After verifying the stagnant point flow model with the previous numerical work in the presence of a shrinking and stretching sheet using HNFs, the patterns of the velocity outline, temperature outline , temperature distribution for a special case, and Nusselt number at the starting point will now be discussed. Herein is an examination of how the variables such as , , , and estimations impacted these patterns and which parameters could be effective in addressing any application. Careful observation of these numerical results was performed to gain control over the convection and conduction processes.
4.1. Velocity Outline
In
Figure 6a,b, we plotted the numerically calculated velocity profile. For each graph, we fixed the estimations of
, and we used only copper nanofluids with a concentration of 0.1. In
Figure 6a, we set the estimation of
to zero, while in
Figure 6b, we set the estimation of
to 1. These two graphs allowed us to observe the patterns of the velocity profile.
In
Figure 6a, one can see that when the estimation of
was negative, the velocity profile was at a minimum at
= 0. As the estimation of
escalated, the velocity of the transport material also escalated. This means that in the shrinking case, the velocity profile escalated. One can also observe that even for the
= 0 case, increasing the estimation of
led to an escalation in the velocity profile. Similarly, in the
= 2 case in
Figure 6a, one can see that as the estimation of
escalated, the velocity profile decayed. This indicates that in the stretching case, the velocity of the material decayed. In
Figure 6b, graphs were developed for the same cases but with
= 1. Here, one can observe that the minimum and maximum estimations at
= 0 slightly decayed. In the shrinking case, it was noted that when the estimation of
was less than −2, the velocity profile decayed. Similarly, it is evident that when there was an escalation in
, which is a suction parameter, the velocity profile decayed in both the stretching and shrinking cases.
In the shrinking case, the sheet shrank, causing the entire fluid material to accumulate in one place, leading to an escalation in velocity. On the other hand, in the stretching case, the fluid material spread, resulting in a decay in velocity. γ is referred to as the suction parameter because it allows fluid to leave the boundary, thereby causing a decay in the velocity profile.
In
Figure 7a–c, we also wanted to observe the pattern of the velocity profile of HNFs using the slip flow parameter
. For this purpose, we fixed the volume fraction of copper at 0.1 and set
as 0, which represents the suction parameter. In
Figure 7a, it was noted that when the estimation of
was −2, as the slip flow parameter escalated from 0 to 1, the velocity profile at
= 0 decayed. From observing all the subgraphs in
Figure 7a, one can see that increasing the estimations of
also escalated the velocity profile.
In
Figure 7b, we set the estimation of
to zero, indicating a stationary sheet. Here, it was noted that as the estimation of
escalated, the velocity profile at
= 0 escalated. However, if we consider an overall escalation in the estimation of
, it became apparent that the velocity profile was escalating.
In
Figure 7c, one can see that we fixed the estimation of
as 2, indicating a stretching sheet. Here, it was noted that as the slip parameter
escalated, the velocity decayed at
= 0 in the stretching case. It is quite clear in this stretching case that the velocity profile decayed as the estimation of
escalated. From this discussion, it became evident that the pattern of the slip flow parameter was different for the stretching and shrinking cases. Even when the sheet was stationary, i.e.,
= 0, increasing the slip flow parameter led to an escalation in the velocity profile at
= 0.
In
Figure 8a–c, we created graphs to observe the velocity profiles of the HNFs, and this time, we wanted to escalate the volume fraction of copper. We fixed
= 1 and
= 0 (no suction), and then from
Figure 8a–c, we escalated the estimation of
to check its effect.
Looking at
Figure 8a, where
is −2, one can observe a shrinking sheet. In these graphs, it was noted that as the volume fraction of copper escalated, the velocity profile at
= 0 also escalated. Overall, increasing the estimation of
also escalated the velocity profile for each fixed estimation of the volume fraction of the hybrid mixture. It was noted that when the volume fraction was 0.05 or 0.1, there was a nearly negligible difference in the velocity profile, while keeping the volume fraction at 0.01 showed a significant difference with a notable decay in the velocity profile.
In
Figure 8b, where
was set to 0, representing a stationary sheet, it was also clear that the velocity outline escalated at
= 0, when the volume fraction was increased. In
Figure 8a,b, one can observe the cases where the volume fraction was 0.01. In
Figure 8a, the velocity escalated but at a decreasing rate, while in
Figure 8b, the velocity field escalated at an increasing rate.
In
Figure 8c, it is evident that increasing the volume fraction of the nanofluid led to an upgrade in the velocity outline at
= 0, where
was set to 2, representing the stretching case. This indicates that increasing the volume fraction of HNFs provided significant benefits to the velocity outline in the shrinking and stationary cases observed in
Figure 8a and
Figure 8b, respectively. However, in
Figure 8c, which discusses the stretching case, it was noted that for higher estimations of the volume fraction, the velocity outline decayed as the estimations of
escalated. Finally, one can also observe that in both the stretching and shrinking cases, the velocity outline escalated when the volume fraction of the HNF was kept at 0.01.
Physical Reasons: In the shrinking case (negative ), the sheet contracted, causing the fluid to be drawn toward the surface. This accumulation of fluid led to an increase in velocity as (a dimensionless distance parameter) increased. The fluid was compressed, resulting in a higher velocity near the surface. Conversely, in the stretching case (positive ), the sheet expanded, causing the fluid to spread out. This spreading resulted in a decrease in velocity as increased because the fluid was being pulled away from the surface, leading to a lower velocity near the surface. The suction parameter introduces a mechanism that allows fluid to leave the boundary layer. This suction effect causes a decrease in the velocity profile in both shrinking and stretching cases. As increases, the boundary layer thickness decreases, leading to a reduction in velocity.
The slip flow parameter influences the velocity profile differently in shrinking, stationary, and stretching cases. In the shrinking case (negative ), increasing reduces the velocity at equal zero but increases the overall velocity profile with increasing . In the stationary case ( equals zero), increasing increases the velocity at equals zero and the overall velocity profile with increasing . In the stretching case (positive ), increasing reduces the velocity at equals zero and the overall velocity profile with increasing . Increasing the volume fraction of copper nanofluids enhanced the velocity profile in the shrinking and stationary cases. A higher concentration of nanoparticles increased the effective thermal conductivity and viscosity, leading to a more pronounced velocity profile. However, in the stretching case, while increasing the volume fraction initially enhanced the velocity profile, it eventually caused a decrease as increased due to the increased resistance and reduced momentum diffusion caused by the higher nanoparticle concentration.
4.2. Temperature Outline
This section will discuss the temperature outline and analyze its pattern by observing it through various parameters. We produced
Figure 9a–c, in which we wanted to observe the temperature outline
(
) and examine the pattern as we varied the
estimations. In these graphs, we set
= 0 and
= 0.1, while in
Figure 9a–c, we varied the estimations of
.
In
Figure 9a, it was noted that regardless of the
estimations, the temperature outline decayed as the
estimations escalated, and the temperature outline was maximum at
= 0. In
Figure 6a, it was noted that when
= 2, the temperature decrement occurred rapidly. Furthermore, in
Figure 9a, it was noted that when
= 0, the temperature decrement was much slower. In
Figure 9a, it was noted that when
= −2, the temperature decrement was faster than in the case of
= 2. Here,
= 0 indicated a no-suction condition.
Moving on to
Figure 9b, where
had an estimation of 0.7, it was noted that in the stretching case with
= 2, the temperature decayed rapidly, and the least decrement occurred at
= −1, which represented a shrinking case. This implies that adjusting the slip flow parameter can significantly affect the temperature decrement. In
Figure 9b, it was noted that
= 0 had a faster decrement compared to the case after
= 2.
In
Figure 9c, where the slip flow parameter δ was set to 1, and λ estimations ranged from −2 to 2, the temperature outline showed the fastest decay. Additionally, for the stretching case, the temperature decrement was the fastest among all cases, regardless of the estimation of the slip flow parameter.
In
Figure 10a,b, we observed the pattern of the temperature outline, where we set
= 0 and the volume fraction of copper to 0.1. We fixed the estimation of
for each graph. It was noted in both
Figure 10a,b that for almost every graph, the temperature outline decayed as the
estimations escalated.
In
Figure 10a, where
is −2, it was noted that increasing the slip flow parameter
resulted in a significant temperature decrement. This means that the temperature decayed rapidly as the
estimation escalated from 0 to 1. Now let us examine
Figure 10b, where
was 2 and represented the stretching case. It was also evident here that increasing the slip flow parameter
led to a rapid decline in temperature. This indicates that maintaining a high slip condition is essential for developing a cooling process.
It is now clear that whether it is a stretching sheet or a shrinking sheet, increasing the slip flow parameter results in a significant decay in the temperature outline, initiating the cooling process. In
Figure 11a–c, we wanted to observe the temperature outline, where we set
= 0 and kept the estimation of
as 1.
Figure 11a–c were developed specifically to observe the temperature outline
with varying estimations of
. In
Figure 11a, where
= −2, it was noted that the temperature outline decayed for all estimations of
as the estimation of
escalated. However, when the estimation of
was 0.05, the temperature declined at a faster rate compared to other
estimations.
Let us observe
Figure 11b, where
was 0. Here, it was noted that for
= 0.05, the temperature decay was the highest, while for
= 0.01, the temperature outline declined at a slower rate. Moving on to
Figure 11c, it was noted that for
= 0.05, the temperature decline was the most significant, and the cooling process was rapid. Here,
had an estimation of 2. Based on these results, one can understand that in shrinking cases, the temperature decline is faster for almost all
estimations.
4.3. Temperature Distribution
In this section, we will discuss the temperature distribution over the domain. For this purpose, we imposed a cold temperature Tw on the wall, and the ambient temperature Tinf, provided in
Table 2, represents the hot temperature. Here, we examined the temperature distribution with the variation in parameters used in the flow. We will discuss our observations of the temperature patterns here.
In
Figure 12a, it was noted that we kept
as 1 and
as 0.1. In
Figure 12a, it was noted that when v was −2, the temperature increment was slower compared to other
estimations. One can observe a faster temperature increment for the stretching case in almost all other cases compared to the shrinking case. Moving on to
Figure 12b, we set
as 1, and it was noted that this change almost brought the temperature increment close to each estimation of
. This means that by increasing the slip parameter’s estimations, we could control the temperature increment for different
estimations. This discussion also proves that in stretching cases, the temperature increment is the highest. In
Figure 13a–c, we examined the pattern of temperature distribution by changing the suction slip flow parameter
. Here, we kept the estimation of
as 0. In
Figure 13a, it was noted that at
= 0, the temperature increment was very fast compared to other
estimations. As the
estimations escalated, the temperature increment slowed down. Here,
had an estimation of −2, and this represents a shrinking case.
Moving on to
Figure 13b,
was 0 and the sheet was stationary. In this case, it was noted that as the slip flow parameter
was raised, the temperature increment escalated rapidly. This case was completely different from the shrinking case described in
Figure 13a. In
Figure 13c, we developed these graphs by setting
as 2. In these graphs, similar to the shrinking case, an escalation in
estimation had a negative impact on the temperature increment, meaning that the temperature increment declined. In all the graphs of
Figure 13a–c, it was noted that the temperature distribution increment was directly proportional to the
estimations, but it could be easily controlled by
estimations.
In
Figure 14a–c, the temperature distribution along the shrinking or stretching sheet was examined with
and
set to 1. These graphs were developed to analyze the pattern of temperature distribution with varying
estimations. In
Figure 14a, where
was estimated at −2, it was observed that when
was 0.05, the temperature increment was significantly higher compared to other
estimations. Specifically, when
was 0.05, there was a sudden escalation in temperature increment after
equalled 2. In
Figure 14b, where the sheet was stationary,
equaling 0.05 again showed the highest temperature increment. The temperature increment was very slow when
was estimated at 0.01. In
Figure 14c, which represents the stretching case, it can also be observed that the same
estimation provided a significant temperature increment. Thus, it can be concluded that only the shrinking sheet benefits from lower volume fractions, resulting in better temperature increments.
Validation: To validate the results above for the temperature, the numerical results from a recent article [
48] were considered for the percentage change in the temperature due to an ambient temperature. The formula for the percentage change in temperature (T%) is given below and referenced in [
48]:
According to [
48], with a settled
= 323.15 K and
= 293.15 K for the present code, and while setting up that
equals
equals 0 and
equals 0.1, the percentage change in temperature
was calculated using the formula given above. Therefore,
Figure 15 can be seen for a comparison showing that the values of
vs.
correlate for these two studies in a similar pattern. The difference in these results is due to the reason that in [
48], a ternary nanofluid was used, whereas in the present numerical scheme, the fluid flow and heat transfer in the stretching and shrinking sheet were evaluated using hybrid nanofluids. Nevertheless, both numerical formulations showed good agreement with each other.
4.4. Nusselt Number and Skin Friction Coefficient
In
Table 4, we performed calculations for the Nusselt number considering specific cases. As we all know, the Nusselt number is a ratio between the convection process and the conduction process. If the Nusselt number is increasing, it means that the convection process is becoming more dominant, while a decreasing Nusselt number indicates an increase in the conduction process. In
Table 4, we calculated the Nusselt number at the starting point. It was noted that in some cases, as we increased the
estimations, the Nusselt number also increased. However, in cases where the sheet was stationary (
equaled 0), increasing
estimations led to a decrease in the Nusselt number. These cases are highlighted in the table. Additionally, in cases where the volume fraction was 0.01, it was often observed that increasing the slip flow parameter led to a decrease in the Nusselt number. In
Table 4, it can also be seen that for the same estimations of
, whether it was 0.05 or 0.1, increasing
estimations resulted in an increase in the Nusselt number, indicating a rise in the convection process. From this entire discussion, it became evident that
estimations and
estimations can significantly influence the pattern of the Nusselt number with the alteration in
estimations.
Physical Reasons: In the shrinking case, where is negative, the fluid accumulates near the surface due to compression, resulting in slower temperature increments. The increased fluid density near the surface reduces heat transfer efficiency, causing a slower rise in temperature. Conversely, in the stretching case, where is positive, the fluid spreads out, allowing more efficient heat transfer and resulting in a faster temperature increment. The expansion of the fluid enhances convection, promoting a rapid rise in temperature. The suction parameter influences the temperature distribution by drawing fluid away from the boundary layer, which reduces the temperature increment. As increases, the suction effect is enhanced, leading to more uniform temperature profiles across different values by reducing the boundary layer thickness. The slip flow parameter also affects heat transfer. For stationary or stretching sheets, increasing enhances the temperature increment by promoting fluid motion and mixing, which improves heat transfer. In contrast, for shrinking sheets, increasing reduces the temperature increment by weakening the heat transfer due to reduced fluid accumulation near the surface. Regarding the volume fraction of nanofluids , higher volume fractions of copper nanofluids increase thermal conductivity and heat capacity, leading to significant temperature increments. In shrinking cases, lower volume fractions are more effective due to better fluid accumulation. In stretching and stationary cases, higher volume fractions enhance heat transfer, resulting in a notable temperature rise.
4.5. An Application of the Results in PV/T
In this section, we employed the present simulation to investigate the enhancement in electrical efficiency in a photovoltaic thermal (PV/T) system, focusing on a specific test case. The PV/T system, designed for harnessing solar radiation to generate electricity, featured a solar panel equipped with a glass layer, a silicon layer, and a copper absorber, alongside a flow channel for coolant passage.
The efficiency of the PV/T system diminished with an escalation in solar radiation-induced heat. To mitigate this, a nanofluid coolant was introduced into the flow channel to decay the system’s temperature, thereby improving the thermal efficiency of the solar panel. The impetus behind this study was derived from pertinent references [
45,
50,
51].
For the sake of simplicity, we assumed adherence to the construction specifications outlined in the references [
52], encompassing the use of glass, silicon, and copper as the absorber materials. Assuming a reference efficiency
of 12% at a of 0 °C, we permitted HNFs to ingress with an inlet temperature (or wall temperature)
= 298.15, positioned at y = 0, while establishing the ambient temperature
= 45 °C as y tended toward infinity. Let
= 25 °C represent the reference temperature. The ensuing equations govern the dynamic behavior of the system, Equations (18)–(21), encapsulating the intricate interplay of thermal and electrical parameters in this PV/T configuration [
52].
where
is the temperature coefficient, assumed to be 0.0045 [1/K] in this context, and
represents the cell temperature, provided as follows.
After considering Equations (18)–(20), the electrical efficiency for the PV/T channel was determined by the following expression, in which the packing factor p = 0.8.
Electrical efficiency [
52]:
After successfully developing the numerical simulation, the numerical results for the electrical efficiency will be explained as follows.
Figure 16a–d depict the variation in cell efficiency with the non-dimensional parameter
. Each figure maintained a constant suction parameter
, while the stretching and shrinking parameters remained fixed for each curve. The numerical outcomes were generated with fixed estimations of
= 1 and the volume fraction of nanomaterials (0.1). In
Figure 16a, the electrical efficiency of the PV/T system peaked at approximately 10.16% for
= 0, irrespective of the
estimations. Notably, a negative correlation between electrical efficiency and the non-dimensional parameter
was observed. Under
= 0,
Figure 16a highlights that, for
estimations of −2 and −1 in the shrinking case, differences in electrical efficiency were negligible, with maximum efficiency observed at these
estimations. Additionally, a decline in electrical efficiency became asymptotic after reaching
= 3.
Figure 16b explores the electrical efficiency pattern against
with an escalated suction parameter
estimation of 0.4. Here, the impact of
estimations on electrical efficiency became more pronounced, underscoring the role of suction in optimizing a PV/T system’s efficiency. Nevertheless, a decline in electrical efficiency was noted with increasing
estimations, especially in the presence of a stretching sheet. Once again, the maximum electrical efficiency of 10.16 was observed at
= 0.
Figure 16c and
Figure 16d delve into the influence of suction parameters tested at 0.7 and 1, respectively. The impact of
alterations on electrical efficiency diminished in both figures. A consistent decline in electrical efficiency with increasing
estimations was observed, reaching an asymptote at
= 2. The numerical results consistently indicated the optimization of electrical efficiency for the shrinking sheet in both scenarios. Notably,
Figure 16d suggests that in the presence of suction, altering the stretching/shrinking sheet had a minimal impact on electrical efficiency.
In summary, this study concludes that electrical efficiency experiences a decline with increasing . Furthermore, the adoption of a shrinking sheet proves effective in optimizing electrical efficiency. The influence of suction on electrical efficiency is limited when altering estimations in the model.
In
Figure 17a–c, we present a comprehensive analysis of electrical efficiency plotted against the non-dimensional parameter
. In each figure, we maintained fixed estimations for
,
, and
, while systematically varying the suction parameter (
) to observe its impact on the electrical efficiency pattern.
In
Figure 17a, we examined the scenario of a shrinking sheet with a
estimation set to −2. For all fixed estimations of the suction parameter
, the electrical efficiency experienced a decline with increasing
due to elevated temperatures. Interestingly, the absence of suction at the bottom wall enhanced electrical efficiency against
. Notably, the electrical efficiency reached a plateau or became asymptotic when
= 3. At the origin (
= 0), the electrical efficiency peaked at approximately 10.16%, gradually decreasing to 8.7% at
= 3, reflecting a decrement of 14.3%.
Turning our attention to
Figure 17a,b, we explored two
cases (0 and 2) to scrutinize the electrical efficiency pattern against
while incrementing
estimations. Both figures clearly illustrate a decreasing trend in electrical efficiency with rising
. Moreover, an accelerated decline was observed as
, the suction parameter, escalated. In
Figure 17b, where the sheet remained stationary, it was noteworthy that electrical efficiency leveled off or became asymptotic for all
estimations when
= 3. However,
Figure 17c, featuring a stretching sheet with a
estimation of 2, revealed a decline in electrical efficiency against
, reaching an asymptote at
= 2.
In
Figure 18a–c, we conducted a detailed examination of the electrical efficiency pattern vis-à-vis the non-dimensional parameter
, manipulating the slip flow parameter
. In each figure, we maintained fixed estimations for
,
, and
, with each curve representing a constant estimation of
. In
Figure 18a, a discernible trend emerged as electrical efficiency declined with increasing
for each fixed estimation of
. The systematic escalation in
for fixed
estimations led to a consistent decay in electrical efficiency. Notably, the electrical efficiency of the PV/T system exhibited a continuous decline, reaching an asymptotic state at approximately
= 3.
Transitioning to
Figure 18b, we introduced a suction effect of 0.4 to assess the numerical results. Here, it became apparent that, in the absence of slip, electrical efficiency could be optimized through the augmentation of
estimations. In
Figure 18c, a suction effect with
= 1 was incorporated to explore the electrical efficiency pattern against
while varying the
estimations. Intriguingly, a distinct pattern emerged in this figure, showcasing an improvement in electrical efficiency with increasing
estimations. The decline in electrical efficiency became asymptotic around
= 2.5. This observation underscores that, in the presence of suction, electrical efficiency experiences enhancement with an escalation in the slip flow parameter. Conversely, in the absence of suction, there was a decline in electrical efficiency with an escalation in the slip flow parameter. These findings underscore the nuanced influence of slip flow parameters on electrical efficiency under different conditions, contributing valuable insights to the scientific understanding of photovoltaic-thermal systems.
In our investigation of the influence of a copper volume fraction on the performance of photovoltaic thermal (PV/T) systems, we conducted a thorough analysis of the numerical results for electrical efficiency concerning the non-dimensional parameter
. These findings were specific to slipping effects and shrinking sheets, with
= 1 and
= −2. Within each figure, we maintained a fixed estimation for
while systematically varying the parameter
along each curve. In
Figure 19a, where
= 0, representing no suction effect, the electrical efficiency of the PV/T system exhibited stability for lower volume fractions of copper nanomaterials up to
= 2. Subsequently, a rapid decline ensued, becoming asymptotic at
= 4.5. Notably, the decline was more pronounced for
= 0.05. Importantly, the figure highlights that the maximum electrical efficiency was achieved at
= 0 for all volume fraction estimations of copper.
Moving to
Figure 19b, with a slightly escalated suction effect (
= 0.4) at the lower wall, electrical efficiency experienced a decline with increasing
. However, for each fixed estimation of
, there was an improvement in electrical efficiency with an escalation in the volume fraction of copper in the base fluids. The rate of decrement became asymptotic at
= 2.7 for all
estimations. A noteworthy observation is that the introduction of suction at the lower wall accelerated the rate of decrement in electrical efficiency against
compared to the scenario in
Figure 19a. In
Figure 19c, with a suction effect estimation of 0.7, a similar trend persisted. The rise in electrical efficiency remained unaffected by an escalation in the copper volume fraction. For all
estimations, the electrical efficiency continuously declined, reaching an asymptote at
= 2.
Upon comprehensive observation of
Figure 19a–c, a consistent pattern emerged, indicating that the maximum electrical efficiency occurred at
= 0, gradually declining and reaching an asymptotic state at
= 8.7. This entailed a notable 16.8% decrement when the electrical efficiency stabilized or reached a plateau. These findings contribute valuable insights into the intricate interplay between copper volume fractions, suction effects, and non-dimensional parameters in the context of PV/T systems.
Following an in-depth exploration of the electrical efficiency dynamics in photovoltaic/thermal (PV/T) systems, particularly in response to variations in non-dimensional symmetry parameters, we present
Table 5 to highlight combinations of non-dimensional parameters that yielded the highest average electrical efficiency within the studied framework.
Table 1 revealed that opting for
= 0.4,
= 2,
= 0.7, and
= 0.01 resulted in the maximum average electrical efficiency for the PV/T system, reaching an impressive 10%. These discerned non-dimensional parametric combinations served as pivotal insights into optimizing system performance. Moreover, these numerical outcomes opened avenues for comparisons with established references, fostering a comprehensive understanding of the photovoltaic thermal system’s electrical efficiency. These findings also suggest practical recommendations, emphasizing the potential benefits of employing stagnant point flow and HNFs within the system for enhanced efficiency.
Validation of the Work
Although working with stretching sheets for fluid flow and heat transfer through the cooling application of hybrid nanofluids is a new concept, the work to optimize the thermal and electrical performance of PV/T systems is not new. Several studies can be referenced in this regard [
45,
50,
51,
52]. To validate the above results, the thermal efficiency of the PV/T system was also computed and compared with [
52]. The formula for computation is given below and is referenced in [
52].
In Equation (22), all the parameters were defined in the nomenclature compared to the simple cases of and (simplest case).
Assuming the volume fraction of copper nanofluids is equal to zero (i.e.,
= 0) and all other parameters remain zero, conventional water is supposed to flow through a fluid channel. The numerical simulation of the present study was then compared with the available literature [
53]. In [
53], an experimental study was conducted using water as the fluid. In contrast, the present work aimed to enhance the performance of a PV/T system using hybrid nanofluids. To compare both approaches, the thermal efficiency was calculated using the formula mentioned above. In
Figure 20, it can clearly be seen that the thermal efficiency of the PV/T system, as determined numerically using COMSOL Multiphysics, correlated somewhat with the experimental work. The trends in thermal efficiency across these two results indicated the validation of the present simulation through COMSOL Multiphysics. Therefore, this work can be further developed for hybrid nanofluids.