Two-Dimensional Transient Solutions for Crossflow Heat Exchangers With Neither Gas Mixed

1987 ◽  
Vol 109 (2) ◽  
pp. 281-286 ◽  
Author(s):  
G. Spiga ◽  
M. Spiga
Keyword(s):  
Heat Exchangers ◽  
Initial Conditions ◽  
Transient Behavior ◽  
Computational Time ◽  
Two Dimensional ◽  
Transform Method ◽  
Wide Range ◽  
The Laplace Transform ◽  
Inlet Conditions ◽  
Conductance Ratio

The two-dimensional transient behavior of gas-to-gas crossflow heat exchangers is investigated, solving by analytical methods the thermal balance equations in order to determine the transient distribution of temperatures in the core wall and in both the unmixed gases. Assuming large wall capacitance, the general solutions are deduced by the Laplace transform method and are presented as integrals of modified Bessel functions on space and time, for a transient response with any arbitrary initial and inlet conditions, in terms of the number of transfer units, capacity rate and conductance ratio. Specializing the entrance temperature and assuming constant initial conditions, the most meaningful transient conditions (such as step, ramp, and exponential responses) have been simulated and the relevant solutions, expressed by means of either integrals or series, have been accurately computed with extremely low computational time. The temperature responses are then presented in graphic form for a wide range of the number of transfer units.

Transient Response of Gas-to-Gas Crossflow Heat Exchangers With Neither Gas Mixed

1983 ◽  
Vol 105 (3) ◽  
pp. 563-570 ◽  
Author(s):  
F. E. Romie
Keyword(s):  
Heat Exchangers ◽  
Rate Ratio ◽  
Graphical Form ◽  
Laplace Transform Method ◽  
Transform Method ◽  
Unit Step ◽  
The Laplace Transform ◽  
Mixed Mean ◽  
Conductance Ratio ◽  
Temperature Responses

The transient mixed mean temperatures of the two gases leaving a crossflow heat exchanger are found for a unit step increase in the entrance temperature of either gas. The temperature responses are given in graphical form for the range of parameters: Ntu from 1 to 8, capacity rate ratio from 0.6 to 1.67, and conductance, ratio from 0.5 to 2.0. The solutions are found using the Laplace transform method and apply to single-pass crossflow exchangers with neither gas mixed.

scholarly journals Mathematical modeling and algorithm for calculation of thermocatalytic process of producing nanomaterial

2021 ◽  
Vol 23 (3) ◽  
pp. 1590
Author(s):  
Bakhtiyar Ismailov ◽  
Zhanat Umarova ◽  
Khairulla Ismailov ◽  
Aibarsha Dosmakanbetova ◽  
Saule Meldebekova
Keyword(s):  
Mathematical Model ◽  
Differential Equations ◽  
Solid Phase ◽  
Nonlinear Effects ◽  
Operating Characteristics ◽  
Laplace Transform Method ◽  
Transform Method ◽  
Wide Range ◽  
The Laplace Transform ◽  
Complex Mathematical Model

<p>At present, when constructing a mathematical description of the pyrolysis reactor, partial differential equations for the components of the gas phase and the catalyst phase are used. In the well-known works on modeling pyrolysis, the obtained models are applicable only for a narrow range of changes in the process parameters, the geometric dimensions are considered constant. The article poses the task of creating a complex mathematical model with additional terms, taking into account nonlinear effects, where the geometric dimensions of the apparatus and operating characteristics vary over a wide range. An analytical method has been developed for the implementation of a mathematical model of catalytic pyrolysis of methane for the production of nanomaterials in a continuous mode. The differential equation for gaseous components with initial and boundary conditions of the third type is reduced to a dimensionless form with a small value of the peclet criterion with a form factor. It is shown that the laplace transform method is mainly suitable for this case, which is applicable both for differential equations for solid-phase components and calculation in a periodic mode. The adequacy of the model results with the known experimental data is checked.</p>

Second Grade Fluid for Rotating MHD of an Unsteady Free Convection Flow in a Porous Medium

2015 ◽  
Vol 362 ◽  
pp. 100-107 ◽  
Author(s):  
Z. Ismail ◽  
I. Khan ◽  
A.Q. Mohamad ◽  
S. Shafie
Keyword(s):  
Porous Medium ◽  
Free Convection ◽  
Initial Conditions ◽  
Boussinesq Approximation ◽  
Second Grade ◽  
Convection Flow ◽  
Inclined Plate ◽  
Free Convection Flow ◽  
Transform Method ◽  
The Laplace Transform

Rotating effects and magnetohydrodynamic (MHD) free convection flow of second grade fluids in a porous medium is considered in this paper. It is assumed that the bounding infinite inclined plate has ramped wall temperature with the presence of heat and mass diffusion. Based on Boussinesq approximation, the analytical expressions for dimensionless velocity, temperature and concentration are obtained by using the Laplace transform method. All the derived solutions satisfying the involved differential equations with imposed boundary and initial conditions. The influence of various parameters on the velocity has been analyzed in graphs and discussed.

Hitting-Time Densities of a Two-Dimensional Markov Process

1992 ◽  
Vol 6 (4) ◽  
pp. 561-580
Author(s):  
C. H. Hesse
Keyword(s):  
Global Analysis ◽  
First Passage Time ◽  
Type Equation ◽  
Passage Time ◽  
Two Dimensional ◽  
First Passage ◽  
Wide Range ◽  
The Laplace Transform ◽  
Mean And Variance

This paper deals with the two-dimensional stochastic process (X(t), V(t)) where dX(t) = V(t)dt, V(t) = W(t) + ν for some constant ν and W(t) is a one-dimensional Wiener process with zero mean and variance parameter σ2= 1. We are interested in the first-passage time of (X(t), V(t)) to the plane X = 0 for a process starting from (X(0) = −x, V(0) = ν) with x > 0. The partial differential equation for the Laplace transform of the first-passage time density is transformed into a Schrödinger-type equation and, using methods of global analysis, such as the method of dominant balance, an approximation to the first-passage density is obtained. In a series of simulations, the quality of this approximation is checked. Over a wide range of x and ν it is found to perform well, globally in t. Some applications are mentioned.

scholarly journals Two-Dimensional Evaluation of ATHAM-Fluidity, a Nonhydrostatic Atmospheric Model Using Mixed Continuous/Discontinuous Finite Elements and Anisotropic Grid Optimization

2016 ◽  
Vol 144 (11) ◽  
pp. 4349-4372 ◽  
Author(s):  
Julien Savre ◽  
James Percival ◽  
Michael Herzog ◽  
Chris Pain
Keyword(s):  
Optimization Algorithm ◽  
Large Scale ◽  
Atmospheric Model ◽  
Computational Time ◽  
Limited Area ◽  
Test Cases ◽  
Two Dimensional ◽  
Element Discretization ◽  
Wide Range ◽  
Grid Optimization

Abstract This paper presents the first attempt to apply the compressible nonhydrostatic Active Tracer High-Resolution Atmospheric Model–Fluidity (ATHAM-Fluidity) solver to a series of idealized atmospheric test cases. ATHAM-Fluidity uses a hybrid finite-element discretization where pressure is solved on a continuous second-order grid while momentum and scalars are computed on a first-order discontinuous grid (also known as ). ATHAM-Fluidity operates on two- and three-dimensional unstructured meshes, using triangular or tetrahedral elements, respectively, with the possibility to employ an anisotropic mesh optimization algorithm for automatic grid refinement and coarsening during run time. The solver is evaluated using two-dimensional-only dry idealized test cases covering a wide range of atmospheric applications. The first three cases, representative of atmospheric convection, reveal the ability of ATHAM-Fluidity to accurately simulate the evolution of large-scale flow features in neutral atmospheres at rest. Grid convergence without adaptivity as well as the performances of the Hermite–Weighted Essentially Nonoscillatory (Hermite-WENO) slope limiter are discussed. These cases are also used to test the grid optimization algorithm implemented in ATHAM-Fluidity. Adaptivity can result in up to a sixfold decrease in computational time and a fivefold decrease in total element number for the same finest resolution. However, substantial discrepancies are found between the uniform and adapted grid results, thus suggesting the necessity to improve the reliability of the approach. In the last three cases, corresponding to atmospheric gravity waves with and without orography, the model ability to capture the amplitude and propagation of weak stationary waves is demonstrated. This work constitutes the first step toward the development of a new comprehensive limited area atmospheric model.

scholarly journals Analytical Solution of Two-Dimensional Sine-Gordon Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-15
Author(s):  
Alemayehu Tamirie Deresse ◽  
Yesuf Obsie Mussa ◽  
Ademe Kebede Gizaw
Keyword(s):  
Numerical Solutions ◽  
Initial Conditions ◽  
Nonlinear Partial Differential Equations ◽  
Test Problems ◽  
Two Dimensional ◽  
Sine Gordon Equation ◽  
Science And Engineering ◽  
Transform Method ◽  
Differential Transform ◽  
Good Agreement

In this paper, the reduced differential transform method (RDTM) is successfully implemented for solving two-dimensional nonlinear sine-Gordon equations subject to appropriate initial conditions. Some lemmas which help us to solve the governing problem using the proposed method are proved. This scheme has the advantage of generating an analytical approximate solution or exact solution in a convergent power series form with conveniently determinable components. The method considers the use of the appropriate initial conditions and finds the solution without any discretization, transformation, or restrictive assumptions. The accuracy and efficiency of the proposed method are demonstrated by four of our test problems, and solution behavior of the test problems is presented using tables and graphs. Further, the numerical results are found to be in a good agreement with the exact solutions and the numerical solutions that are available in literature. We have showed the convergence of the proposed method. Also, the obtained results reveal that the introduced method is promising for solving other types of nonlinear partial differential equations (NLPDEs) in the fields of science and engineering.

scholarly journals Approximate Analytic Solutions of Two-Dimensional Nonlinear Klein–Gordon Equation by Using the Reduced Differential Transform Method

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Wayinhareg Gashaw Belayeh ◽  
Yesuf Obsie Mussa ◽  
Ademe Kebede Gizaw
Keyword(s):  
Numerical Solutions ◽  
Initial Conditions ◽  
Type Systems ◽  
Analytical Approximation ◽  
Differential Transform Method ◽  
Two Dimensional ◽  
Transform Method ◽  
Differential Transform ◽  
Klein Gordon ◽  
Reduced Differential Transform Method

In this paper, the reduced differential transform method (RDTM) is successfully implemented for solving two-dimensional nonlinear Klein–Gordon equations (NLKGEs) with quadratic and cubic nonlinearities subject to appropriate initial conditions. The proposed technique has the advantage of producing an analytical approximation in a convergent power series form with a reduced number of calculable terms. Two test examples from mathematical physics are discussed to illustrate the validity and efficiency of the method. In addition, numerical solutions of the test examples are presented graphically to show the reliability and accuracy of the method. Also, the results indicate that the introduced method is promising for solving other type systems of NLPDEs.

The Evaluation of an Integral Equation Method for Two-Dimensional Shock-Free Flows

1975 ◽  
Vol 26 (1) ◽  
pp. 59-70 ◽  
Author(s):  
D Nixon ◽  
J Patel
Keyword(s):  
Integral Equation ◽  
Integral Equation Method ◽  
Free Flow ◽  
Equation Method ◽  
Computational Time ◽  
Test Cases ◽  
Two Dimensional ◽  
Exact Methods ◽  
Wide Range ◽  
Steady Shock

SummaryThe numerical aspects of the integral equation method developed by Nixon and Hancock for two-dimensional steady shock-free flow have been rationalised; this numerically refined method is evaluated by calculating the pressure distribution around a wide range of aerofoils. These test cases include aerofoils in supercritical shock-free flow as well as subcritical flow and exact solutions are available for comparison. The computational time in the present method is significantly less than that required by the exact methods. The present results compare satisfactorily with the exact results.

Transient Behavior of Crossflow Heat Exchangers With Longitudinal Conduction and Axial Dispersion

2004 ◽  
Vol 126 (3) ◽  
pp. 425-433 ◽  
Author(s):  
Manish Mishra ◽  
P. K. Das ◽  
Sunil Sarangi
Keyword(s):  
Heat Transport ◽  
Heat Exchangers ◽  
Temperature Response ◽  
Transient Behavior ◽  
Axial Dispersion ◽  
Transient Temperature ◽  
Inlet Temperature ◽  
Conductive Heat ◽  
Two Dimensional ◽  
Transient Temperature Response

Transient temperature response of the crossflow heat exchangers with finite wall capacitance and both fluids unmixed is investigated numerically for step, ramp and exponential perturbations provided in hot fluid inlet temperature. Effect of two-dimensional longitudinal conduction in separating sheet and axial dispersion in fluids on the transient response has been investigated. Conductive heat transport due to presence of axial dispersion in fluids have been analyzed in detail and shown that presence of axial dispersion in both of the fluid streams neutralizes the total conductive heat transport during the energy balance. It has also been shown that the presence of axial dispersion of high order reduces the effect of longitudinal conduction.

A Generalized Prediction Method for Rotordynamic Coefficients of Annular Gas Seals

2015 ◽  
Vol 137 (9) ◽  
Author(s):  
Xin Yan ◽  
Kun He ◽  
Jun Li ◽  
Zhenping Feng
Keyword(s):  
Reaction Force ◽  
Excitation Frequency ◽  
Prediction Method ◽  
Experimental Tests ◽  
Computational Time ◽  
Transform Method ◽  
Rotordynamic Coefficients ◽  
The Laplace Transform ◽  
Computational Fluid Dynamics Cfd ◽  
Gas Seals

The improvement in rotordynamic performance of the annular gas seal requires efficient and accurate prediction methods of rotordynamic coefficients. Although the existed transient computational fluid dynamics (CFD) methods in published literature have excellent numerical accuracy, most of them face the challenge due to rotordynamic coefficients at every excitation frequency to be solved by a separate transient CFD prediction thus much time-consuming. In this paper, a generalized prediction method is proposed to address this difficulty. Based on the Laplace transform method, the solution procedures for the reaction force/motion equation of the annular gas seal are deduced. With the specified excitations (rotor motion), the rotordynamic coefficients at all excitation frequencies can be solved by only one or two transient CFD solutions. To verify the present generalized method, the rotordynamic coefficients of two typical hole-pattern seals are computed and compared to the available experimental data. The results show that the predicted rotordynamic coefficients are in good agreement with the experimental tests. Compared to the previous transient CFD methods, the computational time of the present generalized method is reduced significantly while the accuracy is still maintained.

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