Mathematical analysis of a non-Newtonian polymer in the forward roll coating process

2020 ◽  
Vol 40 (8) ◽  
pp. 703-712 ◽  
Author(s):  
Muhammad Zahid ◽  
Muhammad Zafar ◽  
Muhammad A. Rana ◽  
Muhammad S. Lodhi ◽  
Abdul S. Awan ◽  
...  
Keyword(s):  
Pressure Distribution ◽  
Flow Rate ◽  
Coating Thickness ◽  
Power Input ◽  
Control Flow ◽  
Roll Coating ◽  
Material Parameters ◽  
Engineering Parameters ◽  
Profile Flow ◽  
Forward Roll Coating

AbstractThis article describes the development of a mathematical model of forward roll coating of a thin film of a non-Newtonian material when it passes through a small gap between the two counter-rotating rolls. The conservation equations of mass, momentum, and energy in the light of LAT (lubrication approximation theory) are non-dimensionalized and solutions for the velocity profile, flow rate, pressure distribution, pressure, forces, stresses, power input to the roller, and temperature distribution are calculated analytically. It is found that by changing (increasing/decreasing) the value of material parameters, one can really control the engineering parameters like, stress and the most important the coating thickness and is a quick reference for the engineer working in coating industries. Some results are shown graphically. From the present study, it has been established that the material parameter is a device to control flow rate, coating thickness, separation points, and pressure distribution.

Numerical analysis of the forward roll coating of a Rabinowitsch fluid

2019 ◽  
Vol 36 (2) ◽  
pp. 191-208 ◽  
Author(s):  
Zahid M ◽  
M Zafar ◽  
MA Rana ◽  
MTA Rana ◽  
MS Lodhi
Keyword(s):  
Pressure Distribution ◽  
Flow Rate ◽  
Coating Thickness ◽  
Equations Of Motion ◽  
Power Input ◽  
Roll Coating ◽  
Point Separation ◽  
Velocity Flow ◽  
Engineering Parameters ◽  
Separation Force

This paper analyses an isothermal roll coating for a Rabinowitsch fluid. Equations of motion are simplified using lubrication approximation theory. Analytical solutions of velocity, flow rate and pressure-gradient are calculated. Outcomes of coating thickness, pressure distribution, split position, stresses, forces, the power input to the roller and the temperature rise between the coating rollers and substrate are obtained. The effects of some involved parameters are displayed through graphs and tables. It is noted that the Rabinowitsch parameter is a controlling quantity for coating thickness, flow rate, separation point, separation force, pressure distribution, and power input. Moreover, an exponential and monotonic change in the above-mentioned engineering parameters is a function of the Rabinowitsch fluid parameter.

Roll coating analysis of a second-grade material

2017 ◽  
Vol 34 (3) ◽  
pp. 232-255 ◽  
Author(s):  
M Zahid ◽  
MA Rana ◽  
AM Siddiqui
Keyword(s):  
Coating Thickness ◽  
Fluid Model ◽  
Power Input ◽  
Control Flow ◽  
Second Grade ◽  
Adiabatic Temperature Rise ◽  
Roll Coating ◽  
Grade Fluid ◽  
Grade Material ◽  
Separation Force

In this study, the lubrication approximation theory is used to provide numerical results in roll coating over a moving flat porous web. The rate of injection at the roll surface is assumed equal to the rate of suction on the web. The second-grade fluid is used, which reduces with appropriate modifications to the Newtonian fluid model. Results are obtained in such quantities as coating thickness, split location, pressure distribution, stresses, forces, the power input to the roll, and adiabatic temperature rise between the coating roll and the coated web. Some of these results are shown graphically. It is found that material parameters β and Reynolds number are the parameters to control flow rate, coating thickness separation points, separation force, power input, and pressure.

A theoretical analysis of the calendering of Ellis fluid

2016 ◽  
Vol 33 (2) ◽  
pp. 207-226 ◽  
Author(s):  
Muhammad Asif Javed ◽  
Nasir Ali ◽  
Muhammad Sajid
Keyword(s):  
Theoretical Analysis ◽  
Velocity Profile ◽  
Pressure Gradient ◽  
Pressure Distribution ◽  
Sheet Thickness ◽  
Power Input ◽  
Lubrication Approximation ◽  
Material Parameters ◽  
Operating Variables ◽  
Roll Separating Force

We present a theoretical analysis of calendering of Ellis fluid based on lubrication approximation. The equations governing the flow are nondimensionalized and solved to get closed form expressions of velocity and pressure gradient. Runge–Kutta algorithm is employed to compute the pressure distribution. The operating variables which are used in the calendering process, i.e. roll-separating force, power input to the rolls and exiting sheet thickness are calculated. The influence of the material parameters on the velocity profile, pressure gradient, pressure distribution and operating variables is shown graphically and discussed in detail.

scholarly journals Mathematical Analysis of Pseudoplastic Polymers during Reverse Roll-Coating

Polymers ◽  
2020 ◽  
Vol 12 (10) ◽  
pp. 2285
Author(s):  
Fateh Ali ◽  
Yanren Hou ◽  
Muhammad Zahid ◽  
Muhammad Afzal Rana
Keyword(s):  
Pressure Distribution ◽  
Flow Rate ◽  
Perturbation Technique ◽  
Power Input ◽  
Weissenberg Number ◽  
Point Of View ◽  
Governing Equations ◽  
Velocity Flow ◽  
Separation Force ◽  
Physical Quantities

This article presents a mathematical model and theoretical analysis of coating of a thin film of non-Newtonian polymers as they travel through a small space between two reverse-rotating rolls. The dimensionless forms of the governing equations are simplified with the help of the lubrication approximation theory (LAT). By using the perturbation technique, the analytical solutions for velocity, flow rate and pressure gradient were obtained. From an engineering point of view, some significant results such as thickness of the coated web, pressure distribution, separation points, separation force and power input were computed numerically. The effect of velocities ratio k and Weissenberg number We on these physical quantities is presented graphically; others are shown in tabular form. It is noted that the involved material parameters provide a mechanism to control the flow rate, pressure distribution, the thickness of coating, separation force and power input. Moreover, the separation point is shifted toward the nip region by increasing velocities ratio k.

Analysis of the lubrication approximation theory in the calendering/sheeting process of upper convected Jeffery’s material

2020 ◽  
pp. 875608792096254
Author(s):  
M Zahid ◽  
NZ Khan ◽  
AM Siddiqui ◽  
S Iqbal ◽  
A Muhammad ◽  
...  
Keyword(s):  
Pressure Distribution ◽  
Flow Rate ◽  
Approximation Theory ◽  
Material Parameter ◽  
Sheet Thickness ◽  
Power Input ◽  
Lubrication Approximation ◽  
Flow Equations ◽  
Velocity Flow ◽  
Roll Separating Force

This paper analyses an isothermal calendering for an upper convected Jeffery’s Material. Lubrication Approximation Theory (LAT) is applied to simplify the flow equations. Analytical solutions of velocity, flow rate, and pressure gradient are carried out. Outcomes of sheet thickness, detachment point, roll separating force, power input to the roll, and pressure distribution are obtained. The effects of some involved parameters are displayed through graphs and tables. It is noted that the material parameter is a controlling device for sheet thickness, flow rate, detachment point, roll separating force, power input, and the pressure distribution. We observed that as the material parameter increases, the detachment point increases which results in increased sheet thickness.

Influence of magnetohydrodynamics and heat transfer on the reverse roll coating of a Jeffrey fluid: A theoretical study

2021 ◽  
pp. 875608792110296
Author(s):  
Fateh Ali ◽  
Yanren Hou ◽  
Muhammad Zahid ◽  
MA Rana ◽  
Muhammad Usman
Keyword(s):  
Heat Transfer ◽  
Flow Rate ◽  
Coating Thickness ◽  
Gradient Flow ◽  
Power Input ◽  
Jeffrey Fluid ◽  
Governing Equations ◽  
Controlling Parameter ◽  
Separation Force ◽  
Velocity Pressure

The purpose of this article is to provide a mathematical model of magnetohydrodynamic (MHD) non-isothermal flow of an incompressible Jeffrey fluid as it goes through a minimal gap between the two counter rotating rolls. The dimensionless forms of governing equations are obtained by using appropriate dimensionless parameters. The LAT (lubrication approximation theory) is utilized to simplify the dimensionless form of governing equations. Analytical solutions for the velocity, pressure gradient, flow rate, Nusselt number and temperature distribution are presented. How the Jeffrey parameters, MHD and velocities ratio influence on the flow patterns and heat transfer rate are explored. Outcomes of some significant engineering quantities such as flow rate, power input, pressure distribution and roll separation force are obtained numerically in tabular form and some are displayed graphically. We found that the MHD parameter served as a controlling parameter for different engineering quantities like velocity, temperature, flow rate, and coating thickness. Moreover, the coating thickness on the web decreases by increasing the values of velocities ratio.

Ribbing Instability Analysis of Forward Roll Coating

2009 ◽  
Vol 25 (2) ◽  
pp. 167-175
Author(s):  
K. N. Lie ◽  
Y. M. Chiu ◽  
J. Y. Jang
Keyword(s):  
Surface Coating ◽  
Velocity Ratio ◽  
Rolling Process ◽  
Linear Stability Theory ◽  
Wave Number ◽  
Roll Coating ◽  
Critical Capillary Number ◽  
Roll Velocity ◽  
Forward Roll Coating ◽  
Numerical Program

AbstractThe ribbing instability of forward roll coating is analyzed numerically by linear stability theory. The velocity ratio of two rolls is fixed to be 1/4 for practical surface coating processes. The base flows through the gap between two rolls are solved by use of powerful CFD-RC software package. A numerical program is developed to solve the ribbing instability for the package is not capable of solving the eigenvalue problem of ribbing instability. The effects of the gap between two rolls, flow viscosity, surface tension and average roll velocity on ribbing are investigated. The criterion of ribbing instability is measured in terms of critical capillary number and critical wave number. The results show that the surface coating becomes stable as the gap increases or as the flow viscosity decreases and that the surface coating is more stable to the ribbing of a higher wave number than to the ribbing of a lower wave number. The effect of average roll velocity is not determinant to the ribbing instability. There are optimum and dangerous velocities for each setup of rolling process.

Mathematical model of pressure distribution in a main pipeline during pumping with the use of drug reducing agents, taking into account their degradation

2021 ◽  
pp. 396-406
Author(s):  
Мурсалим Мухутдинович Гареев ◽  
Марат Иозифович Валиев ◽  
Филипп А. Карпов
Keyword(s):  
Experimental Data ◽  
Mathematical Model ◽  
Pressure Distribution ◽  
Flow Rate ◽  
Pressure Head ◽  
Petroleum Products ◽  
Reducing Agents ◽  
Main Pipeline ◽  
Main Mode ◽  
Pipeline Pressure

Путевая деградация противотурбулентных присадок (ПТП) может стать причиной изменения основных параметров режима магистрального трубопровода - давления и расхода - относительно установившихся значений и осложнить контроль их отклонений от нормативных показателей. При этом до настоящего момента отсутствовала методика расчета режимов перекачки при использовании ПТП с учетом деградации. Авторами была поставлена цель по разработке методики для математического описания распределения давления в трубопроводе с учетом путевой деградации присадки, а также при различных концентрациях ПТП. Для достижения указанной цели предлагается дополнить уравнение баланса напоров с учетом эмпирической зависимости эффективности присадки от длины трубопровода. При расчетах давления в промежуточных точках трассы предлагается использовать данные опытно-промышленных испытаний по изменению эффективности ПТП. Для иллюстрации применения методики рассматриваются примеры перекачки нефти и нефтепродуктов с добавлением присадок в различных концентрациях. На основании экспериментальных данных получена адекватная математическая модель снижения эффективности ПТП по длине магистрального трубопровода для различных концентраций вводимой присадки. Path degradation of drug reducing agents (DRA) can cause changes in the main mode parameters of the main pipeline; pressure and flow rate, relative to the stable values, and complicate the adjustment of their deviations from the standard indicators. At the same time, up until now there has been no methodology for calculating pumping modes when using DRA that takes degradation into account. The authors set a goal to develop a methodology to mathematically describe the pressure distribution in the pipeline, taking into account the path degradation of the agent, as well as the parameters at different DRA concentrations. To achieve this goal, it is proposed to supplement the equation of the pressure head balance with the empirical dependency of agent efficiency on the length of the pipeline. When calculating the pressure at intermediate points of the route, it is proposed to use the pilot run data on the change in the DRA efficiency. To illustrate the application of the methodology, examples of pumping oil and petroleum products with added agents in various concentrations are discussed. On the basis of the experimental data, an adequate mathematical model of the DRA efficiency reduction along the length of the main pipeline for different concentrations of introduced agent was obtained.

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