Classical Graetz Problem Research Articles

Thermal entrance problem for blood flow inside an axisymmetric tube: The classical Graetz problem extended for Quemada's bio-rheological fluid with axial conduction.

The heat-conducting nature of blood is critical in the human circulatory system and features also in important thermal regulation and blood processing systems in biomedicine. Motivated by these applications, in the present investigation, the classical Graetz problem in heat transfer is extended to the case of a bio-rheological fluid model. The Quemada bio-rheological fluid model is selected since it has been shown to be accurate in mimicking physiological flows (blood) at different shear rates and hematocrits. The steady two-dimensional energy equation without viscous dissipation in stationary regime is tackled via a separation of variables approach for the isothermal wall temperature case. Following the introduction of transformation variables, the ensuing dimensionless boundary value problem is solved numerically via MATLAB based algorithm known as bvp5c (a finite difference code that implements the four-stage Lobatto IIIa collocation formula). Numerical validation is also presented against two analytical approaches namely, series solutions and Kummer function techniques. Axial conduction in terms of Péclet number is also considered. Typical values of Reynolds number and Prandtl number are used to categorize the vascular regions. The graphical representation of mean temperature, temperature gradient, and Nusselt numbers along with detail discussions are presented for the effects of Quemada non-Newtonian parameters and Péclet number. The current analysis may also have potential applications for the development of microfluidic and biofluidic devices particularly which are used in the diagnosis of diseases in addition to blood oxygenation technologies.

A note on classical Graetz problem based on Cattaneo–Christov heat flux model

A note on classical Graetz problem based on Cattaneo–Christov heat flux model

Heat transport in shear-driven flow with axial conduction

The classical Graetz problem of fluid entering a long channel possessing different temperature from that of fluid is extended here to the case where the upper wall of the channel is not stationary. Formulation of the current problem incorporates the simple Couette flow driven by the channel movement and taking into account the axial heat conduction, besides the usual convection. Analytical solutions in the thermally developing regime regarding the temperature distribution as well as the heat transfer rates from both the upper moving and the lower fixed channel walls are then obtained by means of the series expansion method making use of the proper orthogonal family of eigenfunctions. Simple approximate formulae for the rates of heat transfer at the channel entrance valid for large Peclet numbers and at the far downstream locations valid for all Peclet numbers are further derived. More simplistic outcomes are also provided when the axial pressure gradient is assumed to be constant. Results are then discussed from low to high Peclet numbers, which are in compliant with the well-known results available in the literature for the channel flow within stationary walls. On the other hand, the rate of heat transfer from the moving wall is shown to be of high importance when Peclet number increases, which requires special attention in technological applications.

Mathematical modelling of classical Graetz–Nusselt problem for axisymmetric tube and flat channel using the Carreau fluid model: a numerical benchmark study

Abstract The thermal entrance problem (also known as the classical Graetz problem) is studied for the complex rheological Carreau fluid model. The solution of two-dimensional energy equation in the form of an infinite series is obtained by employing the separation of variables method. The ensuing eigenvalue problem (S–L problem) is solved for eigenvalues and corresponding eigenfunctions through MATLAB routine bvp5c. Numerical integration via Simpson’s rule is carried out to compute the coefficient of series solution. Current problem is also tackled by an alternative approach where numerical solution of eigenvalue problem is evaluated via the Runge–Kutta fourth order method. This problem is solved for both flat and circular confinements with two types of boundary conditions: (i) constant wall temperature and (ii) prescribed wall heat flux. The obtained results of both local and mean Nusselt numbers, fully developed temperature profile and average temperature are discussed for different values of Weissenberg number and power-law index through graphs and tables. This study is valid for typical range of Weissenberg number W e ≤ 1 $\left(We\le 1\right)$ and power-law index n < 1 $\left(n{< }1\right)$ for shear-thinning trend while n > 1 $\left(n{ >}1\right)$ for shear-thickening behaviour. The scope of the present study is broad in the context that the solution of the said problem is achieved by using two different approaches namely, the traditional Graetz approach and the solution procedure documented in M. D. Mikhailov and M. N. Ozisik, Unified Analysis and Solutions of Heat and Mass Diffusion, New York, Dover, 1994.

Toward an Improved Hybrid Model for Simulating Continuous Flow Microwave Heating of Water

This paper introduces an improved hybrid (numerical-analytical) model for simulating microwave (MW) heating of laminar duct flow. The proposed procedure links numerical results to analytical calculations, providing a tool for accurate prediction of the bulk temperature distribution in a relatively reduced computation time, enhancing the design of MW heating of continuous flow water systems. The hybrid solution was obtained by first numerically solving the Maxwell equations in correspondence of an average dielectric permittivity; discrete values of the cross-section averaged heat generation arising from the numerical solution were first corrected by a suitable weighting function and then interpolated by a function resulting from the discrete Fourier series. The momentum and the energy equations fed by the above calculated heat generation distribution were uncoupled from Maxwell equations. The problem being linear, the analytical thermal solution was sought as the sum of two partial solutions, each one affected by a single non-homogeneity. The former solution turned out to be the classical Graetz problem, while the second one, driven by the heat generation, was solved in closed form by the variation of parameters method. On the other hand, the same problem was solved by a complete numerical approach in order to have a reference solution; thus, the Maxwell equations and the energy balance for the flowing fluid were simultaneously solved considering temperature dependent dielectric permittivity. Fully developed velocity, thermally developing conditions and no phase transition during the heating process were assumed for both the hybrid and the numerical solution. The availability of the reference solution allowed to prove the substantial enhancement of the hybrid solution in describing the bulk temperature distribution along the pipe when compared to the one related to the classical constant properties approach. Results, presented and discussed for different inlet velocities, show that increasing velocities provide a better agreement due to the smoothing effect realized by higher frequencies fluctuations in heat generation distribution felt by the flowing fluid.

Thermal and rheological effects in a classical Graetz problem using a nonlinear Robertson‐Stiff fluid model

AbstractThe present article elaborates the Graetz problem for the Robertson‐Stiff fluid model with imposed iso‐thermal conditions. The closed‐form expression of Robertson‐Stiff fluid velocity is obtained. Employing the classical separation of variables approach, the energy equation of the said problem is reduced into an eigenvalue problem. The solution of the eigenvalue problem is developed numerically via the MATLAB built‐in algorithm BVP4C. The constants appearing in series solutions are computed by Simpson's rule. The special case of this analysis with appropriate scaling is also applicable for the Bingham, power‐law, and Newtonian fluid models. The impact of the dissipation function on Nusselt numbers and mean temperature is also considered. The pictorial representation of average temp7erature and Nusselt number are discussed in the presence of the plug radius, power‐law index, and Brinkman number. It is observed that the presence of the plug radius and power‐law index delay the prevalence of fully developed conditions for the Nusselt number. Moreover, the local Nusselt number for channel confinement attains higher values as compared with tube confinement. The present investigation has numerous applications in the field of engineering, nanotechnology, biomedical sciences, and development of several thermal types of equipment or microfluidic devices.

Extended lumped formulation for laminar forced convection in tubes exposed to forced convection normal flows: Application to aerial and submerged pipelines

The present study addresses forced convection heat transfer of an internal viscous fluid entering a tube with fully developed laminar velocity and uniform entrance temperature. The internal viscous fluid exchanges heat with an external viscous fluid flowing with constant velocity normal to the tube. Both fluids are assumed to have constant or nearly constant thermophysical properties. The description corresponds to a generalized Graetz problem with heat convection boundary condition. Contrary to the tradition in vogue, the differential formulation of the two-dimensional energy equation combined with the method of separation of variables and the ensuing Sturm–Liouville theory is not employed to solve analytically the dual convection problem. Rather, the objective of the study is to derive, develop and appraise a novel approximate, analytical method entitled extended lumped formulation, which involves a first order differential equation. The primary input in the formulation is the length averaged, mean Nusselt number distribution for a diminished problem consisting in a viscous fluid with fully developed velocity and uniform temperature with constant tube wall temperature, namely the classical Graetz problem. The approximate, analytical solution of the extended lumped equation is expressed by a single exponential function that contains the generalized Stanton number, which is easily evaluated by hand on the back-of-the-envelope with an inexpensive calculator. This step brings forth the developing mean bulk temperature of the internal viscous fluid for the entire spectrum of the modified Biot numbers (0 < Bi < ∞). Subsequently, the total heat transfer is easily determined with an algebraic equation that shares the same simplicity. The agreement between the approximate, analytical mean bulk temperature with the counterpart exact, analytical mean bulk temperature based on the generalized Graetz series is excellent for all Bi values ranging from 0.1 to 100.

Theoretical analysis of thermal entrance problem for blood flow: An extension of classical Graetz problem for Casson fluid model using generalized orthogonality relations

An analysis of classical Graetz problem is carried out for the case of fluid obeying the Casson constitutive equation. The considered model physically corresponds to the thermal entry flow of blood in a duct. The governing equation for the premeditated problem is investigated by employing the separation of variables approach in conjunction with the MATLAB built in package bvp4c for the calculation of the eigenvalues and related numerical solution of the eigenvalue problem. The solution is computed for the case of uniform surface temperature boundary condition for both flat and circular geometries. The axial diffusion and viscous dissipation effects on temperature field are also taken into account. The expressions of bulk mean temperature and Nusselt number are presented and discussed in terms of the main effect brought by the yield stress parameter, Peclet number and Brinkman number. It is found that both local and mean Nusselt numbers for blood enhance considerably with the increase of dimensionless plug radius and Brinkman number. In contrast, the thermal entrance length reduces with the rise of Peclet number. The results of present analysis have potential applications in development of nano fluidic, microfluidic and bio-medical devices used in haemodialysis and oxygenation.

The Graetz Problem for the Ellis Fluid Model

Abstract The determination of temperature and auxiliary quantities such as local and average Nusselt numbers for thermally developing flow is referred as the Graetz problem. In the classical Graetz problem, the fluid entering the tube or channel is Newtonian in nature. Here, an extension of the classical Graetz problem is presented by assuming that the fluid entering the tube or channel obeys the Ellis constitutive equation. The energy equation for the considered problem is solved using the separation of variables technique supplemented with the MATLAB routine bvp4c for computation of the eigenvalues and numerical solution of the associated Sturm-Liouville boundary value problem. The problem is solved for two types of thermal boundary conditions, namely, uniform surface temperature and uniform surface heat flux for both flat and circular geometries. Expressions for bulk mean temperature and local and average Nusselt numbers are presented and discussed through tables and graphs.

Exact Graetz problem solution by using hypergeometric function

This paper proposes an exact solution of the classical Graetz problem in terms of an infinite series represented by a nonlinear partial differential equation considering two space variables, two boundary conditions and one initial condition. The mathematical derivation is based on the method of separation of variables whose several stages were illustrated to reach the solution of the Graetz problem.A MATLAB code was used to compute the eigenvalues of the differential equation as well as the coefficient series. In addition, the analytical solution was compared to the numerical values obtained previously by Shah and London. It is important to note that the analytical solution is in good agreement with published numerical data.

An analytical method for solving exact solutions of the convective heat transfer in fully developed laminar flow through a circular tube

AbstractThis paper proposes an exact solution to the classical Graetz problem in terms of an infinite series represented by a nonlinear partial differential equation considering two space variables, two boundary conditions, and one initial condition. The mathematical derivation utilized the separation of variables method where several stages were completed to reach the solution to the Graetz problem.

Laminar forced convective slip flow in a microduct with a sinusoidally varying heat flux in axial direction

In this study, steady laminar forced convection slip flow in a microtube subjected to an axially varying heat flux is investigated numerically using the finite volume method. The classical Graetz problem is considered, which can be named as micro-Graetz problem for the microscale condition. The viscous dissipation effect is included in the analysis. The problem studied can be also named as micro-Graetz–Brinkman problem. The slip flow regime is considered by incorporating the velocity slip and temperature jump conditions at wall. The effects of rarefaction, the viscous dissipation and the dimensionless amplitude of the axially varying or periodic heat flux on the local and mean Nusselt numbers as well as on the wall and bulk temperatures are obtained for some specific ranges of corresponding parameters. These effects are found to be interactive. It is disclosed that the mean Nusselt number decreases with an increase in the amplitude.

Analysis of heat transfer and entropy generation for a low-Peclet-number microtube flow using a second-order slip model: an extended-Graetz problem

The classical Graetz problem, which is the problem of the hydrodynamically developed, thermally developing laminar flow of an incompressible fluid inside a tube neglecting axial conduction and viscous dissipation, is one of the fundamental problems of internal-flow studies. This study is an extension of the Graetz problem to include the rarefaction effect, viscous dissipation term and axial conduction with a constant wall temperature thermal boundary condition. The energy equation is solved to determine the temperature field analytically using general eigenfunction expansion with a fully developed velocity profile. To analyze the low-Peclet-number nature of the flow, the flow domain is extended from $$-\infty $$ to $$+\infty $$ . To model the rarefaction effect, a second-order slip model is implemented. The temperature distribution, local Nusselt number, and local entropy generation are determined in terms of confluent hypergeometric functions. This kind of theoretical study is important for a fundamental understanding of the convective heat transfer characteristics of flows at the microscale and for the optimum design of thermal systems, which includes convective heat transfer at the microscale, especially operating at low Reynolds numbers.

Numerical Study of Fluid Flow and Heat Transfer in Partially Heated Microchannels Using the Explicit Finite Volume Method

AbstractFluid flow behavior and heat transfer characteristics of pressure driven flow in partially heated microchannels have been investigated using the explicit Finite Volume Method (FVM). The developed flow solver utilizes the compressible Navier‐Stokes (NS) system of equations with Maxwell's first order slip boundary condition and the corresponding temperature jump relation derived by von Smoluchowski to achieve the solutions. The effects of Knudsen number and gas species have been studied for partially heated microchannels similar to the classical Graetz problem. The existence of nonlinear pressure distribution and dominant heat diffusion for higher Knudsen number regime are predicted by the solution. The fluid flow and heat transfer characteristics of the partially heated microchannels greatly depend on the transport properties of the medium under similar flow conditions.

The effect of gaseous slip on microscale heat transfer: An extended Graetz problem

On the basis of Langmuir’s theory of adsorption of gases on solids, the effect of temperature jump on microscale heat transfer is investigated. A mathematical model, extended from the classical Graetz problem, is developed to analyze convective heat transfer in a microtube in various slip-flow regimes. The surface slip corrections are made by employing the Langmuir model, as well as the conventional Maxwell model. The effects of axial heat conduction are also investigated by extending the finite integral transform technique to the slip-flow case. We show that the Langmuir model always predicts a reduction in heat transfer with increasing rarefaction, as does the Maxwell model, except when the energy accommodation coefficient is relatively much smaller than that for momentum accommodation. This implies that, for most physical applications, the Reynolds analogy between heat transfer and momentum transfer is preserved in slip-flow regimes with low Mach numbers.

Heat and mass transfer coefficients in catalytic monoliths

We analyze the classical Graetz problem in a tube with an exothermic surface reaction and show that the heat(mass) transfer coefficient is not a continuous function of the axial position and jumps from one asymptote to another at ignition/extinction points. We show that the steady-state heat(mass) transfer coefficient is not a unique function of position in parameter regions in which the Graetz problem with surface reaction has multiple solutions. We also analyze the more general two-dimensional model (with axial conduction/diffusion included and Danckwerts boundary conditions) and show that for fixed values of the reaction parameters, the heat(mass) transfer coefficient has three asymptotes. Unlike the Graetz problem, in this case the heat(mass) transfer coefficient is always finite and bounded at the inlet and is given by a new asymptote. We present analytical expressions for all three asymptotes for the case of flat and parabolic velocity profiles. It is also shown that in catalytic monoliths, ignition/extinction may often occur in the entry region and hence the local transfer coefficients and not the average values proposed in the literature determine the ignition/extinction behavior. Finally, we use the new results to develop and analyze an accurate one-dimensional two-phase model of a catalytic monolith with position dependent heat and mass transfer coefficients and determine analytically the dependence of the ignition/extinction locus on various design and operating parameters.

Slip-flow heat transfer in circular tubes

The influence of rarefaction on heat transfer in circular tubes is studied. A spatial rescaling factor ρ s, which is a measure of rarefaction through its dependence on the Knudsen number, is introduced to identify similarities with the classical Graetz problem. It is found that heat transfer depends both on the degree of rarefaction and on the surface accommodation coefficients. The temperature jump at the wall, ignored in recent investigations, is found to be of essential importance in the heat transfer analysis. A novel uniform asymptotic approximation to high-order eigenfunctions is derived that allows an efficient and accurate determination of the region close to the entrance.

PDE boundary conditions from minimum reduction of the PDE

Convection-diffusion partial differential equations (PDEs) are second order in the direction of flow, and therefore require two boundary conditions with respect to the spatial independent variable in that direction. The entering or “inflow” boundary condition is usually obvious. The exiting or “outflow” boundary condition is not so obvious. We propose reducing the PDE by one order in the direction of flow to produce the outflow boundary condition. This procedure is illustrated with a dynamic version of the classical Graetz problem in heat transfer. The proposed procedure has the advantages of physically meaningful boundary conditions which produce numerical solutions of good accuracy, and ease of implementation in a method of lines code.

ASYMPTOTIC AND NUMERICAL SOLUTIONS FOR THERMALLY DEVELOPING FLOWS OF NEWTONIAN AND NON-NEWTONIAN FLUIDS IN CIRCULAR TUBES WITH UNIFORM WALL TEMPERATURE

Abstract Methods that predict heat transfer rates in thermally developing flows, important in engineering design, are often compared with the classical Graetz problem. Surprisingly, numerical solutions to this problem generally do not give accurate results in the entrance region. This inaccuracy stems from the existence of a singularity at the tube inlet. By adopting a fundamental approach based upon singular perturbation theory, the heat transfer process in the tube entrance has been analyzed to bring out the asymptotic boundary layer structure of the generalized problem with non-Newtonian flows. Using a standard finite difference method with only 21 radial nodes, results within 0.3% of the exact solution to the Graetz problem (Newtonian limit of generalized power law fluid flows) are obtained. Compared with previous numerical solutions reported in the literature, these results are an order of magnitude improvement in the accuracy with an order of magnitude decrease in the required number of radial nodes...

Adaptive grid method for convection-diffusion equations

Numerical solutions to the classical Graetz problem in the entrance region of a pipe produce estimates of the Nusselt number which are greatly in error except when the uniform grid is extremely fine, leading to huge storage requirements. The use of adaptive grids, it is proposed, will resolve the twin problem of excessive storage space and inaccuracy of results. Therefore, two adaptive grids using different methods of generating the weighting function are developed and tested in this work. One of the adaptive grids produced results that compared favourably with the exact solution with as few as 11 grid points. With the uniform grid, as many as 301 grid points would have been required.