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In the present work, heat transfer and fluid flow characteristics for both Newtonian and non-Newtonian fluids in tube-in-tube helical coil (TTHC) heat exchangers have been investigated numerically. The various TTHC heat exchanger configurations studied are (1) parallel and (2) counter flow, with and without baffles. The power law index (n) and Dean number (NDe) are varied from 0.5 to 1.25 and 50 to 500, respectively. Further, two different models have been proposed to predict the friction factor (f) and Nusselt number (Nu) in TTHC. It is observed that f and Nu in the TTHC heat exchanger with baffles in the annulus is higher as compared to without baffles. Furthermore, at low Prandtl number the baffles have significant influence on heat transfer, while at high Prandtl number flow configuration has high significance. Copyright © 2020 American Chemical Society.

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Department of Chemical Engineering

Industrial \& Engineering Chemistry Research

Heat transfer and fluid flow characteristics for Newtonian and non-Newtonian fluids in a tube-in-tube helical coil heat exchanger

Physics of Fluids

Flow and stability of a gravity-driven thin film over a locally heated porous wall

Abstract.: Two-dimensional steady-state solutions and their stability analysis are presented for a gravity-driven thin film of a thermoviscous liquid. The governing equations and boundary conditions are simplified using the lubrication approximation. The analytically obtained film thickness evolution equation consists of various dimensionless parameters such as the Marangoni number, Biot number and thermoviscosity number. The viscosity of the liquid is assumed as an exponential function of temperature. The viscosity decreases within the liquid film as the temperature increases. Due to localized heating interfacial temperature gradients generate surface tension gradient which results into thermocapillary or Marangoni stress. The Marangoni stress opposes the fluid flow at the leading edge of heater leading to an increase in the film thickness locally. This locally thick structure becomes unstable beyond critical values of the parameters that leads to formation of rivulets in the transverse direction. Using the linear stability analysis it is found that the Marangoni stress and the thermoviscous effect have a destabilizing effect on the thin-film flow. At much higher values of the thermoviscosity number another mode of instability appears which is known as thermocapillary instability which leads to oscillating film profiles. For streamwise perturbations, the destabilizing effect of the thermoviscosity number for localized and uniform heating remains consistent. Graphical abstract: [Figure not available: see fulltext.]. © 2019, EDP Sciences, Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature.

The European Physical Journal E

Stability analysis of a gravity-driven thermoviscous liquid film flowing over a heated flat substrate

Steady two-dimensional solutions and their stability analysis are presented for thin film of a thermoviscous liquid flowing inside a cylinder rotating about its horizontal axis. The inner surface of the cylinder is either uniformly hotter or colder than the enveloping air. The mass, momentum, and energy equations are simplified using thin-film approximation. The analytically obtained film thickness evolution equation consists of various dimensionless parameters such as gravitational number, Bond number, Biot number, thermoviscosity number, and Marangoni number. The viscosity of the liquid is considered as an exponential function of temperature. The viscosity increases (decreases) within the film thickness away from the inner surface of the cylinder when the surface is uniformly hotter (colder) than the atmosphere. For hotter (colder) surface, the film thickness on the rising side decreases (increases) when convective heat transfer at the free surface is increased. The surface tension gradient at the free surface generates Marangoni stress that has a destabilizing (stabilizing) effect on the thin film flow in the case of a hotter (colder) cylinder. The thermoviscosity number stabilizes (destabilizes) the flow on a heating (cooling) surface and this effect increases with an increase in the heat transfer at the free surface. For a hotter surface and in the presence of Marangoni stress, the convective heat transfer at the interface has the destabilizing effect for small values of the Biot number and assumes a stabilizing role for larger values. Non-linear simulations show consistency with the linear stability analysis. © 2018 Author(s).

Hydrodynamic stability of thermoviscous liquid film inside a rotating horizontal cylinder: Heating and cooling effects

APS March Meeting Abstracts 2018, E17.

Effect of Thermocapillary stress on the Stability of a Non-isothermal Thin Liquid Film Flowing Inside a Heated Rotating Horizontal Cylinder

Two-dimensional base state solutions for rimming flows and their stability analysis to small axial perturbations are analyzed numerically. A thin liquid film which is uniformly covered with an insoluble surfactant flows inside a counterclockwise rotating horizontal cylinder. In the present work, a mathematical model is obtained which consists of coupled thin film thickness and surfactant concentration evolution equations. The governing equations are obtained by simplifying the momentum and species transport equations using the thin-film approximation. The model equations include the effect of gravity, viscosity, capillarity, inertia, and Marangoni stress. The concentration gradients generated due to flow result in the surface tension gradient that generates the Marangoni stress near the interface region. The oscillations in the flow due to inertia are damped out by the Marangoni stress. It is observed that the Marangoni stress has stabilizing effect, whereas inertia and surface tension enhance the instability growth rate. In the presence of low diffusion of the surfactant or large value of the Péclet number, the Marangoni stress becomes more effective. The analytically obtained eigenvalues match well with the numerically computed eigenvalues in the absence of gravity. © 2017 Author(s).

Stability analysis of rimming flow inside a horizontally rotating cylinder in the presence of an insoluble surfactant

Abstract.: The stability of a thin electrolyte liquid film driven by gravity over a vertical substrate is presented. A film thickness evolution equation is derived and solved numerically. The substrate is non-uniformly heated from below which is modeled by imposing a temperature profile at the liquid-solid interface. The electrohydrodynamics is included in the model with Maxwell’s stress tensor. The governing flow and energy equations are simplified using the lubrication approximation. The Poisson-Boltzmann equation with Debye-Hückel approximation is used for the potential which is generated inside the film due to a charged layer at the liquid-solid interface. The positive temperature gradient at the substrate leads to the formation of a thermocapillary ridge due to an opposing Marangoni stress. This thermocapillary ridge becomes unstable beyond critical parameters related to Marangoni stress and convective energy loss at the free surface. The electroosmotic flow has no effect on the base profile of the film, but enhances its instability. A parameter space is presented delineating the stable and unstable regimes for the film dynamics. Graphical abstract: Figure not available: see fulltext. © 2017, EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg.

Stability of a gravity-driven thin electrolyte film flowing over a heated substrate

Response to "Comment on 'Stability analysis of the rimming flow inside a uniformly heated rotating horizontal cylinder'" [Phys. Fluids 29, 089101 (2017)]

The stability analysis is presented for a thin viscous liquid film flowing inside a uniformly heated horizontal cylinder that is rotating about its axis. The free surface evolution equation for the liquid-gas interface is obtained by simplifying the Navier-Stokes and energy equations within the lubrication approximation.Various dimensionless numbers are obtained that quantify the effect of gravity, viscous drag, inertia, surface tension, and thermocapillary stress. The film thickness evolution equation is solved numerically to obtain two-dimensional, steady state solutions neglecting axial variations. A liquid pool forms at the bottom of the cylinder when gravity dominates other forces. This liquid pool is shifted in the direction of rotation when inertia or viscous drag is increased. Small axial perturbations are then imposed to the steady solutions to study their stability behavior. It is found that the inertia and capillary pressure destabilize whereas the gravity and thermocapillary stress stabilize the rimming flow. The influence of Marangoni number is reported by computing the stable and unstable parametric regions. Thicker films are shown to be more susceptible to become unstable. Published by AIP Publishing.

Stability analysis of the rimming flow inside a uniformly heated rotating horizontal cylinder

The dynamics and linear stability of a gravity drive thin film flowing over non-uniformly heated porous substrate are studied. A governing equation for the evolution of film-thickness is derived within the lubrication approximation. Darcy-Brinkman equation is used to model flow in the porous medium along with a tangential stress-jump condition at the interface of the porous layer and the fluid film. A temperature profile is imposed at the solid wall to model an embedded heater beneath the porous layer. At the upstream edge of the heater, an opposing thermocapillary stress at the liquid-air interface leads to the formation of a thermocapillary ridge. The ridge becomes unstable beyond a critical Marangoni number leading to the formation of rivulets that are periodic in the spanwise direction. Increase in the values of parameters such as Darcy number, stress jump coefficient, and porosity is shown to have stabilizing effect on the film dynamics. The critical Marangoni number is shown to increase monotonically with Darcy number for various values of porosity. At large values of stress-jump coefficient, a non-monotonic variation in critical Marangoni number versus Darcy number is shown. A correlation is developed numerically for the ratio of critical Marangoni number at large Darcy number to that for a non-porous substrate as a function of porosity and thickness of the porous substrate. A transient growth analysis is carried out followed by non-linear stability analysis. The non-modal growth is found to be negligible thus indicating that the eigenvalues are physically determinant. © 2016 AIP Publishing LLC.