굴곡관에서 박동성 층류에 의한 열전달 증진시

Strouhal 수의  영향

이범석, 강인석

포항공과 대학교 화학공학과

Effect of the Strouhal number on heat-transfer enhancement

by pulsatile laminar flow in a wavy channel

Beom Seok Lee, In Seok Kang

Department of Chemical Engineering, POSTECH

Introduction

Heat-transfer enhancement by use of a well-defined laminar flow is of theoretical importance as well as the importance in applications. As one way of enhacement technique, pulsatile flows has been used in the systems of periodic geometries. Since a pulsatile flow can be driven by a periodic time dependent pressure gradient, effective mixing occurs in the channel and it is expected that heat transfer is enhanced. But it is not yet clear whether the heat-transfer enhancement really occurs or not.

In the present work, we analyze the problem of heat-transfer enhancement due to pulsatile laminar flow in a wavy channel with special attention given to the effects of geometric frequency (□w) and the Strouhal number (St) on the heat transfer rate.

Problem Formulation

An axisymmetric wavy channel is used as the basic geometry. In Fig.1, the model system is shown and the important geometric parameters are the amplitude ([eqn: gamma``]) and the wave frequency ([eqn: rm Omega_w``]). The amplitude is defined as the ratio of the height of valley to the radius of channel and the wave frequency is defined in the equation,

[eqn: y``=``R`[ (1-gamma)` + `gamma` cos(rm {Omega_w}it`pi`x)]``]

which describes the shape of the channel.

Fig. 1. The model system in which [eqn: rm Omega_w``]is the wave frequency and[eqn: gamma``] is defined by H/R.

The governing equations are the continuity and the Navier-Stokes equations, and the heat-transfer equations. To put the time-dependent governing equations of flow and temperature fields into the nondimensional form, we first define the characteristic scales as:

[eqn: t_c``=``1 over Omega_h`,```l_c``=``R,```del_c``=``1 over l_c`,```u_c``=``u_mean`,``` p_c``=``rho{u_c}^2```rm and` it T^*``=``{T-T_w}over{T_i`-T_w}] ,

where [eqn: Omega_h`] is the hydrodynamic or oscillatory frequency, [eqn: u_{mean}`] the mean velocity at the inlet, [eqn: T_i`] the inlet temperature and [eqn: T_w`] the wall temperature. The nondimensionalized equations are

[eqn: {bold del^*} cdot {bold u^*}``=``0,]	(1)

[eqn: St{partial bold u^*}over {partial t^*}+ bold{u^* cdot del^* u^*}``=`` - {{bold del} p^*} + 1 over Re {del^*}^2{bold u^*},``]             (2)

[eqn: St{partial T^*}over{partial t^*} + {bold u^*}cdot{bold nabla}^* T^*` `=``1 over {Pe}{del^*}^2 T^*.``]                     (3)

As a result, we have three important flow parameters. One is the Reynolds number [eqn: Re`], and another is the Peclet number [eqn: Pe`]. The last number is an important parameter which is called the Strouhal number [eqn: St``]. It represents the ratio of the unsteady acceleration to the steady acceleration and it is the measure of the unsteadiness caused by pulsatile flow. The number is defined as [eqn: St``=``(Omega_h`l_c`)/ u_c``].

Fig. 2. The boundary conditions of the heat-transfer equation.

Our system has four boundaries ([eqn: partial Omega_1`,``partial Omega_2`,``partial Omega_3```and``partial Omega_4`]) as shown in Fig.2. At the solid surface ([eqn: partial Omega_1``]), the no slip condition holds. At the inlet ([eqn: partial Omega_2``]), the pulsatile velocity profile is imposed. The symmetry condition at the center line ([eqn: partial Omega_3``]) is due to the axisymmetry of the flow in the channel. For the outlet ([eqn: partial Omega_4``]), we assume that the velocity field is periodically fully developed in the sense that the velocity profile at the outlet is identical to that at the one wavelength preceding section. Four boundary conditions of heat-transfer equation are given in Fig.2. At [eqn: partial Omega_1``]and [eqn: partial Omega_2``], constant wall and constant inlet temperature conditions are assumed. On [eqn: partial Omega_3``], we can apply axisymmetry condition and on [eqn: partial Omega_4``], the fully-developed condition is used.

We use the unsteady pulsatile velocity profile imposed at the vessel inlet as same as that given in Jeong and Kang[1].

Numerical Analysis

The ADI (Alternating Direction Implicit) method is used as a time-marching method for both the governing equations of flow and temperature field. We need only to solve [eqn: m times m`] tridiagonal system if ADI method is applied. The 3-point central difference is used for spatial derivatives. It is necessary to define a tool to measure heat-transfer enhancement. First we define the local Nusselt number [eqn: N`u_x`] which represents the local heat transfer rate. From [eqn: N`u_x`], it is possible to compute the spatio-temporal mean Nusselt number, namely, [eqn: <N`u>_t`] which is defined as

[eqn: <N`u>_s``=``1 over A int_A``N`u_x`dA]     (4)       [eqn: <N`u>_t``=``1 over T{int_0}^T`<N`u>_s`dt`],       (5)

where A is the area of heat transfer surface and T is the period of the imposed sinusoidal pressure gradient (here [eqn: T``=``2 pi`]).

Using the enhancement factor [eqn: E`], we compare the enhancement of heat transfer for the unsteady flow with for the steady flow field and prove the existence of heat transfer enhancement by the Strouhal number and the geometric wave length. The enhacement factor is defined as

[eqn: E``=``{N`u_{unsteady}}over{N`u_{steady}}``=``{N`u(Pe,St,epsilon)}over{N`u(Pe,epsilon=0)}],	(6)

where [eqn: epsilon`] is modulatory amplitude of flow.

Results and Discussions

First, we solved the equations for the flow field and heat transfer in the case of steady laminar flow to compare the results with those for the pulsatile laminar flow cases. It is known that the steady laminar flow in a wavy channel does not increase the heat transfer rate. Thus, it is necessary to introduce pulsatile flow to introduce unique mixing characteristics.

We solved the problem for the system with amplitude [eqn: gamma``=``0.2`] and the geometric wave length [eqn: Omega_w``=``1.0`]. Fig. 3 shows the isotherms and the flow field at a typical time (t=2[eqn: pi`]) with Re=250, Pr=1 and St=2. At that time the reverse flow is noticeable and it is expected that the thermal gradient at the wavy wall is large.

As shown in Fig. 4, the pulsatile laminar flow can enhance heat transfer as much as 80% at Re=250, St=2 for the small values of the modulatory amplitude ([eqn: epsilon=0.2`]) and we observe the existence of the optimal Strouhal number ([eqn: St vert_{rm opt}]) or the optimal hydrodynamic frequency ([eqn: Omega_h vert_{rm opt}`]) to maximize the heat transfer enhancement. This means that heat transfer depends on the value of the Strouhal number sensitively. It is so called 'resonant heat-transfer enhancement' (see the reference [2].) As the Strouhal number goes to the order of [eqn: O(`10^1`)] or [eqn: O(10^-1`)], heat transfer rate approaches the values of the steady states.

As Re increases, the maximum value of enhancement factor [eqn: E vert_max] becomes larger and the maximum point shifts to the left in Fig.4. Therefore the heat-transfer enhancement is also a function of Re as well as of St. According to the change of [eqn: Omega_w`], there is a maximum value of enhancement factor which has the optimal Strouhal number for each case in Fig. 5. [eqn: E vert_max] occurs at [eqn: Omega_w`]=1.5 and St=3.0. It shows that the more unsteadiness should be given to flow field to get the maximum value of E as [eqn: Omega_w`] becomes larger. However when we compare the values of [eqn: <N`u>_t`] for each wave frequency in Fig. 5, the optimum wave frequency is [eqn: Omega_w vert_{rm opt}] =1.0 at St=2 rather than [eqn: Omega_w`]=1.5 at St=3 as shown in Table 1.

Although we have seen only the limited results, the numerical solutions clearly show that the heat transfer can be enhanced by pulsatile flows. But the numerical results also show that the heat transfer enhancement can be achieved only by selecting the flow parameters very carefully.

Literature cited

[1] Y. S. Jeong and I. S. Kang, Pulsatile flow in stenosed blood vessels, KJChE, 12,  No. 5, 540-550 (1995).

[2] N. K. Ghaddar, M. Magen, B. B. Mikic and A. T. Patera, Numerical investigation of incompressible flow in grooved channels. Part 2, Resonance and oscillatory heat-transfer enhancement, J. Fluid Mech., 168, 541-567 (1986).

[eqn: Omega_w``]	<Nu>st	[eqn: St vert_{rm opt}``]	[eqn: <N`u>_t vert_{rm opt}``]
0.25	3.416	2	3.861
0.50	3.297	1	4.921
0.75	3.187	2	5.404
1.00	3.017	2	5.497
1.25	2.951	2	5.350
1.50	2.838	3	5.348

Table 1. The steady Nusselt number (<Nu>st) and the optimum spatio-temporal mean Nusselt number ([eqn: <N`u>_t vert_{rm opt}``]) for several wave frequencies.

Fig. 3. Isotherms and flow field at Re=250 and St=2.

Fig.4. The effect of St on heat transfer	Fig. 5. The effect of St on heat transfer for

when [eqn: Omega_w`]=1.0.	various geometric wave frequencies.