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Natural Convection Flow in a Vertical Tube Inspired by Time-Periodic Heating

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Natural Convection Flow in a Vertical Tube Inspired by Time-Periodic Heating Alexandria Engineering Journal (2016) xxx, xxx–xxx HOSTED BY Alexandria University Alexandria Engineering Journal www.elsevier.com/locate/aej www.sciencedirect.com ORIGINAL ARTICLE Natural convection flow in a vertical tube inspired by time-periodic heating Basant K. Jha, Michael O. Oni * Department of Mathematics, Ahmadu Bello University, Zaria, Nigeria Received 15 May 2016; revised 15 August 2016; accepted 25 August 2016 KEYWORDS Abstract This paper theoretically analyzes the flow formation and heat transfer characteristics of Natural convection; fully developed natural convection flow in a vertical tube due to time periodic heating of the surface Fully developed flow; of the tube. The mathematical model responsible for the present physical situation is presented Time-periodic heating; under relevant boundary conditions. The essential features of natural convection flow formation Vertical tube and associated heat transfer characteristics through the vertical tube are clearly highlighted by the variation in the dimensionless velocity, dimensionless temperature, skin-friction, mass flow rate and rate of heat transfer. Moreover, the effect of Prandtl number and Strouhal number on the momentum and thermal transport characteristics is discussed thoroughly. The study reveals that flow formation, rate of heat transfer and mass flow rate are appreciably influenced by Prandtl num- ber and Strouhal number. Ó 2016 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). 1. Introduction difference approach. From miniaturization of electrical and electronic panels, Bar-Cohen and Rohsenow [3] analyzed fully In the course of modeling a real life situation of fluid flow, developed natural convection between two periodically heated periodic heat input in a channel has captured the attention parallel plates. Wang [4] investigated free convection flow of many researchers due to its everyday applications in electri- between vertical plates with periodic heat input. He presented cal and electronic appliances such as oven, drying machines, the solutions for the temperature and velocity field of the fluid thermostat component, automatic control and microchips. A as function of Prandtl number, Strouhal number and indirectly number of researchers have studied the effect of periodic heat Rayleigh number. He concluded that increasing Strouhal num- input on time dependent free convection flow in a channel. ber decreases the unsteady temperature and velocity of the Chung and Anderson [1] examined unsteady laminar free con- fluid. vection in a channel. Yang et al. [2] studied laminar natural In recent past, Jha and Ajibade [5] analyzed free convection convection with oscillatory surface temperature using the finite flow between vertical porous plates with periodic heat input being an extension of [4] by including suction/injection on the vertical porous plates. They concluded that inclusion of * Corresponding author. suction/injection has distorted the symmetric nature of the E-mail addresses: [email protected] (B.K. Jha), michaeloni29@ flow considered by [4]. Other research work on periodic heat yahoo.com (M.O. Oni). input includes Jha and Ajibade [6] who investigated the effects Peer review under responsibility of Faculty of Engineering, Alexandria of heat generating/absorbing fluid between vertical porous University. http://dx.doi.org/10.1016/j.aej.2016.08.025 1110-0168 Ó 2016 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article in press as: B.K. Jha, M.O. Oni, Natural convection flow in a vertical tube inspired by time-periodic heating, Alexandria Eng. J. (2016), http:// dx.doi.org/10.1016/j.aej.2016.08.025 2 B.K. Jha, M.O. Oni Nomenclature a radius of the tube T1 steady temperature at the tube A steady velocity profile T2 amplitude of periodic temperature at the tube B periodic velocity profile u velocity of fluid F steady temperature profile G periodic temperature profile Greek alphabets g acceleration due to gravity a thermal diffusivity In modified Bessel’s function of first kind of order n. b coefficient of thermal expansion Where n ¼ 0; 1; 2; 3; ... j thermal conductivity M mass flow rate q fluid density Pr Prandtl number w phase of temperature r dimensional radial coordinate v phase of velocity R dimensionless radial coordinate s skin friction St Strouhal number m kinematic viscosity t time x frequency T temperature of the fluid T0 initial temperature plates with periodic heat input. Sparrow and Gregg [7] studied 2. Mathematical analysis nearly Quasi-steady free convection heat transfer in gases. Menold and Yang [8], Nanda and Sharma [9] and Muhuri Consider a natural convection flow of a viscous incompressible and Gupta [10] investigated the effect of periodic heating on fluid in a vertical tube as shown in Fig. 1. Initially, the surface a single vertical plate with no edge on the boundary layer temperature of the tube and fluid is at T and then assumed to development. A coupled stress fluid modeling on free convec- 0 be heated periodically to Tðr; tÞ¼T1 þ T2 cosðxtÞ. The natu- tion oscillatory hydromagnetic flow in inclined rotating chan- ral convection flow formation is due to the temperature gradi- nel was studied by Ahmed et al. [11] to see the effect of periodic ent between the surface of the tube and fluid. Every other heat input on microchannel. Adesanya [12] studied free con- parameters are assumed constant except otherwise stated and vective flow of heat generating fluid through a porous vertical are presented in nomenclature. The flow is assumed to be fully channel with velocity slip and temperature jump. He con- developed and the viscous dissipation term in the energy equa- cluded that increase in slip and temperature jump parameters tion is also assumed to be negligible. increases the flow velocity and fluid temperature respectively. For small temperature difference, the density of the fluid in Also, Adesanya et al. [13] investigated the effect of a transverse the buoyancy term in the momentum equation considered var- magnetic field on the flow of viscous incompressible fluid flow- ies with temperature whereas the density appearing elsewhere ing through a channel subjected to periodic heating using Ado- in these equations is considered constant (Boussinesq’s mian decomposition method. They found that both skin approximation). Under the usual Boussinesq’s approximation, friction and Nusselt number decrease at the wall with increas- the equation of state is assumed to be ing value of magnetic field parameter. Hossain and Floryan [14] studied pressure-driven flow in a horizontal channel exposed to thermal modulations and concluded that the heat- ing results in a significant reduction in the pressure gradient required to drive the flow when compared to a similar flow in an isothermal channel. In other related works, Shiu and Wu [15] discussed tran- = sient heat transfer in annular fins of various shapes with their bases subjected to a heat flux varying as a sinusoidal function of time. Furthermore, Cole and Crittenden [16] investigated Steady-Periodic heating of cylinder using Green’s function ⃗ approach. =0 The objective of the present investigation was to present a theoretical analysis of natural convection flow in a vertical ( , )= 1 + 2 cos( ) tube inspired by time-periodic heating of the surface of the = =0 tube. Analytical solution of the momentum and energy =0 equations is derived in terms of modified Bessel’s function of first kind. Line and contour graphs are plotted to inves- tigate the effect of periodic heating as well as Prandtl Figure 1 Schematic diagram of the problem. number. Please cite this article in press as: B.K. Jha, M.O. Oni, Natural convection flow in a vertical tube inspired by time-periodic heating, Alexandria Eng. J. (2016), http:// dx.doi.org/10.1016/j.aej.2016.08.025 Natural convection flow in a vertical tube 3 q ¼ q ½ À bð À Þ ð Þ ð Þ¼ ð Þ 0 1 T T0 1 F R 1 12 Using Boussinesq’s approximation, the governing continu- "#ÀÁpffiffiffiffiffiffi ÀÁpffiffiffiffiffiffiffiffiffiffiffi I R iSt I R iPrSt ity, momentum and energy equations describing the present ð Þ¼ 1 0 ÀÁpffiffiffiffiffiffi À 0 ÀÁpffiffiffiffiffiffiffiffiffiffiffi ð Þ B R ð À Þ 13 physical situation can be written respectively in dimensional iSt Pr 1 I0 iSt I0 iPrSt form as follows: 1 @u AðRÞ¼ ½1 À R2ð14Þ ¼ 0 ð2Þ 4 @r where I0 is the modified Bessel’s function of first kind of order @u 1 @ @u zero. ¼ m r þ gbðT À T Þð3Þ @t r @r @r 0 The phase ðwÞ of the periodic temperature and the phase ðvÞ of the periodic velocity are obtained by [5] @T 1 @ @T ¼ a r ð4Þ À ImagðGðRÞÞ @t r @r @r w ¼ tan 1 ð15Þ RealðGðRÞÞ subject to the following relevant boundary conditions: À ImagðBðRÞÞ @ @ v ¼ 1 ð Þ u ¼ 0 T ¼ 0 r ¼ 0 tan ð ð ÞÞ 16 @r @r ð5Þ Real B R u ¼ 0 T ¼ T þ T cosðxtÞ r ¼ a 1 2 While the amplitudes of the periodic temperature and peri- In order to solve Eqs. (2)–(5), we use the following expres- odic velocity are given respectively by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sions to separate the velocity and temperature respectively into j j¼ ð ð ð ÞÞ2 þ ð ð ÞÞ2Þ ð Þ steady and periodic parts respectively by Wang [4]: G Real G R Imag G R 17 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi gba x ð ; Þ¼ ½ð À Þ ð Þþ ð Þ i tðÞ 2 2 u r t m T1 T0 A R T2B R e 6 jBj¼ ðRealðBðRÞÞ þ ImagðBðRÞÞ Þ ð18Þ ixt The phase of the velocity v is the angle between the veloc- Tðr; tÞ¼T0 þ½ðT1 À T0ÞFðRÞþT2GðRÞe ð7Þ ities Bi and BR while the amplitude jBj denotes the absolute where AðRÞ; FðRÞ represent the steady parts and BðRÞ; GðRÞ velocity at any given point in the tube.
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