Mathematical Problems in Engineering

Volume 2015, Article ID 953604, 11 pages

http://dx.doi.org/10.1155/2015/953604

## Mathematical and Numerical Analysis of Heat Transfer Enhancement by Distribution of Suction Flows inside Permeable Tubes

Mechanical Engineering Department, King Abdulaziz University, P.O. Box 80204, Jeddah 21589, Saudi Arabia

Received 1 February 2015; Revised 24 March 2015; Accepted 27 March 2015

Academic Editor: Rama S. R. Gorla

Copyright © 2015 A.-R. A. Khaled. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Heat transfer enhancement in permeable tubes subjected to transverse suction flow is investigated in this work. Both momentum and energy equations are solved analytically and numerically. Both solutions based on negligible entry regions are well matched. Two different suction velocity distributions are considered. A parametric study including the influence of the average suction velocity and the suction velocity profile is conducted for various Peclet numbers. It is found that enhancement of heat transfer over that in impermeable tubes is only possible with large Peclet numbers. This enhancement increases as suction velocities towards the tube outlet increase and as those towards the tube inlet decrease simultaneously. The identified enhancement mechanisms are expanding the entry regions, increasing the transverse advection, and increasing the downstream excess temperatures under same transverse advection. The average suction velocity that produces maximum enhancement increases as the Peclet number increases until it reaches asymptotically its uppermost value at large Peclet numbers. The maximum reported enhancement ratios for the exponential and linear suction velocity distributions are 17.62-fold and 14.67-fold above those for impermeable tubes, respectively. This work demonstrates that significant heat transfer enhancement is attainable when the suction flow inside the permeable tubes is distributed properly.

#### 1. Introduction

The study of fluid flow and heat transfer inside permeable tubes or channels exposed to surface suction is important to many industrial applications. These applications include transpiration cooling where channel surfaces are cooled by passing cooled fluid through the pores of these surfaces, controlling boundary layers over surfaces of airplane wings or turbine blades by injection or suction of fluid at theses surfaces, lubrication of permeable bearings, fluid filtration processes, cooling of combustion chambers exhaust ports, and cooling of porous walled reactors [1]. Accordingly, the aim of the present work is to investigate heat transfer enhancement inside permeable tubes subjected to nonuniform surface suction.

Bergles [2] indicated that surface suction is an effective technique that can be used to enhance heat transfer [2, 3]. He indicated that improvement in heat transfer coefficient is expected to reach several hundred percent for laminar flow with suction at the solid boundary [2–5]. He pointed out that this enhancement is due to reduction in the boundary layer thickness [2]. This can be clearly seen for external flows [4–8]. However, surface suction within internal flows tends to reduce the mean velocity inside the tube or the channel. This effect may thicken the boundary layer and causes impediments of both flows near the boundary and heat transfer rate. Therefore, the novelty of the present work is to explore the various conditions that may reveal heat transfer enhancement inside permeable tubes exposed to surface suction.

Among initial works that analyzed the problem of fluid flow and heat transfer inside permeable tubes or channels with surface suction are the works of Kinney [9], Pederson and Kinney [10], Raithby [11], and Sorour and Hassab [12]. These works illustrated the variations of the temperature profile and Nusselt number with wall Reynolds number in porous tubes or channels. These works and many others were the motivations behind recent works that accounted for all possible hydrodynamic conditions on heat transfer inside channels subjected to wall suction. For example, Sorour et al. [13] and Bubnovich et al. [14] analyzed dynamically and thermally the developing flow inside a channel subjected to nonuniform suction at one wall. Hwang et al. [15] investigated numerically forced laminar convection in the entrance region of a square duct subjected to uniform mass transpiration. Makinde et al. [1] analyzed heat transfer in channels exposed to wall suction in presence of nanofluids with both wall slip and viscous dissipation effects. All of the aforementioned works and many similar ones in the literature did not explore the heat transfer enhancement due to surface suction inside permeable tubes. Thus, the motivation behind the present work is to enrich the literature with a study about the role of surface suction and its profile inside permeable tubes on heat transfer enhancement.

Promoting flow close to the energy exchanging boundary usually results in heat transfer enhancement [2, 16]. Meanwhile, the internal flow impedance near this boundary increases as suction Reynolds number () increases [9, 11]. For permeable circular tubes, the internal flow starts to separate from the boundary when due to the adverse pressure gradient caused by reduction in the mean fluid velocity. To avoid this instability condition and to sustain maximum velocity close to tube surface, the suction Reynolds number must be . This constraint can be shown using Kinney [9] and Raithby [11] works to yield universal surface friction coefficient to be at least equal to 95% of its maximum value when . Accordingly, the flow close to the energy exchanging boundary can be kept maximally promoted under this constraint. Therefore, the present work is concerned with heat transfer enhancement inside permeable tubes exposed to surface suction with .

In the next sections, flow and heat transfer inside a preamble tube subjected to internal suction flow are modelled and analyzed. The surface suction velocity is considered to have either linear or exponential profile distributions. Both momentum and thermal energy transfer equations are solved using various analytical and numerical methods. Different heat transfer enhancement indicators are computed. Both analytical and numerical computations of these indicators are validated under an applicable constraint and using early studies. An extensive parametric study has been conducted in order to identify and explore the influence of average suction velocity, suction velocity profile, and Peclet number on the heat transfer enhancement indicators.

#### 2. Problem Formulation

##### 2.1. Modeling of Flow and Heat Transfer inside the Permeable Tube

Consider a tube of length and inner diameter . The tube wall is permeable so as to allow fluid suction at its boundary as shown in Figure 1. Both the flow inside the tube and that through the permeable boundary are considered to be laminar flows. The density, specific heat, thermal conductivity, and the dynamic viscosity of the fluid are , , , and , respectively. The dimensionless continuity, momentum, and energy equations of the fluid are given by [17–19]:where and are the dimensionless axial and radial velocities, respectively. and are the dimensionless axial and radial positions, respectively. and are the tube aspect ratio and the reference Peclet number, respectively. and are the dimensionless pressure and dimensionless temperature fields, respectively. The dimensionless variables and parameters used in (1)–(3) are given bywhere , , , and are reference axial velocity, inlet pressure, outlet pressure, and inlet temperature, respectively. is the constant heat flux applied at the inner surface of the tube. The boundary conditions of (1)–(3) are given bywhere , is the dimensional local suction velocity at the tube inner surface. is the mean dimensionless axial velocity at any given cross section. Using Kinney [9] and Raithby [11] works, it can be shown that with , the convective terms in (2) can be neglected. The aforementioned range results in less than 5.0% relative error associated with calculating the universal wall friction coefficient by neglecting the convective terms. The previous constraint can practically be satisfied for high aspect ratio tubes (), viscous fluids, or with small suction velocities. Accordingly, the solution of (2) in absence of the convective terms with boundary conditions given by (10a) and (11a) is the following:where . Substituting (12) in (1) and solving the resulting equation yield to the following distribution of the dimensionless radial velocity:where . Applying the boundary condition given by (11b) results into the following differential equation: