# Analysis of Natural Convection in Nanofluid Flow through a Channel with Source/Sink Effect

Authors: Imran Siddique; Kashif Sadiq; Fahd Jarad; Mohammed K. Al Mesfer; Mohd Danish; Sonia Yaqoob
Journal: Journal of Nanomaterials
 (2022)

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## Abstract

In this study, the natural convection nanofluids flow through a channel formed by two vertical parallel plates having distanced between them has been examined under the influence of the ramped velocity. Sodium alginate is considered as base fluid, and nanoparticles of titania (TiO2) and alumina (Al2O3) are added to it. Analytical and semianalytical results for temperature and velocity profiles are obtained with Laplace transform and inverse Laplace algorithms (Tzou, Stehfest, Talbot, Honig and Hirdes, and Fourier series), respectively. Furthermore, the impacts of nanoparticles, Prendtl number, heat absorption, and time on velocity and temperature are drawn graphically and discussed. The outcomes show that the high thermal conductivity of particles increases the temperatures, and the high density of particles decreases the velocities of the nanofluids. The current findings are compared to previous findings in the literature. In the tables, the effect of volume fraction on Nusselt numbers and skin frictions is explored.


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## Body

## 1. Introduction

The study of viscous fluid between parallel plates is significant due to its vast applications in the science and engineering fields. Sahebi et al. [1] presented the analysis of the free convection non-Newtonian nanofluid flow in a vertical channel numerically and described its significance. Many engineers studied the flow of non-Newtonian fluids with constant physical properties and heat transport, manipulating problems related to fluid mechanics and heat transfer. Adesanya [2] studied the unsteady free convection flow with absorbing generation between two infinite parallel plates with a temperature jump and velocity slip in the slip flow regime. Boulama and Galanis [3] provided the exact results for mixed convection nanofluid flow between two plates with mass and heat transfer. Ajibade and Bichi [4] studied an unsteady incompressible convective fluid flow that is optically dense through an upright channel due to the collective effect of thermal radiation and variable viscosity and concluded that by increasing the viscosity variation parameters and thermal radiation, the velocity of fluid increases and temperature also increases with a boost in thermal radiation. Rajkumar et al. [5] investigated the numerical results of the inner convection of heat sources in tandem planar. Nada [6] studied the heat transport rate of free convection flow in horizontally and vertically closed narrow heated finned base plates, and the outcomes confirmed that the fins increase the rate of heat exchange with fin array geometries. Usually, the concentration disparity in the mass transfer influences the rate of heat relocation. The buoyancy effects are the driving forces for natural convection.Several researchers discussed their work on nanofluids. The idea of nanofluid was given by Eastman and Choi [7]. Nanofluid allocates the fluid in which the nanoparticles are hanging in the traditional fluid. Because of the rapid advancement of nanotechnology [8], various models of nanofluids are being implemented in the field of thermal engineering. So nanofluid shows more effective thermal conductivity as compared with the base fluid. Suspended components raise thermal conduction and heat conveyance processes as the solid bimetallic particles bear more caloric conductivity than the base fluid. High viscosity and more static with better diffusion, wetting, and propagation through solid aerofoils, even for minor nanoparticle addition, are significant features of nanofluids [9]. Nanofluids are comprised of super-fine nanoparticles (size<100nm) mixed in water or organic solvent [10].Generally, nanoparticles of chemically stable materials like copper (Cu), gold (Au), silicon oxide (SiO2) or silica, zirconium oxide (ZrO2), titania (TiO2), copper oxide (CuO), alumina (Al2O3), metallic nitrides (SiN, AlN), and carbon nanotubes (CNTs) are used. These solid-liquid specks quickly drop down, filled the flow ducts, serious pressure failure, and causing erosion of pipelines. Therefore because of these defects, ordinary solid-fluid fusions for heat change at micro levels are used instead of nanofluids. Nanofluids can increase critical temperatures and surfactants, or standard emulsifiers cannot increase thermal conductivity. Cooling plays important role in providing comfort for required functioning and well-founded results of developing products especially computers, electronic circuits, X-ray generators, automobile engines, high energy lasers, etc. Improvement of heat transferal characteristics of nanofluids stimulates the attainable development in the heating system or consignment and heat flows caused by power in small-scaled products elevated its applications in defensive structure, microelectronics, fabricating, transportation, metrology, and engine cooling system.The researchers have conducted extensive studies in this field. Some investigations are experimental, while others are computational, and only a small amount of research has been undertaken on the analytical side. The efficacy of carbon nanostructures and water-based nanoliquids as coolants was investigated by Halelfadl et al. [11]. They looked at how low nanoparticle volume fractions (varying from 0.0055% to 0.278%) affected nanofluid density, thermal conductivity, and viscosity.Solar thermal devices’ efficiency and performance could be improved. The use of nanotechnology in solar collectors has been the subject of extensive research. Solar cells are heat engines that capture sunlight and transmit the heat to a liquid running past them. Tyagi et al. [12] discovered that adding nanoparticles to a collector improves its efficiency. His findings reveal that by changing the volume fraction from 0.1% to 2% and the size of the fraction, the efficiency increases dramatically. When compared to water, Yousefi et al. [13] discovered that nanofluid (with 0.2% wt.) had a higher efficiency. When they added surfactant to their trials, they saw a 15.63% improvement [14, 15], and the references therein include examples of nanoparticles in solar energy applications.Fluid flow and linked mass and energy transmission through a channel have received less attention than the situation of a single plate. This design may be found in a wide range of fields, such as petroleum reservoirs, fire engineering, combustion modeling, and nuclear energy, to name a few. Many engineering systems show transport phenomena that combine the effects of concentration and thermal buoyancy. Modern thermal protection devices, chemical distilleries, building ventilation systems, solar panels, heat exchangers, and electric circuits all contain them [16, 17]. Gupta et al. [18] used Marangoni convection to study the flow of two separate nanofluids over a stretched surface in a porous medium. Gohar et al. [19] investigated a Darcy-Forchheimer flow of Casson hybrid nanofluid via a curved surface that was constantly growing. The viscous fluid flow in a porous medium is expressed by the Darcy-Forchheimer effect. Adnan et al. [20] investigated the flow of Cu-water and Cu-kerosene oil through two Riga plates, taking into account surface convection and radiation effects. Zaka Ullah et al. [21] studied the flow of a hybrid nanofluid in a diverging and a converging channel. The effects of ramped temperature, ramped concentration, chemical reaction, heat production, and magnetic force on Casson nanofluid flow via a conduit were studied by Sadiq et al. [22]. The influence of various fluid dynamical processes and flow geometry is the focus of the bulk of the study. In addition, the bulk of previous research was conducted using either experimental or numerical methods.The energy storage devices are beneficial for regulating power and energy demand in concentrated solar power facilities. It is expected that increasing the capacity of materials used in total energy storage will increase their performance. As a result, the leading objective of this dissertation is to establish a solution to the problem of natural convection flow of two different nanofluids in a vertical channel under the influence of ramped velocity. Ramped wall velocity is useful to control the flow of the fluid. To the best of the author’s knowledge, ramped velocity is not considered for this model. Sodium alginate (SA) is taken as a base fluid having nanoparticles of titania (TiO2) and alumina (Al2O3) is studied. Analytical and semianalytical results for velocity field and temperature distribution are obtained by using the Laplace transform method and inverse numerical algorithms (Stehfest’s [23], Tzou’s [24], Talbot [25], Honig and Hirdes [26], and Fourier series [27]). Finally, the effects of nanoparticles, Prandtl number, heat absorption, and time on temperature and velocity profiles are graphically illustrated and discussed. The current findings are compared to previous findings in the literature. In the tables, the effect of volume fraction on Nusselt numbers and skin frictions is explored. The findings of this study are predicted to have a significant impact on solar thermal devices.


## 2. Mathematical Formulation

Consider the convective flow of two different nanofluids with ramped velocity in a vertical channel formed by two parallel infinite plates separated by a distanced in the presence of the source/sink effect. The left plate is considered along the x’-axis as shown in Figure 1. At t′=0, the nanofluid and plates are at rest at a moderate temperature T0. At t′>0, the right plate which is situated at y′=d begins to accelerate along x’-direction with A0t′, and temperatures of left and right plate are remained constant T1 and T2 (T2>T1), respectively. The slippage between nanoparticles and base fluid is neglected.Figure 1
Flow geometry.The heat and momentum are only the functions ofy′ and t′ as the walls of the channel are extended infinitely. Sodium alginate (SA) is considered as a conventional base fluid having nanoparticles of titania (TiO2) and alumina (Al2O3). The thermophysical characteristics of SA and nanoparticles are given in Table 1.Table 1
Thermophysical characteristics of nanoparticles and SA [28, 29].
MaterialkW/m.Kβ×105K−1CpJ/Kg.KρKg/m3Sodium alginate C6H9NaO7 (SA)0.63760.994175989Titania (TiO2)8.95380.90686.24250Alumina (Al2O3)400.857653970Under the above assumptions, the governing equations are [30]
(1)ρnf∂u′y′,t′∂t′=μnf∂2u′y′,t′∂y′2+gρβnfT′y′,t′−T1,(2)ρcpnf∂T′y′,t′∂t′=knf∂2T′y′,t′∂y′2−Q0T′y′,t′−T1,with corresponding conditions
(3)u′y′,0=0,T′y′,0=T1,0≤y′≤d,(4)u′0,t′=0,u′d,t′=A0t′,t′>0,(5)T′0,t′=T1,T′d,t′=T2,t′>0.The thermal physical features of nanofluid are described by(6)ρnf=φρs+1−φρf,μnf1−φ2.5=μf,knf=kf−ksφ+2kf+ksks−2kf−ksφ+2kfkf,ρβnfρβf=1−φ+φρβsρβf,ρcpnfρcpf=1−φ+φρcpsρcpf.Introducing the following non-dimensional parameters into Eqs. (1)–(5).
(7)u=u′νfA0d2,y=y′d,t=νfd2t′,T=T′−T1T2−T1,we get
(8)∂uy,t∂t=p1∂2uy,t∂y2+p2Ty,t,(9)∂Ty,t∂t=p3∂2Ty,t∂y2−p4Ty,t,with corresponding conditions,
(10)Ty,0=0,uy,0=0;0≤y≤1,(11)T0,t=0,u0,t=0;t>0,(12)T1,t=1,u1,t=t;t>0,where
(13)p1=μnfρnfvf,p2=Grβnfβf,p3=1Prknfkfρcpfρcpnf,p4=Qρcpfρcpnf,Gr=gβfT2−T1A0,Pr=vfρcpfkf,Q=Q0d2vfρcpf.


## 3. Solution of the Problem

### 3.1. Temperature Profile

Applying the Laplaceitransformito Eqs. (9), (11)1, and (12)1, using Eq. (10)1, we obtain
(14)sT¯y,s=p3∂2T¯y,s∂y2−p4T¯y,s,(15)T0,s=0,T¯1,s=1s.Solution of Eq. (14) with the conditions in Eq. (15) is as
(16)T¯y,s=sinhs+p4/p3ys.sinhs+p4/p3,or
(17)T¯y,s=∑n=0∞1sexp−2n+1p3−yp3s+p4−1sexp−2n+1p3+yp3s+p4.The inverse Laplace transform of Eq. (17) is
(18)Ty,t=12∑n=0∞expp4p32n+1−yerfc2n+1−y2p3t+p4t+erfc2n+1−y2p3t+p4t−12∑n=0∞expp4p32n+1+yerfc2n+1+y2p3t+p4t+erfc2n+1+y2p3t+p4t.


### 3.2. Velocity Field

Applying the Laplace transform to Eqs. (8), (11)2, and (12)2, using Eq. (10)2, we obtain
(19)su¯y,s=p1∂2u¯y,s∂y2+p2T¯y,s.(20)u¯0,s=0,u¯1,s=1s2.Putting the value ofT¯y,s from Eq. (16) in Eq. (19), we have
(21)su¯y,s=p1∂2u¯y,s∂y2+p2sinhs+p4/p3ys.sinhs+p4/p3.Solution of Eq. (21) with the conditions in Eq. (20) is as
(22)u¯y,s=p2p3s+p1p4−p3−p1sp1p4−p3−p1ss2sinhys/p1sinhs/p1−p2p3p1p4−p3−p1sssinhys+p4/p3sinhs+p4/p3,or
(23)u¯y,s=1ssinhy/p1ss.sinhs/p1−a1s−a2sinhy/p1ss.sinhs/p1+a1s−a2sinhs+p4/p3ys.sinhs+p4/p3,where a1=p2p3/p3−p1,a2=p1p4/p3−p1.The inverse Laplace transform of Eq. (23) is
(24)uy,t=1−a1ea2t∗∑n=0∞ψnyp1,t,0,1p1−ψn−yp1,t,0,1p1+a1ea2t∗∑n=0∞ψnyp3,t,p4,1p3−ψn−yp3,t,p4,1p3,where
(25)ψnx,τ,y,z=12e−2n+1z−xyerfc2n+1z−x2τ−yτ+e2n+1z−xyerfc2n+1z−x2τ+yτ.


## 3.1. Temperature Profile

Applying the Laplaceitransformito Eqs. (9), (11)1, and (12)1, using Eq. (10)1, we obtain
(14)sT¯y,s=p3∂2T¯y,s∂y2−p4T¯y,s,(15)T0,s=0,T¯1,s=1s.Solution of Eq. (14) with the conditions in Eq. (15) is as
(16)T¯y,s=sinhs+p4/p3ys.sinhs+p4/p3,or
(17)T¯y,s=∑n=0∞1sexp−2n+1p3−yp3s+p4−1sexp−2n+1p3+yp3s+p4.The inverse Laplace transform of Eq. (17) is
(18)Ty,t=12∑n=0∞expp4p32n+1−yerfc2n+1−y2p3t+p4t+erfc2n+1−y2p3t+p4t−12∑n=0∞expp4p32n+1+yerfc2n+1+y2p3t+p4t+erfc2n+1+y2p3t+p4t.


## 3.2. Velocity Field

Applying the Laplace transform to Eqs. (8), (11)2, and (12)2, using Eq. (10)2, we obtain
(19)su¯y,s=p1∂2u¯y,s∂y2+p2T¯y,s.(20)u¯0,s=0,u¯1,s=1s2.Putting the value ofT¯y,s from Eq. (16) in Eq. (19), we have
(21)su¯y,s=p1∂2u¯y,s∂y2+p2sinhs+p4/p3ys.sinhs+p4/p3.Solution of Eq. (21) with the conditions in Eq. (20) is as
(22)u¯y,s=p2p3s+p1p4−p3−p1sp1p4−p3−p1ss2sinhys/p1sinhs/p1−p2p3p1p4−p3−p1sssinhys+p4/p3sinhs+p4/p3,or
(23)u¯y,s=1ssinhy/p1ss.sinhs/p1−a1s−a2sinhy/p1ss.sinhs/p1+a1s−a2sinhs+p4/p3ys.sinhs+p4/p3,where a1=p2p3/p3−p1,a2=p1p4/p3−p1.The inverse Laplace transform of Eq. (23) is
(24)uy,t=1−a1ea2t∗∑n=0∞ψnyp1,t,0,1p1−ψn−yp1,t,0,1p1+a1ea2t∗∑n=0∞ψnyp3,t,p4,1p3−ψn−yp3,t,p4,1p3,where
(25)ψnx,τ,y,z=12e−2n+1z−xyerfc2n+1z−x2τ−yτ+e2n+1z−xyerfc2n+1z−x2τ+yτ.


## 4. Nusselt Numbers and Skin Frictions

The Nusselt numbers and skin frictions on both walls of the channel can express as(26)Nusseltnumbers=Nu0,1=−knfkfL−1∂θ¯y,s∂yy=0,1,skinfrictions=Sk0,1=−μnfμfL−1∂u¯y,s∂yy=0,1.


## 5. Numerical Results and Discussions

The flow of two different SA-based nanofluids with natural convection and ramped velocity is compared in this section. The influence of volume fraction and time on flow and temperature is highlighted graphically and discussed. Furthermore, the results of this problem are compared by using different numerical inversion algorithms in Tables2–3. In the graphical comparison, all parameters Q=1.1,Pr=6.2,φ=0.04,t=0.5, and Gr=3.8 are fixed.Table 2
Variation of Nusselt numbers.
φtTiO2Al2O3y=0y=1y=0y=100.5-0.118-2.989-0.118-2.9890.010.5-0.129-3.024-0.131-3.0310.050.5-0.185-3.164-0.197-3.1980.100.5-0.273-3.344-0.305-3.413Table 3
Variation of skin frictions.
φtTiO2Al2O3y=0y=1y=0y=100.5-0.52-0.085-0.52-0.0850.010.5-0.53-0.108-0.532-0.1060.050.5-0.576-0.207-0.583-0.1970.100.5-0.647-0.344-0.659-0.327Figure2(a) illustrates the variation and comparison of temperature profiles of two different nanofluids. The temperatures increase by increasing φ. The inclusion of nanoparticles increases the heat transport rate of nanofluids. Figure 2(b) depicts that the temperatures of nanofluids rise as time rises. The temperature of nanofluid-containing particles of Al2O3 is higher due to greater thermal conductivity. Figures 2(c) and 2(d) show the influence of Prandtl number and heat absorption on temperatures. The temperature fields of the nanofluids are shown to be lower when Pr and Q are increased.Figure 2
Variation of temperature fields.
(a)(b)(c)(d)Figure3(a) indicates that the thickness of nanofluids increases with an increase of φ due to the higher density of nanoparticles as a result velocities reduce. The velocity of SA+TiO2 is higher than SA+Al2O3 due to the low density of TiO2. Figure 3(b) illustrates that the velocities of both fluids increase with increasing time. The influence of Prandtl number and heat absorption on the velocity profiles is seen in Figures 3(c) and 3(d). The velocities of nanofluids decrease with an increase in Pr and Q.Figure 3
Variation of the velocity fields.
(a)(b)(c)(d)The authenticity of our results obtained for temperature and velocity is presented in Figure4 by comparing them to the results of Hajizadeh et al. [30]. These figures show that for t=1, our outcomes are equivalent to those found in [30]. The coinciding curves demonstrate the veracity of our findings.Figure 4
Comparison of results.
(a)(b)Table2 shows that when the volume fraction of nanoparticles rises, the heat transfer rate decreases on both plates. Table 3 displays the quantitative data of skin friction, which is decreased when the volume fraction values on both plates rise for two nanofluids.Tables4 and 5 show the comparison of our analytical solutions (18) and (24), with the semianalytical solutions obtained by different numerical inverse Laplace transform algorithms (Stehfest’s [23], Tzou’s [24], Fourier series [25], Talbot [26], and Honig and Hirdes [27]). From these figures and tables, it can be seen that all the numerical inverse Laplace transform algorithms have good agreement with our obtained results.Table 4
Comparison of temperature profile with different Laplace inversion algorithms.
yResults Eq. (18)Stehfest’sTzou’sHoning and HirdesFourier seriesTalbot00000000.13×10−105.2×10−82.4×10−61.1×10−83×10−93×10−100.21.1×10−73.3×10−76×10−61.59×10−71.3×10−71×10−70.34.1×10−64.6×10−61.8×10−54.06×10−64×10−64×10−60.47.7×10−59×10−51.1×10−47.75×10−57.7×10−57×10−50.59.9×10−49×10−41×10−39.1×10−49.8×10−49×10−40.69×10−38.4×10−38.5×10−38.41×10−38.4×10−39×10−30.70.0480.0480.0480.0480.0480.0480.80.1880.1880.1880.1880.1880.1880.90.510.510.510.510.510.51111.0011.0030.94911Table 5
Comparison of velocity profile with different Laplace inversion algorithms.
yResults Eq. (24)Stehfest’sTzou’sHoning and HirdesFourier seriesTalbot00000000.10.9640.9640.9650.9760.9670.9640.21.931.931.931.9541.9341.930.32.8982.8982.8992.9342.9042.8980.43.8693.8693.873.9183.8783.8690.54.8454.8454.8474.9064.8564.8450.65.8275.8275.8295.95.845.8270.76.8166.8166.8196.9016.8316.8160.87.817.817.8127.9067.8267.810.98.7588.7588.768.8668.7768.75819.3799.3799.3819.4979.3999.379


## 6. Conclusion

The focus of this work is to examine the results of the convective flow of two nanofluids in an upright channel with ramped velocity. Analytical results of temperature and velocity fields are attained by using the Laplace transform technique. Sodium alginate is considered a base fluid, and nanoparticles ofTiO2 and Al2O3 are added to it. Analytical and semianalytical results are compared. The effects of time, Prandtl number, heat absorption, and volume fraction are discussed in detail. The current findings are compared to previous findings in the literature. In the tables, the effect of volume fraction on Nusselt numbers and skin frictions is explored. The main observations are as follows:
(i)
The temperature profiles increase for higher values ofφ due to greater thermal conductivities(ii)
The velocity fields decrease for greater values ofφ due to high densities(iii)
The velocity and temperature fields are increasing function of timet(iv)
The nanoparticles (Al2O3) increase temperature much more than the nanoparticles (TiO2)(v)
The velocity is less when the nanoparticles alumina (Al2O3) is added in base fluid than by adding the nanoparticles titania (TiO2)(vi)
The velocity can be controlled and predicted with ramped velocity conditions(vii)
The thickness of nanofluids increases due to higher viscosity caused by greater values ofPr in return velocity and temperature reduces(viii)
The momentum and energy of nanofluids are reduced for higher values ofQ(ix)
Nusselt numbers and skin frictions decrease on both walls of channel for both nanofluids by increasingφ(x)
The solutions obtained by different methods are in good agreement



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