Journal of Mechanical Science and Technology 30 (7) (2016) 3357~3364 www.springerlink.com/content/1738-494x(Print)/1976-3824(Online) DOI 10.1007/s12206-016-0645-0 Modelling and experimental validation for off-design performance of the helical heat exchanger with LMTD correction taken into account† Nguyen Minh Phu* and Nguyen Thi Minh Trinh Faculty of Mechanical Engineering, University of Technology, Vietnam National University - Ho Chi Minh City, Vietnam (Manuscript Received August 27, 2015; Revised March 15, 2016; Accepted March 27, 2016) Abstract Today the helical coil heat exchanger is being employed widely due to its dominant advantages In this study, a mathematical model was established to predict off-design works of the helical heat exchanger The model was based on the LMTD and e-NTU methods, where a LMTD correction factor was taken into account to increase accuracy An experimental apparatus was set-up to validate the model Results showed that errors of thermal duty, outlet hot fluid temperature, outlet cold fluid temperature, shell-side pressure drop, and tube-side pressure drop were respectively ±5%, ±1%, ±1%, ±5% and ±2% Diagrams of dimensionless operating parameters and a regression function were also presented as design-maps, a fast calculator for usage in design and operation of the exchanger The study is expected to be a good tool to estimate off-design conditions of the single-phase helical heat exchangers Keywords: Effectiveness-NTU; Experimental validation; Helical heat exchanger; LMTD; Off-design; Pressure drop Introduction Heat exchangers are core components of thermal systems Their improvements will allow efficient use of energy Therefore researches on the heat exchangers have been often paid attentions to and published with a high density for the past few decades There are two problems which are often mentioned in previous studies Those are to enhance heat transfer rate of heat exchangers [1, 2] and predict off-design conditions of the available heat exchangers [3, 4] Mostly the studies concentrated on the straight tube heat exchangers as confirmed by Wongwises and Polsongkram [5, 6] However, the helical coil heat exchangers show dominant advantages in comparison with the straight tube heat exchangers Prabhanjan et al [7] showed experimentally that heat transfer rate of helical heat exchangers is higher than that of straight tube heat exchangers due to centrifugal forces to act on the moving fluid, causing the formulation of secondary flow Besides, the helical capillary in a refrigeration system has a length which is 14% shorter than the straight capillary as showed by Zareh et al [8] Furthermore, the helical heat exchangers offer compactness, compensation of thermal expansion, vibration reduction, easy construction and low capital cost In recent years, there has been a remarkable consideration on applications of helical heat exchangers for thermal systems Seara et al [9] formed * Corresponding author Tel.: +84 906 929498, Fax.: +84 838 653823 E-mail address: nmphu@hcmut.edu.vn † Recommended by Associate Editor Ji Hwan Jeong © KSME & Springer 2016 an analytical model to investigate the helical coil rectifier in an ammonia-water absorption chiller Xiaowen and Lee [10] studied experimentally the helical heat exchanger for heat recovery air-conditioners Sogni and Chiesa [11] developed a model to calculate heat recovery boiler using a helical tube Also, helical heat exchangers are regularly used for the liquidto-suction heat exchanger and liquid subcooler in refrigeration cycles [12] Enhanced heat transfer by adding fins to the gas side of the helical exchanger was also noted by researchers Crimped spiral fins are usually used because of their durability and reliability Srisawad and Wongwises [13] provided the experimental data of the crimped finned heat exchangers Boonsri and Wongwises [14] formed the rigorous mathematical model for the helical exchanger with crimped spiral fins The theoretical results from their model coincided well with the experimental data Hardik et al [15] conducted experiments in order to investigate local temperature and Nusselt number distribution in helical coils Correlations of local Nusselt numbers such as functions of diameter ratio, Reynolds number, and Prandtl number were given with maximum deviation of 21% Bezyan et al [16] analyzed the results of numerical simulation by using CFD software package for helical heat exchangers The best configuration was indicated in their study Mukesh Kumar et al [17] performed CFD simulation for flow and thermal behavior of nanofluid in helical heat exchangers to estimate heat transfer coefficient and pressure drop Generally, the previous studies were to find out the charac- 3358 N M Phu and N T M Trinh / Journal of Mechanical Science and Technology 30 (7) (2016) 3357~3364 Tube-side fluid 1 = + UA hi Ai he Ae (1) The above equation can be rewritten as follows: h A + hi Ai = e e UA hi Ai he Ae Shell-side fluid (2) Let the subscript “ref” be reference parameters corresponding to known conditions The known conditions, for example, can be obtained from a manufacturer’s catalogue or experiment From Eq (2) a ratio can be created between the operating conditions and the reference conditions of the same heat exchanger as follows: Fig Helical coil heat exchanger teristics and design of helical heat exchangers But estimation of off-design conditions (i.e temperature, pressure drop, thermal duty) of an available helical heat exchanger has not been noted In order to estimate those conditions, the geometrical parameters inside the exchanger should be given and inputted to heat transfer and pressure drop models Unfortunately, the geometrical parameters are sometimes missed from manufacturers Few parameters are known from manufacturers’ catalogues This causes obstacles in prediction of operating conditions different from design conditions In practice, heat exchangers usually run in part-load or overload modes To overcome such a difficulty, Garcia et al [18] developed a model for the straight tube condenser and evaporator of a refrigeration system Errors of the predicted temperatures and capacities are from ±1 to ±7% in comparison to the measured values However, the pressure drop model is somewhat complicated and geometry of tube bundles has to be known Furthermore, experimental validation of the pressure drop model was not performed In this paper, a similar model to that of Garcia et al [18] is formed for the helical heat exchanger LMTD correction factor is taken into account so that the predicted results are closer to those of the experiment Moreover the results of pressure drop are also presented regarding both the modelling and experimental approaches Model formulation 2.1 Heat transfer model Fig presents the schematic diagram of a helical heat exchanger A fluid is traveling inside a helical tube Another fluid is passing across the helical tube The fluids carry different thermal energies; therefore, heat is transferred from hot fluid to cold fluid through the surface of the helical tube The general ideal of the mathematical model can be seen in Ref [18] Some equations are presented here for the sake of easy understanding Overall conductance of a heat exchanger can be written as the equation below if fouling and wall resistances are neglected: UA hi he he ,ref Ae + hi ,ref Ai = UAref hi ,ref he ,ref he Ae + hi Ai (3) We can define ratios as: bi = hi hi ,ref (4) be = he he ,ref (5) In heat transfer design, thermal resistances should be equal in order to gain an optimum design Thus an approximation can be done as the following equation: (6) he ,ref Ae = hi ,ref Ai Since the Eq (3) can be rewritten as: 2hi ,ref Ai UA bi b e = bi b e = UAref he ,ref b e Ae + hi ,ref bi Ai b e + bi (7) The heat transfer coefficient for the fluid flowing inside helical tube can be computed by means of the Rogers and Mayhew’s correlation [19] as: 0.1 ổdử k hi = 0.023Re0.85 Pr 0.4 ỗ ÷ èDø d (8) for Re £ 50000 [15] where Reynolds number and Prandtl number are, respectively: Re = Pr = & md Ac m cpm k (9) (10) D and d are respectively the outer and inner diameters of the helical tube N M Phu and N T M Trinh / Journal of Mechanical Science and Technology 30 (7) (2016) 3357~3364 Therefore hi can be rewritten as follows: C= hi = ( 0.023) k 0.6 m -0.45 m& 0.85c 0.4 ( D -0.1d -0.05 ) p ( ) ÷÷ ø 0.25 FN k D (12) where FN is a correction factor whose values are dependent on the number of rows of tube bundle Neglecting the influence of temperature-dependent properties, i.e (Pr/Prw)0.25 = 1, the coefficient he can be rearranged by: he = ( 0.27 ) k 0.64 m -0.27 m& 0.63c 0.36 ( FN D -0.37 ) p ( ) (13) As can be seen in the above-mentioned equations, the heat transfer coefficients are functions of three terms including constant coefficient, properties of fluid, and geometry of the helical heat exchanger The terms of constant coefficient, and geometry will be eliminated in the ratio of heat transfer coefficients Thus, the ratios of tube side and shell side heat transfer coefficients are respectively given by: 0.6 ổ k hi =ỗ hi ,ref ỗố kref ữ ữ ứ ổ k h be = e = ỗ he ,ref ỗố kref ữ ữ ứ bi = ổ m ỗ ỗ mref ố 0.64 ổ m ỗ ç m ref è ÷ ÷ ø -0.45 ữ ữ ứ ổ m& ỗ ỗ m& ref ố -0.27 ữ ữ ứ ổ m& ỗ ỗ m& ref ố 0.85 ữ ữ ứ ổ cp ỗ ç c p ,ref è 0.63 æ cp ç ç c p ,ref è (18) (11) Heat transfer and pressure drop of cross-flow straight tube bundles can be used to model helical heat exchanger as shown in previous studies [11, 20] Therefore the shell side heat transfer coefficient is obtained from the Zukauskas’s correlation (1000 < Re < 200000) for in-line tube bundles [21]: æ Pr he = 0.27 Re0.63 Pr 0.36 ỗỗ ố Prw Cmin Cmax 3359 ữ ÷ ø 0.4 where Cmin and Cmax are the smaller and the larger of m& i c p ,i and m& ec p ,e , respectively To enhance reliability of the heat transfer model, a LMTD correction factor is considered in the model The factor is that of a cross-flow heat exchanger where the flow in the helical tube is considered to be unmixed and the flow outside the tube is mixed [22] In this research tube-side flow is assigned as hot fluid and shell-side flow is cold fluid Therefore, the factor can be found by means of: F= r= r0 = p= q= r r0 q ln q 1 - ln p 1- p p-q 1- q ln 1- p Ti ,in - Ti ,out Ti ,in - Te ,in Te ,out - Te ,in Ti ,in - Te ,in Pressure drop inside a tube of length L and inner diameter d is given by: Dp = f 0.36 2.2 Pressure drop inside helical tube (14) ÷ ÷ ø (19) L m& d r Ac2 (20) (15) The effectiveness and number of transfer units (e-NTU) relation of the helical heat exchanger are similar to those of a cross-flow heat exchanger (with one fluid mixed and the other unmixed) if the number of turns of the helical tube is equal or more than six as pointed out by San et al [20] Therefore the relation is: The Fanning friction factor inside a helical tube can be used correlation of Srinivasan et al [23] as follows: ổ 2R fỗ ữ ố d ø 0.5 é ỉ R -2 ù = 0.084 Re ỗ ữ ỳ ởờ ố d ứ ûú -0.2 (21) -2 2R ỉ 2R ÷ < 700 and < d < 104 d è ø for Re ỗ e= 1 - exp ộở -C (1 - exp(- NTU ) ) ùû C { } (16) ì ü éë1 - exp(- NTU C )ựỷý ợ C ỵ (17) Cmax mixed, Cmin unmixed e = - exp í- Cmax unmixed, Cmin mixed where R is curvature radius of helical coil Similarly to Eqs (14) and (15), the pressure drop ratio of operating conditions to reference conditions can be correlated as: æ m Dp =ỗ Dpref ỗố m ref ữ ữ ứ 0.2 r ref r ổ m& ỗ ỗ m& ref è 1.8 ÷ ÷ ø (22) 3360 N M Phu and N T M Trinh / Journal of Mechanical Science and Technology 30 (7) (2016) 3357~3364 Table Reference data Input: - Reference data: Q& ref , m& i ,ref , m& e ,ref , Ti,in,ref, Te,in,ref, Dph ,ref , Dpc ,ref Heat transfer rate Q& ref = 6.2 kW Tube-side pressure drop Dpi ,ref = 93 kPa Shell-side pressure drop Dpe ,ref = 20 kPa Tube-side volumetric flow rate 0.278 l/s Shell-side volumetric flow rate 0.194 l/s Inlet tube-side fluid temperature Ti,in,ref = 59.5°C Inlet shell-side fluid temperature Te,in,ref = 31.5°C - Operating data: m& i , m& e , Ti,in, Te,in ¯ Calculate the remain parameters of reference data: Ti,out,ref= Ti,in,ref - Q& ref /( m& i ,ref cp,i,ref) cp,e,ref) Te,out,ref= Te,in,ref + Q& /( m& ref DTlm ,ref = e , ref Fref from Eq (19) (Ti ,in ,ref - Te ,out ,ref ) - (Ti ,out ,ref - Te ,in ,ref ) ln UAref = Q& max Ti ,in ,ref - Te ,out ,ref Ti ,out ,ref - Te ,in ,ref Qref Fref DTlm ,ref ¯ Assume e ¯ = Cmin (Ti ,in - Te,in ) Q& = e Q& ¬ ­ Test section e Dpi and Dpe from Eqs (22) and (24) bi and be from Eqs (14) and (15) UA from Eq (7) NTU=UA/Cmin enew from Eqs (16) or (17) ¯ |e -enew|< pre-specified value ¯ Yes Output: Q& , Ti,out, Te,out, Dpi , Dpe Cool water tank Air-cooled heat exchanger t max Ti,out= Ti,in - Q& /( m& i cp,i) Te,out= Te,in + Q& /( m& cp,e) p Flow meter Pump t Valve No t p t ® Flow meter Fig Program flow chart 2.3 Shell-side pressure drop Hot water tank Pump Heater From Ref [24] the Fanning friction factor of the fluid across helical tube bundle can be expressed as: Fig Experimental apparatus f = 0.26 Py Re -0.117 (23) where Py is shell-side porosity which depends on geometry of the bundle Finally, the shell-side pressure drop ratio can be computed from the following relation: æ m Dp =ỗ Dpref ỗố m ref ữ ữ ứ 0.117 r ref r ổ m& ỗ ỗ m& ref ố 1.8883 ÷ ÷ ø mal duty at operating conditions Q& max is evaluated in the next step From these parameters Q& can be found After that the outlet fluid temperatures at operating conditions are computed Next bi, be, UA, and NTU are calculated Thereafter, new effectiveness is determined and compared to the assumed value A new loop is carried out if the error is greater than a given tolerance (24) The key equations are Eqs (14), (15), (22) and (24) As can be seen they are independent on geometry of the exchanger The models above should be programed by using a computer program The system of equations has a lot of temperaturedependent properties Therefore the EES software [25] is the pertinent candidate for the current study The properties of fluids are evaluated at bulk temperature The procedure for solving the system of equations is summarized by Fig From reference data the remaining temperatures and UAref can be calculated Effectiveness is then assumed Maximum ther- Experimental validation The experimental apparatus shown in Fig was performed to determine whether the present model could be validated In the apparatus, water is used as the working fluids for both sides of the tested helical heat exchanger Hot water is heated by a three-phase electrical heater in a hot water tank The tank’s temperature can be adjusted to set various experiments The hot water is pumped to the tube-side of the exchanger Here the hot water is decreased in temperature by cold water The cold water almost at room temperature enters the shellside of the exchanger The cold water rejects heat to the envi- 3361 N M Phu and N T M Trinh / Journal of Mechanical Science and Technology 30 (7) (2016) 3357~3364 Table Constant coefficients for Eq (30) Constant coefficients C1 C2 C3 C4 0.968806 0.382933 0.420696 -0.729444 2.050495 +5% Modelled Experimental -5% 4 97.84% 40 6 Q Modelled (kW) Fig Experimental vs theoretical heat transfer rate Shell-side pressure drop [kPa] QMeasured (kW) R2 C0 30 20 10 200 43 42 600 800 1000 Fig Shell-side pressure drop 41 100 +1% 40 Modelled Experimental -1% 39 38 37 36 36 37 38 39 40 41 42 43 44 Modelled outlet cold fluid temperature (oC) Fig Experimental vs theoretical cold fluid temperature 58 Measured outlet hot fluid temperature (oC) 400 Shell-side flow rate [lit/h] Tube-side pressure drop [kPa] Measured outlet cold fluid temperature (oC) 44 80 60 40 20 450 550 650 750 850 950 1050 Tube-side flow rate [lit/h] 56 Fig Tube-side pressure drop 54 +1% 52 -1% 50 48 46 46 48 50 52 54 56 58 Modelled outlet hot fluid temperature (oC) Fig Experimental vs theoretical outlet hot fluid temperature ronment by an air-cooled heat exchanger right after the test section It then travels to a large tank and mixes water in the tank Water volumetric flow rates are measured by floating flow meters with 0.0028 l/s resolution Inlet and outlet temperatures of both sides are measured by thermocouples with Fig Comparison of theoretical results 0.1°C precision Tube-side and shell-side pressure differences are processed by differential pressure transducers with an ac- 3362 N M Phu and N T M Trinh / Journal of Mechanical Science and Technology 30 (7) (2016) 3357~3364 1.4 ratioMi=0.75 ratioMi=0.5 ratioT i=1 ratioT i=1 1.2 ratioT e=0.75 ratioT e=0.75 0.8 ratioQ ratioQ ratioT e=1 ratioT e=1 0.8 0.6 0.6 ratioT e=1.25 0.4 0.4 0.6 0.8 1.2 1.4 0.4 0.4 1.6 ratioT e=1.25 0.6 0.8 1.2 1.4 1.6 1.4 1.6 ratioM e ratioM e 2.4 ratioMi=0.75 ratioMi=1.25 ratioTi=1.25 2.2 1.8 ratioTe=0.75 ratioTi=1.25 ratioTe=0.75 1.6 1.8 ratioQ ratioQ ratioTe=1 1.4 ratioTe=1 1.6 1.2 0.8 0.4 ratioTe=1.25 1.4 ratioTe=1.25 1.2 0.6 0.8 1.2 1.4 1.6 0.4 0.6 0.8 ratioM e 1.2 ratioM e Fig 10 Effect of operating conditions on thermal duty curacy of ±0.075% of the measured value Water flow rates are adjusted by ball valves Table presents reference data used in the current work Figs 4-6 show the calculated results and experimental results of thermal duty and outlet water temperatures Error of the thermal duty is less than ±5%, and it can be noted that errors of the outlet water temperatures are lower than ±1% This confirmed that the heat transfer model is good Figs and show the pressure drops between two approaches, modelling and experiment It can be concluded that the results coincide well with each other The relative error of shell-side pressure drop is no greater than ±5% The difference of tube-side pressure drop is within ±2%, except the difference up to ±8% at low flow rate Fig compares the analytical result of the present study with that from previous study [14] Reference data used in the comparison was marked in the Fig The previous study formed the analytic model for helical heat exchanger with crimped spiral fin on outer tube The shell-side fluid was air like a cold fluid It can be clearly seen that the trends are almost the same and the deviation is moderate The difference is due to the fact that the present model is formed for smooth pipes However, geometrical parameters are eliminated Therefore, the prediction in case of finned tube may be applicable Parametric study In this section, the effects of operating conditions on heat transfer rate are presented Results are displayed in the form of non-dimensional parameters so that they can be used as design-maps in design or operation of arbitrary helical heat exchangers Only the results of heat transfer rate are presented because pressure drop model is straightforward as shown by Eqs (22) and (24) If the fluids are not changed for an existing heat exchanger, the predicted pressure drops are much simpler because in that case the pressure drops are nearly proportional to the square of flow rate The dimensionless parameters can be defined as follows: ratioTe = ratioTi = Te ,in Te ,in ,ref Ti ,in Ti ,in ,ref (25) (26) ratioM e = m& e m& e ,ref (27) ratioM i = m& i m& i ,ref (28) N M Phu and N T M Trinh / Journal of Mechanical Science and Technology 30 (7) (2016) 3357~3364 Q& ratioQ = & Qref (29) Four first ratios were selected as independent parameters to explore their effects on the heat transfer rate, i.e ratioQ The results can be seen in Fig 10 Generally ratioQ increases when flow rate increases, the inlet hot water temperature increases, or the inlet cold water temperature decreases, as expected These are obvious because increase in flow rates leads to increase in heat transfer coefficients, and increase in hot fluid temperature or decrease in cold fluid temperature results in an increase in logarithm mean temperature difference To facilitate use in design and rating of the helical heat exchanger, a correlation of heat transfer rate and operating conditions is developed The heat transfer rate is a function of flow rates and inlet temperatures in the following form: ratioQ = C0 ´ ratioM eC1 ´ ratioM iC2 ´ ratioTeC3 ´ ratioTi C4 L & mp NTU p Pr Q& R Re T UA 3363 : Length of a pipe (m) : Mass flow rate (kg/s) : Number of transfer units : Pressure (Pa) : Prandtl number : Heat transfer rate (W) : Curvature radius (m) : Reynolds number : Temperature (°C) : Overall conductance (W/K) Greek symbols b D e m r : Heat transfer coefficient ratio : Difference : Effectiveness : Dynamic viscosity (N.s/m2) : Density (kg/m3) (30) Subscripts Table shows the constant coefficients of the above equation The equation is valid for water as the working fluids, ratioM e (0.9:1.4), ratioM i (0.9:1.4), ratioTe (0.7:1.2), ratioTi (0.9:1.4) corresponding to heat transfer rate ratio ratioQ in the relatively large range of 0.5 to Conclusions The single phase heat transfer model and pressure drop model were formulated to predict off-design conditions of the helical heat exchanger The heat transfer model considered LMTD correction factor to reach a better prediction The models could evaluate outlet fluid temperatures, thermal duty, and pressure drops for various operating conditions without geometrical information of heat transfer surface An experiment was set-up to determine the reliability of the models Results showed that the differences between calculation and experiment are from ±1 to ±5% The dimensionless figures of the operating parameters were presented as the design-maps Moreover a power regression model equation of dimensionless heat transfer rate as a function of four parameters was supplied The valid working fluid for the equation is water The equation can be used in a range of heat transfer rates from 0.5 to times reference data with the coefficient of determination R2 of 97.84% Nomenclature -A cp d D F h k : Area (m2) : Specific heat at constant pressure (J/kg.K) : Internal diameter of helical coil (m) : External diameter of helical coil (m) : LMTD correction factor : Heat transfer coefficient (W/m2.K) : Thermal conductivity (W/m.K) c e i in lm max out ref w : Cross-sectional : External : Internal : Inlet : Logarithm mean : Maximum value : Outlet : Reference : Wall References [1] F Vitillo et al., An innovative plate heat exchanger of enhanced compactness, Applied Thermal Engineering, 87 (2015) 826-838 [2] Y Yujie et al., Performance evaluation of heat transfer enhancement in plate-fin heat exchangers with offset strip fins, Physics Procedia, 67 (2015) 543-550 [3] A Rovira et al., Thermoeconomic optimisation of heat recovery steam generators of combined cycle gas turbine power plants considering off-design operation, Energy Conversion and Management, 52 (4) (2011) 1840-1849 [4] N Kayansayan, Thermal behavior of heat exchangers in offdesign conditions, Heat Recovery Systems and CHP, (3) (1989) 265-273 [5] S Wongwises and M Polsongkram, Evaporation heat transfer and pressure drop of HFC-134a in a helically coiled concentric tube-in-tube heat exchanger, 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[22] R A Bowman, A C Mueller and W M Nagle, Mean temperature difference in design, Trans Am Soc Mech Engrs., 62 (1940) 283-294 [23] S Kakaỗ and H Liu, Heat Exchangers - Selection, Rating, and Thermal Design, Second Ed., CRC Press (2002) [24] E M Smith, Advances in Thermal Design of Heat Exchangers, John Wiley & Sons (2005) [25] S A Klein, Engineering Equation Solver, F-Chart Software (2013) Nguyen Minh Phu received B.E in 2006, and M.E in 2009 from Ho Chi Minh city University of Technology (HCMUT), Vietnam, and Ph.D from University of Ulsan, Korea in 2012 He had been with the Arizona State University at Tempe during the summer 2014 as exchange visitor He has been a lecturer of Mechanical Engineering Faculty in HCMUT since 2006 His research interests include the design of thermal systems and the applied renewable energy Nguyen Thi Minh Trinh received B.E in 2002, and M.E in 2009 from Ho Chi Minh city University of Technology (HCMUT), Vietnam She is a lecturer of Mechanical Engineering Faculty in HCMUT At the same time, she is a lecturer of Energy management and Energy audit training project, organized by Japan International Cooperation Agency (JICA) in Vietnam through the Ministry of Industry and Trade of Vietnam Her research interests include energy efficiency and economics, heat exchanger, refrigeration and air conditioning engineering  ... can be created between the operating conditions and the reference conditions of the same heat exchanger as follows: Fig Helical coil heat exchanger teristics and design of helical heat exchangers... units (e-NTU) relation of the helical heat exchanger are similar to those of a cross-flow heat exchanger (with one fluid mixed and the other unmixed) if the number of turns of the helical tube is... properties Therefore the EES software [25] is the pertinent candidate for the current study The properties of fluids are evaluated at bulk temperature The procedure for solving the system of equations