero (m). sw wall shear stress (N m2) sh wall shear stress in the h direction (N m2) su wall shear stress in the u direction (N m2)
Flows in helical tubes also have been investigated numerically. Austin and Seader [17] solved the Navier–Stokes equation in the stream-function form using the finite difference method to simulate steady, incompressible and fully developed flows in helical tubes. They presented a correlation for pressure drops in terms of the Dean number. Soeberg [18] simulated laminar flow in helical tubes based on the symmetry of the secondary flow and investigated the velocity profile changes which were influenced by the Dean number. Liu and Masliyah [19] and Yamamoto et al. [20] simulated the flows in helical tubes with finite pitches and found the friction factors would increase as the curvature ratios increase. However, the factors would decrease as the pitches increase. Yanase et al. [21] used the Forier–Chebyshev spectral method to analyze the stability of the two-vortex and the four-vortex solutions for flows in slightly curved tubes and found that the twovortex solution was stable in response to any small disturbances, while the four-vortex solution was unstable to asymmetric disturbances. Lai et al. [22] simulated turbulent flows in curved tubes and found that there were three vortex pairs in the cross section. One was the Dean-type vortex pair. Another existed in the tube core and was caused by the pressure imbalance. The third was near the outer wall and was the turbulence driven secondary flow. Manlapaz and Churchill [23] simulated laminar flows in coiled tubes with a finite pitch and developed a friction factor equation for the laminar flow region. Ivan and Michele [24] simulated turbulent flows using k e, SST k—x and RSM—x models. Pressure drops were in excellent agreement with experimental data when using SST k—x and RSM—x models, but results were unsatisfactory when using k e model with wall functions. To the authors’ knowledge, most previous studies did not consider the roughness effects on friction factors in helical tubes. There are no reported equations for calculating the friction factors in rough helical tubes. However, most helical tubes used for industries are commercial tubes with different height of roughness. Many investigations have been done on flows in rough straight tubes which can be taken as the bases for investigation on rough helical tubes. Nikuradse [25] experimentally investigated turbulent flows in rough straight tubes by roughening the tubes with uniform sand grains. At high Reynolds numbers, the results agreed
Greek letters a; b; c constants in the friction factor equation d thickness of boundary layer (m) e relative roughness, e = Ks/r0 (dimensionless) h angels in the cross section l dynamic viscosity (kg s1 m1) q density (kg m3) u angles in the axial direction Subscripts 1 at the edge of the boundary layer c curved rough tubes c, r helical rough tubes c, s helical smooth tubes rough fully rough regime str, r straight rough tubes trans transitionally rough regime w near the wall
well with others’ experimental data for commercial tubes. However, the results deviated from the performances of commercial tubes at moderate Reynolds number. Colebrook and White [26] investigated turbulent flows in roughened straight tubes experimentally and found Nikuradse’s [25] deviations from the commercial tubes were due to the uniform roughness in tubes. Then Colebrook [27] presented an implicit equation to calculated friction factors in rough straight tubes. The equation was widely used to calculate friction factors for turbulent flow in rough straight tubes. Many researchers [28–32] obtained explicit equations to calculate the friction factors in rough straight tubes with specific limits. Moody [33] presented a composite friction factor chart of all flow regions for rough straight tubes which was widely used in engineering. For rough helical tubes, Clancy [1] measured the pressure drops in rough helical tubes and straight tubes and found friction factors were influenced by Reynolds numbers, helical diameters and roughness heights. McElligott [2] compared friction factors in rough helical tubes and friction factors in rough straight tubes with comparable roughness. The results showed that friction factors in helical tubes depended more on roughnesses than on curvature ratios. This paper describes a theoretical investigation on turbulent flows in rough helical tubes and derived friction factor equations considering the roughness effects, curvature ratios and Reynolds numbers. The equations can predict the friction factors accurately compared with literature data. The comprehensive effects of roughnesses and curvature ratios on the friction factor are also discussed based on the equations.
2. Equation derivation 2.1. Physical model and mathematical description Fig. 1 shows a helical tube schematic with the main parameters and a secondary flow illustration. The helical radius is defined as R, the tube inner radius is defined as r 0 and the helical pitch is defined as b. When the helical pitch is small compared with the helical diameter, torsion effects can be neglected at moderate Reynolds
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527
Fig. 1. Schematic of a helical tube and secondary flow in the cross section.
Fig. 2. The curved tube and velocity vectors in the orthogonal coordinates.
number [19]. Moreover, the effects of pitch on friction factors can be eliminated if the helical radius is modified by Eq. (1) [14].
"
Rc ¼ R 1 þ
b
2pR
2 #
ð1Þ
For this analysis, the helical tube shown in Fig. 1 is simplified to be a helical tube with zero pitch as shown in Fig. 2(a). The toroidal orthogonal coordinate system shown in Fig. 2(b) is adopted, where r denotes the distance from the center of the tube in the cross
section, h denotes the angle between r and the symmetry of the cross section, and u denotes the angular distance from a fixed cross section. V r , V h and V u are taken as the velocity magnitudes in directions of r, h and u. The continuity and momentum equations, Eqs. (2)–(5), are obtained based on the following assumptions: (1) The helical tubes are simplified to be curved tubes with zero pitch. (2) The flows are incompressible and steady.
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(3) The flows are fully developed, so, Vr, Vh and Vu are independent of u. (4) The helical radius is much larger than the tube radius, i.e. R r0.
@V r V r 1 @V h þ þ ¼0 r @h @r r
ð2Þ
Z
@ ðV 2h Þdr þ ½V h V r rr00 d þ @h r0 d r @V h 0 m @r r0 d r0
Z
r0
V 2u sin h
r 0 d
R
Z dr ¼
r0 r0 d
1 @P
qr0 @h
dr
ð8Þ
According to Ito [36],
@P r0 ¼ q V 2u1 sin h @h R
2 @V r V h @V r V 2h V u cos h Vr þ @r r @h r R 1 @P m @ @V h V h 1 @V r ¼ þ r @h q @r r @h @r r
V r ¼ V h ¼ 0 r ¼ r0 ð3Þ
V h1 ¼
2 @V h V h @V h V h V r V u sin h þ þ þ Vr @r r @h r R 1 @P @ @V h V h 1 @V r þm þ ¼ @r @r r @h qr @h r
ð4Þ
2
2
@V u V h @V u @ V u 1 @V u 1 @ V u 1 @P Vr þm þ þ ¼ þ 2 r @r r @ u2 @r r @h qR @ u @r 2
! ð5Þ
Vr O(d), @=@r Oðd1 Þ, and d r0. In the boundary layer, the value of r is approximate to r0. Then Eqs. (2) and (4) are reduced to,
@V r 1 @V h þ ¼0 r 0 @h @r
ð6Þ 2
@V h V h @V h V u sin h 1 @P @2V h m 2 þ þ ¼ @r r 0 @h qr0 @h @r R
ð7Þ
The pressure gradient in h direction in Eq. (7) is set as independent of r and equal to the pressure gradient at the edge of the boundary layer. Eq. (7) is integrated by using Eq. (6).
@V h @r
¼ 0 r ¼ r0 d 1
where V u1 and V h1 are the velocity magnitudes at the edge of the boundary layer in u direction and h direction, respectively. Then Eq. (8) is reduced to,
q
According to Adler [34], the boundary layer will become thinner when the Dean number increases. The friction factor equation is derived based on the boundary layer approximation method. This method divides the cross section of curved tubes into two regions, the central core in the central region and the thin boundary layer near the wall region. The detailed information of the boundary layer is shown in Fig. 3. In the boundary layer, the viscous effect is significant. The thickness of the boundary layer is denoted as d. The axial velocities, V u , will gradually fall to zero when approaching the wall because of the no slip wall conditions. Then V h and V u are comparable in the boundary layer. Order-of-magnitude analysis was done to reduce Eqs. (2)–(5). Similar to the boundary layer theory for plates [35], we know that in the boundary layer, Vh O(1), Vu O(1),
Vr
1 r0
Z
@ @h
0
d
V 2h dy ¼ sh r 0 þ q
dr 0 2 r0 sin h q V sin h R u1 R
Z 0
d
V 2u dy
ð9Þ
where y is the distance normal to the wall near the wall region. As the velocities in r direction in the boundary layer are neglected. The velocities in the curved tubes are composed by velocities in axial direction, V u , and velocities in circumferential direction, V h . Then the velocities are assumed to be:
Vu ¼
V ð1 þ K 2 Þ
1=2
Vh ¼
KV ð1 þ K 2 Þ
1=2
ð10Þ
where K is the ratio of V h and V u . V is the velocity magnitude in curved tubes. According to the relationship between velocities and wall stresses,
su ¼ l
@V u @r w
sh ¼ l
@V h @r w
ð11Þ
Then wall stresses in u direction and h direction can be written as,
su ¼
sw 2 1=2
ð1 þ K Þ
sh ¼
K sw ð1 þ K 2 Þ
1=2
ð12Þ
Y is defined to have the same order-of-magnitude as V m =V .
Y
Vm V
ð13Þ
where V m is the mean axial velocity magnitude and V is the friction velocity magnitude, which is defined by,
Fig. 3. Illustration of boundary layer in the cross section.
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H. Zhao et al. / International Journal of Heat and Mass Transfer 95 (2016) 525–534
rffiffiffiffiffiffi
sw q
V ¼
ð14Þ
In fully rough regimes, y0 ¼ jK s , where j is constant [35]. y0 in Eq. (24) is replaced to give,
As K is the ratio of V h and V u , we can get,
Vu Vm
V h kV m
ð15Þ
Eq. (13) is combined with Eqs. (12) and (14) to get,
su
qV 2m Y2
sh
K qV 2m Y2
ð16Þ
Z p 0
su dh
The friction factor of a helical tube, f c is defined by,
@P 1 R ¼ f c qV 2m @u 2 2r 0
ð19Þ
ð20Þ
A Y2
divided by qV 2m r 0 .
R
R
K 2 eY K s r0
eY K s R
K Y
2
eY K s R
K ðr 0 =RÞ1=2
ð21Þ
ð28Þ
e ¼ K s =r0 as the relative roughness. Comparing K=Y 2
Define e, 2 Y
with K e K s =r0 , we can get that,
1 Ke
ð29Þ
The relationships in Eqs. (28) and (29) can be rewritten as,
where A is constant.
r 1=2 0 K¼D R
2.2. Logarithmic law for turbulent flow near walls
Y 2 eY ¼
The logarithmic velocity distribution law used in this analysis is Prandtl’s velocity-distribution law which was derived from the mixing length theory [35]. It is widely accepted and used for wall turbulence, usually it is called law of the wall. The law can be written as,
where D and E are constants. Eq. (31) can be solved to get Y,
Y ¼ ln
ð30Þ
E Ke
E K eY 2
ð31Þ
¼ ln K eY 2 þ ln E
ð32Þ
Eq. (32) can be rewritten by using Eqs. (21) and (30).
V ¼ aV ln y þ b where V is the velocity magnitude, a and b are constants, V is the friction velocity and y is the distance from the wall. Then for the boundary layer in the helical tube, we can get,
V 1 V ¼ aV ln ðd=yÞ
ð27Þ
K 2 eY K s =r0 with eY K s =R gives,
Y 2 eY
Define that,
fc ¼
ð26Þ
Each term must have the same order-of-magnitude. Comparing
Substitute Eq. (19) into Eq. (18) to get the relationship between Y and f c .
f c Y2
where B and C are constants.Eq. (25) is then combined with Eqs. (13) and (15), we can obtain the relationship between Y and d=K s .
d q @h@ 0d V 2h dy sh r0 q drR0 V 2u1 sin h q rR0 sin h 0 V 2u dy ¼ þ qV 2m r0 qV 2m r0 qV 2m r0 qV 2m r0
ð18Þ
ð25Þ
The order-of-magnitude of each term in Eq. (9) is obtained after
ð17Þ
Eq. (16) is combined with Eq. (17) to get,
@P RqV 2m @u r0 Y 2
V u1 ¼ B ln ðd=K s Þ þ C V
d eY K s
The pressure gradient in u direction is,
@P 2R ¼ @u pr0
2.3. Friction factor equation in the fully rough regime
ð22Þ
1 1=2
fc
¼ F ln
Ge ðr 0 =RÞ1=2 fc
2.4. Friction factor in the transitionally rough regime
y0 ¼ HK s þ Il=qV
V 1 ¼ aV ln ðd=y0 Þ
V u1 ¼ AV ln
According to Colebrook [27], in the transitionally rough regime,
d HK s þ Il=qV
Eq. (35) is combined with Eqs. (13) and (15),
V u1 ¼ aV ln ðd=y0 Þ
d eY ðHK s þ Il=qV Þ
The relationships between y0 and the mean roughness, K s , in the fully rough regime and the transitionally rough regime are different. The following analysis will be divided into two sections, friction factors in the fully rough regime and friction factors in the transitionally rough regime.
ð34Þ
where H and I are constants. Eq. (24) can be rewritten as,
At the edge of the boundary layer, V h1 ¼ 0 and V 1 ¼ V u1 . Eq. (23) can be rewritten as,
ð24Þ
ð33Þ
where F and G are constants. r 0 =R is defined as the curvature ratio.
where V 1 is the velocity magnitude at the edge of the boundary layer in the cross section. For rough helical tubes, the velocity magnitude is zero at a point, just like that for the straight rough tubes near the wall. The distance normal to the wall at this point is set to be y0 [35]. V 1 can be expressed as,
ð23Þ
ð35Þ
ð36Þ
Each side of Eq. (36) is divided by r 0 ,
d LeY Y JeY e þ r0 Re
ð37Þ
Re ¼ 2qV m r0 =l; is the Reynolds number. J and L are constants.
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H. Zhao et al. / International Journal of Heat and Mass Transfer 95 (2016) 525–534
The order-of-magnitude of each term in Eq. (9) is obtained after divided by qV 2m r0 ,
R
R
d q @h@ 0d V 2h dy sh r0 q drR0 V 2u1 sin h q rR0 sin h 0 V 2u dy ¼ þ qV 2m r0 qV 2m r0 qV 2m r0 qV 2m r0
d 2 K r0
K Y2
d R
ð38Þ
d R
Comparing d=R and dK 2 =r0 , we can get the relationship shown in Eq. (39), which is the same as Eq. (28).
K ðr0 =RÞ1=2
ð39Þ 2
Comparing dK =r 0 with K=Y , we can get,
ð40Þ f ¼
Then substitute Eq. (37) into Eq. (40), we can get,
JeY e þ
LeY Y 1 Re KY 2
ð41Þ
Eq. (41) can be rewritten as,
ð42Þ
where M is constant. The friction factor equation is obtained by solving Eq. (42), using Eqs. (21) and (30),
1=2
fc
"
be r0 1=2 c r0 1=2 ¼ a ln þ 3=2 Re fc R R fc
#
ð43Þ
The thickness of the viscous sub-layer decreases as the Reynolds number increases. The roughness will dominates the wall stresses when the roughnesses are all out of the viscous sub-layer. Turbulent flows in the transitionally rough regime will transform to the fully rough regime. With very high Reynolds numbers, Eq. (43) can be written as Eq. (44) by eliminating the term including the Reynolds number. 1=2
fc
be r 0 1=2 fc R
ð46Þ
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 1 2 5 1 2 2 2 2 ðDDÞ þ ðDðDPÞÞ þ ðDM Þ þ ðDLÞ M D L DP
The experimental errors for friction factors are below 5%, which is calculated by Eq. (47). 3.2. Determination of the equation parameters As Eq. (43) is a multiple parameter equation and the friction factor is in implicit expression. Eq. (43) is transformed to Eq. (48) for regression analysis to get the parameters.
c
2.5. Equation compatibility for different flow regimes
¼ a ln
Df ¼ f
," 1=2 # be eð1=a=f c Þ Re ¼ 3=2 þ fc ðr 0 =RÞ1=2 fc
where a, b and c are constants.
1
DP D 2DP D5 p 2 ¼ q 4 0:5qV 2m L M2 L
ð45Þ
ð47Þ
LeY Y M ¼ Je e þ Re KY 2 Y
1
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Xn @ ln u2 ¼ ð Dxi Þ 2 i¼1 u @xi
Du
2
d 1 r 0 KY 2
decreasing when the Reynolds number increased, the flows were in transitionally regime. The parameters in Eq. (43) can be obtained by the regression analysis of experimental data. The differential pressures were measured by water manometer at low flow rates and by mercury manometer at high flow rates. Flow rates were obtained by weighting a mass of water during a specific time interval. The experimental errors were not presented in reference [1]. The relative errors for pressure drops are set as 2% based on the minimum calibration of 1 mm. The maximum relative errors for mass flow rates and length are set as 2% and 0.1%, respectively, based on the significant digits of the experimental data. The relative errors are calculated by the following equations.
In Clancy’s experiments [1], pressure drops are measured in 12 helical tubes for different curvature ratio and different roughness. The Reynolds number varied from 1 103 to 1 105 . The experimental data for each tube are defined as one group of data and is marked by the tube number. As the equation is derived based on the logarithmic velocity distribution law, the friction factors in the laminar flow region should not be used. The critical Reynolds number is calculated by Eq. (49) [15].
ð44Þ
Eq. (44) is the same as Eq. (31), which shows that the derived friction factor equations for transitionally and fully rough regimes are compatible. 3. Equation parameter determination Friction factor equations for transitionally and fully rough regime turbulent flows have been derived. However, the constants in Eqs. (33) and (43) need to be determined using experimental data.
Recri ¼ 2 104
bers varied from 1 103 to 1 105 . As the friction factors kept
r 0:32 0 R
ð49Þ
Several groups of data are used for the regression analysis. The data with Reynolds number below 1 104 are not used. Then about 5 points in each group are left. The total points used are about 30. The curvature ratios and the relative roughnesses are set as constant in Eq. (48) for each group. The results of the regression analysis are shown in Table 1 and the confidence level is 95%. The values of a, b and c shown in Table 1 are averaged, then gives,
a = 0.923033
3.1. The experiments of Clancy The experimental data from Clancy [1] are used to approximate the derived equations. Fig. 4 shows a schematic diagram of the test apparatus of Clancy [1]. The pressure drops in helical tubes were measured with the curvature ratio varied from 0.056 to 0.063, mean roughness varied from about 0.6–4 lm and Reynolds num-
ð48Þ
b = 0.104455
c = 1.142495 Then Eq. (43) can be expressed as,
1 1=2
fc
¼ 0:923 ln
" # 0:104e r0 1=2 1:142 r 0 1=2 þ 3=2 fc R f c Re R
ð50Þ
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H. Zhao et al. / International Journal of Heat and Mass Transfer 95 (2016) 525–534
Fig. 4. Schematic diagram of the test apparatus of Clancy [1].
Table 1 Regression analysis results. Tube number
2
a
0.92589 b 0.10158 c 1.19671 Relative roughness 0.00016 Curvature ratio 0.06214 Adj. R-square 0.99877
Table 2 Equation verification. 3
4
5
9
10
0.92879 0.09756 1.18591 0.00018 0.06101 0.99915
0.91636 0.09101 0.98235 0.00013 0.06115 0.99411
0.92058 0.12272 1.19694 0.00010 0.06186 0.99840
0.92567 0.09693 1.11030 0.00013 0.06101 0.99858
0.92091 0.11693 1.18276 0.00011 0.06158 0.99750
Re 1 10
Relative roughness 5
0:315 10
3
Experiment
Eq. (50)
Relative error (%)
0.0257
0.025916
0.8
1 105
0:2849103
0.0247
0.024705
0.01
2 104
0:315 103
0.035
0.034533
1.3
2 104
0:302 103
0.033
0.032989
0.03
4
0:284 103
0.0324
0.032957
1.7
2 10
According to the discussion in Section 2.5, when Re is high enough, the turbulent flows in the transitionally rough regime will convert to the fully rough regime. Influence of Re will diminishes, left only the relative roughness’s influence. We set c as zero in Eq. (50). The equation for the fully rough regime is obtained.
1 1=2
fc
¼ 0:923 ln
0:104e r 0 1=2 ð Þ fc R
ð51Þ
The coefficients in Eq. (50) are obtained by regression analysis of experimental data with random roughnesses. But the uniform roughness effects and random roughness effects on friction factors are different [26,27]. Then coefficients of the equations to predict friction factors in uniform roughened helical tubes [37–40] in the transitionally rough regime and fully rough regime will be different. 3.3. Equation verification The equations are verified by the experimental data from McElligott [2]. In Ref. [2], the Reynolds numbers are 2 104 and 1 105. The relative roughnesses are from 0.284 103 to 0.315 103 which are different from those of Ref. [1]. Friction factors measured in the experiments, factors calculated by Eq. (50) and the relative errors are shown in Table 2 and Fig. 5, which shows a good prediction by using Eq. (50). 4. Discussion A good friction factor equation for rough helical tubes should be compatible with that for smooth helical tubes, which means when the roughness diminishes, the equation for rough tubes should
Fig. 5. Equation verification.
become equivalent to that for smooth tubes. We will discuss this in Section 4.1. Eq. (50) shows that the friction factors of rough helical tubes are influenced by the combined effects of the Reynolds number, the curvature ratio and the relative roughness. We will discuss these effects in Sections 4.2, 4.3 and 4.4. 4.1. Compatibility with Ito’s equations Equation (43) is derived on the bases of Ito’s theoretically investigation on turbulent flows in smooth curved tubes. Ito [15] investigated turbulent flow pressure drops in smooth curved tubes experimentally and theoretically. Two equations are derived based on the 1/7th power velocity distribution law and the logarithmic
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H. Zhao et al. / International Journal of Heat and Mass Transfer 95 (2016) 525–534
velocity distribution law, respectively, which are shown in Eqs. (52) and (53). The equations for smooth tubes have been widely used.
fc
1=2 R 0:316 ¼ h i1=5 2 r0 Re rR0 "
1 1=2
fc
¼
3=2 2:11log10 Ref c
1=2 # R 0:23 r0
ð52Þ
ð53Þ
We set the relative roughness as zero in Eq. (50) to calculate the friction factors in smooth helical tubes to see whether the equation is consistent with Ito’s equations. Friction factors are also calculated by Eqs. (52) and (53) with the same curvature ratio and the same Reynolds number. The results are shown in Fig. 6. The relative errors are less than 4%. The curvature ratios are 0.1 and 0.01. Cur means the curvature ratio.
Fig. 7. Comparison of friction factors in helical tubes with different roughnesses and the same curvature ratios.
4.2. Roughness effects on the friction factors Friction factors are calculated using Eq. (50) for the same curvature ratio but different relative roughnesses. The Reynolds numbers varies from 1 104 to 6 105. The relative roughness varies from 4 104 to 8 103 . The curvature ratio is 0.1, where the helical radius is 50 mm and the tube inner radius is 5 mm. The results are shown in Fig. 7. Fig. 7 shows that the friction factors decrease as the Reynolds numbers increase in a range of Re with relatively low values. The range is different for tubes with different roughness, like the transitionally rough regime in Moody chart. In the transitionally rough regime, the factors depend both on the roughness and the Reynolds number. When Re is high enough, friction factor keeps unchanged as Reynolds numbers increase, like the fully rough regime in Moody chart. 4.3. Roughness and curvature ratio effects on the transition Friction factors in the transitionally rough regime, ftrans, are calculated by Eq. (50) and friction factors in fully rough regime, f rough , are calculated by Eq. (51). The curvature ratios are 0.1 and 0.05. The relative roughnesses are 0.001, 0.002 and 0.004 for each curvature ratio. Re varies from 1 104 to 6 105 . Ratios, f trans =f rough are calculated and the results are shown in Fig. 8. When the flows in the transitionally rough regime transfer to the fully rough regime, the ratio approaches to unit. The trends of the ratio as the Reynolds number increases reflect the curvature ratio effects and roughness
Fig. 6. Comparison of friction factors calculated by Eqs. (50), (52) and (53).
effects on the transition progress. Fig. 8 shows that the curvature ratio has little effect on the transition. Roughness dominates the transition process, like that in straight tubes. 4.4. Combined effects of roughness and curvature ratio on friction factors Eq. (50) shows that the roughness effects on friction factors are influenced by Reynolds numbers and curvature ratios. The factors, f c;s , in smooth helical tubes are calculated by Eq. (50) with the relative roughness set as zero. The friction factors, f c;r , in rough helical tubes are calculated by Eq. (50) with the relative roughness set as 0.004 and the Reynolds number varies from 1 105 to 6 105. The ratio, f c;r =f c;s reflects the roughness effects on the friction factors in helical tubes. f c;r =f c;s is calculated with different curvature ratio and the results are shown in Fig. 9. The results in Fig. 9 shows that f c;r =f c;s increases as the curvature ratio increases or the Reynolds number increases. That means roughness effects on friction factors increase with increasing curvature ratios and Reynolds numbers. When the curvature ratios increase or the Reynolds numbers increase, the centrifugal force increases and the secondary flow intensities increase, which will cause more pressure drops in the helical tube. Eq. (50) also shows that the curvature ratio effects on friction factors are influenced by relative roughness and the Reynolds
Fig. 8. Comparison of f trans =f rough with different curvature ratios and different roughnesses.
H. Zhao et al. / International Journal of Heat and Mass Transfer 95 (2016) 525–534
533
rough helical tubes are theoretically derived in this paper. The parameters in the equations are determined by regression analysis of experimental data of Clancy [1] and verified by experimental data of McElligott [2]. The influences of curvature ratio, relative roughness and the Reynolds number on friction factors in rough helical tubes are analyzed. The main conclusions are, (1) Two equations are theoretically derived for calculating the friction factors for turbulent flows in rough helical tubes. One is for the transitionally rough regime and the other is for the fully rough regime. The friction factors are influenced by the Reynolds number, curvature ratio and the relative roughness. (2) The parameters of the equations are obtained by regression analysis of experimental data. (3) The roughness has similar effects on friction factors for turbulent flow in rough helical tubes as that in rough straight tubes. When the roughness and the Reynolds number are high enough, the friction factors become independent of the Reynolds number. (4) For turbulent flow in rough helical tubes, the roughness effects on friction factors increase as the curvature ratio increases. The roughness effects and curvature ratio effects increase as the Reynolds number increase.
Fig. 9. Comparison of f c;r =f c;s with different curvature ratios.
Acknowledgments This work was financially supported by the National Natural Science Foundation of China (51576103) and the National S&T Major Project (Grant No. ZX06901). References
Fig. 10. Comparisons of f c;r =f str; r with different Reynolds numbers.
number. Friction factors in helical tubes, f c;r , are calculated by Eq. (50) with curvature ratio of 0.1, and the relative roughness varies from 0.0008 to 0.0166. Friction factors in rough straight tubes with the same relative roughness, f str;r , are calculated by Eq. (54) presented by Churchill [30].
( f str; r ¼
2log10
"
e
7:4
þ
0:9 #)2 7 Re
ð54Þ
The ratio, f c; r =f str; r , reflects the curvature ratio effects on frication factors. f c; r =f str; r is calculated with different Reynolds number and the results are shown in Fig. 10. The results in Fig. 10 shows that the ratio, f c;r =f str; r , increases with the increasing roughness height, which means the curvature ratio effects on friction factors in rough helical tubes increase with the increasing roughness.
5. Conclusions Helical tubes are extensively used as heat exchangers in chemical, petroleum and nuclear industries. Turbulent flows in helical tubes are more complex than those in straight tubes. However, previous investigations on friction factors in helical tubes are all for smooth helical tubes. All the commercially used helical tubes are rough tubes. Two equations for calculating friction factors in
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