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Computational role of homogeneous–heterogeneous chemical reactions and a mixed convective ternary hybrid nanofluid in a vertical porous microchannel

  • Ajjanna Roja , Rania Saadeh , Javali Kotresh Madhukesh , MD. Shamshuddin EMAIL logo , Koushik Vijaya Prasad , Umair Khan , Chander Prakash and Syed Modassir Hussain
Published/Copyright: November 15, 2024
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High Temperature Materials and Processes
From the journal High Temperature Materials and Processes Volume 43 Issue 1

Abstract

This article mainly scrutinizes the heat transfer and flow characteristics of a mixed convection ternary hybrid nanofluid in a porous microchannel considering the catalytic chemical reaction and nonuniform heat absorption/generation. Using appropriate similarity transformations, the modeled equations are converted into reduced ones and then solved via the Runge–Kutta–Fehlberg 4th/5th order method. To strengthen this analysis, the convection mechanism has been deployed. The effect of pertinent physical parameters on the fluid motion and thermal field is displayed, including some important engineering variables like the Nusselt number, Sherwood number, and drag force. The novel outcomes display that the flow reduces with porous permeability and nanoparticle volume fraction. The temperature of the nanofluid improves with nonuniform heat absorption/generation. The concentration decreases in the presence of both homogeneous and heterogeneous reaction intensities. The heat transfer rate enhances for the Eckert number, and a similar influence on the mass transfer rate is noticed for homogeneous reaction parameters. Further, the drag force declines for the Grashof number. The outcomes show that, in all cases, the ternary hybrid nanofluid shows a greater impact than the nanofluid. The attained findings represent applications in the era of cooling and heating systems, thermal engineering, and energy production.

Keywords: ternary hybrid nanofluid; mixed convection flow; non-uniform heat source/sink; vertical microchannel; homogeneous–heterogeneous reaction

Nomenclature

A

Dimensionless pressure differentials

A 1 , A 2 , A 3 , A 4

Expression of thermophysical properties

B i 1 , B i 2

Biot numbers

C f

Skin friction coefficient

C p

Heat capacity ( J · k g − 1 · k − 1 )

D A , D B

Diffusion coefficients of species S 1 and S 2

Ec

Eckert number

F

Dimensionless velocity

g

Acceleration due to gravity ( m · s − 2 )

Gr

Grashof number

h

Channel width ( m )

h 1 , h 2

Convective heat transfer coefficients

KC

Heterogeneous parameter

KS

Homogeneous parameter

k

Thermal conductivity ( W · m − 1 · k − 1 )

k *

Porous medium permeability

k 1

Porous media shape factor

Ki

Rate quantities

Nu

Nusselt number

p

Pressure term ( Pa )

Pr

Prandtl number

q ‴

Non-uniform heat source/sink

Re

Reynolds number

R *

Exponential space heat source/sink term

S *

Temperature-dependent heat source/sink term

s 1 , s 2

Concentration of chemical species S 1 and S 2

s 0

Constant concentration

Sc

Schmidt number

T 1

Ambient temperature ( K )

T

Fluid temperature ( K )

T 0

Hot fluid temperature ( K )

u

Dimensional velocity ( m · s − 1 )

w 1

Uniform suction/injection

x , y

Co-ordinate axes

Greek letters

ρ

Density ( kg · m − 3 )

μ

Dynamic viscosity ( kg · m − 1 · s − 1 )

ν

Kinematic viscosity ( m 2 · s − 1 )

β

Thermal expansion coefficient

η

Dimensionless co-ordinate axes

θ

Dimensionless temperature

ϕ

Nanoparticle volume fraction

χ 1 , χ 2

Dimensionless concentrations

1 Introduction

Extremely high heat fluxes are the consequence of recent developments in micro-scale systems. Operating temperatures are critical to the functioning of these devices. Therefore, maintaining the appropriate operation of electric devices depends in large part on their cooling efficiency. Heat transfer enhancement techniques fall into two general categories: passive and active techniques. The external power is essential to increase the overall process of heat transportation, whereas it is not required for passive techniques. Application of these microdevices can be seen in multidisciplinary fields such as physics, chemistry, and biology, including microheat exchangers, microelectromechanical systems, and cooling systems for microdevices. The thermophysical properties of coolants have an impact on micro-scale devices’ cooling capability. The heat transfer capacity is restricted in conventional coolants, including ethylene glycol, oils, and water, due to their lower thermal conductivities. Therefore, research groups find it very interesting to search for new effective coolants. Nano-sized particles were endowed with enhanced thermal conductivity by dispersing them in convectional coolants, and these coolants are called nanofluids. Pahlevaninejad et al. [1] examined the transfer of heat characteristics of nanofluids in a microchannel by incorporating the impact of heat flux. Shen et al. [2] discussed the impact of material walls and the number of layers on nanofluid flow in a nanochannel and a microchannel. The heat transport characteristics of a single-phase nanofluid flow in a channel under the influence of convection heat transport were analyzed by Pinar et al. [3]. A numerical investigation of a carbon nanotube nanofluid flow in a microchannel was illustrated by Basha et al. [4]. Some significant works on nanofluids of distinct geometries are found in the studies of Krishna et al., Ramesh et al., Madhukesh et al., and Roja et al. [5,6,7,8,9,10].

Novel fluids with numerous features and exceptional thermal performance in devices have been created as a result of recent developments in nanofluid technology. These fluids include hybrid and ternary hybrid nanofluids. In comparison to mono nanofluids and convectional coolants, binary nanofluids show greater promise as effective heat transport agents in micro-/minichannel heat sinks. Because of the systems’ high thermal efficiency, hybrid nanofluids offer researchers and scientists a great deal of potential. In contrast to nanofluids, it has high engineering and industrial significance. However, there is a notable increase in thermal efficiency, time savings, and cost savings when two nanoparticles are added to the base fluid. Researchers are finding it increasingly appealing to include because of heat transport applications, including solar heating, machining coolants, refrigerators, nuclear system generator cooling, and so on. In this context, many authors have explored their work associated with various applications [11,12,13,14,15,16,17]. In comparison to binary hybrid nanofluids, triple nanofluids are thought to represent a new category of binary nanofluids that can enhance the thermal conductivity even more. As anticipated, ternary hybrid nanofluids exhibit superior heat transport properties in comparison to binary and normal nanofluids. The dual or hybrid nanofluid has produced noticeably superior outcomes than the mono nanofluids in each of these studies. Due to the success of dual nanofluids, scientists have extended this technology to consider ternary hybrid nanofluid design.

Ternary hybrid nanofluids, as they are called, offer even more potential in a variety of fields, such as energy, biomedicine, power generation, and the production of functional coatings. Three different nanoparticles immersed in a conventional fluid exhibit the best efficiency in dynamic viscosity, thermal conductivity, and larger stability than dual or mono nanofluids in the aforementioned applications. Mohamed Souby et al. [18] elucidated the entropy and heat transport characteristics of ternary hybrid nanofluid flows with microchannel heat sinks. A comparison analysis between a hybrid nanofluid and a ternary hybrid nanofluid was performed, and they confirmed that the ternary hybrid nanofluid provides the best efficiency to the thermal systems. Sharma et al. [19] studied the thermophoretic particle deposition on a ternary hybrid nanofluid flow in a microchannel with employed convective boundary conditions (BCs). Ishak et al. [20] analyzed an electrically conducting triple nanofluid flow that contains gyrotactic microorganisms in the shrinking/stretching of a horizontal plate. Rajput et al. [21] examined the ternary hybrid nanofluid between the two parallel plates under the impact of magnetism. Bhagya Swetha Latha et al. [22] investigated the ternary hybrid nanofluid flow with Cattaneo Christov heat flux with magnetism. Adun et al. [23] investigated the impacts of thermal field, volume concentration, and ratio mixture on the development of a ternary hybrid nanofluid.

Magnetohydrodynamics (MHD) is the study of charged fluids in a magnetic field. MHD yields close control of fluid flow in microchannels by modulating the dynamic equilibrium among magnetic pulls and dynamics of liquids. Applications range from microfluidic pumps, drug delivery systems, and electronic cooling through the use of MHD for dynamic flow control as well as reducing mechanical complexity while increasing heat dissipation and mixing efficiency in miniature devices. Krishna et al. [24] investigated the combined consequence of Soret, MHD, and Joule impacts over rotating vertical permeable plates with mixed convection. Ramesh et al. [25] explored the Maxwell liquid circulation across an extended surface in the presence of MHD and thermal radiation using numerical simulations. Krishna and Chamka [26] examined the time-dependent MHD ion slip impacts over a rotating nanofluid movement in a semi-infinite flat surface. Some noticeable works on MHD are listed in Ramesh et al., Krishna et al., and Cyriac et al. [27,28,29].

Chemical reactions inherently occur in homogeneous–heterogeneous (H-H) forms. Heterogeneous as well as homogeneous reactions are involved in many chemically reacting systems. Regarding the catalytic boundaries, heterogeneous reactions exist, while homogeneous reactions happen in the fluid’s main body. These reactions occur in various processes, including burning, biochemical frameworks, catalysis, and combustion. Only in the presence of a catalyst can heterogeneous reactions take place, such as those that happen during food preparation, distillation, fog formation, hydrometallurgical devices, and polymer production. A substance’s physical characteristics will alter during these stages, but its chemical makeup would not change. The initial work was developed on a boundary layer mathematical model of fluid flow with these reactions over a flat sheet by Merkin [30]. Hashmi et al. [31] analyzed the impact of an electrically conducting fluid flow on H-H chemical reactions. Naveed [32] examined the irreversibility scrutinization of nanofluid flow under the impacts of Joule heating and magnetism. Various authors have studied the H-H reactions on different aspects under different conditions associated with various applications [33,34,35,36].

Energy is an extremely essential resource that enables a society’s economic growth. Heat transfer can occur in a variety of ways, including Ohmic heating (Joule heating), thermal radiation (linear or nonlinear), and heat generation/absorption (uniform or non-uniform). Thermal generation/absorption is of essential importance to electronics, plastic manufacturing, rubber, nuclear fuel debris removal, nuclear reactors, waste material disposal, and various businesses. The system is heated by heat sources and cooled by heat sinks. Heat sources or sinks have a significant impact on heat transfer if there is a noticeable temperature difference between the surface and the surrounding fluid. Research studies have looked at the relationship between a heat source and sink parameters and between fluid flow and heat transmission properties. Sandeep and Sulochana [37] examined the dual solution for an electrically conducting fluid flow in a stretching/shrinking sheet under the impact of a non-uniform heat source/sink. Pattnaik et al. [38] investigated the impact of a uniform heat source/sink on micropolar fluid flow with H-H reactions. Madhukesh et al. [39] illustrated an electrically conducting ternary hybrid nanofluid flow in a horizontal microchannel.

From the abovediscussed review of literature, it is declared that the importance of non-uniform heat absorption/generation in various applications, such as energy production, thermal engineering, and geophysics, is associated with chemical reactions (H-H reactions). Because of this, the objective of the current examination was to explore a new field of inquiry by analyzing the properties of a mixed convection ternary hybrid nanofluid flow in a porous vertical microchannel affected by nonuniform heat absorption/generation. The modeled equations were reduced to ordinary differential ones by incorporating suitable dimensionless variables. The well-known Runge–Kutta–Fehlberg 4th/5th (RKF-45) method was utilized to solve the equation, and the impacts of all pertinent parameters are discussed with graphs. The current investigation was carried out to find the answers to the following research questions.

  1. What is the impact of changing values of Grashof number (Gr) on thermal and velocity profiles?

  2. How are the concentration profiles affected by changing the H-H reaction parameters?

  3. How does the rate of thermal distribution deflect toward nanofluids and ternary hybrid nanofluids?

2 Mathematical formulation

Consider a steady, incompressible, and laminar flow of a ternary hybrid nanofluid composed of engine oil through a porous medium, confined between vertical microchannel plates on either side, as illustrated in Figure 1. Further,

  • The flow is laminar and steady.

  • Channel plates are coated with porous media, and the distance between the plates is “a”.

  • At y = 0 , the lower plate is held at a temperature T 1 , while the upper plate, positioned at y = h , is maintained at a temperature T 0 .

  • The flow direction is in the x-axis, and the y-axis is perpendicular to it.

  • A ternary hybrid nanofluid consisting of GO , Au , C o 3 O 4 , and e ngine oil is considered.

  • Assuming the channel plates are infinitely long, the flow and thermal fields are considered to be fully developed.

  • Non-uniform heat source/sink constraints are incorporated into the energy equation.

  • The convective conditions for temperature are considered at both the plates.

  • The isothermal cubic autocatalytic model has been employed for H-H reactions.

Figure 1 Flow problem geometry.
Figure 1

Flow problem geometry.

Under these assumptions, the governing equations for momentum, temperature, and concentration can be expressed as follows [30,34,39]:

2.1 Momentum equation

(1) ρ thnf W 1 ∂ u ∂ y = − ∂ p ∂ x + μ thnf ∂ 2 u ∂ y 2 + ( ρ β ) thnf g ( T − T 0 ) − μ thnf k ⁎ u .

To simplify the momentum equation, apply the following dimensionless transformations [39]:

(2) p = P ⁎ μ f 2 h 2 ρ f , x = X h , y = h η , u = F h υ f .

By substituting equation (2) into equation (1), we obtain the following simplified equation:

(3) A 1 A 2 ∂ 2 F ∂ η 2 − Re ∂ F ∂ η + A A 2 + A 4 Gr A 2 θ − A 1 A 2 k 1 F = 0 .

Here, Re = W 1 h υ f , is the Reynolds number, k 1 = h 2 k ⁎ is the porous media shape factor, Gr = h 3 β f g ( T 1 − T 0 ) υ f 2 is the Grashof number, A = − ∂ P ⁎ ∂ X is the pressure term, A 1 = μ thnf μ f = 1 ( 1 − ϕ 1 ) 2.5 ( 1 − ϕ 2 ) 2.5 ( 1 − ϕ 3 ) 2.5 , and A 2 = ρ f ρ thnf , A 4 = ( ρ β ) thnf ( ρ β ) f .

The relevant BCs are provided as:

(4) u = 0 : y = 0 u → 0 : y = h .

The simplified BCs for equation (4) are as follows:

(5) F ( 0 ) = 0 F ( 1 ) = 0 .

2.2 Energy equation [30,34,39]

(6) ( ρ C p ) thnf W 1 ∂ T ∂ y = k thnf ∂ 2 T ∂ y 2 + q ‴ + μ thnf k ⁎ u 2 .

To simplify the thermal equation, apply the following dimensionless transformations [39]:

(7) θ = T − T 0 T 1 − T 0 , y = h η .

The term q ‴ represents a non-uniform heat source or sink, and it is expressed as:

(8) q ‴ = k f W 1 h υ f [ R ⁎ ( T 1 − T 0 ) F ′ + S ⁎ ( T − T 0 ) ] .

Here, R * is the space-dependent heat source/sink and S * is the temperature-dependent heat source/sink coefficient. The estimation R * & S * > 0 represents internal heat generation and R * & S * < 0 represents internal heat absorption.

Furthermore, by substituting equations (7) and (8) into equation (6), we obtain the following simplified equation:

(9) k thnf k f ∂ 2 θ ∂ η 2 − A 3 Re Pr ∂ θ ∂ η + Ec Pr ( A 1 k 1 ) F 2 + Re ( R × F ′ + S × θ ) = 0 .

Here, Pr = υ f α f is the Prandtl number and Ec = υ f 2 ( C p ) f ( T 1 − T 0 ) h 2 is the Eckert number.

The relevant BCs are provided as follows:

(10) k thnf ∂ T ∂ y = h 1 ( T − T 1 ) : y = 0 k thnf ∂ T ∂ y = − h 2 ( T − T 0 ) : y = h .

The simplified BCs for equation (10) are expressed in the following form:

(11) k thnf k f ∂ θ ∂ η η = 0 = Bi 1 ( θ ( η ) η = 0 − 1 ) k thnf k f ∂ θ ∂ η η = 1 = − Bi 2 ( θ ( η ) η = 1 ) ,

where Bi i = 1 , 2 = h h i = 1 , 2 k f denotes the Biot number.

2.3 Mass equation

The isothermal cubic catalyst model for H-H reactions involving two chemical species is described as follows [30,34]:

(12) S 1 + 2 S 2 → 3 S 2 , rate = K c s 1 s 2 2 ,

(13) S 1 → S 2 , rate = K s s 1 .

In this model, the concentration of chemical species S 1 & S 2 is represented by s 1 and s 2 , respectively. The rate quantities are denoted by K i ( i = c , s ) . The term isothermal reaction applies to both types of reactions. This leads to the following formulation of the mass equation [30,34]:

(14) W 1 ∂ s 1 ∂ y = D A ∂ 2 s 1 ∂ y 2 − K c s 1 s 2 2 ,

(15) W 1 ∂ s 2 ∂ y = D B ∂ 2 s 2 ∂ y 2 + K c s 1 s 2 2 .

To simplify the mass equation, apply the following dimensionless transformations [30,34]

(16) χ 1 = s 1 s 0 , χ 2 = s 2 s 0 , y = h η .

By substituting equation (16) in equations (14) and (15), we obtain the following equations:

(17) 1 Sc ∂ 2 χ 1 ∂ η 2 − Re ∂ χ 1 ∂ η − KC χ 1 χ 2 2 = 0 ,

(18) β Sc ∂ 2 χ 2 ∂ η 2 − Re ∂ χ 2 ∂ η + KC χ 1 χ 2 2 = 0 .

The respective BCs are provided as follows:

(19) D A ∂ s 1 ∂ y = h K s s 1 D B ∂ s 2 ∂ y = − h K s s 1 : y = 0 s 1 → s 0 s 2 → 0 : y = h .

The BCs, equation (19), are reduced as follows:

(20) ∂ χ 1 ∂ η η = 0 = KS ( χ 1 ( η ) η = 0 ) ∂ χ 2 ∂ η η = 0 = − KS ( χ 1 ( η ) η = 0 ) χ 1 ( η ) η = 1 = 1 χ 2 ( η ) η = 1 = 0 .

The prediction of diffusion coefficients S 1 & S 2 is nearly equal in most situations. Consequently, assuming that the diffusion factors D A and D B are identical, it follows the result χ 2 = 1 − χ 1 . Therefore, equations (17) and (18) and BCs, equation (20), take the following form:

(21) 1 Sc ∂ 2 χ 1 ∂ η 2 − Re ∂ χ 1 ∂ η − KC χ 1 ( 1 − χ 1 ) 2 = 0 ,

(22) ∂ χ 1 ∂ η η = 0 = KS ( χ 1 ( η ) η = 0 ) χ 1 ( η ) η = 1 = 1 .

Here, Sc = υ f D A is the Schmidt number, KC = Kc h 2 s 0 2 υ f is the homogeneous reaction strength, KS = h 2 Ks D A is the heterogeneous reaction strength, and β = D A D B is the ratio of diffusion coefficient.

The effective thermophysical properties of the nanofluid are as follows [22,23]:

(23) ρ thnf = ρ f ( 1 − ϕ 3 ) ( ( 1 − ϕ 2 ) ( ( 1 − ϕ 1 ) ρ f + ϕ 1 ρ 1 ) + ϕ 2 ρ 2 ) + φ 3 ρ 3 ,

(24) ( ρ C p ) thnf = ( 1 − ϕ 3 ) × ( ( 1 − ϕ 2 ) ( ( 1 − ϕ 1 ) ( ρ C p ) f + ϕ 1 ( ρ C p ) 1 ) + ϕ 2 ( ρ C p ) 2 ) + φ 3 ( ρ C p ) 3 ,

(25) ( ρ β ) thnf = ( 1 − ϕ 3 ) × ( ( 1 − ϕ 2 ) ( ( 1 − ϕ 1 ) ( ρ β ) f + ϕ 1 ( ρ β ) 1 ) + ϕ 2 ( ρ β ) 2 ) + φ 3 ( ρ β ) 3 ,

k thnf = k 13 + 2 k hnf − 2 ϕ 3 ( k hnf − k 13 ) k 13 + 2 k hnf + ϕ 3 ( k hnf − k 13 ) k hnf , k hnf = k 12 + 2 k nf − 2 ϕ 2 ( k nf − k 12 ) k 12 + 2 k nf + ϕ 2 ( k nf − k 12 ) k nf , k nf = k f k 11 − 2 ϕ 1 ( k f − k 11 ) + 2 k f k 11 + ϕ 1 ( k f − k 11 ) + 2 k f .

The thermophysical property values for the nanofluid and base fluid are as follows (Table 1):

Table 1

Values of thermophysical characteristics for the nanofluid and base fluid [22]

SI no. Property GO Au Co 3 O 4 Engine oil
01 ρ 1800 8908 8862 884
02 k 5000 91 99.2 0.144
03 C p 717 445 421 1910
04 β 0.000284 0.0000134 0.0000185 0.0007

The following are the significant engineering factors, along with their simplified forms [17,39]

(26) Cf = h 2 μ thnf ρ f υ f 2 ∂ u ∂ y y = h → Cf = μ thnf μ f ∂ F ∂ η η = 1 ,

(27) Nu = − h T 1 − T 0 k thnf ∂ T ∂ y y = h → Nu = − k thnf k f ∂ θ ∂ η η = 1 .

3 Numerical methodology

The boundary value problems as outlined by equations (3), (4), (9), (10), (21), and (22) were converted into a system of initial value problems using the shooting method. To solve these initial value problems, the RKF-45 method was employed, with a step size of 0.001 specified for this numerical analysis. The following formulas illustrate the technique used for this computation:

R 1 = If ( j n , k n ) ,

R 2 = If j n + 1 4 I , k n + 1 4 R 1 ,

R 3 = If j n + 3 8 I , k n + 3 32 R 1 + 9 32 R 2 ,

R 4 = If j n + 12 13 I , k n + 1 , 932 2 , 197 R 1 − 7 , 200 2 , 197 R 2 + 7 , 296 2 , 197 R 3 ,

R 5 = If j n + I , k n + 429 216 R 1 − 8 R 2 + 3 , 680 5 , 137 R 3 − 845 4 , 104 R 4 ,

R 6 = If j n + I 2 , k n − 8 27 R 1 + 2 R 2 − 3 , 544 2 , 565 R 3 + 2 , 197 4 , 101 R 4 − 1 5 R 5 .

An approximate result was obtained using the fourth-order Runge–Kutta scheme.

K i + 1 = k n + 25 216 R 1 + 1 , 408 2 , 565 R 3 + 2 , 197 4 , 101 R 4 − 1 5 R 5 .

The fifth-order Runge–Kutta method was employed to achieve a greater accuracy:

k i + 16 ′ = k n + 16 135 R 1 + 6 , 656 12 , 825 R 3 + 28 , 561 56 , 430 R 4 − 9 50 R 5 + 2 55 R 6 .

To compute the error term, the difference between two previously obtained values was calculated. If the error term was large, the process was adjusted by reducing the step size to meet the desired accuracy.

4 Results and discussion

This analysis examined the effects of isothermal H-H reactions on the flow of a ternary hybrid nanofluid with a non-uniform heat source/sink in a vertical microchannel. This study investigated a ternary hybrid nanofluid composed of nanofluids GO (graphene oxide), A u (gold), and Co3O4 (cobalt oxide) mixed with engine oil. Profiles were plotted for both the ternary hybrid nanofluid GO + Au + Co3O4/engine oil and nanofluid Au/engine oil for all relevant physical parameters. As shown in Table 2, the results demonstrate good agreement with reported results.

Table 2

Present scheme is validated against the reported results with A = Re = 1 in the absence of non-uniform heat absorption/generation, nanoparticles, and porous permeability parameters

η → 0 0.2 0.4 0.6 0.8 1
[40] Numerical 0 0.07114875 0.11376948 0.12154600 0.08676372 0
Analytical 0 0.07114875 0.09639032 0.12154600 0.08676372 0
Present work RKF-45 0 0.711487336843350249 × 10−1 0.113769451841154057 0.121545973417565345 0.867636994395631218 × 10−1 0

Figure 2 illustrates the thermal field changes for both ternary hybrid nanofluids and nanofluids with varying Ec estimations. The thermal distribution amplifies with a rise in the Ec. The impact of viscous dissipation becomes more significant with an increase in Ec. Essentially, when the Ec is elevated, the additional energy generated from viscous effects contributes to the overall thermal energy of the fluid, leading to an increase in its temperature. In ternary hybrid nanofluids, the combined effects of increased viscous dissipation, complex nanoparticle interactions, and higher particle concentration are more dominant compared to nanofluids for higher Ecs. This results in better thermal distribution in ternary hybrid nanofluids. The Ec is a crucial parameter for cooling the industrial machinery, electronic equipment, material processing, chemical reactors, and energy systems. Figures 3 and 4 report the significance of Gr in flow and thermal fields. Figure 3 shows that velocity increases with an increase in Gr. A higher Gr signifies the stronger buoyancy-driven forces in the flow, and as a result the velocity increases. The convection currents become stronger due to improved buoyancy effects. This intensifies the heat transfer within the fluid, which can lead to a higher temperature, as shown in Figure 4. In ternary hybrid nanofluids, both the velocity and temperature are more significant than in nanofluids due to enhanced buoyancy effects, greater effective nanoparticle concentration, and altered fluid properties that collectively optimize the heat transfer and fluid dynamics. Groffers valuable insights into natural convection effects, aiding in the optimization of design and performance for effective heat transfer and fluid management in applications such as cooling of electronics, biomedical devices, and energy systems.

Figure 2 Repercussion of Ec on the thermal field.
Figure 2

Repercussion of Ec on the thermal field.

Figure 3 Repercussion of Gr on velocity.
Figure 3

Repercussion of Gr on velocity.

Figure 4 Repercussion of Gr on the thermal field.
Figure 4

Repercussion of Gr on the thermal field.

A permeable space provides enhanced protection against the liquid stream, as it reduces the fluid velocity and affects the boundary layer. The flow of the liquid is significantly influenced by the permeable media. Figures 5 and 6 present the variation of velocity and temperature with increasing porous permeability parameter. An increment in k 1 declines the fluid velocity and amplifies the thermal field in a system. The drag force and frictional forces increase with a higher porous parameter due to increased flow resistance and localized flow disruptions, which influence the fluid flow paths. Additionally, this increment reduces convective heat transfer but enhances heat retention due to complex flow paths. Consequently, an increase in temperature is observed, as shown in Figure 7. Improving the efficiency and achieving a thorough understanding of controlling, optimizing, and selecting the system’s thermal exchange performance are facilitated by using this constraint. This is significant in the applications of cooling systems, chemical reactions providing a high surface area, microfluidic systems, phase change materials, and heat exchangers. Figures 7 and 8 illustrate the variation of concentration for homogeneous and heterogeneous strength parameters. The concentration gradient decreases with an increase in both homogeneous and heterogeneous strength. An increase in KC can lead to the formation of products that alter the concentration profile. These products may affect the physical properties of the fluid or interact with the nanofluids, further modifying the concentration distribution. Consequently, the concentration profile decreases as shown in Figure 7. An increase in the heterogeneous reaction parameter can alter the flow dynamics by increasing the resistance or changing flow patterns, which may decrease the mass transfer rate. In heterogeneous reactions, efficient mass transfer between solid and liquid phases is critical. Enhanced reaction parameters can affect the contact efficiency between reactants and the catalyst, leading to reduced chemical distribution within the flow system, as shown in Figure 8. It provides a significant role in chemical reactors, energy systems, cooling of high-power systems, and heat exchangers.

Figure 5 Repercussion of k 1 on velocity.
Figure 5

Repercussion of k 1 on velocity.

Figure 6 Repercussion of k 1 on the thermal field.
Figure 6

Repercussion of k 1 on the thermal field.

Figure 7 Repercussion of KC on concentration.
Figure 7

Repercussion of KC on concentration.

Figure 8 Repercussion of KS on concentration.
Figure 8

Repercussion of KS on concentration.

Figures 9 and 10 depict the variation of velocity and temperature with higher nanoparticle volume fraction. A higher nanoparticle volume fraction increases the viscosity. This greater viscosity leads to higher flow resistance, resulting in reduced nanofluid velocity through the microchannel. As shown in Figure 10, the thermal performance exhibits dual behavior at higher values of ϕ . It is clear that, up to the midpoint of the channel, the impact decreases, while it increases in the remaining portion. An increase in ϕ leads to a combination of enhanced thermal conductivity and increased viscosity, which affects heat transfer differently along the length of the channel. As a result, the temperature exhibits dual behavior. This approach can be employed to achieve better thermal management, enhanced heat transfer, and improved efficiency in systems with complex heat distribution and reactions. It is particularly useful in applications such as automotive cooling systems, chemical processing, biomedical devices, microelectronics cooling, and thermal management in renewable energy systems. Figures 11 and 12 demonstrate the variation of the thermal profile for both space and internal heat generation/absorption parameters. The analysis encompasses both internal heat generation and heat absorption. An increase in the internal heat generation parameter leads to a more pronounced thermal field in the fluid flow due to greater heat input and enhanced heat transfer, as displayed in Figure 11. Additionally, it is evident that the temperature distribution is higher for ternary nanoliquids compared to single nanofluids. Figure 12 illustrates the decrease in thermal dispersion under conditions of internal heat absorption. This occurs because greater heat absorption results in increased heat removal and more efficient redistribution of thermal energy. The space-dependent heat generation/absorption constraint significantly impacts thermal dispersion in the system, with higher values leading to improved thermal dispersion. By controlling these constraints, engineers can enhance the thermal transport efficiency, temperature control, and overall system performance in applications such as thermal energy storage, power production, manufacturing processes, chemical reactors, electronic device cooling, and HVAC (heating, ventilation, and air conditioning) systems.

Figure 9 Repercussion of ϕ on velocity.
Figure 9

Repercussion of ϕ on velocity.

Figure 10 Repercussion of ϕ on the thermal field.
Figure 10

Repercussion of ϕ on the thermal field.

Figure 11 Repercussion of R * on the thermal field.
Figure 11

Repercussion of R * on the thermal field.

Figure 12 Repercussion of S * on the thermal field.
Figure 12

Repercussion of S * on the thermal field.

Figures 13 and 14 present the variation in the drag force and heat transfer rate for various physical parameters. A decrease in surface drag force is observed with higher values of porous permeability and Gr (Figure 13). An increase in porous media reduces flow resistance, resulting in a smoother, more streamlined flow. This minimizes frictional forces and decreases drag on the channel walls. On the other hand, a higher Gr boosts buoyancy-driven convection, which reduces viscous drag by improving fluid mixing and lowering wall shear stress. Consequently, the drag force is reduced. Figure 14 shows that the heat transfer rate is enhanced with larger values of the Ec and Re. Increasing the Ec enhances a larger proportion of kinetic energy to thermal energy transformation. Additionally, a higher Re leads to a more pronounced flow velocity profile and better thermal mixing, further boosting the convective heat transfer rate. Consequently, this results in improved heat transfer on the channel walls. Additionally, the heat transfer rate is higher for ternary hybrid nanofluids than for single nanofluids due to improved thermal mixing and optimized flow characteristics. This will benefit practical applications such as heat transfer devices, energy-efficient technologies, ecosystems, and thermal management systems.

Figure 13 Variation in skin friction coefficient.
Figure 13

Variation in skin friction coefficient.

Figure 14 Variation in Nu.
Figure 14

Variation in Nu.

5 Conclusions

In this study, the effects of H-H reactions, in conjunction with non-uniform heat generation/absorption, on mixed convection of a ternary hybrid nanofluid flow in a vertical microchannel were investigated. The problem was numerically solved using the RKF-45 method, along with the shooting method. The key findings of the study are as follows:

  • The fluid velocity decreases with increasing porous parameters and nanoparticle volume fraction. However, the effect is reversed for the Gr.

  • The dual impact on the thermal field is observed with an increase in ϕ .

  • Significant improvements in the thermal distribution are observed with larger values of Ec, Gr, and porous parameter.

  • In the case of internal heat generation, the space-dependent heat generation/absorption constraint will enhance the thermal circulation.

  • An increase in KC and KS reduces the concentration gradient, which can enhance the reaction efficiency, optimize the heat transfer, and improve the process control.

  • In all these cases, ternary hybrid nanofluids exhibit a more prominent role than single nanofluids in affecting the drag force, heat transfer, and mass transfer rates.

5.1 Limitations and future work

This investigation was confined to the study of GO , Au , and C o 3 O 4 nanoparticle accumulation using engine oil. It focused solely on thermal enhancement in the flow of ternary hybrid nanofluids (GO + Au + Co3O4/engine oil) through a vertical porous microchannel under the influence of non-uniform heat source/sink constraints and H-H reactions. Future work could extend this research to explore various Newtonian and non-Newtonian fluids combined with different nanoparticles, diverse BCs, as well as time-dependent impact.

5.2 Applications of the present work

The outcomes of the present work can be found useful in various engineering aspects such as electronic devices, cooling gadgets, and HVAC systems. The outcomes are also useful in improving the thermal distribution in solar collector devices and thermoelectric generators. Chemical processing industries benefit from this work in controlling the temperature and mass distribution, further improving the overall system performance and waste thermal recovery process.

Acknowledgements

The authors thank the Deanship of Scientific Research, Islamic University of Madinah, Madinah, Saudi Arabia, for supporting this research work. Also, this research was funded by the Scientific Deanship of Zarqa University, Jordan.

  1. Funding information: The authors thank the Deanship of Scientific Research, Islamic University of Madinah, Madinah, Saudi Arabia, for supporting this research work. Also, this research is funded by the Scientific Deanship of Zarqa University, Jordan.

  2. Author contributions: A.R., R.S., and J.K.M.: conceptualization, methodology, software, formal analysis, validation, writing – original draft. K.V.P. and U.K.: writing – original draft, data curation, investigation, visualization, validation. MD.S.: conceptualization, writing – original draft, writing – review and editing, supervision, resources. S.M.H. and C.P.: validation, investigation, writing – review and editing, formal analysis, project administration, funding acquisition.

  3. Conflict of interest: The authors state no conflict of interest.

  4. Data availability statement: The datasets used and/or analyzed during the current study are available from the corresponding author upon reasonable request.

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Received: 2024-07-21
Revised: 2024-08-21
Accepted: 2024-09-26
Published Online: 2024-11-15

© 2024 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/htmp-2024-0057
Keywords for this article
ternary hybrid nanofluid; mixed convection flow; non-uniform heat source/sink; vertical microchannel; homogeneous–heterogeneous reaction
Creative Commons
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  1. Research Articles
  2. De-chlorination of poly(vinyl) chloride using Fe2O3 and the improvement of chlorine fixing ratio in FeCl2 by SiO2 addition
  3. Reductive behavior of nickel and iron metallization in magnesian siliceous nickel laterite ores under the action of sulfur-bearing natural gas
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  55. Stretch-forming characteristics of austenitic material stainless steel 304 at hot working temperatures
  56. Work hardening and X-ray diffraction studies on ASS 304 at high temperatures
  57. Study of phase equilibrium of refractory high-entropy alloys using the atomic size difference concept for turbine blade applications
  58. A novel intelligent tool wear monitoring system in ball end milling of Ti6Al4V alloy using artificial neural network
  59. A hybrid approach for the machinability analysis of Incoloy 825 using the entropy-MOORA method
  60. Special Issue on Recent Developments in 3D Printed Carbon Materials - Part II
  61. Innovations for sustainable chemical manufacturing and waste minimization through green production practices
  62. Topical Issue on Conference on Materials, Manufacturing Processes and Devices - Part I
  63. Characterization of Co–Ni–TiO2 coatings prepared by combined sol-enhanced and pulse current electrodeposition methods
  64. Hot deformation behaviors and microstructure characteristics of Cr–Mo–Ni–V steel with a banded structure
  65. Effects of normalizing and tempering temperature on the bainite microstructure and properties of low alloy fire-resistant steel bars
  66. Dynamic evolution of residual stress upon manufacturing Al-based diesel engine diaphragm
  67. Study on impact resistance of steel fiber reinforced concrete after exposure to fire
  68. Bonding behaviour between steel fibre and concrete matrix after experiencing elevated temperature at various loading rates
  69. Diffusion law of sulfate ions in coral aggregate seawater concrete in the marine environment
  70. Microstructure evolution and grain refinement mechanism of 316LN steel
  71. Investigation of the interface and physical properties of a Kovar alloy/Cu composite wire processed by multi-pass drawing
  72. The investigation of peritectic solidification of high nitrogen stainless steels by in-situ observation
  73. Microstructure and mechanical properties of submerged arc welded medium-thickness Q690qE high-strength steel plate joints
  74. Experimental study on the effect of the riveting process on the bending resistance of beams composed of galvanized Q235 steel
  75. Density functional theory study of Mg–Ho intermetallic phases
  76. Investigation of electrical properties and PTCR effect in double-donor doping BaTiO3 lead-free ceramics
  77. Special Issue on Thermal Management and Heat Transfer
  78. On the thermal performance of a three-dimensional cross-ternary hybrid nanofluid over a wedge using a Bayesian regularization neural network approach
  79. Time dependent model to analyze the magnetic refrigeration performance of gadolinium near the room temperature
  80. Heat transfer characteristics in a non-Newtonian (Williamson) hybrid nanofluid with Hall and convective boundary effects
  81. Computational role of homogeneous–heterogeneous chemical reactions and a mixed convective ternary hybrid nanofluid in a vertical porous microchannel
  82. Thermal conductivity evaluation of magnetized non-Newtonian nanofluid and dusty particles with thermal radiation
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