Numerical investigation on flow, heat and mass transfer performance of fractional Oldroyd-B hybrid nanofluid as a coolant for power battery

This paper introduced for the first time a viscoelastic hybrid nanofluid as the coolant for direct contact cooling power battery. The governing boundary layer equations were established by adopting fractional Oldroyd-B model and fractional Buongiorno’s model. Second-order velocity slip boundary conditions were also considered. Then the solutions were numerically acquired by finite difference coupled with L1 algorithm. Impact of main physical parameters on the flow, heat and mass transfer of the viscoelastic hybrid nanofluid on the cylindrical battery was graphically presented and detailly discussed. Outcomes show that the heat transfer is improved by both Brownian motion(Nb) and thermophoresis(Nt) to different degrees. When Nb grows from 0.05 to 0.1, the average Nusselt number increases by 2.2%, higher than 0.027% of Nt. The slip behavior only affects the velocity distribution near the individual cell and slightly enhances heat and mass transfer. The velocity relaxation fractional derivative contributes to convection, heat and mass transfer on the cell wall, while velocity retardation fractional derivative behaves just the opposite. The proposed viscoelastic hybrid nanofluid with appropriate volume fractions of nanoparticles enhances heat transfer on the cell wall and is strongly recommended as a candidate for power battery coolant.

Due to significant advantages such as energy conservation and environmental protection, new energy electric vehicles are greatly supported by various countries and becoming increasingly popular among consumers.Owning to the high energy demand of the power source, however, the battery heats up severely during operation.If the generated heat is not dissipated in a timely manner, the resultant local high temperature can easily cause premature failure of individual battery and even endangers the surrounding battery packages.Therefore, thermal management of the battery is one of the key technical challenges that need to be overcome for ever increasing electric vehicles.
The cooling media for battery packages include air 1 , liquid 2 , phase change materials(PCM) 3 , heat pipe 4 , etc. Air cooling is relatively simple in structure and low in cost, which is suitable for small energy supply devices such as electric bicycles and mobile robots 5 .However, due to low thermal conductivity and instability, air cooling may fail for endurance electric vehicles that require higher battery energy density 6 .PCM has the features of exchanging large energy in phase change without additional energy consumption, but its relatively poor thermal conductivity may lead to thermal instability of the battery package when the car has run for a long time 7 .Heat pipe cooling possesses excellent thermal conductivity, but its high cost limits its application in large-scale battery packages 8 .In current battery thermal management system, liquid cooling has become the most popular method due to its relatively low cost and thermal stability.
In practical applications of liquid cooling, battery may contact with the liquid directly or indirectly, see Fig. 1.Direct contact cooling method allows for direct heat exchange between the liquid and the battery.Indirect contact cooling method needs additional pipelines embedded in a cooling plate.The heat produced by the power battery is first transmitted to the plate, and then taken away by the liquid in the pipeline.Direct contact cooling method has absolute advantages in terms of heat exchange efficiency and overall temperature uniformity.
With the increasing demand for energy density and fast charging rate of power batteries, traditional coolants such as water, oil, and ethylene glycol may not be able to meet the requirements for fast cooling of power batteries

Mathematical model
Consider the problem of direct contact cooling for cylindrical cell shown in Fig. 1a.The viscoelastic hybrid nanofluid flows at a velocity of u e from the inlet, perpendicular to the axis of the cylindrical cell, and then flows around the cell and out through the outlet.In this process, the heat on the cell surface is taken away.Note that the variation of temperature in the viscoelastic hybrid nanofluid would cause change in density, resulting in heat transfer accompanied by mass transfer, both of which can be described by the renovated fractional Buongiorno's model proposed by Shen et al. 21.
The above problem can be abstracted as an unsteady boundary layer flow around a cylinder as shown in Fig. 2a, and further extended to the general case of stationary point flow on a flat plate 20 , see Fig. 2b.Since nonlinear wall slip behavior occurs in most viscoelastic fluid flow, second-order velocity slip between the nanofluid and the plate is also considered for this problem.In the Cartesian coordinate system, set x-axis along the plate and y-axis perpendicular to it.u and v are two velocity components along x-axis and y-axis, respectively.The temperature is T and the concentration is C.
Assuming that the viscoelastic hybrid nanofluid is a chemically stable fluid, nanoparticles Cu and Al 2 O 3 are sphericity with uniform size, the base fluid and nanoparticles are in thermal equilibrium state, particles are dispersed evenly through the base fluid, nanofluid is incompressible, and radiation heat transfer on the cell wall is neglected.Then the governing equations are established as 10 : (1) where S xy is shear stress, h p = (c p0 ) p (T-T ∞ ) is the specific enthalpy of the nanoparticles, q is heat flux, j p is the diffusion mass flux for the nanoparticles, ρ is the density, c p0 is specific heat at constant pressure, the subscripts hnf, p denote hybrid nanofluid and solid particles, respectively.j p can be obtained by adding Brownian diffusion and thermophoresis: where D B and D T are diffusion coefficients of Brownian and thermophoresis, respectively.Introducing the fractional constitutive equation of shear stress for Oldroyd-B hybrid nanofluid 22 : where α and β denote relaxation and retardation fractional derivatives of velocity, respectively, λ 1 and λ 2 indicate relaxation time and retardation time of velocity, respectively, D α t and D β t are Caputo fractional derivative operators, D α t is defined as 23 : where Ŵ(•) is Gamma function, 0 ≤ α ≤ 1.Using Eqs. ( 6) and (7), the momentum Eq. ( 2) for fractional Oldroyd-B hybrid nanofluid can be rewritten as: where μ is viscosity.Adopt the modified fractional Buongiorno's model 21 : where λ 3 is relaxation time of temperature, γ is fractional derivative of temperature, k is thermal conductivity.Substituting Eqs. ( 5) and (9) into Eqs.(3) and (4), the energy and concentration equations are acquired as: (4) where the subscripts w and ∞ represent the cell wall and the infinity, respectively, h is convective heat transfer coefficient, b and c are positive constants.The second-order slip velocity 24 K n is Knudsen number, ς = min{1/k n ,1}, ξ(0 ≤ ξ ≤ 1) is the momentum adaptation coefficient, χ is the molecular mean free path, a 1 > 0, a 2 > 0.
The skin friction coefficient, Nusselt number (represents heat transfer efficiency) and Sherwood number (represents mass transfer efficiency) are important physical quantities, which are defined as 21,25 : The hybrid nanofluid employed as the direct contact cooling medium for cylindrical battery is prepared by dispersing 2 vol.%Cu and 2 vol.%Al 2 O 3 nanoparticles in a viscoelastic surfactant solution, aqueous solution of cetyltrimethylammonium chloride (CTAC) and sodium salicylate (NaSal).The thermophysical properties of CTAC/NaSal-water and nanoparticles Cu and Al 2 O 3 are shown in Table 1.
Introducing dimensionless variables as follows: where L is the circumference of the cell, υ f is kinematic viscosity of CTAC/NaSal-water.Substituting the above dimensionless variables into Eqs.(1), ( 8), ( 10), (11) and ignoring the dimensionless superscript "*"for simplicity, we get: where Pr is Prandtl number, Nb and Nt are parameters of Brownian motion and thermophoresis, respectively, Sc is Schmidt number.The prominent parameters are symbolized as 27,28 : The subscripts f,nf,hnf,p,p1,p2 indicate base fluid, nanofluid, hybrid nanofluid, hybrid solid particles, nanoparticle Cu and nanoparticle Al 2 O 3 , respectively.ϕ 1 and ϕ 2 are volume fractions of Cu and Al 2 O 3 , respectively.
Using Eq. ( 15), the corresponding initial and boundary conditions become: where ) .The average skin friction coefficient, Nusselt number and Sherwood number in dimensionless forms turn into:

Calculation process
When t = 0, the values of u, v, θ and φ are acquired from Eq. (34).At (k-1)-level time step, the values of u, v, θ and φ in Eqs. ( 30), ( 31), ( 32), (33) are regarded as constants.At each inner node of i-layer in the computation region, the linear iterative equation forms a tridiagonal system, which can be solved by the pursuit method.
When the absolute values of the difference between two consecutive time steps for u, v, θ and φ are less than 10 -5 , the iterative process stops and reaches a stable state.The computation region is a rectangle with X max = 1 and Y max = 10.Here Y max lies far away from the boundary layer.Taking into account both numerical precision and computing time, the spatial steps are finally selected as ∆x = 0.01, ∆y = 0.05, and the time step as ∆t = 0.02. Figure 3 depicts the comparison of different numerical solutions for dimensionless temperature and concentration when the spatial grids are doubled in x and y directions.It is found that the numerical solutions maintain good consistency.Therefore, the selected grid size is suitable for the following calculations.To further prove the effectiveness of the numerical solutions, comparison of the present numerical results of − y=0 with similarity solution -θ'(0) in previous work is carried out when The results are in good agreement.

Effects of nanoparticles
Figure 4 shows the changes in heat and mass transfer by adding different volume fractions (ϕ 1 ) of nanoparticles Cu.It is seen from Fig. 4b that the average Nusselt number ANu(= k f /k hnf Nu ) is an increasing function of time, while the average Sherwood number Ash(= Sh ) is a decreasing function of time.Higher ϕ 1 slightly increases the concentration and Ash due to larger density in the hybrid nanofluid.ϕ 1 also enhances the temperature because of greater viscosity produced by nanoparticles.Note that ANu is no longer directly related to the temperature gradient on the wall, but rather a comprehensive function of the temperature gradient calculated from the initial moment.ANu slightly decreases with higher ϕ 1 , indicating that excessive addition of nanoparticles can actually reduce heat transfer efficiency.Considering the two factors of viscosity and thermal conductivity, low percentages of 2 vol.%Cu + 2 vol.%Al 2 O 3 are added in the base solution of CTAC/NaSal-water for this problem.Note that the addition of nanoparticles not only increases the thermal conductivity, but also enhances convective heat transfer through Brownian motion and.thermophoresis, which will be discussed later.

Effects of Brownian motion and thermophoresis parameters
Figure 5a discloses that both temperature and concentration and their boundary layer thicknesses decline with higher Nb.Physically, higher Nb indicates more intense in haphazard movement and more frequent in collision among nanoparticles, promoting dynamic balance in heat and mass transfer.Nb intensifies both average Nusselt number and average Sherwood number shown in Fig. 5c.As compared to Nb, Nt has a similar effect on temperature and average Nusselt number, but an opposite effect on concentration and average Sherwood number.That is to say, the thermophoresis parameter Nt increases heat transfer efficiency, but reduces mass transfer efficiency.By migrating the hybrid nanoparticles from the hot cell wall to cooler potential stream, Nt also helps in thermal equilibrium but to a lesser extent than Nb, see Figs. 5b and d. Figure 5 points out that the Brownian motion and thermophoresis can practically increase the heat transfer efficiency on the cell wall, especially Brownian motion.www.nature.com/scientificreports/

Effects of slip parameters
Figure 6 exposes the influence of first-order slip parameter (b 1 ) and second-order slip parameter (b 2 ) on the flow, heat and mass transfer of the hybrid nanofluid within the boundary region.It is observed that the effects of both slip parameters are similar, but that of first-order slip parameter is much more obvious.Due to the slip behavior, the velocity at the stagnation point(y = 0) on the cylindrical cell is no longer zero.As expected, the stagnation point velocity increases against an increment in both slip parameters, see Figs. 6a and b.In the vicinity of the cell wall, the velocity profiles intersect each other for different b 1 and b 2 , indicating that the slip parameters only affect the hybrid nanofluid velocity near the cell wall.The absolute values of average skin friction coefficient ACf(= (1-ϕ 1 ) 2.5 (1-ϕ 2 ) 2.5 u 2 e C f ) decrease with both b 1 and b 2 as time goes on, see Figs. 6e and f.The temperature and concentration and their boundary layer thicknesses reduce via both slip parameters, see Figs. 6c and d, which means that the slip parameters are to some extent beneficial for heat and mass transfer.This is further proved by the increment of average Nusselt number and Sherwood number through b 1 and b 2 , as shown in Figs.6g and h.It is observed from Figs. 7a and b that the velocity near the cell wall increases via α, but decreases through β.That's because the absolute values of average skin friction coefficient decline with α but grow with β, as indicated in Figs.7e and f.Therefore, α promotes nanofluid convection near the cell wall, while β obstructs it.Note that the velocity profiles intersect each other for different α and β as one moves away from the wall.That's because viscoelastic nanofluids possess transient memory properties and exhibit delayed reactions to external forces.In addition, the momentum boundary layer thickness declines with α, but increases with β, indicating that β enhances the viscoelastic effect, while α weakens it.

Effects of velocity fractional derivatives
From Figs. 7c,d,g and h, we can derive that the relaxation fractional derivative α slightly promotes heat and mass transfer, while the retardation fractional derivative β slightly suppresses them.

Effects of temperature fractional derivative
Figure 8 depicts the impact of temperature fractional derivative (γ) on flow, heat and mass transfer.It is seen from Fig. 8a that the velocity significantly reduces with increasing γ.Hence, higher γ hinders hybrid nanofluid convection.Both the temperature and concentration and their boundary layer thicknesses are reduced by swelling the value of γ, see Fig. 8b.It is noticed from Fig. 8c that the curves of average Nusselt number intersect each other for different γ, reflecting the thermoelastic effect in the heat conduction process.The heat transfer efficiency first reduces but then greatly increases through γ as time goes on.The average Sherwood number slightly declines as γ increases.All this indicates that, the temperature fractional derivative γ significantly improves heat transfer efficiency (though slightly delayed), but slightly reduces mass transfer efficiency.

Estimation of flow, heat and mass transfer for varying parameters
Table 3 reveals influence degree of relevant parameters on flow, heat and mass transfer.The intensity of heat and mass transfer is respectively characterized by average Nusselt number and average Sherwood number.Greater skin coefficient means larger resistance on the flow.It is obvious that the velocity fractional derivates α and β have great effects on average skin coefficient AC f , followed by Brownian motion parameter Nb and thermophoresis parameter Nt on average Sherwood number ASh.When β changes from 0.5 to 0.7, AC f increases by 311.89%.Moreover, the temperature fractional derivate γ affects the average Nusselt number and average Sherwood number more than α and β.When Nb grows from 0.05 to 0.1, the average Nusselt number increases by 2.2%, higher than 0.027% of Nt.Nb improves heat transfer more than Nt, which is consistent with the findings in Fig. 5.

Conclusion
This study introduces for the first time a viscoelastic hybrid nanofluid as the coolant for direct contact cooling power battery.The fractional Oldroyd-B model and fractional Buongiorno's model are adopted to establish the boundary layer governing equations, followed by Numerical simulation on the flow, heat and mass transfer of hybrid nanofluid on the cylindrical cell.The key observations are listed below: (1) Brownian motion enhances heat transfer more than thermophoresis.When the Brownian motion and thermophoresis parameters independently grows from 0.05 to 0.1, the average Nusselt number increases by 2.2% and 0.027%, respectively.(2) Brownian motion improves mass transfer, while thermophoresis obstructs it.
(3) The slip behavior makes the velocity on the cell wall no longer zero, but only affects the velocity distribution near the individual cell.(4) The slip behavior slightly promotes heat and mass transfer.(5) The velocity fractional derivatives describe the short memory characteristic of viscoelastic nanofluid.While temperature fractional derivative explains the thermoelastic effect in the heat conduction process.(6) The velocity relaxation fractional derivative contributes to convection, heat and mass transfer on the cell wall, while velocity retardation fractional derivative behaves just the opposite.(7) The velocity fractional derivates α and β have great effects on average skin coefficient.When β changes from 0.5 to 0.7, the average skin coefficient increases by 311.89%.(8) The temperature fractional derivative improves heat transfer efficiency but reduces mass transfer efficiency.Therefore, the proposed viscoelastic hybrid nanofluid with appropriate volume fractions of nanoparticles enhances heat transfer on the cell wall and is strongly recommended as a candidate for power battery coolant.

Figure 1 .
Figure 1.Schematic diagram of liquid cooling structure for power battery.

Figure 2 .
Figure 2. Physical model of direct contact cooling for cylindrical cell.

Figure 5 .
Figure 5. Effects of Brownian motion and thermophoresis parameters.

Figure 7
Figure 7 is designed to investigate the impact of velocity relaxation fractional derivative (α) and retardation fractional derivative (β) on flow, heat and mass transfer.It is detected that the effects of α and β are contrary.It is observed from Figs. 7a and b that the velocity near the cell wall increases via α, but decreases through β.That's because the absolute values of average skin friction coefficient decline with α but grow with β, as indicated in Figs.7e and f.Therefore, α promotes nanofluid convection near the cell wall, while β obstructs it.Note that the velocity profiles intersect each other for different α and β as one moves away from the wall.That's because viscoelastic nanofluids possess transient memory properties and exhibit delayed reactions to external forces.In addition, the momentum boundary layer thickness declines with α, but increases with β, indicating that β enhances the viscoelastic effect, while α weakens it.From Figs.7c,d,g and h, we can derive that the relaxation fractional derivative α slightly promotes heat and mass transfer, while the retardation fractional derivative β slightly suppresses them.

Table 3 .
Estimation of average skin friction coefficient AC f , average Nusselt number ANu, average Sherwood number ASh at t = 0.5 for varying parameters α, β, γ, Nb, Nt.