Introduction

Nanoliquids, a novel thermo-fluids class engineered through stable adjournment of nanosized metallic/nonmetallic objects (droplets, tubes, fibers, particles) in base materials with improved conductivity, exhibit the effective thermal management advantages with compactness. Nonetheless the intensification of thermal conductivity is not the main component accountable for nanofluids developed thermal efficiency, other aspects like sedimentation, gravity, inter-phase resistive force, dispersion, gravity, airborne phonon advection, invariable shear rate, migration of nanoparticles persuaded via layering and viscosity gradient at solid–liquid boundary also have a noteworthy contribution. The hydrothermal features of nanoliquids are revealed through the net impact of relative alterations in nanofluids thermophysical characteristics which are sympathetic regarding multiple factors comprising particle morphology (shape and size), pH value and base liquid characteristics, material temperature and additives. No doubt the amalgamation of disseminated solid elements in base liquid demonstrates extraordinary rise in thermophysical characteristics and maintains vast utilizations for illustration functioning of heat exchanger, thermal defiance, nuclear reactors cooling, etc. Choi [1] elaborates that nanoparticles size ought to be under \(50\,{\text{nm}}\) to convalesce heat transportation of ordinary materials. Later, with an innovation of technology in contemporary industry the nanoliquids with size under \(100\,{\text{nm}}\) have been established for distinct engineering functionalities. Various nanoliquid models (single-phase, two-phase, CNTs, Tiwari and Das) are reported to investigate nanotechnology [2,3,4,5,6,7]. Some recent works in this direction can be seen in Refs. [8,9,10,11,12,13,14,15,16,17,18,19].

Investigation regarding entropy generation and utilization of energy introduces a significant methodology in the thermal design and turns out to be the prime concern in numerous engineering demands for illustration (microelectronics, nuclear reactors chilling, heat exchangers, boiler and solar collectors). Precision estimation regarding entropy generation elucidates energy dissipation of a system effectively and consequently is a trustworthy device to acquire an optimum design in several industries. No doubt heat transportation intimate links with the thermodynamics first relation which is the only thermodynamic principle utilized in traditional heat transportation analysis. Thermodynamics second relation delivers an amount regarding rate of entropy generation or irreversibility in a system or procedure and accordingly influences the effectiveness of heat transportation process. Recently, increasing interest about irreversibility influence on energy interactions is noted. Initial analysis about entropy generation is elaborated by Bejan [20,21,22]. He revealed that entropy in convective liquid flow is owing to shear stresses and heat transportation. Numerous investigators employed entropy concept to minimize irreversibility of heat transport problems like boiling phenomenon and steam turbines [23]. Further, recent investigations covering entropy generation concept can be consulted through the Refs. [24,25,26,27,28,29,30,31,32].

The researchers at present have considerable concern about improvement for energy storage and heating/cooling rate in various modern and advanced technologies. In this direction, numerous flow problems have been formulated considering different kinds of distributions (wall to ambient temperature) utilizing various fluid models. Nonetheless, major problem for researchers is to introduce extraordinary storing of energy through minimal price. Some advanced technologies like waste heat retrieval, solar energy and simultaneous heat/power plants necessitate economically more apposite and effectual approach to store energy. In general, three approaches are utilized regarding energy storing, that is sensible heat, latent heat and chemical/thermal energy. Among these approaches, the latent heat is regarded more apposite and well organized (by altering the materials phase). Consequently, thermal energy in a material is stowed via latent heat through melting procedure. Melting process is extremely vital in processes comprising soil melting, the soil freezing around heat exchanger loops of a field-based pump, permafrost melting, magma solidification, frozen grounds thawing, freeze cure of sewage, semiconductor material preparation, welding and molding of an industrial process [33]. Various researches have elaborated melting aspect under different flow configurations (see Refs. [34,35,36,37,38,39]).

In this paper, we have studied entropy generation in dissipative flow of carbon nanotubes. Melting heat is accounted. Analysis is carried out between two infinite plates. Systems of differential equations are obtained via implementing suitable variables. Shooting method (bvp4c) is used for solutions development. Physical quantities are discussed.

Formulation

Unsteady squeezed flow of carbon nanomaterial is assumed between two parallel plates. Both plates are distant \(h(t)\) apart. Lower plate (at \(y = 0\)) is stretched with velocity \(U_{{\text{w}}} = \tfrac{ax}{{1 - {ct}}}\), while the plate at \(y = h(t)\) is set in motion toward the plate at \(y = 0\) with a squeezing velocity \(v_{{{\text{h}}}} = \tfrac{{\text{d}h}}{{{\text{d}}t}} = \sqrt {\tfrac{{\upsilon_{{{\text{f}}}} (1 - {{ct}})}}{a}}\). Melting heat is considered for heat transfer. Entropy generation effect is considered. Flow is parallel to \(x\) axis, while \(y\) axis is normal to it. Expressions under interest are [11]:

$$\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0,$$
(1)
$$\rho_{{{\text{nf}}}} \left( {\frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y}} \right){ = } - \frac{{\partial p_{1} }}{\partial x} + \mu_{{{\text{nf}}}} \, \left( {\frac{{\partial^{2} u}}{{\partial x^{2} }} + \frac{{\partial^{2} u}}{{\partial y^{2} }}} \right),$$
(2)
$$\rho_{{{\text{nf}}}} \left( {\frac{\partial v}{\partial t} + u\frac{\partial v}{\partial x} + v\frac{\partial v}{\partial y}} \right){ = } - \frac{{\partial p_{1} }}{\partial y} + \mu_{{{\text{nf}}}} \, \left( {\frac{{\partial^{2} v}}{{\partial x^{2} }} + \frac{{\partial^{2} v}}{{\partial y^{2} }}} \right),$$
(3)
$$\frac{\partial T}{\partial t} + u\frac{\partial T}{\partial x} + v\frac{\partial T}{\partial y} = \alpha_{{{\text{nf}}}} \left( {\frac{{\partial^{2} T}}{{\partial x^{2} }} + \frac{{\partial^{2} T}}{{\partial y^{2} }}} \right) + \frac{{\mu_{{{\text{nf}}}} }}{{(\rho c_{{{\text{p}}}} )_{{{\text{nf}}}} }}\left[ {{{4}}\left( {\frac{\partial u}{\partial x}} \right)^{2} + \left( {\frac{\partial v}{\partial x} + \frac{\partial u}{\partial y}} \right)^{2} } \right],$$
(4)

with boundary conditions

$$u = U_{{{\text{w}}}} \left( x \right) = \frac{{ax}}{{{{1}} - {{ct}}}},\quad T = T_{{{\text{m}}}}\,\, {{ at }}\,y = {{0}},\quad u = {{0}},\quad v = v_{{{\text{h}}}} = \frac{{{{d}}h}}{{{{d}}t}},\quad T = T_{{{\text{h}}}}\, {{ at }}\,\,y = h(t).$$
(5)

Melting condition is [5]:

$$k_{{{\text{nf}}}} \left( {\frac{\partial T}{\partial y}} \right) = \rho_{{{\text{nf}}}} (\lambda_{{{1}}} + c_{{{\text{s}}}} (T_{{{\text{m}}}} - T_{{{0}}} ))v\,\,{{ at }}\,\,y = {{0}}.$$
(6)

The expressions used in theoretical model proposed by Xue [40] are

$$ \begin{aligned} \mu_{{{\text{nf}}}} = & \frac{{\mu_{{{\text{f}}}} }}{{\left( {1 - \phi } \right)^{2.5} }},\quad \nu_{{{\text{nf}}}} = \frac{{\mu_{{{\text{nf}}}} }}{{\rho_{{{\text{nf}}}} }},\quad \rho_{{{\text{nf}}}} = \left( {1 - \phi } \right)\,\rho_{{{\text{f}}}} + \phi \uprho_{{{\text{CNT}}}} ,\, \\ \alpha_{{{\text{nf}}}} = & \frac{{k_{{{\text{nf}}}} }}{{\rho_{{{\text{nf}}}} \left( {c_{{{\text{p}}}} } \right)_{{{\text{nf}}}} }},\quad \frac{{k_{{{\text{nf}}}} }}{{k_{{{\text{f}}}} }} = \frac{{\left( {1 - \phi } \right) + 2\phi \tfrac{{k_{{{\text{CNT}}}} }}{{k_{{{\text{CNT}}}} - k_{{{\text{f}}}} }}\ln \tfrac{{k_{{{\text{CNT}}}} + k_{{{\text{f}}}} }}{{2k_{{{\text{f}}}} }}}}{{\left( {1 - \phi } \right) + 2\phi \tfrac{{kf}}{{k_{{{\text{CNT}}}} - k_{{{\text{f}}}} }}\ln \tfrac{{k_{{{\text{CNT}}}} + k_{{{\text{f}}}} }}{{2k_{{{\text{f}}}} }}}}, \\ \end{aligned} $$
(7)

We use the transformations [38]:

$$\eta = \frac{y}{h(t)},\quad u = \left( {\frac{{ax}}{{({{1}} - {{ct}})}}} \right)f^{{\prime }} \left( \eta \right),\quad v = - \sqrt {\frac{{a\upsilon_{{{\text{f}}}} }}{{{{1}} - {{ct}}}}} f\left( \eta \right),\quad \theta \left( \eta \right) = \frac{{T - T_{{{\text{m}}}} }}{{T_{{{\text{f}}}} - T_{{{\text{m}}}} }},\,$$
(8)

where

$$h(t) = \sqrt {\frac{{\upsilon_{{{\text{f}}}} (1 - {{ct}})}}{a}.}$$
(9)

Equation (1) is verified, while the other equations reduced to

$$\left( {\frac{1}{{^{{\left( {1 - \phi } \right)^{2.5} (1 - \phi + \phi \tfrac{{\rho_{{\text{CNT}}} }}{{\rho_{{\text{f}}} }})}} }}} \right)f^{{\left( {iv} \right)}} - f^{{\prime }} f^{{\prime \prime }} + ff^{{{\prime \prime \prime }}} - \frac{Sq}{2}(3f^{{\prime \prime }} + \eta f^{{{\prime \prime \prime }}} ) = 0,$$
(10)
$$\left( {\frac{{k_{{\text{nf}}} /k_{{\text{f}}} }}{{(1 - \phi + \phi \tfrac{{\left( {\rho c_{{\text{p}}} } \right)_{{\text{CNT}}} }}{{\left( {\rho c_{{\text{p}}} } \right)_{{\text{f}}} }})}}} \right)\theta^{{\prime \prime }} + \mathop {Pr}\limits (f\theta^{{\prime }} - \frac{Sq}{2}\eta \theta^{{\prime }} ) + \frac{Pr}{{\left( {1 - \phi } \right)^{2.5} }{{(1 - \phi + \phi \tfrac{{\left( {\rho c_{{\text{p}}} } \right)_{{\text{CNT}}} }}{{\left( {\rho c_{{\text{p}}} } \right)_{{\text{f}}} }})}}}(4Ecf^{\prime 2} + Ec_{1} f^{\prime \prime 2} ) = 0.$$
(11)
$$f^{\prime}(0) = 1,\quad \theta (0) = 0,\quad \left( {\frac{{k_{{\text{nf}}} }}{{k_{{\text{f}}} }}} \right)M\theta^{{\prime }} \left( 0 \right) + \mathop {Pr}\limits (1 - \phi + \phi \frac{{\rho_{{\text{CNT}}} }}{{\rho_{{\text{f}}} }})f(0) = 0,\quad f^{{\prime }} (1) = 0,\quad \, f\left( 1 \right) = 1,\quad \theta (1) = 1.$$
(12)

The involved dimensionless parameters are

$${Sq} = \frac{c}{a},\quad Pr = \frac{\upsilon }{\alpha },\quad Ec = \frac{{\nu_{{\text{f}}}^{2} }}{{h^{2} (c_{{\text{p}}} )_{{\text{f}}} (T_{{\text{m}}} - T_{{\text{h}}} )}},\quad Ec_{1} = \frac{{U_{{\text{w}}}^{2} }}{{(c_{{\text{p}}} )_{{\text{f}}} (T_{{\text{m}}} - T_{{\text{h}}} )}},$$
$$M = \frac{{c_{{\text{pf}}} \left( {T_{\infty } - T_{{\text{m}}} } \right)}}{{\lambda + c_{{\text{s}}} \left( {T_{{\text{m}}} - T_{0} } \right)}},\quad \eta = \frac{y}{h(t)}.$$
(13)

Expressions of skin friction coefficient (\(C_{{\text{f}}} \sqrt {Re_{{\text{x}}} }\)) and Nusselt number \(( {\tfrac{{Nu_{{\text{x}}} }}{{\sqrt {Re_{{\text{x}}} } }}} )\)

Expressions for \(C_{{\text{f}}} \sqrt {Re_{{\text{x}}} }\) and \(( {\tfrac{{Nu_{{\text{x}}} }}{{\sqrt {Re_{{\text{x}}} } }}} )\) in dimensional and non-dimensional forms are [11]:

$$C_{{\text{f}}} = \frac{{\left( {\tau_{{\text{xy}}} } \right)_{{\text{y}} = 0} }}{{\rho_{{\text{f}}} U_{{\text{w}}}^{2} }},\quad Nu_{{\text{x}}} = \frac{{{x}}{\mathbf{q}_{{\text{w}}} }}{{k_{{\text{nf}}}(T_{{\text{f}}} - T_{{\text{h}}} )}},$$
(14)

where

$${\varvec{\uptau}}_{{\text{xy}}} = \mu_{{\text{nf}}} \left( {\frac{\partial u}{\partial y}} \right)_{{\text{y}} = 0} ,\quad {\mathbf{q}}_{{\text{w}}} = - k_{{\text{nf}}} \left( {\frac{\partial T}{\partial y}} \right)_{{{y} = 0}}$$
(15)
$${or}\,C_{{\text{f}}} \sqrt {Re_{{\text{x}}}} = \frac{1}{{\left( {1 - \phi_{1} } \right)^{2.5} \left( {1 - \phi_{2} } \right)^{2.5} }}f^{{\prime \prime }} (0),\quad \frac{{Nu_{{\text{x}}} }}{{\sqrt {Re_{{\text{x}}} } }} = - \frac{{k_{{\text{nf}}} }}{{k_{{\text{f}}} }}\theta^{{\prime }} (0),$$
(16)

where \(Re_{{\text{x}}} = \sqrt {\tfrac{{\left( {1 - {ct}} \right)\,\upsilon_{{\text{f}}} }}{a}}\) represents local Reynolds number.

Entropy analysis

Entropy generation rate is [32]:

$$S_{{\text{G}}} = S_{{\text{H}}} ({entropy\, via\, heat\, transfer}) + S_{{\text{F}}} ({entropy\, via\, fluid\, friction}).$$
(17)

Or

$$S_{{\text{G}}} = \frac{{k_{{\text{nf}}} }}{{T_{{\text{h}}}^{2} }}\left( {\left( {\frac{\partial T}{\partial x}} \right)^{2} + \left( {\frac{\partial T}{\partial y}} \right)^{2} } \right) + \frac{{\mu_{{\text{nf}}} }}{{T_{{\text{h}}} }}\left( {4\left( {\frac{\partial u}{\partial x}} \right)^{2} + \left( {\frac{\partial v}{\partial x} + \frac{\partial u}{\partial y}} \right)^{2} } \right).$$
(18)

Non-dimensional entropy generation is

$${N{\text{s}}} = N_{{{\text{H}}}} ({entropy\, via\, heat\, transfer}) + N_{{\text{F}}} ({entropy\, via\, fluid\, friction}),$$
(19)
$${N{\text{s}}} = \frac{{S_{{\text{G}}} }}{{S_{{{{\text{G}}}_{0} }} }} = \theta^{\prime 2} + \frac{Pr}{{\varOmega (k_{{\text{nf}}} /k_{{\text{f}}} )\left( {1 - \phi } \right)^{2.5} }}(4Ecf^{\prime 2} + Ec_{1} f^{\prime \prime 2} )$$
(20)
$${where}\, S_{{{{\text{G}}}_{0} }} = \frac{{k_{{\text{nf}}} (T_{{\text{m}}} - T_{{\text{h}}} )}}{{T_{{\text{h}}}^{2} h^{2} }}.$$
(21)

Bejan number is [32]:

$$Be = \frac{{N_{{\text{H}}} ({{entropy\, via\, heat\, transfer}})}}{{N_{{\text{G}}} ({Total\, entropy})}},$$
(22)
$$Be = \frac{{\theta^{\prime 2} }}{{\theta^{\prime 2} + \tfrac{Pr}{{\varOmega (k_{\text{nf}} /k_{\text{f}} )\left( {1 - \phi } \right)^{2.5} }}(4Ecf^{\prime 2} + Ec_{1} f^{\prime \prime 2} )}} .$$
(23)

Note that Bejan number (\(Be\)) lies between 0 and 1. \(N_{{\text{H}}}\) dominates over \(N_{{\text{F}}}\) when \(0.5 < Be \le 1.0\), and \(N_{{\text{F}}}\) overrides \(N_{{\text{H}}}\) for \(0 \le Be < 0.5\), while for \(Be = 1.0\) both \(N_{{\text{H}}}\) and \(N_{{\text{F}}}\) are equal.

Solution via shooting method

Governing equations for flow are solved via shooting method using the fifth-order Runge–Kutta algorithm (bvp4c). First, we have an interest to write the first-order initial value problem as follows [4, 38, 41, 42]:

$$f = s_{1} ,$$
(24)
$$f^{{\prime }} = s_{2} ,$$
(25)
$$f^{{\prime \prime }} = s_{3} ,$$
(26)
$$f^{{{\prime \prime \prime }}} = s_{4} ,$$
(27)
$$s_{4}^{\prime } = f^{(iv)} = \left( {1 - \phi } \right)^{2.5} \left( {1 - \phi + \phi \frac{{\rho_{CNT} }}{{\rho_{f} }}} \right)\left( {\frac{Sq}{2}(\eta s_{4} + 3s_{3} ) - s_{2} s_{3} + s_{{1{s}_{4} }} } \right),$$
(28)
$$\theta = s_{5} ,$$
(29)
$$\theta^{{\prime }} = s_{6} ,$$
(30)
$$s_{6}^{{\prime }} = \frac{{\left( {1 - \phi + \phi \tfrac{{\left( {\rho c_{{{\text{p}}}} } \right)_{{{\text{CNT}}}} }}{{\left( {\rho c_{{{\text{p}}}} } \right)_{{{\text{f}}}} }}} \right)}}{{((k_{{{\text{nf}}}} /k_{{{\text{f}}}} ))}}\left( { - \mathop {Pr}\limits (s_{1} s_{6} - \frac{{Sq}}{2}\eta s_{6} } \right) - \frac{Pr}{{\left( {1 - \phi } \right)^{2.5} }}(Ec_{1} s_{3}^{2} + 4Ecs_{2}^{2} ).$$
(31)

along with

$$s_{2} (0) = 1,\quad \left( {\frac{{k_{{{\text{nf}}}} }}{{k_{{{\text{f}}}} }}} \right)Ms_{6} (0) + \mathop {Pr}\limits \left( {1 - \phi + \phi \frac{{\rho_{{{\text{CNT}}}} }}{{\rho_{{{\text{f}}}} }}} \right)s_{1} (0) = 0,\quad s_{5} (0) = 0,$$
$$s_{2} (1) = 1,\, \, s_{1} (1) = 1,\, \, s_{5}^{\prime } (1) = 1.$$
(32)

Analysis

Analysis for flow and temperature

In this subsection, influences of involved parameters on flow and temperature are analyzed graphically. Figures 1–3 are plotted in order to study variations of flow under \({\text{Sq,}}\)\(\phi\) and \(M\). Intensification in flow is seen for larger \({\text{Sq}}\). Variation in flow via \({\text{Sq}}\) can be studied in two cases: (1) \({\text{Sq}} > 0\), (2) \({\text{Sq}} < 0\). Here, \({\text{Sq}} > 0\) is associated with motion of the squeezing plate (upper plate) toward stretchable plate (lower plate), while \({\text{Sq}} < 0\) corresponds to motion of the squeezing plate (upper plate) away from stretchable plate (lower plate). For higher \({\text{Sq}}\) (\({\text{Sq}} > 0\)), the velocity of fluid increases due to influence of a force (squeezing force) felt by fluid particles. Also the increments in \(\phi\) and \(M\) lead to an enhancement in flow. Physically, an increment in \(M\) is associated with more rapid flow of heated fluid toward melting surface which intensifies flow. Interestingly, single-walled CNTs show overriding trends when compared with multiple-walled CNTs. Variations in temperature via \({\text{Sq}}\), \(\phi\) and \(M\) are labeled in Figs. 4–6. Reduction in temperature is observed for an increment in \({\text{Sq}}\) (\({\text{Sq}} > 0\)). Higher \({\text{Sq}}\) leads to small collision among the fluid particles. Hence, temperature decays while the associated penetration depth rises. Decay in temperature is noted with the variations in \(\phi\) and \(M\), while opposite trend is seen for associated penetration depth. Physically, higher \(M\) leads to more flow from hot fluid toward the melting surface. Hence, fluid temperature decays. Furthermore, overriding impact is observed for single-walled CNTs.

Fig. 1
figure 1

\(f^{\prime}(\eta )\) via \({\text{Sq}}\)

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Fig. 2
figure 2

\(f^{\prime}(\eta )\) via \(\phi\)

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Fig. 3
figure 3

\(f^{\prime}(\eta )\) via \(M\)

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Fig. 4
figure 4

\(\theta (\eta )\) via \({\text{Sq}}\)

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Fig. 5
figure 5

\(\theta (\eta )\) via \(\phi\)

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Fig. 6
figure 6

\(\theta (\eta )\) via \(M\)

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Analysis for entropy generation and Bejan number

Variations in Ns and Be via \({\text{Sq}}\), \(\phi\) and \(M\) are depicted in this subsection. Figures 7, 9 and 11 illustrate variations in Ns via \({\text{Sq}}\), \(\phi\) and \(M\), respectively. It is noticed that Ns reduces with an increment in \({\text{Sq}}\). Furthermore, production of entropy is maximum at the both walls, while it is minimum at the center of channel. Higher \(\phi\) leads to smaller Ns, while Ns intensifies for larger \(M\). At both walls, production of entropy is maximum, while at center, the entropy production is minimum. Further, single-walled CNTs show overriding trend comparatively with multiple-walled CNTs. In order to analyze the dominance of entropy due to fluid friction over entropy due to heat transfer or vice versa, Be is labeled via \(\eta\) in Figs. 8, 10 and 12. Intensification in Be is noted with an increment in \({\text{Sq}}\). It is observed that at the lower wall the entropy due to fluid friction is prominent for higher \({\text{Sq}}\). Reduction in Be is noted for higher \(\phi\). It is also observed that entropy via fluid friction shows overriding behavior at both walls. Be decays with an increment in \(M\), and dominance in entropy via fluid friction is noticed at both walls over entropy via heat transfer. Furthermore, single-walled CNTs show prominent behavior (Figs. 9–12).

Fig. 7
figure 7

\(Be\) via \({\text{Sq}}\)

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Fig. 8
figure 8

\(Be\) via \({\text{Sq}}\)

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Fig. 9
figure 9

\({\text{Ns}}\) via \(\phi\)

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Fig. 10
figure 10

\(Be\) via \(\phi\)

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Fig. 11
figure 11

\({\text{Ns}}\) via \(M\)

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Fig. 12
figure 12

\(Be\) via \(M\)

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Table 1 presents constructed thermophysical features of nanoparticles and base liquid, while numerical values of \(C_{\text{f}}\) and \(Nu_{\text{x}}\) under \({\text{Sq}},\)\(\phi\) and \(M\) are presented in Tables 2 and 3.

Table 1 Thermophysical features of base liquid and CNTs [40]
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Table 2 Analysis of \(C_{\text{f}}\) via \({\text{Sq}},\)\(\phi\) and \(M\)
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Table 3 Analysis of \(Nu_{\text{x}}\) via \({\text{Sq}},\)\(\phi\) and \(M\) when \(Ec = Ec_{1} = 0.2.\)
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Final remarks

The key points of the presented analysis are

  • Intensification in flow is observed with the increment in \({\text{Sq}},\)\(\phi\) and \(M\).

  • Decay in temperature is noted against \({\text{Sq}},\)\(\phi\) and \(M\).

  • Single-walled CNTs show overriding behavior than multiple-walled CNTs in terms of both flow and temperature.

  • Intensification in heat transfer rate is analyzed for larger \({\text{Sq}},\)\(\phi\) and \(M\).

  • Entropy production rate decays with an increment in \({\text{Sq}}\) and \(\phi\).

  • Bejan number is an increasing function of \({\text{Sq}}\), while it reduces for \(\phi\) and \(M\).

  • Role of multiple-walled CNTs is prominent than single-walled CNTs for both entropy generation and Bejan number.