Control Volume-based Finite Difference Method Research Articles

Modelling and simulation of heat conduction in 1-D polar spherical coordinates using control volume-based finite difference method

Purpose – The purpose of this paper is to obtain a finite difference method (FDM) solution using control volume for heat transport by conduction and the heat absorption by the enthalpy model in the sand mixture used in casting manufacturing processes. A mixture of sand and different chemicals (binders) is used as moulding materials in the casting processes. The presence of various compounds in the system improve the complexity of the heat transport due to the heat absorption as the binders are decomposing and transformed into gaseous products due to significant heat shock. Design/methodology/approach – The geometrical domain were defined in a 1D polar coordinate system and adapted for numerical simulation according to the control volume-based FDM. The simulation results were validated by comparison to the temperature measurements under laboratory conditions as the sand mould mixture was heated by interacting with a liquid alloy. Findings – Results of validation and simulation methods were about high correspondence, the numerical method presented in this paper is accurate and has significant potential in the simulation of casting processes. Originality/value – Both numerical solution (definition of geometrical domain in 1D polar coordinate system) and verification method presented in this paper are state-of-the-art in their kinds and present high scientific value especially regarding to the topic of numerical modelling of heat flow and foundry technology.

Numerical Simulation of Macrosegregation Formation in Binary Alloy Solidification Processing Based on Modified Projection Method

A simplified mathematical model for describing the solidification processing of fluid flow, heat and solute transfer in binary alloys is developed. The conservation equations are numerically solved on a staggered grid by using the control volume-based finite difference method. As the interdendritic flow in the mushy zone is governed by Darcy’s law, the Modified Projection Method is presented for solving the momentum and continuity equations. The solidification processing of Fe-C alloy in rectangular domains is simulated and discussed. The numerical result shows that, by using the Modified Projection Method technique, the presented model is able to predict the macrosegregation formation effectively.

Fluid Flow and Heat Transfer Modeling of AC Arc in Ferrosilicon Submerged Arc Furnace

A two-dimensional mathematical model was developed to describe the heat transfer and fluid flow in an AC arc zone of a ferrosilicon submerged arc furnace. In this model, the time-dependent conservation equations of mass, momentum, and energy in the specified domain of plasma zone were numerically solved by coupling with the Maxwell and Laplace equations for magnetic filed and electric potential, respectively. A control volume-based finite difference method was used to solve the governing equations in cylindrical coordinates. The reliability of the developed model was checked by experimental data from the previous available literature. The results of present model were in good agreement with the given data comparing with other models, because of solving the Maxwell and Laplace equations simultaneously in order to calculate current density. In addition, parametric studies were carried out to evaluate the effects of electrical current and arc length on flow field and temperature distribution within the arc. According to the computed results, a lower power input led to a higher arc efficiency.

Numerical study of thermal stress during different stages of silicon Czochralski crystal growth

In this paper, the influence of various crystal heights to the crystal/melt interface shape and thermal stresses distribution in the large diameter (300 mm) of the silicon single crystal growth in a Czochralski process was studied numerically. A tow dimensional fluid flow and heat transfer with solidification model was developed. The Navier-Stoks and energy equations in melt and the heat conduction equation in crystal are solved using the control volume-based finite difference method. The thermal elastic stress fields for different stages are calculated from the temperature field by adopting the plane strain model in an axi-symmetric geometry of a cylindrical crystal. It was found that the melt/crystal interface shape becomes more concave and the maximum value of thermal stress in the crystal reduces as the crystal grows. A good agreement between our numerical simulations and those found in the literature is obtained.

NUMERICAL STUDY FOR A SECONDARY CIRCULAR CLARIFIER WITH DENSITY EFFECT

A computer program is developed for the prediction of the flow pattern and the removal efficiency of suspended solid (SS) in a circular secondary clarifier. In this study the increased density effect by SS on hydrodynamics has been systematically investigated in terms of Froude Number (Fr), baffle existence, and a couple of important empirical models associated with the particle settling and Reynolds stresses. A control-volume based-finite difference method by Patankar is employed together with the SIMPLEC algorithm for the resolution of pressure-velocity coupling. The k-e turbulence and its modified version are incorporated for the evaluation of Reynolds stresses. The calculation results predicts well the overall flow pattern such as the waterfall phenomenon at the front end of the clarifier and the bottom density current with the formation of strong recirculation especially for the case of decrease of Fr. Even if there are some noticeable differences in the prediction of two turbulence models, the calculated results of the radial velocity profiles are generally in good agreement against experimental data appeared in open literature. Parametric investigation has been systematically made with the Fr and baffle condition with detailed analysis.

Computational analysis of species transport and electrochemical characteristics of a MOLB-type SOFC

A multi-physics model coupling electrochemical kinetics with fluid dynamics has been developed to simulate the transport phenomena in mono-block-layer built (MOLB) solid oxide fuel cells (SOFC). A typical MOLB module is composed of trapezoidal flow channels, corrugated positive electrode-electrolyte-negative electrode (PEN) plates, and planar inter-connecters. The control volume-based finite difference method is employed for calculation, which is based on the conservation of mass, momentum, energy, species, and electric charge. In the porous electrodes, the flow momentum is governed by a Darcy model with constant porosity and permeability. The diffusion of reactants follows the Bruggman model. The chemistry within the plates is described via surface reactions with a fixed surface-to-volume ratio, tortuosity and average pore size. Species transports as well as the local variations of electrochemical characteristics, such as overpotential and current density distributions in the electrodes of an MOLB SOFC, are discussed in detail.

Thermal Energy Storage Module Design Using Energy and Exergy Analysis

In this paper, irreversibility of a thermal energy storage system is numerically investigated. The system consists of two concentric cylinders. The outer cylinder is filled with phase change material (PCM), while working fluid flows inside the inner pipe. The system works periodically. The related governing equations are solved by a control volume-based finite difference method. The effects of different parameters such as PCM size and melting point temperature are examined on the irreversibility of the system. The results show that the irreversibility of thermal storage module is strongly affected by the size of PCM (diameter and length of the external cylinder) and melting temperature. Based on the obtained results, the irreversibility of the system can be reduced by proper selection of PCM size and melting temperature.

Calculation of turbulent flow and heat transfer in a porous-baffled channel

This study presents the numerical predictions on the turbulent fluid flow and heat transfer characteristics for rectangular channel with porous baffles which are arranged on the bottom and top channel walls in a periodically staggered way. The turbulent governing equations are solved by a control volume-based finite difference method with power-law scheme and the k– ε turbulence model associated with wall function to describe the turbulent structure. The velocity and pressure terms of momentum equations are solved by SIMPLE (semi-implicit method for pressure-linked equation) method. The parameters studied include the entrance Reynolds number Re (1×10 4–5×10 4), the baffle height ( h=10, 20 and 30 mm) and kind of baffles (solid and porous); whereas the baffle spacing S/ H are fixed at 1.0 and the working medium is air. The numerical calculations of the flow field indicate that the flow patterns around the porous- and solid-type baffles are entirely different due to different transport phenomena and it significantly influences the local heat transfer coefficient distributions. Relative to the solid-type baffle channel, the porous-type baffle channel has a lower friction factor due to less channel blockage. Concerning the heat transfer effect, both the solid-type and porous-type baffles walls enhanced the heat transfer relative to the smooth channel. It is further found that at the higher baffle height, the level of heat transfer augmentation is nearly the same for the porous-type baffle, the only difference being the Reynolds number dependence. As expected, the centerline-averaged Nusselt number ratio increases with increasing the baffle height because of the flow acceleration.

A Two-Dimensional Numerical Study of Heat Transfer in a Finned Tube Assembly During Axisymmetric Dehumidification

This paper presents the analysis of two-dimensional heat transfer in an annular finned tube assembly during the process of dehumidification. All possible fin surface conditions, namely, dry, fully wet, and partially wet, have been studied and fin efficiency under these conditions have been modeled. Computations have been carried out using a control volume-based finite-difference method and compared with past one-dimensional analytical studies and available experimental data. The parameters that influenced the heat transfer rate in the finned tube structure are ratio of fin and wall thermal conductivities, ratio of fin thickness to fin pitch, ratio of wall thickness to fin pitch, ratio of fin length to fin pitch, cold fiuid Biot number, ambient Blot number, the relative humidity and dry bulb temperature of the incoming air, and the cold fluid temperature inside the coil. It was found that the heat transfer increased with increment in both dry bulb temperature and the relative humidity of the air. The fin efficiency changed rapidly with relative humidity under partially wet condition. The results suggest that coil performance can be very significantly altered by the condensation phenomenon on the fin surface and designs with dry fin data may not be adequate. The present results are expected to be very useful for the design of dehumidifier (cooling) coils for air conditioning applications.

A control volume-based finite difference method for solving the equilibrium equations in terms of displacements

This paper presents a novel control volume-based finite difference method for solving the equilibrium equations in terms of displacements, i.e., the generalized Navier equations. The method is based on the widely used cv-FDM solution method of heat conduction and fluid flow problems. When solving fluid flow problems with cv-FDM, the staggered grid is often used since it gives more stable solutions and helps to avoid numerical difficulties regarding the first derivatives in pressure and velocity. Due to the close connection between the equilibrium equations and the momentum equations, the staggered grid is also introduced in the new method, and it turns out to give a rather elegant formulation when nonconstant material properties are involved. Once the mesh has been established, the basic fluxes and equations are discretized. This is done in detail in order to establish a concise numerical foundation. The formulation of prescribed stresses is explained in detail since it turns out to be rather complex. The resulting linear algebraic equations are solved iteratively by line Gauss-Seidel. The relative nature of the displacement fields, and specifically what that means with regard to boundary conditions, is also discussed. Finally, the numerical model is validated and discussed together with suggestions for future work.

Laminar flow and heat tranfer from multiple impinging slot jets with an inclined confinement surface

Results of numerical simulation of two-dimensional flow field and heat transfer due to laminar heated multiple slot jets discharging normally into a converging confined channel are presented. A control volume-based finite difference method was employed to solve the governing mass, momentum and energy equations. To handle the complexity of the geometry a program was developed using a non-orthogonal body-fitted coordinate system. A fully staggered grid system was generated using the solution to a set of elliptic grid generation equations. The parameters studied were: the jet Reynolds number (600 < Re < 1000), and the angle of inclination of the upper surface (0° < θ < 20°). Inclination of the confined surface so as to accelerate the exhaust flow was found to level the Nusselt number distribution on the impingement surface.

Transient conductive, radiative heat transfer coupled with moisture transport in attic insulations

A transient, one-dimensional thermal model that incorporates combined conduction, radiation heat transfer, and moisture transport for residential attic insulations has been developed. The governing equations are the energy equation, the radiative transport equation for volumetric radiation within the insulation batt, and the species equations for bound H2O and vapor H2O. A simultaneous solution procedure with a Eulerian control volume-based finite difference method was used to solve the energy equation and the species equations. The method of discrete ordinates was used in solving the radiative transport equation. For H2O transport, both diffusion of vapor H2O and bound H2O and moisture adsorption/desorption within the insulation binder are included in the model. The experimental data measured at an occupied North Mississippi residence for R19STD (standard R19 fiberglass insulation batt without a foil radiant barrier) were used to validate the model which predicted heat fluxes for summer, spring, winter, and fall seasonal conditions. These predictions were compared with the measured heat flux data and the predictions from the dry model (without the moisture transport). Various profiles such as temperature-time histories, relative humidity time histories, spatial H2O concentrations, spatial temperatures, and spatial heat fluxes are presented to explain the overall heat transfer behavior. 18 refs.

Fluid Flow and Heat Transfer in the Crescent-Shaped Lumen Catheter

This paper presents a numerical investigation of fluid flow, frequency response in the fully developed region, and convective heat transfer in the entrance region of the crescent-shaped lumen catheter. The catheter is commonly used in the biomedical field to clinically diagnose heart disease and also to treat vessel blockage in surgery. The catheter is subjected to a constant wall temperature. The solution to discretization of the momentum and energy equations is obtained by using the numerically generated boundary fitted coordinate system. According to this method, the complex domain in the physical plane is transformed into a regular square domain in the computational plane. The control volume-based finite difference method is then used to discretize the transformed governing equations. Results for the thermal entry region flow, frequency response, and heat transfer are presented in graphical form. The representative curves illustrating variations of the flow rate, frequency response, damping coefficient, bulk temperature, and the Nusselt numbers with pertinent parameters in the entire thermal entry region are plotted. The optimized catheter design for diagnostic use in the medical industry is also presented graphically.

Effects of Buoyancy on Laminar Flow in Curved Elliptic Ducts

This paper numerically investigates the effects of buoyancy on fully developed laminar flow in a curved duct with an elliptic cross section. The flow of Newtonian fluids is assumed steady in terms of Boussinesq approximation. The curved elliptic duct is subjected to thermal boundary conditions of axially uniform heat flux and peripherally uniform wall temperature. The numerically generated boundary-fitted coordinate system is applied to discretize the solution domain of the elliptic duct, and the Navier-Stokes equations and the energy equation, including the curvature ratio, are solved by use of the control volume-based finite difference method. The solution covers a wide range of curvature ratios, and Dean and Grashof numbers. The results presented are displayed graphically and in tabular form to illustrate the buoyancy effect. It is further shown that buoyancy acts to increase both the Nusselt number and the friction factor and changes the distribution of the velocity and the temperature. The results for the curved circular duct with and without buoyancy are compared with the data available in the open literature for all cases. Also compared with the published data are the results of laminar flow in a curved elliptic duct, and very good agreement is obtained.

Numerical Analysis of Laminar Flow in Curved Elliptic Ducts

The complete form of the Navier-Stokes equations is solved in this paper for a steady, incompressible, fully developed laminar flow in a curved duct of elliptic cross section. This is achieved by the use of the control volume-based finite difference method via the numerically generated boundary fitted coordinate system. The curvature ratio is included in the primitive variable governing equations, which are solved based on the SIMPLE algorithm. Solutions are obtained for the minor-axis to major-axis ratios of the elliptic duct, 0.2, 0.5, and 0.8, and for Dean numbers ranging from 11.41 to 635.7. It is found that only one pair of vortices appears on the cross-section, even at a Dean number of 635.7. The friction factor and the ratios of the curved duct to straight duct are tabulated and the correlation equation is developed. Furthermore, the distribution of the axial velocity is displayed graphically to illustrate its variations with the Dean number and the minor-axis to major-axis ratio of the elliptic duct on the horizontal symmetry line and on the half-vertical symmetry line. The present method is also applied to solve for a fully developed laminar flow in a curved square flow. The results are compared with the data available in the literature and very close agreement is observed.

Convective heat transfer in the entrance region of a rectangular duct with two indented sides

This study presents a numerical solution to convective heat transfer in laminar flow in the thermal entrance region of a rectangular duct with two indented sides. The flow is considered to be hydrodynamically fully developed and thermally developing laminar flow of incompressible, Newtonian fluids with constant thermal properties. The ducts are subjected to a constant wall temperature. An algebraic technique is used to discretize the solution domain and a boundary-fitted coordinate system is numerically developed. The governing equations in the boundary-fitted coordinates are solved by the control volume-based finite difference method. Distribution of the bulk temperature and the Nusselt number along the direction of flow is calculated and presented graphically. Also calculated is the thermal entrance length of the rectangular ducts with two indented sides. The parameters, such as the friction factor times the Reynolds number, and the Nusselt number for the fully developed flow and thermally developing flow are obtained.