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Numerical studies of a tube-in-tube helically coiled heat exchanger
Available online at www.sciencedirect.com
Chemical Engineering and Processing 47 (2008) 2287–2295
Numerical studies of a tube-in-tube helically coiled heat exchanger Vimal Kumar, Burhanuddin Faizee, Monisha Mridha, K.D.P. Nigam ∗ Department of Chemical Engineering, Indian Institute of Technology, Delhi, Hauz Khas, New Delhi-110016, India Received 6 August 2007; received in revised form 29 December 2007; accepted 2 January 2008 Available online 11 January 2008
Abstract In the present study a tube-in-tube helically coiled (TTHC) heat exchanger has been numerically modeled for fluid flow and heat transfer characteristics for different fluid flow rates in the inner as well as outer tube. The three-dimensional governing equations for mass, momentum and heat transfer have been solved using a control volume finite difference method (CVFDM). The renormalization group (RNG) k–ε model is used to model the turbulent flow and heat transfer in the TTHC heat exchanger. The fluid considered in the inner tube is compressed air at higher pressure and cooling water in the outer tube at ambient conditions. The inner tube pressure is varied from 10 to 30 bars. The Reynolds numbers for the inner tube ranged from 20,000 to 70,000. The mass flow rate in the outer tube is varied from 200 to 600 kg/h. The outer tube is fitted with semicircular plates to support the inner tube and also to provide high turbulence in the annulus region. The overall heat transfer coefficients are calculated for both parallel and counter flow configurations. The Nusselt number and friction factor values in the inner and outer tubes are compared with the experimental data reported in the literature. New empirical correlations are developed for hydrodynamic and heat-transfer predictions in the outer tube of the TTHC. © 2008 Elsevier B.V. All rights reserved. Keywords: Tube-in-tube helical heat exchanger; Heat transfer; Helical tube; RNG k–
1. Introduction Helical coil heat exchangers are one of the most common equipment found in many industrial applications ranging from chemical and food industries, power production, electronics, environmental engineering, manufacturing industry, air-conditioning, waste heat recovery, cryogenic processes, and space applications. Helical coils are extensively used as heat exchangers and reactors due to higher heat and mass transfer coefficients, narrow residence time distributions and compact structure. The modification of the flow in the helically coiled tubes is due to the centrifugal forces (Dean roll cells, [4,5]). The curvature of the tube produces a secondary flow field with a circulatory motion, which causes the fluid particles to move toward the core region of the tube. The secondary flow enhances heat transfer rates as it reduces the temperature gradient across the cross-section of the tube. Thus there is an additional convective Abbreviations: CVFDM, control volume finite difference method; HVAC, heating, ventilating and air conditioning; LMTD, log-mean temperature difference; TTHC, tube-in-tube helical coil. ∗ Corresponding author. Tel.: +91 11 26591020; fax: +91 11 26591020. E-mail address: [email protected] (K.D.P. Nigam). 0255-2701/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.cep.2008.01.001
heat transfer mechanism, perpendicular to the main flow, which does not exist in conventional heat exchangers. An extensive review of fluid flow and heat transfer in helical pipes has been presented in the literature [1–3]. There is considerable amount of work reported in the literature on heat transfer in coiled tubes; however, very less attention has been paid to study the outer heat transfer coefficient. Figueiredo and Raimunda [6], Haraburda [7], Prasad et al. [8] and Patil et al. [9] have discussed the design procedure for coil-in-shell heat exchangers considering helical coiled tubes as a bank of straight tubes for calculating outer heat transfer coefficients. In the coil-in-shell heat exchangers poor circulation is observed in shell regions near the coil which could be avoided by using a tube-in-tube helical coil (TTHC) configuration. A tube-in-tube helical coil heat exchanger requires the knowledge of the heat transfer rates for the two flowing fluid, i.e., the flow in the helical tube as well as in the helical annulus. Karahalios [10] and Petrakis and Karahalios [11–13] reported the fluid flow and heat transfer in a curved pipe with a solid core. They showed that the size of the core affects the flow in the annulus with flows approaching parabolic for large cores [13]. Karahalios [10] studied the heat transfer in a curved annulus with a constant temperature gradient on both inner and outer walls of the annulus as
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the thermal boundary conditions. All the above reported studies for helical coils were confined with one of two major boundary conditions, constant wall heat flux or constant wall temperature [2,14]. However in industrial applications of heat exchangers where one is interested in fluid-to-fluid heat exchanger the use of constant wall temperature or constant wall flux conditions does not appear to be physically realistic. This complicates the design of tube-in-tube helical coil heat exchangers, where either the heating or cooling is supplied by a secondary fluid, with the two fluids separated by the wall of the coil. Garimella et al. [15] reported average heat transfer coefficients of laminar and transition flows for forced convection heat transfer in coiled annular ducts. Two different coil diameters and two annular radius ratios were used in the experiment. Hot and cold waters were used as working fluids. They reported that the heat transfer coefficients obtained from the coiled annular ducts were higher than those obtained from a straight annulus, especially in the laminar region. Xin et al. [16] studied the effects of coil geometries and the flow rates of air and water on pressure drop in both annular vertical and horizontal helicoidal pipes. Experiments were performed for the superficial water Reynolds number from 210 to 23,000 and superficial air Reynolds number from 30 to 30,000. Their results showed that the transition from laminar to turbulent flow covers a wide Reynolds number range. Rennie and Raghavan [17] experimentally reported the heat transfer in a coil-in-coil heat exchanger comprised of one loop. This configuration results in secondary flows in both the inner tube and in the annulus, as both sections are curved and subject to centrifugal forces. The flows of both the fluids were co-current and counter-current. They also reported that increasing the tube or annulus Dean numbers resulted in an increase in the overall heat transfer coefficient. The heat transfer characteristics of a double-pipe helical heat exchanger for both parallel and counter flow using water in the inner as well as outer tubes was numerically studied by Rennie and Raghavan [18]. Overall heat transfer coefficients were calculated for inner Dean numbers in the range of 38–350 for the boundary conditions of constant wall temperature and constant heat flux. The results showed an increasing overall heat transfer coefficients as the inner Dean number increased; however, flow conditions in the annulus had a stronger influence on the overall heat transfer coefficient. Furthermore, increasing the size of the inner tube resulted in lower thermal resistances in the annulus, though the thermal resistance in the inner tube remained fairly constant. From the literature it can be seen that no study has been reported which considers the fluid at high pressures and temperature, though in most of the industrial applications the process streams are available at higher pressure and temperature (e.g., chemical reactions; heating, ventilating and air conditioning (HVAC) systems; and heat exchangers). Therefore in the present work the performance of a tube-in-tube helical heat exchanger for a compressed air–water counter-current and co-current flow system is studied numerically over a wide range of Reynolds number considering turbulent flow regime. The hydrodynamics and heat transfer of compressed air flowing in the inner tube of tube-in-tube helical coil heat exchanger with high pressure (i.e., 10–30 bars) are being reported the first time, which has not been
Fig. 1. Tube-in-tube helical heat exchanger.
considered in the previous literature. Fluid-to-fluid heat transfer has been studied using physically realistic boundary conditions. The effect of the fluid flow on the heat transfer and hydrodynamics have been studied in the tube as well as in the annulus. In the present work baffles have been introduced in the annulus area of the tube-in-tube helical heat exchanger. All the computations have been carried out on a SUN Fire V880 computer in the Chemical Reaction Engineering Laboratory at Indian Institute of Technology, Delhi. 2. Mathematical modeling of TTHC The geometry considered and the systems of the coordinates are illustrated in Fig. 1 (where, di,inner is the diameter of inner coiled tube; Rc is the coil radius; H is the distance between the two turn; and do,inner is the inner diameter of the outer tube). The TTHC geometry was developed with inner and outer diameters of 0.023 and 0.0508 m, respectively, with coil diameter of 0.762 m and with a pitch of 0.020 m. The TTHC was comprised of four full turns. As the tube side fluid was under highly turbulence flow regime therefore it was assumed that the flow will get fully developed within four turns. The TTHC details are given in Table 1 for both inner as well as for outer tubes, respectively. The coils were orientated in the vertical position. Inlets and outlets were located at each end of the coil. The boundary conditions associated with the inlets specified the inlet velocities in the axial direction. Coil properties were set to those of stainless steel, with a thermal conductivity of 16 W m−1 K−1 , density of 7882 kg.m−3 and a specific heat of 502 J kg−1 K−1 . The outer coil was set to be adiabatic (representing an insulated tube) and the inner coil was set to allow conductive heat flow through the tube.
V. Kumar et al. / Chemical Engineering and Processing 47 (2008) 2287–2295 Table 1 Geometrical and flow parameters for the inner and outer tube of TTHC
Outer diameter (m) Inner diameter (m) Coil diameter (m) Pitch (m) Number of turns Flow rate (kg/h) Pressure (bars) Prandtl number
Inner tube
Outer tube
0.0254 0.023 0.762 0.100 4 40–85 10, 20 and 30 1
0.0508 0.0484 0.762 0.100 4 200–600 1 0.74, 7, 33 and 150
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Ideal gas law: P = ρRT
(1)
Continuity equation: ∂ui =0 ∂xi
(2)
Momentum equation: ∂ui 2 ∂p ∂uk ∂(ρui uj ) ∂ ∂uj μeff − μeff − = + ∂xj ∂xj ∂xj ∂xi 3 ∂xk ∂xi (3)
The pressure drop and heat transfer in the TTHC is studied for five different Reynolds number (30,000 < NRe < 70,000) in the inner tube. The outer tube mass flow rates is varied from 200 to 600 kg/h. The Reynolds number in the outer tube is different for pressure drop and heat transfer calculations as it depends upon the equivalent diameter of the outer tube. The detailed description of the equivalent diameter calculation is reported in Kumar et al. [19]. The simulations were carried out at three different levels of pressure values, i.e., form 10–30 bar. The annulus side fluid properties were calculated at average temperature i.e., mean of inlet and outlet temperature in the annulus. Compressed air density in the inner tube was calculated using ideal gas law. The total number of simulations performed were 150 (5 inner tube flow rates, 5 outer tube flow rates, 3 operating pressure values and 2 flow arrangement systems). The output of the simulations included the inlet and outlet velocities, mass flow rates and enthalpy rates, as well as velocity, pressure, and temperature fields at various specified cross-sections. 2.1. Governing equations The geometry, system of coordinates and details of TTHC considered for the present work are discussed in our previous work [19]. In the TTHC the cold and hot fluids enter from their corresponding inlets and heat transfer take place between the two fluids due to conduction and forced convection. Heat transfer by radiation can be neglected because temperature in the TTHC studied is quite low (max. 353 K). The Cartesian coordinate system (x, y, z) was used to represent flow in numerical simulation. The flow was considered to be steady, and incompressible. At the inlet (φ = 0◦ ), fluid with turbulence intensity I and temperature T0 enters the TTHC heat exchanger at a velocity of u0 . Turbulent flow and heat transfer develop simultaneously downstream in the tubes. For the turbulent flow and heat transfer modeling the RNG k– model proposed by Yakhot and Orszag [20] was used in the TTHC because the RNG model included an additional term in its equation that significantly improve the accuracy for rapidly strained flows, such as those in curved pipes. The effect of swirl on turbulence is included in the RNG model, enhancing accuracy for swirling flows. The three dimensional governing equations of turbulent flow and the heat transfer in the TTHC can be written in tensor form in the master Cartesian system as follows:
Energy equation:
dui ∂ui ∂ ∂uj ∂T ∂(ρui cp T ) αT μeff + μeff = + ∂xi ∂xi ∂xi dxj ∂xj ∂xi 2 ∂uk − μeff (4) δij 3 ∂xk Turbulent kinetic energy equation: ∂k ∂ ∂(ρui k) αk μeff + μt S 2 + Gb − ρε = ∂xi ∂xi ∂xi
(5)
Dissipation rate of turbulent kinetic energy equation: ∂ε ε ∂ ∂(ρui ε) αε μeff + C1ε μt S 2 = ∂xi ∂xi ∂xi k − C2ε ρ
ε2 − Ra k
The effective viscosity, μeff can be defined as 2 Cμ k √ μeff = μmol 1 + μmol ε
(6)
(7)
where μmol is the molecular viscosity. The coefficients αT , αk and αε in Eqs. (4)–(6) are the inverse effective Prandtl numbers for T, k, and ε, respectively. The values of inverse effective Prandtl numbers, αT, αk and αε are computed using the following formula derived analytically by the RNG theory: α − 1.3929 0.6321 α + 2.3929 0.3679 μmol = (8) α − 1.3929 α + 2.3929 μeff 0 0 where α0 is equal to 1/NPr , 1.0, and 1.0, for the computation of αT , αk and αε , respectively. When a non-zero gravity field and temperature gradient are present simultaneously, the k–ε model account for the generation of k (kinetic energy) due to buoyancy (Gb in Eq. (5)) and the corresponding contribution to the production of ε (energy dissipation) in Eq. (6). The effects of buoyancy are also included despite the fact that the effect of buoyancy is not so significant at very high Reynolds number. The generation of turbulence due to buoyancy is given by Gb = βgi
μt ∂T NPr,t ∂xi
(9)
where NPr,t is the turbulent Prandtl number for energy, gi is the component of the gravitational vector in the ith direction. In the
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case of the RNG k– model, NPr,t = 1/αT , and β, the coefficient of thermal expansion, is defined as β = −1/ρ(∂p/∂T)T . In Eq. (6), R is given by R=
Cμ ρη3 (1 − η/η0 ) ε2 1 + ζη3 k
(10)
where η = S k/ε, η0 ≈ 4.38, ζ = 0.012. The model constants C , C1 , and C2 are equal to 0.085, 1.42 and 1.68, respectively. The term S in Eq. (5) and Eq. (6) is
the modulus of the mean rate-of-strain tensor, defined as S = 2Sij Sij , where 1 ∂ui ∂uj (11) + Sij = 2 ∂xj ∂xi
0.07(di,inner /2) in the present study. The factor of 0.07 is based on the maximum value of the mixing length in fully developed turbulent pipe flow. The computation domain (4 turns) used in this study can ensure that the outflow condition (i.e., fully developed flow and thermal assumed) can be satisfied at the exit plane of the curved pipe. Therefore, at the outlet, the diffusion term for all dependent variables were set to zero in the exit direction: ∂φ =0 ∂n
(16)
where φ represents the variables ui , T, k and ε. 3. Numerical computation
The two-layer based, non-equilibrium wall function was used for the near-wall treatment of flow in the given geometry. The non-equilibrium wall functions are recommended for complex flows because of the capability to partly account for the effects of pressure gradients and departure from equilibrium. The numerical results for turbulent flow tend to be more susceptible to grid dependency than those for laminar flow due to the strong interaction of the mean flow and turbulence. The distance from the wall at the wall-adjacent cells must be determined by considering the range over which the log-law is valid. The size of wall adjacent cells can be estimated from yp ( yp+ ν/uτ ), where u ≡(τ w /ρ)0.5 = u (cf /2)0.5 . In the present study, the yp+ was taken in the range of 30–60. The wall heat flux was computed using qw = ρCp u T/T+ , where T+ is obtained from ⎧ u ⎪ ⎨ NPr , y+ ≤ yv+ u∗ T+ = (12) u ⎪ ⎩ NPr,w − NPr , y+ > yv+ u∗ NPr =
1/2 NPr π/4 NPr,w 1/4 AT −1 sin (π/4) κ NPr,w NPr
(13)
where NPr is the molecular Prandtl number, NPr,w , the turbulent wall Prandtl number (NPr,w = 1.2) and AT , the Van Driest constant (AT = 26). At the inlet, uniform profiles for all the dependent variables were employed: us = u0 ,
T = T0 ,
k = k0 ,
ε = ε0 .
(14)
For the inner and outer walls of the heat exchanger, heat conduction in solid wall was considered. Ambient temperature was considered across the outer wall of the TTHC heat exchanger, i.e., T = 300 K. Turbulent kinetic energy at the inlet, k0 , and the dissipation rate of turbulent kinetic energy at the inlet, ε0 , were estimated by 3 k0 = (u0 I)2 2
ε0 =
Cμ3/4
k3/2 L
3.1. Numerical method The governing equations for flow and heat transfer in the TTHC were solved in the master Cartesian coordinate system with a control volume finite difference method (CVFDM) similar to that introduced by Patankar [21]. The grid topology and density considered was similar as used in our previous work [19]. The convection term in the governing equations was modeled with the bounded second-order upwind scheme and the diffusion term was computed using multi linear interpolating polynomial nodes. The SIMPLEC algorithm was used to resolve the coupling between velocity and pressure [22]. To accelerate the convergence, the under-relaxation factor for the pressure, p, was 0.3; that for temperature, T, was 0.9; that for the velocity component in the i-direction, ui , k and ε was 0.7; and that for body force was 0.8. In order to predict the performance of the TTHC, the double precision solver was used for simulations. The numerical computation was considered converged when the residual summed over all the computational nodes at the nth iteration was less than or equal to 10−6 . 3.2. Calculation of heat transfer coefficient The overall heat-transfer coefficient, Uo , was calculated from the inlet and outlet temperature data and the flow rates, using the following equation: Uo =
q Ao TLMTD
where TLMTD is the log mean temperature difference. The overall heat transfer rates were based on the outer surface area, Ao , of the inner tube. The TLMTD was calculated based on the temperature difference, T1 and T2 using following equation:
TLMTD =
(15)
In Eq. (14), the turbulence intensity level, I, is defined as u/ /u × 100%. As the turbulent eddies cannot be larger than the pipe, the turbulence characteristic length scale, L, is set to be
(17)
T2 − T1 ln( T2 / T1 )
(18)
Heat transfer coefficients were calculated for both the inner and outer tubes. For these calculations, average bulk temperatures and average temperature of the coil at each cross-section were used. The inner and outer heat transfer coefficients are
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usually obtained from the overall thermal resistance consisting of three resistances in series: the convective resistance in the inner surface, the conductance resistance of the pipe wall and the convective resistance on the outer surface by the following equation: Ao Ao ln(do /di ) 1 1 = + + Uo A i hi 2πkL ho
(19)
where do is the outer diameter of the tube; di is the inner diameter of the tube; k is the thermal conductivity of the wall; and L is the length of the tube. Heat transfer coefficients for the shell side, ho , and for the tube side, hi , were calculated using traditional Wilson plot technique [17,19]. For the calculation of inner heat transfer coefficient, the mass flow rate in the shell side was kept constant; and assumed that the outer heat transfer coefficient is constant. The inner heat transfer coefficient was assumed to behave in the following manner with the fluid velocity in the tube side, ui : hi = Cuni
lowing two equations [23], respectively:
(20)
Eq. (20) was placed into Eq. (19) and the values for the constant C and the exponent n were determined through curve fitting. The inner and outer heat transfer coefficients could then be calculated. This procedure was repeated for each outer tube flow rate. Similar procedure was adopted for the calculation of annulus heat transfer coefficient (by keeping tube side flow rate constant and annulus side flow rate varying). 4. Results and discussion The TTHC heat exchanger was analyzed in terms of heat transfer in downstream length and heat exchanger efficiency before discussing the friction factor and fully developed heattransfer coefficient in the inner and outer tube, respectively. 4.1. Effectiveness-NTU The effectiveness is defined as the ratio of the actual amount of heat transfer to the maximum possible heat transfer for the given heat exchanger. The maximum amount of heat transferred, qmax , is calculated by: qmax = Cmin (Tin,h − Tin,c )
Ao Uo Cmin
ε=
1 − exp(1 − NTU(1 + Cmin /Cmax )) 1 + Cmin /Cmax
, for co-current flow ε=
(23)
1 − exp(−NTU(1 − Cmin /Cmax )) 1 − Cmin /Cmax exp(−NTU(1 − Cmin /Cmax ))
, for counter-current flow
(24)
Fig. 2 shows the plot of effectiveness vs. NTU for both cocurrent and counter-current flow in the TTHC heat exchanger for ms = 500 kg/h and P = 30 bars. It can be seen from the figure that the simulated data fits well with the model values (Eqs. (23) and (24)). In the present work the effectiveness was calculated considering the average heat transfer coefficient which causes negligible deviation from the theoretical predictions (Eqs. (23) and (24)). Fig. 2 also shows that with increasing NTU the effectiveness increases. 4.2. Friction factor in TTHC The inner and outer Fanning friction factors were determined as
(21)
˙ c Cp,c and m ˙ h Cp,h (mass where Cmin is the minimum between m flow rate multiplied by specific heat), Tin,h is the inlet temperature of the hot fluid, and Tin,c is the inlet temperature of the cold fluid. The effectiveness is generally plotted vs. the NTUmax to obtain graphs that contain curves of constant Cmin to Cmax ratios. NTUmax is defined as: NTUmax =
Fig. 2. Effectiveness vs. number of transfer units in TTHC.
(22)
Furthermore, the theoretical effectiveness, ε, for a co-current and counter-current flow heat exchangers are given by the fol-
f =
P × De 2ρu20 × L
(25)
where L is the length of the heat exchanger and De is the equivalent diameter for the inner and outer tubes and u0 is the inlet velocity for the inner and outer tubes, respectively. The friction factor measurements in the present work were compared with the experimental data of Mishra and Gupta [24]. Mishra and Gupta’s experimental data were presented by the following empirical correlation: fc − fs =
0.0075 √ λ
(26)
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[24] for helically coiled tubes. It can be seen from the figure that the pressure drop in the annulus section is higher as compared to the data of Mishra and Gupta [24] for the similar process conditions. This may be due to friction generated by outer wall of the inner-coiled tube, inner wall of the outer-coiled tube as well as semicircular plate installed in the annulus region. Based on the outer tube pressure drop data a new model is proposed which accounts the curvature ratio and can also be used to predict pressure drop in straight tube when λ = 0. The proposed model is given below: fc 0.551 = 1 + 0.0927NDe fs
Fig. 3. Friction factor vs. inner Reynolds number in TTHC.
for 4500 < NRe < 105, 6.7 < < 346 and 0 < (H/2Rc ) < 25.4, where fs is the friction factor in straight tube fs =
0.079 0.25 NRe
(27)
4.2.1. Inner friction factor Fig. 3 shows the numerical predictions in the inner tube of the TTHC at various inner tube flow rates. It can be seen from the figure that as the inner tube Reynolds number increases the friction factor decreases. Fig. 3 also illustrates the comparison of present predictions of the friction factor with the experimental data reported by Mishra and Gupta [24]. It can be observed that the numerical results in the inner tube of TTHC agree fairly well with the experimental measurements of Mishra and Gupta [24]. 4.2.2. Outer friction factor The numerical predictions of friction factor in the outer tube are shown in Fig. 4. The fanning friction factor in the annulus side is compared with the data reported by Mishra and Gupta
Fig. 4. Friction factor vs. outer Reynolds number in TTHC.
, for 300 < NDe < 900
(28)
The accuracy of the proposed model was within ±2%. The proposed model for outer friction factor is based on the numerical computations carried out without baffles in the TTHC. The friction factor in inner and outer tubes were also compared with the friction factor in straight tubes and also reported in Figs. 3 and 4. It can be seen from both the figures that the friction factor is higher in both inner as well as outer tubes of the TTHC heat exchanger as compared to the straight tube. 4.3. Fully developed heat transfer The fully developed heat transfer predictions in the present work were compared with the experimental data reported in the literature for straight and curved tubes. During the numerical computations the energy balances of inner and outer tubes have been calculated and it was observed that the energy balance was well within 5% for all the computations. Fig. 5 shows the overall heat transfer coefficient (Uo ) for different values of flow rate in the inner-coiled tube and in the annulus region for both cocurrent and counter-current flow systems. It can be seen from the figure that the overall heat transfer coefficient increases with increase in the inner tube flow rate for a constant flow rate in the annulus region. About 30% increase in the overall heat transfer coefficient was observed over two-fold change in the inner Reynolds number for both co-current and counter-current flow systems. Similar trends in the variation of overall heat transfer coefficient were observed for different flow rates in the annulus region. It can also be concluded from Fig. 5 that the overall heat transfer coefficient increases with increase in the flow rate in the annulus region for a constant inner tube flow rate. As expected the results from the counter-current flow were similar to the cocurrent flow configuration (changing the flow direction should have negligible effect on the heat transfer coefficients). Heat transfer rates, however, are much higher in the counterflow configuration, due the increased log mean temperature difference. Further it can be seen from the figure that with the increase of operating pressure in the inner tube the overall heat transfer coefficient increases. This may be due to the change in the fluid properties when the operating pressure increases (during the computations the properties of the compressed air were calculated at the corresponding inlet temperature and pressure, e.g., density, viscosity, thermal conductivity and specific heat). The fully developed heat transfer predictions were compared with the measurements of Mori and Nakayama [25] for inner
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and outer tubes. The model proposed by Mori and Nakayama [25] is as follows: Mori and Nakayama [25]: 1/10 NPr 4/5 dt NNu,C = NRe 2/3 dc 26.2(NPr − 0.074) 0.098 for NPr ≈ 1 × 1+ 1/5 (NRe (dt /dc )2 ) and NRe (dt /dc )2 > 0.1.
−0.4 NNu,C NPr =
1/12 1 5/6 dt NRe 41.0 dc 0.061 × 1+ 1/6 {NRe (dt /dc )2.5 } and NRe (dt /dc )2.5 > 0.4.
(29a)
for NPr < 1 (29b)
Fig. 6. Nusselt number vs. inner Reynolds number in TTHC for NPr = 7.
4.3.1. Inner Nusselt number The arithmetic mean temperature (average of inlet and outlet temperatures) value was taken to compute the physical and thermal properties to calculate Reynolds and Prandtl numbers. Fig. 6 illustrates the comparison of fully developed Nusselt number in the inner-coiled tube with the data of Mori and Nakayama [25]. It can be observed, that the present predictions are slightly higher as compared to the data of Mori and Nakayama [25]. The maximum difference between the present numerical results and the experimental data is approximately 5–10%. This discrepancy can be attributed to the distinction in the boundary conditions (i.e., constant wall flux for correlations and fluid-to-fluid heat transfer for present scenario). Further the difference may also be due to the homogenization of thermal gradients in the outer tube due to the higher turbulence. However, the nature of the plot is similar to that reported in the literature. 4.3.2. Outer Nusselt number The outer Nusselt numbers were evaluated at the modified Reynolds number that uses the concept of equivalent diameter. Fig. 7 shows the annulus Nusselt number vs. annulus Reynolds number. It can be seen from the figure that the Nusselt number
Fig. 5. Overall heat transfer coefficient vs. inner tube flow rate in TTHC for (a) QS = 300 kg/h and (b) QS = 500 kg/h.
Fig. 7. Nusselt number vs. outer Reynolds number in TTHC for NPr = 7.
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Fig. 8. Predicted vs. simulated Nusselt number in the outer tube of the TTHC for different Reynolds and Prandtl numbers.
values increase with increase in the annulus Reynolds number. This is due to increase in turbulence between the fluid elements (semicircular baffles were placed in the outer tube which provides turbulence to the annulus fluid). Fig. 7 also illustrates the comparison of present predictions of fully developed Nusselt number in the outer tube with the experimental data of Mori and Nakayama [25]. The outer Nusselt number in the present work is 2–3 times higher as compared to the Mori and Nakayama’s measurement. This may be due to the application of physically realistic boundary condition in the heat exchanger (Mori and Nakayama [25] used the constant wall temperature while in the present work a direct fluid to fluid heat transfer was considered). The present experimental prediction for the fully developed heat transfer for inner and outer tubes were also compared with the heat transfer in straight tube and it was observed that the heat transfer in the present heat exchanger is higher as compared to the straight tubes (Figs. 6 and 7). For the outer tube heat transfer a new empirical model was developed from the numerical data considering all the process conditions used during the computations 0.817 0.3 −0.1 NNu = 0.0509NRe NPr λ ,
for 5000 < NRe < 15, 000 and 0.74 < NPr < 150
(30)
The maximum deviation between the model results and the numerical predictions was approximately ±3%. The proposed model for outer heat transfer is based on the numerical computations carried out without baffles in the TTHC. The heat transfer predictions from present numerical simulations and Eq. (30) are reported in Fig. 8 for both counter-current vs. the co-current flow configurations for Reynolds number varying from 5000 to 15,000 and Prandtl number 0.74–150. There is a reasonable agreement between the two values. 5. Conclusions Numerical study of a tube-in-tube helical coiled heat exchanger was performed considering compressed air in the
inner tube at various operating pressures and cooling water in the outer tube. The mass flow rate in the inner tube and the annulus were both varied and the counter-current and co-current flow configurations were tested. The present numerical model considers physically realistic situations, such as heat conduction within the tube walls, which was not considered in the literature. The numerically obtained overall heat transfer coefficient (Uo ) for different values of flow rate in the inner-coiled tube and in the annulus region were reported. It was observed that the overall heat transfer coefficient increases with increase in the inner-coiled tube flow rate for a constant flow rate in the annulus region. Similar trends in the variation of overall heat transfer coefficient were observed for different flow rates in the annulus region for a constant flow rate in the inner-coiled tube. It was also observed that with the increase in operating pressure in the inner tube the overall heat transfer coefficient also increases. The friction factor value in the inner-coiled tube was in agreement with the literature data. The annulus side friction factor values were higher as compared to the values reported in the literature. The heat transfer in the inner and outer tubes was higher as compared to the data reported in the literature, which may be due to the distinction in the boundary conditions. New correlations have been proposed for pressure drop and heat transfer predictions in the outer tube of TTHC. Appendix A. Nomenclature
a A AT C Cp d D h H k L n p P
q qw Rc ui u0 U T
T1
T2 x xi
radius of the helical pipe (m) area (m2 ) Van Driest constant constant in Eq. (20) specific heat (kJ/kg K) tube diameter (m) coil diameter (m) heat transfer coefficient (W/m2 K) pitch (m) thermal conductivity (W/m K) length of heat exchanger (m) exponent in Eq. (20) pressure (N/m2 ) parameter to calculate the generation of turbulent kinetic energy due to the mean velocity gradient and buoyancy heat transfer rate (J/s) wall heat flux (W/m2 ) radius of the coil (m) velocity component in i-direction (i = 1, 2, 3) (m/s) inlet velocity (m/s) overall heat transfer coefficient (W/m2 K) temperature (K) temperature difference at inlet (K) temperature difference at outlet (K) spatial position (m) master Cartesian coordinate in i-direction (i = 1, 2, 3) (m)
V. Kumar et al. / Chemical Engineering and Processing 47 (2008) 2287–2295
Greek letters α angle (◦ ) ε turbulent energy dissipation ϕ axial angle k turbulent kinetic energy λ curvature ratio (d/Dc ) ρ density of fluid (kg/m3 ) μ dynamic viscosity (kg/ms) στ turbulent Prandtl number in energy equation σk diffusion Prandtl number for k σε turbulent Prandtl number for ε wall shear stress (N/m2 ) τw Subscripts 0 inlet conditions b bulk quantity i inside/inner o outside/outer w wall condition + standard wall coordinates. Dimensionless numbers NRe Reynolds number (=ρU √ 0 dh /μ) Dean number (=NRe / λ) NDe NPr Prandtl number (=Cp × λ/k) NPr,w Turbulent wall Prandtl number NNu Nusselt number (=h × d/k) References [1] S.A. Berger, L. Talbot, L.S. Yao, Flow in curved pipes, Ann. Rev. Fluid Mech. 15 (1983) 461–512. [2] R.K. Shah, S.D. Joshi, Convective heat transfer in curved ducts, in: S. Kakac, R.K. Shah, W. Aung (Eds.), Handbook of Single-Phase Convective Heat Transfer, Wiley, New York, 1987 (Chapter 5). [3] P. Naphon, S. Wongwises, A review of flow and heat transfer characteristics in curved tubes, Renewable Sustainable Energy Rev. 10 (2006) 463–490. [4] W.R. Dean, Note on the motion of fluid in a curved pipe, Philos. Mag. 4 (1927) 208–223. [5] W.R. Dean, The streamline motion of fluid in a curved pipe (second paper), Philos. Mag. 7 (1928) 673–695. [6] A.R. Figueiredo, A.M. Raimundo, Analysis of the performances of heat exchangers used in Hot-water stores, Appl. Thermal Eng. 16 (1996) 605–611.
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[7] S. Haraburda, Consider helical-coil heat exchangers, Chem. Eng. 102 (1995) 149–151. [8] B.V.S.S.S. Prasad, D.H. Das, A.K. Prabhakar, Pressure drop, heat transfer and performance of a helically coiled tubular exchanger, Heat Recovery Syst. CHP 9 (1989) 249–256. [9] R.K. Patil, B.W. Shende, P.K. Ghosh, Designing a helical-coil heat exchanger, Chem. Eng. 92 (1982) 85–88. [10] G.T. Karahalios, Mixed convection flow in a heated curved pipe with core, Phys. Fluids A 2 (1990) 2164–2175. [11] M.A. Petrakis, G.T. Karahalios, Technical note: steady flow in a curved pipe with a coaxial core, Int. J. Num. Methods Fluids 22 (1996) 1231–1237. [12] M.A. Petrakis, G.T. Karahalios, Exponential-decaying flow in a gently curved annular pipe, Int. J. Non-Linear Mech. 32 (1997) 823–835. [13] M.A. Petrakis, G.T. Karahalios, Fluid flow behavior in a curved annular conduit, Int. J. Non-Linear Mech. 34 (1999) 13–35. [14] K. Nandakumar, J.H. Masliyah, Swirling flow and heat transfer in coiled and twisted pipes, Advances in Transport Process, in: A.S. Mujumdar, R.A Masliyah (Eds.), vol. 4, Wiley Eastern. [15] S. Garimella, D.E. Richart, R.N. Christensen, Experimental investigation of heat transfer in coiled annular ducts, ASME J. Heat Transfer 110 (1988) 329–336. [16] R.C. Xin, M.A. Ebadian, The effects of Prandtl numbers on local and average convective heat transfer characteristic in helical pipes, J. Heat Transfer 119 (1997) 467–473. [17] T.J. Rennie, V.G.S. Raghavan, Experimental studies of a doublepipe helical heat exchanger, Exp. Thermal Fluid Sci. 29 (2005) 919–924. [18] T.J. Rennie, V.G.S. Raghavan, Numerical studies of a double-pipe helical heat exchanger, Appl. Thermal Eng. 26 (2006) 1266–1273. [19] V. Kumar, S. Saini, M. Sharma, K.D.P. Nigam, Pressure drop and heat transfer study in tube-in-tube helical heat exchanger, Chem. Eng. Sci. 61 (2006) 4403–4416. [20] V. Yakhot, S.A. Orszag, Renormalization group analysis of turbulence. I. Basic theory, J. Sci. Comp. 1 (1986) 1–51. [21] S.V. Patankar, Numerical Heat Transfer and Fluid Flow, Hemisphere, Washington, DC, 1980. [22] J.P. Van Doormaal, G.D. Raithby, Enhancements of the SIMPLE method for predicting incompressible flow problems, Numer. Heat Transfer 7 (1984) 147–158. [23] F.M. White, Heat Transfer, Addison-Wesley Publishing Company, Inc., Reading, Massachusetts, 1984, 588 pp. [24] P. Mishra, S.N. Gupta, Momentum transfer in curved pipes: Newtonian fluids, Ind. Eng. Chem. Des. Dev. 18 (1979) 130–137. [25] Y. Mori, W. Nakiyama, Study on forced convective heat transfer in curved pipes: Part – III, Int. J. Heat Mass Transfer 10 (1967) 681–695. [26] E.N. Sieder, G.E. Tate, Heat transfer and pressure drop of liquids in tubes, Ind. Eng. Chem. 28 (1936) 1429–1435.
Chemical Engineering and Processing 47 (2008) 2287–2295
Numerical studies of a tube-in-tube helically coiled heat exchanger Vimal Kumar, Burhanuddin Faizee, Monisha Mridha, K.D.P. Nigam ∗ Department of Chemical Engineering, Indian Institute of Technology, Delhi, Hauz Khas, New Delhi-110016, India Received 6 August 2007; received in revised form 29 December 2007; accepted 2 January 2008 Available online 11 January 2008
Abstract In the present study a tube-in-tube helically coiled (TTHC) heat exchanger has been numerically modeled for fluid flow and heat transfer characteristics for different fluid flow rates in the inner as well as outer tube. The three-dimensional governing equations for mass, momentum and heat transfer have been solved using a control volume finite difference method (CVFDM). The renormalization group (RNG) k–ε model is used to model the turbulent flow and heat transfer in the TTHC heat exchanger. The fluid considered in the inner tube is compressed air at higher pressure and cooling water in the outer tube at ambient conditions. The inner tube pressure is varied from 10 to 30 bars. The Reynolds numbers for the inner tube ranged from 20,000 to 70,000. The mass flow rate in the outer tube is varied from 200 to 600 kg/h. The outer tube is fitted with semicircular plates to support the inner tube and also to provide high turbulence in the annulus region. The overall heat transfer coefficients are calculated for both parallel and counter flow configurations. The Nusselt number and friction factor values in the inner and outer tubes are compared with the experimental data reported in the literature. New empirical correlations are developed for hydrodynamic and heat-transfer predictions in the outer tube of the TTHC. © 2008 Elsevier B.V. All rights reserved. Keywords: Tube-in-tube helical heat exchanger; Heat transfer; Helical tube; RNG k–
1. Introduction Helical coil heat exchangers are one of the most common equipment found in many industrial applications ranging from chemical and food industries, power production, electronics, environmental engineering, manufacturing industry, air-conditioning, waste heat recovery, cryogenic processes, and space applications. Helical coils are extensively used as heat exchangers and reactors due to higher heat and mass transfer coefficients, narrow residence time distributions and compact structure. The modification of the flow in the helically coiled tubes is due to the centrifugal forces (Dean roll cells, [4,5]). The curvature of the tube produces a secondary flow field with a circulatory motion, which causes the fluid particles to move toward the core region of the tube. The secondary flow enhances heat transfer rates as it reduces the temperature gradient across the cross-section of the tube. Thus there is an additional convective Abbreviations: CVFDM, control volume finite difference method; HVAC, heating, ventilating and air conditioning; LMTD, log-mean temperature difference; TTHC, tube-in-tube helical coil. ∗ Corresponding author. Tel.: +91 11 26591020; fax: +91 11 26591020. E-mail address: [email protected] (K.D.P. Nigam). 0255-2701/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.cep.2008.01.001
heat transfer mechanism, perpendicular to the main flow, which does not exist in conventional heat exchangers. An extensive review of fluid flow and heat transfer in helical pipes has been presented in the literature [1–3]. There is considerable amount of work reported in the literature on heat transfer in coiled tubes; however, very less attention has been paid to study the outer heat transfer coefficient. Figueiredo and Raimunda [6], Haraburda [7], Prasad et al. [8] and Patil et al. [9] have discussed the design procedure for coil-in-shell heat exchangers considering helical coiled tubes as a bank of straight tubes for calculating outer heat transfer coefficients. In the coil-in-shell heat exchangers poor circulation is observed in shell regions near the coil which could be avoided by using a tube-in-tube helical coil (TTHC) configuration. A tube-in-tube helical coil heat exchanger requires the knowledge of the heat transfer rates for the two flowing fluid, i.e., the flow in the helical tube as well as in the helical annulus. Karahalios [10] and Petrakis and Karahalios [11–13] reported the fluid flow and heat transfer in a curved pipe with a solid core. They showed that the size of the core affects the flow in the annulus with flows approaching parabolic for large cores [13]. Karahalios [10] studied the heat transfer in a curved annulus with a constant temperature gradient on both inner and outer walls of the annulus as
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the thermal boundary conditions. All the above reported studies for helical coils were confined with one of two major boundary conditions, constant wall heat flux or constant wall temperature [2,14]. However in industrial applications of heat exchangers where one is interested in fluid-to-fluid heat exchanger the use of constant wall temperature or constant wall flux conditions does not appear to be physically realistic. This complicates the design of tube-in-tube helical coil heat exchangers, where either the heating or cooling is supplied by a secondary fluid, with the two fluids separated by the wall of the coil. Garimella et al. [15] reported average heat transfer coefficients of laminar and transition flows for forced convection heat transfer in coiled annular ducts. Two different coil diameters and two annular radius ratios were used in the experiment. Hot and cold waters were used as working fluids. They reported that the heat transfer coefficients obtained from the coiled annular ducts were higher than those obtained from a straight annulus, especially in the laminar region. Xin et al. [16] studied the effects of coil geometries and the flow rates of air and water on pressure drop in both annular vertical and horizontal helicoidal pipes. Experiments were performed for the superficial water Reynolds number from 210 to 23,000 and superficial air Reynolds number from 30 to 30,000. Their results showed that the transition from laminar to turbulent flow covers a wide Reynolds number range. Rennie and Raghavan [17] experimentally reported the heat transfer in a coil-in-coil heat exchanger comprised of one loop. This configuration results in secondary flows in both the inner tube and in the annulus, as both sections are curved and subject to centrifugal forces. The flows of both the fluids were co-current and counter-current. They also reported that increasing the tube or annulus Dean numbers resulted in an increase in the overall heat transfer coefficient. The heat transfer characteristics of a double-pipe helical heat exchanger for both parallel and counter flow using water in the inner as well as outer tubes was numerically studied by Rennie and Raghavan [18]. Overall heat transfer coefficients were calculated for inner Dean numbers in the range of 38–350 for the boundary conditions of constant wall temperature and constant heat flux. The results showed an increasing overall heat transfer coefficients as the inner Dean number increased; however, flow conditions in the annulus had a stronger influence on the overall heat transfer coefficient. Furthermore, increasing the size of the inner tube resulted in lower thermal resistances in the annulus, though the thermal resistance in the inner tube remained fairly constant. From the literature it can be seen that no study has been reported which considers the fluid at high pressures and temperature, though in most of the industrial applications the process streams are available at higher pressure and temperature (e.g., chemical reactions; heating, ventilating and air conditioning (HVAC) systems; and heat exchangers). Therefore in the present work the performance of a tube-in-tube helical heat exchanger for a compressed air–water counter-current and co-current flow system is studied numerically over a wide range of Reynolds number considering turbulent flow regime. The hydrodynamics and heat transfer of compressed air flowing in the inner tube of tube-in-tube helical coil heat exchanger with high pressure (i.e., 10–30 bars) are being reported the first time, which has not been
Fig. 1. Tube-in-tube helical heat exchanger.
considered in the previous literature. Fluid-to-fluid heat transfer has been studied using physically realistic boundary conditions. The effect of the fluid flow on the heat transfer and hydrodynamics have been studied in the tube as well as in the annulus. In the present work baffles have been introduced in the annulus area of the tube-in-tube helical heat exchanger. All the computations have been carried out on a SUN Fire V880 computer in the Chemical Reaction Engineering Laboratory at Indian Institute of Technology, Delhi. 2. Mathematical modeling of TTHC The geometry considered and the systems of the coordinates are illustrated in Fig. 1 (where, di,inner is the diameter of inner coiled tube; Rc is the coil radius; H is the distance between the two turn; and do,inner is the inner diameter of the outer tube). The TTHC geometry was developed with inner and outer diameters of 0.023 and 0.0508 m, respectively, with coil diameter of 0.762 m and with a pitch of 0.020 m. The TTHC was comprised of four full turns. As the tube side fluid was under highly turbulence flow regime therefore it was assumed that the flow will get fully developed within four turns. The TTHC details are given in Table 1 for both inner as well as for outer tubes, respectively. The coils were orientated in the vertical position. Inlets and outlets were located at each end of the coil. The boundary conditions associated with the inlets specified the inlet velocities in the axial direction. Coil properties were set to those of stainless steel, with a thermal conductivity of 16 W m−1 K−1 , density of 7882 kg.m−3 and a specific heat of 502 J kg−1 K−1 . The outer coil was set to be adiabatic (representing an insulated tube) and the inner coil was set to allow conductive heat flow through the tube.
V. Kumar et al. / Chemical Engineering and Processing 47 (2008) 2287–2295 Table 1 Geometrical and flow parameters for the inner and outer tube of TTHC
Outer diameter (m) Inner diameter (m) Coil diameter (m) Pitch (m) Number of turns Flow rate (kg/h) Pressure (bars) Prandtl number
Inner tube
Outer tube
0.0254 0.023 0.762 0.100 4 40–85 10, 20 and 30 1
0.0508 0.0484 0.762 0.100 4 200–600 1 0.74, 7, 33 and 150
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Ideal gas law: P = ρRT
(1)
Continuity equation: ∂ui =0 ∂xi
(2)
Momentum equation: ∂ui 2 ∂p ∂uk ∂(ρui uj ) ∂ ∂uj μeff − μeff − = + ∂xj ∂xj ∂xj ∂xi 3 ∂xk ∂xi (3)
The pressure drop and heat transfer in the TTHC is studied for five different Reynolds number (30,000 < NRe < 70,000) in the inner tube. The outer tube mass flow rates is varied from 200 to 600 kg/h. The Reynolds number in the outer tube is different for pressure drop and heat transfer calculations as it depends upon the equivalent diameter of the outer tube. The detailed description of the equivalent diameter calculation is reported in Kumar et al. [19]. The simulations were carried out at three different levels of pressure values, i.e., form 10–30 bar. The annulus side fluid properties were calculated at average temperature i.e., mean of inlet and outlet temperature in the annulus. Compressed air density in the inner tube was calculated using ideal gas law. The total number of simulations performed were 150 (5 inner tube flow rates, 5 outer tube flow rates, 3 operating pressure values and 2 flow arrangement systems). The output of the simulations included the inlet and outlet velocities, mass flow rates and enthalpy rates, as well as velocity, pressure, and temperature fields at various specified cross-sections. 2.1. Governing equations The geometry, system of coordinates and details of TTHC considered for the present work are discussed in our previous work [19]. In the TTHC the cold and hot fluids enter from their corresponding inlets and heat transfer take place between the two fluids due to conduction and forced convection. Heat transfer by radiation can be neglected because temperature in the TTHC studied is quite low (max. 353 K). The Cartesian coordinate system (x, y, z) was used to represent flow in numerical simulation. The flow was considered to be steady, and incompressible. At the inlet (φ = 0◦ ), fluid with turbulence intensity I and temperature T0 enters the TTHC heat exchanger at a velocity of u0 . Turbulent flow and heat transfer develop simultaneously downstream in the tubes. For the turbulent flow and heat transfer modeling the RNG k– model proposed by Yakhot and Orszag [20] was used in the TTHC because the RNG model included an additional term in its equation that significantly improve the accuracy for rapidly strained flows, such as those in curved pipes. The effect of swirl on turbulence is included in the RNG model, enhancing accuracy for swirling flows. The three dimensional governing equations of turbulent flow and the heat transfer in the TTHC can be written in tensor form in the master Cartesian system as follows:
Energy equation:
dui ∂ui ∂ ∂uj ∂T ∂(ρui cp T ) αT μeff + μeff = + ∂xi ∂xi ∂xi dxj ∂xj ∂xi 2 ∂uk − μeff (4) δij 3 ∂xk Turbulent kinetic energy equation: ∂k ∂ ∂(ρui k) αk μeff + μt S 2 + Gb − ρε = ∂xi ∂xi ∂xi
(5)
Dissipation rate of turbulent kinetic energy equation: ∂ε ε ∂ ∂(ρui ε) αε μeff + C1ε μt S 2 = ∂xi ∂xi ∂xi k − C2ε ρ
ε2 − Ra k
The effective viscosity, μeff can be defined as 2 Cμ k √ μeff = μmol 1 + μmol ε
(6)
(7)
where μmol is the molecular viscosity. The coefficients αT , αk and αε in Eqs. (4)–(6) are the inverse effective Prandtl numbers for T, k, and ε, respectively. The values of inverse effective Prandtl numbers, αT, αk and αε are computed using the following formula derived analytically by the RNG theory: α − 1.3929 0.6321 α + 2.3929 0.3679 μmol = (8) α − 1.3929 α + 2.3929 μeff 0 0 where α0 is equal to 1/NPr , 1.0, and 1.0, for the computation of αT , αk and αε , respectively. When a non-zero gravity field and temperature gradient are present simultaneously, the k–ε model account for the generation of k (kinetic energy) due to buoyancy (Gb in Eq. (5)) and the corresponding contribution to the production of ε (energy dissipation) in Eq. (6). The effects of buoyancy are also included despite the fact that the effect of buoyancy is not so significant at very high Reynolds number. The generation of turbulence due to buoyancy is given by Gb = βgi
μt ∂T NPr,t ∂xi
(9)
where NPr,t is the turbulent Prandtl number for energy, gi is the component of the gravitational vector in the ith direction. In the
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case of the RNG k– model, NPr,t = 1/αT , and β, the coefficient of thermal expansion, is defined as β = −1/ρ(∂p/∂T)T . In Eq. (6), R is given by R=
Cμ ρη3 (1 − η/η0 ) ε2 1 + ζη3 k
(10)
where η = S k/ε, η0 ≈ 4.38, ζ = 0.012. The model constants C , C1 , and C2 are equal to 0.085, 1.42 and 1.68, respectively. The term S in Eq. (5) and Eq. (6) is
the modulus of the mean rate-of-strain tensor, defined as S = 2Sij Sij , where 1 ∂ui ∂uj (11) + Sij = 2 ∂xj ∂xi
0.07(di,inner /2) in the present study. The factor of 0.07 is based on the maximum value of the mixing length in fully developed turbulent pipe flow. The computation domain (4 turns) used in this study can ensure that the outflow condition (i.e., fully developed flow and thermal assumed) can be satisfied at the exit plane of the curved pipe. Therefore, at the outlet, the diffusion term for all dependent variables were set to zero in the exit direction: ∂φ =0 ∂n
(16)
where φ represents the variables ui , T, k and ε. 3. Numerical computation
The two-layer based, non-equilibrium wall function was used for the near-wall treatment of flow in the given geometry. The non-equilibrium wall functions are recommended for complex flows because of the capability to partly account for the effects of pressure gradients and departure from equilibrium. The numerical results for turbulent flow tend to be more susceptible to grid dependency than those for laminar flow due to the strong interaction of the mean flow and turbulence. The distance from the wall at the wall-adjacent cells must be determined by considering the range over which the log-law is valid. The size of wall adjacent cells can be estimated from yp ( yp+ ν/uτ ), where u ≡(τ w /ρ)0.5 = u (cf /2)0.5 . In the present study, the yp+ was taken in the range of 30–60. The wall heat flux was computed using qw = ρCp u T/T+ , where T+ is obtained from ⎧ u ⎪ ⎨ NPr , y+ ≤ yv+ u∗ T+ = (12) u ⎪ ⎩ NPr,w − NPr , y+ > yv+ u∗ NPr =
1/2 NPr π/4 NPr,w 1/4 AT −1 sin (π/4) κ NPr,w NPr
(13)
where NPr is the molecular Prandtl number, NPr,w , the turbulent wall Prandtl number (NPr,w = 1.2) and AT , the Van Driest constant (AT = 26). At the inlet, uniform profiles for all the dependent variables were employed: us = u0 ,
T = T0 ,
k = k0 ,
ε = ε0 .
(14)
For the inner and outer walls of the heat exchanger, heat conduction in solid wall was considered. Ambient temperature was considered across the outer wall of the TTHC heat exchanger, i.e., T = 300 K. Turbulent kinetic energy at the inlet, k0 , and the dissipation rate of turbulent kinetic energy at the inlet, ε0 , were estimated by 3 k0 = (u0 I)2 2
ε0 =
Cμ3/4
k3/2 L
3.1. Numerical method The governing equations for flow and heat transfer in the TTHC were solved in the master Cartesian coordinate system with a control volume finite difference method (CVFDM) similar to that introduced by Patankar [21]. The grid topology and density considered was similar as used in our previous work [19]. The convection term in the governing equations was modeled with the bounded second-order upwind scheme and the diffusion term was computed using multi linear interpolating polynomial nodes. The SIMPLEC algorithm was used to resolve the coupling between velocity and pressure [22]. To accelerate the convergence, the under-relaxation factor for the pressure, p, was 0.3; that for temperature, T, was 0.9; that for the velocity component in the i-direction, ui , k and ε was 0.7; and that for body force was 0.8. In order to predict the performance of the TTHC, the double precision solver was used for simulations. The numerical computation was considered converged when the residual summed over all the computational nodes at the nth iteration was less than or equal to 10−6 . 3.2. Calculation of heat transfer coefficient The overall heat-transfer coefficient, Uo , was calculated from the inlet and outlet temperature data and the flow rates, using the following equation: Uo =
q Ao TLMTD
where TLMTD is the log mean temperature difference. The overall heat transfer rates were based on the outer surface area, Ao , of the inner tube. The TLMTD was calculated based on the temperature difference, T1 and T2 using following equation:
TLMTD =
(15)
In Eq. (14), the turbulence intensity level, I, is defined as u/ /u × 100%. As the turbulent eddies cannot be larger than the pipe, the turbulence characteristic length scale, L, is set to be
(17)
T2 − T1 ln( T2 / T1 )
(18)
Heat transfer coefficients were calculated for both the inner and outer tubes. For these calculations, average bulk temperatures and average temperature of the coil at each cross-section were used. The inner and outer heat transfer coefficients are
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usually obtained from the overall thermal resistance consisting of three resistances in series: the convective resistance in the inner surface, the conductance resistance of the pipe wall and the convective resistance on the outer surface by the following equation: Ao Ao ln(do /di ) 1 1 = + + Uo A i hi 2πkL ho
(19)
where do is the outer diameter of the tube; di is the inner diameter of the tube; k is the thermal conductivity of the wall; and L is the length of the tube. Heat transfer coefficients for the shell side, ho , and for the tube side, hi , were calculated using traditional Wilson plot technique [17,19]. For the calculation of inner heat transfer coefficient, the mass flow rate in the shell side was kept constant; and assumed that the outer heat transfer coefficient is constant. The inner heat transfer coefficient was assumed to behave in the following manner with the fluid velocity in the tube side, ui : hi = Cuni
lowing two equations [23], respectively:
(20)
Eq. (20) was placed into Eq. (19) and the values for the constant C and the exponent n were determined through curve fitting. The inner and outer heat transfer coefficients could then be calculated. This procedure was repeated for each outer tube flow rate. Similar procedure was adopted for the calculation of annulus heat transfer coefficient (by keeping tube side flow rate constant and annulus side flow rate varying). 4. Results and discussion The TTHC heat exchanger was analyzed in terms of heat transfer in downstream length and heat exchanger efficiency before discussing the friction factor and fully developed heattransfer coefficient in the inner and outer tube, respectively. 4.1. Effectiveness-NTU The effectiveness is defined as the ratio of the actual amount of heat transfer to the maximum possible heat transfer for the given heat exchanger. The maximum amount of heat transferred, qmax , is calculated by: qmax = Cmin (Tin,h − Tin,c )
Ao Uo Cmin
ε=
1 − exp(1 − NTU(1 + Cmin /Cmax )) 1 + Cmin /Cmax
, for co-current flow ε=
(23)
1 − exp(−NTU(1 − Cmin /Cmax )) 1 − Cmin /Cmax exp(−NTU(1 − Cmin /Cmax ))
, for counter-current flow
(24)
Fig. 2 shows the plot of effectiveness vs. NTU for both cocurrent and counter-current flow in the TTHC heat exchanger for ms = 500 kg/h and P = 30 bars. It can be seen from the figure that the simulated data fits well with the model values (Eqs. (23) and (24)). In the present work the effectiveness was calculated considering the average heat transfer coefficient which causes negligible deviation from the theoretical predictions (Eqs. (23) and (24)). Fig. 2 also shows that with increasing NTU the effectiveness increases. 4.2. Friction factor in TTHC The inner and outer Fanning friction factors were determined as
(21)
˙ c Cp,c and m ˙ h Cp,h (mass where Cmin is the minimum between m flow rate multiplied by specific heat), Tin,h is the inlet temperature of the hot fluid, and Tin,c is the inlet temperature of the cold fluid. The effectiveness is generally plotted vs. the NTUmax to obtain graphs that contain curves of constant Cmin to Cmax ratios. NTUmax is defined as: NTUmax =
Fig. 2. Effectiveness vs. number of transfer units in TTHC.
(22)
Furthermore, the theoretical effectiveness, ε, for a co-current and counter-current flow heat exchangers are given by the fol-
f =
P × De 2ρu20 × L
(25)
where L is the length of the heat exchanger and De is the equivalent diameter for the inner and outer tubes and u0 is the inlet velocity for the inner and outer tubes, respectively. The friction factor measurements in the present work were compared with the experimental data of Mishra and Gupta [24]. Mishra and Gupta’s experimental data were presented by the following empirical correlation: fc − fs =
0.0075 √ λ
(26)
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[24] for helically coiled tubes. It can be seen from the figure that the pressure drop in the annulus section is higher as compared to the data of Mishra and Gupta [24] for the similar process conditions. This may be due to friction generated by outer wall of the inner-coiled tube, inner wall of the outer-coiled tube as well as semicircular plate installed in the annulus region. Based on the outer tube pressure drop data a new model is proposed which accounts the curvature ratio and can also be used to predict pressure drop in straight tube when λ = 0. The proposed model is given below: fc 0.551 = 1 + 0.0927NDe fs
Fig. 3. Friction factor vs. inner Reynolds number in TTHC.
for 4500 < NRe < 105, 6.7 < < 346 and 0 < (H/2Rc ) < 25.4, where fs is the friction factor in straight tube fs =
0.079 0.25 NRe
(27)
4.2.1. Inner friction factor Fig. 3 shows the numerical predictions in the inner tube of the TTHC at various inner tube flow rates. It can be seen from the figure that as the inner tube Reynolds number increases the friction factor decreases. Fig. 3 also illustrates the comparison of present predictions of the friction factor with the experimental data reported by Mishra and Gupta [24]. It can be observed that the numerical results in the inner tube of TTHC agree fairly well with the experimental measurements of Mishra and Gupta [24]. 4.2.2. Outer friction factor The numerical predictions of friction factor in the outer tube are shown in Fig. 4. The fanning friction factor in the annulus side is compared with the data reported by Mishra and Gupta
Fig. 4. Friction factor vs. outer Reynolds number in TTHC.
, for 300 < NDe < 900
(28)
The accuracy of the proposed model was within ±2%. The proposed model for outer friction factor is based on the numerical computations carried out without baffles in the TTHC. The friction factor in inner and outer tubes were also compared with the friction factor in straight tubes and also reported in Figs. 3 and 4. It can be seen from both the figures that the friction factor is higher in both inner as well as outer tubes of the TTHC heat exchanger as compared to the straight tube. 4.3. Fully developed heat transfer The fully developed heat transfer predictions in the present work were compared with the experimental data reported in the literature for straight and curved tubes. During the numerical computations the energy balances of inner and outer tubes have been calculated and it was observed that the energy balance was well within 5% for all the computations. Fig. 5 shows the overall heat transfer coefficient (Uo ) for different values of flow rate in the inner-coiled tube and in the annulus region for both cocurrent and counter-current flow systems. It can be seen from the figure that the overall heat transfer coefficient increases with increase in the inner tube flow rate for a constant flow rate in the annulus region. About 30% increase in the overall heat transfer coefficient was observed over two-fold change in the inner Reynolds number for both co-current and counter-current flow systems. Similar trends in the variation of overall heat transfer coefficient were observed for different flow rates in the annulus region. It can also be concluded from Fig. 5 that the overall heat transfer coefficient increases with increase in the flow rate in the annulus region for a constant inner tube flow rate. As expected the results from the counter-current flow were similar to the cocurrent flow configuration (changing the flow direction should have negligible effect on the heat transfer coefficients). Heat transfer rates, however, are much higher in the counterflow configuration, due the increased log mean temperature difference. Further it can be seen from the figure that with the increase of operating pressure in the inner tube the overall heat transfer coefficient increases. This may be due to the change in the fluid properties when the operating pressure increases (during the computations the properties of the compressed air were calculated at the corresponding inlet temperature and pressure, e.g., density, viscosity, thermal conductivity and specific heat). The fully developed heat transfer predictions were compared with the measurements of Mori and Nakayama [25] for inner
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and outer tubes. The model proposed by Mori and Nakayama [25] is as follows: Mori and Nakayama [25]: 1/10 NPr 4/5 dt NNu,C = NRe 2/3 dc 26.2(NPr − 0.074) 0.098 for NPr ≈ 1 × 1+ 1/5 (NRe (dt /dc )2 ) and NRe (dt /dc )2 > 0.1.
−0.4 NNu,C NPr =
1/12 1 5/6 dt NRe 41.0 dc 0.061 × 1+ 1/6 {NRe (dt /dc )2.5 } and NRe (dt /dc )2.5 > 0.4.
(29a)
for NPr < 1 (29b)
Fig. 6. Nusselt number vs. inner Reynolds number in TTHC for NPr = 7.
4.3.1. Inner Nusselt number The arithmetic mean temperature (average of inlet and outlet temperatures) value was taken to compute the physical and thermal properties to calculate Reynolds and Prandtl numbers. Fig. 6 illustrates the comparison of fully developed Nusselt number in the inner-coiled tube with the data of Mori and Nakayama [25]. It can be observed, that the present predictions are slightly higher as compared to the data of Mori and Nakayama [25]. The maximum difference between the present numerical results and the experimental data is approximately 5–10%. This discrepancy can be attributed to the distinction in the boundary conditions (i.e., constant wall flux for correlations and fluid-to-fluid heat transfer for present scenario). Further the difference may also be due to the homogenization of thermal gradients in the outer tube due to the higher turbulence. However, the nature of the plot is similar to that reported in the literature. 4.3.2. Outer Nusselt number The outer Nusselt numbers were evaluated at the modified Reynolds number that uses the concept of equivalent diameter. Fig. 7 shows the annulus Nusselt number vs. annulus Reynolds number. It can be seen from the figure that the Nusselt number
Fig. 5. Overall heat transfer coefficient vs. inner tube flow rate in TTHC for (a) QS = 300 kg/h and (b) QS = 500 kg/h.
Fig. 7. Nusselt number vs. outer Reynolds number in TTHC for NPr = 7.
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Fig. 8. Predicted vs. simulated Nusselt number in the outer tube of the TTHC for different Reynolds and Prandtl numbers.
values increase with increase in the annulus Reynolds number. This is due to increase in turbulence between the fluid elements (semicircular baffles were placed in the outer tube which provides turbulence to the annulus fluid). Fig. 7 also illustrates the comparison of present predictions of fully developed Nusselt number in the outer tube with the experimental data of Mori and Nakayama [25]. The outer Nusselt number in the present work is 2–3 times higher as compared to the Mori and Nakayama’s measurement. This may be due to the application of physically realistic boundary condition in the heat exchanger (Mori and Nakayama [25] used the constant wall temperature while in the present work a direct fluid to fluid heat transfer was considered). The present experimental prediction for the fully developed heat transfer for inner and outer tubes were also compared with the heat transfer in straight tube and it was observed that the heat transfer in the present heat exchanger is higher as compared to the straight tubes (Figs. 6 and 7). For the outer tube heat transfer a new empirical model was developed from the numerical data considering all the process conditions used during the computations 0.817 0.3 −0.1 NNu = 0.0509NRe NPr λ ,
for 5000 < NRe < 15, 000 and 0.74 < NPr < 150
(30)
The maximum deviation between the model results and the numerical predictions was approximately ±3%. The proposed model for outer heat transfer is based on the numerical computations carried out without baffles in the TTHC. The heat transfer predictions from present numerical simulations and Eq. (30) are reported in Fig. 8 for both counter-current vs. the co-current flow configurations for Reynolds number varying from 5000 to 15,000 and Prandtl number 0.74–150. There is a reasonable agreement between the two values. 5. Conclusions Numerical study of a tube-in-tube helical coiled heat exchanger was performed considering compressed air in the
inner tube at various operating pressures and cooling water in the outer tube. The mass flow rate in the inner tube and the annulus were both varied and the counter-current and co-current flow configurations were tested. The present numerical model considers physically realistic situations, such as heat conduction within the tube walls, which was not considered in the literature. The numerically obtained overall heat transfer coefficient (Uo ) for different values of flow rate in the inner-coiled tube and in the annulus region were reported. It was observed that the overall heat transfer coefficient increases with increase in the inner-coiled tube flow rate for a constant flow rate in the annulus region. Similar trends in the variation of overall heat transfer coefficient were observed for different flow rates in the annulus region for a constant flow rate in the inner-coiled tube. It was also observed that with the increase in operating pressure in the inner tube the overall heat transfer coefficient also increases. The friction factor value in the inner-coiled tube was in agreement with the literature data. The annulus side friction factor values were higher as compared to the values reported in the literature. The heat transfer in the inner and outer tubes was higher as compared to the data reported in the literature, which may be due to the distinction in the boundary conditions. New correlations have been proposed for pressure drop and heat transfer predictions in the outer tube of TTHC. Appendix A. Nomenclature
a A AT C Cp d D h H k L n p P
q qw Rc ui u0 U T
T1
T2 x xi
radius of the helical pipe (m) area (m2 ) Van Driest constant constant in Eq. (20) specific heat (kJ/kg K) tube diameter (m) coil diameter (m) heat transfer coefficient (W/m2 K) pitch (m) thermal conductivity (W/m K) length of heat exchanger (m) exponent in Eq. (20) pressure (N/m2 ) parameter to calculate the generation of turbulent kinetic energy due to the mean velocity gradient and buoyancy heat transfer rate (J/s) wall heat flux (W/m2 ) radius of the coil (m) velocity component in i-direction (i = 1, 2, 3) (m/s) inlet velocity (m/s) overall heat transfer coefficient (W/m2 K) temperature (K) temperature difference at inlet (K) temperature difference at outlet (K) spatial position (m) master Cartesian coordinate in i-direction (i = 1, 2, 3) (m)
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Greek letters α angle (◦ ) ε turbulent energy dissipation ϕ axial angle k turbulent kinetic energy λ curvature ratio (d/Dc ) ρ density of fluid (kg/m3 ) μ dynamic viscosity (kg/ms) στ turbulent Prandtl number in energy equation σk diffusion Prandtl number for k σε turbulent Prandtl number for ε wall shear stress (N/m2 ) τw Subscripts 0 inlet conditions b bulk quantity i inside/inner o outside/outer w wall condition + standard wall coordinates. Dimensionless numbers NRe Reynolds number (=ρU √ 0 dh /μ) Dean number (=NRe / λ) NDe NPr Prandtl number (=Cp × λ/k) NPr,w Turbulent wall Prandtl number NNu Nusselt number (=h × d/k) References [1] S.A. Berger, L. Talbot, L.S. Yao, Flow in curved pipes, Ann. Rev. Fluid Mech. 15 (1983) 461–512. [2] R.K. Shah, S.D. Joshi, Convective heat transfer in curved ducts, in: S. Kakac, R.K. Shah, W. Aung (Eds.), Handbook of Single-Phase Convective Heat Transfer, Wiley, New York, 1987 (Chapter 5). [3] P. Naphon, S. Wongwises, A review of flow and heat transfer characteristics in curved tubes, Renewable Sustainable Energy Rev. 10 (2006) 463–490. [4] W.R. Dean, Note on the motion of fluid in a curved pipe, Philos. Mag. 4 (1927) 208–223. [5] W.R. Dean, The streamline motion of fluid in a curved pipe (second paper), Philos. Mag. 7 (1928) 673–695. [6] A.R. Figueiredo, A.M. Raimundo, Analysis of the performances of heat exchangers used in Hot-water stores, Appl. Thermal Eng. 16 (1996) 605–611.
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