Numerical evaluation of laminar heat transfer enhancement in nanofluid flow in coiled square tubes
 Agus Pulung Sasmito^{1, 2},
 Jundika Candra Kurnia^{1}Email author and
 Arun Sadashiv Mujumdar^{1, 2}
DOI: 10.1186/1556276X6376
© Sasmito et al; licensee Springer. 2011
Received: 30 October 2010
Accepted: 9 May 2011
Published: 9 May 2011
Abstract
Convective heat transfer can be enhanced by changing flow geometry and/or by enhancing thermal conductivity of the fluid. This study proposes simultaneous passive heat transfer enhancement by combining the geometry effect utilizing nanofluids inflow in coils. The two nanofluid suspensions examined in this study are: waterAl_{2}O_{3} and waterCuO. The flow behavior and heat transfer performance of these nanofluid suspensions in various configurations of coiled square tubes, e.g., conical spiral, inplane spiral, and helical spiral, are investigated and compared with those for water flowing in a straight tube. Laminar flow of a Newtonian nanofluid in coils made of square cross section tubes is simulated using computational fluid dynamics (CFD)approach, where the nanofluid properties are treated as functions of particle volumetric concentration and temperature. The results indicate that addition of small amounts of nanoparticles up to 1% improves significantly the heat transfer performance; however, further addition tends to deteriorate heat transfer performance.
Introduction
Convective heat transfer can be enhanced by active as well as passive methods. While the former usually provide better enhancement, it requires additional external forces and/or equipment which can increase the complexity, capital, and operating costs of the system. In contrast, passive heat transfer enhancement can be achieved by changing flow geometry or modifying thermophysical properties of working fluid. Hence, it is generally a more desirable approach when compared to an active method. In our previous study [1–3] (Sasmito AP, Kurnia JC, Mujumdar AS: Numerical evaluation of transport phenomena in a Tjunction microreactor with coils of square cross section tubes, submitted), we have shown that coiled tubes provide better heat transfer performance relative to straight tubes under certain conditions. In this study, the potential application of coiled tubes using nanofluids to improve heat transfer performance is investigated.
Coiled tubes have been known as one of the passive heat transfer enhancement techniques in heat and mass transfer applications due to the presence of secondary flows which improve heat and mass transfer rates. They have been widely used in process industries, e.g., heat exchangers and chemical reactors, due to their compact design, high heat transfer rate, and ease of manufacture. Aside from their industrial applications, studies of the transport phenomena in coiled duct have also attracted many attention from engineering researchers. The presence of secondary flows induced by coil curvature and the complex temperature profiles caused by curvatureinduced torsion are among significant phenomena which can be observed in coiled tubes. Numerous experimental [4–8] and numerical [1–3, 9–13] investigations on heat transfer and flow characteristics inside coiled tubes have already been reported. Furthermore, reviews on the flow and heat transfer characteristics and potential application of coiled tubes in process industries and heat transfer application can be found in [14, 15].
It is well known that conventional heat transfer fluids including water, oil, and ethylene glycol mixtures have poor heat transfer rate due to their low thermal conductivity. Therefore, over the past decade, extensive research have been conducted to improve thermal conductivity of these fluids by suspending nanoparticles of diverse materials in heat transfer fluids, called nanofluids [16]. Modern technology provides opportunities to process and produce particles below 50 nm. It is also expected that nanofluids should provide not only higher heat transfer rate, but also good stability of the suspension by eliminating possible agglomeration and sedimentation to permit longterm application [17]. To date, several experimental (see for example [18–23]) and numerical (see for example [24–28]) investigations to characterize heat transfer performance of nanofluids have been already reported. Choi et al. [18] showed that addition of small amounts of less than 1% nanoparticles can double the thermal conductivity of working fluids. Vajjha et al. [24] showed that heat transfer rate increases up to 94% by adding 10% Al_{2}O_{3} nanofluid and increase up to around 89% by adding 6% CuO nanofluid. In addition, the comprehensive reference on nanofluids can be found in the book of Das et al. [29], while several reviews of nanofluids are available in the literature [30–42].
It has been shown that coiled tubes geometry and nanofluids can passively enhanced heat transfer performance. Now, to maximize the advantages of the heat transfer enhancement, we propose to combine both techniques simultaneously; i.e., employing the combination of coiled tubes filled with nanofluids. Therefore, the aim of the study presented here is threefold: (i) to investigate the heat transfer performance of various configurations of coils of square tubes, e.g., conical spiral, inplane spiral, and helical spiral, relative to the straight pipe; (ii) to evaluate simultaneous passive heat transfer enhancementchannel geometry and fluid thermophysical propertiesin coiled tubes filled with nanofluids; (iii) to study the heat performance of two different nanofluids, waterAl_{2}O_{3} and waterCuO, in coiled tubes at various nanoparticle concentrations. The most significant aspect of this study is to determine the potential advantages and limitations of heat transfer enhancement of coiled of square tubes filled with nanofluids and provide design guidelines for their applications through mathematical modeling.
The layout of the article is as follows. First, the mathematical model is introduced; it comprises conservation equations for mass, momentum, and energy. The nanofluid thermophysical properties are treated as functions of particle volumetric concentration and temperature. The mathematical model is then solved numerically utilizing finitevolumebased CFD software Fluent 6.3.26, the UserDefined Function written in C language is used extensively to capture the nanofluid properties. The model is further validated against experimental data by Anoop et al. [19] in terms of heat transfer performance for both basefluid and nanofluid. Fluid flow and heat transfer performance of various coiled tube designs filled with nanofluids is evaluated in terms of a figure of Merit Defined later. Parametric studies for particle concentration and nanofluid type are then carried out. Finally, conclusions are drawn and possible extensions of the study are highlighted.
Mathematical model
Governing equations
In the above equations, ?_{nf} is the nanofluid fluid density, u is the fluid velocity, p is the pressure, µ_{nf} is the dynamic viscosity of the nanofluid, c_{p,nf} is the specific heat of the nanofluid and k_{nf} is thermal conductivity of the nanofluid.
Constitutive relations
Thermophysical properties of nanofluids
Base case and operating parameters
Parameter  Value  Unit 

c _{p,np,Al2O3}  765  J · kg^{1} · K 
c _{p,np, CuO}  540  J · kg^{1} · K 
d _{np, Al2O3}  59 × 10^{9}  m 
d _{np, CuO}  29 × 10^{9}  m 
k _{np, Al2O3}  36  W · m^{1} · K^{1} 
k _{np, CuO}  18  W · m^{1} · K^{1} 
 5 × 10^{4}  ? 
?  1.381 × 10^{23}  J · K^{1} 
? _{np, Al2O3}  3600  kg · m^{3} 
? _{np, CuO}  6510  kg · m^{3} 
 9 × 10^{3}  kg · s^{1} 
p _{out}  101325  Pa 
T _{0}  298.15  K 
T _{in}  298.15  K 
T _{wall}  323.15  K 
 2.8217 × 10^{2}  ? 
 3.917 × 10^{3}  ? 
 3.0669 × 10^{2}  ? 
 3.91123 × 10^{3}  ? 
(Al_{2}O_{3})  0.9830  ? 
(Al_{2}O_{3})  12.959  ? 
(CuO)  0.9197  ? 
(CuO)  22.8539  ? 
ß_{1} (Al_{2}O_{3})  8.4407  ? 
ß_{2} (Al_{2}O_{3})  1.07304  ? 
ß_{1} (CuO)  9.881  ? 
?_{2} (CuO)  0.9446  ? 
where ß_{1}, ß_{2}, , , and , are constants (see Table 1).
Thermophysical properties of basefluids
Properties of nanoparticles are given in Table 1.
Heat transfer performance
Boundary conditions
The boundary conditions for the flow inside the channel are prescribed as follows

Inlet At the inlet, we prescribe inlet mass flow rate and inlet temperature.(19)

Outlet At the outlet, we specify the pressure and streamwise gradient of the temperature is set to zero; the outlet velocity is not known a priori but needs to be iterated from the neighboring computational cells.(20)

Walls At walls, we set no slip condition for velocities and constant wall temperature.(21)
In this article, a constant mass flow rate at a Reynolds number (Re = ?UD_{h}/µ) of approximately 1000 is prescribed at the inlet for comparison purposes.
Numerics
Equations 13 together with appropriate boundary conditions and constitutive relations comprising of five dependent variables, u, v, w, p, and T, were solved using the finite volume solver Fluent 6.3.26. UserDefined functions (UDF) were written in C language to account for particle volumetric concentration and temperaturedependence of the thermophysical properties of the nanofluids.
The equations were solved with the wellknown SemiImplicit PressureLinked Equation (SIMPLE) algorithm, firstorder upwind discretization and Algebraic Multigrid (AMG) method. As an indication of the computational cost, it is noted that on average, around 200500 iterations and 500 MB of Random Access Memory (RAM) are needed for convergence criteria for all relative residuals of 10^{6}, this takes 530 min on a workstation with a quadcore processor (1.83 GHz) and 8 GB of RAM.
Results and discussion
Geometric parameters
Parameter  Value  Unit 

w  1.00 × 10^{2}  m 
s  1.00 × 10^{2}  m 
R _{pi}  2.00 × 10^{2}  m 
R _{po}  9.00 × 10^{2}  m 
R _{ci}  2.00 × 10^{2}  m 
R _{co}  9.00 × 10^{2}  m 
R _{h}  4.00 × 10^{2}  m 
L  1.20  m 
Validation
When developing and implementing mathematical model to predict the behavior of nanofluid heat transfer, one needs to pay special attention to validation of the model due to inherent complexity of coupled physical phenomena and interaction between basefluid and nanoparticle. In this study, we aim to validate our model with an experimental nanofluid heat transfer by Anoop et al. [19], which has error of approximately 4%. The heat transfer performance of nanofluid flows in circular tube with diameter 4.75 × 10^{3} m and length of 1.2 m is approximated with 2D axisymmetric model, see Anoop et al. [19] for details of the experimental setup.
Effect of geometry
Basefluid
Nanofluids
Four square cross section tube geometries were examined for flow of nanofluid suspensions of waterAl_{2}O_{3} with nanoparticle concentration of 1%. The results are depicted in Figure 6 where the mixed mean temperature and total heat transfer of basefluid and nanofluids are shown. It is noted that adding 1% concentration of Al_{2}O_{3} in water improves the heat transfer performance. The total heat transfer for straight tube increases up to 50% as compared to that for water, whereas for coiled tubes, the heat transfer improves by about 50% in the nearinlet region and then decreases toward the outlet. Furthermore, among the coiled tube geometries, inplane spiral gives the highest heat transfer improvement, followed by helical spiral and conical spiral tubes. This implies that inplane spiral tube may have potential application to be used along with nanofluid due to its higher heat transfer performance. Therefore, the most of the following results refer to inplane spiral coils.
Effect of nanoparticle concentration
Effect of nanofluid type
Overall heat transfer performance
Aside from higher heat transfer performance, keeping pressure drop at a minimum is of interest for reducing the operating cost and saving energy. Figure 12b shows a summary of the pressure drop required for all cases studied. Note that the mass flow rate is kept constant in all cases; hence, it can be used directly to represent the pumping power required. The straight channel requires the lowest pressure drop among all cases; whereas the coiled tube designs require more than double the pressure drop of the straight channel. Among the coiled tubes, helical spiral tube needs the highest pressure drop, followed by inplane spiral and conical spiral tubes. An interesting phenomenon is observed at a nanofluid concentration of 1% when the pressure drop for coiled tubes is slightly lower than that for water. This is due to the fact that at low particle concentrations, the particle volumetric concentration affects the nanofluid viscosity negligibly while the effect of temperature increases in the nanofluid thermophysical properties.
With respect to the heat transfer performance and pressure drop required in the system, the "Figure of Merit" concept is introduced as a measure of the heat transferred per unit pumping power (see Equation 14 for details). Figure 12c presents the computed figures of merit for various tube geometries, nanofluid concentrations and nanofluid types. It is found that apart from the higher heat transfer rate, the coiled tubes have lower figures of merit than those of the straight tube. This can be explained by the higher pressure drops required in the coiled systems (see Figure 12b). Among all coiled tubes tested, the conical spiral tube gives the highest figure of merit, followed by inplane spiral and conical spiral tubes. Furthermore, for the straight tube, the addition of nanoparticles improves the figure of merit significantly, albeit it decreases with increasing concentration. For coiled tubes filled with nanofluids, on the other hand, the improvement of figure of merit is only shown at low particle concentration of 1% and then it drops lower than that of water when more nanoparticles are added. Clearly, these results suggest that one can add nanoparticle up to 1% volumetric concentration to water to enhance heat transfer performance in coiled tubes; higher nanoparticle concentrations are not recommended.
Concluding remarks
A computational study was conducted to investigate the laminar flow heat transfer performance of square cross section tubes, i.e., straight, conical spiral, inplane spiral, and helical spiral, with water and two nanofluids. It is found that adding 1% nanoparticle volumetric concentration improves heat transfer performance and the figure of merit for all tubes. However, higher amounts of nanoparticles is not recommended. Inplane spiral tubes give better performance than other coiled tubes for nanofluids. Furthermore, Al_{2}O_{3} nanofluid gives slightly better heat transfer performance than CuO nanofluid in coiled tubes. Future study will evaluate various modeling approaches for nanofluid heat transfer, e.g., singlephase, twophase mixture, EulerEuler, and EulerLagrange models, in coils with respect to the effect of secondary flow to the nanoparticle concentration.
Abbreviations
 AMG:

algebraic multigrid
 CFD:

computational fluid dynamics
 RAM:

random access memory
 SIMPLE:

semiimplicit pressurelinked equation
 UDF:

userdefined functions. List of symbols: A_{c}: Cross section area (m^{2})
 c _{p} :

Specific heat (J · kg^{1} · K^{1})
 :

Viscosity parameter
 d _{p} :

Particle diameter (m)
 D _{h} :

Hydraulic diameter (= 4A_{c}/P_{c}) (m)
 FoM:

Figure of merit
 h :

Heat transfer coefficient (W · m^{2} · K^{1})
 k :

Thermal conductivity (W · m^{1} · K^{1})
 ? :

Boltzmann constant (J · K^{1})
 :

Brownian motion constant
 L :

Total length channel (m)
 :

Mass flow rate (kg · s^{1})
 p :

Pressure (Pa)
 P _{ c } :

Cross section perimeter (m)
 R :

Radius of coil (m)
 Re Reynolds number (= ?U D:

_{h}/µ)
 s :

Spacing (m)
 T :

Temperature (K)
 u :

u, v, w, U: Velocity (m · s^{1})
 V :

Mean velocity (m · s^{1})
 w :

Channel width
 W :

Total heat transfer (J · s^{1})
 W _{pump} :

Pumping power (W). Greek: ß: Brownian motion parameter
 ? :

Fluid density (kg · m^{3})
 ? :

Particle volumetric concentration (%)
 ? :

Efficiency (%)
 µ :

Dynamic viscosity (Pa · s). Subscripts: c: Conical spiral
 h:

Helical spiral
 i:

Inner
 in:

Inlet
 L:

Length
 mean:

Mean value
 norm:

Normalized value
 nf:

Nanofluids
 np:

Nanoparticle
 o:

Outer
 out:

Outlet
 p:

Inplane spiral
 pump:

Pump
 w:

Water
 wall:

Wall.
Declarations
Authors’ Affiliations
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