EGA for a Convective Regime Over a Vertical Cylinder Stretching Linearly

Theoretical/experimental studies on heat transfer in fluids due to stretching surfaces are available in the literature. These studies were prompted by a wide array of industrial applications such as fibre production, wire drawing, metallic plates cooling, extrusion processes, manufacture of plastic, glass fibre production, and so on. The thermo-fluidics involving a stretching surface has been much investigated for a variety of assumptions.

Flow in the boundary-layer on a continuous solid surface was first pioneered by Sakiadis [1] considering the two-dimensional, axisymmetric boundary-layer flow over a flat surface moving with a constant velocity. Due to the entrainment of ambient fluid, this phenomenon represented a boundary-layer problem differing from Blasius flow over a semi-infinite flat plate. Crane [2] considered flow over a moving strip with velocity proportional to the distance from the slit and obtained closed-form exponential solution. Afzal and Varshney [3] examined the cooling of a low heat resistance stretching sheet moving through a fluid. Chiam [4] examined micropolar fluid flow over a stretching sheet. Gupta and Gupta [5] examined heat and mass transfer over an isothermal stretching sheet with suction or blowing. Banks [6] devised a similarity solution of the boundary-layer equations for the stretching wall. Grubka and Bobba [7] extracted solutions in terms of Kummer's function for the phenomena constrained to prescribed wall temperature and heat flux. Chen and Chen [8] investigated the problem for viscoelastic fluid and reported solutions in terms of Kummer's function. Chauhan and Vyas [9] examined heat transfer in MHD viscous flow due to the stretching of boundary in the presence of a naturally porous bed. Chiam [10] studied hydromagnetic flow over a surface stretching with power-law velocity. Anderson and Valnes [11] investigated ferro-fluid flow over a stretching sheet in the presence of a magnetic dipole to explore the effects of the magneto-thermo-mechanical interaction on skin friction and heat transfer. Boutros et al. [12] considered a steady two-dimensional boundary-layer stagnation point flow towards a heated stretching sheet placed in a porous medium and exploited Lie group method for solving the problem. Mjankwi et al. [13] investigated variable properties effects on unsteady nanofluid flow over a stretching sheet. Vyas and Rai [14], Vyas and Ranjan [15], Vyas and Srivastava [16] and Vyas and Srivastava [17] examined various configurations involving the stretching surface. Grosan and Pop [18] reported an axisymmetric mixed convection boundary-layer flow of nanofluid past a vertical cylinder. Hayat et al. [19] investigated mixed convection flow of a Casson nanofluid over a stretching sheet with convective heated reaction and heat source/sink.

It is pertinent to record that these studies and references contained therein pivoted on the first law of thermodynamics only. That is, these systems got treated primarily for heat transfer aspects and the thermodynamic irreversibility involved was not looked into. However, from physical understanding we know that the quantification of entropy is instrumental in optimal design. The quantification of entropy is facilitated through a second law analysis that helps to understand the energy losses in the system.

Bejan [20,21], in his seminal works, created a breakthrough in entropy management in thermo-fluidics. This opened a new horizon in thermal systems where entropy generation analysis was well received, given its wide spectrum of utility. Butt and Ali [22] reported entropy generation due to a stretching cylinder. Das et al. [23] discussed entropy due to convective heating of the stretching cylinder. Vyas and Khan [24] reported entropy generation distribution for Casson fluid flow caused by a stretching cylinder. Vyas and Soni [25] analysed entropy generation for boundary-layer flow of fluid due to the melting stretching sheet. Vyas and Ranjan [26] examined the entropy analysis of radiative MHD forced convection flow in a porous medium channel with weakly temperature-dependent convection coefficient. Vyas and Srivastava [27] studied entropy analysis of generalized MHD Couette flow inside a composite duct with asymmetric convective cooling. Mukhopadhaya and Ishak [28] discussed mixed convection flow in a thermally stratified medium along with a stretching cylinder. Kumam et al. [29] presented entropy generation in the MHD flow of CNTs Casson nanofluid in rotating channels.

It is expected that the work presented here would be a basis for future explorations wherein the configuration may be a part of larger systems to be designed and simulated.

We consider an axisymmetric, steady 2-D heat-generating fluid flow over a stretching cylinder placed vertically in fluid-saturated porous medium. The fluid is uniformly heat generating. We assume that the surface of the cylinder remains at constant temperature T_W and the temperature far away from the cylinder's surface is T_∞ where T_W > T_∞. The cylinder stretches with a velocity in a fashion that its surface velocity U(x) varies with distance from the origin and the far away fluid is at rest. We consider a coordinate system such that (r, x) axes are in radial direction and along the cylinder axis respectively.

The governing equations under boundary-layer assumptions read: (1) ∂(ru)∂x+∂(rv)∂r=0 {{\partial (ru)} \over {\partial x}} + {{\partial (rv)} \over {\partial r}} = 0 (2) u∂u∂x+v∂u∂r=ϑr∂∂r(r∂u∂r)+gβ(T−T∞)−ϑuK0 u{{\partial u} \over {\partial x}} + v{{\partial u} \over {\partial r}} = {\vartheta \over r}{\partial \over {\partial r}}\left( {r{{\partial u} \over {\partial r}}} \right) + g\beta (T - {T_\infty }) - {{\vartheta u} \over {{K_0}}} (3) u∂T∂x+v∂T∂r=1ρcp1r∂∂r(r∂T∂r)+Q0ρcp(T−T∞) u{{\partial T} \over {\partial x}} + v{{\partial T} \over {\partial r}} = {1 \over {\rho {c_p}}}{1 \over r}{\partial \over {\partial r}}\left( {r{{\partial T} \over {\partial r}}} \right) + {{{Q_0}} \over {\rho {c_p}}}(T - {T_\infty }) The boundary conditions are: (4) r=R; u=U(x), v=0, T=Tw r = R;\;\;u = U(x),\;\;v = 0,\;\;T = {T_w} (5) r→∞: u→0, T→T∞ r \to \infty :\;\;u \to 0,\;\;T \to {T_\infty } where (u, v) constitute velocity in (x, r) direction respectively; ν is ‘kinematic viscosity’; ρ is ‘fluid density’; μ is ‘fluid viscosity coefficient’, k is ‘thermal conductivity ’, T is ‘fluid temperature ’, R is ‘radius of the cylinder’, U(x)=U0xL U(x) = {{{U_0}x} \over L} is the stretching velocity, U_o is the characteristic velocity, and L is the characteristic length.

We prescribe the similarity transformation (6) η=r2−R22R(Uϑx)12 ψ=(Uϑx)12Rf(η) θ(η)=T−T∞Tw−T∞ \eta = {{{r^2} - {R^2}} \over {2R}}{\left( {{U \over {\vartheta x}}} \right)^{{1 \over 2}}}\;\;\psi = (U\vartheta x{)^{{1 \over 2}}}Rf(\eta )\;\;\theta (\eta ) = {{T - {T_\infty }} \over {{T_w} - {T_\infty }}} where the stream function ψ is defined as (7) u=r−1∂ψ∂r, v=−r−1∂ψ∂x u = {r^{ - 1}}{{\partial \psi } \over {\partial r}},\;\;v = - {r^{ - 1}}{{\partial \psi } \over {\partial x}} We see that continuity Eq. (1) gets identically satisfied. In the light of Eqs (6) and (7), Eqs (1) through Eqs (4) and (5) take the following form: (8) (1+2Mη)f‴+2Mf″+ff″−f′2+Sθ−M2Kf′=0 (1 + 2M\eta )f''' + 2Mf'' + ff'' - {f'^2} + S\theta - {{{M^2}} \over K}f' = 0 (9) (1+2Mη)θ″+2Mθ′+Prfθ′+M2Qθ=0 (1 + 2M\eta )\theta '' + 2M\theta ' + Prf\theta ' + {M^2}Q\theta = 0 When (10) η=0: f=0,f′=1,θ=1 \eta = 0:\;\;\;f = 0,f' = 1,\theta = 1 (11) η→∞: f′→0,θ→0 \eta \to \infty :\;\;\;f' \to 0,\theta \to 0 Where prime denotes differentiation with respect to η, S=gβL2(Tw−T∞)U02x S = {{g\beta {L^2}({T_w} - {T_\infty })} \over {U_0^2x}} describes local mixed convection parameter, M=(ϑLU0R)1/2 M = {\left( {{{\vartheta L} \over {{U_0}R}}} \right)^{1/2}} stands for curvature parameter, K=K0R2 K = {{{K_0}} \over {{R^2}}} is the permeability parameter, Pr=μCpk Pr = {{\mu {C_p}} \over k} is the Prandtl number and Q=Q0R2k Q = {{{Q_0}{R^2}} \over k} is the heat absorption/generation parameter

The non-linear boundary value problem (BVP) described by Eqs (8)–(11) has been solved numerically using the fourth-order Runge-Kutta method along with the shooting method. To employ the shooting method, the BVP is reduced to following initial value problems: (12) dfdη=p {{df} \over {d\eta }} = p (13) dpdη=q {{dp} \over {d\eta }} = q (14) dqdη=1(1+2Mη)(p2−fq−2Mq−Sθ+M2Kp) {{dq} \over {d\eta }} = {1 \over {(1 + 2M\eta )}}({p^2} - fq - 2Mq - S\theta + {{{M^2}} \over K}p) (15) dθdη=Z {{d\theta } \over {d\eta }} = Z (16) dzdη=−1(1+2Mη)(2Mz+Prfz+M2Qθ) {{dz} \over {d\eta }} = {{ - 1} \over {(1 + 2M\eta )}}(2Mz + Prfz + {M^2}Q\theta ) Subject to initial conditions (17) f(0)=0, p(0)=1, θ(0)=1 f(0) = 0,\;\;p(0) = 1,\;\;\theta (0) = 1 (18) p→0, θ→0 as η→∞ p \to 0,\;\;\theta \to 0\;\;{\rm{as}}\;\;\eta \to \infty

The aforementioned system of IVP is solved with the proper guess values s₀ and a₀ for the unknown quantities f ^″(0) and θ ^′(0) respectively, such that the end conditions f ^′(∞) = 0 and θ (∞) = 0 are satisfied. If the end conditions are not satisfied with these guess values, then the system of IVPs is solved with another guess values s₁ and a₁ for f ^′(∞) and θ (∞) = 0, respectively. The convergence of the shooting method is ensured by suitable guess values of unknown quantities. Here, it is to be noted that these guess values are searched purely on hit and trial basis and are refined further as per prescribed error tolerance. In the present case prescribed, error tolerance was taken of 10⁻⁶ magnitude together with grid space Δη = 0.001. The refinement of these guess values in this problem has been taken care of by secant method. Furthermore, it is submitted that the present problem can also be handled by other methods available in the literature [30,31,32,33,34,35].

The local volumetric rate of entropy generation S_G for the configuration under consideration is given as: (19) SG=kT∞2(∂T∂r)2+μT∞(∂u∂r)2+μT∞u2K0 {S_G} = {k \over {T_\infty ^2}}{\left( {{{\partial T} \over {\partial r}}} \right)^2} + {\mu \over {{T_\infty }}}{\left( {{{\partial u} \over {\partial r}}} \right)^2} + {\mu \over {{T_\infty }}}{{{u^2}} \over {{K_0}}} We prescribe the following quantities (20) SG0=kT∞2(TW−T∞)2L2, ω=TW−T∞T∞, X=R2L2, Br=μU2k(Tw−T∞) {S_{{G_0}}} = {k \over {T_\infty ^2}}{{{{({T_W} - {T_\infty })}^2}} \over {{L^2}}},\;\;\omega = {{{T_W} - {T_\infty }} \over {{T_\infty }}},\;\;X = {{{R^2}} \over {{L^2}}},\;\;Br = {{\mu {U^2}} \over {k({T_w} - {T_\infty })}} which are characteristic entropy, characteristic temperature, characteristic length and Brinkman number respectively.

Consequently, entropy generation number N_S using Eqs (19) and (20) reads as (21) Ns=SGSG0=(1+2Mη)M2Xθ′2+BrωX(1Mf″2+1Kf′2) {N_s} = {{{S_G}} \over {{S_{{G_0}}}}} = {{(1 + 2M\eta )} \over {{M^2}X}}{\theta '^2} + {{{B_r}} \over {\omega X}}\left( {{1 \over M}{{f''}^2} + {1 \over K}{{f'}^2}} \right) (22) =N1+N2 = {N_1} + N2 where (23) N1=(1+2Mη)M2Xθ′2 and N2=BrωX(1Mf″2+1Kf′2) {N_1} = {{(1 + 2M\eta )} \over {{M^2}X}}{\theta '^2}\;\;{\rm{and}}\;\;{N_2} = {{{B_r}} \over {\omega X}}\left( {{1 \over M}{{f''}^2} + {1 \over K}{{f'}^2}} \right) stands for heat transfer irreversibility and dissipative irreversibility term, respectively.

The Bejan number Be is defined as (24) Be=N1N1+N2 Be = {{{N_1}} \over {{N_1} + {N_2}}}

As evident from the expression for entropy number N_S we need distributions for velocity and temperature and respective gradients to compute N_S. The BVP governing momentum and thermal regimes have been solved numerically by the fourth-order R-K integration scheme supplemented with the shooting method. In this strategy, the BVP is reoriented to a system of initial problems where unknown gradients are guessed in an iterative procedure until the convergence is achieved.

The numerically computed values were then utilized to compute the entropy generation number. The plots for the quantities of interest were portrayed graphically to have an insight of the phenomenon.

Figure 1 displays that increasing Brinkmann number Br (which is a measure of dissipative effect) leads to a rise in entropy generation number. Here, it should be noted from the expression of entropy generation number N_S that Br features in N₂ with curvature parameter M and permeability K. Thus a combination of Br, M and K may be arrived at for the desired situation while taking on entropy optimization. Figure 2 exhibits the effect of curvature parameter M on entropy generation number. We observe that a rise in curvature parameter M leads to a decay in entropy generation number. Figure 3 demonstrates that with the rise in Q, the entropy generation number also rises. Larger values of Q are indicative of more heat being poured into the system. This additional heat changes the thermal regime to the effect that heat transfer gradient and dissipation go side by side and which significantly modify entropy distribution. Figure 4 demonstrates the effect of mixed convection parameter S on entropy generation number. We find that how even smaller changes in S do have a qualitative and quantitative bearing on entropy distribution. A slight increase in the mixed convection parameter increases the entropy generation. Figure 5 exhibits that the entropy generation number increases with the increase in the characteristic temperature ratio ω. Thus, a characteristic temperature that is conveniently at our choice can play a significant role in entropy generation management without disturbing the setup! Figure 6 displays that an increase in permeability parameter K results in the lowering of entropy generation number.

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Before analysing Bejan number Be (0 ≤ Be ≤ 1), it is not out of place to recall that Be is the ratio of heat transfer irreversibility and irreversibility due to heat transfer and dissipation. When Bejan number vanishes, it means that there is no thermodynamic irreversibility due to heat transfer. Be being unity signifies the absence of irreversibility due to dissipation. However, Be ≥ 1/2 signifies the situation when heat transfer irreversibility contribution to total entropy is greater than or equal to that of fluid friction irreversibility. Figures 7–12 display that Bejan number has one thing common, that is, Bejan number Be attains maxima and or minima at various spatial distances depending upon the parameters values. These figures also reveal that Bejan number Be is larger at far away distances from the stretching surface compared to that of close to the region adjacent to the stretching surface. Furthermore, we also observe that the choice of selected parameters values Be ≤ 0.5.

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Figure 7 displays that increasing values of Brinkman number reduce Bejan number at any spatial distance. However, Be becomes zero at η = 7.7 for any choice of Br. Figure 8 gives the scenario between the curvature parameter M and Bejan number. The Bejan number variation versus η for M does not follow a unifying trend but rather it has fluctuation in the middle. The figure reveals that the Bejan number Be increases with increasing M until the spatial distance η = 4.6, and then the trend is reversed till the Bejan number Be attains vanishing value. The same trend is seen for Q in Figure 9. Figure 10 underscores the qualitative impact of S on Bejan number Be. Figure 11 demonstrates the quantitative and qualitative effects of characteristic temperature ratio ω on Bejan number Be. Figure 12 shows the qualitative effect of permeability values on Bejan number Be. Here, the chosen values of K are striking to note that how even rather small variation in K has a bearing on Bejan number Be.

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The entropy generation analysis for mixed convection in a vertical stretching cylinder embedded in a porous medium has been undertaken. Momentum and thermal regimes were obtained for velocity and temperature distributions and respective gradients. These quantities were utilised to compute the entropy generation number and Bejan number. The plots of quantities of interest show the qualitative and quantitative impact of parameters entering into the problem which has been discussed in the preceding section. The values of parameters chosen here are just representative and the analysis is very much an attempt to display that entropy management can be achieved by selecting parameters wisely. We expect that the work would be a formidable baby model to be extended for a much larger configuration wherein the findings could provide substantial insight.