Network Simulation Model for Free Convective Flow from a Vertical Cone

An axisymmetric, mathematical models is obtainable for the laminar natural convective laminar flow with effects of heat as well as mass transmission on an incompressible viscid liquid past a cone by uniform wall concentration, variable surface temperature including with effect of chemical reaction. The solutions of non-dimensional leading boundary layer equation of flow which are coupled, unsteady and nonlinear PDE’s were obtained using NSM, a robust statistical method which demonstrate elevated effectiveness and accurateness and the computer code Pspice. The different profiles of temperature, velocity and concentration were analyzed for different parameter, namely λ (chemical reaction), Sc (Schmidt number), Pr, N (buoyancy ratio constraint) and power law exponential of wall temperature n are discussed graphically. The current experiment result is compare with the existing outcome in the open literature and is found to be in outstanding conformity.


Introduction
The problem of natural convective boundary layer flow and mass transfer over a cone get a great deal of attention in various branches of science and engineering. The cooling of nuclear reactors is often used to study the structure of stars and planets through free convection studies. Intensive research effort, both theoretical and experimental, has been devoted to problems of free convection heat transfer in view not only of their own interest but also of the application to astro/geophysics and engineering during the last decades. Due to the influence of temperature variation and concentration variation or combination of both induces natural convection. Similarity or non-similarity solutions for 2D axi-symmetrical problems for natural laminar convective flow over vertical cone in steady state have developed by many authors [1][2][3][4] since 1953. Natural convective flow past over a vertical cone surface immersed in an infinite, compressible and viscous fluid combined with the effects of mass transfer is analyzed by Kafoussias [5]. Many authors [6][7][8][9][10][11][12][13][14][15][16] have analyzed the natural convective flow from a cone /rotating cone/ truncated cone with permeable/saturated permeable /non-Darcian porous regime for various boundary condition with Magnetohydrodynamic field, chemical reaction, radiation and absorption/generation etc., Network Simulation Solutions of present problem is well-tested, extremely compatible quantitative computational technique was applied. This computational procedure was invented by Nagel [17] in the application of semiconductors (1975) at University of California, Berkeley. Consequently it was

Mathematical Formulation
After the following assumptions, the transient heat flux of a viscous incompressible liquid with an improved layered natural size through a vertical cone is obtained, which has uneven wall temperature and uniform surface concentration, and generates. The effects of viscous dissipation and pressure gradient along the boundary layer are neglected and there exists only chemical reaction of first order in combination of species concentration and liquid. Further chemical species concentration of the diffusing substance is treated to be very small which is far away from the wall of the cone. Hence the Dufour and Soret effects are ignored. Also cone surface and surrounding fluid is in equal temperature with concentration. Then at time , the cone surface temperature is raised to where and n is the power law exponents variable in wall temperature . Concentration around surface the cone is raised to and both are retained in the consistent level. The physical model system is considered such that represents the length of the interval along cone surface from the apex ( = 0) and y represents the length of the interval along exterior perpendicular. The properties of the fluid are considered as stable except the density difference which induces body force in boundary layer equation of momentum. And the governing boundary layer equation with respect to Boussinesq approximation is given by: where u, v are velocities towards x and y direction, r is the local radius of the cone, x and y are the spatial coordinates, g is the acceleration due to gravity, t ' is the time, β is the volumetric coefficient of thermal expansion, βc is the volumetric coefficient of concentration expansion, T ' is the temperature, C ' is the concentration, µ is the dynamic viscosity, υ is the kinematic viscosity, α is the thermal diffusivity, ρ is density, Q0 is the dimensional heat generation/absorption coefficient, cp is the specific heat in constant pressure, D is the mass diffusivity, k1 is the dimensional parameter due to chemical reaction.
Shear stress  , , The following quantities are used dimensionlessly: , , Equations (1), (2), (3), (4) and (5) are transformed into the following dimensionless form: where U and V are dimensionless velocity in X and Y direction, R is the non-dimensional local radius, N is the non-dimensional buoyancy ratio, L is the reference length, Pr is the Prandtl number, Sc is the Schmidt number, t is the dimensionless time, ∆ is the non-dimensional heat generation and absorption parameter, λ is the dimensionless chemical reaction parameter.
Local Shear stress X  , heat transfer rate X Nu and local Sherwood number X Sh in dimensionless quantities are respectively Average Shear stress  , average Sherwood number Sh and average heat transfer rate Nu in nondimensional quantities are respectively Use the new method to solve nonlinear partially coupled differential equations (8) to (11) with initial and boundary conditions (12), Network Simulation Method. The estimation of the boundary layer equation is based on the finite difference discretization formula and the estimation of spatial coordinates An electrical network circuit design is formulated for each boundary layer equations. Electric analogy is applied in which the variable voltage (V) is similar to velocities (U, V), temperature (T) and concentration (C); the variable electric current (J) is equivalent to the velocity fluxes For each dimensionless boundary layer equation, three circles are developed. The entire network is converted into a suitable program, which is solved by the Pspice circuit simulator [17]. The time interval required for convergence is not necessary, because the Pspice code is used with the calculation method of a highly advanced mathematical algorithm, which is common to most numerical methods currently used.

Design of Network Model
Set the inclination height of the semi-infinite vertical cone to L = 1, which can be considered as a rectangular field, where X ranges from 0 to 1, and Y ranges from 0 to Ymax = 20, where X = L corresponds to the inclination of the vertical cone Height, and is considered Ymax as ∞, where Ymax is located in the external thermal category, momentum and species boundary. The integration area is regarded as a rectangle with grid size ∆X = 0.25 and Y = 0.25.
The network model is designed as follows. The finite difference differential equations formulated from non-dimensional continuity, momentum balance, energy balance and mass balance Equations (8) -(11) by implementing electrical equivalence together with Kirchhoff's law is  (22)-(24) is considered as an electrical current, and is written as the combinations of Resistors, Capacitors and Generators as derived in Zueco [20]. More rigorous The numerical analysis carried out by excellent procedure described by Gonzalez-Fernandez and Alhama [19], Zueco [20].  The finite difference scheme corresponding to equation (10) is , , , In order to achieve temperature, speed and concentration, the terminal conditions (when X = 0 and Y→) are used for the grounded unit. The constant voltage and constant current at Y = 0 are used to reproduce uniform wall temperature and uneven wall concentration, respectively.

Result and Discussion
In order to prove the validity of the current results, the obtained experimental values are compared with the current results of Chamkha [8], and the mutual compatibility is very good. In particular, the numerical solution of local skin friction X  with different Nusselt values

X Nu and Prandtl
Pr values in Table 1 was compared with the results of Chamkha [8].     Figure 2a that when heat is generated, the buoyancy increases, which leads to an increase in the flow velocity to the velocity distribution, and its volume increases, but the momentum boundary layer becomes thinner due to the rise and Pr.. It can be seen from Figure 2b that the boundary layer thickness and velocity are reduced to a higher Sc and . Figure 2c depicts that velocity raises steadily with time and attain a temporal maximum and consequently it attains the steady state. However, time taken to attain the steady state based on buoyancy ratio parameter N. An raise in N leads to an raise in the velocity near the cone surface. It can be seen from Figure 2d that the velocity of the entire boundary layer has increased to a lower value of n.    Its noted from Figure 3a that the value of the temperature increase is greater or less relative to the higher value of Pr and Δ., the thermal boundary layer becomes thinner. Figure 3b reveals that temperature raises for higher values of Sc and lower values of  and thermal boundary layer thickness reduced for lower values of Sc and .From Figure 3c noticed that as we move far away from the cone surface, the temperature reduces for all the values of N, thus for larger value of N the fluid cools rapidly. From Figure 3d one can observe that the temperature is maximised for lower values of n From Figure 4a it is seen that the concentration reduces and raise in Δ and Pr increased the time required to attain the steady state. Further concentration boundary layer thickness reduced for larger values of Pr and ∆. Figure 4b depicts that concentration reduces for small values of λ and larger values Schmidt number Sc. This reduces the effect of focal buoyancy. Figure 4c one can observe that concentration field reduces with higher values of Figure 4d shows that concentration field of the species raises for lower values of n.  Figure 5b shows the local skin friction behaviour for different values λ of and Sc . Figure 5c shows that an increase in N is accompanied by an increase in auxiliary buoyancy. It greatly accelerates the flow and enhances the shear stress. As N increases, the time required to reach steady state decreases Figure 6a shows that when the value of Pr is lower and higher, the local Nselt number increases; that is, the magnitude of the local Nusselt number increases with the heating/absorption coefficient Δ. Evident from from Fig. 6b that as Sc increases, the local Nusselt number keeps decreasing Inspection of Figure 6c shows that an increase in N strongly boosts Nusselt number X Nu , that is, it enhances the heat transfer gradient at the cone surface. Figure 6d illustrate that when the value of n is low, the local Nusselt number It can be seen from Figure 7a above that the local Sherwood number is increased to a value higher Δ and a value lower than Pr.. Figure 7b indicates that as  and Sc increase, the gradient of surface species (ie, the mass transfer rate on the surface of the cone) is very high. Figure 7c shows that the increase of N greatly enhances the Sherwood number. In other words, it enhances the gradient of the quality of the cone surface. Moreover, the number of Sherwood increases as N increases. The physical reason is that for liquids with a lower Prandtl number (Pr = 0.71), the positive force produces obvious overflow near the surface in the boundary layer, but for liquids with a higher Prandtl number (Pr = 6.7), the increase Speed is not obvious.. At the same time, the time required to reach the steady state decreases as the value of N increases. Figure 7d shows that for higher values of n, the local Sherwood number increases. For brevity, the graphical results of average skin contact, Neslett number and Sherwood number are not included.

Conclusion
From the uneven wall temperature, the vertical cone with uniform wall concentration, combined with the chemical reaction coefficient and the comprehensive effect of heat generation/absorption, a mathematical model of natural convection is established. Parametric studies were carried out to clarify the influence of thermo physical factors on temperature, velocity and concentration. Have been observed 1.
The time require to attain steady state raises with higher values of Δ, Pr, , Sc , N and n.