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

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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 © 2020 Published by Semnan University Press. All rights reserved. 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] 86 A. Kardgar/ JHMTR 7 (2020) 85-94 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.
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