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Optimal and critical values of geometrical parameters of shell and helically coiled tube heat exchangers
Author’s Accepted Manuscript Optimal and critical values of geometrical parameters of shell and helically coiled tube heat exchangers Ashkan Alimoradi www.elsevier.com/locate/csite
PII: DOI: Reference:
S2214-157X(16)30193-9 http://dx.doi.org/10.1016/j.csite.2017.03.003 CSITE176
To appear in: Case Studies in Thermal Engineering Received date: 19 December 2016 Revised date: 23 January 2017 Accepted date: 19 March 2017 Cite this article as: Ashkan Alimoradi, Optimal and critical values of geometrical parameters of shell and helically coiled tube heat exchangers, Case Studies in Thermal Engineering, http://dx.doi.org/10.1016/j.csite.2017.03.003 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Optimal and critical values of geometrical parameters of shell and helically coiled tube heat exchangers Ashkan Alimoradi* Mechanical Engineering Department, Faculty of Engineering, Razi University, Kermanshah, Iran [email protected] [email protected]
*Corresponding author. Tel.: +989381875326; fax: +988346132250.
Abstract In the present study, calculations of the heat transfer and entropy generation were performed for steady state forced convection heat transfer in shell and helically coiled tube heat exchangers. The effect of geometrical parameters of the heat exchanger including: tube diameter (dt), coil diameter (dc), diameter of the inlet of shell (dv), shell diameter (dsh), height of coil (Hc), height of shell (Hsh), pitch (p) and the distance between the inlet and outlet of the shell (f) on the heat transfer rate and entropy generation was investigated Simultaneously. The critical and optimal values of this parameters were obtained which minimize and maximize the COD (heat transfer rate per entropy generation), respectively.
Keywords: Heat exchanger, Heat transfer, Entropy generation, Optimal, Critical
Nomenclature Q
Heat transfer rate (W)
c
specific heat capacity (J/kg°C)
COD
Coefficient of design (K)
d
Diameter (m)
f
Distance between inlet and outlet of the shell (m)
H
Height (m)
m°c
Mass flow rate (kg/s)
p
Pitch (m)
S°gen
Entropy generation rate (W/K)
T
Temperature (K)
Subscripts c
Coil
in
Inlet
out
Outlet
sh
Shell
v
Shell inlet
1. Introduction Heat exchangers are one of the key components in many industrial application like: HVAC, petroleum process, refrigeration, food preparation and etc. shell and helically coiled tube heat exchangers which usually consist of helical coiled tube and a cylindrical shell, are one of the most widely used heat exchangers in that mentioned applications. There are numerous studies about the heat transfer process in these types of heat exchangers. Many researchers [1-10] proposed equations for calculation of the heat transfer rates of inner and outer side of the coil as functions of geometrical as well as operational parameters of the heat exchanger. Also there are some studies which investigate the second law of thermodynamics and entropy generation in these types of heat exchangers. Ko [11] studied steady, laminar, fully developed forced convection heat transfer in a shell and helically coiled tube heat exchangers. He suggested an equation which determine the optimal curvature ratio based on the minimum entropy generation principle. Sasmito et al. [12] numerically investigated the effect of various cross sections geometries (i.e. circular, ellipse and square) on the heat transfer rate and entropy generation in a shell and helically coiled tube heat exchanger. Ahadi et al. [13] analyzed combined effects of length and heat flux of the coil as well as the effects of temperature dependence of thermo physical properties on the entropy generation rates and optimal operation in shell and helically coiled tube heat exchangers. Then, by using the minimal entropy generation principle, the inlet Reynolds number is optimized. Also they found that the entropy generation rates are highly dependent on the combined effects of length and heat flux of the coil. Huminic et al. [14] studied the laminar flow regime heat transfer and entropy generation inside a helically coiled tube-intube heat exchanger by using two different types of nano-fluids. Results show that, the use of nano-fluids in a helically coiled tube-in-tube heat exchanger improves the heat transfer performances and leads to reduction of the entropy generation. M. A. Abdous et al. [15] studied the entropy generation in the helically coiled tube under flow boiling. They found the optimum tube and coil diameters. The effect of different flow conditions such as mass velocity, inlet vapor quality, saturation temperature, and heat flux on contributions of pressure drop and heat transfer in entropy generation was discussed. The entropy generation analysis shows that there is a favorable region to use the helically coiled tube with respect to the straight one. A. Arabkoohsar et al. [16] found the optimal geometry and operational conditions of tube-in-tube helically coiled heat exchangers for both laminar and turbulent flows based on the second law of
thermodynamics. First, they derived a dimensionless function for entropy generation number as a function of four dimensionless variables, i.e. Prandtl number, Dean number, the ratio of helical pipe diameter to the tube diameter and the duty parameter of heat exchanger. Then entropy generation number was minimized to develop analytical expressions for the optimal values of that mentioned parameters. In this study, the effect of all geometrical parameters of the cylindrical shell and helically coiled tube heat exchangers on the heat transfer and entropy generation rate will be obtained. Then the optimum values of these parameters which maximize the heat transfer rate and minimize the entropy generation rate are found. According to literature review, this investigation was not performed for these special types of shell and helically coiled tube heat exchangers. Also there are some new geometrical parameters in this study which their effect and optimum values have not been obtained already. 2. Heat exchanger specification and applied equations The heat exchanger with its geometrical parameters has been shown in Fig .1. The range of change of these parameters has been shown in Table 1. The reason for selecting these ranges is that the most producer companies design these types of heat exchangers, in that mentioned range. For example, SENTRY Company produce models (TLR and FLR) of shell and helically coiled tube heat exchanger for cooling applications. The dimensions of these heat exchanger are as follows: dt=6.35 and 9.5 mm, dc=108 and 133 mm, dsh=124 and 143 mm, dv=12.7 and 19.05 mm, f=283 and 337 mm, p=10 and 14.92 mm, Hc=169 and 205 mm and Hsh=283 and 337 mm, respectively [17]. As it can be seen from Table 2, most dimensionless geometrical parameters of these two models are in the range of the present work. In the previous work [1], the heat transfer phenomena was studied by using a numerical code with the following specification: a) The standard K-ε model for simulation of turbulence. b) The SIMPLE algorithm as the pressure and velocity coupling scheme. c) First order Upwind as discretization scheme for momentum, K-ε and energy equations. d) Water as the working fluid with its temperature dependent properties.
Also this model was validated by use of an experimental set-up for the following ranges of the geometrical parameters: 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 The data of that work including: mass flow rates and inlet/outlet temperatures of both shell and coil sides, are used in the present study for calculation of heat transfer and entropy generation rate. 3. Calculation of heat transfer and entropy generation rate The heat transfer and entropy generation rate will be obtained according to the following equations: ̇ ̇
( ̇
) (
)
̇
̇
( (
) (1) )
(2)
The designer of the heat exchangers desire to model a heat exchanger which has the maximum heat transfer rate and minimum entropy generation. So the definition of the following parameters as the coefficient of design (COD) is reasonable: ̇
(3)
The geometrical parameters can be effective on this parameters. So to investigate the effect of geometrical parameters on the COD, they will be changed according to Table 1 while for all cases the inlet velocity of both sides was considered 1 m/s and the inlet temperature of coil side
and shell side was considered 90 and 20 °C, respectively. These parameters can be reduced to the following dimensionless parameters (which COD is a function of them): (4) The optimal and critical values of these dimensionless parameters will be obtained which maximize and minimize the heat COD, respectively. 4. Results and discussions Figs .2-3 show the effect of the dimensionless geometrical parameters on the heat transfer rate (Q), entropy generation rate (S°gen) and coefficient of design (COD). The following results can be obtained from these figures: Fig .2a shows that, with increase of (dsh) the heat transfer rate decreases (this is probably because, with increase of dsh, the existing gap between coil and shell increases. This leads to decrease of the effective velocity of the fluid adjacent the outer side of the coil which results in lower heat transfer). The heat transfer also decreases with increase of Hsh according to this figure. Also it can be found from Fig. 2c that, with increase of shell height, the entropy generation rate decreases while, it has not a regular changes with increase of shell diameter. According to these results, it seems that, an optimum point may be exist for COD as a function of the ratio of these two parameters. As it can be seen from Fig 3a, when Hsh/dsh is equal to 2.4, the COD is optimum. According to Fig. 2a-2b it can be found that, (dv) and (f) have a regular effect on the heat transfer rate. With increase of both of them, the heat transfer rate increases. The reason of enhancement of heat transfer rate with increasing of dv is that with increase of dv, the mass flow rate of the shell side increases (because the inlet velocity is kept constant) which leads to higher heat transfer rate. Also the reason of enhancement of heat transfer rate with increasing of f is that with increase of f, the fluid particles have more time for transferring energy). Furthermore, Fig. 2c-2d show that, the entropy generation rate will generally increase with increase of (dv) while, with increase of (f), first decreases until a minimum point (at f=0.22 m) and then increases. It can be found from Fig. 3b that, the optimum and critical values of COD will occur at (dv/f) equal to 0.107 and 0.083, respectively.
With referring again to Fig. 2b and 2d, the effect of (p) and (dt) on the heat transfer and entropy generation rate can be obtained. According to Fig. 2b, with increase of pitch and decrease of tube diameter, the heat transfer rate decreases uniformly. The reason of enhancement of heat transfer rate with decreasing of dt is that with decrease of dt, the mass flow rate of the coil side decreases (because the inlet velocity is kept constant) which leads to lower heat transfer rate. Also increase of pitch size results in decrease of the effective velocity of the fluid adjacent the outer side of the coil and thus decrease of heat transfer rate. It can be seen from Fig. 2d that, with increase of dt, the entropy generation rate increases uniformly (as mentioned with increase of tube diameter, the mass flow rate increases and thus the entropy generation rate increases according to Eq. 2 ). Also it can be found from this figure that, the maximum entropy generation rate will occur at p=22 mm. The effect of (p/dt) on the COD has been shown in Fig. 3c. As it can be seen, the optimum value of (p/dt) is equal to 1.54 while the critical value will occur in the following range: 3.077< (p/dt)< 4. A similar process is done to investigate the effect of (Hc) and (dc) on the heat transfer and entropy generation rate. Based on Fig. 2a and 2c, it is obvious that, the heat transfer rate increases uniformly with increase of (dc) (this is probably because, with increase of dc, the existing gap between coil and shell decreases, which results in higher heat transfer rate) and entropy generation rate is maximum when (dc) is equal to 7 cm. So according to Eq. 3, the COD is probably minimum (critical point) when (dc/dt) is equal to 7 which can be seen in Fig. 3d. Also the value of optimum (dc/dt) can be obtained from this figure which is equal to 10. A similar situation exists for (Hc). According to Fig. 2a and 2c, it can be seen that, the heat transfer rate increases uniformly with increase of (Hc) (this is because, with increase of the height of the coil, the heat transfer surface area increases which leads to higher heat transfer rate) and entropy generation rate is maximum when (Hc) is equal to 18 cm. After testing several parameters for dimensionless out the Hc, it was found that (f) is the best parameter which results more logical optimum and critical point than others. As it can be seen from Fig. 3e, the optimum value of COD will occur at Hc/f equals to 0.909. These results are summarized in Table 3. In this table, the optimal and critical values of dimensionless geometrical parameters (for the studied ranges) can be seen. The designer of this heat exchanger can use this useful table during the designing process.
5. Conclusions In this work, by using numerical and experimental data, the heat transfer and entropy generation rate were calculated for forced convection heat transfer in shell and helically coiled tube heat exchangers (which water is selected as the working fluid for its both sides). The optimal and critical values of dimensionless geometrical parameters (i.e. p/dt, Hc/f, Hsh/dsh, dc/dt and dv/f) which maximize the hat transfer and minimize entropy generation rate (which are desired in design process) were obtained.
References [1] Ashkan Alimoradi, Farzad Veysi, Prediction of heat transfer coefficients of shell and coiled tube heat exchangers using numerical method and experimental validation, International Journal of Thermal Sciences 2016; 107:196-208. http://dx.doi.org/10.1016/j.ijthermalsci.2016.04.010 [2] Ashkan Alimoradi, Study of thermal effectiveness and its relation with NTU in shell and helically coiled tube heat exchangers, Case Studies in Thermal Engineering 9 (2017) 100–107. http://dx.doi.org/10.1016/j.csite.2017.01.003 [3] Ashkan Alimoradi, Investigation of exergy efficiency in shell and helically coiled tube heat exchangers, Case Studies in Thermal Engineering 10 (2017) 1–8. http://dx.doi.org/10.1016/j.csite.2016.12.005 [4] 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: 430-46. http://dx.doi.org/10.1016/j.compchemeng.2009.11.008. [5] Pawar SS, Sunnapwar VK. Experimental studies on heat transfer to Newtonian and nonNewtonian fluids in helical coils with laminar and turbulent flow. Exp Therm Fluid Sci 2013; 44:792-804. http://dx.doi.org/10.1016/j.expthermflusci.2012.09.024. [6] Mirgolbabaei H, Taherian H, Domairy G, Ghorbani N. Numerical estimation of mixed convection heat transfer in vertical helically coiled tube heat exchangers. Inte J Numer Method Fluids 2011; 66:805-19. http://dx.doi.org/10.1002/fld.2284.
[7] Xin RC, Ebadian MA. The effects of Prandtl numbers on local and average convective heat transfer characteristics in helical pipes. J Heat Transf 1997; 119:467-73. http://dx.doi.org/10.1115/1.2824120. [8] 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, International Journal of Heat and Mass Transfer, 2012; 55:4295-300. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2012.03.074. [9] 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; 86:221-32. http://dx.doi.org/10.1016/j.cherd.2007.10.021. [10] N.Ghorbani, H.Taherian, M.Gorji, H.Mirgolbabaei, An experimental study of thermal performance of shell-and-coil heat exchangers, International Communications in Heat and Mass Transfer 2010; 37:775–781. [11] T.H. Ko, Thermodynamic analysis of optimal curvature ratio for fully developed laminar forced convection in a helical coiled tube with uniform heat flux, International Journal of Thermal Sciences 2006; 729–737. http://dx.doi.org/10.1016/j.ijthermalsci.2005.08.007. [12] J. C. Kurnia, A. P. Sasmito, T. Shamim, A. S. Mujumdar, Numerical investigation of heat transfer and entropy generation of laminar flow in helical tubes with various cross sections, Applied Thermal Engineering 2016; 849–860. http://dx.doi.org/10.1016/j.applthermaleng.2016.04.037. [13] Mohammad Ahadi, Abbas Abbassi, Entropy generation analysis of laminar forced convection through uniformly heated helical coils considering effects of high length and heat flux and temperature dependence of thermo physical properties, Energy Volume 82, 2015, Pages 322–332. http://dx.doi.org/10.1016/j.energy.2015.01.041 [14] Gabriela Huminic, Angel Huminic, Heat transfer and entropy generation analyses of nanofluids in helically coiled tube-in-tube heat exchangers, International Communications in Heat and Mass Transfer Volume 71, February 2016, Pages 118–125. http://dx.doi.org/10.1016/j.icheatmasstransfer.2015.12.031
[15] Mohammad Ali Abdous, Hamid Saffari, Hasan Barzegar Avval, Mohsen Khoshzat, Investigation of entropy generation in a helically coiled tube in flow boiling condition under a constant heat flux, International Journal of Refrigeration, Volume 60, December 2015, Pages 217–233. http://dx.doi.org/10.1016/j.ijrefrig.2015.07.026 [16] M. Farzaneh-Gord, H. Ameri, A. Arabkoohsar, Tube-in-tube helical heat exchangers performance optimization by entropy generation minimization approach, Applied Thermal Engineering,Volume 108, 5 September 2016, Pages 1279–1287. http://dx.doi.org/10.1016/j.applthermaleng.2016.08.028. [17] http://sentry-equip.com/Products/helical-coil-HX.htm.
Fig. 1: Geometry of the heat exchanger.
Fig. 2: Effect of geometrical parameters on the heat transfer and entropy generation rate.
Fig. 3: Effect of geometrical parameters on the COD.
Table 1 Ranges of the geometrical parameters [1]. Parameters dt dc dsh dv Hc Hsh p f
Values(m) 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
Table 2 Ranges of geometrical parameters of SENTRY heat exchangers versus the present work. Dimensionless parameters p/dt dc/dt Hc/f Hsh/dsh dv/f
TLR model 1.57 17 0.597 2.282 0.045
FLR model 1.57 14 0.608 2.356 0.057
Present work ranges 1.54-5.714 6-10 0.479-1.667 1.091-3.333 0.04-0.129
Models that are in the present Ranges: Both None Both Both Both
Table 3 Optimum and critical values of dimensionless geometrical parameters. Dimensionless geometrical parameters Hsh/dsh dv/f p/dt Hc/f dc/dt
Lower limit
Upper limit
Optimum value
Critical value
1.091 0.04 1.54 0.479 6
3.333 0.129 5.714 1.667 10
2.4 0.107 1.54 0.909 10
1.091 0.083 3.077< p/dt< 4 0.479 7
PII: DOI: Reference:
S2214-157X(16)30193-9 http://dx.doi.org/10.1016/j.csite.2017.03.003 CSITE176
To appear in: Case Studies in Thermal Engineering Received date: 19 December 2016 Revised date: 23 January 2017 Accepted date: 19 March 2017 Cite this article as: Ashkan Alimoradi, Optimal and critical values of geometrical parameters of shell and helically coiled tube heat exchangers, Case Studies in Thermal Engineering, http://dx.doi.org/10.1016/j.csite.2017.03.003 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Optimal and critical values of geometrical parameters of shell and helically coiled tube heat exchangers Ashkan Alimoradi* Mechanical Engineering Department, Faculty of Engineering, Razi University, Kermanshah, Iran [email protected] [email protected]
*Corresponding author. Tel.: +989381875326; fax: +988346132250.
Abstract In the present study, calculations of the heat transfer and entropy generation were performed for steady state forced convection heat transfer in shell and helically coiled tube heat exchangers. The effect of geometrical parameters of the heat exchanger including: tube diameter (dt), coil diameter (dc), diameter of the inlet of shell (dv), shell diameter (dsh), height of coil (Hc), height of shell (Hsh), pitch (p) and the distance between the inlet and outlet of the shell (f) on the heat transfer rate and entropy generation was investigated Simultaneously. The critical and optimal values of this parameters were obtained which minimize and maximize the COD (heat transfer rate per entropy generation), respectively.
Keywords: Heat exchanger, Heat transfer, Entropy generation, Optimal, Critical
Nomenclature Q
Heat transfer rate (W)
c
specific heat capacity (J/kg°C)
COD
Coefficient of design (K)
d
Diameter (m)
f
Distance between inlet and outlet of the shell (m)
H
Height (m)
m°c
Mass flow rate (kg/s)
p
Pitch (m)
S°gen
Entropy generation rate (W/K)
T
Temperature (K)
Subscripts c
Coil
in
Inlet
out
Outlet
sh
Shell
v
Shell inlet
1. Introduction Heat exchangers are one of the key components in many industrial application like: HVAC, petroleum process, refrigeration, food preparation and etc. shell and helically coiled tube heat exchangers which usually consist of helical coiled tube and a cylindrical shell, are one of the most widely used heat exchangers in that mentioned applications. There are numerous studies about the heat transfer process in these types of heat exchangers. Many researchers [1-10] proposed equations for calculation of the heat transfer rates of inner and outer side of the coil as functions of geometrical as well as operational parameters of the heat exchanger. Also there are some studies which investigate the second law of thermodynamics and entropy generation in these types of heat exchangers. Ko [11] studied steady, laminar, fully developed forced convection heat transfer in a shell and helically coiled tube heat exchangers. He suggested an equation which determine the optimal curvature ratio based on the minimum entropy generation principle. Sasmito et al. [12] numerically investigated the effect of various cross sections geometries (i.e. circular, ellipse and square) on the heat transfer rate and entropy generation in a shell and helically coiled tube heat exchanger. Ahadi et al. [13] analyzed combined effects of length and heat flux of the coil as well as the effects of temperature dependence of thermo physical properties on the entropy generation rates and optimal operation in shell and helically coiled tube heat exchangers. Then, by using the minimal entropy generation principle, the inlet Reynolds number is optimized. Also they found that the entropy generation rates are highly dependent on the combined effects of length and heat flux of the coil. Huminic et al. [14] studied the laminar flow regime heat transfer and entropy generation inside a helically coiled tube-intube heat exchanger by using two different types of nano-fluids. Results show that, the use of nano-fluids in a helically coiled tube-in-tube heat exchanger improves the heat transfer performances and leads to reduction of the entropy generation. M. A. Abdous et al. [15] studied the entropy generation in the helically coiled tube under flow boiling. They found the optimum tube and coil diameters. The effect of different flow conditions such as mass velocity, inlet vapor quality, saturation temperature, and heat flux on contributions of pressure drop and heat transfer in entropy generation was discussed. The entropy generation analysis shows that there is a favorable region to use the helically coiled tube with respect to the straight one. A. Arabkoohsar et al. [16] found the optimal geometry and operational conditions of tube-in-tube helically coiled heat exchangers for both laminar and turbulent flows based on the second law of
thermodynamics. First, they derived a dimensionless function for entropy generation number as a function of four dimensionless variables, i.e. Prandtl number, Dean number, the ratio of helical pipe diameter to the tube diameter and the duty parameter of heat exchanger. Then entropy generation number was minimized to develop analytical expressions for the optimal values of that mentioned parameters. In this study, the effect of all geometrical parameters of the cylindrical shell and helically coiled tube heat exchangers on the heat transfer and entropy generation rate will be obtained. Then the optimum values of these parameters which maximize the heat transfer rate and minimize the entropy generation rate are found. According to literature review, this investigation was not performed for these special types of shell and helically coiled tube heat exchangers. Also there are some new geometrical parameters in this study which their effect and optimum values have not been obtained already. 2. Heat exchanger specification and applied equations The heat exchanger with its geometrical parameters has been shown in Fig .1. The range of change of these parameters has been shown in Table 1. The reason for selecting these ranges is that the most producer companies design these types of heat exchangers, in that mentioned range. For example, SENTRY Company produce models (TLR and FLR) of shell and helically coiled tube heat exchanger for cooling applications. The dimensions of these heat exchanger are as follows: dt=6.35 and 9.5 mm, dc=108 and 133 mm, dsh=124 and 143 mm, dv=12.7 and 19.05 mm, f=283 and 337 mm, p=10 and 14.92 mm, Hc=169 and 205 mm and Hsh=283 and 337 mm, respectively [17]. As it can be seen from Table 2, most dimensionless geometrical parameters of these two models are in the range of the present work. In the previous work [1], the heat transfer phenomena was studied by using a numerical code with the following specification: a) The standard K-ε model for simulation of turbulence. b) The SIMPLE algorithm as the pressure and velocity coupling scheme. c) First order Upwind as discretization scheme for momentum, K-ε and energy equations. d) Water as the working fluid with its temperature dependent properties.
Also this model was validated by use of an experimental set-up for the following ranges of the geometrical parameters: 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 The data of that work including: mass flow rates and inlet/outlet temperatures of both shell and coil sides, are used in the present study for calculation of heat transfer and entropy generation rate. 3. Calculation of heat transfer and entropy generation rate The heat transfer and entropy generation rate will be obtained according to the following equations: ̇ ̇
( ̇
) (
)
̇
̇
( (
) (1) )
(2)
The designer of the heat exchangers desire to model a heat exchanger which has the maximum heat transfer rate and minimum entropy generation. So the definition of the following parameters as the coefficient of design (COD) is reasonable: ̇
(3)
The geometrical parameters can be effective on this parameters. So to investigate the effect of geometrical parameters on the COD, they will be changed according to Table 1 while for all cases the inlet velocity of both sides was considered 1 m/s and the inlet temperature of coil side
and shell side was considered 90 and 20 °C, respectively. These parameters can be reduced to the following dimensionless parameters (which COD is a function of them): (4) The optimal and critical values of these dimensionless parameters will be obtained which maximize and minimize the heat COD, respectively. 4. Results and discussions Figs .2-3 show the effect of the dimensionless geometrical parameters on the heat transfer rate (Q), entropy generation rate (S°gen) and coefficient of design (COD). The following results can be obtained from these figures: Fig .2a shows that, with increase of (dsh) the heat transfer rate decreases (this is probably because, with increase of dsh, the existing gap between coil and shell increases. This leads to decrease of the effective velocity of the fluid adjacent the outer side of the coil which results in lower heat transfer). The heat transfer also decreases with increase of Hsh according to this figure. Also it can be found from Fig. 2c that, with increase of shell height, the entropy generation rate decreases while, it has not a regular changes with increase of shell diameter. According to these results, it seems that, an optimum point may be exist for COD as a function of the ratio of these two parameters. As it can be seen from Fig 3a, when Hsh/dsh is equal to 2.4, the COD is optimum. According to Fig. 2a-2b it can be found that, (dv) and (f) have a regular effect on the heat transfer rate. With increase of both of them, the heat transfer rate increases. The reason of enhancement of heat transfer rate with increasing of dv is that with increase of dv, the mass flow rate of the shell side increases (because the inlet velocity is kept constant) which leads to higher heat transfer rate. Also the reason of enhancement of heat transfer rate with increasing of f is that with increase of f, the fluid particles have more time for transferring energy). Furthermore, Fig. 2c-2d show that, the entropy generation rate will generally increase with increase of (dv) while, with increase of (f), first decreases until a minimum point (at f=0.22 m) and then increases. It can be found from Fig. 3b that, the optimum and critical values of COD will occur at (dv/f) equal to 0.107 and 0.083, respectively.
With referring again to Fig. 2b and 2d, the effect of (p) and (dt) on the heat transfer and entropy generation rate can be obtained. According to Fig. 2b, with increase of pitch and decrease of tube diameter, the heat transfer rate decreases uniformly. The reason of enhancement of heat transfer rate with decreasing of dt is that with decrease of dt, the mass flow rate of the coil side decreases (because the inlet velocity is kept constant) which leads to lower heat transfer rate. Also increase of pitch size results in decrease of the effective velocity of the fluid adjacent the outer side of the coil and thus decrease of heat transfer rate. It can be seen from Fig. 2d that, with increase of dt, the entropy generation rate increases uniformly (as mentioned with increase of tube diameter, the mass flow rate increases and thus the entropy generation rate increases according to Eq. 2 ). Also it can be found from this figure that, the maximum entropy generation rate will occur at p=22 mm. The effect of (p/dt) on the COD has been shown in Fig. 3c. As it can be seen, the optimum value of (p/dt) is equal to 1.54 while the critical value will occur in the following range: 3.077< (p/dt)< 4. A similar process is done to investigate the effect of (Hc) and (dc) on the heat transfer and entropy generation rate. Based on Fig. 2a and 2c, it is obvious that, the heat transfer rate increases uniformly with increase of (dc) (this is probably because, with increase of dc, the existing gap between coil and shell decreases, which results in higher heat transfer rate) and entropy generation rate is maximum when (dc) is equal to 7 cm. So according to Eq. 3, the COD is probably minimum (critical point) when (dc/dt) is equal to 7 which can be seen in Fig. 3d. Also the value of optimum (dc/dt) can be obtained from this figure which is equal to 10. A similar situation exists for (Hc). According to Fig. 2a and 2c, it can be seen that, the heat transfer rate increases uniformly with increase of (Hc) (this is because, with increase of the height of the coil, the heat transfer surface area increases which leads to higher heat transfer rate) and entropy generation rate is maximum when (Hc) is equal to 18 cm. After testing several parameters for dimensionless out the Hc, it was found that (f) is the best parameter which results more logical optimum and critical point than others. As it can be seen from Fig. 3e, the optimum value of COD will occur at Hc/f equals to 0.909. These results are summarized in Table 3. In this table, the optimal and critical values of dimensionless geometrical parameters (for the studied ranges) can be seen. The designer of this heat exchanger can use this useful table during the designing process.
5. Conclusions In this work, by using numerical and experimental data, the heat transfer and entropy generation rate were calculated for forced convection heat transfer in shell and helically coiled tube heat exchangers (which water is selected as the working fluid for its both sides). The optimal and critical values of dimensionless geometrical parameters (i.e. p/dt, Hc/f, Hsh/dsh, dc/dt and dv/f) which maximize the hat transfer and minimize entropy generation rate (which are desired in design process) were obtained.
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Fig. 1: Geometry of the heat exchanger.
Fig. 2: Effect of geometrical parameters on the heat transfer and entropy generation rate.
Fig. 3: Effect of geometrical parameters on the COD.
Table 1 Ranges of the geometrical parameters [1]. Parameters dt dc dsh dv Hc Hsh p f
Values(m) 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
Table 2 Ranges of geometrical parameters of SENTRY heat exchangers versus the present work. Dimensionless parameters p/dt dc/dt Hc/f Hsh/dsh dv/f
TLR model 1.57 17 0.597 2.282 0.045
FLR model 1.57 14 0.608 2.356 0.057
Present work ranges 1.54-5.714 6-10 0.479-1.667 1.091-3.333 0.04-0.129
Models that are in the present Ranges: Both None Both Both Both
Table 3 Optimum and critical values of dimensionless geometrical parameters. Dimensionless geometrical parameters Hsh/dsh dv/f p/dt Hc/f dc/dt
Lower limit
Upper limit
Optimum value
Critical value
1.091 0.04 1.54 0.479 6
3.333 0.129 5.714 1.667 10
2.4 0.107 1.54 0.909 10
1.091 0.083 3.077< p/dt< 4 0.479 7