Abstract
The homogeneous-heterogeneous reaction in the boundary layer flow of a water-based nanofluid in the stagnation-point region of a plane surface is investigated. The type of small particles explored here is the single-walled carbon nanotubes. The homogeneous nanofluid model is employed for description of behaviours of nanofluids. Here, the homogeneous (bulk) reaction is isothermal cubic autocatalytic, while the heterogeneous (surface) reaction is single, isothermal, and first order. The steady state of this system is analysed in detail, with equal diffusion coefficients being considered for both reactants and autocatalysts. Multiple solutions of the reduced system are captured for some particular sets of physical parameters, which seem to be overlooked in all previous published works with regard to studies of homogeneous-heterogeneous reactions modeled by homogeneous nanofluid models. Besides, we discover the significant limitation of previous conclusion about that the solutions by homogeneous nanofluid flow models can be recovered from those by regular fluids.
1. Introduction
Chemically reacting systems in liquids or biochemical systems such as catalysis and burning admit both homogeneous and heterogeneous reactions. However, the correlation between homogeneous and heterogeneous reactions is rather complex to handle. The reaction heat that is generated or absorbed during the chemical reactions has strong influence on the flow and heat transfer in the surrounding fluid that in turn affects the local concentrations of the reactants and the products and eventually the performance of the chemical reactions.
Motivated by experimental observations of Williams et al. [1] and Song et al. [2] with regard to homogeneous-heterogeneous reactions on methane/ammonia and propane oxidation over a platinum, Chaudhary and Merkin [3, 4] developed a mathematical model on homogeneous-heterogeneous reactions in which the bulk reaction is expressed by the isothermal cubic autocatalytic kinetics and the surface reaction is governed by the first-order kinetics. Xu [5] made an extension to the reaction model of Chaudhary and Merkin [3, 4] to introduce the effect of the homogeneous-heterogeneous reactions into the energy equation.
It has been known that heat transfer rate of base fluids could be enhanced significantly as highly conductive solid particles are added into them [6–8]. For example, Choi et al. [6] discovered the prominent heat transfer enhancement of working fluid when copper nanotubes are in suspension of water. As such, if the nanoparticles are added in the working fluid to make it become a nanofluid, since the thermophysical properties of the fluid have been changed, the chemical reaction happening in that fluid can be affected. When assuming that the suspended nanoparticles are not involved in the chemical reactions of interest, Xu and his collaborators [9, 10] further extended the homogeneous-heterogeneous reaction model to nanofluids by using Buongiorno’s approach [11]. Hayat et al. [12] applied the homogeneous-heterogeneous reaction model to three-dimensional flow of a nanofluid in the suspension of carbon nanotubes over a nonlinear stretchable surface. Bachok et al. [13] studied a stagnation region flow towards a stretching sheet with homogeneous-heterogeneous reaction effects. Other works engaged in the application of the homogeneous-heterogeneous model to different flow and heat transfer problems can be found in the literature studies [14–17].
The suspended nanoparticles in nanofluids can be either metal, metal oxide, carbon nanotubes, or the combinations of them. Recently, carbon nanotubes (CNTs) have attracted much attention since they were discovered by Iijima [18] through discharging carbon electrodes to develop the new multiwalled carbon nanotubes (MWCNTs). Bethune et al. [19] then reported the development of single-walled carbon nanotubes (SWCNTs) that are structurally different from the multiwalled carbon nanotubes (MWCNTs). Since then, carbon nanotubes have become an attractive topic owing to their unique mechanical and thermal properties. Ramasubramaniam et al. [20] fabricated homogeneous single-wall carbon nanotubes (SWCNTs) for electrical conductivity applications. Xue [21] established a relation for effective thermal conductivity of carbon nanotubes based on composites. The high diffusion conductivity of the single-wall carbon nanotubes was then checked by some researchers such as Ma et al. [22], Khoshmehr et al. [23], and Park and Kim [24].
As aforementioned, chemical reactions in liquids are complex, and different reaction rates can happen even under the same conditions. From the modelling point of view, this means that the solutions of the theoretical model for the chemical reaction of interest may not be unique. Though a few studies on homogeneous-heterogeneous chemical reactions in the boundary layer flow and heat transfer based on the homogeneous nanofluid model [8] have been done, no solution multiplicities yet have been reported. Based on the discussions by Chaudhary and Merkin [3, 4] as well as Xu and his collaborators [5, 9, 10], multiple solutions could be possible for homogeneous-heterogeneous chemical reactions with the homogeneous nanofluid model. In this paper, we investigated the homogeneous-heterogeneous reactions in a nanofluid in suspension of the single-walled carbon nanotubes (SWCNTs) when equal diffusion coefficients are considered for both autocatalyst and reactants. We found the existence of solution bifurcations which, to the best of our knowledge, has not been reported before. We also testified the validation of the previous conclusion [25] on that the solutions by homogeneous nanofluid flow models can be recovered from those for regular fluids and commented that it is only valid for a few special cases. This work can be deemed as a significant improvement of previous studies on homogeneous-heterogeneous reactions in a homogeneous nanofluid.
2. Mathematical Modeling
2.1. Problem Description
Consider a steady boundary layer flow and heat transfer of a SWCNT-nanofluid in the stagnation-point region towards a plane surface in the presence of homogeneous-heterogeneous chemical reactions. The homogeneous-heterogeneous reaction model proposed by Chaudhary and Merkin [3, 4] is applied to describe the behaviours of chemical reactions. In this model, the homogeneous reaction takes place in the bulk, which is isothermal cubic autocatalytic and governed bywhile the heterogeneous reaction occurs on the catalyst surface, which is single, isothermal, and the first order and written by
In the aforementioned equations, and denote concentrations of the species and , respectively, and and are the homogeneous and the heterogeneous reaction rates, respectively. Note that Williams et al. [1] and Song et al. [2] have confirmed the existence of homogeneous-heterogeneous reactions via experiments about methane/ammonia and propane oxidation over platinum.
It is known that the homogeneous flow model [8] has been used to predict the behaviours of homogeneous-heterogeneous chemical reactions in nanofluids’ flow. By this approach, the thermal and physical properties of a nanofluid are derived from the base fluid and the suspended solid nanoparticles [21] aswhere is the volume fraction of nanoparticles, is the dynamic viscosity, is the kinematic viscosity, is the density, is the thermal diffusivity, is the heat capacity, and is the thermal conductivity. The subscripts , , and CNT denote nanofluid, fluid, and carbon nanotube (SWCNT), respectively. The thermophysical properties of water and SWCNTs used in this study are listed in Table 1.
Correspondingly, the governing equations describing the conservations of total mass, momentum, and energy, as well as chemical reaction diffusion, in the presence of single-wall carbon nanotubes (SWCNTs), in the framework of the boundary-layer approximations, are written assubject to the boundary conditionswhere and are the fluid velocity components in the - and -directions, respectively, is the incoming velocity of external flow, and are the diffusion coefficient of species and , respectively, is the surface temperature, is the reference temperature, is a constant, and is the concentration of species at the far field.
2.2. Scaling Transformation
In the studies of flow and heat transfer in the boundary layers, it is a common practice to introduce the boundary layer assumptions [26] to reduce the governing partial differential equations to ordinary differential equations via similarity transformation. The resulting equations contain self-similar solutions which are similar to themselves as independent and dependent variables are appropriately scaled.
We utilize the same idea to handle the problem considered in this work. In doing so, we introduce the following similarity variables:
Substituting the similarity variables (6) into equation (4), continuity equation (4a) is automatically satisfied, and the rest equations are reduced tosubject to the boundary conditionswhere is the Prandtl number, is the strength coefficient of the homogeneous reaction, and are the Schmidt numbers, is the ratio between the diffusion coefficients, and is the strength coefficient due to the heterogeneous reaction. Note that, here, all those physical parameters are defined based on the properties of the base fluid so that the influence of nanoparticles can be clearly reflected.
In many practical applications, the diffusion of species and is in comparable sizes, which indicates that the diffusion coefficients and can be deemed as equal [3]. As a result, we are able to choose . In this case, the following relation holds:
Using equations (9), (7c), and (7d) which are combined to obtain
2.3. Remarks on the Homogeneous Flow Model
It is worth mentioning to this end that Magyari [25] once argued that the solutions obtained by the homogeneous nanofluid flow model could be recovered by the corresponding regular fluid’s results, and no further efforts need to be paid for investigation of homogeneous modelling nanofluid flows. Here, we testify the validity of his conclusion via equations (7a), (7b), and (10). In doing so, we introduce the following scaling transformations:where , , , , and are positive constants.
Substituting equation (11) into equations (7a), (7b), and (10), we obtain
To resume the governing equations to Newtonian fluid’s ones, it must hold
From equation (13a), we obtain
However, from equation (13c), we have
If only the flow field is considered, then the fixed relation between homogeneous modelling nanofluid and regular fluid’s solutions is readily discovered from correlation (14). However, as multiple physical fields are taken into account, such fixed relation is no longer in existence. This is to say, it is impossible to recover homogeneous modelling nanofluid’s solutions via simple scaling transformations. Therefore, we are able to conclude that Magyari’s comment [25] is only available for a few specific problems. For most fluid flow problems, especially those involving into multiphysical fields, the homogeneous nanofluid model could not be replaced by scaling transformations so that it plays a significant role on investigation of flow behaviours of dilute nanofluids.
2.4. Computational Method
Solutions of governing equations (7a), (7b), and (10) subject to the boundary conditions (8) are solved by means of the shooting technique [27]. The integration distances are chosen from a range of values such as with the step size being set to , respectively. If all computational results for a prescribed set of physical parameters with different integration distances and step sizes remain unchanged, then we think the solutions are stable. In this way, it is found that, for and , the stability of multiple solutions can be kept. The Newtonian iteration is executed with the convergence criteria of for all considered cases to examine the accuracy of results.
3. Solution Dependence on
It has been already known from Chaudhary and Merkin [3] that the strength coefficients of homogeneous and heterogeneous reactions are key factors to generate solution bifurcations. We therefore testify the roles of and on the solution behaviours with various nanoparticle volume fractions. In our computation, the Prandtl number is fixed, , for the base fluid (water) throughout the study. Since the homogeneous nanofluid model is only suitable for the dilute nanofluid, the range of the volume fraction of nanoparticles is chosen from 0 to 0.2 . Particularly, from the definitions of nanofluids’ parameters denoted in equation (3), it is readily known that the case corresponds to the regular fluid.
3.1. Solutions for Small and Moderate
We start our discussion by considering the variation of due to the jointly effects of and in the presence of SWCNT nanoparticles. Clearly, the multiple solutions still exist as the nanoparticles are considered. For example, for and , multiple solutions are found for and as in particular regions. However, for , only unique solution can be found, as shown in Figure 1. It is also observed from this figure that the hysteresis point is obtained for the decreasing values of (say ). For all values, , multiple solution are possible.
The bifurcation points for possible ranges of are listed in Table 2. It is seen in this table that multiple solutions are found in the ranges for and for , while only one unique solution is discovered for .
We then consider the effect of the nanoparticle volume fraction on for different values of . In doing so, we choose , , and as an illustrative case where multiple solutions are available. As shown in Figure 2, as increases, the bifurcation points for both maximum and minimum shift towards the left side. This indicates that the addition of nanoparticles is helpful to keep the stability of chemical reaction for large , while it could lead to the instability of chemical reaction for small . The maximum and minimum points for solution multiplicities are listed in Table 3. It is clearly shown that multiple solution regions are for , for , and for .
The influence of on for different values of at , , and is presented in Figure 3. It is observed that only one unique solution exists as the homogeneous reaction rate , while multiple solutions are captured for and provided that is properly chosen. It is worth mentioning that, in our model assumption, the heterogeneous reaction rate is expected to be always positive. In the meantime, has to be equal or greater than zero and less than or equal to 1 (owing to ). This means that all solutions for and (or ) are physically meaningless. Particularly, the system can keep the self-sustained surface reaction without the homogeneous reaction being involved when nanoparticles are appended. As the hysteresis point is reached for increasing value of (say ), multiple solutions are possible for all values of . Figure 4 shows the effect of on for some values of nanoparticle volume fraction . From the figure, it is observed that the variation of could not alter the behavior of solution multiplicity. In other words, the existence of multiple solutions is not affected by changes of nanoparticle volume fraction. The role of the nanoparticle volume fraction is that it can accelerate or postpone the chemical reaction rate.
3.2. Solutions for Large K
It has been reported by Chaudhary and Merkin [3] that there is a shear-layer-type structure developed by the concentration profile as continuously increases. The similar structure is found to be still in existence when nanofluids are considered. As shown in Figure 5(a), the reaction gradually compacts to a relatively thin region that increasingly departs from the surface as evolves. This figure also shows that, as enlarges, the thickness of the fully reacted-state region grows larger, which leads to the thickness of the unreacted-state region which becomes thinner.
(a)
(b)
From equation (10), it is expected that the asymptotic behavior for large is in the order of (1). In the reaction zone, this equation suggestswhere is a constant and denotes large . Here, the reaction front has a thickness of and is centered on . Therefore, is expanded aswhere , , , are constants to be determined.
Using equation (16) in equation (10), the following equation is obtained:subject to the boundary condition
Here, the prime denotes the differentiation with respect to .
Equation (18) is expected to have a solution of the form
Substituting equation (20) into equation (18) and then omitting the high-order terms, at the first-order, it reads
Taking boundary condition (19) into account, it is expected that is exponentially small in in the inner wall region such that
As briefly discussed in [28], equation (21) subject to boundary condition equation (22) is the cubic-Fisher wave equation that is essentially an eigenvalue problem for . Its solution has exponential decaying property at infinity as expected. The reason is that the solutions to equation (21) have algebraic properties of as , where , which in turn bring about the perturbation of to in the external flow.
Using the method proposed by Billingham and Nadeem [28], the unique solution for equation (21) is given as
At , it holdssubject to the boundary conditionswhere prime denotes the differentiation with respect to .
Since equation (24) includes a complementary function which satisfies the required boundary conditions, an additional boundary condition has to be introduced to give a solution to the full equation. To meet this requirement, the following correlation is introduced (refer to [3]):which gives . Therefore, can be expressed as
However, as concluded by Chaudhary and Merkin [3], equation (27) is not a complete solution since the condition at does not match to that in equation (23). Therefore, they suggested to consider the condition at the inner edge of the reaction front where the inner solution decays exponentially small in . As such, the following relations hold:where is in the order of (1) for the region. Due to equation (28), it is readily known that possesses exponentially decaying behaviors for large . Therefore, we have
Note that this relation still holds when the effect of nanofluids is considered.
It is worth mentioning to this end that the nanofluid is considered in our case; therefore, the nanoparticle volume fraction plays a role on . From equation (27), we notice that is closely related to which is equal to 0.647, 0.685, and 0.744 for , , and , respectively. This indicates that enhances as evolves for prescribed values of , as shown in Figure 5(b).
4. Autoignition
Autoignition takes place at . To decide the occurrence time of the autoignition, equation (10) subject to the following boundary conditions needs to be taken into account:
As shown in Figure 6(a), the nanoparticle volume fraction has perceptible effects on . For each , there is a critical value of . For , two different branches are available when other parameters are prescribed. These solutions correspond to the two points in Figure 3, where for the case , , and . Though not shown in the figures, it is expected that the lower and the upper branches for approach to 0 and 1, respectively, as approaches to infinity.
(a)
(b)
(c)
Particularly, for , equation (10) suggests that the upper branch has the following form:where is a scaling transformation function. If we let be unscaled, then the following leading order problem is derived:subject to the boundary conditions
By solving this equation numerically, we obtain for , for , and for when and , respectively. As a result, we obtain the asymptotic expressions of for as
For example, asymptotic expression (35) is shown in Figure 6(a) by the dashed curve which is in good agreement with the numerical solution, especially for large . Also, from our previous analysis, the behavior of the lower solution branch becomes exponentially small as grows considerably large.
As expected, the critical value is dependent on and as well. The dependence of on is shown in Figure 6(b), with the corresponding values of being shown in Figure 6(c). From these figures, we notice that reduces monotonously as evolves. On the contrary, for prescribed values of , decreases gradually as enlarges. Some critical values for various values of are listed in Table 4 in which the comparison with the results of Chaudhary and Merkin [3] is presented, and good agreement is found.
5. Solution Dependance on
It is obvious that can alter solution behaviours. We, therefore, discuss its influence on when is small and large, respectively.
5.1. Solution for Large
To check the significance of , we take and are both in the order of (1). From equation (10), it is known that solutions could be possible for large , provided thatwhere and are the scaling function and variable, respectively.
Equation (37) clearly shows that as approaches to infinity. The reaction region therefore has the thickness of for considerably large . Substituting equation (37) in (10), it yieldssubject to the boundary conditions
can be expanded, based on equations (38) and (39), as
From equation (37), we further obtainwhich indicates that there is no possibility for the existence of multiple solutions at large when and are both in the order of O(1).
However, equation (41) will lose its valid range when and are not small. To handle this situation, we writewhere and both are and . Equation (10) is then reduced to the following leading order equation:subject to the boundary conditions
The hysteresis bifurcation point can then be calculated by solving equation (43). For example, we obtain and at , and at , and and at which further leads to
As shown in Figure 7, the increase in causes the decrease in (Figure 7(a)) but the increase in (Figure 7(b)).
(a)
(b)
(c)
5.2. Solution for Small
In this case, the reaction region is thicker than the region of the boundary layer flow. Since the thick outer region is of extent , we introduce new variable and leave unscaled, and then write
Substituting equation (46) into (10) and using the new scalings, at the leading order, we obtain
In the inner boundary layer region, we expand at the leading order, obtainingwhere are constants which can be determined successively. Particularly, using equation (47), we obtain .
At the order of , is linear and satisfies ; we therefore expand it, as approaches to 0, bywhere are constants.
On the contrary, based on equations (48) and (49), it suggests
Note that the inner region has to satisfy the boundary conditions (8b) at . Therefore, the leading order equation givessubject to the boundary conditions
The solution of this equation is . At the order of , it holdssubject to the boundary conditionswhich give . As a result, we obtain
Thus, the appropriate inner condition for equation (47) is
The solutions of equation (47) subject to the inner condition (56) determine that from which we obtain
Equation (57) indicates that there is a singularity at . To determine the nature of the hysteresis bifurcation for small , we put
The solution at the leading order is simplified as
However, at the order of , equation (53) still holds and satisfieswhich give . So, we have
To balance the outer region with the inner boundary condition, we obtain
We, therefore, are able to find the solution of equation (46) by solving leading order equation (47) subject to the inner boundary condition
It is now expected that equation (47) subject to the inner boundary conditions (63) has multiple solutions. The hysteresis bifurcated point can be determined in a similar way. We obtain the critical values and for , and for , and and for , which further give
6. Conclusions
The homogeneous-heterogeneous reactions in a nanofluid with the suspended single-walled carbon nanotubes at the stagnation-point region of a plane surface have been investigated. The homogeneous (bulk) reaction is assumed to be isothermal cubic autocatalytic, while the heterogeneous (surface) reaction is single, isothermal, and first order. The homogeneous nanofluid model is employed with equal diffusion coefficients being considered for both reactants and autocatalysts. By doing similarity transformation, multiple solutions of the model have been captured for particular sets of physical parameters, which are overlooked in previous publications on modelling homogeneous-heterogeneous reactions. Furthermore, we have discovered the limitation of Magyari’s conclusion [25] about that the solutions of the homogeneous nanofluid model can be recovered from those of the regular fluid model. This statement is only valid for a few special situations.
In summary, the main novel aspects of this study are as follows:(i)Multiple solutions for modelling homogeneous-heterogeneous reactions are found for some sets of physical parameters, and their origins are discussed(ii)Homogeneous and heterogeneous reaction rates and , which are the obvious key factors to affect the chemical reactions, are important parameters to give multiple solutions(iii)Nanoparticle volume fraction plays a significant role on altering solution stability, and it also can affect the stability of chemical reactions(iv)Magyari’s conclusion [25] on that the solutions of the homogeneous nanofluid model can be recovered from those of the regular fluid model is found to be limited and only valid for a few special cases
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was partially supported by the National Natural Science Foundation of China (Grant no. 11872241). This work was supported in part by the Australian Research Council through the Centre of Excellence grant CE140100003 to QS.