- Email: [email protected]
Prediction of heat transfer coefficients of shell and coiled tube heat exchangers using numerical method and experimental validation
International Journal of Thermal Sciences 107 (2016) 196e208
Contents lists available at ScienceDirect
International Journal of Thermal Sciences journal homepage: www.elsevier.com/locate/ijts
Prediction of heat transfer coefficients of shell and coiled tube heat exchangers using numerical method and experimental validation Ashkan Alimoradi, Farzad Veysi* Mechanical Engineering Department, Faculty of Engineering, Razi University, Kermanshah, Iran
a r t i c l e i n f o
a b s t r a c t
Article history: Received 7 October 2015 Received in revised form 17 March 2016 Accepted 11 April 2016
In this work, the heat transfer of shell and helically coiled tube heat exchangers was investigated. Numerical and experimental methods were used to investigate the effect of physical properties of fluid (i.e. viscosity, thermal conductivity, specific heat capacity and density), operational parameters (i.e. the velocity and temperature of fluid) and geometrical parameters (i.e. pitch, diameter of the tube, diameter of shell's inlet, diameters of coil and shell, heights of coil and shell, and the distance between the inlet and outlet of the shell) on Nusselt numbers of both sides. Totally 42 cases and 15 tests were investigated in the numerical analysis and experimental work, respectively. Measurements and analysis were performed, when the steady state was attained. The working fluid of both sides is water, which its viscosity and thermal conductivity were assumed to be dependent on temperature, in the numerical analysis. Results indicate that if the pitch size is doubled, the shell side Nusselt number increases by 10%, while the coil side Nusselt numbers increases by only 0.8%. Also it was found that an increase of 50% in the height and diameter of the shell causes a decrease of 34.1% and 28.3% in the Nusselt number of the shell side, respectively. Based on the results, two correlations were developed to predict Nusselt numbers of coil side and shell side for wide ranges of Reynolds and Prandtl numbers (1000 < Rec < 27,000, 2000 < Resh < 49,000 and 1.9 < Prc and Prsh < 7.1). These correlations were compared with the experimental data of the present study and previous works. It was found that these correlations are in good agreement with the experimental data for wide ranges of operational and geometrical parameters. © 2016 Published by Elsevier Masson SAS.
Keywords: Helical coil Heat exchanger Numerical Experimental Nusselt number
1. Introduction Helically coiled tube heat exchangers are widely used in various industries such as piping systems, air conditioning, storage tanks, and chemical reactors. In petroleum units, the heat exchanger, which is used to cool the lubricating oil of the mechanical seal of pumps, is a shell and coiled tube heat exchanger. These types of exchangers are one of the compact heat exchangers types used to increase heat transfer rate, require less volume and weight compared with other types of heat exchangers. Modeling of the heat transfer characteristics of this type of heat exchanger, was considered in different literatures. Most studies are about the coil side heat transfer characteristics, while the shell side heat transfer was not investigated in more details.
* Corresponding author. E-mail addresses: [email protected], [email protected]. ir (A. Alimoradi), [email protected], [email protected] (F. Veysi). http://dx.doi.org/10.1016/j.ijthermalsci.2016.04.010 1290-0729/© 2016 Published by Elsevier Masson SAS.
Salimpour [1] investigated heat transfer coefficients of shell and helically coiled tube heat exchangers experimentally. He found that the shell side heat transfer coefficient increases, with increase of the pitch size. Two correlations were developed to predict the inner and outer heat transfer coefficients as follows:
Nuc ¼ 0:152De0:431 Pr1:06 g0:277
(1)
Nush ¼ 19:64Re0:513 Pr0:129 g0:938 sh
(2)
M. Moawed [2] experimentally studied the forced convection from outside surfaces of helical coiled tubes with constant wall heat flux. Experiments were performed in an open circuit airflow wind tunnel system, operated in suction mode. A general correlation for the average Nusselt number was obtained as follows:
A. Alimoradi, F. Veysi / International Journal of Thermal Sciences 107 (2016) 196e208
Nomenclature
u
area m2 specific heat capacity J/kg c diameter m dean number ¼ Re(dt/dc)0.5 distance between inlet and outlet of the shell m height m heat transfer coefficient W/m2 K Fanning friction factor turbulence kinetic energy J/kg thermal conductivity W/m K coil's length m flow rate kg/s Nusselt number pitch m Prandtl number ¼ m C/k heat transfer rate W heat flux W/m2 Rayleigh number Reynolds number uncertainty temperature K or C
A C d De f H h j K k l m_ Nu p Pr Q q Ra Re S T
0:914 0:281 Nuo ¼ 0:0345Re0:48 dc dt;o p dt;o
Subscripts b bulk c coil cr critical Deq equivalent diameter h hydraulic i inner k stage Ln normalized length o outer sh shell t tube v shell's inlet w wall
(3)
6:6 102 Re 2:3 103 7:086 dc dt;o 16:142 1:81 p dt;o 3:205 Beigzadeh et al. [3] developed Artificial Neural Network (or ANN) models to predict the heat transfer and friction factor of the coil side, in helically coiled tubes. The predicted Nusselt numbers of the coil side, were compared with the proposed equation by Xin and Ebadian [4] and a reasonable agreement was observed. Xin and Ebadian [4] Studied effects of the Prandtl number and geometric parameters on the local and average convective heat transfer characteristics in helical pipes and suggested the following equation for the coil side Nusselt number: 0:643
Nuc ¼ 2:153 þ 0:318De
0:177
Pr
average velocity m/s
Greek symbols g dimensionless pitch p/pdc d curvature ratio dt,i/dc ε turbulent dissipation J/kg m viscosity Pa.s r density kg/m3
Which is applicable for:
197
Jayakumar et al. [5,6] numerically and experimentally studied the coil side of shell and helically coiled tube heat exchangers and found that the use of temperature dependent properties of working fluids results in prediction of more accurate heat transfer coefficients. They also found that arbitrary boundary conditions, such as constant wall temperature and constant heat flux are not applicable for prediction of heat transfer, when fluid-to-fluid heat transfer occurs in a heat exchanger. Correlations were developed to calculate the coil side heat transfer coefficient of the heat exchanger as follows:
Nuc ¼ 0:025De0:9112 Pr0:4
(6)
Which is applicable for [5]:
2000 < De < 12000 And:
(4) Nuc ¼ 0:085De0:74 Pr0:4 d0:1
(7)
Nuc ¼ 0:116Re0:71 Pr0:4 d0:11
(8)
Which is applicable for:
20 De 2000 0:7 Pr 175 0:0267 dt;i =dc 0:0884
Which are applicable for [6]:
And:
Nuc ¼ 0:00619Re0:92 pr0:4 Which is applicable for:
5000 Re 10000 0:7 Pr 5 0:0267 dt;i =dc 0:0884
dt;i 1 þ 3:455 dc
(5)
14000 < Re < 70000 3000 < De < 22000 3 < Pr < 5 0:05 < d < 0:2 Genic' et al. [7] experimentally studied the shell side heat transfer of helically coiled tube heat exchangers. They investigated three heat exchangers (with different geometrical parameters) and proposed the following correlation for the shell side Nusselt number (the Reynolds number is based on the hydraulic diameter):
198
A. Alimoradi, F. Veysi / International Journal of Thermal Sciences 107 (2016) 196e208
1
Nush ¼ 0:5Re0:55 Pr3
mb mw
0:14 (9)
shell side volume surface contact with fluid
(10)
Pawar et al. [8] studied isothermal steady state and nonisothermal unsteady state conditions in helical coils for Newtonian and non-Newtonian fluids experimentally. The following correlation was proposed for the coil side Nusselt number:
Nuc ¼ 0:0472De0:8346 Pr0:4
(15)
Which is applicable for:
They defined the hydraulic diameter as follows:
dh ¼ 4
0:3 0:4533 NuDeq ¼ 0:0041Re0:2 Deq Prsh RaDeq
(11)
120 < ReDeq < 1200 1:2 107 < RaDeq < 3:2 108 Kalb and Seader [13] theoretically studied the fully developed heat transfer in curved tubes with circular cross section at steady state, under the thermal boundary condition of axially uniform wall heat flux with peripherally uniform wall temperature. They proposed the following equation, for coil side Nusselt number:
Nuc ¼ 0:913De0:476 Pr0:2
(16)
Which is applicable for: Which is applicable for:
586 De 4773 0:055 d 0:0757 3:83 Pr 7:3
80 De 1200 0:7 Pr 5
Hardik et al. [9] experimentally investigated the effect of the curvature of the helical coil, Reynolds number and Prandtl number on the friction factor and local Nusselt number. They used water as the working fluid in their experiments. A correlation was suggested for fully developed coil side Nusselt number as follows:
dc Nuc ¼ 0:0456 dt;i
!0:16 0:8
Re
0:4
Pr
(12)
Pimenta et al. [10] experimentally studied the heat transfer coefficients of Newtonian and non-Newtonian fluids, at constant wall temperature as a boundary condition, in fully developed laminar flow inside a helical coil and suggested the following equation:
Nuc ¼ 0:5De0:481 0:465 Pr0:367
(13)
Janssen et al. [14] studied convective heat transfer in coiled tubes experimentally and numerically. For the boundary condition of a constant wall temperature, the following correlation was suggested:
0 Nuc ¼ @
d
0:32 þ 3 dt;i c
0:86
d 0:8 dt;i c
1 ARe0:5 Pr0:33
dt;i dc
0:14þ0:8ddt;i c
(17)
Dravid et al. [15] studied the heat transfer of helically coiled tube heat exchangers numerically and experimentally. They suggested the following equation:
pffiffiffiffiffiffi Nuc ¼ 0:76 þ 0:65 De Pr0:175
(18)
Which is applicable for:
Which is applicable for:
50 < De < 2000 5 < Pr < 175
15 < De < 1020 10 < Pr < 353 dt;i dc ¼ 0:0263 p ¼ 0:01134m dt;i ¼ 0:004575m
Bai et al. [16] performed an experiment to study the turbulent heat transfer in a horizontal helically coiled tube, over a wide range of experimental parameters. The following correlation was proposed:
Beigzadeh et al. [11] applied Adaptive Neuro-Fuzzy Inference System (or ANFIS) and Genetic Algorithm techniques (or GA) to model and predict the thermal and fluid flow characteristics, in helically coiled tubes. The experimental data were employed as inputs of the model. The Nusselt number was determined as follows:
Nuc ¼ 0:328Re0:58 Pr0:4
Nuc ¼ 0:359Re0:781 Pr0:016 d0:933 g0:172
(14)
Ghorbani et al. [12] studied the mixed convection heat transfer in shell and coiled tube heat exchangers for various Reynolds numbers, tube to coil diameter ratio and dimensionless coil pitch experimentally. It was found that the tube diameter has negligible influence on the shell-side heat transfer coefficient and increasing the coil pitch increases it. They also found that the equivalent diameter of the shell which has the same definition with the hydraulic diameter that, Genic' et al. [7] have been used (i.e. Equation (10)) is the best characteristic length for the Nusselt number of the shell side. Also they proposed a correlation for the shell side Nusselt number as follows:
mb mw
0:11 (19)
Which is applicable for:
4:5 104 < Re < 19 104 Prasad [17] proposed correlations for two special situations of geometrical parameters for shell side Nusselt number as follows:
Nush ¼0:057Re0:8 dh For dc dt;o ¼ 17:24
(20)
Nush ¼0:110Re0:8 dh For dc dt;o ¼ 34:9
(21)
Cioncolini et al. [18] experimentally investigated the turbulence emergence process in adiabatic coiled pipes. Results were compared with equation proposed by ESDU [19] and a reasonable agreement was observed. Equation is as follows:
A. Alimoradi, F. Veysi / International Journal of Thermal Sciences 107 (2016) 196e208
199
0 !0:68 1 RePr 0:11 d c @1 þ 0:059Re0:34 A mb qffiffiffi Nuc ¼ dt;i mw 1:07 þ 12:7 2f Pr0:667 1 j 2
j ¼ ðlnðReÞ 3:28Þ2
(23)
Which is applicable for:
1:5 104 Re 3 105 0:7 Pr 11:6 4:9 dc dt;i 271 Mirgolbabaei et al. [20] numerically investigated the mixed convection heat transfer from vertical helically coiled tubes in a cylindrical shell at various Reynolds and Rayleigh numbers, various coil to tube diameter ratios and non-dimensional coil pitches. While most studies, by the time, was focusing on constant wall temperature or constant heat flux, in this study it was a fluid-tofluid heat exchanger to emulate the real shell-to-coil side fluids thermal mechanism. They studied the influence of the tube diameter, coil pitch and shell side mass flow rate on the shell side heat transfer coefficient. They found that the normalized length of the shell is the best characteristic length for the Nusselt number of the shell side. Also they proposed a correlation for the shell side Nusselt number as follows:
NuLn ¼ 0:073
Dt Dc
0:769
0:231 Re0:005 Pr0:15 Ln Ln RaLn
(24)
Which is applicable for:
40 < ReLn < 205 8:1 106 < RaLn < 2:2 108 Jamshidi et al. [21] determined coil and shell side heat transfer coefficients using Wilson's plots experimental apparatus. Taguchi method was used to investigate the effect of fluid flow and geometrical parameters on the heat transfer rate. Results indicate that the higher coil diameter, coil pitch and mass flow rate in shell and tube can enhance the heat transfer rate in these types of heat exchangers. As can be seen, in correlations proposed for the coil side Nusselt number, the effect of the velocity of fluid, diameter of the tube and coil, changes significantly from one equation to another. Also correlations proposed for the shell side's Nusselt number do not contain all geometrical parameters of the heat exchanger. In this study the effect of operational and geometrical parameters of helically coiled tube heat exchangers on the Nusselt numbers of both sides was investigated using numerical method and experimental validation. At the end of this study, two correlations are proposed to predict Nusselt numbers of the coil side and the shell side for wide ranges of Reynolds numbers and geometrical parameters.
(22)
consists of two parts: helical coil and shell. The helical coil used in the heat exchanger, has been made from an Aluminum tube. Care was taken to locate the coil into the middle of the shell. A pipe (with a specified length and diameter) made from PE, was used as the shell of the heat exchanger. The constructed heat exchanger is shown in Fig. 3b. To prevent heat losses from outer surfaces of the shell, it was well insulated using fiberglass (4 cm thickness). The heat exchanger is positioned horizontally during experiments. Forced convection was considered as the boundary condition for the inner and outer surfaces of the coil. In addition to the heat exchanger, There are different components else in the set up. A Rotameter (Model: LZM-15, 0.2e2 GPM) was used to measure the flow rate inside the coil, while the flow rate inside the shell was measured by use of a calibrated pot (which has a capacity of 7.5 L) and a timer. K type thermocouples were installed in: 1) Inlet and outlet of the coil (i.e. T1 and T2). These thermocouples were inserted in the small holes made in the inlet and outlet plastic tubes of the coil side and sealed to prevent any leakage. 2) Three points on the coil's surface (i.e. T5, T6 and T7). First, three small grooves were made on the outer surface of the coil carefully. Then each thermocouples are imbedded in the grooves and attached to the coil's surface 3) Inlet, outlet and two different points of the shell (i.e. T3, T4, T8 and T9, respectively). The process for installation of T3 and T4 is like that used for T1 and T2. Also T8 and T9 are easily inserted in the two small holes made in the shell. The distribution of thermocouples can be seen in Fig. 2. These temperatures are recorded by a Data logger (Model: BTM-4208SD, 12 channels, 0.1 c precision). All connections of the heat exchanger, were carefully sealed to prevent any leakage. An electric heater (2000 w) has been placed in the storage tank for heating the water. During experiments, the input electric power to the heater is not changed. When the temperature of the water inside the storage
2. Experimental work 2.1. Materials and set up The experimental set up and its schematic diagram are shown in Figs. 1 and 2, respectively. The heat exchanger used for these experiments is like the heat exchanger shown in Fig. 3a, which
Fig. 1. Experimental set up.
200
A. Alimoradi, F. Veysi / International Journal of Thermal Sciences 107 (2016) 196e208
Fig. 2. Flow diagram of the experimental set up.
tank reaches to a prescribed temperature (65e85 c), the centrifugal pump is started to transfer the hot water to the helical coiled tube. Then the cooled water will come back from the coil to the storage tank to be reheated. The cooling water, which is supplied from the city water, will flow into the shell side through the bottom of the heat exchanger (to ensure that, all air inside the heat exchanger is discharged), but will not be recovered after exit. Two ball valves are used to control the flow rate of both sides. Experiments were performed in Razi University of Kermanshah, Iran. It seems, there are different parameters affect the thermal performance and heat transfer coefficients of these types of heat exchangers. In addition to operational parameters and physical properties of fluids, geometrical properties of the heat exchanger can also influence the thermal performance and heat transfer coefficients. Geometrical parameters, which may influence the thermal performance, were shown in Fig. 4a. In this work, effects of these mentioned parameters were investigated. Dimensions of tested heat exchanger have been shown in the Table 1.
2.2. Calculations of Nusselt numbers After a few minutes (often 10e15 min), temperatures will be fixed and the heat transfer rates of coil side and shell side will be approximately equal. These suggest that, the flow is at steady state. Once steady state is attained, values of flow rates and temperatures will be noted. By changing flow rates of both sides, different test runs have been created (totally 15 test runs). Range of the operational parameters is given in Table 2. As can be seen, wide ranges of temperatures and flow rates were covered for both sides. To calculate Nusselt numbers of the coil side and shell side, following equations have been used: Fig. 3. a: The typical heat exchanger. b: The constructed heat exchanger.
Qc ¼ m_ c CðT1 T2 Þ
(25)
A. Alimoradi, F. Veysi / International Journal of Thermal Sciences 107 (2016) 196e208
201
Fig. 4. (a): Geometrical parameters of the heat exchanger. (b): Control volumes created on the coil. (c): Control volumes created on the heat exchanger.
Table 1 Dimensions of heat exchangers for the numerical and experimental works. Parameters (m)
dc
dsh
Hc
Hsh
dt,i
dt,o
dv
f
p
l
Numerical Experimental
0.08 0.08
0.12 0.1
0.2 0.23
0.24 0.34
0.01 0.0076
0.01 0.01
0.01 0.009
0.25 0.338
0.02 0.03
2.64 1.92
Table 2 Ranges of operational parameters. Parameters
Range
Flow rate of the coil side Flow rate of the shell side Ti,c To,c Ti,sh To,sh
1.1e7 Lpm 1e6.48 Lpm 48.3e80.6 c 45.2e65.1 c 25.2e30.3 c 31.5e56.6 c
Qsh ¼ m_ sh CðT4 T3 Þ qw;c
Qc Q ¼ ; qw;sh ¼ sh Ai Ao
hc ¼
qw;c
jTc Tw
hsh ¼
qw;sh
jTw Tsh
Nuc ¼
hc dt;i
Nush ¼
kc hsh dt;o ksh
(30)
(31)
Mean temperatures of the wall, coil and shell (i.e. Tw ; Tc and Tsh ) are obtained by averaging (T5, T6 and T7), (T1 and T2) and (T3, T4, T8 and T9), respectively.
(26) 2.3. Uncertainty analysis
(27)
(28)
(29)
There are always errors in measurements so absolute measurements do not exist. The calculations of uncertainty analysis are based on the method proposed by Abernethy et al. [22]. The maximum uncertainty for coil side flow rate, shell side flow rate, temperatures, inner diameter of the tube, outer diameter of the tube, inner area of the coil, outer area of the coil and length of the coil are 5%, 4%, 0.2%, 1.31%, 1%, 1.31%, 1% and 0.05%, respectively. Also the uncertainty for specific heat capacity and thermal conductivity were assumed to be negligible. Based on Equations (25) and (31), following equations were used to calculate the maximum uncertainty for Nusselt numbers:
202
A. Alimoradi, F. Veysi / International Journal of Thermal Sciences 107 (2016) 196e208
SQ ¼
Sqw ¼ Sh ¼
S2m
S2Q
S2qw
þ
S2C
þ
S2A 2
2
þ SDT
0:5
0:5
þ SDT
0:5
SNu ¼ S2h þ S2dt þ S2k
(32)
b) Residuals of energy equation was considered less than 107. c) Residuals of K and ε equations were considered less than 103.
(33)
Nuc and Nush will be obtained based on Equation (25) until 31. It seems that they are functions of following parameters:
(34) 0:5
(35)
Based on Equations (32)e(35), the maximum uncertainty for coil side and shell side Nusselt numbers are 5.36% and 4.28% respectively. 3. Numerical work A numerical model was constructed in order to predict the heat transfer phenomena in the shell and coiled tube heat exchanger, which is shown in Fig. 3a. It was divided into several small control volumes represented in Fig. 4b and c. An attempt was made to use regular mesh (involving pentahedral and hexahedral elements) in the most regions of the heat exchanger. But in some regions (around the outer surface of the coil and the inlet and outlet regions of the shell) the irregular mesh (involving tetrahedral elements) was used inevitably. A computer code was developed for the model analysis. In addition to the primary governing equations (i.e. continuity, momentum and energy), a model must be used for turbulence. It will be shown that the standard K-ε as the turbulence model and the SIMPLE (Semi Implicit Method for Pressure Linked Equations) algorithm as the pressure and velocity coupling scheme are good choices for the numerical analysis. Also first order Upwind was used as discretization scheme for momentum, energy and K-ε equations. The temperatures and flow rates of entering fluids are known for both sides. In addition to these two variables, values of turbulence intensity and hydraulic diameters for inlets and outlets of both sides must be determined. It was assumed that the hydraulic diameter of the coil side is equal to the inner diameter of the tube (i.e. dt,i) and for the shell side is equal to the diameter of the shell's inlet (dsh). The turbulence intensity (i.e. I) is defined as the ratio of the root-mean-square of the velocity fluctuations to the mean flow velocity. The turbulence intensity in the core of a fully developed duct flow can be estimated from the following formula derived from an empirical correlation for pipe flows [23]: 1=8
I ¼ 0:16ReDh
(36)
It was observed that the use of constant values for the thermal and transport properties of the heat transport mediums, results in prediction of an inaccurate heat transfer coefficient [5]. So the viscosity and thermal conductivity are considered to be dependent on temperature as follows [5]:
mðTÞ ¼ 2:1897e 11T4 3:055e 8T3 þ 1:6028e 5T2 0:0037524T þ 0:33158
(37)
kðTÞ ¼ 1:5362e 8T3 2:261e 5T2 þ 0:010879T 1:0294 (38) Residuals of governing equations were considered as follows: a) Residuals of continuity and momentum equations were considered less than 104.
Nuc ¼ F Rec ; Prc ; dc ; dsh ; Hc ; Hsh ; dv ; dt;i ; p; f
(39)
Nush ¼ G Resh ; Prsh ; dc ; dsh ; Hc ; Hsh ; dv ; dt;o ; p; f
(40)
After collecting geometrical parameters into non-dimensional combination of parameters (dimensionless groups), Equations (39) and (40) will be simplified as follows:
Nuc ¼ F Rec ; Prc ;
dc dsh Hc Hsh dv f p ; ; ; ; ; ; dt;i dt;i dt;i dt;i dt;i dt;i dt;i
! (41)
dc dsh Hc Hsh dv f p ; ; ; ; ; ; Nush ¼ G Resh ; Prsh ; dt;o dt;o dt;o dt;o dt;o dt;o dt;o (42) To obtain the effect of these parameters on the Nusselt numbers, each parameter will be changed on average five times, while other parameters are held constant. So the numerical analysis was performed for different cases (totally 42 cases). Reynolds numbers will be obtained from the following equations:
Rec ¼
rc uc dt;i mc
Resh ¼
rsh ush dt;o msh
(43)
(44)
4. Results and discussions 4.1. Mesh independency study Mesh independency study was performed on 5 different grids, which are like the grid shown in Fig. 4b and c (but with different cell's density), for a special case. Results can be seen in Table 3. As seen, changes of outlet fluids temperatures will decrease to less than 0.1 c from grid 4 to 5, so more refinement of mesh does not decrease the temperatures errors in any appreciable way. 4.2. Operational parameters effects With changes in operational parameters (i.e. inlet velocities and temperatures of fluids), Reynolds and prandtl number will change. So to investigate the effect of operational parameters, Reynolds and Prandtl numbers must be changed. To investigate the effect of Re, inlet velocities are changed, but when they have been changed, temperatures of outlet fluids will change too and therefore, mean temperatures of fluids will change. Change of mean temperatures of the fluids cause the viscosity and thermal conductivity change too (according to Equations (37) and (38)). So the Prandtl number will change that is not desirable. So it seems that it is better to investigate the effect of Pr number first. For this purpose, temperatures of inlet fluids must be changed, but in this case average temperatures of fluids and thus viscosities will change. With change of viscosities, Reynolds numbers will change too that is undesirable. So the problem must be solved simultaneously by trial and error because the Reynolds and Prandtl numbers are
A. Alimoradi, F. Veysi / International Journal of Thermal Sciences 107 (2016) 196e208
203
Table 3 Mesh independency study. Grid
1
2
3
4
5
Number of coil side cells Number of shell side cells To,c (c ) To,sh (c )
156,576 464,923 81.49 66.65
312,970 666,001 82.1 63.67
350,637 749,444 82.6 61.18
381,962 814,457 82.65 60.97
472,364 1,056,211 82.74 60.95
interdependent by the viscosity. It was assumed that the Nusselt number changes with Re and Pr as follows:
Nu ¼ c1 Rem Pr n
(45)
Where c1 is a constant. To obtain m and n the following procedure was performed: First, a power is guessed for the Prandtl number (for example n ¼ 0.4). Then, by changing inlet temperatures of fluids (at constant inlet velocities), the viscosity and then the Reynolds number will changed. With change of Reynolds number, its power (i.e. m) will be obtained. In this case, Equation (45) will be simplified as follows:
Nu$Pr n ¼ c1 Rem
(46)
On the other hand, by changing inlet velocities (at constant inlet temperatures), the new power for the Reynolds number will be obtained (i.e. m0 ). In this case, Equation (45) will be simplified as follows:
Nu$Pr n ¼ c2 Rem
0
(47)
Where c2 is a constant. If m and m0 be equal, the solution procedure will be stopped otherwise, the mentioned procedure must be repeated. The new power of Pr for the next stage (i.e. nkþ1) will be obtained from the following equation:
nkþ1 ¼ m0 k þ ðnk mk Þ
(48)
These processes have been shown in Table 4 for the both sides. Fig. 5a and b represent change of Nusselt numbers versus Reynolds numbers for the final stage. Dimensions of the heat exchanger, which is used to investigate the effect of mentioned parameters (i.e. Re and Pr) have been shown in Table 1. As seen, the power of Pr for coil side is 0.315 that is very close to 0.33 that Janssen et al. [14] have predicted, while most of previous works (e.g. [4e6,8,9,16]) have estimated this power equal to 0.4. This power is equal to 0.717 for the shell side, which in no previous work has been obtained. Also, the power of Re for coil side is 0.685 that is very close to 0.71, that was reported in [6]. These results have been obtained for following ranges:
1000 < Rec < 27000 2000 < Resh < 49000 1:9 < Prc and Prsh < 7:1
Recr ¼ 2300ð1 þ 8:6
dt;i dc
0:45 ! (49)
4.3. Geometrical parameters effects and final formulations
Coil shell
n
m
m0
Stage 1
0.4 0.8 0.319 0.716 0.315 0.717
0.779 0.815 0.690 0.722 0.685 0.723
0.698 0.731 0.686 0.723 0.685 0.723
Stage 3
Different correlations have been suggested to calculate the critical Reynolds number for flow inside helically coiled tubes. Schmidet [24] has proposed a correlation, which is determining the critical Reynolds number as follows:
In this work, the fluid flow inside the helical coil is mainly in the range of turbulent flow, according to (49).
Table 4 Trial and error for estimating the power of Re and Pr.
Stage 2
Fig. 5. (a): Nusselt number of coil side versus Reynolds number. (b): Nusselt number of shell side versus Reynolds number.
It was assumed that the change of Nusselt number with each dimensionless geometrical parameter, is as follows:
Nu ¼ c3 Rem Pr n xr
(50)
Where c3 is a constant and x is one of these dimensionless geometrical parameters: dc/dt, Hc/dt, f/dt or p/dt. m and n also have been obtained for both side in previous section. To obtain the effect
204
A. Alimoradi, F. Veysi / International Journal of Thermal Sciences 107 (2016) 196e208
Fig. 6. (a): Nusselt number of coil side versus dimensionless dc, f, Hc and p. (b): Nusselt number of shell side versus dimensionless dc, f, Hc and p. (c): Nusselt number of coil side versus dimensionless dv, Hsh, dsh. (d): Nusselt number of shell side versus dimensionless dv, Hsh, dsh.
of dimensionless geometrical parameters, each parameter on average has been changed five times, while other parameters are held constant according to Table 1. In all cases, inlet velocities for both coil side and shell side have been considered 1 m/s, and temperatures of entering fluids into the coil and shell have been considered 20 c and 90 c , respectively. Results have been shown in Fig. 6a and b. According to Fig. 6a, the power of dc for the coil side Nusselt number is 0.216, which is exactly equal to the power that Salimpour [1] has obtained and it is very close to 0.238 that Kalb et al. [13] have obtained. It can be seen from Fig. 6a and b, that if the coil diameter increases by 50% then the Nusselt number of the coil side decreases by 8.4% and the Nusselt number of the shell side increases by 16.6%. So it can be found that increase of the diameter of the coil, causes increase of the overall heat transfer coefficient. According to Fig. 6a, the effect of pitch size on the coil side's Nusselt number is negligible. If the pitch size is doubled, the Nusselt number of the coil side increases by only 0.8%. Equations (4)e(8) and (11)e(13) and (16)e(19) confirm this conclusion but Equations (1) and (14) don't confirm it. According to Fig. 6b, increase of the dimensionless pitch from 2 to 4, causes increase of the shell side Nusselt number. The similar results have been obtained by Mirgolbabaei et al. [20], Taherian [25] and Ghorbani et al. [12] for 1.8 < p/dt,o < 2, 1.89 < p/dt,o < 5.96 and 1.31 < p/dt,o < 1.87, respectively. According to Fig. 6b, if the pitch
size is doubled, the Nusselt number of the shell side increases by 10%. So increase of the pitch size leads to increase of the overall heat transfer coefficient. The similar result has been obtained by Jamshidi et al. [21]. With change of dv, Hsh or dsh, the distance between inlet and outlet of the shell (i.e. f) will also changes that is undesirable. So for this cases, Equation (50) will be modified as follows:
NuRem Pr n
z f ¼ c4 xr dt
(51)
Table 5 Ranges of geometrical parameters. Parameters
Values (m)
dt dc dsh dv Hc Hsh p f
0.007, 0.008, 0.01, 0.011, 0.013, 0.016 0.06, 0.07, 0.08, 0.10 0.10, 0.12, 0.14, 0.16, 0.22 0.01, 0.015, 0.02, 0.025, 0.03 0.12, 0.14, 0.16, 0.18, 0.20 0.24, 0.28, 0.32, 0.36, 0.40 0.0154, 0.02, 0.022, 0.025, 0.033, 0.04 0.12, 0.144, 0.20, 0.22, 0.251
A. Alimoradi, F. Veysi / International Journal of Thermal Sciences 107 (2016) 196e208
205
Fig. 7. (a): Obtaining overall correlation for Nuc. (b): Obtaining overall correlation for Nush. (c): Nusselt numbers ratio.
When the effect of all parameters were obtained, following correlations can be proposed to predict the coil side and shell side Nusselt numbers:
Where c4 is a constant, x is dimensionless dv, Hsh or d sh and z is equal to 0.013 and 0.561 for coil side and shell side, respectively (based on Fig. 6a and b). Results have been shown
Nuc ¼
c5 Re0:685 c
dc dt;i
Nush ¼ c6 Re0:723 sh
!0:216
dc dt;o
0:378
dv dt;i
dv dt;o
!0:024
dsh dt;i
0:556
!0:012
dsh dt;o
0:82
Hc dt;i
Hc dt;o
!0:03
0:043
Hsh dt;i
Hsh dt;o
in Fig. 6c and d. According to these figures, the most effective geometrical parameters on the shell side Nusselt number are the height and the diameter of the shell. It can be seen from Fig. 6d that, increase of the height and the diameter of the shell and thereby increase of its volume, leads to a substantial decrease of the Nusselt number of the shell side. For example if the diameter and height of the shell are increased by 50%, then the Nusselt number of the shell side decreases by 28.3% and 34.1%, respectively.
!0:045
1:03
f dt;i
f dt;o
!0:013
0:561
p dt;i
p dt;o
!0:011
0:138
Prc0:315
0:717 Prsh
(52)
(53)
Results of this section have been obtained for ranges shown in Table 5. Coefficients of Equations (52) and (53) will be obtained Based on all studied cases, represented in Fig. 7a and b, respectively. The Nusselt numbers ratio (i.e. Nuc/Nush) was obtained for all cases. Results have been shown in Fig. 7c. According to this Figure, it can be seen that in helically coiled tube heat exchangers, the Nusselt number of the coil side is usually more than twice (on average 2.36 times) of the Nusselt number of the shell side. After obtaining coefficients of Equations (52) and (53), final form
206
A. Alimoradi, F. Veysi / International Journal of Thermal Sciences 107 (2016) 196e208
of equations, which predict the coil side and shell side Nusselt numbers will be obtained as follows:
Nuc ¼
0:255Re0:685 c
dc dt;i
Nush ¼ 0:247Re0:723 sh
!0:216
dc dt;o
0:378
dv dt;i
dv dt;o
!0:024
dsh dt;i
0:556
dsh dt;o
!0:012
0:82
Hc dt;i
Hc dt;o
!0:03
0:043
Which are obtained for:
1000 < Rec < 27000 2000 < Resh < 49000 1:9 < Prc and Prsh < 7:1 6 < dc =dt < 10 10 < dsh =dt < 22 12 < Hc =dt < 20 24 < Hsh =dt < 40 1 < dv =dt < 3 12 < f =dt < 25:1 2 < p=dt < 4
respectively) and results of the present work is good. Also Equation (55) has been compared with proposed correla-
Hsh dt;i
Hsh dt;o
!0:045
1:03
f dt;i
f dt;o
!0:013
0:561
p dt;i
p dt;o
!0:011
0:138
Prc0:315
0:717 Prsh
(54)
(55)
tion for shell side Nusselt number by Ghorbani et al. [12] (i.e. Equation (15)) in Fig. 8b. As can be seen, for ReDeq < 600 (or Resh < 7500), there is an excellent agreement between Equations (55) and (15), while for ReDeq > 600 (or Resh > 7500), there is a substantial difference, because Equation (15) was obtained for mixed convection boundary condition, while when the Reynolds number increases, boundary condition will change from mixed convection to forced convection, where Equation (15) may not be applicable for. They had tested three heat exchanger
Based on Equations (54) and (55), it can be found that, Nusselt numbers of the coil side and shell side are proportional to dt0.94 and dt0.897, respectively. So if the diameter of the tube increases, while inlet velocities and other geometrical parameters are held constant, then Nusselt numbers of both sides will increase. Based on this result and Equations (30) and (31), it can be found that the heat transfer of coil side and shell side are proportional to dt-0.06 and dt-0.103, respectively. So if the diameter of the tube increases, while inlet velocities and other geometrical parameters are held constant, then the heat transfer of both sides will decrease and thereby the overall heat transfer coefficient will decrease too. The similar results have been obtained by Mirgolbabaei et al. [20]. They used coils with the same dimensionless pitch but different tube diameters (dt ¼ 9.52 and 12.5 mm), and concluded with increasing the tube diameter, the heat transfer coefficient of shell side decreases.
4.4. Comparison with experimental results In this section, Equations (54) and (55) are compared with the experimental data obtained from the present study and previous works. Equation (54) has been compared with proposed correlations for coil side Nusselt number in previous works (which are based on the experimental data or validated experimentally) in Fig. 8a. As can be seen, agreement between equation represented by Jayakumar et al. [5,6] (i.e. Equation (8)) and results of the present work is excellent. They had tested a heat exchanger with: dt,i ¼ 10 mm, dt,o ¼ 12.7 mm, p ¼ 30 mm, dc ¼ 300 mm. Also agreement between Proposed correlations by Xin et al. [4], Pawar et al. [8], Hardik et al. [9] (i.e. Equations (5), (11) and (12),
Fig. 8. (a): Comparison between coil side's Nusselt number of present study and previous works. (b): Comparison between shell side Nusselt number of the present study and Equation (15).
A. Alimoradi, F. Veysi / International Journal of Thermal Sciences 107 (2016) 196e208
207
Similarly, it was found that Equation (55) is in a good agreement with experimental results obtained by Ghorbani et al. [12]. According to Fig. 9b and ranges of geometrical parameters of their work and dimensions of the tested heat exchanger of the present study, it can be found that Equation (55) is validated experimentally for the following ranges:
900 <Resh < 5000 8 < dc dt;o < 13:27 10 < dsh dt;o < 16:58 23 < Hc dt;o< 40:44 30:42 < Hsh dt;o < 40:44 1:31 < p dt;o < 3
4.5. Comparison of turbulence models As mentioned, in this work the standard K-ε model was used to consider the effect of turbulence and the SIMPLE algorithm was used as the pressure and velocity coupling scheme. Lin et al. [26] and Yang et al. [27] also used this model to analyze the fully developed turbulent convective heat transfer in a circular cross section helical pipes with finite pitch. They found that, the results
Fig. 9. (a): Comparison between numerical and experimental Nusselt numbers for coil side. (b): Comparison between numerical and experimental Nusselt numbers for shell side.
experimentally in the following geometrical ranges:
10:19 < dc dt;o < 13:27 12:47 < dsh dt;o < 16:58 30:42 < Hc dt;o < 40:44 30:42 < Hsh dt;o < 40:44 1:31 < p dt;o < 1:87 Also numerical results (i.e. Equations (54) and (55)) were compared with experimental data obtained in the present study. Fig. 9a and b represent numerical and experimental Nusselt numbers of the coil side and the shell side versus Reynolds numbers of the coil side and the shell side, respectively. It can be seen that the experimental results and suggested equations based on the numerical work are in good agreement (According to Fig. 8a and b, Fig. 9a and b it can be found that the average errors for Equations (54), (55), (1), (4), (8), (11), (12) and (15) are 3.77%, 13.13%, 52.51%, 30.16%, 1.52%, 29.40%, 15.92% and 21.92%, respectively. Therefore, it can be asserted that Equations (54) and (55) are more accurate than others.). Also it was found that Equation (54) is in a good agreement with experimental results obtained by Jayakumare et al. [6]. So based on the Fig. 9a and dimensions of the heat exchanger of their work and dimensions of the tested heat exchanger of the present study, it can be found that the (54) is validated experimentally for the following ranges:
1000 < Rec< 9500 10:53< dc dt;i < 30 3 < p dt;i < 3:95
Fig. 10. (a): Temperature of coil side outlet fluids for different turbulence models. (b): Temperature of shell side outlet fluids for different turbulence models.
208
A. Alimoradi, F. Veysi / International Journal of Thermal Sciences 107 (2016) 196e208
obtained from the model are in good agreement with the experimental data. Also in this section this model was compared with three other schemes: 1) Realizable K-ε model with SIMPLEC algorithm (as the pressure and velocity coupling scheme) 2) Standard K-u model with SIMPLE algorithm 3) Spalart allmaras model with SIMPLE algorithm. For this purpose, four cases were studied. For all case, the inlet fluids temperature into the coil and shell, have been considered 5 c and 90 c , respectively, while inlet velocities for both side were considered 0.25, 0.5, 0.75 and 1 m/s for case 1, 2, 3 and 4 respectively. Results have been shown in Fig. 10a and b. As can be seen, the difference between the temperatures of outlet fluids is negligible (maximum differences for coil side and shell side are 3.69% and 2.49% respectively). As a result, it has been concluded that by the use of any of these models, results changes are negligible. 5. Conclusions In this study, the effect of operational and geometrical parameters of helically coiled tube heat exchangers on Nusselt numbers of both sides were investigated using numerical method and experimental validation. Based on the results, two correlations were suggested for calculating the coil side and the shell side Nusselt numbers for wide ranges of Reynolds and Prandtl numbers. Also it was found that: 1) The effect of pitch size on the coil side's Nusselt number is negligible, while if the pitch size is doubled, the Nusselt number of the shell side increases by 10%. Therefore increase of the pitch size leads to increase of the overall heat transfer coefficient. 2. If the tube diameter is increased, while inlet velocities and other geometrical parameters are held constant, the Nusselt numbers of both sides will increase and all heat transfer coefficients (i.e. coil side, shell side and overall heat transfer coefficients) will decrease. 3. %50 increase in coil diameter results in %8.4 decrease in the coil side Nusselt number and 16.6% increase in the Nusselt number of the shell side. Therefore, the increase of the coil diameter leads to increase of the overall heat transfer coefficient 4. Increase of the height and the diameter of the shell and thereby increase of its volume, results in a substantial decrease of the shell side Nusselt number. If the diameter and the height of the shell increase by 50%, the Nusselt number of the shell side decreases by 28.3% and 34.1%, respectively. 5. In helically coiled tube heat exchangers, the Nusselt number of the coil side is usually more than twice the Nusselt number of the shell side. References [1] Salimpour MR. Heat transfer coefficients of shell and coiled tube heat exchangers. Exp Therm Fluid Sci 2009;33:203e7. http://dx.doi.org/10.1016/ j.expthermflusci.2008.07.015. [2] Moawed M. Experimental study of forced convection from helical coiled tubes with different parameters. Energy Convers Manag 2011;52:1150e6. http:// dx.doi.org/10.1016/j.enconman.2010.09.009. [3] Beigzadeh R, Rahimi M. Prediction of heat transfer and flow characteristics in helically coiled tubes using artificial neural networks. Int Commun Heat Mass Transf 2012;39:1279e85. http://dx.doi.org/10.1016/
j.icheatmasstransfer.2012.06.008. [4] Xin RC, Ebadian MA. The effects of Prandtl numbers on local and average convective heat transfer characteristics in helical pipes. J Heat Transf 1997;119:467e73. http://dx.doi.org/10.1115/1.2824120. [5] Jayakumar JS, Mahajani SM, Mandal JC, Vijayan PK, Bhoi R. Experimental and CFD estimation of heat transfer in helically coiled heat exchangers. Chem Eng Res Des 2008;86:221e32. http://dx.doi.org/10.1016/j.cherd.2007.10.021. [6] Jayakumar JS, Mahajani SM, Mandal JC, Vijayan PK, Bhoi R. CFD analysis of single-phase flows inside helically coiled tubes. Comput Chem Eng 2010;34: 430e46. http://dx.doi.org/10.1016/j.compchemeng.2009.11.008. [7] Genic SB, Jac'imovic' BM, Jaric' MS, Budimir NJ, Dobrnjac MM. Research on the shell-side thermal performances of heat exchangers with helical tube coils. Int J Heat Mass Transf 2012;55:4295e300. http://dx.doi.org/10.1016/ j.ijheatmasstransfer.2012.03.074. [8] Pawar SS, Sunnapwar VK. Experimental studies on heat transfer to Newtonian and non-Newtonian fluids in helical coils with laminar and turbulent flow. Exp Therm Fluid Sci 2013;44:792e804. http://dx.doi.org/10.1016/ j.expthermflusci.2012.09.024. [9] Hardik BK, Baburajan PK, Prabhu SV. Local heat transfer coefficient in helical coils with single phase flow. Int J Heat Mass Transf 2015;89:522e38. http:// dx.doi.org/10.1016/j.ijheatmasstransfer.2015.05.069. [10] Pimenta a TA, Campos JBLM. Heat transfer coefficients from Newtonian and non-Newtonian fluids flowing in laminar regime in a helical coil. Int J Heat Mass Transf 2013;58:676e90. http://dx.doi.org/10.1016/ j.ijheatmasstransfer.2012.10.078. [11] Beigzadeh R, Rahimi M. Prediction of thermal and fluid flow characteristics in helically coiled tubes using ANFIS and GA based correlations. Int Commun Heat Mass Transf 2012;39:1647e53. http://dx.doi.org/10.1016/ j.icheatmasstransfer.2012.10.011. [12] Ghorbani N, Taherian H, Gorji M, Mirgolbabaei H. Experimental study of mixed convection heat transfer in vertical helically coiled tube heat exchangers. Exp Therm Fluid Sci 2010;34:900e5. http://dx.doi.org/10.1016/ j.expthermflusci.2010.02.004. [13] Kalb CE, Seader JD. Heat and mass transfer phenomena for viscous flow in cured circular tubes. Int J Heat Mass Transf 1972;15:801e17. http:// dx.doi.org/10.1016/0017-9310(72)90160-3. [14] Janssen LAM, Hoogendoorn CJ. Laminar convective heat transfer in helical coiled tubes. Int J Heat Mass Transf 1978;21:1197e206. http://dx.doi.org/ 10.1016/0017-9310(78)90138-2. [15] Dravid AN, Smith KA, Merrill EW, Brian PLT. Effect of Secondary fluid motion on laminar flow heat transfer in helically coiled tubes. AIChE J 1971;17: 1114e22. http://dx.doi.org/10.1002/aic.690170517. [16] Bai B, Guo L, Feng Z, Chen X. Turbulent heat transfer in a horizontal helically coiled tube. Heat Transf Asian Res 1999;28:395e403. DOI: 10.1002/(SICI) 1523 1496(1999)28:5<395: AID-HTJ5>3.0.CO; 2-Y. [17] Prasad BVSSS, Das DH, Prabhakar AK. Pressure drop, heat transfer and performance of a helically coiled tubular exchanger. Heat Recovery Syst CHP 1989;9:249e56. http://dx.doi.org/10.1016/0890-4332(89)90008-2. [18] Cioncolini A, Santini L. On the laminar to turbulent flow transition in diabatic helically coiled pipe flow. Exp Therm Fluid Sci 2006;30:653e61. http:// dx.doi.org/10.1016/j.expthermflusci.2006.01.004. [19] ESDU. Internal forced convective heat transfer in coiled pipes. Eng Sci Data Unit 2001. Item No. 78031 London. [20] Mirgolbabaei H, Taherian H, Domairy G, Ghorbani N. Numerical estimation of mixed convection heat transfer in vertical helically coiled tube heat exchangers. Int J Numer Method Fluids 2011;66:805e19. http://dx.doi.org/ 10.1002/fld.2284. [21] Jamshidi N, Farhadi M, Ganji DD, Sedighi K. Experimental analysis of heat transfer enhancement in shell and helical tube heat Exchangers. Appl Therm Eng 2013;51:644e52. http://dx.doi.org/10.1016/ j.applthermaleng.2012.10.008. [22] Abernethy RB, Benedict RP, Dowdell RB. ASME measurement uncertainty, ASME. 1983. paper 83-WA/FM-3. [23] ANSYS FLUENT 12.0 User's Guide, cell zone and boundary conditions. ANSYS Inc; 2009. p. 58e9. [24] Schmidt EF. Wfirmeiibergang und druckverlust in rohrschlangen. G’zemieJng Tech 1967;39:781e9. [25] Taherian H. Natural convection heat transfer in heat exchangers with vertical helical coils. Ph.D. Thesis. Halifax, Nova Scotia, Canada: Dalhousie University; 1998. [26] Yang G, Ebadian MA. Turbulent forced convection in a helicoidal pipe with substantial pitch. Int J Heat Mass Transf 1996;39:2015e22. [27] Lin CX, Ebadian MA. Developing turbulent convective heat transfer in helical pipes. Int J Heat Mass Transf 1997;40:3861e73.
Contents lists available at ScienceDirect
International Journal of Thermal Sciences journal homepage: www.elsevier.com/locate/ijts
Prediction of heat transfer coefficients of shell and coiled tube heat exchangers using numerical method and experimental validation Ashkan Alimoradi, Farzad Veysi* Mechanical Engineering Department, Faculty of Engineering, Razi University, Kermanshah, Iran
a r t i c l e i n f o
a b s t r a c t
Article history: Received 7 October 2015 Received in revised form 17 March 2016 Accepted 11 April 2016
In this work, the heat transfer of shell and helically coiled tube heat exchangers was investigated. Numerical and experimental methods were used to investigate the effect of physical properties of fluid (i.e. viscosity, thermal conductivity, specific heat capacity and density), operational parameters (i.e. the velocity and temperature of fluid) and geometrical parameters (i.e. pitch, diameter of the tube, diameter of shell's inlet, diameters of coil and shell, heights of coil and shell, and the distance between the inlet and outlet of the shell) on Nusselt numbers of both sides. Totally 42 cases and 15 tests were investigated in the numerical analysis and experimental work, respectively. Measurements and analysis were performed, when the steady state was attained. The working fluid of both sides is water, which its viscosity and thermal conductivity were assumed to be dependent on temperature, in the numerical analysis. Results indicate that if the pitch size is doubled, the shell side Nusselt number increases by 10%, while the coil side Nusselt numbers increases by only 0.8%. Also it was found that an increase of 50% in the height and diameter of the shell causes a decrease of 34.1% and 28.3% in the Nusselt number of the shell side, respectively. Based on the results, two correlations were developed to predict Nusselt numbers of coil side and shell side for wide ranges of Reynolds and Prandtl numbers (1000 < Rec < 27,000, 2000 < Resh < 49,000 and 1.9 < Prc and Prsh < 7.1). These correlations were compared with the experimental data of the present study and previous works. It was found that these correlations are in good agreement with the experimental data for wide ranges of operational and geometrical parameters. © 2016 Published by Elsevier Masson SAS.
Keywords: Helical coil Heat exchanger Numerical Experimental Nusselt number
1. Introduction Helically coiled tube heat exchangers are widely used in various industries such as piping systems, air conditioning, storage tanks, and chemical reactors. In petroleum units, the heat exchanger, which is used to cool the lubricating oil of the mechanical seal of pumps, is a shell and coiled tube heat exchanger. These types of exchangers are one of the compact heat exchangers types used to increase heat transfer rate, require less volume and weight compared with other types of heat exchangers. Modeling of the heat transfer characteristics of this type of heat exchanger, was considered in different literatures. Most studies are about the coil side heat transfer characteristics, while the shell side heat transfer was not investigated in more details.
* Corresponding author. E-mail addresses: [email protected], [email protected]. ir (A. Alimoradi), [email protected], [email protected] (F. Veysi). http://dx.doi.org/10.1016/j.ijthermalsci.2016.04.010 1290-0729/© 2016 Published by Elsevier Masson SAS.
Salimpour [1] investigated heat transfer coefficients of shell and helically coiled tube heat exchangers experimentally. He found that the shell side heat transfer coefficient increases, with increase of the pitch size. Two correlations were developed to predict the inner and outer heat transfer coefficients as follows:
Nuc ¼ 0:152De0:431 Pr1:06 g0:277
(1)
Nush ¼ 19:64Re0:513 Pr0:129 g0:938 sh
(2)
M. Moawed [2] experimentally studied the forced convection from outside surfaces of helical coiled tubes with constant wall heat flux. Experiments were performed in an open circuit airflow wind tunnel system, operated in suction mode. A general correlation for the average Nusselt number was obtained as follows:
A. Alimoradi, F. Veysi / International Journal of Thermal Sciences 107 (2016) 196e208
Nomenclature
u
area m2 specific heat capacity J/kg c diameter m dean number ¼ Re(dt/dc)0.5 distance between inlet and outlet of the shell m height m heat transfer coefficient W/m2 K Fanning friction factor turbulence kinetic energy J/kg thermal conductivity W/m K coil's length m flow rate kg/s Nusselt number pitch m Prandtl number ¼ m C/k heat transfer rate W heat flux W/m2 Rayleigh number Reynolds number uncertainty temperature K or C
A C d De f H h j K k l m_ Nu p Pr Q q Ra Re S T
0:914 0:281 Nuo ¼ 0:0345Re0:48 dc dt;o p dt;o
Subscripts b bulk c coil cr critical Deq equivalent diameter h hydraulic i inner k stage Ln normalized length o outer sh shell t tube v shell's inlet w wall
(3)
6:6 102 Re 2:3 103 7:086 dc dt;o 16:142 1:81 p dt;o 3:205 Beigzadeh et al. [3] developed Artificial Neural Network (or ANN) models to predict the heat transfer and friction factor of the coil side, in helically coiled tubes. The predicted Nusselt numbers of the coil side, were compared with the proposed equation by Xin and Ebadian [4] and a reasonable agreement was observed. Xin and Ebadian [4] Studied effects of the Prandtl number and geometric parameters on the local and average convective heat transfer characteristics in helical pipes and suggested the following equation for the coil side Nusselt number: 0:643
Nuc ¼ 2:153 þ 0:318De
0:177
Pr
average velocity m/s
Greek symbols g dimensionless pitch p/pdc d curvature ratio dt,i/dc ε turbulent dissipation J/kg m viscosity Pa.s r density kg/m3
Which is applicable for:
197
Jayakumar et al. [5,6] numerically and experimentally studied the coil side of shell and helically coiled tube heat exchangers and found that the use of temperature dependent properties of working fluids results in prediction of more accurate heat transfer coefficients. They also found that arbitrary boundary conditions, such as constant wall temperature and constant heat flux are not applicable for prediction of heat transfer, when fluid-to-fluid heat transfer occurs in a heat exchanger. Correlations were developed to calculate the coil side heat transfer coefficient of the heat exchanger as follows:
Nuc ¼ 0:025De0:9112 Pr0:4
(6)
Which is applicable for [5]:
2000 < De < 12000 And:
(4) Nuc ¼ 0:085De0:74 Pr0:4 d0:1
(7)
Nuc ¼ 0:116Re0:71 Pr0:4 d0:11
(8)
Which is applicable for:
20 De 2000 0:7 Pr 175 0:0267 dt;i =dc 0:0884
Which are applicable for [6]:
And:
Nuc ¼ 0:00619Re0:92 pr0:4 Which is applicable for:
5000 Re 10000 0:7 Pr 5 0:0267 dt;i =dc 0:0884
dt;i 1 þ 3:455 dc
(5)
14000 < Re < 70000 3000 < De < 22000 3 < Pr < 5 0:05 < d < 0:2 Genic' et al. [7] experimentally studied the shell side heat transfer of helically coiled tube heat exchangers. They investigated three heat exchangers (with different geometrical parameters) and proposed the following correlation for the shell side Nusselt number (the Reynolds number is based on the hydraulic diameter):
198
A. Alimoradi, F. Veysi / International Journal of Thermal Sciences 107 (2016) 196e208
1
Nush ¼ 0:5Re0:55 Pr3
mb mw
0:14 (9)
shell side volume surface contact with fluid
(10)
Pawar et al. [8] studied isothermal steady state and nonisothermal unsteady state conditions in helical coils for Newtonian and non-Newtonian fluids experimentally. The following correlation was proposed for the coil side Nusselt number:
Nuc ¼ 0:0472De0:8346 Pr0:4
(15)
Which is applicable for:
They defined the hydraulic diameter as follows:
dh ¼ 4
0:3 0:4533 NuDeq ¼ 0:0041Re0:2 Deq Prsh RaDeq
(11)
120 < ReDeq < 1200 1:2 107 < RaDeq < 3:2 108 Kalb and Seader [13] theoretically studied the fully developed heat transfer in curved tubes with circular cross section at steady state, under the thermal boundary condition of axially uniform wall heat flux with peripherally uniform wall temperature. They proposed the following equation, for coil side Nusselt number:
Nuc ¼ 0:913De0:476 Pr0:2
(16)
Which is applicable for: Which is applicable for:
586 De 4773 0:055 d 0:0757 3:83 Pr 7:3
80 De 1200 0:7 Pr 5
Hardik et al. [9] experimentally investigated the effect of the curvature of the helical coil, Reynolds number and Prandtl number on the friction factor and local Nusselt number. They used water as the working fluid in their experiments. A correlation was suggested for fully developed coil side Nusselt number as follows:
dc Nuc ¼ 0:0456 dt;i
!0:16 0:8
Re
0:4
Pr
(12)
Pimenta et al. [10] experimentally studied the heat transfer coefficients of Newtonian and non-Newtonian fluids, at constant wall temperature as a boundary condition, in fully developed laminar flow inside a helical coil and suggested the following equation:
Nuc ¼ 0:5De0:481 0:465 Pr0:367
(13)
Janssen et al. [14] studied convective heat transfer in coiled tubes experimentally and numerically. For the boundary condition of a constant wall temperature, the following correlation was suggested:
0 Nuc ¼ @
d
0:32 þ 3 dt;i c
0:86
d 0:8 dt;i c
1 ARe0:5 Pr0:33
dt;i dc
0:14þ0:8ddt;i c
(17)
Dravid et al. [15] studied the heat transfer of helically coiled tube heat exchangers numerically and experimentally. They suggested the following equation:
pffiffiffiffiffiffi Nuc ¼ 0:76 þ 0:65 De Pr0:175
(18)
Which is applicable for:
Which is applicable for:
50 < De < 2000 5 < Pr < 175
15 < De < 1020 10 < Pr < 353 dt;i dc ¼ 0:0263 p ¼ 0:01134m dt;i ¼ 0:004575m
Bai et al. [16] performed an experiment to study the turbulent heat transfer in a horizontal helically coiled tube, over a wide range of experimental parameters. The following correlation was proposed:
Beigzadeh et al. [11] applied Adaptive Neuro-Fuzzy Inference System (or ANFIS) and Genetic Algorithm techniques (or GA) to model and predict the thermal and fluid flow characteristics, in helically coiled tubes. The experimental data were employed as inputs of the model. The Nusselt number was determined as follows:
Nuc ¼ 0:328Re0:58 Pr0:4
Nuc ¼ 0:359Re0:781 Pr0:016 d0:933 g0:172
(14)
Ghorbani et al. [12] studied the mixed convection heat transfer in shell and coiled tube heat exchangers for various Reynolds numbers, tube to coil diameter ratio and dimensionless coil pitch experimentally. It was found that the tube diameter has negligible influence on the shell-side heat transfer coefficient and increasing the coil pitch increases it. They also found that the equivalent diameter of the shell which has the same definition with the hydraulic diameter that, Genic' et al. [7] have been used (i.e. Equation (10)) is the best characteristic length for the Nusselt number of the shell side. Also they proposed a correlation for the shell side Nusselt number as follows:
mb mw
0:11 (19)
Which is applicable for:
4:5 104 < Re < 19 104 Prasad [17] proposed correlations for two special situations of geometrical parameters for shell side Nusselt number as follows:
Nush ¼0:057Re0:8 dh For dc dt;o ¼ 17:24
(20)
Nush ¼0:110Re0:8 dh For dc dt;o ¼ 34:9
(21)
Cioncolini et al. [18] experimentally investigated the turbulence emergence process in adiabatic coiled pipes. Results were compared with equation proposed by ESDU [19] and a reasonable agreement was observed. Equation is as follows:
A. Alimoradi, F. Veysi / International Journal of Thermal Sciences 107 (2016) 196e208
199
0 !0:68 1 RePr 0:11 d c @1 þ 0:059Re0:34 A mb qffiffiffi Nuc ¼ dt;i mw 1:07 þ 12:7 2f Pr0:667 1 j 2
j ¼ ðlnðReÞ 3:28Þ2
(23)
Which is applicable for:
1:5 104 Re 3 105 0:7 Pr 11:6 4:9 dc dt;i 271 Mirgolbabaei et al. [20] numerically investigated the mixed convection heat transfer from vertical helically coiled tubes in a cylindrical shell at various Reynolds and Rayleigh numbers, various coil to tube diameter ratios and non-dimensional coil pitches. While most studies, by the time, was focusing on constant wall temperature or constant heat flux, in this study it was a fluid-tofluid heat exchanger to emulate the real shell-to-coil side fluids thermal mechanism. They studied the influence of the tube diameter, coil pitch and shell side mass flow rate on the shell side heat transfer coefficient. They found that the normalized length of the shell is the best characteristic length for the Nusselt number of the shell side. Also they proposed a correlation for the shell side Nusselt number as follows:
NuLn ¼ 0:073
Dt Dc
0:769
0:231 Re0:005 Pr0:15 Ln Ln RaLn
(24)
Which is applicable for:
40 < ReLn < 205 8:1 106 < RaLn < 2:2 108 Jamshidi et al. [21] determined coil and shell side heat transfer coefficients using Wilson's plots experimental apparatus. Taguchi method was used to investigate the effect of fluid flow and geometrical parameters on the heat transfer rate. Results indicate that the higher coil diameter, coil pitch and mass flow rate in shell and tube can enhance the heat transfer rate in these types of heat exchangers. As can be seen, in correlations proposed for the coil side Nusselt number, the effect of the velocity of fluid, diameter of the tube and coil, changes significantly from one equation to another. Also correlations proposed for the shell side's Nusselt number do not contain all geometrical parameters of the heat exchanger. In this study the effect of operational and geometrical parameters of helically coiled tube heat exchangers on the Nusselt numbers of both sides was investigated using numerical method and experimental validation. At the end of this study, two correlations are proposed to predict Nusselt numbers of the coil side and the shell side for wide ranges of Reynolds numbers and geometrical parameters.
(22)
consists of two parts: helical coil and shell. The helical coil used in the heat exchanger, has been made from an Aluminum tube. Care was taken to locate the coil into the middle of the shell. A pipe (with a specified length and diameter) made from PE, was used as the shell of the heat exchanger. The constructed heat exchanger is shown in Fig. 3b. To prevent heat losses from outer surfaces of the shell, it was well insulated using fiberglass (4 cm thickness). The heat exchanger is positioned horizontally during experiments. Forced convection was considered as the boundary condition for the inner and outer surfaces of the coil. In addition to the heat exchanger, There are different components else in the set up. A Rotameter (Model: LZM-15, 0.2e2 GPM) was used to measure the flow rate inside the coil, while the flow rate inside the shell was measured by use of a calibrated pot (which has a capacity of 7.5 L) and a timer. K type thermocouples were installed in: 1) Inlet and outlet of the coil (i.e. T1 and T2). These thermocouples were inserted in the small holes made in the inlet and outlet plastic tubes of the coil side and sealed to prevent any leakage. 2) Three points on the coil's surface (i.e. T5, T6 and T7). First, three small grooves were made on the outer surface of the coil carefully. Then each thermocouples are imbedded in the grooves and attached to the coil's surface 3) Inlet, outlet and two different points of the shell (i.e. T3, T4, T8 and T9, respectively). The process for installation of T3 and T4 is like that used for T1 and T2. Also T8 and T9 are easily inserted in the two small holes made in the shell. The distribution of thermocouples can be seen in Fig. 2. These temperatures are recorded by a Data logger (Model: BTM-4208SD, 12 channels, 0.1 c precision). All connections of the heat exchanger, were carefully sealed to prevent any leakage. An electric heater (2000 w) has been placed in the storage tank for heating the water. During experiments, the input electric power to the heater is not changed. When the temperature of the water inside the storage
2. Experimental work 2.1. Materials and set up The experimental set up and its schematic diagram are shown in Figs. 1 and 2, respectively. The heat exchanger used for these experiments is like the heat exchanger shown in Fig. 3a, which
Fig. 1. Experimental set up.
200
A. Alimoradi, F. Veysi / International Journal of Thermal Sciences 107 (2016) 196e208
Fig. 2. Flow diagram of the experimental set up.
tank reaches to a prescribed temperature (65e85 c), the centrifugal pump is started to transfer the hot water to the helical coiled tube. Then the cooled water will come back from the coil to the storage tank to be reheated. The cooling water, which is supplied from the city water, will flow into the shell side through the bottom of the heat exchanger (to ensure that, all air inside the heat exchanger is discharged), but will not be recovered after exit. Two ball valves are used to control the flow rate of both sides. Experiments were performed in Razi University of Kermanshah, Iran. It seems, there are different parameters affect the thermal performance and heat transfer coefficients of these types of heat exchangers. In addition to operational parameters and physical properties of fluids, geometrical properties of the heat exchanger can also influence the thermal performance and heat transfer coefficients. Geometrical parameters, which may influence the thermal performance, were shown in Fig. 4a. In this work, effects of these mentioned parameters were investigated. Dimensions of tested heat exchanger have been shown in the Table 1.
2.2. Calculations of Nusselt numbers After a few minutes (often 10e15 min), temperatures will be fixed and the heat transfer rates of coil side and shell side will be approximately equal. These suggest that, the flow is at steady state. Once steady state is attained, values of flow rates and temperatures will be noted. By changing flow rates of both sides, different test runs have been created (totally 15 test runs). Range of the operational parameters is given in Table 2. As can be seen, wide ranges of temperatures and flow rates were covered for both sides. To calculate Nusselt numbers of the coil side and shell side, following equations have been used: Fig. 3. a: The typical heat exchanger. b: The constructed heat exchanger.
Qc ¼ m_ c CðT1 T2 Þ
(25)
A. Alimoradi, F. Veysi / International Journal of Thermal Sciences 107 (2016) 196e208
201
Fig. 4. (a): Geometrical parameters of the heat exchanger. (b): Control volumes created on the coil. (c): Control volumes created on the heat exchanger.
Table 1 Dimensions of heat exchangers for the numerical and experimental works. Parameters (m)
dc
dsh
Hc
Hsh
dt,i
dt,o
dv
f
p
l
Numerical Experimental
0.08 0.08
0.12 0.1
0.2 0.23
0.24 0.34
0.01 0.0076
0.01 0.01
0.01 0.009
0.25 0.338
0.02 0.03
2.64 1.92
Table 2 Ranges of operational parameters. Parameters
Range
Flow rate of the coil side Flow rate of the shell side Ti,c To,c Ti,sh To,sh
1.1e7 Lpm 1e6.48 Lpm 48.3e80.6 c 45.2e65.1 c 25.2e30.3 c 31.5e56.6 c
Qsh ¼ m_ sh CðT4 T3 Þ qw;c
Qc Q ¼ ; qw;sh ¼ sh Ai Ao
hc ¼
qw;c
jTc Tw
hsh ¼
qw;sh
jTw Tsh
Nuc ¼
hc dt;i
Nush ¼
kc hsh dt;o ksh
(30)
(31)
Mean temperatures of the wall, coil and shell (i.e. Tw ; Tc and Tsh ) are obtained by averaging (T5, T6 and T7), (T1 and T2) and (T3, T4, T8 and T9), respectively.
(26) 2.3. Uncertainty analysis
(27)
(28)
(29)
There are always errors in measurements so absolute measurements do not exist. The calculations of uncertainty analysis are based on the method proposed by Abernethy et al. [22]. The maximum uncertainty for coil side flow rate, shell side flow rate, temperatures, inner diameter of the tube, outer diameter of the tube, inner area of the coil, outer area of the coil and length of the coil are 5%, 4%, 0.2%, 1.31%, 1%, 1.31%, 1% and 0.05%, respectively. Also the uncertainty for specific heat capacity and thermal conductivity were assumed to be negligible. Based on Equations (25) and (31), following equations were used to calculate the maximum uncertainty for Nusselt numbers:
202
A. Alimoradi, F. Veysi / International Journal of Thermal Sciences 107 (2016) 196e208
SQ ¼
Sqw ¼ Sh ¼
S2m
S2Q
S2qw
þ
S2C
þ
S2A 2
2
þ SDT
0:5
0:5
þ SDT
0:5
SNu ¼ S2h þ S2dt þ S2k
(32)
b) Residuals of energy equation was considered less than 107. c) Residuals of K and ε equations were considered less than 103.
(33)
Nuc and Nush will be obtained based on Equation (25) until 31. It seems that they are functions of following parameters:
(34) 0:5
(35)
Based on Equations (32)e(35), the maximum uncertainty for coil side and shell side Nusselt numbers are 5.36% and 4.28% respectively. 3. Numerical work A numerical model was constructed in order to predict the heat transfer phenomena in the shell and coiled tube heat exchanger, which is shown in Fig. 3a. It was divided into several small control volumes represented in Fig. 4b and c. An attempt was made to use regular mesh (involving pentahedral and hexahedral elements) in the most regions of the heat exchanger. But in some regions (around the outer surface of the coil and the inlet and outlet regions of the shell) the irregular mesh (involving tetrahedral elements) was used inevitably. A computer code was developed for the model analysis. In addition to the primary governing equations (i.e. continuity, momentum and energy), a model must be used for turbulence. It will be shown that the standard K-ε as the turbulence model and the SIMPLE (Semi Implicit Method for Pressure Linked Equations) algorithm as the pressure and velocity coupling scheme are good choices for the numerical analysis. Also first order Upwind was used as discretization scheme for momentum, energy and K-ε equations. The temperatures and flow rates of entering fluids are known for both sides. In addition to these two variables, values of turbulence intensity and hydraulic diameters for inlets and outlets of both sides must be determined. It was assumed that the hydraulic diameter of the coil side is equal to the inner diameter of the tube (i.e. dt,i) and for the shell side is equal to the diameter of the shell's inlet (dsh). The turbulence intensity (i.e. I) is defined as the ratio of the root-mean-square of the velocity fluctuations to the mean flow velocity. The turbulence intensity in the core of a fully developed duct flow can be estimated from the following formula derived from an empirical correlation for pipe flows [23]: 1=8
I ¼ 0:16ReDh
(36)
It was observed that the use of constant values for the thermal and transport properties of the heat transport mediums, results in prediction of an inaccurate heat transfer coefficient [5]. So the viscosity and thermal conductivity are considered to be dependent on temperature as follows [5]:
mðTÞ ¼ 2:1897e 11T4 3:055e 8T3 þ 1:6028e 5T2 0:0037524T þ 0:33158
(37)
kðTÞ ¼ 1:5362e 8T3 2:261e 5T2 þ 0:010879T 1:0294 (38) Residuals of governing equations were considered as follows: a) Residuals of continuity and momentum equations were considered less than 104.
Nuc ¼ F Rec ; Prc ; dc ; dsh ; Hc ; Hsh ; dv ; dt;i ; p; f
(39)
Nush ¼ G Resh ; Prsh ; dc ; dsh ; Hc ; Hsh ; dv ; dt;o ; p; f
(40)
After collecting geometrical parameters into non-dimensional combination of parameters (dimensionless groups), Equations (39) and (40) will be simplified as follows:
Nuc ¼ F Rec ; Prc ;
dc dsh Hc Hsh dv f p ; ; ; ; ; ; dt;i dt;i dt;i dt;i dt;i dt;i dt;i
! (41)
dc dsh Hc Hsh dv f p ; ; ; ; ; ; Nush ¼ G Resh ; Prsh ; dt;o dt;o dt;o dt;o dt;o dt;o dt;o (42) To obtain the effect of these parameters on the Nusselt numbers, each parameter will be changed on average five times, while other parameters are held constant. So the numerical analysis was performed for different cases (totally 42 cases). Reynolds numbers will be obtained from the following equations:
Rec ¼
rc uc dt;i mc
Resh ¼
rsh ush dt;o msh
(43)
(44)
4. Results and discussions 4.1. Mesh independency study Mesh independency study was performed on 5 different grids, which are like the grid shown in Fig. 4b and c (but with different cell's density), for a special case. Results can be seen in Table 3. As seen, changes of outlet fluids temperatures will decrease to less than 0.1 c from grid 4 to 5, so more refinement of mesh does not decrease the temperatures errors in any appreciable way. 4.2. Operational parameters effects With changes in operational parameters (i.e. inlet velocities and temperatures of fluids), Reynolds and prandtl number will change. So to investigate the effect of operational parameters, Reynolds and Prandtl numbers must be changed. To investigate the effect of Re, inlet velocities are changed, but when they have been changed, temperatures of outlet fluids will change too and therefore, mean temperatures of fluids will change. Change of mean temperatures of the fluids cause the viscosity and thermal conductivity change too (according to Equations (37) and (38)). So the Prandtl number will change that is not desirable. So it seems that it is better to investigate the effect of Pr number first. For this purpose, temperatures of inlet fluids must be changed, but in this case average temperatures of fluids and thus viscosities will change. With change of viscosities, Reynolds numbers will change too that is undesirable. So the problem must be solved simultaneously by trial and error because the Reynolds and Prandtl numbers are
A. Alimoradi, F. Veysi / International Journal of Thermal Sciences 107 (2016) 196e208
203
Table 3 Mesh independency study. Grid
1
2
3
4
5
Number of coil side cells Number of shell side cells To,c (c ) To,sh (c )
156,576 464,923 81.49 66.65
312,970 666,001 82.1 63.67
350,637 749,444 82.6 61.18
381,962 814,457 82.65 60.97
472,364 1,056,211 82.74 60.95
interdependent by the viscosity. It was assumed that the Nusselt number changes with Re and Pr as follows:
Nu ¼ c1 Rem Pr n
(45)
Where c1 is a constant. To obtain m and n the following procedure was performed: First, a power is guessed for the Prandtl number (for example n ¼ 0.4). Then, by changing inlet temperatures of fluids (at constant inlet velocities), the viscosity and then the Reynolds number will changed. With change of Reynolds number, its power (i.e. m) will be obtained. In this case, Equation (45) will be simplified as follows:
Nu$Pr n ¼ c1 Rem
(46)
On the other hand, by changing inlet velocities (at constant inlet temperatures), the new power for the Reynolds number will be obtained (i.e. m0 ). In this case, Equation (45) will be simplified as follows:
Nu$Pr n ¼ c2 Rem
0
(47)
Where c2 is a constant. If m and m0 be equal, the solution procedure will be stopped otherwise, the mentioned procedure must be repeated. The new power of Pr for the next stage (i.e. nkþ1) will be obtained from the following equation:
nkþ1 ¼ m0 k þ ðnk mk Þ
(48)
These processes have been shown in Table 4 for the both sides. Fig. 5a and b represent change of Nusselt numbers versus Reynolds numbers for the final stage. Dimensions of the heat exchanger, which is used to investigate the effect of mentioned parameters (i.e. Re and Pr) have been shown in Table 1. As seen, the power of Pr for coil side is 0.315 that is very close to 0.33 that Janssen et al. [14] have predicted, while most of previous works (e.g. [4e6,8,9,16]) have estimated this power equal to 0.4. This power is equal to 0.717 for the shell side, which in no previous work has been obtained. Also, the power of Re for coil side is 0.685 that is very close to 0.71, that was reported in [6]. These results have been obtained for following ranges:
1000 < Rec < 27000 2000 < Resh < 49000 1:9 < Prc and Prsh < 7:1
Recr ¼ 2300ð1 þ 8:6
dt;i dc
0:45 ! (49)
4.3. Geometrical parameters effects and final formulations
Coil shell
n
m
m0
Stage 1
0.4 0.8 0.319 0.716 0.315 0.717
0.779 0.815 0.690 0.722 0.685 0.723
0.698 0.731 0.686 0.723 0.685 0.723
Stage 3
Different correlations have been suggested to calculate the critical Reynolds number for flow inside helically coiled tubes. Schmidet [24] has proposed a correlation, which is determining the critical Reynolds number as follows:
In this work, the fluid flow inside the helical coil is mainly in the range of turbulent flow, according to (49).
Table 4 Trial and error for estimating the power of Re and Pr.
Stage 2
Fig. 5. (a): Nusselt number of coil side versus Reynolds number. (b): Nusselt number of shell side versus Reynolds number.
It was assumed that the change of Nusselt number with each dimensionless geometrical parameter, is as follows:
Nu ¼ c3 Rem Pr n xr
(50)
Where c3 is a constant and x is one of these dimensionless geometrical parameters: dc/dt, Hc/dt, f/dt or p/dt. m and n also have been obtained for both side in previous section. To obtain the effect
204
A. Alimoradi, F. Veysi / International Journal of Thermal Sciences 107 (2016) 196e208
Fig. 6. (a): Nusselt number of coil side versus dimensionless dc, f, Hc and p. (b): Nusselt number of shell side versus dimensionless dc, f, Hc and p. (c): Nusselt number of coil side versus dimensionless dv, Hsh, dsh. (d): Nusselt number of shell side versus dimensionless dv, Hsh, dsh.
of dimensionless geometrical parameters, each parameter on average has been changed five times, while other parameters are held constant according to Table 1. In all cases, inlet velocities for both coil side and shell side have been considered 1 m/s, and temperatures of entering fluids into the coil and shell have been considered 20 c and 90 c , respectively. Results have been shown in Fig. 6a and b. According to Fig. 6a, the power of dc for the coil side Nusselt number is 0.216, which is exactly equal to the power that Salimpour [1] has obtained and it is very close to 0.238 that Kalb et al. [13] have obtained. It can be seen from Fig. 6a and b, that if the coil diameter increases by 50% then the Nusselt number of the coil side decreases by 8.4% and the Nusselt number of the shell side increases by 16.6%. So it can be found that increase of the diameter of the coil, causes increase of the overall heat transfer coefficient. According to Fig. 6a, the effect of pitch size on the coil side's Nusselt number is negligible. If the pitch size is doubled, the Nusselt number of the coil side increases by only 0.8%. Equations (4)e(8) and (11)e(13) and (16)e(19) confirm this conclusion but Equations (1) and (14) don't confirm it. According to Fig. 6b, increase of the dimensionless pitch from 2 to 4, causes increase of the shell side Nusselt number. The similar results have been obtained by Mirgolbabaei et al. [20], Taherian [25] and Ghorbani et al. [12] for 1.8 < p/dt,o < 2, 1.89 < p/dt,o < 5.96 and 1.31 < p/dt,o < 1.87, respectively. According to Fig. 6b, if the pitch
size is doubled, the Nusselt number of the shell side increases by 10%. So increase of the pitch size leads to increase of the overall heat transfer coefficient. The similar result has been obtained by Jamshidi et al. [21]. With change of dv, Hsh or dsh, the distance between inlet and outlet of the shell (i.e. f) will also changes that is undesirable. So for this cases, Equation (50) will be modified as follows:
NuRem Pr n
z f ¼ c4 xr dt
(51)
Table 5 Ranges of geometrical parameters. Parameters
Values (m)
dt dc dsh dv Hc Hsh p f
0.007, 0.008, 0.01, 0.011, 0.013, 0.016 0.06, 0.07, 0.08, 0.10 0.10, 0.12, 0.14, 0.16, 0.22 0.01, 0.015, 0.02, 0.025, 0.03 0.12, 0.14, 0.16, 0.18, 0.20 0.24, 0.28, 0.32, 0.36, 0.40 0.0154, 0.02, 0.022, 0.025, 0.033, 0.04 0.12, 0.144, 0.20, 0.22, 0.251
A. Alimoradi, F. Veysi / International Journal of Thermal Sciences 107 (2016) 196e208
205
Fig. 7. (a): Obtaining overall correlation for Nuc. (b): Obtaining overall correlation for Nush. (c): Nusselt numbers ratio.
When the effect of all parameters were obtained, following correlations can be proposed to predict the coil side and shell side Nusselt numbers:
Where c4 is a constant, x is dimensionless dv, Hsh or d sh and z is equal to 0.013 and 0.561 for coil side and shell side, respectively (based on Fig. 6a and b). Results have been shown
Nuc ¼
c5 Re0:685 c
dc dt;i
Nush ¼ c6 Re0:723 sh
!0:216
dc dt;o
0:378
dv dt;i
dv dt;o
!0:024
dsh dt;i
0:556
!0:012
dsh dt;o
0:82
Hc dt;i
Hc dt;o
!0:03
0:043
Hsh dt;i
Hsh dt;o
in Fig. 6c and d. According to these figures, the most effective geometrical parameters on the shell side Nusselt number are the height and the diameter of the shell. It can be seen from Fig. 6d that, increase of the height and the diameter of the shell and thereby increase of its volume, leads to a substantial decrease of the Nusselt number of the shell side. For example if the diameter and height of the shell are increased by 50%, then the Nusselt number of the shell side decreases by 28.3% and 34.1%, respectively.
!0:045
1:03
f dt;i
f dt;o
!0:013
0:561
p dt;i
p dt;o
!0:011
0:138
Prc0:315
0:717 Prsh
(52)
(53)
Results of this section have been obtained for ranges shown in Table 5. Coefficients of Equations (52) and (53) will be obtained Based on all studied cases, represented in Fig. 7a and b, respectively. The Nusselt numbers ratio (i.e. Nuc/Nush) was obtained for all cases. Results have been shown in Fig. 7c. According to this Figure, it can be seen that in helically coiled tube heat exchangers, the Nusselt number of the coil side is usually more than twice (on average 2.36 times) of the Nusselt number of the shell side. After obtaining coefficients of Equations (52) and (53), final form
206
A. Alimoradi, F. Veysi / International Journal of Thermal Sciences 107 (2016) 196e208
of equations, which predict the coil side and shell side Nusselt numbers will be obtained as follows:
Nuc ¼
0:255Re0:685 c
dc dt;i
Nush ¼ 0:247Re0:723 sh
!0:216
dc dt;o
0:378
dv dt;i
dv dt;o
!0:024
dsh dt;i
0:556
dsh dt;o
!0:012
0:82
Hc dt;i
Hc dt;o
!0:03
0:043
Which are obtained for:
1000 < Rec < 27000 2000 < Resh < 49000 1:9 < Prc and Prsh < 7:1 6 < dc =dt < 10 10 < dsh =dt < 22 12 < Hc =dt < 20 24 < Hsh =dt < 40 1 < dv =dt < 3 12 < f =dt < 25:1 2 < p=dt < 4
respectively) and results of the present work is good. Also Equation (55) has been compared with proposed correla-
Hsh dt;i
Hsh dt;o
!0:045
1:03
f dt;i
f dt;o
!0:013
0:561
p dt;i
p dt;o
!0:011
0:138
Prc0:315
0:717 Prsh
(54)
(55)
tion for shell side Nusselt number by Ghorbani et al. [12] (i.e. Equation (15)) in Fig. 8b. As can be seen, for ReDeq < 600 (or Resh < 7500), there is an excellent agreement between Equations (55) and (15), while for ReDeq > 600 (or Resh > 7500), there is a substantial difference, because Equation (15) was obtained for mixed convection boundary condition, while when the Reynolds number increases, boundary condition will change from mixed convection to forced convection, where Equation (15) may not be applicable for. They had tested three heat exchanger
Based on Equations (54) and (55), it can be found that, Nusselt numbers of the coil side and shell side are proportional to dt0.94 and dt0.897, respectively. So if the diameter of the tube increases, while inlet velocities and other geometrical parameters are held constant, then Nusselt numbers of both sides will increase. Based on this result and Equations (30) and (31), it can be found that the heat transfer of coil side and shell side are proportional to dt-0.06 and dt-0.103, respectively. So if the diameter of the tube increases, while inlet velocities and other geometrical parameters are held constant, then the heat transfer of both sides will decrease and thereby the overall heat transfer coefficient will decrease too. The similar results have been obtained by Mirgolbabaei et al. [20]. They used coils with the same dimensionless pitch but different tube diameters (dt ¼ 9.52 and 12.5 mm), and concluded with increasing the tube diameter, the heat transfer coefficient of shell side decreases.
4.4. Comparison with experimental results In this section, Equations (54) and (55) are compared with the experimental data obtained from the present study and previous works. Equation (54) has been compared with proposed correlations for coil side Nusselt number in previous works (which are based on the experimental data or validated experimentally) in Fig. 8a. As can be seen, agreement between equation represented by Jayakumar et al. [5,6] (i.e. Equation (8)) and results of the present work is excellent. They had tested a heat exchanger with: dt,i ¼ 10 mm, dt,o ¼ 12.7 mm, p ¼ 30 mm, dc ¼ 300 mm. Also agreement between Proposed correlations by Xin et al. [4], Pawar et al. [8], Hardik et al. [9] (i.e. Equations (5), (11) and (12),
Fig. 8. (a): Comparison between coil side's Nusselt number of present study and previous works. (b): Comparison between shell side Nusselt number of the present study and Equation (15).
A. Alimoradi, F. Veysi / International Journal of Thermal Sciences 107 (2016) 196e208
207
Similarly, it was found that Equation (55) is in a good agreement with experimental results obtained by Ghorbani et al. [12]. According to Fig. 9b and ranges of geometrical parameters of their work and dimensions of the tested heat exchanger of the present study, it can be found that Equation (55) is validated experimentally for the following ranges:
900 <Resh < 5000 8 < dc dt;o < 13:27 10 < dsh dt;o < 16:58 23 < Hc dt;o< 40:44 30:42 < Hsh dt;o < 40:44 1:31 < p dt;o < 3
4.5. Comparison of turbulence models As mentioned, in this work the standard K-ε model was used to consider the effect of turbulence and the SIMPLE algorithm was used as the pressure and velocity coupling scheme. Lin et al. [26] and Yang et al. [27] also used this model to analyze the fully developed turbulent convective heat transfer in a circular cross section helical pipes with finite pitch. They found that, the results
Fig. 9. (a): Comparison between numerical and experimental Nusselt numbers for coil side. (b): Comparison between numerical and experimental Nusselt numbers for shell side.
experimentally in the following geometrical ranges:
10:19 < dc dt;o < 13:27 12:47 < dsh dt;o < 16:58 30:42 < Hc dt;o < 40:44 30:42 < Hsh dt;o < 40:44 1:31 < p dt;o < 1:87 Also numerical results (i.e. Equations (54) and (55)) were compared with experimental data obtained in the present study. Fig. 9a and b represent numerical and experimental Nusselt numbers of the coil side and the shell side versus Reynolds numbers of the coil side and the shell side, respectively. It can be seen that the experimental results and suggested equations based on the numerical work are in good agreement (According to Fig. 8a and b, Fig. 9a and b it can be found that the average errors for Equations (54), (55), (1), (4), (8), (11), (12) and (15) are 3.77%, 13.13%, 52.51%, 30.16%, 1.52%, 29.40%, 15.92% and 21.92%, respectively. Therefore, it can be asserted that Equations (54) and (55) are more accurate than others.). Also it was found that Equation (54) is in a good agreement with experimental results obtained by Jayakumare et al. [6]. So based on the Fig. 9a and dimensions of the heat exchanger of their work and dimensions of the tested heat exchanger of the present study, it can be found that the (54) is validated experimentally for the following ranges:
1000 < Rec< 9500 10:53< dc dt;i < 30 3 < p dt;i < 3:95
Fig. 10. (a): Temperature of coil side outlet fluids for different turbulence models. (b): Temperature of shell side outlet fluids for different turbulence models.
208
A. Alimoradi, F. Veysi / International Journal of Thermal Sciences 107 (2016) 196e208
obtained from the model are in good agreement with the experimental data. Also in this section this model was compared with three other schemes: 1) Realizable K-ε model with SIMPLEC algorithm (as the pressure and velocity coupling scheme) 2) Standard K-u model with SIMPLE algorithm 3) Spalart allmaras model with SIMPLE algorithm. For this purpose, four cases were studied. For all case, the inlet fluids temperature into the coil and shell, have been considered 5 c and 90 c , respectively, while inlet velocities for both side were considered 0.25, 0.5, 0.75 and 1 m/s for case 1, 2, 3 and 4 respectively. Results have been shown in Fig. 10a and b. As can be seen, the difference between the temperatures of outlet fluids is negligible (maximum differences for coil side and shell side are 3.69% and 2.49% respectively). As a result, it has been concluded that by the use of any of these models, results changes are negligible. 5. Conclusions In this study, the effect of operational and geometrical parameters of helically coiled tube heat exchangers on Nusselt numbers of both sides were investigated using numerical method and experimental validation. Based on the results, two correlations were suggested for calculating the coil side and the shell side Nusselt numbers for wide ranges of Reynolds and Prandtl numbers. Also it was found that: 1) The effect of pitch size on the coil side's Nusselt number is negligible, while if the pitch size is doubled, the Nusselt number of the shell side increases by 10%. Therefore increase of the pitch size leads to increase of the overall heat transfer coefficient. 2. If the tube diameter is increased, while inlet velocities and other geometrical parameters are held constant, the Nusselt numbers of both sides will increase and all heat transfer coefficients (i.e. coil side, shell side and overall heat transfer coefficients) will decrease. 3. %50 increase in coil diameter results in %8.4 decrease in the coil side Nusselt number and 16.6% increase in the Nusselt number of the shell side. Therefore, the increase of the coil diameter leads to increase of the overall heat transfer coefficient 4. Increase of the height and the diameter of the shell and thereby increase of its volume, results in a substantial decrease of the shell side Nusselt number. If the diameter and the height of the shell increase by 50%, the Nusselt number of the shell side decreases by 28.3% and 34.1%, respectively. 5. In helically coiled tube heat exchangers, the Nusselt number of the coil side is usually more than twice the Nusselt number of the shell side. References [1] Salimpour MR. Heat transfer coefficients of shell and coiled tube heat exchangers. Exp Therm Fluid Sci 2009;33:203e7. http://dx.doi.org/10.1016/ j.expthermflusci.2008.07.015. [2] Moawed M. Experimental study of forced convection from helical coiled tubes with different parameters. Energy Convers Manag 2011;52:1150e6. http:// dx.doi.org/10.1016/j.enconman.2010.09.009. [3] Beigzadeh R, Rahimi M. Prediction of heat transfer and flow characteristics in helically coiled tubes using artificial neural networks. Int Commun Heat Mass Transf 2012;39:1279e85. http://dx.doi.org/10.1016/
j.icheatmasstransfer.2012.06.008. [4] Xin RC, Ebadian MA. The effects of Prandtl numbers on local and average convective heat transfer characteristics in helical pipes. J Heat Transf 1997;119:467e73. http://dx.doi.org/10.1115/1.2824120. [5] Jayakumar JS, Mahajani SM, Mandal JC, Vijayan PK, Bhoi R. Experimental and CFD estimation of heat transfer in helically coiled heat exchangers. Chem Eng Res Des 2008;86:221e32. http://dx.doi.org/10.1016/j.cherd.2007.10.021. [6] Jayakumar JS, Mahajani SM, Mandal JC, Vijayan PK, Bhoi R. CFD analysis of single-phase flows inside helically coiled tubes. Comput Chem Eng 2010;34: 430e46. http://dx.doi.org/10.1016/j.compchemeng.2009.11.008. [7] Genic SB, Jac'imovic' BM, Jaric' MS, Budimir NJ, Dobrnjac MM. Research on the shell-side thermal performances of heat exchangers with helical tube coils. Int J Heat Mass Transf 2012;55:4295e300. http://dx.doi.org/10.1016/ j.ijheatmasstransfer.2012.03.074. [8] Pawar SS, Sunnapwar VK. Experimental studies on heat transfer to Newtonian and non-Newtonian fluids in helical coils with laminar and turbulent flow. Exp Therm Fluid Sci 2013;44:792e804. http://dx.doi.org/10.1016/ j.expthermflusci.2012.09.024. [9] Hardik BK, Baburajan PK, Prabhu SV. Local heat transfer coefficient in helical coils with single phase flow. Int J Heat Mass Transf 2015;89:522e38. http:// dx.doi.org/10.1016/j.ijheatmasstransfer.2015.05.069. [10] Pimenta a TA, Campos JBLM. Heat transfer coefficients from Newtonian and non-Newtonian fluids flowing in laminar regime in a helical coil. Int J Heat Mass Transf 2013;58:676e90. http://dx.doi.org/10.1016/ j.ijheatmasstransfer.2012.10.078. [11] Beigzadeh R, Rahimi M. Prediction of thermal and fluid flow characteristics in helically coiled tubes using ANFIS and GA based correlations. Int Commun Heat Mass Transf 2012;39:1647e53. http://dx.doi.org/10.1016/ j.icheatmasstransfer.2012.10.011. [12] Ghorbani N, Taherian H, Gorji M, Mirgolbabaei H. Experimental study of mixed convection heat transfer in vertical helically coiled tube heat exchangers. Exp Therm Fluid Sci 2010;34:900e5. http://dx.doi.org/10.1016/ j.expthermflusci.2010.02.004. [13] Kalb CE, Seader JD. Heat and mass transfer phenomena for viscous flow in cured circular tubes. Int J Heat Mass Transf 1972;15:801e17. http:// dx.doi.org/10.1016/0017-9310(72)90160-3. [14] Janssen LAM, Hoogendoorn CJ. Laminar convective heat transfer in helical coiled tubes. Int J Heat Mass Transf 1978;21:1197e206. http://dx.doi.org/ 10.1016/0017-9310(78)90138-2. [15] Dravid AN, Smith KA, Merrill EW, Brian PLT. Effect of Secondary fluid motion on laminar flow heat transfer in helically coiled tubes. AIChE J 1971;17: 1114e22. http://dx.doi.org/10.1002/aic.690170517. [16] Bai B, Guo L, Feng Z, Chen X. Turbulent heat transfer in a horizontal helically coiled tube. Heat Transf Asian Res 1999;28:395e403. DOI: 10.1002/(SICI) 1523 1496(1999)28:5<395: AID-HTJ5>3.0.CO; 2-Y. [17] Prasad BVSSS, Das DH, Prabhakar AK. Pressure drop, heat transfer and performance of a helically coiled tubular exchanger. Heat Recovery Syst CHP 1989;9:249e56. http://dx.doi.org/10.1016/0890-4332(89)90008-2. [18] Cioncolini A, Santini L. On the laminar to turbulent flow transition in diabatic helically coiled pipe flow. Exp Therm Fluid Sci 2006;30:653e61. http:// dx.doi.org/10.1016/j.expthermflusci.2006.01.004. [19] ESDU. Internal forced convective heat transfer in coiled pipes. Eng Sci Data Unit 2001. Item No. 78031 London. [20] Mirgolbabaei H, Taherian H, Domairy G, Ghorbani N. Numerical estimation of mixed convection heat transfer in vertical helically coiled tube heat exchangers. Int J Numer Method Fluids 2011;66:805e19. http://dx.doi.org/ 10.1002/fld.2284. [21] Jamshidi N, Farhadi M, Ganji DD, Sedighi K. Experimental analysis of heat transfer enhancement in shell and helical tube heat Exchangers. Appl Therm Eng 2013;51:644e52. http://dx.doi.org/10.1016/ j.applthermaleng.2012.10.008. [22] Abernethy RB, Benedict RP, Dowdell RB. ASME measurement uncertainty, ASME. 1983. paper 83-WA/FM-3. [23] ANSYS FLUENT 12.0 User's Guide, cell zone and boundary conditions. ANSYS Inc; 2009. p. 58e9. [24] Schmidt EF. Wfirmeiibergang und druckverlust in rohrschlangen. G’zemieJng Tech 1967;39:781e9. [25] Taherian H. Natural convection heat transfer in heat exchangers with vertical helical coils. Ph.D. Thesis. Halifax, Nova Scotia, Canada: Dalhousie University; 1998. [26] Yang G, Ebadian MA. Turbulent forced convection in a helicoidal pipe with substantial pitch. Int J Heat Mass Transf 1996;39:2015e22. [27] Lin CX, Ebadian MA. Developing turbulent convective heat transfer in helical pipes. Int J Heat Mass Transf 1997;40:3861e73.