A Simple Model for Prediction of Preheating and Pyrolysis Time of a Thermally Thin Charring Particle

2012 ◽  
Author(s):  
Y. Haseli ◽  
J. A. van Oijen ◽  
L. P. H. de Goey
Keyword(s):  
Differential Equation ◽  
Simple Model ◽  
Solid Particle ◽  
Time Integration ◽  
Quadratic Function ◽  
Conservation Equation ◽  
Virgin Material ◽  
Algebraic Equations ◽  
Energy Conservation Equation ◽  
Pyrolysis Model

The aim of this paper is to present a simple model, based on a time and space integral method, for prediction of preheating and conversion time of a charring solid particle exposed to a non-oxidative hot environment. The main assumptions are 1) thermo-physical properties remain constant throughout the process; 2) temperature profile within the particle is assumed to obey a quadratic function with respect to the space coordinate; 3) pyrolysis initiates when the surface temperature reaches a characteristic pyrolysis temperature; 4) decomposition of virgin material occurs at an infinitesimal thin layer dividing the particle into char and virgin material regions; 5) the volume of the particle remains unaltered; 6) volatiles escape through the pores immediately after formation. Employing assumption (2) allows one to convert the energy conservation equation of the particle, which is basically described in the form of a partial differential equation (PDE), into an ordinary differential equation (ODE) by performing space integration. Next, by applying approximate time integration the ODE is transformed into an algebraic equation. Applying this approach to the preheating and pyrolysis stages of a thermally thin charring solid particle leads to a set of algebraic equations which provides reactor designers with a convenient means for computation of the heating up time, mass loss history and total conversion of particle. The accuracy of the simple model is assessed by comparing its prediction with that of a one-dimensional detailed pyrolysis model. Overall, good agreement is achieved indicating that this new model can be used for engineering and design purposes.

scholarly journals INDOOR AIR TEMPERATURE DISTRIBUTION – AN ALTERNATIVE APPROACH TO BUILDING SIMULATION

2003 ◽  
Vol 2 (1) ◽  
Author(s):  
A. T. Franco ◽  
C. O. R. Negrão
Keyword(s):  
Temperature Distribution ◽  
Air Temperature ◽  
Indoor Air ◽  
Linear Equations ◽  
Three Dimensional ◽  
Momentum Conservation ◽  
Conservation Equation ◽  
Algebraic Equations ◽  
Conservation Of Energy ◽  
Energy Conservation Equation

The current paper presents a model to predict indoor air temperature distribution. The approach is based on the energy conservation equation which is written for a certain number of finite volumes within the flow domain. The magnitude of the flow is estimated from a scale analysis of the momentum conservation equation. Discretized two or three-dimensional domains provide a set of algebraic equations. The resulting set of non-linear equations is iteratively solved using the line-by-line Thomas Algorithm. As long as the only equation to be solved is the conservation of energy and its coefficients are not strongly dependent on the temperature field, the solution is considerably fast. Therefore, the application of such model to a whole building system is quite reasonable. Two case studies involving buoyancy driven flows were carried out and comparisons with CFD solutions were performed. The results are quite promising for cases involving relatively strong couplings between heat and airflow.

scholarly journals An ordinary differential equation approach for nonlinear programming and nonlinear complementary problem

2020 ◽  
Vol 1 (1) ◽  
pp. 1
Author(s):  
Irfan Nurhidayat ◽  
Zijun Hao ◽  
Chu-chin Hu ◽  
Jein-Shan Chen
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Time Integration ◽  
Nonlinear Dynamical System ◽  
Special Technique ◽  
Algebraic Equations ◽  
Approximation Solution ◽  
Nonlinear Dynamical ◽  
Group Preserving Scheme ◽  
Original Time

We consider an ordinary differential equation (ODE) approach for solving non- linear programming (NLP) and nonlinear complementary problem (NCP). The Karush- Kuhn Tucker (KKT) optimality conditions can be converted to NCP. Based on the Fischer-Burmeister (FB) function and the Natural-Residual (NR) function are obtained the new NCP-functions. A special technique is employed to reformulate of the NCP as the system of nonlinear algebraic equations (NAEs) later on reformulated once more by force. of an original time-like function into an ODE. Afterwards, a group preserving scheme (GPS) is a package to reformulate an ODE into the new numerical equation in a way the ODEs system is designed into a nonlinear dynamical system (NDS) and is continued to a discovery the new numerical equation through activating the Lorentz group SO0(n, 1) and its Lie algebra so(n, 1). Lastly, the fictitious time integration method (FTIM) is utilized into this new numerical equation to determine an approximation solution at the numerical experiments area.

scholarly journals INDOOR AIR TEMPERATURE DISTRIBUTION – AN ALTERNATIVE APPROACH TO BUILDING SIMULATION

2003 ◽  
Vol 2 (1) ◽  
pp. 19
Author(s):  
A. T. Franco ◽  
C. O. R. Negrão
Keyword(s):  
Temperature Distribution ◽  
Air Temperature ◽  
Indoor Air ◽  
Linear Equations ◽  
Three Dimensional ◽  
Momentum Conservation ◽  
Conservation Equation ◽  
Algebraic Equations ◽  
Conservation Of Energy ◽  
Energy Conservation Equation

The current paper presents a model to predict indoor air temperature distribution. The approach is based on the energy conservation equation which is written for a certain number of finite volumes within the flow domain. The magnitude of the flow is estimated from a scale analysis of the momentum conservation equation. Discretized two or three-dimensional domains provide a set of algebraic equations. The resulting set of non-linear equations is iteratively solved using the line-by-line Thomas Algorithm. As long as the only equation to be solved is the conservation of energy and its coefficients are not strongly dependent on the temperature field, the solution is considerably fast. Therefore, the application of such model to a whole building system is quite reasonable. Two case studies involving buoyancy driven flows were carried out and comparisons with CFD solutions were performed. The results are quite promising for cases involving relatively strong couplings between heat and airflow.

Heat Transfer Analysis of a Novel Pressurized Air Receiver for Concentrated Solar Power via Combined Cycles

2009 ◽  
Vol 1 (4) ◽  
Author(s):  
I. Hischier ◽  
D. Hess ◽  
W. Lipiński ◽  
M. Modest ◽  
A. Steinfeld
Keyword(s):  
Heat Transfer ◽  
Thermal Efficiency ◽  
Porous Ceramic ◽  
Conservation Equation ◽  
Heat Transfer Analysis ◽  
Operational Parameters ◽  
Small Aperture ◽  
Solar Absorber ◽  
Energy Conservation Equation ◽  
Combined Cycles

A novel design of a high-temperature pressurized solar air receiver for power generation via combined Brayton–Rankine cycles is proposed. It consists of an annular reticulate porous ceramic (RPC) bounded by two concentric cylinders. The inner cylinder, which serves as the solar absorber, has a cavity-type configuration and a small aperture for the access of concentrated solar radiation. Absorbed heat is transferred by conduction, radiation, and convection to the pressurized air flowing across the RPC. A 2D steady-state energy conservation equation coupling the three modes of heat transfer is formulated and solved by the finite volume technique and by applying the Rosseland diffusion, P1, and Monte Carlo radiation methods. Key results include the temperature distribution and thermal efficiency as a function of the geometrical and operational parameters. For a solar concentration ratio of 3000 suns, the outlet air temperature reaches 1000°C at 10 bars, yielding a thermal efficiency of 78%.

Energy conservation, time-reversal invariance and reciprocity in ducts with flow

2001 ◽  
Vol 431 ◽  
pp. 223-237 ◽  
Author(s):  
WILLI MÖHRING
Keyword(s):  
Energy Conservation ◽  
Sound Wave ◽  
Conservation Equation ◽  
Smooth Transition ◽  
Rotational Flow ◽  
Flow Energy ◽  
Energy Conservation Equation ◽  
Reversal Invariance ◽  
Edge Conditions ◽  
Piecewise Continuous

A sound wave propagating in an inhomogeneous duct consisting of two semi-infinite uniform ducts with a smooth transition region in between and which carries a steady flow is considered. The duct walls may be rigid or compliant. For an irrotational sound wave it is shown that the three properties of the title are closely related, such that the validity of any two implies the validity of the third. Furthermore it is shown that the three properties are fulfilled for lossless locally reacting duct walls provided the impedance varies at most continuously. For piecewise-continuous wall properties edge conditions are essential. By an analytic continuation argument it is shown that reciprocity remains true for walls with loss. For rotational flow, energy conservation theorems have been derived only with the help of additional potential-like variables. The inter-relation between the three properties remains valid if one considers these additional variables to be known. If only the basic gasdynamic variables in both half-ducts are known, one cannot formulate an energy conservation equation; however, reciprocity is fulfilled.

scholarly journals New Properties of Modified Second Kind Chebyshev Polynomials

2020 ◽  
Vol 55 (3) ◽  
Author(s):  
Semaa Hassan Aziz ◽  
Mohammed Rasheed ◽  
Suha Shihab
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Chebyshev Polynomial ◽  
Chebyshev Polynomials ◽  
Digital Computer ◽  
Higher Order ◽  
Algebraic Equations ◽  
Equation Problem ◽  
Higher Order Differential Equations

Modified second kind Chebyshev polynomials for solving higher order differential equations are presented in this paper. This technique, along with some new properties of such polynomials, will reduce the original differential equation problem to the solution of algebraic equations with a straightforward and computational digital computer. Some illustrative examples are included. The modified second kind Chebyshev polynomial is calculated using only a small number of the modified second kind Chebyshev polynomials, which leads to attractive results.

scholarly journals A Two-Layer Model for Steady-Amplitude Gravity Waves and Convection Generated by a Thermal Forcing

2018 ◽  
Vol 75 (7) ◽  
pp. 2199-2216 ◽  
Author(s):  
A. A. M. Sayed ◽  
L. J. Campbell
Keyword(s):  
Gravity Waves ◽  
Lower Layer ◽  
Stable Stratification ◽  
Internal Gravity Waves ◽  
Conservation Equation ◽  
Lower Atmosphere ◽  
Thermal Forcing ◽  
Energy Conservation Equation ◽  
Vertical Location ◽  
Convective Cells

Abstract A two-dimensional two-layer mathematical model is described representing internal gravity waves and convection generated by a thermal forcing in the lower atmosphere. The model consists of an upper layer with stable stratification, a lower layer with unstable stratification, and a thermal forcing in the form of a nonhomogeneous term in the energy conservation equation. Exact analytical solutions are derived for some simple configurations. Depending on the vertical location and depth of the thermal forcing, the model can be used to represent different configurations in which gravity waves are generated by diabatic heating. When the thermal forcing is centered in the lower layer, convective cells are generated in the lower layer, and gravity waves are forced and propagate upward from the interface between the two layers. When the thermal forcing is centered at the interface, the convection in the lower layer is weaker, and gravity waves are forced by the direct effect of the thermal forcing in the upper layer and the influence of the convective cells below. Steady-amplitude solutions for the vertical profile of the gravity waves and convection are derived and generalized to include cases where there is a spectrum of horizontal wavenumbers or vertical wavenumbers or frequencies present.

Mathematical Model of In-Flight Oxidation of Metallic Particles in Thermal Spraying

2000 ◽  
Author(s):  
A.M. Ahmed ◽  
R.H. Rangel ◽  
V.V. Sobolev ◽  
J.M. Guilemany
Keyword(s):  
Mathematical Model ◽  
Thermal Behavior ◽  
Spray Deposition ◽  
Particle Surface ◽  
Thermal Spraying ◽  
Deposition Process ◽  
Conservation Equation ◽  
Spherical Particles ◽  
Energy Conservation Equation ◽  
Acceleration And Deceleration

Abstract This paper presents a mathematical model of the in-flight oxidation of spherical particles during thermal spray deposition process. The model includes analysis of the mechanical and thermal behavior of the powder particles. The former accounts for acceleration and deceleration of the particles at the spray distance under different fluid velocities. The thermal behavior takes into account heating, melting, cooling and possible solidification as the particle travel towards the substrate. A finite-difference method is used to solve the thermal energy conservation equation of the particles. The model includes nonequilibrium calculations of the phase change phenomena in the liquid-solid (mushy) zone. The growth of the oxide layer at the particle surface is represented by a modified boundary condition, which includes finite-rate oxidation. The results obtained give the interrelations between various process parameters and the oxidation phenomenon and agree with experimental observation.

Modeling of Oxidation of Plasma-Sprayed Molybdenum Coatings

2000 ◽  
Author(s):  
Y.P. Wan ◽  
X.Y. Jiang ◽  
H. Zhang ◽  
S. Sampath ◽  
V. Prasad ◽  
...  
Keyword(s):  
Spray Deposition ◽  
Mechanical Energy ◽  
Particle Surface ◽  
Conservation Equation ◽  
Type Model ◽  
Energy Conservation Equation ◽  
Thin Oxide Film ◽  
Molten Particle ◽  
Plasma Sprayed ◽  
Substrate Temperatures

Abstract A model for oxidation of molybdenum particles during plasma spray deposition is developed. The diffusion of metal an-ions or oxygen cat-ions through a thin oxidized film, chemical reactions on the surface, and diffusion of oxidant in gas phase are considered as possible rate-controlling mechanisms with controlling parameters as the temperature of the particle surface, and local oxygen concentration and flow field surrounding the particle. The deposition of molten particle and its rapid solidification and deformation is treated using a Madejski-type model, in which the mechanical energy conservation equation is solved to determine the splat deformation and one-dimensional heat conduction equation with phase change is solved to predict the solidification and temperature evolution. Calculations are performed for a single molybdenum particle sprayed under the Sulzer Metco-9MB spraying conditions. Results show that the mechanism that controls the oxidation of this droplet is the diffusion of metal/oxygen ions through a very thin oxide film. A higher substrate temperature results in a larger rate of oxidation at the splat surface, and hence, a larger oxygen content in the coating layer. Compared to the oxidation of droplet during m-flight, the oxidation during deposition is not weak and can become dominant at high substrate temperatures.

Sign in / Sign up

Export Citation Format

Share Document