Effect of velocity and rheology of nanofluid on heat transfer of laminar vibrational flow through a pipe under constant heat flux
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Abstract
Transverse vibration creates strong vorticity to the plane perpendicular to flow direction which leads to the radial mixing of fluid and, therefore, the results of heat transfer are significantly improved. Comparative studies of effects on heat transfer were investigated through a wellvalid CFD model. Water and waterbased nanofluid were selected as working substances, flowing through a pipe subjected to superimposed vibration applied to the wall. To capture the vibration effect in all aspects; simulations were performed for various parameters such as Reynolds number, solid particle diameter, volume fraction of nanofluid, vibration frequency, and amplitude. Temperature, solid particle diameter and volume fractiondependent viscosity have been considered; whereas, the thermal conductivity of nanofluid has been defined to the function of temperature, particle diameter and Brownian motion. Due to transverse vibrations, the thermal boundary layer is rapidly ruined. It increases the temperature in the axial direction for low Reynolds number flow that results in high heat transfer. As the Reynolds number increases, vibration effect is reduced for pure liquid, while there is noticeable increase for nanofluid. The rate of increment of heat transfer by varying volume fraction and particle diameter shows the usual feature as nanofluid under steadystate flow, but when subjected to vibration is much higher than pure liquid. As the frequency increases, the vibration effects are significantly reduced, and in amplitude they are profounder than frequency. The largest increase of about 540% was observed under the condition of vibrational flow compared to a steadystate flow.
Keywords
CFD Volume fraction Nanoparticle Vibration Heat enhancementList of symbols
 A
Vibration amplitude, mm
 \(C_{\text{p}}\)
Specific heat \({\text{J kg}}^{  1} {\text{K}}^{  1}\)
 \(d_{\text{f}}\)
Equivalent diameter of a base fluid molecule, nm
 \(d_{\text{s}}\)
Diameter of the nanoparticle, nm
 \(f\)
Vibration frequency, Hz
 \(h\)
Heat transfer coefficient, \({\text{W m}}^{  2} {\text{K}}^{  1}\)
 \(k\)
Thermal conductivity, \({\text{W m}}^{  1} {\text{K}}^{  1}\)
 \(K_{\text{b}}\)
Boltzmann’s constant, \(1.38066 \times 10^{  23} \;\;{\text{J K}}^{  1}\)
 \(L\)
Length of the pipe, m
 \(M\)
Molecular weight of the base fluid, \({\text{Kg mol}}^{  1}\)
 \(N\)
Avogadro number, \(6.022 \times 10^{23} {\text{mol}}^{  1}\)
 \(Pr\)
Prandtl number of the base fluid \(\frac{{\left( {C_{\text{p}} } \right)_{\text{f}} \mu_{\text{f}} }}{{k_{\text{f}} }},\left( \cdot \right)\)
 \(q_{\text{w}}\)
Constant wall surface heat flux, \({\text{W m}}^{  2}\)
 \(R\)
Pipe radius, m
 \(Re_{\text{b}}\)
Brownian motion Reynolds number
 \(Re\)
Reynolds number \(\left( \cdot \right)\)
 \(\bar{w}\)
Axial velocity (z direction) \({\text{m s}}^{  1}\)
Greek symbols
 \(\emptyset\)
Volume fraction of the nanoparticles
 \(\mu\)
Dynamic viscosity, \({\text{Pa s}}\)
 \(\rho\)
Fluid density, \({\text{kg m}}^{  3}\)
 \(\lambda_{\text{f}}\)
Molecular free path, nm
Subscripts
 \({\text{nf}}\)
Nanofluid
 \({\text{f}}\)
Base fluid
 \({\text{in}}\)
Inlet
 \({\text{out}}\)
Outlet
 \({\text{s}}\)
Solid particle
 \({\text{sf}}\)
Steadystate flow
 \({\text{vf}}\)
Vibrational flow
 \({\text{w}}\)
Wall
Abbreviations
 CFD
Computational fluid dynamics
 FVM
Finite volume method
Introduction
Suspension of solid nanoparticles (< 100 nm diameter) of very low concentrations (1–5% volume) in conventional heat transfer fluid is called nanofluid [7]. This increases the heat transfer significantly because the thermal conductivity of nanoparticles is higher than that of the liquid, which increases the thermal conductivity of nanofluid [1, 10, 20, 22, 25, 31]. The dispersion of particles of mini or microsize into liquids significantly increases heat transfer compared to nanofluid. But some major deficiencies, such as erosion of surface, clogging the flow which is not suitable for small channels, sedimentation of particles, and more pumping power, limit their industrial applications when compared to nanofluid. Many researchers found that the increased value of thermal conductivity of nanofluid is not the sole reason for heat transfer boost but some other factors such as Reynolds number, particle diameter, and volume fraction also affect the heat transfer rate [20, 29, 30]. It was concluded that the convective heat transfer coefficient increased by increasing volume fraction of nanoparticles and flow rate, and decreased by increasing particle diameter. Nanofluid suffers from all the problems of mini or microsized suspended fluid when the volume fraction and particle diameter increased from its maximum value suggested by [7]. Nanofluids are insignificant for low flow rate applications, whereas its performance increases with the Reynolds number. Static mixing equipment or strip twisted tape is usually used to increase the heat transfer performance in comparatively low Reynolds number flows. These devices promote radial mixing. Sharma et al. [24] examined the effect of inclusion of strip twisted tape on heat transfer through transition flow of nanofluid and found a better heat transfer than the flow without twisted tapes.
Nanofluid has a great potential to carry heat in wide area of applications, which is not limited to heat exchanger, engine cooling, etc., but also in magnetohydrodynamic application, which are mainly used in bioengineering, paint technology and medical sciences. Many researches have used nanofluid of Newtonian as well as nonNewtonian types to investigate the effect of solid particle concentration on magnetohydrodynamic flow over stretching sheet [5, 21, 28]. It was concluded that nanofluid with metallic nanoparticle shows better performance than that of base fluid.
It has been proven by many researchers that the induced vibration on the flow system increases heat transfer rate by producing chaotic motion, which leads to mixing of fluid [6, 16, 18]. Sufficient radial mixing of fluid can be achieved either by turbulent flow, which requires higher pumping power or by use of static mixture, but these types of devices have manufacturing complexity and the problem of cleaning as well.
Lee and Chang [18] applied transverse vibration of 0–70 Hz frequency and 0.1–1.0 mm amplitude on the 8mmdiameter pipe to investigate the effects on critical heat flux. Noteworthy increment in critical heat flux was reported, that was the function of both amplitude and frequency. Chen et al. [6] had selected a copper heat pipe with internal groove and imposed vibration in longitudinal direction. Their study found that heat transfer increment of heat pipe was directionally proportional to vibration energy applied to this mode of vibration. Easa and Barigou [13] numerically evaluated the performance of transverse vibration on heat pipe. It has been shown by contour plot of temperature and vorticity plot that vibration produces strong spiraling motion due to the secondary component of velocity, which enhances the radial mixing of fluid and great addition in heat transfer comparatively in short length pipe. Such methods are capable of reducing the length of the pipe because it reduces the hydrodynamic entrance length and the thermal entrance length considerably. It also performs cleaning action on the walls as its strong chaotic motion reduces fouling of the pipe.
Compared with longitudinal and rotational direction, the transverse vibration produces more chaotic motion when applied in the transverse direction. It was also proved that radial mixing is much better than the recognized Kenics helical staticmixer, which has the disadvantage of unhygienic fluid processing and more pressure drop, if transverse vibration is applied with the change in the orientation of pipe with step rotation [27].
Zhang et al. [32] conducted an experiment to investigate the effect of vibration in a circular straight pipe in transient flow condition. Comparisons were made among the flow of base fluid and SiO_{2}–water nanofluid under steady state and unsteady (vibration) state. Sharp increment in heat transfer was achieved when the base fluid flows under vibration condition than the steadystate flow. The Reynolds number and vibration frequencies have greater influence on the enhancement effect. The maximum increase of 182% was found when the vibrational flow of nanofluid was compared to the steadystate flow of base fluid and it augmented further with the vibration amplitude.
A large number of studies has been done by researchers for the improvement of heat transfer by dispersing nanoparticles into conventional liquids used for heat transfer process with/or use of static mixture devices, or use of superimposed vibration at pipe wall. Nanofluids have its limitation of particle diameter and volume fraction as discussed earlier. Heat transfer through nanofluids can be increased by increasing Reynolds number but requires higher pumping power. Vibrating the flow of nanofluid through a pipe can solve all the problems described. The studies reviewed above publicized that no methodical study has been conceded out to appraise the effect of vibration on nanofluid flow with varying rheological properties and flow parameters. The work described in this paper is a CFD investigation of effects of vibration on laminar forced convection thermal flow of pure water and of Al_{2}O_{3}–water nanofluid through pipe. Simulations were carried out for different Reynolds numbers. Constant heat flux boundary condition was applied at pipe wall. The viscosity of nanofluid has been calculated using Corcione [9] correlation which is the function of temperature, solid particle diameter and volume fraction of nanoparticle; while the thermal conductivity of nanofluid was calculated from Chou et al. [8] correlation. Three average particle sizes of 25 nm, 50 nm, and 100 nm and four volume concentration of 0% (base liquid), 1.0, 1.5, and 2.0% were used. The effects of vibration parameters on pure liquid and nanofluid have also been investigated for different frequencies and amplitudes, and results have been presented through the ratio of heat transfer coefficient of vibrational flow to steadystate flow.
Theory
Thermophysical properties of nanofluid
Dispersion of nanoparticles of very low concentration in the carrier fluid behaves like a singlephase fluid of uniform properties. To achieve accurate results with this model, it is very necessary to use the most suitable correlations for nanofluid properties. The following equations are used to represent mathematical formulation of nanofluid as a singlephase model.

Density [26]

Specific heat

Viscosity [9]
Corcione [9] has developed an empirical relation with the experimental data taken from literature with 1.84% deviation.

Thermal conductivity [8]
Chou et al. [8] has developed correlation of thermal conductivity of nanofluid. All possible factors to represent nanofluid as a singlephase fluid such as solid particle diameter, volume fraction, and Brownian motion were considered. With a linear regression for experimental results, Buckinghampi theorem was used to develop empirical correlation with a 95% confidence level.
CFD modelling
Consider the case of incompressible thermal laminar flow through the pipe, all the properties of the base fluid are assumed to be constant except the temperaturedependent viscosity. While for nanofluid, the viscosity and thermal conductivity are considered to be the function of temperature. The governing equations [15] are the equation of continuity:
In CFX, different terms of generic transport equations have been discretized separately according to the physics involved in the problem via finite volume method and converted it into system of linear algebraic equation for individual subdomain which was then solved by numerical technique.
Discretization of advection term: the integration of field variable \(\varphi\) is approximated by its neighboring values then its value (\(\varphi )\) can be cast in the form (called advection scheme):
Discretization of transient term: The secondorder backward Euler scheme was used to discretize unsteady term. In this scheme, current time step and last two time steps were used to calculate the transport variable at the start and at the end of time step. This scheme is of secondorder accurate, robust, implicit, and conservative in time, and has no time step limitation.
Flow velocities in both the direction were selected such that the flow remained laminar for all the cases considered.
CFD simulation
Commercially available CFD software package ANSYS CFX 16.2 was used to simulate the steadystate and unsteadystate (vibrational) flow of nanofluid and compared it with the base fluid flow. The effective viscosity and thermal conductivity of nanofluid were described by Eqs. (3) and (5), respectively. The flow geometry was created and meshed using ICEM CFD 16.2. software.
Straight pipe of 6 mm diameter is considered for the analysis. For evaluating nanofluid flow behavior through the horizontal pipe, this much diameter of the pipe is adequate as reported by many researchers [1, 10]. Sufficient 1000 mm length of pipe for comparatively low Reynolds number with three surface boundaries: wall, inlet and outlet were considered to capture the effect of heat transfer.
Initial and/or boundary conditions for different flow conditions
Flow condition  Initial and/or boundary conditions  Description 

Isothermal steadystate flow  \(\nabla .\eta \dot{\gamma } = 0 \,\,{\,\text{at}\,} \,\,r = 0; v_{\text{r}} = 0 \,\,{\,\text{at}\,} \,\,r = R;\,\,v\left( {r,z} \right) = {\text{constant and\,\,}}\) \(T\left( {r,z} \right) = T_{\text{in}}\,\, {\,\text{at}\,} \,\,z = 0;\,\, \frac{\partial T}{{\partial S_{n} }} = 0\,\, {\,\text{at}\,}\,\, r = R\)  Shear stress at pipe center; noslip wall; constant velocity depending upon the Reynolds number; constant \(T_{\text{in}} = 20\;^{ \circ } {\text{C}}\) inlet fluid temperature; zero heat flux at wall 
Steadystate thermal flow  \(\nabla .\eta \dot{\gamma } = 0 {\,\text{at}\,} r = 0; v_{\text{r}} = 0 {\,\text{at}\,} r = R;v\left( {r,z} \right) = {\text{constant and}}\,\,\) \(T\left( {r,z} \right) = T_{\text{in}} {\,\text{at}\,} z = 0; \frac{\partial T}{{\partial S_{n} }} =  \frac{{q_{\text{w}} }}{\lambda } {\,\text{at}\,} r = R\)  Shear stress at pipe center; noslip wall; velocity profile obtained from pervious step used as velocity inlet; \(T_{\text{in}} = 20\;^{ \circ } {\text{C}}\) inlet fluid temperature; \(q_{\text{w}} = 10,500\; {\text{W/m}}^{2}\) heat flux normal to wall 
Unsteadystate thermal flow  \(v_{\text{r}} = 0\,\, {\,\text{at}\,}\,\, r = R.\) When \(t = 0\) \(v\left( {r,z} \right) = {\text{constant and}}\,\, T\left( {r,z} \right) = T_{\text{in}}\,\, {\,\text{at}\,} \,\,z = 0; \frac{\partial T}{{\partial S_{n} }} =  \frac{{q_{\text{w}} }}{\lambda }\,\, {\,\text{at}\,}\,\, r = R\) When \(t > 0\) \(x\left( {x,t} \right) = A{ \sin }\left( {2\pi ft} \right);\,\, \dot{x}\left( {x,t} \right) = A2\pi f{ \cos }\left( {2\pi ft} \right) \,\,{\,\text{at}\,} \,\,r = R\)  Noslip wall. When \(t = 0\) Velocity profile at inlet; \(T_{\text{in}} = 20\;^{ \circ } {\text{C}}\) inlet fluid temperature; \(q_{\text{w}} = 10,500 \;{\text{W}}/{\text{m}}^{2}\) heat flux normal to wall. When \(t > 0\) In addition to above, sinusoidal displacement and its first derivative applied at wall boundary to produce transverse vibration 
The thermophysical properties of the base fluid, nanoparticles, and nanofluid
Material  \(\rho\) \(\left( {{\text{kg m}}^{3} } \right)\)  \(C_{\text{p}}\) \(\left( {{\text{J kg}}^{  1} {\text{K}}^{  1} } \right)\)  \(\mu\) \(\left( {\text{Pa s}} \right)\)  \(k\) \(\left( {{\text{W m}}^{  1} {\text{K}}^{  1} } \right)\) 

Newtonian fluid  998  4180  \(\mu_{\text{f}} = D \times 10^{{\frac{B}{{\left( {T  C} \right)}}}}\)  0.613 
Nanoparticle (Al_{2}O_{3})  3970  765  –  40 
\(d_{\text{s}} = 25\; {\text{nm}}, \;50\; {\text{nm}}, 100\, {\text{nm}}\)  
Nanofluid (base fluid + \(1\%\) nanoparticles)  1027.72  4048  Empirical correlation [9] Eq. (3)  Empirical correlation [8] Eq. (5) 
Nanofluid (base fluid + \(1.5\%\) nanoparticles)  1042.58  3985  
Nanofluid (base fluid + \(2\%\) nanoparticles)  1057.44  3924 
Range of parameters used for simulation of flow
\(D\) \(\left( {\text{mm}} \right)\)  \(L\) \(\left( {\text{mm}} \right)\)  \(f\) \(\left( {Hz} \right)\)  \(A\) \(\left( {\text{mm}} \right)\)  \(Re\) \(\left(  \right)\)  \(\emptyset\) (%)  \(d_{\text{s}}\) \(\left( {\text{nm}} \right)\) 

6  1000  50–100  1–2  400–800  0–2.0  25–100 
Unsteadystate simulations were run for the residence time which is the time taken by fluid to come out from the pipe. Residence time was calculated by dividing the length of pipe with velocity. Depending upon the frequency, one oscillation time was divided into 12 equal numbers to get the value of time step. For example, \(0.2 \;{\text{s}}\) required to complete one cycle for the frequency of 50 Hz and then the time step would be \(0.2/12 = 1.6667 \times 10^{  3} \;{\text{s}}\). Time step chosen was such that the pipe attended two minimum and two maximum positions in a cycle. Simulations were executed for different values of time steps and found that by reducing time step size, simulation time increases with negligible increment in solution accuracy. 12 iterations/time was set by performing a number of simulations with the residual accuracy of \(10^{  4}\) and \(10^{  5}\). For such complex flow, when the residuals of continuity, momentum, and energy reached to the value of \(10^{  4}\) of root mean square, then it assumes that solution is converged for each time step. Solution was continued for next time step until simulation time reached its value to residence time.
Validation of CFD model
Validation of convective heat transfer coefficient for steadystate flow
The flow and heat parameters selected for validation were such that the dimensionless length is always greater than \(x^{*} \ge 0.001\). It is because of this reason that the range of Reynolds number was taken such that \(x^{*}\) always lies in the third condition of Shah equation.
Unsteadystate isothermal flow validation
Thermal steadystate internal flow validation
Simulation model is validated for nonisothermal steadystate flow through pipe in the following two sections: in the first section, validation of radial temperature profile has been done because it gives the assurance of correct radial profile with the same setting used for unsteadystate flow and second validation is done for nonisoviscous flow because temperaturedependent viscosity of nanofluid is considered here.
 1.
 2.Temperaturedependent viscosity is considered for base fluid and nanofluid, and so it is necessary to validate CFD model for nonisoviscous fluid. Kwant et al.’s [17] experimental results have been used to validate the CFDpredicted results shown in Fig. 5. Radial profile of isoviscous fluid shows the usual feature, whereas velocity profile of nonisoviscous fluid becomes flattened compared to that of isothermal condition.
Deshpande and Barigou [11] have confirmed that CFDpredicted results for isothermal Newtonian and nonNewtonian fluid flow under forced vibration are consistent with experimental data. It has also proved that CFD codes are good enough to predict the flow behavior of such complex fluid flow within approximately \(\pm \;\;10\%\), for a variety of rheological behaviors and under a wide range of vibration conditions [13, 27]. CFD codes are also capable for internment of the flow (isoviscous and nonisoviscous) behavior in view of radial profile and heat transfer, etc., as discussed in aforementioned sections. The statement could be made based on the validation that CFD codes are robust, reliable, and upright for the purpose of studying the effect of vibration on heat transfer characteristics of the flow considered here, as there are no experimental data and numerical data available in the literature to validate the codes for nonisothermal unsteady flow.
Results and discussion
CFD simulations have been carried out for different Reynolds numbers to evaluate the effects of vibration frequency and vibration magnitude on the thermal developing laminar flow through pipe. In pipe flow, the effect of rheology (i.e. solid particle concentration and its size) of nanofluid has also been compared with the base fluid under vibration condition. Local and average heat transfer coefficients were calculated using the following equations:
Conclusion
In this paper, the comparison has been made among the convective heat transfer coefficient of steadystate and unsteadystate (vibration) laminar flow of Al_{2}O_{3}–waterbased nanofluid and of pure water. A wellvalidated CFD model used to simulate the flow of water and nanofluid through a horizontal pipe subjected to uniform constant heat flux. Numerical simulation has been conducted for a different range of Reynolds number and for different nanofluid parameters. Considerable enhancement of convective heat transfer coefficient has been reported for pure water at low Reynolds number and further, the ratio of heat transfer coefficient decreased with Reynolds number. In comparison with the base fluid, by adding nanoparticles with different concentration about 51% maximum enhancement was obtained for \({\varnothing} = 2.0 \%\) and this continued with a slight fall of values with the Reynolds number.
For a typical value of Reynolds number, ratio of heat transfer coefficient can be maximized with increment in volume fraction of nanoparticles and the use of solid particles with a moderate diameter. But a considerable increment of volume fraction causes the sedimentation of particles. In addition, heat transfer rate can be optimized between volume fraction and solid particle diameter for comparatively low Reynolds number that would lead to low power consumption.
Mechanical vibration with different frequencies produced substantial enhancements in heat transfer than that of amplitude. Heat transfer rate for nanofluid flow greatly influenced by amplitude and frequency compared with the base fluid and much enhancement than that of base fluid were achieved under vibration. Further, this study can be extended for twophase flow CFD model so that the influence of solid particles concentration and its behavior under vibration flow condition can be evaluated and can be extended to verify the effect of higher concentration of volume fraction on friction factor and distribution of nanoparticles.
Notes
Compliance with ethical standards
Conflict of interest
The authors declare that they have no conflict of interest.
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