Conjugate heat transfer in a vertical channel filled with a nanofluid adjacent to a heat generating solid domain

Flavius Pătrulescu$^\ast $ and Teodor Groşan$^\S $

April 15, 2010.

$^\ast $“Tiberiu Popoviciu” Institute of Numerical Analysis, Romanian Academy, Cluj-Napoca, Romania, e-mail:fpatrulescu@ictp.acad.ro. The work of this author has been partially supported by contract POSDRU/88/1.5/S/60185 with “Babeş-Bolyai” University.

$^\S $Faculty of Mathematics and Computer Science, “Babeş-Bolyai” University, Cluj-Napoca, Romania, e-mail:tgrosan@math.ubbcluj.ro.

The effect of thermal dispersion in the conjugate steady free convection flow of a nanofluid in a vertical channel is investigated numerically using a single phase model. Considering the laminar and fully developed flow regime, a simplified mathematical model is obtained. In the particular cases when solid phase and thermal dispersion effects are neglected the problem was solved analytically. The numerical solution is shown to be in excellent agreement with the close form analytical solution. Nusselt number enhancement with the Grashof number, volume fraction, aspect ratio parameter and thermal diffusivity constant increasing has been found.

MSC. 82D80, 76R10

Keywords. nanofluid, vertical channel, free convection, conjugate heat transfer, heat generation

1 Introduction

Heat transfer in channels occurs in many industrial processes and natural phenomena. It has been the subject of many studies for different flow configurations. We mention some practical applications of convective heat transfer in channels: design of cooling systems for electronic devices, insulation, ventilation, grain storage, geothermal energy recover, solar energy collection, etc. Some classical papers, such as by Aung [ 1 ] , Aung et al. [ 2 ] , Barletta [ 3 ] , Kumar et al. [ 8 ] , Vajravelu and Sastri [ 12 ] , are concerned with the evaluation of the temperature and velocity profiles for the vertical parallel-flow fully developed regime. Enhancement of heat transfer is essential in improving performances and compactness of electronic devices. Usual cooling agents (water, oil, etc.) have relatively small thermal conductivities and therefore heat transfer is not very efficient. Suspensions of nanoparticles in fluids improve physical properties and increase the heat transfer. Small fraction of nanoparticles added in a base fluid leads to a large increase of the fluid thermal conductivity. Good description and classification of the nanofluids characteristics can be found in papers such as: Daungthongsuk and Wongwises [ 5 ] , Wang and Mujumdar [ 14 ] and Kumar et al. [ 9 ] . The chaotic movement of the nanoparticles and
sleeping between the fine particles and fluid generate the thermal dispersion effect and this leads to an increase in the energy exchange rates in fluid. Xuan and Roetzel [ 13 ] proposed a thermal dispersion model for a single phase nanofluid. Thermal dispersion effects in nanofluids flow in enclosure using a single phase model were analyzed by Khanafer et al. [ 7 ] and Kumar et al. [ 9 ] for a differentially heated rectangular cavity, Khaled and Vafai [ 6 ] studied the heat transfer enhancement through control of thermal dispersion effects in a horizontal channel, while Mokmeli and Saffar-Avval [ 10 ] numerically studied nanofluid heat transfer in a straight tube. In all these paper the enhancement of heat transfer due to nanofluids special properties was reported. In the present paper, the effect of the thermal dispersion on the steady free convection flow in a long vertical channel, using the fully developed flow assumptions, is investigated using a single phase thermal dispersion model similar with the model considered by Khanafer et al. [ 7 ] .

2 Basic equations

Consider the fully developed steady flow of an incompressible nanofluid in vertical channel. The left wall of the channel have a thickness $b$ and thus we have to consider a conjugate heat transfer problem. The geometry of the problem, the boundary conditions, and the coordinate system is shown in
Fig. $1$.

$\includegraphics[height=3.3572in, width=4.0776in]{figura_matlab.eps}$

Figure 1 Geometry of the problem and the coordinate system.

The fluid flows up in the channel driven by buoyancy forces, so that the flow is due only to the difference in temperature gradient. The flow being fully developed the following relations apply here $v=0$ and $\partial v/\partial y=0$, where $v$ is the velocity in the transversal direction. Thus, from the continuity equation, we get $\partial u/\partial x=0$ so that the velocity along the channel is $u=u(y)$. Based on the fact that the flow is fully developed we can assume that the temperature depends only by $y$, i.e., $T=T(y)$. The physical properties of the nanofluid are considered constant except for density, which is given by the Boussinesq approximation.

We use in this study the heat capacity and the thermal expansion coefficient of the nanofluid given in Kanafer et al. [ 7 ] as:

\begin{equation} (\rho C_p)_{nf}=(1-\phi )(\rho C_p)_{f}+\phi (\rho C_p)_{s},\label{heat-cap} \end{equation}

\begin{equation} (\rho \beta )_{nf}=(1-\phi )\rho _f\beta _f+\phi \rho _s\beta _s,\label{term-exp} \end{equation}

where $\rho $ is the density, $c_p$ is the specific heat at constant pressure, $\phi $ is the volume fraction of suspension particles, $\beta $ is the expansion coefficient, while subscripts $nf$, $f$ and $s$ stand for nanofluid, fluid and solid, respectively.

For the effective viscosity we consider the model proposed by Brinkman [ 4 ] , which is valid for high volume fraction ($\phi {\gt}0.05$):

\begin{equation} \mu _{nf}=\frac{\mu _f}{(1-\phi )^{2.5}},\label{vasc-nan} \end{equation}

where $\mu $ is the dynamic viscosity.

The effective stagnant thermal conductivity is approximated by the Maxwell-Garnetts model, see Wang and Mujumdar [ 14 ] , which applies for spherical type particles:

\begin{equation} \frac{k_{nf}}{k_f}=\frac{k_s+2k_f-2\phi (k_f-k_s)}{k_s+2k_f+\phi (k_f-k_s)},\label{cond-term-nan} \end{equation}

where $k$ is the thermal conductivity.

The effective thermal conductivity includes also the thermal dispersion enhancement

\begin{equation} k_{eff}=k_{nf}+k_d, \label{cond-term-eff} \end{equation}

where the term due to thermal dispersion, $k_d$, is given by, see Khaled and Vafai [ 6 ] :

\begin{equation} k_d=C(\rho C_p)_{nf}|u|\phi L,\label{term-disp} \end{equation}

where $L$ is the thickness of the channel and $C$ is a constant depending on the diameter of the nanoparticle and its surface geometry.

We limit the study in this paper to water based nanofluids containing $Cu$ nanoparticles. Nanofluids thermo-physical properties are shown in the Table $1$, see Oztop and Abu-Nada [ 11 ] .

Property	$H_2O$	$Cu$
$C_p(J/kgK)$	4179	385
$\rho (kg/m^3)$	997.1	8933
$k(W/mK)$	0.613	400
$\alpha \times 10^7(m^2/s)$	1.47	1163.1
$\beta \times 10^{-5}(1/K)$	21	1.67

Table 1 Physical properties of fluid and $Cu$ nanoparticles.

In the assumption of the fully developed flow the governing equations for the flow and heat transfer have the following form:

\begin{equation} \alpha _s\frac{{\rm d}^2 T_s}{{\rm d}y^2}+\frac{q'''_0}{(\rho C_p)_s}=0;\label{en-solid} \end{equation}

\begin{equation} \mu _{nf}\frac{{\rm d}^2 u}{{\rm d}y^2}+(\rho \beta )_{nf}g(T_f-T_0)=0;\label{en-fluid} \end{equation}

\begin{equation} \frac{{\rm d}}{{\rm d}y}(k_{eff}\frac{{\rm d}T_f}{{\rm d}y})=0;\label{moment-fluid} \end{equation}

subject to the boundary conditions:

\begin{equation} T_s|_{y=0}=T_H;\quad T_f|_{y=L}=T_C;\label{fr-lat} \end{equation}

\begin{equation} T_f|_{y=b}=T_s|_{y=b};\label{temp-b} \end{equation}

\begin{equation} k_s\frac{\partial T_s}{\partial y}|_{y=b}=k_{nf}\frac{\partial T_f}{\partial y}|_{y=b};\label{flux-b} \end{equation}

\begin{equation} u(b)=u(L)=0;\label{vit-b} \end{equation}

where $g$ is the gravitational acceleration, $T$ is the temperature, $u$ is the velocity, $q'''_0$ is the heat generation and $\alpha $ is the thermal diffusivity.

In order to solve equations (7)–(9), subject to the boundary conditions (10)-(13), we introduce the following non-dimensional variables used also by Kumar et al. [ 8 ] :

\begin{equation} \Theta _s=\frac{k_s(T_s-T_0)}{q'''_0 L^2},\quad \Theta _f=\frac{k_{s}(T_f-T_0)}{q'''_0 L^2},\quad Y=\frac{y}{L},\quad U=\frac{u}{U_c},\label{adim} \end{equation}

where $U_c$ and $T_0$ are the characteristic velocity and temperature given by:

\begin{equation} T_0=\frac{T_H+T_C}{2},\quad U_c=\frac{g\beta _f(\frac{q'''_0L^2}{k_{s}})L^2}{\nu _{f}}.\label{marimi-caract} \end{equation}

Using (14) in equations (7)-(9) we obtain the following dimensionless ordinary differential equations:

\begin{equation} \frac{{\rm d}^2 \Theta _s}{{\rm d} Y^2}+1=0;\label{en-adim-solid} \end{equation}

\begin{equation} \frac{{\rm d}^2U}{{\rm d} Y^2}+\lambda _{\phi }\Theta _f=0;\label{en-adim-fluid} \end{equation}

\begin{equation} \frac{{\rm d}}{{\rm d}Y}[(k_{\phi }+C_{\phi }|U|)\frac{{\rm d}\Theta _f}{{\rm d}Y}]=0;\label{mom-adim-fluid} \end{equation}

subject to

\begin{equation} \Theta _s|_{Y=0}=q; \end{equation}

\begin{equation} \Theta _s|_{Y=r}=\Theta _f|_{Y=r}; \end{equation}

\begin{equation} \frac{{\rm d}\Theta _s}{{\rm d}Y}|_{Y=r}=K\frac{{\rm d}\Theta _f}{{\rm d}Y}|_{Y=r}; \end{equation}

\begin{equation} \Theta _f|_{Y=1}=-q; \end{equation}

\begin{equation} U(r)=U(1)=0;\label{fr-viteza} \end{equation}

where:

\begin{equation} r=\frac{b}{L},\quad \lambda _{\phi }=(1-\phi )^{2.5}[(1-\phi )+\phi \frac{\rho _s\beta _s}{\rho _f\beta _f}],\quad q=\frac{k_s(T_H-T_C)}{2q'''_0L^2}, \end{equation}

\begin{equation} K=\frac{k_{nf}}{k_s},\quad k_{\phi }=\frac{k_{nf}/k_{f}}{1-\phi +\phi \frac{(\rho C_p)_s}{(\rho C_p)_f}},\quad C_{\phi }=C\phi Pr Gr \end{equation}

are constants depending on the properties of the nanofluid and $Pr=\nu _f/\alpha _f$, $Gr=g\beta _f q'''_0L^3/\nu ^2_f$ are Prandtl number and Grashof number, respectively.

The physical quantity of interest in this problem is the Nusselt number, which for the conjugate wall is defined as:

\begin{equation} Nu=\frac{hL}{k_f}|_{y=b}\label{nusselt} \end{equation}

where the convective heat transfer coefficient, $h$, is obtained from the relation:

\begin{equation} -k_{eff}\frac{{\rm d}T}{{\rm d}y}|_{y=b}=h(T|_{y=b}-T_0). \label{nusselt2} \end{equation}

Substituting (27) in (26) the dimensionless form of the Nusselt number becomes:

\begin{equation} Nu=-\frac{k_{nf}}{k_f}\frac{1}{\Theta _f|_{Y=r}}\frac{{\rm d}\Theta _f}{{\rm d}Y}|_{Y=r} \label{nusselt-adim} \end{equation}

3 Results and discussions

In the case when thermal dispersion effect is neglected, i.e. $C=0$, the problem has an analytical solution, which is given by

\begin{equation} \Theta _s(Y)=-\tfrac {Y^2}{2}+(r+Ka_1)Y+q;\label{analitic-temp-solid} \end{equation}

\begin{equation} \Theta _f(Y)=a_1Y-a_1-q;\label{analitic-temp-fluid} \end{equation}

\begin{equation} U(Y)=-\lambda _{\phi }[a_1\tfrac {Y^3}{6}-(a_1+q)\tfrac {Y^2}{2}]+a_2Y+a_3;\label{analitic-viteza} \end{equation}

where:

\[ a_1=\tfrac {\frac{1}{2}r^2+2q}{r(1-K)-1}; a_2=\tfrac {\lambda _{\phi }}{6}[a_1(r^2-2r-2)-3q(r+1)]; \]

\[ a_3=\tfrac {\lambda _{\phi }}{6}r[-a_1(r-2)+3q]. \]

In this particular case, the Nusselt number, has the form $Nu=-ak_{nf}/k_f$, where $a=a_1/(a_1r-a_1-q)$, and depends only by thermal characteristics of the nanofluid.

Equations (16)–(23) were solved numerically using finite difference discretization for different volume fractions of $Cu$ nanoparticles, $\phi =0$, $0.05$, $0.1$ and $0.2$, and thermal conductivity ratio parameter, $K =0.001$, $0.01$ and $0.1$. In this study we consider fixed values for $q$ and $r$ (i.e. $q=1$, $r=0.1$) and, following Khaled and Vafai [ 6 ] , the values for constant $C_{\phi }$ were taken $0$, $100$, $250$, $500$, $1000$, $5000$ and $10000$.

We compared the numerical method with the analytical solutions (29)–(31) and a very good agreement was found. In Figs. $2$ to $4$ the analytical solutions are also presented using a dot marker. Thus, we are confident that the numerical method works fine.

Tables $2$ to $4$ show the Nusselt number for different values of the above parameters. We mention that the value of Nusselt number increases with the increase of constant $C_{\phi }$ and thermal conductivity parameter $K$. Due to the conjugate heat transfer and thermal dispersion Nusselt number does not present a monotone behavior in respect with volume fraction $\phi $.

$C_{\phi }$		$\phi $
	$0.05$	$0.1$	$0.2$
$0$	$2.593686$	$2.984831$	$3.912962$
$100$	$4.383934$	$4.209832$	$4.597601$
$250$	$8.156642$	$7.365368$	$6.165330$
$500$	$12.796259$	$11.836095$	$9.605016$
$1000$	$19.102887$	$18.263184$	$15.442523$
$5000$	$31.875747$	$34.480343$	$37.022310$
$10000$	$34.522407$	$38.455023$	$45.131964$

Table 2 Values of Nusselt number for $K=0.1$

$C_{\phi }$		$\phi $
	$0.05$	$0.1$	$0.2$
$0$	$2.567849$	$2.955098$	$3.873984$
$100$	$3.878139$	$3.912190$	$4.532541$
$250$	$5.801081$	$5.747647$	$5.559927$
$500$	$7.698974$	$7.757198$	$7.510472$
$1000$	$9.637866$	$10.024039$	$10.138831$
$5000$	$12.575802$	$14.017470$	$16.581315$
$10000$	$13.106638$	$14.830093$	$18.361529$

Table 3 Values of Nusselt number for $K=0.01$

$C_{\phi }$		$\phi $
	$0.05$	$0.1$	$0.2$
$0$	$2.565294$	$2.952158$	$3.870129$
$100$	$3.885839$	$3.920925$	$4.526447$
$250$	$5.813239$	$5.758804$	$5.571575$
$500$	$7.716117$	$7.774421$	$7.525006$
$1000$	$9.655428$	$10.044972$	$10.161326$
$5000$	$12.575822$	$14.022163$	$16.600783$
$10000$	$13.100287$	$14.825970$	$18.367821$

Table 4 Values of Nusselt number for $K=0.001$

Table $5$ presents the variation of the maximum temperature in solid with $C_{\phi }$ and $K$. The maximum of the temperature in solid increases with the decrease of $C_{\phi }$ and $K$.

$C_{\phi }$		$K$
	$0.001$	$0.01$	$0.1$
$0$	$1.004777$	$1.003020$	$1.000000$
$100$	$1.004739$	$1.002726$	$1.000000$
$250$	$1.004678$	$1.002283$	$1.000000$
$500$	$1.004564$	$1.001548$	$1.000000$
$1000$	$1.004403$	$1.000760$	$1.000000$
$5000$	$1.003992$	$1.000000$	$1.000000$
$10000$	$1.003870$	$1.000000$	$1.000000$

Table 5 Maximum value for the temperature in solid

Figs. $2$ to $4$ present the velocity and temperature profiles for $\phi =0.2$ and different values of $C_{\phi }$ and $K$. The reversed character of the flow becomes less important with the increasing of parameter $C_{\phi }$ for $K=0.001$ (see Fig. $2a$) while for $K=0.1$ the flow is down for large values of $C_{\phi }$ (see Fig. $2b$). Figs. $3$ and $4$ present the temperature profiles for $K=0.1$ and $K=0.001$ in solid and nanofluid. We observe a decrease of the temperature in solid and an increase of the temperature near the cold wall with the increase of $C_{\phi }$ for both values of $K$.

$\includegraphics[height=1.8688in, width=4.0776in]{viteza_subplot.eps}$

Figure 2 Velocity profile for different values of parameter $C_{\phi }$.

$\includegraphics[height=1.8688in, width=4.0776in]{temperatura_K_01.eps}$

Figure 3 Temperature profile in solid (left) and fluid (right) for different values of parameter $C_\phi $ and $K=0.1$.

$\includegraphics[height=1.8688in, width=4.0776in]{temperatura_K_0001.png}$

Figure 4 Temperature profile in solid (left) and fluid (right) for different values of parameter $C_\phi $ and $K=0.001$.

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This work is licensed under a Creative Commons Attribution 4.0 International License.

\(C_{\phi }\)		\(\phi \)
	\(0.05\)	\(0.1\)	\(0.2\)
\(0\)	\(2.593686\)	\(2.984831\)	\(3.912962\)
\(100\)	\(4.383934\)	\(4.209832\)	\(4.597601\)
\(250\)	\(8.156642\)	\(7.365368\)	\(6.165330\)
\(500\)	\(12.796259\)	\(11.836095\)	\(9.605016\)
\(1000\)	\(19.102887\)	\(18.263184\)	\(15.442523\)
\(5000\)	\(31.875747\)	\(34.480343\)	\(37.022310\)
\(10000\)	\(34.522407\)	\(38.455023\)	\(45.131964\)

\(C_{\phi }\)		\(\phi \)
	\(0.05\)	\(0.1\)	\(0.2\)
\(0\)	\(2.567849\)	\(2.955098\)	\(3.873984\)
\(100\)	\(3.878139\)	\(3.912190\)	\(4.532541\)
\(250\)	\(5.801081\)	\(5.747647\)	\(5.559927\)
\(500\)	\(7.698974\)	\(7.757198\)	\(7.510472\)
\(1000\)	\(9.637866\)	\(10.024039\)	\(10.138831\)
\(5000\)	\(12.575802\)	\(14.017470\)	\(16.581315\)
\(10000\)	\(13.106638\)	\(14.830093\)	\(18.361529\)

\(C_{\phi }\)		\(\phi \)
	\(0.05\)	\(0.1\)	\(0.2\)
\(0\)	\(2.565294\)	\(2.952158\)	\(3.870129\)
\(100\)	\(3.885839\)	\(3.920925\)	\(4.526447\)
\(250\)	\(5.813239\)	\(5.758804\)	\(5.571575\)
\(500\)	\(7.716117\)	\(7.774421\)	\(7.525006\)
\(1000\)	\(9.655428\)	\(10.044972\)	\(10.161326\)
\(5000\)	\(12.575822\)	\(14.022163\)	\(16.600783\)
\(10000\)	\(13.100287\)	\(14.825970\)	\(18.367821\)

\(C_{\phi }\)		\(K\)
	\(0.001\)	\(0.01\)	\(0.1\)
\(0\)	\(1.004777\)	\(1.003020\)	\(1.000000\)
\(100\)	\(1.004739\)	\(1.002726\)	\(1.000000\)
\(250\)	\(1.004678\)	\(1.002283\)	\(1.000000\)
\(500\)	\(1.004564\)	\(1.001548\)	\(1.000000\)
\(1000\)	\(1.004403\)	\(1.000760\)	\(1.000000\)
\(5000\)	\(1.003992\)	\(1.000000\)	\(1.000000\)
\(10000\)	\(1.003870\)	\(1.000000\)	\(1.000000\)

Property	\(H_2O\)	\(Cu\)
\(C_p(J/kgK)\)	4179	385
\(\rho (kg/m^3)\)	997.1	8933
\(k(W/mK)\)	0.613	400
\(\alpha \times 10^7(m^2/s)\)	1.47	1163.1
\(\beta \times 10^{-5}(1/K)\)	21	1.67