Theoretical study of vibrating effect on heat transfer in laminar flow

Abstract

Green’s function method was adopted to study the problem of vibrating effect on heat transfer in laminar flow with constant flux and the influence of Prandtl number and the vibrating frequency on the heat transfer characteristics was investigated. The results show that the variation of the frequency leads to a different distribution of the unsteady velocity and temperature; with a lower frequency, the vibrating will weaken the heat transfer, but the heat transfer will be enhanced with a higher frequency. A lower Prandtl number leads to a strenuous variation of heat transfer.

Keywords

vibrating / full developed / heat transfer

Introduction

Vibration widely exists in engineering applications such as a tube vibrating in heat exchangers. Much attention has been paid to the heat transfer of the vibrating cylinder. Lemlich [1] and Cheng et al. [2] have experimentally studied the effect of cylinder vibrating in transverse direction on heat transfer. Cheng et al. [3], Fu and Tong [4], and Gau et al. [5] have conducted numerical investigations on forced convection from a heated cylinder with vibration in a uniform stream. The results show that the vibration can enhance heat transfer and demonstrate the distribution of the velocity field.

There has been much information about the mechanics of heat transfer in the flow out of a vibrating cylinder. However, no study has ever been conducted on the heat transfer characteristics of the flow in a vibrating tube. The aim of this paper is to demonstrate the influence of the Prandtl number and the vibrating frequency on the heat transfer of the flow in a tube with the tube vibrating in the axial direction. The problem is similar to the case of heat transfer in a tube with pulsating flow. Based on the analysis of the pulsating flow field by Hemeada et al. [6], the flow field in the vibrating tube is assumed to consist of a steady flow induced by a constant pressure head and a pure unsteady flow induced by the tube vibrating. Similarly, the temperature field is assumed to consist of a steady temperature and a pure unsteady temperature.

Theoretical analyses

The studied object is shown in Fig. 1. For fully developed laminar flow, the axial vibration of the tube leads to the variation of the velocity component in the axial direction. However, the radial velocity equals zero. The constant heat flux at the tube wall is chosen as the boundary condition to obtain a theoretical solution. The variation of thermophysical properties and viscous dissipation are neglected for simplicity.

For convenience of study, a non-inertial reference frame is established on the vibrating tube. With these analyses, the governing equations and the boundary condition are written as:

(1){\displaystyle \frac{\partial u}{\partial t}}=-{\displaystyle \frac{\partial p}{\rho \partial x}}+\nu \left({\displaystyle \frac{{\partial }^{\mathrm{2}}u}{\partial r^{\mathrm{2}}}}+{\displaystyle \frac{\partial u}{r\partial r}}\right)+\mathrm{i}\omega U{\mathrm{e}}^{\mathrm{i}\omega t},

(2)r=\mathrm{0}:{\displaystyle \frac{\partial u}{\partial r}}=\mathrm{0},r=r_{\mathrm{0}}:u=\mathrm{0},

(3){\displaystyle \frac{\partial T}{\partial t}}+u{\displaystyle \frac{\partial T}{\partial x}}=a{\displaystyle \frac{\mathrm{1}}{r}}{\displaystyle \frac{\partial }{\partial r}}\left(r{\displaystyle \frac{\partial T}{\partial r}}\right),

(4)x=\mathrm{0}:T=T_{\mathrm{0}},r=\mathrm{0}:{\displaystyle \frac{\partial T}{\partial r}}=\mathrm{0},r=r_{\mathrm{0}}:\lambda {\displaystyle \frac{\partial T}{\partial r}}=q.

The variables are nondimensionalized as

r^{\ast }={\displaystyle \frac{r}{r_{\mathrm{0}}}},\beta ={\displaystyle \frac{U}{U_{\mathrm{m}}}},x^{\ast }={\displaystyle \frac{x}{r_{\mathrm{0}}\mathit{P}\mathit{r}\mathit{R}\mathit{e}}},{\theta }^{*}={\displaystyle \frac{T-T_{\mathrm{0}}}{q{r}_{\mathrm{0}}/\lambda }},u^{*}={\displaystyle \frac{u}{U_{\mathrm{m}}}},t^{*}={\displaystyle \frac{ta}{r_{\mathrm{0}}^{\mathrm{2}}}},{\omega }^{*}={\displaystyle \frac{\omega r_{\mathrm{0}}^{\mathrm{2}}}{a}}.

where, U_m=-{\displaystyle \frac{\mathrm{1}}{\mathrm{4}\mu }}{\displaystyle \frac{dp}{dx}}{r}_{\mathrm{0}}^{\mathrm{2}}, Pr={\displaystyle \frac{\nu }{a}}, Re={\displaystyle \frac{U_m{r}_{\mathrm{0}}}{\nu }}.

The velocity distribution can be written as [6]

(5)u^{*}=u_{\mathrm{s}}^{*}+\beta u_{\mathrm{t}}^{*},

where, u_{\mathrm{s}}^{*} represents a steady flow induced by a constant pressure head, and it is a Poiseuille flow,

(6)u_{\mathrm{s}}^{*}=\mathrm{1}-r^{*\mathrm{2}}.

The governing equations and the boundary condition followed by the unsteady flow are

(7){\displaystyle \frac{\partial u_{\mathrm{t}}^{\ast }}{\partial t^{\ast }}}=\mathit{Pr}\left({\displaystyle \frac{{\partial }^{\mathrm{2}}{u}_{\mathrm{t}}^{\ast }}{\partial r^{\ast \mathrm{2}}}}+{\displaystyle \frac{\partial u_{\mathrm{t}}^{\ast }}{r^{\ast }\partial r^{\ast }}}\right)+\mathrm{i}{\omega }^{\ast }{\mathrm{e}}^{\mathrm{i}{\omega }^{\ast }{t}^{\ast }},

(8)r^{\ast }=\mathrm{0}:{\displaystyle \frac{u_t^{\ast }}{\partial r}}=\mathrm{0};r^{\ast }=\mathrm{1}:u_t^{\ast }=\mathrm{0.}

The solution can be written as

(9)u_{\mathrm{t}}^{*}=g(r){\mathrm{e}}^{\mathrm{i}{\omega }^{*}{t}^{*}},

where, g(r)=\mathrm{1}-{\displaystyle \frac{J_{\mathrm{0}}(\sqrt{-i{\omega }^{*}/Pr}{r}^{*})}{J_{\mathrm{0}}(\sqrt{-i{\omega }^{*}/Pr})}}.

In contrast to Eq. (5), the temperature can be written as

(10){\theta }^{*}={\theta }_{\mathrm{s}}^{*}+\beta {\theta }_{\mathrm{t}}^{*}.

The energy equation can be written as

(11)u_{\mathrm{s}}^{*}{\displaystyle \frac{\partial {\theta }_{\mathrm{s}}^{*}}{\partial t}}={\displaystyle \frac{\mathrm{1}}{r^{*}}}{\displaystyle \frac{\partial }{\partial r^{*}}}\left(r^{*}{\displaystyle \frac{\partial {\theta }_{\mathrm{s}}^{*}}{\partial r}}\right),

(12){\displaystyle \frac{\partial {\theta }_{\mathrm{s}}^{*}(x,\mathrm{1})}{\partial r^{*}}}=\mathrm{1},{\theta }_{\mathrm{s}}^{*}(\mathrm{0},r)=\mathrm{0},{\displaystyle \frac{\partial {\theta }_{\mathrm{s}}^{*}(x,\mathrm{0})}{\partial r^{*}}}=\mathrm{0},

(13){\displaystyle \frac{\partial {\theta }_{\mathrm{t}}^{*}}{\partial t^{*}}}+u_{\mathrm{t}}^{*}{\displaystyle \frac{\partial {\theta }_{\mathrm{s}}^{*}}{\partial x^{*}}}={\displaystyle \frac{\mathrm{1}}{r^{*}}}{\displaystyle \frac{\partial }{\partial r^{*}}}\left(r^{*}{\displaystyle \frac{\partial {\theta }_{\mathrm{s}}^{*}}{\partial r}}\right),

(14){\displaystyle \frac{\partial {\theta }_{\mathrm{t}}^{*}(\mathrm{1},t^{*})}{\partial r^{*}}}=\mathrm{0},{\displaystyle \frac{\partial {\theta }_{\mathrm{t}}^{*}(\mathrm{0},t*)}{\partial r^{*}}}=\mathrm{0.}

For fully developed laminar flow in a tube, {\displaystyle \frac{\partial {\theta }_s^{*}}{\partial x}}=\mathrm{4}. Assuming {\theta }_t^{\ast }=\mathrm{4}f(r)e^{i{\omega }^{*}{t}^{*}}, Green’s function satisfy [7]

(15)r^{\ast }{G}^{"}+G^{\prime }-r^{\ast }\mathrm{i}{\omega }^{\ast }G=-\delta (r^{\ast },r^{"}),

(16){{\displaystyle \frac{\partial G}{\partial r^{*}}}|}_{r^{\ast }=\mathrm{0}}=\mathrm{0},{{\displaystyle \frac{\partial G}{\partial r^{*}}}|}_{r^{\ast }=\mathrm{1}}=\mathrm{0},

(17){G|}_{r^{"}-\mathrm{0}}^{r^{"}+\mathrm{0}}=\mathrm{0},{G^{\prime }|}_{r^{"}}={\displaystyle \frac{-\mathrm{1}}{r^{\ast }}}.

The solution of Green’s function is

Therefore,

(18)f=-{\displaystyle {\int }_{\mathrm{0}}^{\mathrm{1}}G(r^{\ast },r^{"})}{r}^{"}g(r^{"})\mathrm{d}{r}^{"}.

The numerical integration method is adopted in the solution procedure because there is an integration of the Bessel function.

Results and discussion

Instantaneous velocity and temperature discussion

Figures 2 and 3 show the distribution of the unsteady velocity and temperature. From Fig. 2, it is observed that the unsteady velocity increases with the radial distance decreasing when {\omega }^{\ast }=\mathrm{1} and {\omega }^{\ast }=\mathrm{10}; when {\omega }^{\ast }=\mathrm{50} the maximum is obtained in the middle space. As shown in Fig. 3, the unsteady temperature has little variation with the radial distance varying in a lower frequency; but in a higher frequency, the temperature at the two ends is evidently higher than that in the middle space.

Average Nusselt number

Average Nusselt number is defined as

(19)\overline{Nu}={\displaystyle \frac{\mathrm{2}}{{\theta }_{\mathrm{w}\mathrm{s}}^{*}+{\beta }^{\mathrm{2}}\overline{{\theta }_{\mathrm{w}\mathrm{t}}^{*}}-{\theta }_{\mathrm{b}\mathrm{s}}^{*}-{\beta }^{\mathrm{2}}\overline{{\theta }_{\mathrm{b}\mathrm{t}}^{*}}}},

where, {\theta }_{\mathrm{ws}}^{*} and {\theta }_{\mathrm{bs}}^{*} represent dimensionless wall temperature and bulk average temperature corresponding to the steady state, respectively. For heat transfer with the steady fully developed laminar flow in a stationary tube,

(20)Nu={\displaystyle \frac{\mathrm{2}}{{\theta }_{\mathrm{w}\mathrm{s}}^{*}-{\theta }_{\mathrm{b}\mathrm{s}}^{*}}}=\mathrm{4.36.}

Therefore,

(21){\theta }_{\mathrm{w}\mathrm{s}}^{*}-{\theta }_{\mathrm{b}\mathrm{s}}^{*}=\mathrm{0.46.}

The wall average temperature corresponding to the tube vibrating is

(22)\overline{{\theta }_{\mathrm{w}\mathrm{t}}^{*}}={\displaystyle \frac{\mathrm{1}}{\mathrm{2}\mathrm{\pi }/{\omega }^{*}}}{\displaystyle {\int }_0^{\mathrm{2}\mathrm{\pi }/{\omega }^{*}}{\theta }_{\mathrm{s}}^{*}}(\mathrm{0},t^{*})\mathrm{d}{t}^{*}=\mathrm{0.}

The average bulk temperature corresponding to the tube vibrating is

(23)\overline{{\theta }_{\mathrm{b}\mathrm{t}}^{*}}={\displaystyle \frac{\mathrm{1}}{\mathrm{2}\mathrm{\pi }/{\omega }^{*}}}{\displaystyle \frac{{\displaystyle {\int }_0^1{\displaystyle {\int }_0^{2\pi /{\omega }^{*}}R{\theta }_t^{*}}(r,t^{*})R(u_t^{*}{e}^{i{\omega }^{*}{t}^{*}})d{t}^{*}d{r}^{*}}}{{\displaystyle {\int }_0^1{u}_s^{*}rd{r}^{*}}}}.

\mathrm{\Delta }Nu is defined as

(24)\Delta Nu={\displaystyle \frac{\overline{N{u}_{\mathrm{2}}}-N{u}_s}{N{u}_s}}.

From Figs. 4 and 5, it is observed that with a lower frequency (about lower than 40), \mathrm{\Delta }Nu is below zero, and a minimum of \mathrm{\Delta }Nu exists for a certain \beta and Pr; with a higher frequency, \mathrm{\Delta }Nu is above zero, that is, the heat transfer is enhanced. The reason for this phenomenon can be seen from Figs. 2 and 3. With a lower frequency, most of the unsteady temperature caused by the tube vibrating is negative and the temperature field is more non-uniform, hence the average temperature of the flow decreases while the wall average temperature in a period does not change, and consequently, the heat transfer is weakened; but for a higher frequency, the increased temperature around the wall is obviously higher than that around the axis, that is, the vibration leads to the uniform distribution of the flow temperature, and as a result, the average flow temperature increases and the heat transfer is enhanced.

It is also observed from Figs. 4 and 5 that the minimum of Nu is lower with a higher \beta and a lower Pr. A lower Pr predicates lower kinematic viscosity which will increase the temperature variation with the same \beta and {\omega }^{*}, and higher thermal diffusivity which will make for the influence of the vibrating wall on the main flow.

Conclusions

The vibrating effect on heat transfer in a tube with constant heat flux is analytically studied, and the unsteady temperature and velocity distribution are obtained. The results are as follows:

1) With a lower frequency, the unsteady velocity decreases with the radial distance increasing, and the heat transfer is weakened.

2) With a higher frequency, the maximum of the unsteady velocity appears in the middle space, and heat transfer is enhanced.

3) A higher \beta and a lower Pr will lead to strenuous variation of Nu.

References

[1] Lemlich R. Effect of vibration on natural convective heat transfer. Industrial and Engineering Chemisty, 1955, 47(6): 1175–1181

[2] Cheng C H, Chen H N, Aung W. Experimental study on the effect of transverse oscillation on convection heat transfer from a circular cylinder. Journal of Heat Transfer1997, 119(3): 474–482

[3] Cheng C H, Hong J L, Aung W. Numerical prediction of lock-on effect on convective heat transfer from a transversely oscillating circular cylinder. International Journal of Heat and Mass Transfer, 1997, 40(8): 1825–1834

[4] Fu Wushung, Tong Baohong. Numerical investigation of heat transfer from a heated oscillating cylinder in a cross flow. International Journal of Heat and Mass Transfer, 2002, 45(14): 3033–3043

[5] Gau C, Wu J M, Liang C Y. Heat transfer enhancement and vortex flow structure over a heated cylinder oscillating in the cross flow direction. Journal of Heat Transfer, 1999, 121(4): 789–795

[6] Hemeada H N, Abdel-Rahim, Manou H. Theoretical analysis of heat transfer in laminar pulsating flow. International Journal of Heat and Mass Transfer, 2002, 45(8): 1667–1680

[7] Sun Dexing. Advanced Heat Transfer. Bejing: China Architecture & Building Press, 2005, 160–163

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