Heat Transfer Engineering, 31(14):1125–1136, 2010 Copyright C Taylor and Francis Group, LLC ISSN: 0145-7632 print / 1521-0537 online DOI: 10.1080/01457631003689211 An Analysis of Heat Conduction Models for Nanofluids ˜ N N QUARESMA,1 EMANUEL N MACEDO, ˆ JOAO HENRIQUE M DA 2 FONSECA, HELCIO R B ORLANDE, and RENATO M COTTA2 School of Chemical Engineering, Universidade Federal Par´a, UFPA Campus Universit´ario Guam´a, Bel´em, PA, Brazil Mechanical Engineering Department–Polit´ecnica/COPPE, Universidade Federal Rio de Janeiro, Rio de Janeiro, RJ, Brazil The mechanism of heat transfer intensification recently brought about by nanofluids is analyzed in this article, in the light of the non-Fourier dual-phase-lagging heat conduction model The physical problem involves an annular geometry filled with a nanofluid, such as typically used for measurements of the thermal conductivity with Blackwell’s line heat source probe The mathematical formulation for this problem is analytically solved with the classical integral transform technique, thus providing benchmark results for the temperature predicted with the dual-phase-lagging model Different test cases are examined in this work, involving nanofluids and probe sizes of practical interest The effects of the relaxation times on the temperature at the surface of the probe are also examined The results obtained with the dualphase-lagging model are critically compared to those obtained with the classical parabolic model, showing that the increase in the thermal conductivity of nanofluids measured with the line heat source probe cannot be attributed to hyperbolic effects INTRODUCTION The constitutive equation that classically relates the heat flux vector to the temperature gradient is Fourier’s law, which considers an infinite speed of propagation of heat in the medium Despite this unacceptable assumption, Fourier’s law provides accurate results for most practical engineering applications However, in applications involving small scales of time and space, the use of other constitutive equations, such as those independently derived by Cattaneo [1] and Vernotte [2], may be required Such models take into account a lag between the heat flux vector and the temperature gradient, resulting in a hyperbolic model for heat conduction [1–13] Thermal conductivity of fluids plays a vital role in the development of energy-efficient heat transfer equipment However, traditional fluids used in those equipments have low thermal The authors acknowledge the financial support provided by CNPq for the postdoctoral fellowship of Professor J N N Quaresma at the Laboratory of Heat Transmission and Technology of the Mechanical Engineering Department of COPPE/UFRJ This work was partially sponsored by CAPES and FAPERJ, with major financial support provided by Petrobras S.A Address correspondence to Professor Helcio R B Orlande, Mechanical Engineering Department–Polit´ecnica/COPPE, Universidade Federal Rio de Janeiro, UFRJ, Cx Postal 68503–Cidade Universit´aria, 21941–972, Rio de Janeiro, RJ, Brazil E-mail: helcio@mecanica.coppe.ufrj.br conductivity [14] On the other hand, metals in the solid form have thermal conductivity larger by orders of magnitude than those of fluids For example, the thermal conductivity of copper at room temperature is about 700 times larger than that of water and around 3000 times larger than that of engine oil Therefore, fluids containing suspended solid metallic particles were expected to display significantly enhanced thermal conductivities relative to conventional heat transfer fluids Numerous theoretical and experimental studies of the effective thermal conductivity of dispersions containing particles have been conducted since Maxwell’s theoretical work on the subject was published more than 100 years ago [14, 15] However, early studies of the thermal conductivity of suspensions have been confined to those containing particles with sizes of the order of millimeters or micrometers In fact, conventional micrometersized particles cannot be used in practical heat-transfer equipment because of severe clogging and sedimentation problems In addition, recent miniaturization, leading to the increasing practical utilization of microchannels and microreactors, also imposed a restriction on the use of micrometer-sized particles [14, 16–18] Modern nanotechnology provides great opportunities to process and produce materials with average sizes below 50 nm [14, 16–18] Recognizing an opportunity to apply this emerging nanotechnology to established thermal energy engineering, 1125 1126 J N N QUARESMA ET AL Choi and coworkers [14, 16–18] proposed that nanometer-sized metallic particles be suspended in industrial heat transfer fluids, such as water or ethylene glycol, to produce a new class of engineered fluids referred to as nanofluids Experiments with nanofluids have indicated significant increases in thermal conductivity, as compared to base liquids without nanoparticles or with larger suspended particles [14, 16–23] Generally, the observed increase in thermal conductivity of nanofluids was substantially larger than that predicted with the available theory On the other hand, some studies reported that such increase in the thermal conductivity could not be detected with optical experimental methods [24, 25] In this context, such a fact resulted in a search for physical phenomena not accounted for in the theoretical predictions for suspensions of micrometer-sized or larger particles, which included Brownian motion, liquid layering, ballistic mechanisms, thermophoresis, aggregation of nanoparticles into clusters, etc [16–26] Due to previous experimental observations that non-Fourier effects are significant at small time and space scales [1–11], hyperbolic heat conduction models were naturally suggested to explain the heat transfer enhancement in nanofluids [12, 13] It was proposed [12, 13] that enhanced heat transfer in nanofluids was caused by a heat transfer mechanism modeled in terms of the dual-phase-lagging constitutive equation [5, 7] In this article we revisit the works of references [12] and [13] and apply the dual-phase-lagging model to a one-dimensional heat conduction problem in cylindrical coordinates The geometry examined here is that typically used for the measurements of thermal conductivity of nanofluids with Blackwell’s heat source probe [27] Such a technique consists of a line heat source, usually taken in the form of a heating wire, which is placed inside the material with unknown properties For large times, the temperature variation of the heat source is shown to be linear with respect to the logarithm of time, so that the thermal conductivity can be computed from the slope of such linear variation The temperature variation of the heat source can be measured through the variation of the heating wire electrical conductivity Alternatively, the heating wire and a temperature sensor, such as a thermocouple or a PT-100, can be encapsulated in a metallic needle that is inserted into the medium with unknown thermal conductivity [27–31] Commercial probes are generally based on this last construction arrangement [31] The heat conduction problem under examination is solved analytically by using the classical integral transform technique (CITT) [32, 33] The controlled accuracy and analytical nature of the solution technique developed in this work allow for the computation of benchmark results for the temperature variation of the probe, based on the dual-phase-lagging model Numerical results are presented in this article for typical configurations of probes and nanofluids, as well as for different values of relaxation times, which result on hyperbolic effects with distinct intensities Such results are critically compared to the classical parabolic heat conduction model, as well as to Blackwell’s largetime solution heat transfer engineering PROBLEM FORMULATION The analysis considered here is similar to that examined in references [12] and [13] and involves a dual-phase-lagging model (DPLM) [5, 7] In such a model, a finite speed of heat propagation in the medium is taken into account through a delay time for the establishment of the heat flux, τq , and a lag between the heat flux vector and the temperature gradient, τT The constitutive equation relating the heat conduction flux vector and the temperature gradient in the dual-phase-lagging model is given by [1–7, 12, 13]: q + τq ∂(∇T f ) ∂q = −K ∇T f + τT ∂t ∂t (1) where q is the heat flux vector and K is the effective thermal conductivity of the medium The energy conservation equation for a purely conducting medium, considered in this work as a nanofluid, is written as (Cs + C f ) ∂Tf = −∇ · q ∂t (2) where Cs is the volumetric heat capacity of the nanoparticles and Cf is the volumetric heat capacity of the base fluid The substitution of Eq (1) into the energy conservation equation (2) results in: τq ∂2T f ∂(∇ T f ) ∂Tf + T + α τ = α ∇ f T ∂t ∂t ∂t (3) with the effective thermal diffusivity given by: α= K Cs + C f (4) Therefore, the use of the constitutive Eq (1) together with the energy conservation Eq (2) leads to a hyperbolic heat conduction model given by Eq (3) Equation (3) can be similarly obtained by considering that the nanoparticles and the base fluid are not in local thermal equilibrium In this case, the energy conservation equation can be written separately for the nanoparticles and the base fluid, respectively, in the following form: Cs ∂Tf ∂ Ts = h (T f − Ts ); C f = K ∇ T f − h (T f − Ts ) (5, 6) ∂t ∂t where h (W/m3oC) is a heat transfer coefficient between the fluid and the nanoparticles Note that it was considered a lumped formulation for the nanoparticles, given by Eq (5) Then, by substituting Ts from Eq (5) into Eq (6), one obtains: Cs C f ∂2T f ∂Tf + h(Cs + C f ) ∂t ∂t = Cs ∂(∇ T f ) K K ∇2T f + (Cs + C f ) (Cs + C f ) h ∂t vol 31 no 14 2010 (7) J N N QUARESMA ET AL A comparison of Eqs (3) and (7) reveals the definition of the effective thermal diffusivity given by Eq (4), as well of the relaxation times given by: τq = Cs C f ; h(Cs + C f ) τT = Cs h (8a, 8b) Therefore, Cs + C f τT = τq Cf (9) τT Cs =1+ >1 τq Cf (10) or, alternatively, 1127 θ(r,τ) = 0; − ∂θ(R,τ) = for ∂τ τ = 0, ∂ ∂θ(R,τ) θ(R,τ) + FoT =1 ∂R ∂τ in A < R < (13b, 13c) at R = A, ∂ ∂θ(R,τ) θ(R,τ) + FoT = at ∂R ∂τ for τ > (13d) R = 1, for τ > (13e) where it was considered that heat flux imposed by the probe is constant in time Considering the case involving a line heat source probe of The classical parabolic heat conduction model, which utiradius a immersed in a cylindrical medium of radius b, Eq (3) lizes Fourier’s law as the constitutive equation that relates the can be rewritten as heat conduction flux vector and the temperature gradient, can ∂ ∂ T (r, t) ∂ T (r, t) ∂ T (r, t) ∂ be directly obtained from Eqs (13a–e) by making the relax=α τq + r T (r, t) + τT , ation times, τq and τT , equal to zero (see Eq (1)) In terms of ∂t ∂t r ∂r ∂r ∂t the nonequilibrium model given by Eqs (5) and (6), τq → in a < r < b, for t > (11a) and τT → can be obtained with h → ∞ (see Eqs (8a) and (8b))—that is, the heat transfer coefficient between the fluid which is subjected to the following initial and boundary condiand the dispersed nanoparticles becomes very large and local tions: thermal equilibrium is attained (Tf = Ts ) ∂ T (r, t) T (r, t) = T0 ; = for t = 0, in a < r < b ∂t (11b, 11c) SOLUTION METHODOLOGY −K ∂ T (r, t) ∂q0 ∂ T (r, t) + τT = q0 + τq ∂r ∂t ∂t at r = a, K for t > (11d) ∂ T (r, t) ∂ T (r, t) + τT = at r = b, ∂r ∂t t >0 for (11e) where q0 is the heat flux resulting from the electrical resistance inside the probe and T is the initial temperature of the medium By defining the following dimensionless variables, θ(R,τ) = FoT = ατq r a αt T (r, t) − T0 ; R = ; A = ; τ = ; Foq = ; (q0 b/K ) b b b b ατT FoT ;β = b Foq (12a–f) the problem given by Eqs (11a–e) can be rewritten in dimensionless form as: Foq = ∂ θ(R,τ) ∂θ(R,τ) + ∂τ2 ∂τ ∂ R ∂R R θ(R,τ) = θav (τ) + θ p (R) + φ(R,τ) (14) where θav (τ) is the average temperature in the medium, which is a priori obtained from Eqs (13a)–(13e); θp (R) is a particular solution and φ(R,τ) is the potential to be solved with the CITT The average temperature θav (τ) is defined as θav (τ) = A Rθ(R,τ)d R A Rd R = − A2 Rθ(R,τ)d R (15) A Now, in order to determine a solution for θav (τ), Eq (13a) is multiplied by [2/(1−A2)]R and integrated over the domain [A,1] in the R-direction The definition given by Eq (15) is then employed and the boundary conditions (13d, 13e) are used to yield Foq ∂ ∂θ(R,τ) θ(R,τ) + FoT ∂R ∂τ in A < R < 1, τ > For the solution of the hyperbolic heat conduction problem given by Eqs (13a)–(13e) we apply the classical integral transform technique (CITT) [32, 33] A split-up procedure [32] is used in order to improve the convergence rate of the final series solution Hence, the solution for the temperature field is written as: d θav (τ) dθav (τ) 2A + = , dτ2 dτ − A2 for τ > (16a) From the initial conditions (13b) and (13c) together with the definition (15), it results that , (13a) θav (τ) = 0; heat transfer engineering vol 31 no 14 2010 dθav (τ) =0 dτ for τ = (16b, 16c) 1128 J N N QUARESMA ET AL Therefore, the solution for θav (τ) is obtained as θav (τ) = 2A τ − Foq − e−τ/Foq − A2 (19) Equation (14) is now introduced into Eqs (13a–e) and the problem for θav (τ) given by Eqs (16a–c) is used in order to obtain the following problems given by Eqs (17a–d) and Eqs (18a–f), for θp (R) and φ(R,τ), respectively: dθ p (R) 2A d R = , R dR dR − A2 − in A < R < (17a) dθ p (R) dθ p (R) = at R = A; = at R = 1(17b, 17c) dR dR The homogeneous problem for the potential φ(R,τ) is now solved with the classical integral transform technique (CITT) [32, 33] For this purpose, the following auxiliary eigenvalue problem is utilized, which shall provide the basis for the eigenfunction expansion of the potential φ(R,τ): d i (R) d R + βi2 R i (R) = 0, in A < R < (20a) dR dR d i (R) i (R) Rθ p (R)d R = = Jo (βi R)Y1 (βi ) − J1 (βi )Y0 (βi R) J1 (βi A)Y1 (βi ) − J1 (βi )Y1 (βi A) = 0, and ∂φ(R,τ) ∂ R φ(R,τ) + FoT ∂R ∂τ in A < R < 1, R i (R) , τ>0 ∂φ(R,τ) =0 ∂τ Ni = φ¯ i (τ) = (18e) (18f) R2 A − ln(R) − 4(1 − A2 )2 × [4A2 ln(A) + (3 + A2 )(1 − A2 )] (21c) (21d) R ˜ i (R)φ(R,τ)d R, transform (22a) A ∞ ˜ i (R)φ¯ i (τ), inverse (22b) i=1 The additional constraints given by Eqs (17d) and (18f) are obtained by substituting Eq (14) into the definition of the average temperature θav (τ) given by Eq (15) The integration of the problem given by Eqs (17a–d) can be readily performed in order to obtain the solution for the potential θp (R) in the form: A − A2 J12 (βi A) − J12 (βi ) π2 βi2 J12 (βi A) φ(R,τ) = A θ p (R) = 0, i = j Ni , i = j where ˜ i (R) = with Rφ(R,τ)d R = satisfy the The auxiliary eigenvalue problem given by Eqs (20a–c) allows the definition of the following integral transform–inverse pair for the potential φ(R,τ): ∂φ(R,τ) ∂ φ(R,τ) + FoT = at R = A, for τ > − ∂R ∂τ (18d) = i (R) where Ni is the normalization integral given by: in A < R < (18b, 18c) ∂ ∂φ(R,τ) φ(R,τ) + FoT = at R = 1, for τ > ∂R ∂τ j (R)d R A (18a) for τ = 0, (21a) i = 1, 2, 3, (21b) It can be shown that the eigenfunctions following orthogonality property [32, 33]: ∂ φ(R,τ) ∂φ(R,τ) Foq + ∂τ2 ∂τ φ(R,τ) = −θ p (R); i (R) (17d) A ∂ = R ∂R d = at R = (20b, 20c) dR dR Equations (20a–c) can be analytically solved to yield the eigenfunctions and the transcendental equation to compute the eigenvalues respectively as [32, 33]: with = at R = A; i (R)/ Ni After the definition of the integral transform-inverse pair with the auxiliary eigenvalue problem (20a–c), the next step in the CITT is thus to accomplish the integral transformation of the original partial differential system given by Eqs (18a–e) For this purpose, Eq (18a) and the initial conditions (18b) and (18c) are multiplied by R ˜ i (R), integrated over the domain [A,1] in the R-direction, and the inverse formula given by Eq (22b) is employed After the appropriate manipulations, the following system of ordinary differential results, for the calculation of the transformed potentials φ¯ i (τ): + FoT βi2 d φ¯ i (τ) β2 d φ¯ i (τ) + i φ¯ i (τ) = 0, + dτ Foq dτ Foq for τ > (23a) (19) heat transfer engineering (22c) vol 31 no 14 2010 J N N QUARESMA ET AL d φ¯ i (τ) =0 φ¯ i (τ) = ¯f i ; dτ for τ = (23b, c) × exp − (23d) ⎧⎡ ⎪ ⎪ ⎨⎢ × ⎢ ⎣1 + ⎪ ⎪ ⎩ where ¯f i = − R ˜ i (R)θ p (R)d R A for i = 1, 2, 3, The infinite system of ordinary differential equations (23a–c) is uncoupled and can be readily solved to yield: ⎡ ⎛ ⎞ ⎤ ¯f i 1+FoT βi2 4Fo β q i ⎝1 − 1− ⎠τ⎦ exp ⎣− φ¯ i (τ) = 2 2Foq 1+FoT β2 i ⎧⎡ ⎪ ⎪ ⎨⎢ × ⎢ ⎣1 + ⎪ ⎪ ⎩ ⎡ ⎤ 4Foq βi2 1− 1+FoT βi2 ⎡ + FoT βi2 × exp ⎣− Foq ⎤ ⎥ ⎢ ⎥ + ⎢1 − ⎦ ⎣ 1− 4Foq βi2 (1+FoT βi2 )2 ⎤⎫ ⎬ ⎦ 1− τ ⎭ + FoT βi2 4Foq βi2 ⎥ ⎥ ⎦ (24) φ(R,τ) = i=1 ¯f i ˜ i (R) ⎡ ⎛ ⎞ ⎤ 1+FoT βi2 4Foq βi2 ⎝1− 1− ⎠ τ⎦ × exp ⎣− 2Foq 1+FoT βi2 ⎧⎡ ⎪ ⎪ ⎨⎢ × ⎢ ⎣1 + ⎪ ⎪ ⎩ ⎡ × exp ⎣− ⎡ ⎤ 1− 4Foq βi2 (1+FoT βi2 )2 (1 + Foq FoT βi2 ) ⎤ ⎥ ⎢ ⎥ + ⎢1 − ⎦ ⎣ 1− 4Foq βi2 (1+FoT βi2 )2 ⎤⎫ ⎬ 4Foq βi 1− τ⎦ ⎭ + FoT βi2 ⎥ ⎥ ⎦ (25) A − A2 R A − ln(R) − 4(1 − A2 )2 ∞ × [4A ln(A)+(3 + A )(1−A )] + 2 i=1 ¯f i ˜ i (R) heat transfer engineering 1− ⎤ 1− 4Foq βi2 (1+FoT βi2 )2 1− 4Foq βi2 (1 + FoT βi2 )2 τ ⎡ ⎤ ⎥ ⎢ ⎥ + ⎢1 − ⎦ ⎣ 1− 4Foq βi2 (1+FoT βi2 )2 4Foq βi2 (1 + FoT βi2 ) 1− τ Foq (1 + FoT βi2 )2 ⎥ ⎥ ⎦ (26) The CITT is also applied in order to obtain the solution for the classical parabolic problem, based on Fouriers’s law In this case, the analytical solution is given by: θ(R,τ) = A 2A τ+ 1− A − A2 − R2 − ln(R) A [4A2 ln(A) + (3 + A2 )(1 − A2 )] 4(1 − A2 )2 + ¯f i ˜ i (R)e−βi2 τ (27) i=1 where the eigenquantities that appear in Eq (27) are the same as those defined earlier for the solution of the problem for the potential φ(R,τ) For large times, Blackwell [27] derived an asymptotic solution for the temperature variation at the surface of the line heat source probe in the parabolic problem, which is shown to be linear with respect to the logarithm of time Such solution is convenient for the measurement of the thermal conductivity of the medium surrounding the probe, which can be computed from the slope of the temperature variation [27–31] Blackwell’s solution [27], in terms of the dimensionless variables given by Eqs (12a), is θa (τ) = Finally, by substituting Eqs (16d), (19), and (25) into Eq (14), the solution for the dimensionless temperature field is obtained as 2A θ(R,τ) = [τ − Foq (1 − e−τ/Foq )] − A2 + × exp − + FoT βi2 2Foq ∞ By substituting Eq (24) into the inverse formula (22b), the analytical solution for the homogeneous potential φ(R,τ) is obtained as ∞ 1129 1 A ln τ + A ln 2 A2 −y (28) where y = 0.5772156649 is Euler’s constant We note in Eq (28) that, in dimensionless terms, the slope of θa (τ) x ln τ is equal to A/2 RESULTS AND DISCUSSION In this session we present numerical results for the dimensionless temperature variation at the surface of the probe, that is, at R = A, obtained with the hyperbolic heat conduction model given by Eq (26) Such results are compared to those obtained with the classical parabolic problem given by Eq (27), as well as to those obtained with Blackwell’s model given by Eq (28) Such analytical solutions were implemented under the Visual vol 31 no 14 2010 1130 J N N QUARESMA ET AL Table Test cases examined 0.055 Dimensions Test case Alumina in water Alumina in water Copper in ethylene glycol Copper in ethylene glycol a (m) b (m) A β 0.045 × 10−5 7.5 × 10−4 × 10−5 7.5 × 10−4 0.025 0.05 0.025 0.05 × 10−3 1.5 × 10−2 × 10−3 1.5 × 10−2 1.71 1.71 2.29 2.29 0.04 Fortran platform Different test cases were examined in this work, involving different configurations of probe diameters and nanofluids With respect to the nanofluids, the following ones were considered in the analysis: (i) alumina nanoparticles in water (K = 0.257 W/mK, Cs = 3.430 × 106 J/m3K, Cf = 2.649 × 106 J/m3-K, β = 1.71) and (ii) copper nanoparticles in ethylene glycol (K = 0.627 W/m-K, Cs = 2.964 × 106 J/m3-K, Cf = 4.183 × 106 J/m3-K, β = 2.29) With respect to the probe geometry, it was considered to be made of a thin resistance wire with diameter a = × 10−5 m inserted into a medium with outer diameter b = 0.025 m, so that A = × 10−3, such as in [12] and [13] Also examined was another probe with diameter a = 7.5 × 10−4 m, typical of those commercially available [31] In this case, the medium was considered with an outer diameter b = 0.05 m, so that A = 1.5 × 10−2 Table summarizes the test cases examined Figure illustrates the convergence behavior of the temperature at the probe surface, obtained with different truncation orders (NT) for the series solution in Eq (26), for A = × 10−3, Foq = × 10−5, and β = 2.29 (test case c) This figure shows that for small dimensionless times (τ ≤ 10−5), convergence at Analytical - Fo q = 1x10-3 FDM-Gear - Fo q = 1x10-3 Analytical - Fo q = 1x10-4 0.03 FDM-Gear - Fo q = 1x10-4 Analytical - Fo q = 1x10-5 0.025 FDM-Gear - Fo q = 1x10-5 0.02 Analytical - Fo q = 1x10-6 FDM-Gear - Fo q = 1x10-6 0.015 Analytical - Fo q = 1x10-10 0.01 FDM-Gear - Fo q = 1x10-10 0.005 1x10 -20 1x10 -15 1x10 -10 1x10 -5 1x10 τ Figure Comparison of analytical solution and finite-difference method solution using Gear’s method (referred to as FDM-Gear) for A = 1.5 × 10−2 and β = 2.29 the graphic scale is obtained with 5000 ≤ NT ≤ 10000 On the other hand, for larger dimensionless times the convergence is reached with approximately 500 terms in the series solution The computation time in a Pentium Intel Dual E2160 1.8-GHz computer was of the order of 9.7 minutes, for NT = 10000 For the results presented next, NT = 10000 was used In order to validate the analytical solution just presented, we compared its results with those obtained numerically with finite 0.011 0.011 Hyperbolic Heat Conduction Dual-Phase-Lagging Model - Fo q = 1x10-5 0.01 0.009 0.01 A = 2x10-3; β = Fo T/Fo q = 2.29 0.008 0.009 Analytical - NT = 100 Analytical - NT = 500 Analytical - NT = 1000 Analytical - NT = 5000 Analytical - NT = 10000 0.006 Hyperbolic Heat Conduction Dual-Phase-Lagging Model A = 2x10-3; β = Fo T/Fo q = 2.29 0.008 Parabolic Model Hyperbolic Model - Fo q = 1x10-10 0.007 θ(A,τ) 0.007 θ(A,τ) A = 1.5x10-2; β = Fo T/Fo q = 2.29 0.035 θ(A,τ) a b c d Nanofluid Hyperbolic Heat Conduction Dual-Phase-Lagging Model 0.05 0.005 0.006 0.005 0.004 0.004 0.003 0.003 0.002 0.002 0.001 0.001 0 1x10 -20 1x10 -15 1x10 -10 1x10 -5 1x10 τ 1.0x10-20 1.0x10-15 1.0x10-10 1.0x10-5 1.0x100 τ Figure Convergence behavior of the analytical solution for A = × 10−3, Foq = × 10−5, and β = 2.29 heat transfer engineering Figure Comparison of the hyperbolic (Foq = 10−10) and parabolic solutions for A = × 10−3 and β = 2.29 vol 31 no 14 2010 J N N QUARESMA ET AL 0.055 0.011 0.01 Hyperbolic Heat Conduction Dual-Phase-Lagging Model 0.05 A = 2x10-3; β = Fo T/Fo q = 1.71 0.009 0.045 Fo q = 1x10-3 0.008 Hyperbolic Heat Conduction Dual-Phase-Lagging Model A = 1.5x10-2; β = Fo T/Fo q = 1.71 Fo q = 1x10-3 0.04 -4 Fo q = 1x10-4 Fo q = 1x10 -5 0.007 0.035 Fo q = 1x10 Fo q = 1x10-6 0.006 θ(A,τ) θ(A,τ) 1131 Fo q = 1x10-10 0.005 Fo q = 1x10-5 Fo q = 1x10-6 0.03 Fo q = 1x10-10 0.025 0.004 0.02 0.003 0.015 0.002 0.01 0.005 0.001 (a) 1x10-20 1x10-15 1x10-10 1x10-5 (b) 1x100 1x10-20 1x10-15 τ 1x100 0.055 Hyperbolic Heat Conduction Dual-Phase-Lagging Model 0.05 A = 2x10-3; β = Fo T/Fo q = 2.29 0.009 0.045 Fo q = 1x10-3 0.008 Hyperbolic Heat Conduction Dual-Phase-Lagging Model A = 1.5x10-2; β = Fo T/Fo q = 2.29 Fo q = 1x10-3 0.04 Fo q = 1x10-4 Fo q = 1x10-4 Fo q = 1x10-5 0.007 0.035 Fo q = 1x10-5 -6 Fo q = 1x10 0.006 θ(A,τ) θ(A,τ) 1x10-5 τ 0.011 0.01 1x10-10 -10 Fo q = 1x10 0.005 0.02 0.003 0.015 0.002 0.01 0.001 0.005 (c) 1x10-20 1x10-15 1x10-10 1x10-5 1x100 Fo q = 1x10-10 0.025 0.004 Fo q = 1x10-6 0.03 (d) 1x10-20 1x10-15 τ 1x10-10 1x10-5 1x100 τ Figure Temperature variation at the probe surface for test cases: (a) A = × 10−3 and β = 1.71; (b) A = 1.5 × 10−2 and β = 1.71; (c) A = × 10−3 and β = 2.29; (d) A = 1.5 × 10−2 and β = 2.29 differences In this case, the problem given by Eqs (13a)–(13e) was discretized in the radial direction with second-order differences and the resulting system of ordinary differential equations was integrated in time with Gear’s method Figure shows a comparison of the results obtained with the analytical solution against those obtained with the finite-difference method solution (referred to in Figure by FDM-Gear) for test case d (A = 1.5 × 10−2, β = 2.29) and different values of Foq The agreement between such solutions is excellent, thus validating the numerical code here developed The finite-difference solution was obtained with 2000 nodes in the spatial grid heat transfer engineering Figure presents a comparison of the hyperbolic and parabolic solutions given by Eqs (26) and (27), respectively, for test case c (A = × 10−3 and β = 2.29) and Foq = × 10−10 We note in this figure that for such a small value of Foq the hyperbolic model behaves exactly as the parabolic one, that is, non-Fourier effects are negligible Such was also the case for other values of A and β examined in this article We now examine the non-Fourier effects of the probe surface temperature variation, for Foq ranging from 10−3 to 10−10 The results obtained for the different test cases described in Table are presented in Figure 4, a–d These figures show vol 31 no 14 2010 1132 J N N QUARESMA ET AL 0.013 0.065 0.012 Hyperbolic Heat Conduction Dual-Phase-Lagging Model 0.011 A = 2x10-3; β = Fo T/Fo q = 1.71 0.01 0.009 Blackwell's Solution Fo q = 1x10-3 0.008 Fo q = 1x10-4 Hyperbolic Heat Conduction Dual-Phase-Lagging Model 0.06 A = 1.5x10-2; β = Fo T/Fo q = 1.71 0.055 0.05 Blackwell's Solution Fo q = 1x10-3 0.045 Fo q = 1x10-4 0.04 -5 Fo q = 1x10-5 θ(A,τ) θ(A,τ) Fo q = 1x10 0.007 -6 Fo q = 1x10 -10 Fo q = 1x10 0.006 0.035 Fo q = 1x10-6 0.03 Fo q = 1x10-10 0.005 0.025 0.004 0.02 0.003 0.015 0.002 0.01 0.005 0.001 (a) -20 -18 -16 -14 -12 -10 -8 -6 -4 -2 (b) 0 -20 -18 -16 -14 -12 ln(τ) 0.013 -8 -6 -4 -2 0.065 0.012 Hyperbolic Heat Conduction Dual-Phase-Lagging Model 0.011 A = 2x10-3; β = Fo T/Fo q = 2.29 0.01 0.009 Blackwell's Solution Fo q = 1x10-3 0.008 Fo q = 1x10-4 Hyperbolic Heat Conduction Dual-Phase-Lagging Model 0.06 A = 1.5x10-2; β = Fo T/Fo q = 2.29 0.055 0.05 Blackwell's Solution Fo q = 1x10-3 0.045 Fo q = 1x10-4 0.04 -5 Fo q = 1x10-5 0.007 θ(A,τ) Fo q = 1x10 θ(A,τ) -10 ln(τ) -6 Fo q = 1x10 -10 Fo q = 1x10 0.006 0.035 Fo q = 1x10-6 0.03 Fo q = 1x10-10 0.005 0.025 0.004 0.02 0.003 0.015 0.002 0.01 0.005 0.001 (c) -20 -18 -16 -14 -12 -10 -8 -6 -4 -2 (d) 0 -20 -18 -16 ln(τ) -14 -12 -10 -8 -6 -4 -2 ln(τ) Figure Comparison of hyperbolic and Blackwell’s [27] solutions for test cases: (a) A = × 10−3 and β = 1.71; (b) A = 1.5 × 10−2 and β = 1.71; (c) A = × 10−3 and β = 2.29; (d) A = 1.5 × 10−2 and β = 2.29 that non-Fourier effects are only noticeable for very small times; as time increases, the temperature variations gradually tend to the parabolic one In fact, even for an extremely large value of Foq such as 10−3, the non-Fourier effects vanish for τ > 10−2 At small times, non-Fourier effects are more pronounced for smaller diameters A On the other hand, the choice of the nanofluid does not seem to affect significantly the temperature behavior Similar conclusions can be obtained by comparing the hyperbolic solution with the asymptotic one deheat transfer engineering veloped by Blackwell for the parabolic formulation, as depicted in Figure 5a–d The results presented in Figures 4a–d and 5a–d permit to examine the suitability of the hyperbolic formulation to the actual heat conduction problem in nanofluids, during thermal conductivity measurements with the line heat source probe For this analysis, we bring into picture the heat transfer coefficient between the base fluid and the particles considered in the thermal nonequilibrium model given by Eqs (5) and (6), which results vol 31 no 14 2010 J N N QUARESMA ET AL Table Dimensional times corresponding to τ = 10−4 a–d, that Blackwell’s solution would not be considered appropriate for the measurement of the thermal conductivity for τ < 1.2 × 10−4 (ln τ = –9) for test cases a and c, and for τ < 2.5 × 10−3 (ln τ = –6) for test cases b and d In other words, the increase generally detected with the line heat source probe for the thermal conductivity of nanofluids cannot be attributed to the non-Fourier heat transfer mechanisms examined earlier In fact, recent theoretical predictions corroborate our findings and demonstrate that nanoparticles and the base fluid are in thermal equilibrium in nanofluids [23, 34] Time τ t (s) 10−4 10−4 10−4 10−4 0.7 2.9 1.5 5.9 Test case a b c d 1133 in the hyperbolic formulation addressed in this article Note in these equations that such heat transfer coefficient is defined in volumetric terms, but can be easily converted to the usual definition of the heat transfer coefficient by using the nanoparticle’s volume to surface area ratio Figure presents the heat transfer coefficient between the base fluid and the particles for different values of Foq , and for spherical particles of different diameters Only test cases a and c are examined in this figure, since they present more significant non-Fourier effects (see also Figures 4a–d and 5a–d) By considering a threshold value for the heat transfer coefficient, it is possible to establish the maximum expected value of Foq for which the system behaves hyperbolically If we assume such a threshold value as W/m2-K, which is indeed extremely small in macroscopic means, we notice in Figure that Foq is actually smaller than 10−5 Figure 4a–d, shows that for Foq = 10−5, non-Fourier effects are negligible for τ > 10−4 Table gives the physical times equivalent to τ = 10−4 for the four test cases addressed in this work Notice in this table that non-Fourier effects would have disappeared for times much smaller than those typically considered for the measurement of the thermal conductivity with Blackwell’s solution for the line heat source probe [27–31] Indeed, notice in Figure 5, CONCLUSIONS In this article we presented an analytical solution based on the classical integral transform technique for the dual-phase-lagging heat conduction model The physical problem examined was representative of that used for the measurement of thermal conductivity with the line heat source probe Results were obtained for the temperature variation at the probe surface, for different combinations of nanofluids and probe diameters Such results were compared to those obtained with the classical parabolic heat conduction model based on Fourier’s law, as well as to the asymptotic solution proposed by Blackwell [27] for the line heat source probe The foregoing analysis reveals that non-Fourier effects are significant only for very small times, generally in the range where Blackwell’s solution is not valid for the measurement of thermal conductivity Therefore, the increase detected with the line heat source probe for the thermal conductivity of nanofluids cannot be attributed to the non-Fourier heat transfer mechanisms addressed in this article NOMENCLATURE 10 Heat Transfer Coefficient (W/m 2K) 103 102 10 a A b Cf Cs ¯f i Foq Test-case a, nm Test-case a, 10 nm Test-case a, 100 nm Test-case c, nm Test-case c, 10 nm Test-case c, 100 nm 104 100 FoT 10-1 10-2 10-3 10-4 10-5 1x10-10 1x10-9 1x10-8 1x10-7 1x10-6 1x10-5 1x10-4 1x10-3 Fo q Figure Heat transfer coefficient for different nanoparticle diameters heat transfer engineering h K Ni NT q q0 r R t T probe radius dimensionless probe radius radius of the cylindrical medium volumetric thermal capacity of the base fluid volumetric thermal capacity of the nanoparticles transformed initial condition dimensionless relaxation time associated with the heat flux dimensionless relaxation time associated with the temperature gradient heat transfer coefficient effective thermal conductivity of the nanofluid normalization integral truncation order in the summations heat flux vector heat flux at the surface of the probe radial variable dimensionless radial variable time variable temperature vol 31 no 14 2010 Heat Transfer Engineering, 31(14):1203–1212, 2010 Copyright C Taylor and Francis Group, LLC ISSN: 0145-7632 print / 1521-0537 online DOI: 10.1080/01457631003732995 A Numerical Study of Entropy Generation in the Entrance Region of Curved Pipes E AMANI and M R H NOBARI Mechanical Engineering Department, Amirkabir University of Technology, Tehran, Iran In this study, developing incompressible viscous flow and heat transfer in curved pipes are studied numerically considering a constant heat flux at the wall to analyze the entropy generation The governing equations including continuity, momentum and energy equations are solved using a second order finite difference method based on the projection algorithm Entropy generation and thermodynamic optimization are investigated through the entrance region of the curved pipes with circular cross section by a general non-dimensional analysis both numerically and analytically Optimal Reynolds number calculation based on the entropy generation minimization are carried out for two cases considering the two groups of non-dimensional parameters The comparison of the numerical results in the entrance region with the analytical ones in the fully developed region indicates that both solutions predict nearly the same optimal Reynolds numbers INTRODUCTION Curved pipes are used for heat transfer enhancement in many industrial applications such as heat exchangers, cooling of gas turbine blades, electrical motors, and so on Detailed information about the flow and heat transfer parameters in circular curved and helical pipes can be found in various studies including Soh and Berger [1] and Nobari and Amani [2] In these pipes, centrifugal forces cause a steep axial velocity gradient near the outer wall with the formation of secondary flow as one or several pairs of counterrotating vortices, depending on the Reynolds number These effects increase both heat transfer rate and pressure drop in the curved pipes, requiring an optimal design by studying the governing parameters involved in the flow characteristics of the curved pipes This can be done by the entropy generation minimization (EGM) method [3] based on the second law of thermodynamics Although the EGM method has been applied in many heat transfer enhancement problems, its usage in designing curved and helical pipes is a relatively new issue In a series of analytical studies, Ko and Ting [4, 5] and Ko [6, 7] have reported different optimal parameters for the helical circular pipes under the fully Address correspondence to M R H Nobari, Associate Professor, Mechanical Engineering Department, Amirkabir University of Technology, 424 Hafez Avenue, PO Box 15875-4413, Tehran, Iran E-mail: mrnobari@cic.aut.ac.ir developed flow assumption with some correlations proposed for the optimal Reynolds numbers, curvature ratios, and mass flow rates Among several numerical studies carried out on the entrance flow, the work by Ko and Ting [8] has investigated the entropy generation to perform optimal Dean number calculation in the rectangular curved ducts Also, Ko [9] has used a longitudinal rib in the curved rectangular ducts to determine its optimum height Best location and optimum number of ribs in these pipes have been investigated in another work [10] For the first time, entropy generation and thermodynamic optimization were investigated by Ko [11] in the entrance region of helical pipes with circular cross section, considering water as a working fluid Here, a three-dimensional developing incompressible viscous flow in curved pipes with a constant heat flux at the wall is studied numerically Governing equations including continuity, full Navier–Stokes, and energy equations are solved by a second-order finite-difference method in a toroidal coordinate system based on the projection algorithm Local entropy generation rates due to friction and heat transfer are studied in detail by employing a thermodynamic optimization using two different groups of nondimensional parameters Accordingly, the entropy generation rates and the optimal Reynolds numbers determined numerically in the entrance region are compared with the analytical prediction in the fully developed region 1203 1204 E AMANI AND M R H NOBARI GOVERNING EQUATIONS + Considering incompressible viscous flow in a curved pipe with constant properties and negligible body forces, the following nondimensional variables and parameters can be defined Nondimensional variables: r= r , a V = τa τ = , µW0 V , W0 p= p ρW02 T − T0 T = ∗ T − T0 (1) Ec = µcp Pr = k ρaW0 Re = , µ W02 , cp (T ∗ − T0 ) Cf = τw /wm , Re Nu = ∂ δ2 r ∂v ∂θ B ∂θ ∂u −v r ∂φ + ∂ B ∂v ∂φ r ∂φ − ∂w δu sin φ δ2 sin φ v sin φ − u cos φ − − rB B ∂θ + + δrw(u cos φ − v sin φ) = − + qw a k (6) δ ∂p + B ∂θ Re (2) where primed quantities are dimensional, a is the pipe radius, V the velocity, R the pipe curvature, W0 and T0 the dimensional axial velocity and temperature at r = and θ = 0, µ the dynamic viscosity, ρ the density, P the pressure, cp the specific heat at constant pressure, k the thermal conductivity, τw the r=1 φ=π nondimensional stress at the walls, wm (= r=0 φ=0 wr dr) the nondimensional mean axial velocity, and T ∗ is a characteristic temperature that is defined based on the thermal boundary conditions at the wall Also, δ denotes the nondimensional curvature, Re the Reynolds number, P r the Prandtl number, Ec the Eckert number, Cf the friction coefficient, N u the Nusselt number, and De the Dean number For a constant heat flux at the wall (qw ), the T ∗ is expressed as T ∗ = T0 + ∂ ∂p ∂v + rB ∂φ Re rB ∂r ∂r ∂w ∂ ∂ ∂ + (rBuw) + (Bvw) + (δrw ) ∂t rB ∂r ∂φ ∂θ k √ De = 2wm Re 2δ (5) ∂ ∂ ∂v ∂ + (rBuv) + (Bv ) + (δrvw) + Buv ∂t rB ∂r ∂φ ∂θ + δrw sin φ = − Nondimensional parameters: a δ= , R ∂w δv sin φ δ2 cos φ v sin φ − u cos φ − + rB B2 ∂θ × + ∂ ∂w rB rB ∂r ∂r + ∂ B ∂w ∂φ r ∂φ + ∂ δ2 r ∂w ∂θ B ∂θ ∂v w 2δ2 ∂u cos φ − sin φ − B ∂θ ∂θ (7) ∂T ∂ ∂ ∂ + (rBuT ) + (BvT ) + (δrwT ) ∂t rB ∂r ∂φ ∂θ = ∂ ∂T rB ReP r rB ∂r ∂r + ∂ δ2 r ∂T ∂θ B ∂θ + Ec + ∂ B ∂T ∂φ r ∂φ ∂p ∂ + (rBup) ∂t rB ∂r (3) ∂ Ec ∂ (Bvp) + (δrwp) + φ (8) + The governing equations including continuity, full Navier– ∂φ ∂θ Re Stokes, and energy equations in a toroidal coordinates system (r, φ, θ) can be written in the nondimensional forms as [2, 12]: where B = + δr cos φ and u, v, and w are the velocity components in r, φ, and θ directions, respectively As is shown∂ ∂ ∂ later in the Results section, the last two terms on the right(rBu) + (Bv) + (δrw) = (4) hand side of the energy equation are negligible The geome∂r ∂φ ∂θ try of the curved pipe in the toroidal coordinates is shown in Figure 1 ∂ ∂ ∂u ∂ + (rBu2 ) + (Buv) + (δruw) − Bv Local entropy generation rate [13] in the nondimensional ∂t rB ∂r ∂φ ∂θ form is defined as ∂p ∂ ∂u a Sgen + rB − δrw cos φ = − = SP + ST Sgen = ∂r Re rB ∂r ∂r k + ∂ B ∂u ∂φ r ∂φ + ∂ δ2 r ∂u ∂θ B ∂θ − ∂v +u r ∂φ heat transfer engineering SP = P rEc φ (T + T ∗∗ ) vol 31 no 14 2010 E AMANI AND M R H NOBARI z 1205 Plane : θ = θ A Plane : θ = θ Φ = Φ1 r Φ=0 θ x a R B B Φ = Φ1 A Outer wall Inner wall (90 ο< Φ < 180ο) (0 ο< Φ < 90ο) Figure (Left) Curved pipe geometry in toroidal coordinate system; (right) different regions of a cross section: Left semicircle is the inner wall and right one is the outer wall These conventions are also used in other figures The angles φ and θ are in degrees in all figures ST = = Figure Grid topology ∇ 2T (T + T ∗∗ )2 (T + T ∗∗ )2 ∂T ∂r + ∂T r ∂φ + δ ∂T B ∂θ to ST [13] By integrating the nondimensional local entropy generation rate over the pipe between θ = and θ = θ1 , the nondimensional total entropy generation rate can be determined as (9) where SP and ST are the local entropy generation rates due to friction and heat transfer, respectively The non-dimensional parameter T ∗∗ and the viscous dissipation φ are T ∗∗ = φ= T∗ T0 − T0 (10) 1 2 2 2 + τθθ ) + τrφ + τrθ + τφθ τ :τ = (τ + τφφ 2 rr (11) φ=π r=1 Si,t (θ1 ) = θ=θ1 Si rdrB d φ r=0 φ=0 θ=0 i = P , T , gen dθ , δ (13) where θ1 is an arbitrary reference angle Note that the fully developed condition occurs at different axial angles (θf d ) for different flow conditions Therefore, θ1 is not selected to be θ1 = θf d , but is selected to have the same value of 360◦ for all cases in our study because the total entropy generation [Eq (13)] will be compared between the cases that have the same heat transfer area; i.e., for constant δ, it means constant θ1 where τrr = ∂u ∂r NUMERICAL METHOD 2δ ∂w + u cos φ − v sin φ B ∂θ An O-type structured orthogonal staggered grid (Figure 2) is generated to solve the governing equations in the toroidal coordinate system employing a second-order finite-difference discretization based on the projection algorithm [14, 15] Briefly, the projection algorithm may be represented by the following three steps: τrφ = τφr = ∂u v ∂v + − ∂r r ∂φ r V∗ −Vn n + [(V ∇)V ]n = ∇ V δt Re τφθ = τθφ = δ ∂v δw sin φ ∂w + + r ∂φ B ∂θ B τrθ = τθr = δ ∂u ∂w δw cos φ + − B ∂θ ∂r B τφφ ∂v = +u r ∂φ τθθ = ∇ pn+1 = (12) Although in many flows the viscous dissipation term in the energy equation is negligible, it can be shown by an order-ofmagnitude analysis that in general SP is not negligible compared heat transfer engineering ∇.V ∗ δt V n+1 − V ∗ + ∇p n+1 = δt (14) (15) (16) In the first step [Eq (14)], the provisional velocity (V ∗ ) is obtained explicitly from the velocity at the previous time step Then, the Poisson equation [(Eq (15)] is solved for the pressure vol 31 no 14 2010 E AMANI AND M R H NOBARI via an iterative method Finally, the velocity at the new time step, V n+1 , is determined by Eq (16) Here, the fully developed velocity profile of the straight pipe, is used as the inlet velocity condition in the curved pipe, but for the temperature, uniform inlet profile is considered, i.e., u(r, φ, 0) = v(r, φ, 0) = 0, w(r, φ, 0) = − r , T (r, φ, 0) = (17) Therefore, the average axial velocity, wm , is 0.5 Other boundary conditions are the no-slip condition for the velocity and the constant heat flux for the temperature at the wall, fully developed conditions at the outlet and symmetry boundary at φ = and π Note that there is a singular point at the center of the O-type grid system (r = 0) To treat the boundary condition for u at this point (in a staggered grid only u nodes are located at r = 0), we simply equate the value of this node to the average value of the surrounding nodes in each cross section, i.e., u(r = 0, θ) = m m < φi < π u(δr, φi , θ), i=1 where m is the number of nodes in the φ direction However, other approaches also can be applied to remove the singularity [1] In our simulations, pipe lengths are selected to be different depending on the flow parameters Note that pipe length should be chosen long enough to ensure fully developed conditions at the outlet This work is done through a trial-and-error approach in which streamwise gradients of all the (nondimensional) velocity components and temperature are checked to be less than 0.01 For example as shown later, for the case of Figures 5, 6, and 7, the outlet angle of the computational domain is chosen to be θout = 660◦ , that is, larger than fully developed angle for this case (θf d = 550◦ ) Also, θf d has been reported in references [1, 2] As will be observed in the range of our simulations, optimal conditions arise in the cases at which both irreversibilities due to friction and heat transfer are of the same order To reach these conditions, the Dean number must be higher, requiring a finer grid resolution and smaller time steps In order to accelerate the convergence rate in these cases, a multigrid method is implemented to solve the Poisson equation for the pressure field obtained in the projection algorithm For this reason, a V-cycle model of the multigrid method with three grids is taken into account Data transfer from the finer to the coarser grid is done by the injection method [16], and a linear interpolation is used for the reverse process The flowchart of the present implemented multigrid algorithm is shown in Figure 3, where A indicates the matrix coefficient, p the unknown vector, and B the constant vector of the system of linear equations derived from the Poisson equation descrtization The solutions of all algebraic systems in each stage of the multigrid algorithm are improved by five iterations of the SOR (successive over-relaxation) method with the relaxation factor of 0.5 Depending on the Dean and Reynolds numbers, the resolutions considered here vary from 30 × 38 to 52 × 52 in the r × φ directions and 23 to 84 in the θ direction For this mesh size, (Cf,m)c / (Cf,m)s 1206 Semi-empirical [17,18] Semi-empirical [19] Computed ( δ < 1/16) Computed ( δ > 1/16) 1 10 10 De 10 10 (a) h h h iterations h n5 A p =B h pb=0 ph0=ph1 h p iterations h p if error< THEN p(t+ ∆t)=p , STOP else rh=Bh- Ahph5 h h r =B - A p Semi-empirical [17,18] Semi-empirical [19] Computed ( δ< 1/16) Computed ( δ> 1/16) h B2h=I2hhrh h h h 2h P =P +I P 2h 2h 2h A p =B 2h p b=0 2h p 0=0 2h 2h n2 iterations 2h p 2h 2h r =B - A p 4h B =I 4h 2h 2h r 2h 2h 2h n4 A p =B 2h p b=0 2h 2h p 0=p 2h 2h p 2=p 2+I 2h iterations 2h p4 2h 4h 4h p n3 A4hp4h=B4h 4h 4h p3 p b=0 iterations 4h p 0=0 (Num)c / (Num)s ∆ h n1 ∆ Ahph=Bh phb=0 h h p0= p (t) Pr = 0.711 1 10 Figure V-cycle multigrid algorithm to solve the system of linear equations [A]{p} = {B} derived from the Poisson equation discretization Here, superscript ih refers to grid resolution and subscripts b and refer to boundary jh and initial values, respectively Aih p ih = [∇ p]ih , B h = [ ∇.δtV ]h , and Iih is the transformation operator [16] (restriction or interpolation operator) between grids heat transfer engineering 10 De (b) Figure Comparison of computed values with semi-empirical correlations of Ishigaki [17, 18] and Manlapaz and Churchill [19]: (a) Average friction factor ratio [(Cf,m )c /(Cf,m )s ] and (b) average Nusselt number ratio [(N um )c /(N um )s ], in fully developed cross section versus Dean number (De) vol 31 no 14 2010 E AMANI AND M R H NOBARI 0.15 0.3 θ = 4ο θ = 12 0 0 0.1 0.2 θ = 30 ο ο 0 0 0.05 to satisfy the stability conditions, time steps are at the order of 10−3 to 10−4 Although the multigrid method is applied here (for high Dean number cases), it takes about days to reach the steady-state solution on a PC with Core II Duo CPU of 1.86 GHz and GB memory The validation of the developed code was performed in a previous paper [2] However, to show the accuracy of the code, the values of the average friction factor ratio [(Cf,m )c /(Cf,m )s ] and the Nusselt number ratio [(N um )c /(N um )s ] are compared with the two semi-empirical correlations proposed by Ishigaki [17, 18] and Manlapaz and Churchill [19] in Figure for the fully developed region As is evident, a very good agreement is obtained It is expected that for the curved pipes, the correlation of Ishigaki [17, 18] should be more accurate than the correlation of Manlapaz and Churchill [19] that is obtained for the general case of the helical pipes 1207 05 θ = 60 ο RESULTS AND DISCUSSION Here, developing incompressible viscous flow in the curved pipes is taken into account to investigate the local entropy generation rates due to dissipation and heat transfer at the constant heat flux at the wall Simulations are carried out for the Dean numbers in the range of 129–1508, the Reynolds numbers of 0 0.2 0 05 θ=4 θ = 12 0 0.4 0.3 0.9 0.2 0.2 ο 0.4 ο 0.05 0.2 0.15 05 0.6 θ = 90 ο 0.35 ο 242–3000, the four different curvature ratios of δ = 13 , 17 , 20 , and , and the Prandtl number of 0.711 30 For the case with De = 791, δ = 20 , and Re = 2500, contours of nondimensional axial velocity and temperature at different cross sections of the curved pipe are shown in Figure As the flow proceeds downstream, the location of the maximum axial velocity and the minimum temperature moves from the center of the curved pipe toward the outer wall, where severe velocity gradients appear Temperature gradient is more severe near the inner wall for constant wall heat flux, as is shown more clearly in the next figures In Figure 6, the corresponding secondary flow field and the secondary kinetic energy coefficient are shown The secondary kinetic energy coefficient, Cske , is defined as Cske = 0.6 0.4 ο 0.4 θ = 60 θ = 30 ο 0.15 θ = 360 0 0.15 θ = 90 ο 0.05 Figure Secondary flow information Streamlines (upper half) and Cske contours (lower half), at different sections (θ is in degrees) for De = 791 (δ = 20 , Re = 2500) 0.15 0.05 0.0 θ = 360 ο Figure Contours of nondimensional axial velocity (upper half) and tem1 perature (lower half) at different sections (θ in degrees) for De = 791 (δ = 20 , Re = 2500) and P r = 0.711 heat transfer engineering u2 + v /wm (18) As the flow developes in the entrance region, the larger Cske shifts from the core region toward the wall due to the secondary flow boundary layer generation at the walls Strong secondary flows and the axial flow gradients on the wall increase the heat transfer rate and pressure drop For the large Dean numbers, similar to the aforementioned case, the development history along the entrance region indicate oscillations in the Cske This oscillatory behavior can be related to the formation of unstable vortices that weaken the secondary flow strength for a while vol 31 no 14 2010 1208 E AMANI AND M R H NOBARI S P*10 2: 0.3 0.6 0.9 1.2 1.5 S T*10 2: 0.6 1.2 1.8 2.4 9 0 θ = 60 θ=4 ο θ = 12 ο θ = 30 ο θ = 60 θ = 90 ο θ = 360 ο θ = 360 ο Figure Contours of radial gradients of axial velocity (upper half) and tem1 perature (lower half) at two sections (θ in degrees) for De = 791 (δ = 20 , Re = 2500) and P r = 0.711 ο ο 0.95 respectively By developing the flow in the entrance region, SP increases on the outer wall where the axial velocity gradient is dominant (Figure 8) On the other hand, ST increases at the inner wall more than at the outer wall Note that constant wall heat flux boundary dictated that the radial gradient of the temperature be uniform overall at the walls But the region of high radial gradient of temperature is larger near the inner walls than the same region near the outer walls (Figure 8) As shown later in Figure 10, the entropy generation rate indicates an oscillatory trend in the entrance region It is observed that the viscous dissipation cannot be neglected in the entropy generation However, this term is negligible in the energy equation, as shown in Figure 9, where the circumferential average Nusselt numbers with and without considering the viscous dissipation are compared along the curved pipe Distinction between these two cases is extremely small even for large De and Ec numbers in the range of the present study 0.9 0.85 11 10 120 θ 240 360 Figure Contours of local entropy generation rates: Sp (upper half), ST (lower half), at different sections (θ in degrees) for De = 791 (δ = 20 , −3 ∗∗ Re = 2500), P r = 0.711, Ec = 0.879 × 10 , and T = The diagram ST ,m shows Sgen,m (Bejan number, Be) versus θ and vanish as the flow develops The oscillatory behavior is also observed in the other parameters such as the average friction factor and the Nusselt number [2] It should be mentioned that there are two pairs of stable(fully developed) Dean vortices in the case concerned (Figure 6, with θ = 360◦ ), comparing to the lower Dean number cases considered in the previous study [2] where only a pair of vortices retain in the fully developed region Figure indicates the contours of the local entropy generation rates at the values of Ec and T ∗∗ of 0.879 × 10−3 and 6, heat transfer engineering Num Be = S T, m /S gen, m 0.1 Case Case 200 400 θ 600 Figure Comparison of average Nusselt number (N um ) versus θ in degrees for Re = 3000, δ = 20 , P r = 0.711, and Ec = 1.82 × 10−3 Case 1: neglecting last two terms on the right hand-side of energy equation [Eq.( 8)] Case 2: considering these terms vol 31 no 14 2010 E AMANI AND M R H NOBARI Re = 1500 Re = 2000 Re = 2500 Re = 3000 Computed: C1 Semi-empirical: A1 2.5 S P, m *10 S P, m *10 1209 2 1.5 1 0 240 θ 480 720 θ 200 400 600 (a) (a) 25 Computed: C1 Semi-empirical: A1 24 Re = 1500 Re = 2000 Re = 2500 Re = 3000 S T, m *10 21 S T, m *10 20 17 16 13 θ 200 12 240 θ 480 Figure 10 Entropy generation rate per unit length of pipe versus θ in degrees for δ = 20 , P r = 0.711, T ∗∗ = 6, η2 = 0.01 × 1020 , and different Re: (a) SP ,m and (b) ST ,m The entropy generation rate per unit length of the pipe in the nondimensional form can represented as dSi,t =2 dθ/δ r=1 φ=π Si rdrBdφ, r=0 600 (b) 720 (b) Si,m = 400 i = P , T , gen Figure 11 Comparison of entropy generation rate per unit length of pipe versus θ in degrees calculated by methods A1 and C1 for Re = 2500, δ = 20 , P r = 0.711, T ∗∗ = 6, and η2 = 0.01 × 1020 : (a) SP ,m and (b) ST ,m Eq (19) (method C1), the analytical relation [13] of the entropy generation in the fully developed region can be expressed in the nondimensional form for a circular pipe with a constant heat flux as follows: Sgen,m = SP ,m + ST ,m φ=0 (19) The entropy generation rate computed from the preceding equation is referred to as method C1 in our text The variations of this parameter along the axial direction of the curved pipe in the entrance region is represented in Figure 10 at the different Re numbers while keeping the other nondimensional parameters constant It is observed that SP ,m and ST ,m take their maximum values in the developing region Furthermore, as Re increases, SP ,m increases while ST ,m decreases, indicating the possibility of the existence of an optimum Reynolds number In addition to heat transfer engineering ST ,m = SP ,m = π wm EcP rRe Cf,m (Tm + T ∗∗ ) 2π N um (Tm + T ∗∗ )(Tw + T ∗∗ ) Tm = Tm,0 + 2θ , wm ReP rδ T w = Tm ± (20) N um (21) where Tm is the nondimensional average (bulk) temperature in each cross section In the preceding equation the minus sign is for cooling and the plus sign is for heating It should be emphasized that the constant properties at the bulk values [13] are taken into account to obtain Eq (20) The fully developed vol 31 no 14 2010 E AMANI AND M R H NOBARI Reopt = 2510 (A1) , 2400 (C1) 2000 S gen,t (C1) (A1) (C1) (A1) (C1) (A1) Re 2500 S 1500 0.15 1000 2000 3000 that depends on the working fluid properties, mass flow rate, and heat transfer rate per unit pipe length Assuming the last four parameters to be constant, the Ec number can be expressed as a function of Reynolds number: (C1) (A1) (C1) (A1) (C1) (A1) 2000 0.18 Re 0.6 1000 0.8 Figure 13 Comparison of Sgen,t in one cycle (0◦ ≤ θ ≤ 360◦ ) versus Re calculated by methods A1 and C1 for P r = 0.711, T ∗∗ = 12, η2 = 0.01×1020 , and different δ 0.8 0.2 0.24 0.6 3000 Reopt = 2320 (A1) , 2280 (C1) 0.4 0.3 0.27 0.7 (a) S P,t S P,t S T,t S T,t S gen,t S gen,t (C1) (A1) (C1) (A1) 0.21 Ec = 64wm 2500 3000 Re (b) Figure 12 Comparison of total entropy generation rates in one cycle (0◦ ≤ θ ≤ 360◦ ) versus Re calculated by the methods A1 and C1 for P r = 0.711, T ∗∗ = 6, and η2 = 0.01 × 1020 : (a) δ = 20 and (b) δ = 17 values for the average friction factor (Cf,m ) and the Nusselt number (Num ) can be substituted into Eq (20) The entropy generation rate computed as mentioned earlier is referred to as method A1 in our text This method was employed in the previous analytical studies [4–7] Here, the semi-empirical correlations proposed by Ishigaki [17, 18] are used for Cf,m and Num in method A1 Figure 11 shows the comparison of SP ,m and ST ,m along the pipe for the different methods explained before As can be deduced from Eqs (20) and (21) and the correlations of Cf,m and N um , the Sgen,m is a function of five nondimensional parameters consisting of Re, δ, P r, Ec, and T ∗∗ Thus, the optimum values of one parameter can be determined by holding the others as constant This has been carried out to obtain the optimum parameters by various studies [4–7] using five different nondimensional parameters The nondimensional Re4 , parameters considered here are 2wm Re, δ, P r, η2 = 64wm EcP r ∗∗ T and The nondimensional parameter η2 is a duty parameter heat transfer engineering Re4 , η2 P r η2 = constant (22) Figures 12 and 13 show the total entropy generation rates based on one cycle (0 ≤ θ ≤ 360◦ ) versus Re number calculated by methods A1 and C1 as explained earlier The results are reported for two curvature (loose and non-loose) and two T ∗∗ values in these figures Since the other cases have the same trend, they are not shown here It is observed that although Sgen,t calculated by method A1 has about 10% deviation with respect to C1, the optimum Reynolds number is predicted by 4% deviation on average Practically, this deviation has less importance, because 20 Absolute relative Error : S gen,t 15 Relative error (%) S S P,t S P,t S T,t S T,t S gen,t S gen,t 1 δ =1/20 δ =1/20 δ =1/7 δ =1/7 0.9 1500 δ =1/20 : Reopt = 2210 (A1) , 2250 (C1) δ =1/7 : Reopt = 2030 (A1) , 1850 (C1) S gen,t 1210 10 Relative Error : S P,t ** -5 -10 T =3.0 Relative Error : S T,t T**=6.0 ** T =12.0 -15 -20 400 800 De 1200 1600 Figure 14 Relative errors in entropy generation rates calculated by method A1 (with respect to C1) versus De for P r = 0.711, η2 = 0.01 × 1020 , and different T ∗∗ Relative error = (Si,t,C1 − Si,t,A1 )/Sgen,t,C1 × 100% vol 31 no 14 2010 E AMANI AND M R H NOBARI ** T = 6.5 : Reopt = 2750 (A1) , 2750 (C1) ** T** = 8.0 : Reopt = 2390 (A1) , 2250 (C1) T = 9.5 : Reopt = 2120 (A1) , 2150 (C1) 3.5 Lines : A1 Symboles : C1 ** T = 6.5 2.5 S gen 8.0 9.5 1.5 0.5 1000 2000 3000 4000 Re Figure 15 Comparison of Sgen,t in one cycle (0◦ ≤ θ ≤ 360◦ ) versus Re calculated by methods A1 and C1 for δ = 20 , P r = 0.711, η3 = 2.048×10−11 , and different T ∗∗ the Sgen curves are almost flat (large curvature radius) around Reopt and at these Re numbers (near the optimal case) the total entropy generation rate (design criterion) differs less than 1% Figure 14 indicates relative errors in the entropy generation rates calculated by method A1 with respect to C1 versus De number Method A1 [Eq (20)] always underpredicts SP ,t and overpredicts ST ,t The SP ,t error increases while ST ,t decreases by increasing the De number A more practical optimization method is to determine the optimum Reynolds (or Dean) numbers for a given heat flux at the wall [8] (a given qkTw a0 = T1∗∗ ) For this purpose, we introduce a different nondimensional parameter as η3 = ρ2 aµ2 kT0 , which depends only on the pipe geometry and fluid properties Assuming δ, P r, and η3 to be constant, the Ec number can written as a function of the Reynolds number: Ec = η3 T ∗∗ Re Pr (23) The optimum Re numbers for the given T ∗∗ values obtained by methods A1 and C1 are shown in Figure 15 As is evident from the figure, both the numerical and analytical methods are in a good agreement It must be mentioned that the conclusions obtained for the first group of nondimensional parameters are also valid here CONCLUSIONS A comparative study of entropy generation is performed by a numerical and an analytical method For numerical simulation, a three-dimensional developing incompressible viscous flow in curved pipes is considered by solving the governing equations, including continuity, momentum, and energy, using a second-order finite-difference method based on the projection algorithm Local entropy generation rates due to the friction heat transfer engineering 1211 and heat transfer are studied in detail by comparing with the approximate analytical solutions [Eq.( 20)] Different entropy generation parameters are calculated by both methods, and the optimizations are carried out for two cases of nondimensional parameters The comparisons show that the values of the total entropy generation rate based on the analytical fully developed calculation differ by about 10% from the numerical ones in the entrance region calculation, but both methods predict nearly the same optimal Reynolds numbers for the two cases of nondimensional parameters Also, the error in the optimal Re prediction by the analytical method leads to less than 1% excess entropy generation Therefore, analytical fully developed calculations can be used as good approximations for designing curved and helical pipes The extension of this point to the helical pipes is affected by the minor influence of the coil pitch on the entropy generation [4, 5] NOMENCLATURE a B Be cp Cf Cske De Ec h k Nu p Pr qw r R Re S t T T∗ T ∗∗ u v V W w pipe radius toroidal coordinate parameter Bejan number specific heat at constant pressure friction factor secondary kinetic energy coefficient Dean number Eckert number convection heat transfer coefficient thermal conductivity Nusselt number pressure Prandtl number heat flux at the wall radial coordinate, Position vector curvature Reynolds number entropy generation rate time temperature characteristic temperature temperature ratio radial velocity circumferential velocity velocity axial velocity nondimensional axial velocity Greek Symbols δ η2 curvature ratio Re4 nondimensional parameter = 64wm EcP r vol 31 no 14 2010 1212 η3 θ µ τ ρ φ φ E AMANI AND M R H NOBARI nondimensional parameter = axial coordinate dynamic viscosity stress density circumferential coordinate viscous dissipation µ3 ρ2 a kT0 [8] [9] Subscripts [10] c fd gen m opt P s t T w at (r = 0, θ = 0) curved pipe fully developed condition entropy generation(all kinds) cross sectional mean value optimum due to friction straight pipe total due to heat transfer pipe wall tensorial quantity [11] [12] [13] [14] [15] Superscripts n → ´ [16] time level vectorial quantity dimensional quantity [17] [18] [19] REFERENCES [1] Soh, W Y., and Berger, S A., Laminar Entrance Flow in a Curved Pipe, Journal of Fluid Mechanics, vol 148, pp 109–135, 1984 [2] Nobari, M R H., and Amani, E., A Numerical Investigation of Developing Flow and Heat Transfer in a Curved Pipe, International Journal of Numerical Methods for Heat and Fluid Flow, vol 19, no 7, pp 847–873, 2009 [3] Bejan, A., Entropy Generation Through Heat and Fluid Flow, Wiley, New York, 1982 [4] Ko, T H., and Ting, K., Entropy Generation and Thermodynamic Optimization of Fully Developed Laminar Convection in a Helical Coil, International Communication in Heat and Mass Transfer, vol 32, pp 214–223, 2005 [5] Ko, T H., and Ting, K., Optimal Reynolds Number for the Fully Developed Laminar Forced Convection in a Helical Coiled Tube, Energy, vol 31, pp 2142–2152, 2006 [6] Ko, T H., Thermodynamic Analysis of Optimal Curvature for Fully Developed Laminar Forced Convection in a Helical Coiled Tube with Uniform Heat Flux, International Journal of Thermal Science, vol 45, pp 729–737, 2006 [7] Ko, T H., Thermodynamic Analysis of Optimal Mass Flow Rate for Fully Developed Laminar Forced Convection in a Helical heat transfer engineering Coiled Tube Based on Minimum Entropy Generation Principle, Energy Conversion and Management, vol 47, pp 3094–3104, 2006 Ko, T H., and Ting, K., Entropy Generation and Optimal Analysis for Laminar Forced Convection in Curved Rectangular Ducts: A Numerical Study, International Journal of Thermal Science, vol 45, pp 138–150, 2006 Ko, T H., Numerical Investigation on Laminar Forced Convection and Entropy Generation in a Curved Rectangular Duct With longitudinal Ribs Mounted in Heated Wall, International Journal of Thermal Science, vol 45, pp 390–404, 2006 Ko, T H., A Numerical Study on Entropy Generation and Optimization for Laminar Forced Convection in a Rectangular Curved Duct With Longitudinal Ribs, International Journal of Thermal Science, vol 45, pp 1113–1125, 2006 Ko, T H., Numerical Investigation of Laminar Forced Convection and Entropy Generation in a Helical Coil With Constant Wall Heat Flux, Numerical Heat Transfer, Part A: Applications, vol 49, no 3, pp 257–278, 2006 Berger, B A., Talbot, L., and Yao, L S., Flow in Curved Pipes, Annual Review of Fluid Mechanics, vol 15, pp 461–512, 1983 Bejan, A., Entropy Generation Minimization, CRC Press, Boca Raton, FL, 1996 Chorin, J A., Numerical Solution of the Navier–Stokes Equations, Mathematics of Computation, vol 22, no 104, pp 745–762, 1968 Temam, R., Navier–Stokes Equations, North-Holland, Amsterdam, The Netherlands, 1978 Hackbusch, W., and Trottenberg, U., Multigrid Methods, Lecture Notes in Mathematics, vol 960, Springer Verlag, New York, 1982 Ishigaki, H., Laminar Flow in Rotating Curved Pipes, Journal of Fluid Mechanics, vol 329, pp 373–388, 1996 Ishigaki, H., Analogy Between Laminar Flows in Curved Pipes and Orthogonally Rotating Pipes, Journal of Fluid Mechanics, vol 268, pp 133–145, 1994 Manlapaz, R., and Churchill, S W., Fully Developed Laminar Convection from a Helical Coil, Chemical Engineering Communications, vol 9, pp 185–200, 1981 E Amani is a Ph.D student in mechanical engineering at Amirkabir University of Technology He received his B.S in mechanical engineering from Ferdowsi University, Mashhad, Iran, in 2004 He received his M.S in energy conversion from Amirkabir University of Technology in 2007 His main areas of research are turbulent combustion modeling via probability density function method and heat transfer in curved pipes M R H Nobari is a professor of thermo-fluids at Amirkabir University of Technology (Tehran Polytechnic) He received his Ph.D in 1993 from the University of Michigan, Ann Arbor His main research interests are in numerical heat transfer and fluid flow, especially in multifluid flow vol 31 no 14 2010 Heat Transfer Engineering, 31(14):1220, 2010 Copyright C Taylor and Francis Group, LLC ISSN: 0145-7632 print / 1521-0537 online DOI: 10.1080/01457632.2010.502814 in memoriam William Begell, 1927–2009 William (Bill) Begell, 82, died on Saturday, July 2009 at Mount Sinai Hospital, New York after a protracted illness Bill had been President of Begell House Inc., the publisher of many journals, books and handbooks in the heat transfer area He had a long history in publishing, starting as a co-founder of Scripta Technica in 1962 and continuing with the founding of Hemisphere Publishing Corporation in 1966 and Begell House Inc., in 1991 The medical and heat transfer and fluid flow areas have always been the main foci of Bill Begell’s publishing activities, the latter reflecting his early career as Engineering Director of the Heat Transfer Research Facility at Columbia University A leading figure in scientific publishing, he was one of the founding members of the Society of Scholarly Publishing and has received many awards He was a Fellow of ASME and received the 2005 ASME Heat Transfer Division Distinguished Service Award Bill Begell will be remembered not only for his many professional achievements but also for his remarkable character He retained his love of life, his sense of humor, and his empathetic nature despite a truly tragic history He was born (as Wilhelm Beigel) in Vilnius (then Vilna) in Lithuania (then Poland) on May 18, 1927 Vilnius was subjected successively to Russian, German and (again) Russian invasions and Bill was the only member of his family who survived Bill’s father (Ferdinand) was shot by the SS on September 4, 1943 A week and a half later, on September 23, Bill, together with his mother and maternal grandmother were sent to the Heereskraffahrpark labor camp (HKP) in a move intended to save them from deportation and death With the encouragement of his family, Bill escaped from the HKP on June 30, 1944 but, tragically, his mother and grandmother were deported four days later and killed After the “liberation” of Vilnius in July 1944, Bill made his way westward and finished his schooling in post-Nazi Germany He eventually immigrated to the USA in June 1947, where he trained and pursued a career as an engineer and, eventually, as a publisher The tragic aspect of Bill’s life was continued in the untimely deaths of his beloved children (Freddie as a result of an accident and Alysia of leukaemia) and of his first wife Ester, who died of heart failure soon after the passing of Alysia Despite the personal devastation that these events brought to Bill, he somehow recovered and, until his final illness he appeared to have returned to something like (though never quite) his original self One of the many memorials for Bill’s life is the legacy of ongoing publications which he founded; these include not only the publications of his company Begell House Inc (which is still going strong under the leadership of Yelena Shafeyeva) but also through the many other publications which started under his guidance in earlier days In particular, Heat Transfer Engineering had its beginnings under the tutelage of Bill Begell during his period at Taylor & Francis On a journey from Tel Aviv to London, Ken Bell and myself were bemoaning the lack of practical journals in heat transfer Rather than just talking about it, we co-opted Jerry Taborek and Ernst Schlunder and made a combined approach to Bill Begell who agreed to launch HTE with Ken Bell as the first Editor-in-Chief I am sure that Bill Begell would be pleased to know that HTE is still meeting its objectives in publishing material of direct relevance to practical heat transfer Bill Begell was a quite remarkable person who, despite the tragedies which beset him, gave so much happiness to those he met He was loved by his many friends, whose lives he enriched and who will miss him dearly 1220 Geoff Hewitt Founding Editor, Heat Transfer Engineering Heat Transfer Engineering, 31(14):1221, 2010 Copyright C Taylor and Francis Group, LLC ISSN: 0145-7632 print / 1521-0537 online DOI: 10.1080/01457632.2010.502815 in memoriam James G Knudsen, 1920–2010 In January, we lost one of our own, James G Knudsen Jim was Professor Emeritus in Chemical Engineering at Oregon State University With heat transfer as his chosen specialty, he lived in, and contributed immeasurably to what could arguably be called its “golden era.” Jim grew up on a farm in Alberta, Canada of Danish immigrant parents His attraction to Chemical Engineering resulted from his association with a family friend who had preceded him into the profession He received his B.S from the University of Alberta in 1943 followed by his M.S in 1944 While teaching at U.A., his mentor encouraged him to go to the University of Michigan for a Ph.D., which he completed in 1950 He then took the position at Oregon State which he held until his retirement During his tenure there, he held the positions of Assistant and Associate Dean of the College of Engineering, and Director of the Engineering Experiment Station He served a term as Director for the Accreditation Board for Engineering and Technology (ABET) A dedicated professional, for a number of years he also was Oregon’s Chemical Engineering member and President of its Board of Examiners for Engineering and Land Surveying In 1958, he co-authored (with Don Katz) the classic text, Fluid Dynamics and Heat Transfer He also was a chapter editor of Perry’s Chemical Engineers Handbook and contributed to a number of sections of the Heat Exchanger Design Manual Concurrent with his academic career, Jim provided consulting support to several industrial companies When Heat Transfer Research, Incorporated (HTRI) came into existence in 1962, he was a key player in formatting and implementing its conceptual and research objectives, remaining as a consultant until his retirement As HTRI later evolved, Jim concentrated his technical focus on exchanger fouling and extended surface heat transfer issues It is fair to say that, at the end of his career, he was the world’s leading authority on cooling water fouling and its control, and one of the top half dozen on fouling in general Jim was a Past President and Fellow of the American Institute of Chemical Engineers (AIChE) He received that institute’s 1977 Founders Award and, in 1983, its D.Q Kern Heat Transfer Award He was a founding officer of AIChE’s first division, the Heat Transfer and Energy Conversion Division (now Transport and Energy Processes) Jim is survived by a daughter, Shelley Lindauer, a faculty member at Utah State University, her husband and two grandchildren He was predeceased by his wife, Joyce, and an older daughter, Kathryn A close personal as well as professional friend, I shall dearly miss Jim (and Joyce), and toast him with a glass of his favorite Oregon Sauvignon Blanc 1221 Emmett R Miller, P.E Heat Transfer Engineering, 31(14):1222–1224, 2010 Copyright C Taylor and Francis Group, LLC ISSN: 0145-7632 print / 1521-0537 online DOI: 10.1080/01457632.2010.494978 book review corner Ralph L Webb Book Review Editor The “Books Received” listing contains several interesting new titles of interest to workers in the field of energy, heat transfer and fluid mechanics, mechanical design, and chemical engineering Brief content description is given of all listed books If you would like to prepare a detailed review one of the books listed below, please e-mail me (Ralph.Webb@psu.edu) The book is yours to keep for preparing the review books received (with brief content description) turbulence Each of these various methods has its own specific performances and limitations, which appear to be complementary rather than competitive After a discussion of the basic concepts, mathematical tools and methods for closure, the book considers second order closure models Emphasis is placed upon this approach because it embodies potentials for clarifying numerous problems in turbulent shear flows Simpler, generally older models are then presented as simplified versions of the more general second order models The influence of extra physical parameters is also considered Finally, the book concludes by examining large Eddy numerical simulations methods Mechanical Engineering Particles in Turbulent Flows, Leonid Zaichik, Vladimir M Alipchenkov, and Emmanuil G Sinaiski, Wiley-VCH, 2008, 318 pages, ISBN-10: 3527407391, $275 Gasoline, Diesel and Ethanol Biofuels from Grasses and Plants, by Ram B Gupta and Ayhan Demirbas, Cambridge University Press, 2010, ISBN-10: 0521763991, 234 pages, $60.00 Brief Content Review Brief Content Review The first-generation biofuels (ethanol from sugar or corn and biodiesel from vegetable oils) are already in the market, with limited success The goal of this book is to introduce readers to second-generation biofuels obtained from non-food biomass, such as forest residue, agricultural residue, switch grass, corn stover, waste wood, municipal solid wastes, and so on Various technologies are discussed, including cellulosic ethanol, biomass gasification, synthesis of diesel and gasoline, bio-crude by hydrothermal liquefaction, bio-oil by fast pyrolysis, and the upgradation of biofuel This book seeks to serve as a comprehensive document presenting various technological pathways and environmental and economic issues related to biofuels Modeling and Simulation of Turbulent Flows, Roland Schiestel, Wiley-ISTE, 2008, ISBN-10: 1848210019, 768 pages, $266 Brief Content Review This book provides the fundamental bases for developing turbulence models on rational grounds The main different methods of approach are considered, ranging from statistical modeling at various degrees of complexity to numerical simulations of The only work available to treat the theory of turbulent flow with suspended particles, this book also includes a section on simulation methods, comparing the model results obtained with the PDF method to those obtained with other techniques, such as DNS, LES and RANS Written by experienced scientists with background in oil and gas processing, this book is applicable to a wide range of industries—from the petroleum industry and industrial chemistry to food and water processing The Scramjet Engine: Processes and Characteristics, by Corin Segal, Cambridge University Press, 2009, 270 pages, ISBN10: 0521838150, $125 Brief Content Review The renewed interest in high-speed propulsion has led to increased activity in the development of the supersonic combustion ramjet engine for hypersonic flight applications In the hypersonic regime the scramjet engine’s specific thrust exceeds that of other propulsion systems This book describes the processes and characteristics of the scramjet engine in a unified manner, reviewing both the theoretical and experimental research The focus is on the phenomena that dictate the thermoaerodynamic processes encountered in the scramjet engine, 1222 book review corner including component analyses and flow path considerations; fundamental theoretical topics related to internal flow with chemical reactions and non-equilibrium effects, hightemperature gas dynamics, and hypersonic effects are included Cycle and component analyses are further described, followed by flowpath examination Finally, the book reviews the current experimental and theoretical capabilities and describes ground testing facilities and computational fluid dynamics facilities developed to date for the study of time-accurate, high-temperature aerodynamics Instrumentation, Measurements, and Experiments in Fluids, by E Rathakrishnan, CRC Press, 2007, 520 pages, ISBN-10: 0849307597, $106.95 Brief Content Review This is an authoritative reference for mechanical engineers that provides the essentials for experimentation in fluids, which provides fundamentals, as well as the details necessary for experimentation on everything from household appliances to hitech rockets The book contains fourteen chapters that provides very detailed information in the field Providing ample detail for self study, it provides insight into all the vital topics and issues associated with the devices and instruments used for fluid mechanics and gas dynamics experiments This work presents easy access to the principles behind the science and goes on to elucidate the current research and findings needed by those seeking to make further advancement The author provides valuable insight into the vital issues associated with the devices used in fluid mechanics and gas dynamics experiments Structured and detailed enough for self study, this volume also provides the backbone for both undergraduate and graduate courses on fluids experimentation Applied Metal Forming: Including FEM Analysis, by Henry Valberg, Cambridge University Press, 2010, 460 pages, ISBN-10: 0521518237, $99.00 Brief Content Review This book outlines metal forming theory and how experimental methods are utilized to understand how metal forming works For each main process, such as forging, rolling, extrusion, wiredrawing, and sheet-metal forming, it demonstrates how FEA (Finite Elements Analysis) can accurately characterize the forming condition and therefore optimize the processes 1223 Brief Content Review This book enables engineers to understand the most important aspects of the vibrations of rotating machines, starting from the most basic explanations and proceeding to more accurate numerical models and analysis Using this book, and the associated MATLAB software, engineers will quickly gain (or restore) the confidence to base their designs on calculations and to understand any dynamic phenomena that might occur Detailed models are developed based on finite element analysis, to enable the accurate simulation of the relevant phenomena for real machines Novices to rotordynamics can expect to make good predictions of critical speeds and rotating mode shapes within days The book is structured more as a learning guide than as a reference book and it provides readers with more than 100 worked examples and more than 100 problems and solutions Thermoelasticity with Finite Wave Speeds, by J´ozef Ignaczak and Martin Ostoja Starzewski, Cambridge University Press, 2009, 496 pages, ISBN-10: 0199541647, $130.00 Brief Content Review Generalized dynamic thermoelasticity is a vital area of research in continuum mechanics, free of the classical paradox of infinite propagation speeds of thermal signals in Fourier-type heat conduction Besides that paradox, the classical dynamic thermoelasticity theory offers either unsatisfactory or poor descriptions of a solid’s response at low temperatures or to a fast transient loading (say, due to short laser pulses) Several models have been developed and intensively studied over the past four decades, yet this book, which aims to provide a point of reference in the field, is the first monograph on the subject since the 1970s The book focuses on dynamic thermoelasticity governed by hyperbolic equations, and, in particular, on the two leading theories: that of Lord-Shulman (with one relaxation time), and that of Green-Lindsay (with two relaxation times) The mathematical aspects of both theories—existence and uniqueness theorems, domain of influence theorems, convolutional variational principles—as well as with the methods for various initial/boundary value problems are explained and illustrated in detail and several applications of generalized thermoelasticity are reviewed Physics Dynamics of Rotating Machines, by Michael I Friswell, John E T Penny, Seamus D Garvey, and Arthur W Lees, Cambridge University Press, 2010, 544 pages, ISBN-10: 0521850169, $115.00 heat transfer engineering Dynamics and Relativity (Paperback), by Jeffrey Forshaw and Gavin Smith, Wiley, 2009, 338 pages, ISBN-10: 0470014601, $60.00 vol 31 no 14 2010 1224 book review corner Brief Content Review This introductory text emphasizes physical principles behind classical mechanics and relativity It assumes little in the way of prior knowledge, introducing relevant mathematics and carefully developing it as need within a physics context Designed to provide a logical development of the subject, the book is divided into four sections, introductory material on dynamics, and special relativity, is then followed by more advanced coverage of dynamics and special relativity Each chapter includes problems ranging in difficulty from simple to challenging with solutions for solving problems The book includes solutions for solving problems, and numerous worked examples included tions, in both monatomic and mixed particle systems The book treats such diverse topics as osmosis, steam engines, superfluids, Bose-Einstein condensates, quantum conductance, light scattering, transport processes, and dissipative structures, all in the framework of the foundations of statistical physics and thermodynamics All classical physics is derived as limiting cases of quantum statistical physics This revised and updated third edition gives comprehensive coverage of numerous core topics and special applications, allowing professors flexibility in designing individualized courses The inclusion of advanced topics and extensive references makes this an invaluable resource for researchers as well as students Diffusion: Mass Transfer in Fluid Systems, by Cussler E L., Cambridge University Press, 2009, 654 pages, ISBN-10: 521871212, $52.50 Text Books and History Thermodynamics: From Concepts to Applications, Second Edition, by Arthur Shavit and Chain Gutfinger, CRC Press, 2008, 1420073680 pages, ISBN-10: 1420073680, $56.88 Brief Content Review The book presents a logical methodology for solving problems in the context of conservation laws and property tables or equations The authors elucidate the terms around which thermodynamics has historically developed, such as work, heat, temperature, energy, and entropy Using a pedagogical approach that builds from basic principles to laws and eventually corollaries of the laws, the text enables students to think in clear and correct thermodynamic terms as well as solve real engineering problems A Modern Course in Statistical Physics (Paperback), by Linda E Reichl, Wiley-VCH, 2008, 427 pages, ISBN-10: 3527407820, $60.00 Brief Content Review This overview of diffusion and separation processes brings clarity to this complex topic Diffusion is a key part of the undergraduate chemical engineering curriculum and at the core of understanding chemical purification and reaction engineering This spontaneous mixing process is also central to our daily lives, with importance in phenomena as diverse as the dispersal of pollutants to digestion in the small intestine For students, Diffusion goes from the basics of mass transfer and diffusion itself, with strong support through worked examples and a range of student questions It also takes the reader right through to the cutting edge of our understanding, and the new examples in this third edition will appeal to professional scientists and engineers Retaining the trademark enthusiastic style, the broad coverage now extends to biology and medicine Remarkable Engineers: From Riquet to Shannon, by Ioan James Cambridge University Press, 2010, 218 pages, ISBN13: 978-0521-73165-2, $34.99 (paperback) and hard cover, ISBN-13: 978-0521-516211, $78.69 Brief Content Review This textbook provides a grounding in the foundations of equilibrium and nonequilibrium statistical physics, and focuses on the universal nature of thermodynamic processes It illustrates fundamental concepts with examples from contemporary research problems One focus of the book is fluctuations that occur due to the discrete nature of matter, a topic of growing importance for nanometer scale physics and biophysics Another focus concerns classical and quantum phase transi- heat transfer engineering Brief Content Review Engineering transformed the world completely between the 17th and 21st centuries Remarkable Engineers tells the stories of 51 of the key pioneers in this transformation, from the designers and builders of the world’s railways, bridges and airplanes, to the founders of the modern electronics and communications revolutions vol 31 no 14 2010