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Analysis of coiled-tube heat exchangers to improve heat transfer rate with spirally corrugated wall
International Journal of Heat and Mass Transfer 53 (2010) 3928–3939
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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Analysis of coiled-tube heat exchangers to improve heat transfer rate with spirally corrugated wall A. Zachár * } H-2103, Hungary Department of Informatics, Szent István University, Páter K. u. 1., Gödöllo
a r t i c l e
i n f o
Article history: Available online 1 June 2010 Keywords: Spirally corrugated helical pipe Heat transfer augmentation Secondary flow
a b s t r a c t Steady heat transfer enhancement has been studied in helically coiled-tube heat exchangers. The outer side of the wall of the heat exchanger contains a helical corrugation which makes a helical rib on the inner side of the tube wall to induce additional swirling motion of fluid particles. Numerical calculations have been carried out to examine different geometrical parameters and the impact of flow and thermal boundary conditions for the heat transfer rate in laminar and transitional flow regimes. Calculated results have been compared to existing empirical formulas and experimental tests to investigate the validity of the numerical results in case of common helical tube heat exchanger and additionally results of the numerical computation of corrugated straight tubes for laminar and transition flow have been validated with experimental tests available in the literature. Comparison of the flow and temperature fields in case of common helical tube and the coil with spirally corrugated wall configuration are discussed. Heat exchanger coils with helically corrugated wall configuration show 80–100% increase for the inner side heat transfer rate due to the additionally developed swirling motion while the relative pressure drop is 10–600% larger compared to the common helically coiled heat exchangers. New empirical correlation has been proposed for the fully developed inner side heat transfer prediction in case of helically corrugated wall configuration. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction Helically coiled-tube heat exchangers are one of the most common equipment found in many industrial applications ranging from solar energy applications, nuclear power production, chemical and food industries, environmental engineering, and many other engineering applications. Heat transfer rate of helically coiled heat exchangers is significantly larger because of the secondary flow pattern in planes normal to the main flow than in straight pipes. Modification of flow is due to the centrifugal forces (Dean roll cells [1]) caused by the curvature of the tube. Several studies have been conducted to analyse the heat transfer rate of coiled heat exchangers in laminar and turbulent flow regimes [2– 6]. Numerical study of laminar flow and forced convective heat transfer in a helical square duct has been carried out by Jonas Bolinder and Sundén [7]. Many authors investigated experimentally the turbulent heat transfer in helical pipes [8,9]. Further enhancement of heat transfer rate in coiled pipes has great importance in several industrial applications mainly where the flow regime is in the laminar or transitional zone like hot water solar energy applications. There are basically two different concepts to increase * Tel.: +36 28 522051; fax: +36 28 410804. E-mail address: [email protected] 0017-9310/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijheatmasstransfer.2010.05.011
the rate of heat transferred, the first one is the active and the other one is the passive method. Many different active techniques exist to increase the heat transfer rate mostly for straight pipes. In case of passive techniques heat transfer enhancement by chaotic mixing in helical pipes has great importance and investigated by Kumar and Nigam [10] and Acharya et al. [11]. Helical screw-tape inserts have been investigated in straight pipes experimentally by Sivashanmugam and Suresh [12]. There is considerable amount of work reported in the literature on heat transfer augmentation in straight pipes with different corrugation techniques, helical tape inserts [13–17]. Experimental investigation of thermosyphon solar water heater with twisted tape inserts has been carried out by Jaisankar et al . [18]. According to the author’s knowledge a few examinations are considered in helically coiled tubes with different passive heat transfer augmentation techniques like inside wall corrugation, helical tape inserts [19] and this question is not studied numerically at all in the available literature. Experimental investigations have been conducted in a helical pipe containing inside springs to study the heat transfer rate and pressure drop by Yildiz et al. [20]. Helical corrugation on the surface of a helically coiled tube possibly increases the heat transfer rate because of the developed swirling motion. Basic aim of this study is to investigate the impact of different geometrical parameters of the corrugation for the inner side heat transfer rate in case of helical tube
A. Zachár / International Journal of Heat and Mass Transfer 53 (2010) 3928–3939
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Nomenclature cp(x, y, z, T) specific heat function of the working fluid (J kg1 K1) diameter of the coil (m) dc dp diameter of the pipe (m) De Dean number (=Re(dp/dc)0.5) h corrugation depth (m) hi inside heat transfer coefficient (W m2 K1) kavg area averaged thermal conductivity at a specific cross section (W m1 K1) Nufd peripherally averaged fully developed Nusselt number (=qwdp/kavg(Tw Tm)) p helical pitch of corrugation (m) pc helical pitch of the coil (m) P(x, y, z) pressure field (Pa) Pr Prandtl number (=cpg/k) DP pressure drop (Pa) qw area averaged wall heat flux at a specific cross section (W m2) Q flow rate (kg/s) Re Reynolds number (=qVavgdp/g) x, y, z Cartesian coordinates (m)
T(x, y, z) temperature function of the water inside the heat exchanger coil (°C) Tw area averaged wall temperature at a specific cross section (°C) Tm mass-flow averaged mean temperature of the fluid at a specific cross section (°C) Tin, Tout area averaged inlet/outlet temperature (°C) U1(x, y, z), U2(x, y, z), U3(x, y, z) velocity component functions into the three possible directions (m/s) Vavg area averaged velocity (m/s) Greek symbols u angular coordinate normal to the tube cross section (°) g(x, y, z, T) dynamic viscosity function of the working fluid (N s m2) k(x, y, z, T) thermal conductivity function of the working fluid (W m1 K1) q(x, y, z, T) density function of the working fluid (kg m3)
decrease the critical Reynolds number significantly as it was proven by Vicente et al. [17] for corrugated straight pipes. In this study most of the investigated flow cases are in laminar flow state because the Reynolds number of the studied flow cases covers a range from Re = 100 to Re = 7000 and perhaps for the largest Reynolds number flow cases (approximately Re = 7000) a small level of turbulence can be occurred. The momentum equation of the fluid is based on the three dimensional Navier–Stokes equations. A SIMPLE like method is applied to solve the momentum and continuity equations. The dependent variables that describe the present flow situation are the temperature, T, velocity components Ui in the x, y and z directions respectively and the pressure field P. 2.1. Conservation equations Fig. 1. Schematic figure of the corrugated coiled-tube heat exchangers with the following geometrical parameters dp = 20 mm, pc = 40 mm, p = 44.5 mm and h = 2 mm.
heat exchangers. Fig. 1 shows the basic configuration of the studied heat exchanger coils with helical corrugation. Helically corrugated coils with two turns have been investigated to examine the fully developed flow state of the corrugated tubes. The corrugation is located on the outer side of the helical pipe which makes a helically spiraling wall on the inner side of the pipe. 2. Mathematical formulation This section provides the basic equations that must be solved to describe the velocity field and the temperature distribution inside the heat exchanger coils. It is well known that, the transition from laminar to turbulent flow in curved pipes occur much higher critical Reynolds number (Recrit) than in straight pipes. The critical Reynolds number for smooth helical pipes can be estimated by the following pffiffiffi formula found in Srinivasan et al. [21] Recrit ¼ 2100ð1 þ 12 dÞ. Applying this formula the values of the critical Reynolds number d in the studied geometrical cases are as follows: d ¼ dpc ¼ 0:015 0:02 0:025 ! Recrit ffi 7393; d ¼ 0:34 ! Recrit ffi 8210; d ¼ 0:34 ! Recrit ffi 8933. 0:34 The physical problem is a steady, three dimensional laminar or transitional flow configuration. Corrugation and surface roughness can
The following set of partial differential equations for U1, U2, U3, P and T as functions of x, y, z describes the flow and temperature field inside a helically coiled heat exchanger. The conservation equations are formulated in the Cartesian coordinate system because the applied flow solver (Ansys CFX 11.0) uses the Cartesian system to formulate the conservation equations for all (vector Ui and scalar T, P) quantities. Description of the entire geometry of the studied problem is incorporated into the generated unstructured numerical grid. 2.1.1. Continuity equation The continuity equation is formulated in the following manner in Cartesian coordinate system
@ ðqU i Þ ¼ 0: @xi
ð1Þ
2.1.2. Momentum equations The following equation system is the representation of the momentum equations in Cartesian coordinate system where i, j e {1, 2, 3},
@ @P @ @U j : ðqU j U i Þ ¼ þ g @xi @xj @xi @xi
ð2Þ
g is the dynamic viscosity and q is the density of the working fluid.
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example viscosity value of the water is two times larger (g 0.001 Pa s) at temperature T = 20 °C compared to the viscosity value (g 0.0005 Pa s) at fluid temperature T = 60 °C. For this reason it is important to consider the variation of the physical properties of the fluids used in the numerical calculations. 2.2. Domain of discretization
Fig. 2. Computational domain p = 22.25 mm and h = 2.5 mm.
with
parameters
dp = 25 mm,
pc = 40 mm,
2.1.3. Heat transport equation The following form of the energy equation is solved to calculate the temperature field
@ @ @T ; ðqcp U i TÞ ¼ k @xi @xi @xi
ð3Þ
where k is the thermal conductivity function and cp is the specific heat function of the working fluid at constant pressure. The transport equations have been formulated in a conservative form according to Patankar [22]. Temperature dependency of different physical properties (k, cp, g, q) of the working fluids has been considered to improve the accuracy of the calculations. Following interpolation polynomials have been fitted to the available numerical data of the physical properties of the water and a water–ethylene glycol mixture used in the numerical calculations
qðTÞ ¼ 998:25 0:123261T 0:00131119T 2 0:0000121406T 3 ; gðTÞ ¼ 0:00166167 0:0000410857T þ 4:64802 107 T 2 1:90559 109 T 3 ; cp ðTÞ ¼ 4222:62 0:694932T þ 0:00624126T 2 þ 8:29448 106 T 3 ; kðTÞ ¼ 0:568733 þ 0:00196461T 9:77855106 T 2 þ 1:2432 108 T 3 ;
qðTÞ ¼ 1082:9 0:460917T 0:00190268T 2 2:23388 106 T 3 ; gðTÞ ¼ 0:00873992 0:000269039T þ 3:29295 106 T 2 1:40149 108 T 3 ; cp ðTÞ ¼ 3180:17 þ 7:18891T 0:0324592T 2 þ 0:0000922688T 3 ; kðTÞ ¼ 0:3756 þ 0:000824786T 2:72145 106 T 2 þ 1:02953 108 T 3 : ð4Þ Eq. (4) are third order polynomials and describe the temperature dependency of the fluid properties of the water and a water–ethylene glycol mixture with 50–50% volumetric ratio. Graphical presentation of these polynomials could show that the additional ethylene glycol makes the mixture more sensitive to the temperature variation. Physical properties of the water are less sensitive for the temperature variation than the pure ethylene glycol or a 50–50% mixture of water- ethylene glycol. It is also important to note that the most sensitive fluid property is the dynamic viscosity. For
Many different geometrical arrangements of the helical tube heat exchanger with helical corrugation have been studied on different grids. A schematic diagram of the studied coil with helical corrugation indicating the computational domain is shown in Fig. 2. The applied grid can be seen in Fig. 3 is strongly non-uniform near the wall of the tube to resolve the wall boundary effects. A careful check for the grid-independence of the numerical solution has been made to ensure the accuracy and validity of the numerical scheme. Gridindependence study has been carried out on several geometrical arrangements. The first study is concentrated on a conventional helical pipe with non-constant fluid properties. The reason to study the grid-independence of the numerical results of the common smooth helical pipes is to ensure the accuracy of the reference calculations used for comparison purposes with the corrugated cases. The area averaged outlet temperature, pressure drop and the Nusselt number have been used to test the independency of the calculation results from the applied grids. The calculations have been carried out on six different grids with the following cell numbers {146,000; 298,099; 766,865; 1,953,825; 2,562,182; 4,488,476} in case of common (without helical corrugation) helical pipes. Relative error are with the formulas of the control quantities calculated T T DP DP Nuc Nuf 100 outTc out f ; 100 Dc P f , and 100 Nu The indexes f and c mean . out f f f the most finest and any other coarser grid resolutions respectively. Geometrical sizes, initial and boundary conditions of the grid-independence test of smooth helical tubes are formulated as follows dp = 20 mm, dc = 370 mm, pc = 40 mm, Tini = 20 °C, Tin = 20 °C, Twall = 80 °C, minlet = 0.05 kg/s, {U, V, W}ini = 0 m/s. Calculated results show that the difference between the outlet temperature on the finest (VI., Tout = 65.77 °C) and on the second finest (V., Tout = 65.78 °C) grids is approximately 0.005%. Larger difference can be found on the same grids between the relative errors of pressure drop, (in grid VI., Dp=15.94 Pa and in grid V., Dp=16.01 Pa) but it is also less than 0.5%. For this reason grid V. has been chosen with 2,562,182 elements to carry out further numerical calculations in case of common helical tube heat exchanger with one turn. Grid-independence study has also been carried out in case of helically corrugated heat exchanger coil with dp = 20 mm, dc = 370 mm, pc = 40 mm, pitch p = 22.25 mm and corrugation depth h = 2 mm with one turn. Initial and boundary conditions of the corrugated helical tubes are formulated as follows Tini = 20 °C, Tin = 20 °C, Twall = 60 °C, minlet = 0.05 kg/s, {U, V, W}ini = 0 m/s. Four different grids have been tested and the results can be seen in Table 1. It has been found that the calculated pressure drop is more sensitive for coarsening the grid than the different heat transfer connected quantities (Tout, Tw, Nufd). It can be concluded that grid resolution larger than 7,200,000 cells in the studied geometrical cases for helical tube with one turn is approximately enough to produce physically realistic results independent from the applied numerical grid. The same basic parameters of grid generation have been applied to create numerical grids for tubes with two turns. The number of finite volumes (total number of cells) is approximately two times larger in case of numerical calculations carried out for tubes with two turns. Fig. 3a shows the generated grid near the outlet region of the corrugated pipe. Fig. 3b presents an enlarged view of the generated grid at the bottom zone of the outlet and the boundary layer grid has been shown. 17 layers have been generated to resolve the boundary region where the physical quantities (velocity and temperature fields) have large gradients normal to the tube wall.
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A. Zachár / International Journal of Heat and Mass Transfer 53 (2010) 3928–3939 Table 1 Relative error of the outlet temperature, pressure drop and the Nusselt number with the studied grid. Studied grid resolutions of one turn configuration [number of cells]
Average outlet temperature (°C) Relative error of outlet temperature (%) Pressure drop (Pa) Relative error of pressure drop (%) Inside Nusselt number (Nu) Relative error of Nu (%)
(I) 6,355,483
(II) 7,775,475
(III) 13,224,881
(IV) 20,143,807
48.41 0.3 645 6.2 100.9 1.7
48.38 0.24 640 5.4 99.8 0.6
48.28 0.04 613 0.98 99.4 0.2
48.26 – 607 – 99.2 –
Fig. 3. One of the applied grids for the calculations and an enlarged view at the bottom of the outlet side of the corrugated coil.
2.3. Initial and boundary conditions The initial velocity field is zero everywhere in the calculation domain. The initial state of the temperature field to be considered is constant everywhere inside the calculation domain. Constant mass flow is assumed at the inlet position of the heat exchanger coil. The gradient of the velocity profile and of the temperature field is assumed to be zero at the end of the outlet. Constant temperature boundary condition is specified for the wall of the heat exchanger coil because this simplification does not significantly modify the heat transfer augmentation effects. 2.4. Numerical solution of the transport equations The corresponding transport equations with the appropriate boundary conditions have been solved a commercially available CFD code (Ansys CFX 11.0). A ‘‘High Resolution Up-Wind like”
scheme is used to discretize the convection term in the transport equations. The resulting large linear set of equations is solved with an algebraic multi-grid solver. The curve-fitting is carried out in Mathematica 3.0 environment. The applied grids for the different geometries have been generated with the Workbench grid generator. Numerical calculations have been carried out on a network grid of 15 personal computers. Each PC contains 4 Gb RAM and a 2.8 GHz CPU with four independent core. 2.5. Calculation of the dimensionless quantities Representing calculated results the following dimensional and nondimensional quantities have been used. In case of Re number calculations the value of the density and dynamic viscosity (q, g) of the working fluids have been calculated by averaging for the entire fluid volume. The thermal conductivity of the fluid (kavg) has been calculated at a specific cross section. It is important to note
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Peripherally Averaged Nu
100
Measured (Dravid) from ref. 1972 [5] Pr ≈ 4 Calculated by Kalb and Seader 1972 [5], Pr=5 Present Study Pr ≈ 5
39.9 19.3
35.5
28.5
10
1 360
500
650
1030
De number Fig. 4. Comparison of the numerical results with measured and calculated results in case of smooth helical tubes.
that the modification of the thermal conductivity between the tube inlet and outlet is not more than 1–2% because of the variation of the fluid temperature. The wall heat flux and the wall temperature (qw, Tw) have been calculated between two nearby (5 mm) prespecified cross sections of the tube on a nearly cylindrical surface. The fluid bulk temperature has been calculated with mass flow is the fluid bulk velocity at the same cross secaveraging, where v tion of the tube
dp qv
dp qw Nu ¼ ; kav g ðT w T m Þ Z Z 1 qdAwall section ; qw ¼ Awall section Z ZA 1 Tw ¼ TdAwall section ; Awall section Z Z 1 Tm ¼ v TdAcross section : v Awall section A
Re ¼
g
current study have been indicated beside the plotted circles. Kalb and Seader [5] give detailed explanations about the small difference arising between the calculated results (present study , Kalb and Seader ) and the measured data ( ). Property variation of the water can modify the Prandtl number more than 30% between the inlet and outlet, and in the measurements the Nusselt numbers have been associated with the mean of the inlet and outlet value of the Prandtl number. The fluid properties in a fully developed state are closer to the state arising near the outlet.
;
3.2. Validation of the numerical calculations in case of helically corrugated straight tubes
ð5Þ
3. Model validation Two completely different problems have been tested based on measurements available in the literature. All of the necessary ingredients of the calculations (applied grid resolutions, temperature dependent physical properties of the applied fluids, boundary conditions, applicability of the laminar or turbulent modeling) are investigated to extend the validity of the numerical results to the corrugated helical tubes. 3.1. Validation of the numerical calculations with experimental results published in the literature in case of common helicallycoiled-tube heat exchangers This subsection contains a series of numerical calculations to investigate the validity of numerical computations for smooth helical tubes. Calculations have been carried out with the following geometrical and material properties, initial and boundary conditions, dp = 20 mm, dc = 370 mm, Tini = 20 °C, Tin = 20 °C, Twall = 60 °C, minlet = {0.05, 0.04, 0.025, 0.0125} kg/s, {U, V, W}ini = 0 m/s, and pure water has been chosen for working fluid with temperature dependent physical properties {cp, k, q, g}. Peripherally averaged fully developed Nusselt number has been calculated to compare the results of the numerical computations (present study) with measured and numerically calculated data available in literature. The following comparison presented in Fig. 4 is based on the graph (Fig. 6 on p. 811 in [5]) published by Kalb and Seader [5]. The results show excellent agreement with the previously measured and numerically calculated data. Because of easier evaluation of Fig. 4 Nusselt number values of the numerical calculation of the
The experimental tests and the presentation of the results can be found in the paper published by Vicente et al. [17]. The measurements have been simulated with a straight corrugated tube with 3000 mm length and dp = 18 mm diameter. The depth of the corrugation is h = 1.03 mm and the pitch is p = 15.95 mm. This tube has been named ‘‘tube 01” in Ref. [17]. It is important to note that in the real experiment the tube was lengthier (>5000 mm) but the measurement point has been located from the upstream electrode at x/d = 120 = 2160 mm, for this reason a shorter pipe is enough to model the flow process numerically because the downstream region is not measured and otherwise it is a fully developed flow domain. The geometry of the numerically modeled pipe consists of four different sections. The first section from the inlet position is a 100 mm length smooth pipe, what is followed by a 638 mm length corrugated tube section. After this a 2233 mm length heated section is followed and finally a short smooth section is located with 29 mm length at the outlet position. The wall of the first, second and fourth (last) sections has been considered thermally adiabatic, and only the third section is heated with a constant heat flux. A side view of the generated numerical grid can be seen in Fig. 5a and a cross sectional view of the same grid presents the boundary layer and the inner region of the generated grid in Fig. 5b. Fifteen layers have been created to resolve the large gradients of the flow variables near the wall inside the corrugated straight tube. A flow field inside the corrugated straight tube at a specific cross section in the fully developed region has been presented in Fig. 6. The induced velocity values of the secondary flow field presented in Fig. 6 are in the range 0 and 0.022 m/s. The average arrow length in Fig. 6 means approximately 0.01 m/s. Dependency of the fully developed heat transfer rate on the Rayleigh number has been investigated and compared to measured data (Fig. 5 on p. 659 in [17]) for a corrugated straight tube (‘‘tube 01”) in the transitional flow regime Re = 2470. The applied initial and boundary conditions are as follows: Tini = 30 °C, Tin = 30 °C, minlet = 0.027638 kg/s 100 l/h, qw = {9000, 10,500} W m2 in case
A. Zachár / International Journal of Heat and Mass Transfer 53 (2010) 3928–3939
Fig. 5a. An unstructured grid of a corrugated straight tube with pitch of corrugation p = 15.95 mm and corrugation depth h = 1.03 mm.
Fig. 5b. A cross sectional view of the same grid shown in Fig. 5a.
Fig. 6. Secondary flow field inside a corrugated straight tube with corrugation parameters (p = 15.95 mm, h = 1.03 mm).
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A. Zachár / International Journal of Heat and Mass Transfer 53 (2010) 3928–3939
30
Fully Dev Nusselt Nu Vicente et al. 2004 [17], Measured Present Study, Numerical calculations
Fully Developed Nu
25 20
26.60107139
23.91187685 19.3776081
19.00439169
15 10 5 0 35000000
42000000
50000000
100000000
Rayleigh Number Ra Fig. 7. Comparison of the numerical results of corrugated straight pipes with experimental data published by Vicente et al. [17].
of water, and qw = {7000, 14,100} W m2 with 50–50% water–ethylene glycol mixture, and the initial value of the velocity field was {U, V, W}ini=0 m/s. Comparison of the numerical results with the measured data can be seen in Fig. 7. It can be clearly seen that the numerical calculations with laminar flow modeling are in an excellent agreement with the measured data. Additionally, a series of numerical calculations have been carried out to investigate further the validity of the numerical computations with testing the dependency of the fully developed heat transfer rate from the Reynolds number in the laminar and transitional flow regime. The same tube geometry (named ‘‘tube 01” in Ref. [17]) has been used to carry out the numerical computations. Two different initial temperature states have been simulated numerically to take into account the Prandtl number dependency of the heat transfer rate. The initial and boundary conditions are as follows: Tini = 20 °C, Tin = 20 °C, minlet = {0.025, 0.032, 0.038} kg/s, and Tini = 50 °C, Tin = 50 °C, minlet = {0.015, 0.017, 0.02, 0.022} kg/s, qw = 10,000 W m2, {U, V, W}ini = 0 m/s. Comparison of the numerical results with the measured data can be seen in Fig. 8. It can be concluded that the numerical calculations with laminar flow modeling (without application of any turbulence model) are in a good agreement with the measured data in the Re number range [0, 2500] in case of Prandtl number value Pr = 6. It can also be seen that above the Reynolds number value 2500 the laminar flow modeling can not produce the results of the experimental tests because of the increasing intensity of the turbulent flow. In the lower Prandtl number case Pr = 3 this limit can be found approximately 10% smaller nearly the value Re = 2250. According
to the result presented by Vicente et al. [17] the flow field inside the corrugated straight pipe is in the transitional flow regime, because the critical Reynolds number is approximately Recrit = 1300 in the case of the studied tube geometry. From this it can be concluded that laminar flow modeling with an adequately refined mesh can produce results are in a good agreement with the physical reality in the transitional flow state. 4. Results Several geometrical parameters of the studied heat exchanger configuration have been investigated numerically. Length of the coil has been specified according to the assumption that the velocity and temperature field is fully developed near the end of the first turn. For this reason 2 turns configuration of the corrugated helical coil has been investigated because it should be enough to test the development of the peripherally averaged Nusselt number. Inner diameter of the helical tubes is dp = {15, 20, 25} mm, helical pitch of the coils is pc = 40 mm, and the helical diameter is dc = 340 mm in the studied cases. The investigated corrugation parameters are the depth and pitch of corrugation. The following values of corrugation pitch are investigated p = {22.25, 44.5, 89} mm to test the impact of this parameter for the heat transfer rate. The depth of corrugation has been varied according to the pipe diameter, in case of dp = 15 mm the values of the corrugation depth are h = {0.75, 1.125, 1.5} mm, in case of dp = 20 mm the depths are h = {1, 1.5, 2} mm and in case of dp = 25 mm the corrugation depths are h = {1.25, 1.875, 2.5} mm. Two different working fluids have been
Fully Dev Nusselt Nu Vicente et al. 2004 [17], Measured Pr = 6 Present Study, Numerical Calculations, Pr ≈ 6 Fully Dev Nusselt Nu Vicente et al. 2004 [17], Measured Pr = 3 Present Study, Numerical Calculations, Pr ≈ 3
60
Fully Developed Nu
50
40
30
20
10
0 1500
1700
1900
2100
2300
2500
2700
2900
3100
Reynolds Number Re Fig. 8. Comparison of the numerical results of a corrugated straight pipe (p = 15.95 mm, h = 1.03 mm) with experimental data published by Vicente et al. [17].
A. Zachár / International Journal of Heat and Mass Transfer 53 (2010) 3928–3939
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Fig. 9. Temperature isotherms at different location of a corrugated coiled heat exchanger configuration for De = 1120.
used in the numerical calculations the first one is common water with temperature dependent flow and thermal properties and the second one is a mixture of the so called TyfocorR ethylene glycol based solar fluid and water with a volumetric ratio 50–50%. Flow and thermal properties of the TyfocorR fluid can be found in a technical reference paper [23]. Temperature of the inlet fluid is 20 °C all of the studied cases and the wall temperature is 60 °C in most of the numerical calculations. In some cases higher and lower surface temperatures are examined but it was found that the different surface temperatures does not significantly modify the heat transfer rate of the studied heat exchangers. The presented results are valid in laminar and transitional flow regime in the Dean number range 30 < De < 1400 and Prandtl number range 3 < Pr < 30. Fig. 9 presents temperature isotherms at different axial location of a corrugated helically coiled heat exchanger. The applied helical pitch is p = 22.25 mm, depth of the corrugation d = 2 mm, dp = 20 mm and the flow rate Q = 0.05 kg/s. A comparison of the isotherms presented in Fig. 9 with the isotherms of a smooth tube coiled heat exchanger indicates the substantial difference between the two temperature fields. It can be concluded that the temperature field of the corrugated case is far more homogeneous than the smooth tube case. This is primarily due to the fact that the additional swirling motion of fluid particles in the secondary flow field induces significantly larger heat transfer rate and mixing in the flow case of the corrugated tubes. Inner and outer sides of a smooth helical coil can easily be identified on a contour plot of a temperature field because the coldest/hottest region ‘‘eyes” is always located near the outer side of the tube cross section with large gradient of the temperature field between the edge of the cold/hot region and the tube wall. This kind of horizontally oriented stratification of the temperature field can be significantly reduced by the presented helical corrugation on the outer side of the tube.
Fig. 10 shows a comparison of the secondary flow field at axial location u = 270° between a helically corrugated coil and a smooth tube coil. The presented flow field has been created the following geometrical and flow parameters where the helical pitch and depth of the corrugation are p = 22.25 mm, h = 2 mm, and the flow rate is Q = 0.05 kg/s. In Fig. 10 the left side indicates the outer wall, and the right side refers to the inner wall of the tube. Because of comparison purposes the same mass flow rate has been specified for the smooth tube coil in the numerical simulation. It is clearly seen that in case of corrugation the secondary flow field contains much larger velocities than the secondary flow of the smooth tube. Approximately the largest velocity is 6 times larger in the corrugated case than the smooth tube case. The induced additional motion in the cross section of the corrugated helical pipe is the main source of the heat transfer augmentation depending on the helical pitch and the depth of the corrugation. The usual secondary flow field of smooth helical pipes has been substantially modified because of the additionally induced swirling motion. A typical secondary flow profile of a smooth helical pipe consists of two oppositely rotating eddies in laminar flow regime. It can be observed in Fig. 10 that the number of eddies has been increased significantly compared to the smooth helical coil case. This process further increase the cross sectional mixing of the temperature field what is important to improve the over all heat transfer rate. Development of the peripherally averaged Nusselt number at different axial locations is presented in Fig. 11. The main geometrical parameters of the heat transfer results depicted in Fig. 11 are as follows dp = 25 mm, dc = 340 mm, p = 22.25 mm and h = 2.5 mm. Fig. 11 contains results of numerical calculations with various inflow rates where the working fluid is water and the Prandtl number approximately equals to Pr 5. The peripherally averaged Nusselt number shows stronger oscillatory behaviour than the smooth helical pipe case. The stronger oscillation of Nusselt
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A. Zachár / International Journal of Heat and Mass Transfer 53 (2010) 3928–3939
Fig. 10. Comparison the tangential (secondary) flow field at u = 270° location of a corrugated and a smooth tube coiled heat exchanger configuration with the same initial and boundary conditions for De = 1120.
number is a consequence of the highly intensified secondary flow field of the corrugated helical pipe. Similar oscillatory behaviour of the Nusselt number has also been found by Lin and Ebadian [3] in their numerical studies of turbulent convective heat transfer in smooth helical pipes. This oscillatory behaviour is also observed in Fig. 12 for the development of the circumferential average Nusselt number. The heat transport simulations depicted in Fig. 12 have been carried out with the following geometrical parameters dp = 20 mm, dc = 340 mm, p = 22.25 mm and h = 2 mm. In this case the working fluid is a mixture of water and ethylene glycol. The flow rates and the corresponding Dean numbers can be seen in Fig. 12. It can be observed that the oscillation of the Nusselt number is significantly smaller than the higher Dean number cases. After the first half of the first turn (u = 180°) the average magnitude of the oscillation has been decreased in higher and lower Dean number cases also. It has also been observed that the oscillation behaviour completely
vanish in case of low Dean number De < 180–200 flows. Above De > 200 a smaller magnitude of oscillation than the oscillations found in the early stage constantly remains. In the range of low Dean number cases the peripherally averaged Nusselt number tends to the usual fully developed heat transfer case similar to flow cases developed inside smooth helical tubes. A correlation between the Nusselt number and the Dean and Prandtl numbers and the basic geometrical parameters of corrugation is proposed to describe the dependency of the heat transfer rate from the flow and geometrical properties
Nu ¼ 0:5855De0:6688 Pr0:408
0:166 0:192 h p : d d
ð6Þ
The formula was obtained via curve-fitting of heat transfer results for the corrugated helical coils. The presented formula is applicable in the following Dean and Prandtl number ranges 30 < De < 1400,
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A. Zachár / International Journal of Heat and Mass Transfer 53 (2010) 3928–3939 0.05kg/s, De=1177 0.025kg/s, De=617
Working Fluid: Water
Peripherally Averaged Nusselt Number
120
0.04kg/s, De=958 0.0125kg/s, De=325
100
80
60
40
20
0 0
100
200
300
400
500
600
700
φ [deg] Fig. 11. Development of the peripherally averaged Nusselt number along the axial direction in case of geometrical parameters dp = 25 mm, p = 22.25 mm, h = 2.5 mm and Pr 5.
Working Fluid: water–ethylene glycol mixture with 50 %–50 %
Peripherally Averaged Nusselt Number
80
0.05 kg/s, De=317
0.04 kg/s,
0.025 kg/s, De=170
0.0125 kg/s, De=92
De=259
70 60 50 40 30 20 10 0 0
100
200
300
400
500
600
700
φ [deg] Fig. 12. Development of the peripherally averaged Nusselt number along the axial direction in case of geometrical parameters dp = 20 mm, p = 22.25 mm, h = 2 mm and Pr 15.
Fig. 13. Nusselt number versus Dean number in case of pure water.
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Fig. 14. Nusselt number versus Dean number in case of water–ethylene glycol mixture.
3 < Pr < 30. A careful checks of the exponents of (h/d) and (p/d) ratios in formula (6) can prove that the heat transfer rate is slightly more sensitive to the modification of the pitch than the modification of the corrugation depth. Fig. 13 shows the curves of Nusselt numbers depending on the Dean number in case of water with different (h/d) ratios where the Prandtl number approximately equals to Pr 5. The heat transport simulations depicted with different symbols in Fig. 13 have been carried out with the following corrugation parameters p = 22.25 mm, h = {2, 1} mm and tube diameter dp = 20 mm. The different symbols represent the numerically calculated configurations and the continuous curves show Eq. (6) at a specific Prandtl number and (h/d) ratios. Because of comparison purposes a curve representing the fully developed heat transfer rate of the smooth helical coil has been depicted in Figs. 13 and 14 also. Fig. 14 presents a comparison of the heat transfer rate at a higher Prandtl number value between the corrugated and smooth tube cases. Following corrugation parameters p = 22.25 mm, h = 2.5 mm and tube diameter dp = 25 mm have been used to get the heat transfer results indicated by the symbol ( ) in Fig. 14. It can be observed from Figs. 13 and 14 that the heat transfer is approximately 80– 100% larger compared to the heat transfer values of smooth helical tubes.
been found that the (p/d) ratio has slightly stronger impact on the heat transfer enhancement than the (h/d) ratio. The results also show that a spirally corrugated helical tube with corrugation parameters (p/d = 1, h/d = 0.1) can increase the heat transfer rate nearly 100% larger than a smooth helical pipe in the Dean number range 30 < De < 1400. From computer simulation point of view it is important to consider the temperature dependency of the working fluid mostly in case of the water–ethylene glycol mixture because the changing of the physical properties of this fluid is nearly ten times larger than the pure water mostly in case of specific heat capacity and the dynamic viscosity. Development of the peripherally averaged Nusselt number is found to be more oscillatory than the oscillatory behaviour observed in case of smooth helical coils. When the flow rates (Dean number) increases the oscillation phenomenon is enhanced. It can be deduced that the oscillation phenomenon is less intense with increasing Prandtl number. The reason behind this process is that the increasing viscosity of the working fluid significantly reduces the intensity of the secondary flow.
References 5. Conclusions Different geometrical parameters of helical corrugation on the outer surface of helically coiled-tube heat exchangers are examined to improve the inside heat transfer rate. Several different inflow rates and temperatures have been studied to test the impact of flow parameters for the efficiency of the heat exchanger. It can be concluded that the heat transfer rate is almost independent from the inlet temperature and the outer surface temperature for this reason the further studies have been conducted with Tin = 20 °C inlet and Tsurf = 60 °C surface temperature. As it was expected the volumetric flow rate significantly influences the performance of a coiled-tube heat exchangers with and without outer helical corrugation. An empirical formula has been suggested to indicate the dependency of the Nusselt number from the Dean and Prandtl numbers. The suggested formula (6) indicating the effect of the basic corrugation parameters for the heat transfer rate. The presented results show that the ratio of the helical pitch and tube diameter (p/d) and the ratio of the corrugation depth and the tube diameter (h/d) nearly increase or decrease in the same way the heat transfer rate of the studied heat exchangers. It has
[1] W.R. Dean, Notes on the motion of fluid in a curved pipe, Philos. Mag. 4 (1927) 208–223. [2] B. Zheng, C.X. Lin, M.A. Ebadian, Combined laminar forced convection and thermal radiation in helical pipe, Int. J. Heat Mass Transfer 43 (2000) 1067– 1078. [3] C.X. Lin, M.A. Ebadian, Developing turbulent convective heat transfer in helical pipes, Int. J. Heat Mass Transfer 40 (1997) 3861–3873. [4] J.J.M. Sillekens, C.C.M. Rindt, A.A. Van Steenhoven, Developing mixed convection heat in coiled heat exchanger, Int. J. Heat Mass Transfer 41 (1998) 61–72. [5] C.E. Kalb, J.D. Seader, Heat and mass transfer phenomena for viscous flow in curved circular tubes, Int. J. Heat Mass Transfer 15 (1972) 801–817. [6] G. Yang, Z.F. Dong, M.A. Ebadian, Laminar forced convection in a helicoidal pipe with finite pitch, Int. J. Heat Mass Transfer 38 (5) (1995) 853–862. [7] C. Jonas Bolinder, Bengt Sundén, Numerical prediction of laminar flow and forced convective heat transfer in a helical square duct with a finite pitch, Int. J. Heat Mass Transfer 39 (1996) 3101–3115. [8] C.E. Kalb, J.D. Seader, Entrance region heat transfer in a uniform walltemperature helical coil with transition from turbulent to laminar flow, Int. J. Heat Mass Transfer 26 (1983) 23–32. [9] B.K. Rao, Turbulent heat transfer to viscoelastic fluids in helical passages, Exp. Heat Transfer 6 (1993) 189–203. [10] Vimal Kumar, K.D.P. Nigam, Numerical simulation of steady flow fields in coiled flow inverter, Int. J. Heat Mass Transfer 48 (2005) 4811–4828. [11] Narasimha Acharya, Mihir Sen, Hsueh-Chia Chang, Analysis of heat transfer enhancement in coiled-tube heat exchangers, Int. J. Heat Mass Transfer 44 (2001) 3189–3199.
A. Zachár / International Journal of Heat and Mass Transfer 53 (2010) 3928–3939 [12] P. Sivashanmugam, S. Suresh, Experimental studies on heat transfer and friction factor characteristics of laminar flow through a circular tube fitted with helical screw-tape inserts, Appl. Therm. Eng. 26 (2006) 1990–1997. [13] P.G. Vicente, A. García, A. Viedma, Experimental investigation on heat transfer and frictional characteristics of spirally corrugated tubes in turbulent flow at different Prandtl numbers, Int. J. Heat Mass Transfer 47 (2004) 671–681. [14] A. García, P.G. Vicente, A. Viedma, Experimental study of heat transfer enhancement with wire coil inserts in laminar-transition-turbulent regimes at different Prandtl numbers, Int. J. Heat Mass Transfer 48 (2005) 4640–4651. [15] V. Zimparov, Prediction of friction factors and heat transfer coefficients for turbulent flow in corrugated tubes combined with twisted tape inserts. Part 1: friction factors, Int. J. Heat Mass Transfer 47 (2004) 589–599. [16] P.G. Vicente, A. García, A. Viedma, Heat transfer and pressure drop for low Reynolds turbulent flow in helically dimpled tubes, Int. J. Heat Mass Transfer 45 (2002) 543–553. [17] P.G. Vicente, A. García, A. Viedma, Mixed convection heat transfer and isothermal pressure drop in corrugated tubes for laminar and transition flow, Int. Commun. Heat Mass Transfer 31 (2004) 651–662.
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[18] S. Jaisankar, T.K. Radhakrishnan, K.N. Seeba, Experimental studies on heat transfer and friction factor characteristics of thermosyphon solar water heater system fitted with spacer at the trailing edge of twisted tapes, Appl. Therm. Eng. (2008) 11–12. [19] L. Li, W. Cui, Q. Liao, X. Mingdao, T. Jen, Q. Chen, Heat transfer augmentation in 3D internally finned and microfinned helical tube, Int. J. Heat Mass Transfer 48 (2005) 1916–1925. [20] Cengiz Yildiz, Yasar Bicer, Dursun Pehlivan, Heat transfer and pressure drop in a heat exchanger with a helical pipe containing inside springs, Energy Convers. Manage. 38 (1997) 619–624. [21] P.S. Srinivasan, S. Nandapurkar, F.A. Holland, Friction factor for coils, Trans. Inst. Chem. Eng. 48 (1970) T156–T161. [22] S.V. Patankar, Numerical Heat Transfer and Fluid Flow, Taylor & Francis, New York, 1980. 1980. [23] TyfocorR Technical information, TYFOROP CHEMIE GmbH, Hamburg, Edition June 2005,.
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Analysis of coiled-tube heat exchangers to improve heat transfer rate with spirally corrugated wall A. Zachár * } H-2103, Hungary Department of Informatics, Szent István University, Páter K. u. 1., Gödöllo
a r t i c l e
i n f o
Article history: Available online 1 June 2010 Keywords: Spirally corrugated helical pipe Heat transfer augmentation Secondary flow
a b s t r a c t Steady heat transfer enhancement has been studied in helically coiled-tube heat exchangers. The outer side of the wall of the heat exchanger contains a helical corrugation which makes a helical rib on the inner side of the tube wall to induce additional swirling motion of fluid particles. Numerical calculations have been carried out to examine different geometrical parameters and the impact of flow and thermal boundary conditions for the heat transfer rate in laminar and transitional flow regimes. Calculated results have been compared to existing empirical formulas and experimental tests to investigate the validity of the numerical results in case of common helical tube heat exchanger and additionally results of the numerical computation of corrugated straight tubes for laminar and transition flow have been validated with experimental tests available in the literature. Comparison of the flow and temperature fields in case of common helical tube and the coil with spirally corrugated wall configuration are discussed. Heat exchanger coils with helically corrugated wall configuration show 80–100% increase for the inner side heat transfer rate due to the additionally developed swirling motion while the relative pressure drop is 10–600% larger compared to the common helically coiled heat exchangers. New empirical correlation has been proposed for the fully developed inner side heat transfer prediction in case of helically corrugated wall configuration. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction Helically coiled-tube heat exchangers are one of the most common equipment found in many industrial applications ranging from solar energy applications, nuclear power production, chemical and food industries, environmental engineering, and many other engineering applications. Heat transfer rate of helically coiled heat exchangers is significantly larger because of the secondary flow pattern in planes normal to the main flow than in straight pipes. Modification of flow is due to the centrifugal forces (Dean roll cells [1]) caused by the curvature of the tube. Several studies have been conducted to analyse the heat transfer rate of coiled heat exchangers in laminar and turbulent flow regimes [2– 6]. Numerical study of laminar flow and forced convective heat transfer in a helical square duct has been carried out by Jonas Bolinder and Sundén [7]. Many authors investigated experimentally the turbulent heat transfer in helical pipes [8,9]. Further enhancement of heat transfer rate in coiled pipes has great importance in several industrial applications mainly where the flow regime is in the laminar or transitional zone like hot water solar energy applications. There are basically two different concepts to increase * Tel.: +36 28 522051; fax: +36 28 410804. E-mail address: [email protected] 0017-9310/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijheatmasstransfer.2010.05.011
the rate of heat transferred, the first one is the active and the other one is the passive method. Many different active techniques exist to increase the heat transfer rate mostly for straight pipes. In case of passive techniques heat transfer enhancement by chaotic mixing in helical pipes has great importance and investigated by Kumar and Nigam [10] and Acharya et al. [11]. Helical screw-tape inserts have been investigated in straight pipes experimentally by Sivashanmugam and Suresh [12]. There is considerable amount of work reported in the literature on heat transfer augmentation in straight pipes with different corrugation techniques, helical tape inserts [13–17]. Experimental investigation of thermosyphon solar water heater with twisted tape inserts has been carried out by Jaisankar et al . [18]. According to the author’s knowledge a few examinations are considered in helically coiled tubes with different passive heat transfer augmentation techniques like inside wall corrugation, helical tape inserts [19] and this question is not studied numerically at all in the available literature. Experimental investigations have been conducted in a helical pipe containing inside springs to study the heat transfer rate and pressure drop by Yildiz et al. [20]. Helical corrugation on the surface of a helically coiled tube possibly increases the heat transfer rate because of the developed swirling motion. Basic aim of this study is to investigate the impact of different geometrical parameters of the corrugation for the inner side heat transfer rate in case of helical tube
A. Zachár / International Journal of Heat and Mass Transfer 53 (2010) 3928–3939
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Nomenclature cp(x, y, z, T) specific heat function of the working fluid (J kg1 K1) diameter of the coil (m) dc dp diameter of the pipe (m) De Dean number (=Re(dp/dc)0.5) h corrugation depth (m) hi inside heat transfer coefficient (W m2 K1) kavg area averaged thermal conductivity at a specific cross section (W m1 K1) Nufd peripherally averaged fully developed Nusselt number (=qwdp/kavg(Tw Tm)) p helical pitch of corrugation (m) pc helical pitch of the coil (m) P(x, y, z) pressure field (Pa) Pr Prandtl number (=cpg/k) DP pressure drop (Pa) qw area averaged wall heat flux at a specific cross section (W m2) Q flow rate (kg/s) Re Reynolds number (=qVavgdp/g) x, y, z Cartesian coordinates (m)
T(x, y, z) temperature function of the water inside the heat exchanger coil (°C) Tw area averaged wall temperature at a specific cross section (°C) Tm mass-flow averaged mean temperature of the fluid at a specific cross section (°C) Tin, Tout area averaged inlet/outlet temperature (°C) U1(x, y, z), U2(x, y, z), U3(x, y, z) velocity component functions into the three possible directions (m/s) Vavg area averaged velocity (m/s) Greek symbols u angular coordinate normal to the tube cross section (°) g(x, y, z, T) dynamic viscosity function of the working fluid (N s m2) k(x, y, z, T) thermal conductivity function of the working fluid (W m1 K1) q(x, y, z, T) density function of the working fluid (kg m3)
decrease the critical Reynolds number significantly as it was proven by Vicente et al. [17] for corrugated straight pipes. In this study most of the investigated flow cases are in laminar flow state because the Reynolds number of the studied flow cases covers a range from Re = 100 to Re = 7000 and perhaps for the largest Reynolds number flow cases (approximately Re = 7000) a small level of turbulence can be occurred. The momentum equation of the fluid is based on the three dimensional Navier–Stokes equations. A SIMPLE like method is applied to solve the momentum and continuity equations. The dependent variables that describe the present flow situation are the temperature, T, velocity components Ui in the x, y and z directions respectively and the pressure field P. 2.1. Conservation equations Fig. 1. Schematic figure of the corrugated coiled-tube heat exchangers with the following geometrical parameters dp = 20 mm, pc = 40 mm, p = 44.5 mm and h = 2 mm.
heat exchangers. Fig. 1 shows the basic configuration of the studied heat exchanger coils with helical corrugation. Helically corrugated coils with two turns have been investigated to examine the fully developed flow state of the corrugated tubes. The corrugation is located on the outer side of the helical pipe which makes a helically spiraling wall on the inner side of the pipe. 2. Mathematical formulation This section provides the basic equations that must be solved to describe the velocity field and the temperature distribution inside the heat exchanger coils. It is well known that, the transition from laminar to turbulent flow in curved pipes occur much higher critical Reynolds number (Recrit) than in straight pipes. The critical Reynolds number for smooth helical pipes can be estimated by the following pffiffiffi formula found in Srinivasan et al. [21] Recrit ¼ 2100ð1 þ 12 dÞ. Applying this formula the values of the critical Reynolds number d in the studied geometrical cases are as follows: d ¼ dpc ¼ 0:015 0:02 0:025 ! Recrit ffi 7393; d ¼ 0:34 ! Recrit ffi 8210; d ¼ 0:34 ! Recrit ffi 8933. 0:34 The physical problem is a steady, three dimensional laminar or transitional flow configuration. Corrugation and surface roughness can
The following set of partial differential equations for U1, U2, U3, P and T as functions of x, y, z describes the flow and temperature field inside a helically coiled heat exchanger. The conservation equations are formulated in the Cartesian coordinate system because the applied flow solver (Ansys CFX 11.0) uses the Cartesian system to formulate the conservation equations for all (vector Ui and scalar T, P) quantities. Description of the entire geometry of the studied problem is incorporated into the generated unstructured numerical grid. 2.1.1. Continuity equation The continuity equation is formulated in the following manner in Cartesian coordinate system
@ ðqU i Þ ¼ 0: @xi
ð1Þ
2.1.2. Momentum equations The following equation system is the representation of the momentum equations in Cartesian coordinate system where i, j e {1, 2, 3},
@ @P @ @U j : ðqU j U i Þ ¼ þ g @xi @xj @xi @xi
ð2Þ
g is the dynamic viscosity and q is the density of the working fluid.
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example viscosity value of the water is two times larger (g 0.001 Pa s) at temperature T = 20 °C compared to the viscosity value (g 0.0005 Pa s) at fluid temperature T = 60 °C. For this reason it is important to consider the variation of the physical properties of the fluids used in the numerical calculations. 2.2. Domain of discretization
Fig. 2. Computational domain p = 22.25 mm and h = 2.5 mm.
with
parameters
dp = 25 mm,
pc = 40 mm,
2.1.3. Heat transport equation The following form of the energy equation is solved to calculate the temperature field
@ @ @T ; ðqcp U i TÞ ¼ k @xi @xi @xi
ð3Þ
where k is the thermal conductivity function and cp is the specific heat function of the working fluid at constant pressure. The transport equations have been formulated in a conservative form according to Patankar [22]. Temperature dependency of different physical properties (k, cp, g, q) of the working fluids has been considered to improve the accuracy of the calculations. Following interpolation polynomials have been fitted to the available numerical data of the physical properties of the water and a water–ethylene glycol mixture used in the numerical calculations
qðTÞ ¼ 998:25 0:123261T 0:00131119T 2 0:0000121406T 3 ; gðTÞ ¼ 0:00166167 0:0000410857T þ 4:64802 107 T 2 1:90559 109 T 3 ; cp ðTÞ ¼ 4222:62 0:694932T þ 0:00624126T 2 þ 8:29448 106 T 3 ; kðTÞ ¼ 0:568733 þ 0:00196461T 9:77855106 T 2 þ 1:2432 108 T 3 ;
qðTÞ ¼ 1082:9 0:460917T 0:00190268T 2 2:23388 106 T 3 ; gðTÞ ¼ 0:00873992 0:000269039T þ 3:29295 106 T 2 1:40149 108 T 3 ; cp ðTÞ ¼ 3180:17 þ 7:18891T 0:0324592T 2 þ 0:0000922688T 3 ; kðTÞ ¼ 0:3756 þ 0:000824786T 2:72145 106 T 2 þ 1:02953 108 T 3 : ð4Þ Eq. (4) are third order polynomials and describe the temperature dependency of the fluid properties of the water and a water–ethylene glycol mixture with 50–50% volumetric ratio. Graphical presentation of these polynomials could show that the additional ethylene glycol makes the mixture more sensitive to the temperature variation. Physical properties of the water are less sensitive for the temperature variation than the pure ethylene glycol or a 50–50% mixture of water- ethylene glycol. It is also important to note that the most sensitive fluid property is the dynamic viscosity. For
Many different geometrical arrangements of the helical tube heat exchanger with helical corrugation have been studied on different grids. A schematic diagram of the studied coil with helical corrugation indicating the computational domain is shown in Fig. 2. The applied grid can be seen in Fig. 3 is strongly non-uniform near the wall of the tube to resolve the wall boundary effects. A careful check for the grid-independence of the numerical solution has been made to ensure the accuracy and validity of the numerical scheme. Gridindependence study has been carried out on several geometrical arrangements. The first study is concentrated on a conventional helical pipe with non-constant fluid properties. The reason to study the grid-independence of the numerical results of the common smooth helical pipes is to ensure the accuracy of the reference calculations used for comparison purposes with the corrugated cases. The area averaged outlet temperature, pressure drop and the Nusselt number have been used to test the independency of the calculation results from the applied grids. The calculations have been carried out on six different grids with the following cell numbers {146,000; 298,099; 766,865; 1,953,825; 2,562,182; 4,488,476} in case of common (without helical corrugation) helical pipes. Relative error are with the formulas of the control quantities calculated T T DP DP Nuc Nuf 100 outTc out f ; 100 Dc P f , and 100 Nu The indexes f and c mean . out f f f the most finest and any other coarser grid resolutions respectively. Geometrical sizes, initial and boundary conditions of the grid-independence test of smooth helical tubes are formulated as follows dp = 20 mm, dc = 370 mm, pc = 40 mm, Tini = 20 °C, Tin = 20 °C, Twall = 80 °C, minlet = 0.05 kg/s, {U, V, W}ini = 0 m/s. Calculated results show that the difference between the outlet temperature on the finest (VI., Tout = 65.77 °C) and on the second finest (V., Tout = 65.78 °C) grids is approximately 0.005%. Larger difference can be found on the same grids between the relative errors of pressure drop, (in grid VI., Dp=15.94 Pa and in grid V., Dp=16.01 Pa) but it is also less than 0.5%. For this reason grid V. has been chosen with 2,562,182 elements to carry out further numerical calculations in case of common helical tube heat exchanger with one turn. Grid-independence study has also been carried out in case of helically corrugated heat exchanger coil with dp = 20 mm, dc = 370 mm, pc = 40 mm, pitch p = 22.25 mm and corrugation depth h = 2 mm with one turn. Initial and boundary conditions of the corrugated helical tubes are formulated as follows Tini = 20 °C, Tin = 20 °C, Twall = 60 °C, minlet = 0.05 kg/s, {U, V, W}ini = 0 m/s. Four different grids have been tested and the results can be seen in Table 1. It has been found that the calculated pressure drop is more sensitive for coarsening the grid than the different heat transfer connected quantities (Tout, Tw, Nufd). It can be concluded that grid resolution larger than 7,200,000 cells in the studied geometrical cases for helical tube with one turn is approximately enough to produce physically realistic results independent from the applied numerical grid. The same basic parameters of grid generation have been applied to create numerical grids for tubes with two turns. The number of finite volumes (total number of cells) is approximately two times larger in case of numerical calculations carried out for tubes with two turns. Fig. 3a shows the generated grid near the outlet region of the corrugated pipe. Fig. 3b presents an enlarged view of the generated grid at the bottom zone of the outlet and the boundary layer grid has been shown. 17 layers have been generated to resolve the boundary region where the physical quantities (velocity and temperature fields) have large gradients normal to the tube wall.
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A. Zachár / International Journal of Heat and Mass Transfer 53 (2010) 3928–3939 Table 1 Relative error of the outlet temperature, pressure drop and the Nusselt number with the studied grid. Studied grid resolutions of one turn configuration [number of cells]
Average outlet temperature (°C) Relative error of outlet temperature (%) Pressure drop (Pa) Relative error of pressure drop (%) Inside Nusselt number (Nu) Relative error of Nu (%)
(I) 6,355,483
(II) 7,775,475
(III) 13,224,881
(IV) 20,143,807
48.41 0.3 645 6.2 100.9 1.7
48.38 0.24 640 5.4 99.8 0.6
48.28 0.04 613 0.98 99.4 0.2
48.26 – 607 – 99.2 –
Fig. 3. One of the applied grids for the calculations and an enlarged view at the bottom of the outlet side of the corrugated coil.
2.3. Initial and boundary conditions The initial velocity field is zero everywhere in the calculation domain. The initial state of the temperature field to be considered is constant everywhere inside the calculation domain. Constant mass flow is assumed at the inlet position of the heat exchanger coil. The gradient of the velocity profile and of the temperature field is assumed to be zero at the end of the outlet. Constant temperature boundary condition is specified for the wall of the heat exchanger coil because this simplification does not significantly modify the heat transfer augmentation effects. 2.4. Numerical solution of the transport equations The corresponding transport equations with the appropriate boundary conditions have been solved a commercially available CFD code (Ansys CFX 11.0). A ‘‘High Resolution Up-Wind like”
scheme is used to discretize the convection term in the transport equations. The resulting large linear set of equations is solved with an algebraic multi-grid solver. The curve-fitting is carried out in Mathematica 3.0 environment. The applied grids for the different geometries have been generated with the Workbench grid generator. Numerical calculations have been carried out on a network grid of 15 personal computers. Each PC contains 4 Gb RAM and a 2.8 GHz CPU with four independent core. 2.5. Calculation of the dimensionless quantities Representing calculated results the following dimensional and nondimensional quantities have been used. In case of Re number calculations the value of the density and dynamic viscosity (q, g) of the working fluids have been calculated by averaging for the entire fluid volume. The thermal conductivity of the fluid (kavg) has been calculated at a specific cross section. It is important to note
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Peripherally Averaged Nu
100
Measured (Dravid) from ref. 1972 [5] Pr ≈ 4 Calculated by Kalb and Seader 1972 [5], Pr=5 Present Study Pr ≈ 5
39.9 19.3
35.5
28.5
10
1 360
500
650
1030
De number Fig. 4. Comparison of the numerical results with measured and calculated results in case of smooth helical tubes.
that the modification of the thermal conductivity between the tube inlet and outlet is not more than 1–2% because of the variation of the fluid temperature. The wall heat flux and the wall temperature (qw, Tw) have been calculated between two nearby (5 mm) prespecified cross sections of the tube on a nearly cylindrical surface. The fluid bulk temperature has been calculated with mass flow is the fluid bulk velocity at the same cross secaveraging, where v tion of the tube
dp qv
dp qw Nu ¼ ; kav g ðT w T m Þ Z Z 1 qdAwall section ; qw ¼ Awall section Z ZA 1 Tw ¼ TdAwall section ; Awall section Z Z 1 Tm ¼ v TdAcross section : v Awall section A
Re ¼
g
current study have been indicated beside the plotted circles. Kalb and Seader [5] give detailed explanations about the small difference arising between the calculated results (present study , Kalb and Seader ) and the measured data ( ). Property variation of the water can modify the Prandtl number more than 30% between the inlet and outlet, and in the measurements the Nusselt numbers have been associated with the mean of the inlet and outlet value of the Prandtl number. The fluid properties in a fully developed state are closer to the state arising near the outlet.
;
3.2. Validation of the numerical calculations in case of helically corrugated straight tubes
ð5Þ
3. Model validation Two completely different problems have been tested based on measurements available in the literature. All of the necessary ingredients of the calculations (applied grid resolutions, temperature dependent physical properties of the applied fluids, boundary conditions, applicability of the laminar or turbulent modeling) are investigated to extend the validity of the numerical results to the corrugated helical tubes. 3.1. Validation of the numerical calculations with experimental results published in the literature in case of common helicallycoiled-tube heat exchangers This subsection contains a series of numerical calculations to investigate the validity of numerical computations for smooth helical tubes. Calculations have been carried out with the following geometrical and material properties, initial and boundary conditions, dp = 20 mm, dc = 370 mm, Tini = 20 °C, Tin = 20 °C, Twall = 60 °C, minlet = {0.05, 0.04, 0.025, 0.0125} kg/s, {U, V, W}ini = 0 m/s, and pure water has been chosen for working fluid with temperature dependent physical properties {cp, k, q, g}. Peripherally averaged fully developed Nusselt number has been calculated to compare the results of the numerical computations (present study) with measured and numerically calculated data available in literature. The following comparison presented in Fig. 4 is based on the graph (Fig. 6 on p. 811 in [5]) published by Kalb and Seader [5]. The results show excellent agreement with the previously measured and numerically calculated data. Because of easier evaluation of Fig. 4 Nusselt number values of the numerical calculation of the
The experimental tests and the presentation of the results can be found in the paper published by Vicente et al. [17]. The measurements have been simulated with a straight corrugated tube with 3000 mm length and dp = 18 mm diameter. The depth of the corrugation is h = 1.03 mm and the pitch is p = 15.95 mm. This tube has been named ‘‘tube 01” in Ref. [17]. It is important to note that in the real experiment the tube was lengthier (>5000 mm) but the measurement point has been located from the upstream electrode at x/d = 120 = 2160 mm, for this reason a shorter pipe is enough to model the flow process numerically because the downstream region is not measured and otherwise it is a fully developed flow domain. The geometry of the numerically modeled pipe consists of four different sections. The first section from the inlet position is a 100 mm length smooth pipe, what is followed by a 638 mm length corrugated tube section. After this a 2233 mm length heated section is followed and finally a short smooth section is located with 29 mm length at the outlet position. The wall of the first, second and fourth (last) sections has been considered thermally adiabatic, and only the third section is heated with a constant heat flux. A side view of the generated numerical grid can be seen in Fig. 5a and a cross sectional view of the same grid presents the boundary layer and the inner region of the generated grid in Fig. 5b. Fifteen layers have been created to resolve the large gradients of the flow variables near the wall inside the corrugated straight tube. A flow field inside the corrugated straight tube at a specific cross section in the fully developed region has been presented in Fig. 6. The induced velocity values of the secondary flow field presented in Fig. 6 are in the range 0 and 0.022 m/s. The average arrow length in Fig. 6 means approximately 0.01 m/s. Dependency of the fully developed heat transfer rate on the Rayleigh number has been investigated and compared to measured data (Fig. 5 on p. 659 in [17]) for a corrugated straight tube (‘‘tube 01”) in the transitional flow regime Re = 2470. The applied initial and boundary conditions are as follows: Tini = 30 °C, Tin = 30 °C, minlet = 0.027638 kg/s 100 l/h, qw = {9000, 10,500} W m2 in case
A. Zachár / International Journal of Heat and Mass Transfer 53 (2010) 3928–3939
Fig. 5a. An unstructured grid of a corrugated straight tube with pitch of corrugation p = 15.95 mm and corrugation depth h = 1.03 mm.
Fig. 5b. A cross sectional view of the same grid shown in Fig. 5a.
Fig. 6. Secondary flow field inside a corrugated straight tube with corrugation parameters (p = 15.95 mm, h = 1.03 mm).
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30
Fully Dev Nusselt Nu Vicente et al. 2004 [17], Measured Present Study, Numerical calculations
Fully Developed Nu
25 20
26.60107139
23.91187685 19.3776081
19.00439169
15 10 5 0 35000000
42000000
50000000
100000000
Rayleigh Number Ra Fig. 7. Comparison of the numerical results of corrugated straight pipes with experimental data published by Vicente et al. [17].
of water, and qw = {7000, 14,100} W m2 with 50–50% water–ethylene glycol mixture, and the initial value of the velocity field was {U, V, W}ini=0 m/s. Comparison of the numerical results with the measured data can be seen in Fig. 7. It can be clearly seen that the numerical calculations with laminar flow modeling are in an excellent agreement with the measured data. Additionally, a series of numerical calculations have been carried out to investigate further the validity of the numerical computations with testing the dependency of the fully developed heat transfer rate from the Reynolds number in the laminar and transitional flow regime. The same tube geometry (named ‘‘tube 01” in Ref. [17]) has been used to carry out the numerical computations. Two different initial temperature states have been simulated numerically to take into account the Prandtl number dependency of the heat transfer rate. The initial and boundary conditions are as follows: Tini = 20 °C, Tin = 20 °C, minlet = {0.025, 0.032, 0.038} kg/s, and Tini = 50 °C, Tin = 50 °C, minlet = {0.015, 0.017, 0.02, 0.022} kg/s, qw = 10,000 W m2, {U, V, W}ini = 0 m/s. Comparison of the numerical results with the measured data can be seen in Fig. 8. It can be concluded that the numerical calculations with laminar flow modeling (without application of any turbulence model) are in a good agreement with the measured data in the Re number range [0, 2500] in case of Prandtl number value Pr = 6. It can also be seen that above the Reynolds number value 2500 the laminar flow modeling can not produce the results of the experimental tests because of the increasing intensity of the turbulent flow. In the lower Prandtl number case Pr = 3 this limit can be found approximately 10% smaller nearly the value Re = 2250. According
to the result presented by Vicente et al. [17] the flow field inside the corrugated straight pipe is in the transitional flow regime, because the critical Reynolds number is approximately Recrit = 1300 in the case of the studied tube geometry. From this it can be concluded that laminar flow modeling with an adequately refined mesh can produce results are in a good agreement with the physical reality in the transitional flow state. 4. Results Several geometrical parameters of the studied heat exchanger configuration have been investigated numerically. Length of the coil has been specified according to the assumption that the velocity and temperature field is fully developed near the end of the first turn. For this reason 2 turns configuration of the corrugated helical coil has been investigated because it should be enough to test the development of the peripherally averaged Nusselt number. Inner diameter of the helical tubes is dp = {15, 20, 25} mm, helical pitch of the coils is pc = 40 mm, and the helical diameter is dc = 340 mm in the studied cases. The investigated corrugation parameters are the depth and pitch of corrugation. The following values of corrugation pitch are investigated p = {22.25, 44.5, 89} mm to test the impact of this parameter for the heat transfer rate. The depth of corrugation has been varied according to the pipe diameter, in case of dp = 15 mm the values of the corrugation depth are h = {0.75, 1.125, 1.5} mm, in case of dp = 20 mm the depths are h = {1, 1.5, 2} mm and in case of dp = 25 mm the corrugation depths are h = {1.25, 1.875, 2.5} mm. Two different working fluids have been
Fully Dev Nusselt Nu Vicente et al. 2004 [17], Measured Pr = 6 Present Study, Numerical Calculations, Pr ≈ 6 Fully Dev Nusselt Nu Vicente et al. 2004 [17], Measured Pr = 3 Present Study, Numerical Calculations, Pr ≈ 3
60
Fully Developed Nu
50
40
30
20
10
0 1500
1700
1900
2100
2300
2500
2700
2900
3100
Reynolds Number Re Fig. 8. Comparison of the numerical results of a corrugated straight pipe (p = 15.95 mm, h = 1.03 mm) with experimental data published by Vicente et al. [17].
A. Zachár / International Journal of Heat and Mass Transfer 53 (2010) 3928–3939
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Fig. 9. Temperature isotherms at different location of a corrugated coiled heat exchanger configuration for De = 1120.
used in the numerical calculations the first one is common water with temperature dependent flow and thermal properties and the second one is a mixture of the so called TyfocorR ethylene glycol based solar fluid and water with a volumetric ratio 50–50%. Flow and thermal properties of the TyfocorR fluid can be found in a technical reference paper [23]. Temperature of the inlet fluid is 20 °C all of the studied cases and the wall temperature is 60 °C in most of the numerical calculations. In some cases higher and lower surface temperatures are examined but it was found that the different surface temperatures does not significantly modify the heat transfer rate of the studied heat exchangers. The presented results are valid in laminar and transitional flow regime in the Dean number range 30 < De < 1400 and Prandtl number range 3 < Pr < 30. Fig. 9 presents temperature isotherms at different axial location of a corrugated helically coiled heat exchanger. The applied helical pitch is p = 22.25 mm, depth of the corrugation d = 2 mm, dp = 20 mm and the flow rate Q = 0.05 kg/s. A comparison of the isotherms presented in Fig. 9 with the isotherms of a smooth tube coiled heat exchanger indicates the substantial difference between the two temperature fields. It can be concluded that the temperature field of the corrugated case is far more homogeneous than the smooth tube case. This is primarily due to the fact that the additional swirling motion of fluid particles in the secondary flow field induces significantly larger heat transfer rate and mixing in the flow case of the corrugated tubes. Inner and outer sides of a smooth helical coil can easily be identified on a contour plot of a temperature field because the coldest/hottest region ‘‘eyes” is always located near the outer side of the tube cross section with large gradient of the temperature field between the edge of the cold/hot region and the tube wall. This kind of horizontally oriented stratification of the temperature field can be significantly reduced by the presented helical corrugation on the outer side of the tube.
Fig. 10 shows a comparison of the secondary flow field at axial location u = 270° between a helically corrugated coil and a smooth tube coil. The presented flow field has been created the following geometrical and flow parameters where the helical pitch and depth of the corrugation are p = 22.25 mm, h = 2 mm, and the flow rate is Q = 0.05 kg/s. In Fig. 10 the left side indicates the outer wall, and the right side refers to the inner wall of the tube. Because of comparison purposes the same mass flow rate has been specified for the smooth tube coil in the numerical simulation. It is clearly seen that in case of corrugation the secondary flow field contains much larger velocities than the secondary flow of the smooth tube. Approximately the largest velocity is 6 times larger in the corrugated case than the smooth tube case. The induced additional motion in the cross section of the corrugated helical pipe is the main source of the heat transfer augmentation depending on the helical pitch and the depth of the corrugation. The usual secondary flow field of smooth helical pipes has been substantially modified because of the additionally induced swirling motion. A typical secondary flow profile of a smooth helical pipe consists of two oppositely rotating eddies in laminar flow regime. It can be observed in Fig. 10 that the number of eddies has been increased significantly compared to the smooth helical coil case. This process further increase the cross sectional mixing of the temperature field what is important to improve the over all heat transfer rate. Development of the peripherally averaged Nusselt number at different axial locations is presented in Fig. 11. The main geometrical parameters of the heat transfer results depicted in Fig. 11 are as follows dp = 25 mm, dc = 340 mm, p = 22.25 mm and h = 2.5 mm. Fig. 11 contains results of numerical calculations with various inflow rates where the working fluid is water and the Prandtl number approximately equals to Pr 5. The peripherally averaged Nusselt number shows stronger oscillatory behaviour than the smooth helical pipe case. The stronger oscillation of Nusselt
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Fig. 10. Comparison the tangential (secondary) flow field at u = 270° location of a corrugated and a smooth tube coiled heat exchanger configuration with the same initial and boundary conditions for De = 1120.
number is a consequence of the highly intensified secondary flow field of the corrugated helical pipe. Similar oscillatory behaviour of the Nusselt number has also been found by Lin and Ebadian [3] in their numerical studies of turbulent convective heat transfer in smooth helical pipes. This oscillatory behaviour is also observed in Fig. 12 for the development of the circumferential average Nusselt number. The heat transport simulations depicted in Fig. 12 have been carried out with the following geometrical parameters dp = 20 mm, dc = 340 mm, p = 22.25 mm and h = 2 mm. In this case the working fluid is a mixture of water and ethylene glycol. The flow rates and the corresponding Dean numbers can be seen in Fig. 12. It can be observed that the oscillation of the Nusselt number is significantly smaller than the higher Dean number cases. After the first half of the first turn (u = 180°) the average magnitude of the oscillation has been decreased in higher and lower Dean number cases also. It has also been observed that the oscillation behaviour completely
vanish in case of low Dean number De < 180–200 flows. Above De > 200 a smaller magnitude of oscillation than the oscillations found in the early stage constantly remains. In the range of low Dean number cases the peripherally averaged Nusselt number tends to the usual fully developed heat transfer case similar to flow cases developed inside smooth helical tubes. A correlation between the Nusselt number and the Dean and Prandtl numbers and the basic geometrical parameters of corrugation is proposed to describe the dependency of the heat transfer rate from the flow and geometrical properties
Nu ¼ 0:5855De0:6688 Pr0:408
0:166 0:192 h p : d d
ð6Þ
The formula was obtained via curve-fitting of heat transfer results for the corrugated helical coils. The presented formula is applicable in the following Dean and Prandtl number ranges 30 < De < 1400,
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Working Fluid: Water
Peripherally Averaged Nusselt Number
120
0.04kg/s, De=958 0.0125kg/s, De=325
100
80
60
40
20
0 0
100
200
300
400
500
600
700
φ [deg] Fig. 11. Development of the peripherally averaged Nusselt number along the axial direction in case of geometrical parameters dp = 25 mm, p = 22.25 mm, h = 2.5 mm and Pr 5.
Working Fluid: water–ethylene glycol mixture with 50 %–50 %
Peripherally Averaged Nusselt Number
80
0.05 kg/s, De=317
0.04 kg/s,
0.025 kg/s, De=170
0.0125 kg/s, De=92
De=259
70 60 50 40 30 20 10 0 0
100
200
300
400
500
600
700
φ [deg] Fig. 12. Development of the peripherally averaged Nusselt number along the axial direction in case of geometrical parameters dp = 20 mm, p = 22.25 mm, h = 2 mm and Pr 15.
Fig. 13. Nusselt number versus Dean number in case of pure water.
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Fig. 14. Nusselt number versus Dean number in case of water–ethylene glycol mixture.
3 < Pr < 30. A careful checks of the exponents of (h/d) and (p/d) ratios in formula (6) can prove that the heat transfer rate is slightly more sensitive to the modification of the pitch than the modification of the corrugation depth. Fig. 13 shows the curves of Nusselt numbers depending on the Dean number in case of water with different (h/d) ratios where the Prandtl number approximately equals to Pr 5. The heat transport simulations depicted with different symbols in Fig. 13 have been carried out with the following corrugation parameters p = 22.25 mm, h = {2, 1} mm and tube diameter dp = 20 mm. The different symbols represent the numerically calculated configurations and the continuous curves show Eq. (6) at a specific Prandtl number and (h/d) ratios. Because of comparison purposes a curve representing the fully developed heat transfer rate of the smooth helical coil has been depicted in Figs. 13 and 14 also. Fig. 14 presents a comparison of the heat transfer rate at a higher Prandtl number value between the corrugated and smooth tube cases. Following corrugation parameters p = 22.25 mm, h = 2.5 mm and tube diameter dp = 25 mm have been used to get the heat transfer results indicated by the symbol ( ) in Fig. 14. It can be observed from Figs. 13 and 14 that the heat transfer is approximately 80– 100% larger compared to the heat transfer values of smooth helical tubes.
been found that the (p/d) ratio has slightly stronger impact on the heat transfer enhancement than the (h/d) ratio. The results also show that a spirally corrugated helical tube with corrugation parameters (p/d = 1, h/d = 0.1) can increase the heat transfer rate nearly 100% larger than a smooth helical pipe in the Dean number range 30 < De < 1400. From computer simulation point of view it is important to consider the temperature dependency of the working fluid mostly in case of the water–ethylene glycol mixture because the changing of the physical properties of this fluid is nearly ten times larger than the pure water mostly in case of specific heat capacity and the dynamic viscosity. Development of the peripherally averaged Nusselt number is found to be more oscillatory than the oscillatory behaviour observed in case of smooth helical coils. When the flow rates (Dean number) increases the oscillation phenomenon is enhanced. It can be deduced that the oscillation phenomenon is less intense with increasing Prandtl number. The reason behind this process is that the increasing viscosity of the working fluid significantly reduces the intensity of the secondary flow.
References 5. Conclusions Different geometrical parameters of helical corrugation on the outer surface of helically coiled-tube heat exchangers are examined to improve the inside heat transfer rate. Several different inflow rates and temperatures have been studied to test the impact of flow parameters for the efficiency of the heat exchanger. It can be concluded that the heat transfer rate is almost independent from the inlet temperature and the outer surface temperature for this reason the further studies have been conducted with Tin = 20 °C inlet and Tsurf = 60 °C surface temperature. As it was expected the volumetric flow rate significantly influences the performance of a coiled-tube heat exchangers with and without outer helical corrugation. An empirical formula has been suggested to indicate the dependency of the Nusselt number from the Dean and Prandtl numbers. The suggested formula (6) indicating the effect of the basic corrugation parameters for the heat transfer rate. The presented results show that the ratio of the helical pitch and tube diameter (p/d) and the ratio of the corrugation depth and the tube diameter (h/d) nearly increase or decrease in the same way the heat transfer rate of the studied heat exchangers. It has
[1] W.R. Dean, Notes on the motion of fluid in a curved pipe, Philos. Mag. 4 (1927) 208–223. [2] B. Zheng, C.X. Lin, M.A. Ebadian, Combined laminar forced convection and thermal radiation in helical pipe, Int. J. Heat Mass Transfer 43 (2000) 1067– 1078. [3] C.X. Lin, M.A. Ebadian, Developing turbulent convective heat transfer in helical pipes, Int. J. Heat Mass Transfer 40 (1997) 3861–3873. [4] J.J.M. Sillekens, C.C.M. Rindt, A.A. Van Steenhoven, Developing mixed convection heat in coiled heat exchanger, Int. J. Heat Mass Transfer 41 (1998) 61–72. [5] C.E. Kalb, J.D. Seader, Heat and mass transfer phenomena for viscous flow in curved circular tubes, Int. J. Heat Mass Transfer 15 (1972) 801–817. [6] G. Yang, Z.F. Dong, M.A. Ebadian, Laminar forced convection in a helicoidal pipe with finite pitch, Int. J. Heat Mass Transfer 38 (5) (1995) 853–862. [7] C. Jonas Bolinder, Bengt Sundén, Numerical prediction of laminar flow and forced convective heat transfer in a helical square duct with a finite pitch, Int. J. Heat Mass Transfer 39 (1996) 3101–3115. [8] C.E. Kalb, J.D. Seader, Entrance region heat transfer in a uniform walltemperature helical coil with transition from turbulent to laminar flow, Int. J. Heat Mass Transfer 26 (1983) 23–32. [9] B.K. Rao, Turbulent heat transfer to viscoelastic fluids in helical passages, Exp. Heat Transfer 6 (1993) 189–203. [10] Vimal Kumar, K.D.P. Nigam, Numerical simulation of steady flow fields in coiled flow inverter, Int. J. Heat Mass Transfer 48 (2005) 4811–4828. [11] Narasimha Acharya, Mihir Sen, Hsueh-Chia Chang, Analysis of heat transfer enhancement in coiled-tube heat exchangers, Int. J. Heat Mass Transfer 44 (2001) 3189–3199.
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[18] S. Jaisankar, T.K. Radhakrishnan, K.N. Seeba, Experimental studies on heat transfer and friction factor characteristics of thermosyphon solar water heater system fitted with spacer at the trailing edge of twisted tapes, Appl. Therm. Eng. (2008) 11–12. [19] L. Li, W. Cui, Q. Liao, X. Mingdao, T. Jen, Q. Chen, Heat transfer augmentation in 3D internally finned and microfinned helical tube, Int. J. Heat Mass Transfer 48 (2005) 1916–1925. [20] Cengiz Yildiz, Yasar Bicer, Dursun Pehlivan, Heat transfer and pressure drop in a heat exchanger with a helical pipe containing inside springs, Energy Convers. Manage. 38 (1997) 619–624. [21] P.S. Srinivasan, S. Nandapurkar, F.A. Holland, Friction factor for coils, Trans. Inst. Chem. Eng. 48 (1970) T156–T161. [22] S.V. Patankar, Numerical Heat Transfer and Fluid Flow, Taylor & Francis, New York, 1980. 1980. [23] TyfocorR Technical information, TYFOROP CHEMIE GmbH, Hamburg, Edition June 2005,