Diffusion coefficient calculated by time-lag using the film-roll method

2021 ◽  
pp. 004051752110278
Author(s):  
Geon Yong Park
Keyword(s):  
Diffusion Coefficient ◽  
Steady State ◽  
Boundary Conditions ◽  
Arrhenius Plot ◽  
Regression Line ◽  
Time Lag ◽  
Steady State Solution ◽  
Transient Solution ◽  
Steady State Condition ◽  
State Concentration

A method for determining the diffusion coefficient by time-lag using the film-roll method for the sublimation diffusion of disperse dye was proposed. A polyethylene terephthalate film-roll coated with dye paste was treated at 170–190°C for various times. A solution consisting of the sum of a steady-state solution and a transient solution was obtained by the homogeneous boundary value problem from a trigonometrical series. The boundary conditions of the steady-state first layer and the steady-state first layer amount of dye were determined from the steady-state concentration distribution. For various diffusion times, the steady-state first layer-passed total amounts of dye that passed through the first layer in the steady-state condition were obtained by subtracting the steady-state first layer amounts from the total amounts. The time-lag was calculated from the linear regression line for the plot of the steady-state first layer-passed total number(X) of positive values against time. The diffusion coefficient was calculated by the boundary conditions of the steady-state first layer and the time-lag. For diffusion at 170°C, 180°C, and 190°C, the correlations of the steady-state first layer-passed total amounts with respect to time were very linear and the reliability of the diffusion coefficients obtained by the time-lag was proved by the good linearity of the Arrhenius plot. The activation energy obtained was 36.8 kcal[Formula: see text]mol−1.

Comparison of Transient and Steady-State Behaviors in Unwinding Mechanism

2011 ◽  
Author(s):  
Wan-Suk Yoo ◽  
Kun-Woo Kim ◽  
Deuk-Man An ◽  
Jae-Wook Lee
Keyword(s):  
Steady State ◽  
Tensile Force ◽  
Equations Of Motion ◽  
Finite Difference Methods ◽  
Nonlinear Partial Differential Equations ◽  
Inertial Force ◽  
Steady State Solution ◽  
State Solution ◽  
Resistance Force ◽  
Transient Solution

In this study, the transient analysis of a cable unwinding from a cylindrical spool package is first studied and compared to experiment. Then, a steady-state solution is also compared to transient solution. Cables are assumed to be withdrawn with a constant velocity through a fixed point which is located along the axis of the package. When the cable is flown out of the package, several dynamic forces, such as inertial force, Coriolis force, centrifugal force, tensile force, and fluid-resistance force are acting on the cable. Consequently, the cable becomes to undergo very nonlinear and complex unwinding behavior which is called unwinding balloon. In this paper, to prevent the problems during unwinding such as tangling or cutting, unwinding behaviors of cables in transient state were derived and analyzed. First of all, the governing equations of motion of cables unwinding from a cylindrical spool package were systematically derived using the extended Hamilton’s principles of an open system in which mass is transported at each boundary. And the modified finite difference methods are suggested to solve the derived nonlinear partial differential equations. Time responses of unwinding cables are calculated using Newmark time integration methods. The transient solution is compared to physical experiment, and then the steady-state solution is compared to transient solution.

The Timoshenko Beam on an Elastic Foundation and Subject to a Moving Step Load, Part 1: Steady-State Response

1996 ◽  
Vol 118 (3) ◽  
pp. 277-284 ◽  
Author(s):  
S. F. Felszeghy
Keyword(s):  
Steady State ◽  
Boundary Conditions ◽  
Elastic Foundation ◽  
Timoshenko Beam ◽  
Equations Of Motion ◽  
Steady State Solution ◽  
State Solution ◽  
Actual Problem ◽  
State Response ◽  
Particular Solution

The response of a simply supported semi-infinite Timoshenko beam on an elastic foundation to a moving step load is determined. The response is found from summing the solutions to two mutually complementary sets of governing equations. The first solution is a particular solution to the forced equations of motion. The second solution is a solution to a set of homogeneous equations of motion and nonhomogeneous boundary conditions so formulated as to satisfy the initial and boundary conditions of the actual problem when the two solutions are summed. As a particular solution, the steady-state solution is used which is the motion that would appear stationary to an observer traveling with the load. Steady-state solutions are developed in Part 1 of this article for all load speeds greater than zero. It is shown that a steady-state solution which is identically zero ahead of the load front exists at every load speed, in the sense of generalized functions, including the critical speeds when the load travels at the minimum phase velocity of propagating harmonic waves and the sonic speeds. The solution to the homogeneous equations of motion is developed in Part 2 where the two solutions in question are summed and numerical results are presented as well.

Flow Field Analysis Around a Lockup Clutch Inside a Torque Converter

2011 ◽  
Author(s):  
Takeshi Yamaguchi ◽  
Shogo Ikeda ◽  
Sho Yamakawa ◽  
Kazuhiro Tanaka
Keyword(s):  
Steady State ◽  
Flow Field ◽  
Automatic Transmission ◽  
Steady State Solution ◽  
Torque Converter ◽  
Field Analysis ◽  
Hydrodynamic Performance ◽  
Transient Solution ◽  
Good Opportunity ◽  
Clutch Engagement

The performance of a torque converter has been one of the most important areas of improvement for an automatic-transmission equipped automobile. Improving the torque converter’s performance and efficiency is key to saving fuel consumption, which is an important consideration with recent environmental awareness. Moreover, the locking up operation or slipping control of an automatic transmission is another good opportunity for improving fuel economy. For this reason, there has been much research carried out to predict hydrodynamic performance and to understand the flow field inside a torque converter either experimentally or analytically using Computational Fluid Dynamics (CFD). Most of the research to date has focused on the inside of a torque converter torus. In recent years, the usage of a lockup clutch has expanded, and the lockup control system has become more complex. Understanding the flow field around the lockup clutch has become a very important issue. Only a few studies have focused on the lockup clutch, and most of the numerical research was solved at steady state conditions. In this paper, not only was an unsteady solution applied to solve the flow field, but also two new techniques were attempted. One was “virtual weight” and the other was “moving mesh.” By using these techniques, the lockup clutch was moved by the balance of its own weight and the opposing pressure acting on its surface. With this approach, the lockup clutch engagement time or the responsiveness of the lockup clutch could be estimated. The flow field calculated by a transient solution was found to be different from the flow field calculated by a steady state solution. The transient solution also revealed that the lockup engagement time and the lockup clutch moving speed were dependent on the lockup engagement pressure and rotation speed.

scholarly journals M[x]/G/1 Multistage Queue with Stand-by Server During Main Server’s Interruptions

2018 ◽  
Vol 11 (1) ◽  
pp. 275 ◽  
Author(s):  
Yuvarani Chandrasekaran ◽  
Vijayalakshmi C
Keyword(s):  
Steady State ◽  
Delay Time ◽  
Steady State Solution ◽  
Single Server ◽  
Supplementary Variable ◽  
Transient Solution ◽  
Variable Technique ◽  
Multiple Vacation ◽  
Server Breakdown ◽  
Two Phases

<p>This paper investigates a multistage batch arrival queue with different server interruptions and a second server replaces the main server during the interruptions. The different server interruptions are assumed to be: multiple vacation, extended vacation, breakdown with delay time and server under two phases of repair. Customers are assumed to arrive in batches according to Poisson process and a single server provides service to the customers. When the main server is inactive due to the interruptions, stand-by server provide service to the arrivals. In addition, customers may renege during server breakdown or during server vacation due to impatience. Transient solution and the corresponding steady state solution is derived using supplementary variable technique.</p>

Effects of suction-injection-combination (SIC) on transient free-convective-radiative flow in a vertical porous channel

2016 ◽  
Vol 231 (2) ◽  
pp. 73-82
Author(s):  
Basant Kumar Jha ◽  
Chibuike Iro ◽  
Sylvester Bakut Joseph
Keyword(s):  
Steady State ◽  
Convective Flow ◽  
Steady State Solution ◽  
Working Fluid ◽  
Value Of Time ◽  
Physical Situation ◽  
Transient Solution ◽  
Free Convective Flow ◽  
Implicit Finite Difference ◽  
Time Required

The effect of suction/injection on transient free-convective flow bounded by two infinite vertical parallel porous plates in the presence of thermal radiation is investigated numerically as well as analytically. The transient mathematical model has been solved using the implicit finite difference method while the steady state version of the physical situation has been solved using perturbation method. During the course of numerical computation, an excellent agreement was found between transient solution at large value of time and steady state solution. In addition, it was found that the time required to reach steady state is directly proportional to the Prandtl number of the working fluid for fixed values of other controlling parameters.

A Scaling Analysis of a CANDU-6 Moderator Tank Scaled-Down Test Facility

2013 ◽  
Author(s):  
Bo W. Rhee ◽  
H. T. Kim ◽  
S. K. Park ◽  
J. E. Cha ◽  
H.-L. Choi
Keyword(s):  
Steady State ◽  
Boundary Conditions ◽  
Power Supply ◽  
State Condition ◽  
Normal Operation ◽  
Source Distribution ◽  
Test Facility ◽  
Dimensionless Numbers ◽  
Steady State Condition ◽  
Scale Down

Several studies have been performed to derive a set of scaling criteria which were thought to be suitable for reproducing thermal-hydraulic phenomena in a scale-down CANDU moderator tank similar to that in a prototype power plant during a full power steady state condition[1,2]. The major variables of interests are moderator flow circulation and temperature inside the moderator tank during a steady state condition. The key phenomena involved include the inlet jet development and impingement, buoyancy force driven by the moderator temperature difference caused by non-uniform heating, and the viscous friction of the flow across the calandria tube array. In these studies, the governing equations were initially transformed into dimensionless equations based on the representative characteristic values of the basic design such as the time, tank diameter, inlet fluid velocity, and average temperature rise, and 3 dimensionless numbers, Re, Pr, Ar, were identified as those characterizing the key phenomena of the system. The relevant boundary conditions were then identified in a dimensionless form, and the compatibility of keeping these 3 dimensionless numbers, the volumetric heat source distribution, and the boundary conditions in dimensionless forms the same for both the prototype and scale-down tanks were examined, and some of them that are less important are relaxed so as to find a practically implementable set of constraints. The size of the scaled-down moderator tank and corresponding inlet velocity is then found for the available power supply size. As an example, an analysis was performed for a power supply capacity of 500 kW as compared to 100MW for the prototype. As a way to confirm the validity of the current work two numerical CFD simulations were carried out with the boundary conditions at the inlet and outlet ports, and on the walls of the solid structures, such as the moderator tank and calandria tubes, which were derived from those of the dimensionless scales to check if the moderator flow and temperature patterns of both the prototype reactor and scaled-down facilities are identical or at least similar. A steady-state solution is first obtained for the candu-6 reactor normal operation. Similar simulation was done for the scaled-down facility and results presented. Comparison results are discussed, and the cause of the potential distortion of the scaling owing to practical limitations and possible solutions is finally discussed.

Statistic thermodynamic analysis of fructans – Part 4: Modeling inulin biosynthesis as a non-equilibium thermodynamic process

2017 ◽  
pp. 206-211
Author(s):  
Roland, H.F. Beck
Keyword(s):  
Steady State ◽  
Boundary Conditions ◽  
Thermodynamic Analysis ◽  
Mathematical Description ◽  
Degree Of Polymerization ◽  
State Condition ◽  
Stable Steady State ◽  
Thermodynamic Process ◽  
Steady State Condition ◽  
Over Time

A model for the mathematical description of inulin biosynthesis with particular focus on the dynamics of the sucrose and 1-kestose concentration has been developed. The model takes into account the specific action of the two involved enzymes SST and FFT. In principle, inulin biosynthesis can thus be described as a thermodynamically stabilizing process leading into a stable nodal sink over time. In the initial phases of the reaction, oscillation of sucrose and 1-kestose concentration is observed, dampening rapidly for both products into a stable steady state condition. Applying the same boundary conditions, a higher number average degree of polymerization will lead to a higher frequency and amplitude in oscillation.

Evaluation of a Diffusion Coefficient Based on Proper Solution Space

2018 ◽  
Vol 11 (1) ◽  
Author(s):  
Yuanlong Wang ◽  
Abdalkaleg Hamad ◽  
Mohsen Tadi
Keyword(s):  
Diffusion Coefficient ◽  
Steady State ◽  
Boundary Conditions ◽  
Heat Conduction Problem ◽  
Solution Space ◽  
Initial Guess ◽  
Proper Solution ◽  
Numerical Examples ◽  
Steady State Heat ◽  
Unknown Diffusion Coefficient

Abstract This note is concerned with the evaluation of the unknown diffusion coefficient in a steady-state heat conduction problem. The proposed method is iterative and, starting with an initial guess, updates the assumed value at every iteration. The updating stage is achieved by generating a set of functions that satisfy some of the required boundary conditions. The correction to the assumed value is then computed by imposing the remaining boundary conditions. Numerical examples are used to study the applicability of this method.

The Timoshenko Beam With a Moving Load

1968 ◽  
Vol 35 (3) ◽  
pp. 481-488 ◽  
Author(s):  
C. R. Steele
Keyword(s):  
Steady State ◽  
Timoshenko Beam ◽  
Moving Load ◽  
Beam Theory ◽  
Steady State Solution ◽  
State Solution ◽  
Asymptotic Results ◽  
Concentrated Load ◽  
Steady State Condition ◽  
Bar Velocity

The problem of a semi-infinite Timoshenko beam of an elastic foundation with a step load moving from the supported end at a constant velocity is discussed. Asymptotic solutions are obtained for all ranges of load speed. The solution is shown to approach the “steady-state” solution, except for three speeds at which the steady state does not exist. Previous investigators have considered only the steady-state solution for the moving concentrated load and have indicated that the three speeds are “critical.” It is shown, however, that only the lowest speed is truly critical in that the response increases with time. For the load speed equal to either the shear or bar velocity, the transients due to the end condition never leave the vicinity of the load discontinuities, so a steady-state condition is never attained. However, the response is shown to be bounded in time for a distributed load. Thus the nonexistence of a steady state does not necessarily indicate a critical condition. Furthermore, the concentrated load solution is shown to have validity at speeds the magnitude of the sonic speeds only for loads of a concentration beyond the limitations of beam theory. Asymptotic results have also been obtained for the beam without a foundation. Since the procedure is similar for beams with, and without, a foundation, only the results are included to show a comparison with the numerical results previously obtained by Florence.

The Timoshenko Beam on an Elastic Foundation and Subject to a Moving Step Load, Part 2: Transient Response

1996 ◽  
Vol 118 (3) ◽  
pp. 285-291 ◽  
Author(s):  
S. F. Felszeghy
Keyword(s):  
Steady State ◽  
Boundary Conditions ◽  
Elastic Foundation ◽  
Transient Response ◽  
Timoshenko Beam ◽  
Equations Of Motion ◽  
Steady State Solution ◽  
State Solution ◽  
Critical Speeds ◽  
Particular Solution

The transient response of a simply supported semi-infinite Timoshenko beam on an elastic foundation to a moving step load is determined. The response is found from summing the solutions to two mutually complementary sets of governing equations. The first solution is a particular solution to the forced equations of motion. The second solution is a solution to a set of homogeneous equations of motion and nonhomogeneous boundary conditions so formulated as to satisfy the initial and boundary conditions of the actual problem when the two solutions are summed. As a particular solution, the steady-state solution is used which is the motion that would appear stationary to an observer traveling with the load. Steady-state solutions were developed in Part 1 of this article for all load speeds greater than zero. The solution to the homogeneous equations of motion is developed here in Part 2. It is shown that the latter solution can be obtained by numerical integration using the method of characteristics. Particular attention is given to the cases when the load travels at the critical speeds consisting of the minimum phase velocity of propagating harmonic waves and the sonic speeds. It is shown that the solution to the homogeneous equations combines with the steady-state solution in such a manner that the beam displacements are continuous and bounded for all finite times at all load speeds including the critical speeds. Numerical results are presented for the critical load speed cases.

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