Wall Thermal Inertia Effects of Pulsatile Flow in a Ribbed Tube: a Numerical Approach Amin Kardgar*,A, Davood Domiri Ganjib

Journal of Heat and Mass Transfer Research 7 (2020) 85-94

Journal of Heat and Mass Transfer Research

Journal homepage: http://jhmtr.semnan.ac.ir

Semnan University Wall Thermal Inertia Effects of Pulsatile Flow in a Ribbed Tube: A Numerical Approach Amin Kardgar*,a, Davood Domiri Ganjib

aDepartment of Mechanical Engineering, University of Mazandaran, Babolsar, Iran. bDepartment of Mechanical Engineering, Babol University of Technology, Babol, Iran.

PAPER INFO ABSTRACT

Paper history: In the present paper, the heat transfer of pulsatile flow in ribbed tube was investigated Received: 2020-02-24 numerically by considering the effect of thermal inertia of solid wall thickness. To this purpose, the CVFV technique with collocated grids arrangement was adopted to discretize Revised: 2020-07-01 momentum and energy equations. In order to avoid checker-board of pressure field in Accepted: 2020-10-16 numerical simulation, Chow and Rhie interpolation scheme was employed. The well- established SIMPLE (Semi-Implicit Method for Pressure Linked Equations) method was Keywords: utilized to handle velocity and pressure coupling in the momentum equation. Stone’s Strongly Pulsatile flow; Implicit Procedure (SIP) was used to solve the set of individual linear algebraic equations. Conjugate heat transfer; Womersley number, Reynolds number, velocity amplitude, and wall thickness ratio are four Womersley number; essential parameters that influence heat transfer and Nusselt number in pulsatile flow in a Wall thickness; ribbed tube. It was deduced by varying Womersley number Nu does not change. Nu enhances ribbed tube. almost 19% by augmentation of wall thickness ratio from 0.125 to 1. It was shown by increasing velocity amplitude from 0.1 to 0.8; Nu reduces almost 4.7%.

DOI: 10.22075/jhmtr.2020.19899.1279

1. Introduction pulsatile flow in the heat exchanger with helical geometry. Their results indicated that enhancement in pressure loss Pulsating flow heat transfer can occur in natural compared to steady one is almost 3–7% in pulsatile flow, phenomena like blood flow in veins [1] and in many while Nu number increased up to 39%. Their findings also industrial application like pulse tube and GM cryocooler, revealed that Nu number increases sharply at small Stirling engine, pulse jets, reciprocating engine, and pulse Reynolds numbers [5]. Li et al. investigated the combustors [2][3]. The flow and heat transfer mechanism of heat transfer of periodic pulsating slot-jet characteristics in pulsating depend on parameters like impingement with different waveforms. Their results frequency, Reynolds number, Prandtl number, oscillating revealed that T-wave is more contributing to heat transfer amplitude. Extensive investigations have been carried out enhancement. Their comprehensive results showed that T- to study these parameters' effect in hydrodynamics and wave at large Re with medium frequency could attain a thermal behavior of such flow by many researchers. high-efficient heat transfer increase [6]. Yuan et al. investigated the effects of wall thickness on Many researchers conducted experimental studies heat transfer of pulsatile laminar flow with the boundary actively on pulsating flow. The experimental mensuration condition of constant heat flux within the tube and channel was appraised for low pulsating amplitudes in a fully wall. They concluded the pulsatile laminar flow could developed region. Heat transfer and the variation of Nu decrease the average Nusselt number and the effects are number in pulsatile flow also was investigated significant with high oscillation amplitude, small experimentally, and the reported results were confusable: frequency, low Prandtl number, and large wall thickness Martinelli et al. claimed that the frequency of oscillation heat capacity [4]. Khosravi et al. conducted an did not influence the heat transfer [7]. Genin et al. and experimental study on pressure loss and heat transfer on

*Corresponding Author: Amin Kardgar, Department of Mechanical Engineering, University of Mazandaran, Babolsar, Iran. Email: [email protected]

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Barker et al. set up an experimental apparatus to measure number in channel flows [16]. Meng et al. performed heat flux and velocity. They demonstrated that no numerical simulation to study solid to fluid thermal enhancement was found in heat transfer coefficient in interaction in pulsating flow with high-frequency inlet pulsating flow [8]. Park and et al. conducted a numerical velocity by using Lattice Boltzmann method with different simulation and found a slight reduction in heat transfer outer wall temperatures. It has been determined the coefficient in their results [9]. Karamercan investigated averaged heat fluxes of the pulsatile flow in one period are oscillation effect on double pipe heat exchanger and nearly equal to pure heat conduction in a condition with concluded that heat transfer and Nu number augmented low-pressure amplitudes and comparatively low- when only the frequency of pulsation of the flow was frequency pulsations [17]. Mathie et al. applied simple greater than a specific value [10]. semi-analytical 1-D model to solve fluid and solid thermal Several numbers of investigations were also carried out coupling in time-varying inlet velocity. They claimed the by considering heat conduction in the wall in pulsatile model could predict fluid flow and heat transfer behavior flow. Kardgar and Jafarian studied the effect of heat in the re-attached and separated flow of a turbulent conduction in axial direction of the solid wall on conjugate boundary layer behind a backward-facing step [18]. M. heat transfer for oscillating flow at the tube section of Sheikholeslami et al. applied Laplace transform to study Pulse Tube Refrigerator (PTRs) via a numerical method. oscillation effect on skin friction coefficient and Nusselt They claimed that the averaged convection heat transfer number. They claimed Nu enhances with volume fraction coefficient enhances from 5.97 to 28.45 by rising Strouhal [19]. Zin et al. investigated the same physics in the number from 0.02 to 1. Additionally, Nu reduces nearly presence of magnetic field. They showed that magnetic 27% by enhancing solid to fluid conductivity ratio from force could suppress fluid flow [20]. 0.5 to 15; however, for solid to fluid conductivity ratio Conjugate heat transfer in pulsatile flow is a real greater than 15, enhancing ksf has no effect on the running condition in many natural and industrial convective heat transfer [11]. Wang and Zhang conducted phenomena. However, up to now, numerous investigations numerical simulation on convective heat transfer in have been conducted to analyze pulsatile flow in tube and turbulent pulsatile flow with large velocity amplitudes of channels with wall heat conduction; there is no study oscillation in a pipe at constant temperature boundary concerned with conjugate pulsating flow in ribbed tube. As condition at the wall. Their results showed that Womersley it was pointed out in literature review, pulsating flow heat number is an essential parameter in pulsatile flow behavior transfer is controversial issue needing special attention. and heat transfer characteristics. Their analysis The aim of this paper is to study the effects of the pulsation demonstrated two essential mechanisms of the Nu number and ribs in heat transfer of tube with solid wall heat augmentation: the larger velocity and the flow setback conduction. To this purpose, 2D axisymmetric geometry during a period of the flow oscillation [12]. Ellahi et al with weakly compressible flow is considered to simulate proposed a model for analysis of oscillating flow in the pulsatile conjugate heat transfer. rotating disk [13]. Jo and Kim analyzed pulsating flow in a channel with 2. Mathematical model parallel-plate, in which a sinusoidal wall temperature Simulations are carried out on two-dimensional boundary condition was imposed on outer surface. They axisymmetric tube. The outer diameter, inner diameter, used spectral method to solve the solid and fluid energy and length of tube are Ro, R¬i and L, respectively. The equations. They claimed that the convection heat transfer physical domain with wall thickness of Ro-Ri is depicted enhances as Re augments with a fixed Peclet number when in Fig. 1. For modeling the problem, the following Reynolds and Peclet number are greater than 1. Their assumptions have been employed: (1) the fluid flow is findings illustrated when the dimensionless thermal incompressible and laminar (2) the gas is ideal (3) thickness is less than 0.01; wall heat conduction can be radiation heat transfer and body forces are neglected. Non- ignored [14]. Zhang et al. studied numerically laminar dimensional form of Navier-Stokes and energy equation in flow heat transfer in pipes with thick wall in fully cylindrical coordinate were used to model the heat transfer developed region. The results show that Nu number is and fluid flow [12]: more affected by the ratio of solid-to-fluid thermal Continuity conductivity and enhancing wall thickness and the 푊표2 휕휌 1 휕(푟푣∗) 휕푢∗ reducing ksf could vary the inner wall surface boundary 푓 (1) ∗ + ∗ ∗ + ∗ = 0 condition to the constant heat flux boundary condition 휋푅푒 휕푡 푟 휕푟 휕푥 when ksf<25 [15]. Thus, Satish and Venkatasubbaiah conducted a numerical analysis of turbulent flow in Momentum-r 푊표2 휕푢∗ 1 휕(푟푢∗푣∗) 휕(푢∗푢∗) channel with moving plate with considering conjugate (2) ∗ + ∗ ∗ + ∗ heat transfer of fluid and solid. They demonstrated that the 휋푅푒 휕푡 푟 휕푟 휕푥 휕푝∗ 2 1 휕 휕푢∗ temperature between solid and fluid interface reduces, and = − + (푟∗ ) average Nu number enhances with increasing Reynolds 휕푥∗ 푅푒 푟∗ 휕푟∗ 휕푟∗ 2 휕 휕푢∗ number. Their results revealed heat conduction in solid + ( ) wall thickness could considerably change the Nusselt 푅푒 휕푥∗ 휕푥∗ A. Kardgar/ JHMTR 7 (2020) 85-94 87

Figure 1. Physical model and boundary condition of numerical simulation

Momentum-x 푊표2 휕푣∗ 1 휕(푟푣∗푣∗) 휕(푢∗푣∗) + + 휋푅푒 휕푡∗ 푟∗ 휕푟∗ 휕푥∗ 휕푝∗ 2 1 휕 휕푣∗ = − + (푟∗ ) (3) 휕푟∗ 푅푒 푟∗ 휕푟∗ 휕푟∗ 2 휕 휕푣∗ 2 2푣∗ + ( ) − 푅푒 휕푥∗ 휕푥∗ 푅푒 푟∗2 Fluid Energy 2 ∗ ∗ ∗ ∗ ∗ 푊표 휕푇 1 휕(푟푣 푇푓 ) 휕(푢 푇푓 ) ∗ + ∗ ∗ + ∗ 휋푅푒 휕푡 푟 휕푟 휕푥 (4) 2 1 휕 휕푇∗ = (푟∗ ) Figure 2. A selected control volume and collocated 푃푟푅푒 푟∗ 휕푟∗ 휕푟∗ arrangement of grid Solid Energy 푊표2 휕푇∗ 2 휕 휕푇∗ = ( ) (5) 휋푅푒 휕푡∗ 푅푒푃푟 휕푥∗ 휕푥∗ passed from east face of control volume illustrated in Fig. 2 1 휕 휕푇∗ 2 can be calculated by: + (푟∗ ) 푐 푃푟푅푒 푟∗ 휕푟∗ 휕푟∗ 퐹푒 = 푚̇ 푒푢푒 The above equations are non-dimensionalized by 푢푒 = (휌푢)푒∆푦 parameters defined as follows: 푚푎푥(푚̇ , 0)푢 + 푚푖푛(푚̇ , 0)푢 퐹푐 = { 푒 푃 푒 퐸 (9) 푟 푥 휔푡 푒 ( ) 푟∗ = , 푥∗ = , 푡∗ = , 푚̇ 푒 1 − 휆푒 푢푃 + 푚̇ 푒휆푒푢퐸 푅 푅 2휋 푥푒 − 푥푃 푝 푣 푢 휆푒 = ∗ ∗ ∗ 푥퐸 − 푥푃 푝 = 2 , 푣 = , 푢 = , (6) 휌푈 푈 푈 Similar expression for the fluxes through other faces of 푈퐷 푇 − 푇 휔 푅푒 = , 푇 = 푤 , 푊표 = 푅√ control volume results in the succeeding coefficients in 휈 푇 − 푇 휐 algebraic equation for central and upwind schemes, 𝑖푛 푤 Control Volume Finite volume method (CVFVM) is respectively. For the case of central scheme [22]: 푐 푐 applied to discretize the continuity, momentum, and 퐴퐸 = 푚̇ 푒휆푒, 퐴푊 = 푚̇ 푤휆푤 푐 푐 energy equations. The detail about the discretization 퐴푁 = 푚̇ 푛휆푛, 퐴푆 = 푚̇ 푠휆푠 (10) technique can be found in [21]. Second-order implicit 푐 푐 푐 푐 푐 퐴푃 = −(퐴퐸 + 퐴푊 + 퐴푁 + 퐴푠) three-time level was implemented to discretize the In addition, for upwind scheme: unsteady term in Navier-Stokes and energy equation, 푐 푐 퐴퐸 = 푚푖푛(푚̇ 푒, 0) , 퐴푊 = 푚푖푛(푚̇ 푤, 0) which leads to following approximation: 퐴푐 = 푚푖푛(푚̇ , 0) , 퐴푐 = 푚푖푛(푚̇ , 0) (11) 휌훥푉 푁 푛 푆 푠 표 표표 퐴푐 = −(퐴푐 + 퐴푐 + 퐴푐 + 퐴푐) (3푢푃 − 4푢푃 + 푢푃 ) (7) 푃 퐸 푊 푁 푠 2훥푡 푐 The relation for 퐴푃 is derived with applying the By rearranging the equation as follows: continuity condition [23]: 퐴푡 푢 − 푄푡 푃 푃 푢 푚̇ + 푚̇ + 푚̇ + 푚̇ = 0 (12) 3휌훥푉 휌훥푉 (8) 푒 푤 푛 푠 퐴푡 = , 푄푡 = (4푢표 − 푢표표) With applying the central difference scheme to 푃 2훥푡 푢 2훥푡 푃 푃 discretize the diffusion term, the following expression is Which 푢표 and 푢표표 in above equations are velocities in 푃 푃 derived: previous and two previous time levels, respectively. 푑 휕푢 휇푒푆푒 Additionally, convective fluxes of x-momentum equation 퐹푒 = (휇 ) 훥푦 = (푢퐸 − 푢푃) (13) 휕푥 푒 푥퐸 − 푥푃

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푑 휇푒푆푒 푑 휇푤푆푤 equations. Bulk temperature, Nusselt number, Reynolds 퐴퐸 = − , 퐴푊 = − 푥퐸 − 푥푃 푥푃 − 푥푊 and Prandtl numbers are defined as follows: 휇 푆 휇 푆 푅푖 푑 푛 푛 푑 푠 푠 ∫ 푢푇푓푟푑푟 퐴푁 = − , 퐴푆 = − 0 푥 − 푥 푥 − 푥 푇푏푓 = 푅 , 푁 푃 푃 푆 ∫ 푖 푢푟푑푟 The system of algebraic equation must be diagonally 0 dominant to avoid divergence in iterative solvers. Central 2 휕푇푓 푁푢푥,푡 = ( ) , difference discretization of convective term in momentum 푇푤 − 푇푓푏 휕푟 (19) 푟=푅푖 and energy equations may result in system of linear 푈퐷 푅푒 = 𝑖, algebraic equations that is not dominant in diagonal. One 휈 approach to compel dominance of diagonal in systems of 푐푝,푓휇푓 푃푟 = linear algebraic equations is usage the deferred correction 푘푓 technique proposed by Kholsa and Rubin [24]. The time-averaged and space averaged Nusselt number 푐 푈퐷푆 퐶퐷푆 푈퐷푆 푚−1 퐹푒 = 푚̇ 푒푢푒 + 푚̇ 푒(푢푒 − 푢푒 ) (14) can be calculated according to the following equations Where superscripts, CDS, and UDS in equation 14 [12]: 푇 denote central differencing and upwind scheme estimates, ∫ |푁푢푥,푡|푑푡 푁푢̅̅̅̅ = 0 , respectively. The rate of convergence in this approach is 푥 푇 ∫0 푑푡 almost similar to upwind scheme [25]. 퐿 The pressure gradient will be treated as a source term ∫ |푁푢푥,푡|푑푥 푁푢̅̅̅̅ = 0 , (20) and is computed as: 푡 퐿 ∫0 푑푥 푢 푚−1 퐿 푇 푄푃 = −(푝푒푆푒 − 푝푤푆푤) (15) ∫ ∫ |푁푢 |푑푡푑푥 ̿̿̿̿ 0 0 푥,푡 The detail about pressure term treatment in momentum 푁푢 = 퐿 푇 ∫ ∫ 푑푡푑푥 equation can be found in [25]. Finally, the system of linear 0 0 equations is as follows: The coupling between solid and fluid energy equations 푢 푢 푢 푢 푢 푢 can be satisfied by two approaches. In the first approach 퐴푃푢푃 + 퐴퐸푢퐸 + 퐴푊푢푊 + 퐴푁푢푁 + 퐴푆 푢푆 = 푄푃 푢 푐 푑 푢 푐 푑 known as conjugate scheme, the whole energy equations 퐴퐸 = 퐴퐸 + 퐴퐸, 퐴푊 = 퐴푊 + 퐴푊 푢 푐 푑 푢 푐 푑 of the solid and fluid is solved as single set of equation and 퐴푁 = 퐴푁 + 퐴푁, 퐴푆 = 퐴푆 + 퐴푆 (16) 푢 푡 푢 푢 푢 푢 the heat flux and temperature equality are satisfied 퐴푃 = 퐴푃 − (퐴퐸 + 퐴푊 + 퐴푁 + 퐴푆 ) 푢 푃 푐 푡 implicitly at the interface. In the second approach known 푄푃 = 푄푢 + 푄푢 + 푄푢 푐 as coupled scheme, the energy equations are solved Where the source term 푄푢 is calculated with: independently. The first method is employed in the 푄푐 = [(퐹푐)푈퐷푆 − (퐹푐)퐶퐷푆]푚−1 (17) 푢 푢 푢 simulation because it is more stable. In the case of cylindrical coordinate system, the term

Δ푉 푣 휇 ⁄ 2 should be added to 퐴푃 in radial momentum 푟푃 3. Results and discussion equation. The boundary condition can be formulated as follows: 3.1. Validation of the numerical simulation 휕푇 푟 = 0, 0 ≤ 푥 ≤ 퐿, 푓 = 0, To model conjugate heat transfer of pulsatile flow in 휕푟 the tube, a code was developed and compiled in Fortran 휕푢 = 0 90. The grid independence check is necessary to confirm 휕푟 the verification and accuracy of the numerical simulation 푟 = 푅 , 0 ≤ 푥 ≤ 퐿, 𝑖 results. Four sets of grid numbers are considered to 휕푇푓 휕푇푠 −푘 = −푘 , 푇 = 푇 , 푣 = 0, investigate the mesh density effect on the numerical 푓 휕푟 푠 휕푟 푓 푠 computation. The mesh number are almost 6250, 9000, 푢 = 0 (18) 16000, and 25000. The velocity on the axis of the tube of 푟 = 푅표, 0 ≤ 푥 ≤ 퐿, 푇푠 = 푇푊 the four sets with different grid numbers is demonstrated 푥 = 0, 0 ≤ 푟 ≤ 푅𝑖, 푇푓 = 푇𝑖푛, in Fig. 3. The implemented mesh number in the 푢 = 푈(1 + 퐴푠푖푛(2휋푓)) computational domain is about 16000 in order to save 휕푇 푥 = 퐿, 0 ≤ 푟 ≤ 푅 , 푓 = 0, computer run time and making a balance between 𝑖 휕푥 휕푢 computational accuracy and costs. = 0 The accuracy of the present method of solution and the 휕푥 numerical model has been verified with the numerical The linear algebraic equation is solved by the Stone’s results, which is published in the scientific literature. For SIP solver. Stone's method, also known as the strongly the verification of the numerical method and the implicit procedure or SIP, is an algorithm for solving a computational model, two numerical simulations at the sparse linear system of algebraic equations. The method different operating conditions and geometric sizes, as utilizes an incomplete LU decomposition, which presented in [26] and [15] are conducted. Zhao and Cheng approximates the exact LU decomposition to find an numerically investigated oscillating flow created with a iterative solution for the system of linear algebraic A. Kardgar/ JHMTR 7 (2020) 85-94 89

Figure 3. Velocity variation in one cycle on the axis of the Figure 6. Averaged Nusselt number versus time at different tube at fully developed region Reynolds number

Figure 4. Comparison of temporal u-velocity with Zhao and Figure 7. Variation of cycled averaged Nusselt number Cheng [26] for L/D=40, Reω=64 and Ao=15 at X/D=6.2 versus Reynolds number

Fig. 5. Variation of temperature at the inner surface of the Figure 8. Variation of averaged Nusselt number versus tube in comparison with Zhang et al. [15] at ksf=1.0, Prandtl number δ/Ri=0.84, Pr=1.0 and Re=50 tube. There is good agreement between the code results pressure wave generator piston in tube with constant wall with Zhang’s numerical simulation. temperature. The comparison with their results is In the proceeding, important parameters will be studied presented in Fig. 4. It can be seen the numerical in the conjugate heat transfer of pulsating flow. These computation of the developed code agrees well with parameters are Reynolds number, Prandlt number, Zhao’s results. The results were also compared to [15], in Womersley number, wall thickness ratio, the geometry of order to get confident about the validation of the results of rib in the tube, and amplitude of velocity. the code by considering the energy equation. Zhang et al. carried out numerical analysis on the conjugate heat 3.2. Effect of Reynolds number on Nusselt transfer of solid and fluid in the steady-state flow at the number

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layer is comparatively less thick than the thermal boundary layer, which in turn means more heat transfer by the bulk motion (also called advection), this means convection is dominant than conduction. 3.4. Effect of Womersley number on Nusselt number The temporal variation of Nu versus Womersley number is depicted in Fig. 9. It seems pulsatile flow at different Wo numbers does not affect convective heat transfer coefficient. Nu in pulsatile flow oscillates around steady-state Nu number. Averaged Nu number in different Wo numbers is also depicted in Fig. 10. As can be seen, Figure 9. Cycled averaged Nusselt number variation versus Nu remains constant by varying Wo number. Based on time at different Womersley number Nandi and Chattopadhyay's works, it was realized that imposing oscillating velocity at the inlet of the tube can provide improved performance of convective heat transfer at different amplitudes of pulsation. However, Mathie et al. [27] found reduction of Nu number in their analysis of heat transfer in unsteady compressible flow. In present paper, it is elucidated that Wo has no effect on Nu number and heat transfer in ribbed-tube which admits Matinelli et al. [7] and Genin et al. [8] results. 3.5. Effect of wall thickness ratio on Nusselt number Nu enhances almost 19% by increasing wall thickness ratio from 0.125 to 1. It is remarkable that Sparrow and Patankar conducted a comprehensive investigation for Figure 10. Cycled averaged Nusselt number variation versus thermally developed flow in duct and tube with different Womersley number boundary conditions for energy equation with negligible wall thickness ratio [28]. They concluded constant wall The variation of Nu number with Re number at temperature and the constant heat flux are two extreme Wo=6.341 and A=0.2 in one period of pulsatile flow is boundary conditions for Nusselt number in fully demonstrated in Fig. 6. The instantaneous Nu number developed flow, and by increasing Bi number, Nu varies changes sinusoidally with time. It is clearly seen that from constant heat flux with Nu=4.36 to constant wall Reynolds number strongly influences the heat transfer temperature Nu=3.66 asymptotically. augmentation in pulsatile flow. Averaged Nusselt number In the present paper, our numerous simulations showed is also depicted in Fig. 7. Nu increases almost 36.8% by that as wall thickness ratio enhances, the convective heat enhancing Re from 100 to 800. The heat transfer rate transfer coefficient or Nu number at inner face of a tube linearly increases since the difference between wall and wall with constant temperature imposed at the outside of bulk temperature is enhancing. In other words, the higher the tube will encounter a boundary condition velocity of fluid flow with increasing Re number causes transformation from Dirichlet to Neumann boundary higher collision among the fluid particles. Due to higher condition. collision among fluid particles, heat transition rate The temperature at the common face of fluid and solid improves. wall reduces with enhancement in wall thickness due to 3.3. Effect of Prandtl number on Nusselt increment of conductive thermal resistance or reduction in number conductive heat transfer at the solid wall. As temperature of the common face between solid and fluid declines, the The variation of averaged Nu on Pr number is depicted convective heat transfer coefficient and the Nu number in Fig. 8. Nu increases sharply by Pr increment. Nu augments to preserve the energy balance between the inner increases almost 35% by Pr enhancement from 0.1 to 5. and outer surfaces of solid region. Hereafter, the Nusselt With the increment in the Prandtl number the Nusselt number improves with thickening the solid wall at a number also increases, so the convection heat transfer rate constant Prandtl Reynolds number. also increases. Prandtl number is defined with the ratio of For evaluating the influence of solid wall heat momentum diffusivity to the thermal diffusivity. So with conduction on Nu number and convective heat transfer in the increase in Prandtl number signifies that the the tube, Guo and Li proposed a non-dimensional momentum diffuses quickly, and the velocity boundary parameter (M). They defined M [29]: A. Kardgar/ JHMTR 7 (2020) 85-94 91

Figure 11. Cycled averaged Nusselt number variation versus Figure 13. Cycled averaged Nusselt number variation versus time at different wall thickness ratio amplitude of pulsating velocity

Figure 12. Cycled averaged Nusselt number variation Figure 14. Cycled averaged Nusselt number variation versus time at different amplitude of pulsating velocity versus wall thickness ratio 0.2, 0.4, 0.6, 0.8 were adopted. Averaged Nu number 휙 (푅2 − 푅2)푘 Δ푇 /퐿 푐표푛푑 표 𝑖 푠 푠 ℎ (21) versus velocity amplitude is demonstrated in Fig. 13. Nu 푀 = = 2 휙푐표푛푣 푅𝑖 휌푐푃푢푎푣푒. Δ푇푙 number decreases by almost 4.7% by increasing velocity The non-dimensional number (M) is redefined by amplitude from 0.1 to 0.8. Because flow reversal increases Maranzana as follow [30]: by enhancing velocity amplitude and hence fluid resident 2 휙푐표푛푑 푟 푁푇푈 time in tube increases and this causes reduction in the 푀 = = (22) 휙푐표푛푣 퐵푖 difference between the bulk temperature and the inner wall It was noted that with M less than 0.01 wall thermal temperature. This will consequence in reduction of the inertia can be neglected. However, they assumed that the temperature gradient at the fluid and solid interface, which heat transfer coefficient is constant at the inner wall of the will result in Nu number decrease. The time variation of tube; therefore, their results is not applicable to the Nu number in one period is depicted in Fig. 14. It can be problems where local heat transfer coefficient is a varying observed by decreasing velocity Nu decreases, and for parameter rather than a given fixed value. Other equations velocity amplitude greater than 0.4, Nu reduces rapidly were proposed by Cotton and Jackson [31] and Faghri and after t=T/2. Sparrow [32]. In all equations it is clear that by increasing 3.7. Effect of height of ribs in the tube on solid wall thickness, axial heat conduction and wall thermal inertia becomes more predominant. It can be Nusselt number deduced from Fig. 11 and Fig. 12 that by enhancing solid Nu number decrease with increasing the height of rib in wall thickness ratio, flow in the tube will not sense outside the tube. As it is depicted in Fig. 15 Nu decreases by wall boundary condition and inner wall boundary 12.7%. Temperature difference between inner surface of condition will approach toward constant wall heat flux. the ribbed tube and bulk temperature increases and 3.6. Effect of amplitude of the inlet velocity on gradient of temperature at the interface of fluid and solid wall also enhances by increasing height ratio of ribs, but Nusselt number the augmentation of bulk temperature difference with In order to investigate the effect of velocity amplitude interface temperature dominants and results in reduction on the heat transfer coefficient five amplitudes of A=0.1, of Nu number.

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Nomenclature A Velocity amplitude cp,f Fluid specific heat cp,s Solid specific heat 푐푓 Coefficient of pressure drop d Distance of ribs in tube Di Inner diameter of tube f Frequency h Height of ribs in tube kf Fluid conductivity ks Solid conductivity l Length of ribs in tube L Tube length Nu Nusselt number Figure 15. Cycled averaged Nusselt number variation versus Pr Prandtl number height of rib ratio p Pressure Re Reynolds number

Ri Inner radius of tube Ro Outer radius of tube Tin Inlet temperature Tw Wall temperature Tf Fluid temperature Ts Solid temperature uin Inlet velocity U Inlet velocity magnitude Wo Womersley number 훿 Wall thickness 휇 Dynamic viscousity 휌푓 Fluid density 휌푠 Solid density 휏푤 Wall shear stress

Figure 16. Cycled averaged Nusselt number variation versus Conflict of Interest time at different rib height ratio The authors declare that They have no conflict of Conclusion interest.

Conjugate heat transfer of pulsatile flow in ribbed tube was investigated by solving non-dimensional Navier- References Stokes and energy equations in cylindrical coordinate [1] Sinha, A. and Misra, J. C. (2012) ‘Numerical system with finite volume technique (FVM). The well- study of flow and heat transfer during known SIMPLE method of Patankar was adopted for oscillatory blood flow in diseased arteries in dealing the coupling between pressure and velocity, and presence of magnetic fields’, Applied checker-board in collocated arrangement of grids was Mathematics and Mechanics (English avoided by implementing Rie and Chow interpolation Edition), 33(5), pp. 649–662. doi: method. It was elucidated that increasing Wo number has 10.1007/s10483-012-1577-8. no effect on the Nu number. Nu enhances almost 35% by [2] Kardgar, A., Jafarian, A. and Arablu, M. augmentation of Pr number from 0.1 to 5, and for Pr (2016) ‘An Eulerian-Lagrangian model to number greater than 1, Nu enhances sharply. Solid wall study the operating mechanism of Stirling thickness is a salient parameter in conjugate heat transfer pulse tube refrigerators’, Scientia Iranica, of pulsating flow. By increasing solid wall thickness ratio 23(1), pp. 277–284. from 0.125 to 1, Nu augmented 19% due to the boundary doi:10.24200/sci.2016.3833. condition transformation from Dirichlet to Neumann. It [3] Kardgar, A. and Jafarian, A. (2020) was also depicted that Nu number decreases almost 4.7% ‘Numerical simulation of turbulent by increasing velocity amplitude from 0.1 to 0.8 because oscillating flow in porous media’, Scientia the difference between the bulk temperature and the inner Iranica. doi: 0.24200/SCI.2020.52521.2788. wall temperature and the temperature gradient at the solid [4] Yuan, H. et al. (2014) ‘Theoretical analysis of and fluid interface reduces. Additionally, increasing height wall thermal inertial effects on heat transfer of ribs in the tube can result in Nu number reduction. of pulsating laminar flow in a channel’, International Communications in Heat and A. Kardgar/ JHMTR 7 (2020) 85-94 93

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