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The purpose of this study is to solve the key problem of the residual stress of strip in buckling and deviation during the continuous annealing process (CAP).

Considering the one-to-one correspondence between deformation and residual stress, the strip was divided into multiple elements. To obtain the total deformation of each element, the influence model of some factors on the deformation of steel strip element was established. Then, according to the constitutive equation of residual stress and deformation, and the deformation coordination relation between strip elements, a set of calculation model, which is suitable for residual stress of strip during CAP was established, the model precision was verified by finite element method simulation, and the influences on residual stress of strip were also analyzed.

The weighted calculation of comparison between analytical results and simulation shows that the accuracy of the analytical method is within 10 per cent, which fully meets the requirements.

The different residual stress of steel strip can be displayed quickly and intuitively in the synchronous process segment according to this model, which is of great importance to the prevention of strip buckling and deviation.

Continuous annealing (CA) units usually lack a physical shapemeter; consequently, real-time display and closed-loop control of the strip shape are impossible to achieve.

A shape model for the CA process is established in this study. Specifically, a virtual shapemeter and closed-loop control system based on the advanced parameter acquisition system and information transmission of CA units are developed in C++ programming language. This system realises real-time dynamic shape display, closed-loop control and shape prediction by collecting raw data of steel coils and parameters during CA.

Field test results show that the shape predicted by the virtual shapemeter coincides with the measured shape by over 90 per cent, which fully meets the precision requirement of industrial applications.

Moreover, shape quality is effectively improved without increasing hardware investments.

The purpose of this paper is to investigate an analytical modeling for the thermoelastic buckling behavior of functionally graded (FG) rectangular plates (FGM) under thermal loadings. The material properties of FGM are assumed to vary continuously through the thickness of the plate, according to the simple power-law distribution. Derivations of equations are based on novel refined theory using a new hyperbolic shear deformation theory. Unlike other theories, there are only four unknown functions involved, as compared to five in other shear deformation theories. The theory presented is variationally consistent and strongly similar to the classical plate theory in many aspects. It does not require the shear correction factor, and gives rise to the transverse shear stress variation so that the transverse shear stresses vary parabolically across the thickness to satisfy free surface conditions for the shear stress. In addition, numerical results for a variety of FG plates with simply supported edge are presented and compared with those available in the literature. Moreover, the effects of geometrical parameters of dimension the length to width aspect ratio (a/b), the plate width to thickness ratio (b/h), and material properties index (k) on the FGM buckling temperature difference are determined and discussed.

In the current paper, the application of the refined theory proposed by Shimpi is based on the assumption that the in-plane and transverse displacements consist of bending and shear components in which the bending components do not contribute toward shear forces and, likewise, the shear components do not contribute toward bending moments. The most interesting feature of this theory is that it accounts for a quadratic variation of the transverse shear strains across the thickness, and satisfies the zero traction boundary conditions on the top and bottom surfaces of the plate without using shear correction factors. It is extended to the analysis of buckling behavior of ceramic-metal FG plates subjected to the three types of thermal loadings, namely; uniform temperature rise, linear temperature change across the thickness, and nonlinear temperature change across the thickness. The material properties of the FG plates are assumed to vary continuously through the thickness of the plate, according to the simple power-law distribution. Numerical results for a variety of FG plates with simply supported edges are given and compared with the available results, wherever possible. Additionally, the effects of geometrical parameters and material properties on the buckling temperature difference of FGM plates are determined and discussed.

Unlike any other theory, the theory presented gives rise to only four governing equations. Number of unknown functions involved is only four, as against five in case of simple shear deformation theories of Mindlin and Reissner (first shear deformation theory). The plate properties are assumed to be varied through the thickness following a simple power-law distribution in terms of volume fraction of material constituents. The theory presented is variationally consistent, does not require shear correction factor, and gives rise to transverse shear stress variation such that the transverse shear stresses vary parabolically across the thickness satisfying shear stress free surface conditions.

To the best of the authors’ knowledge, there are no research works for thermal buckling analysis of FG rectangular plates based on new four-variable refined plate theory (RPT). The novelty of this paper is extended the use of the above-mentioned RPT with the addition of a new function proposed by Shimpi for thermal buckling analysis of plates made of FG materials. Unlike any other theory, the number of unknown functions involved is only four, as against five in the case of other shear deformation theories. The theory takes account of transverse shear effects and parabolic distribution of the transverse shear strains through the thickness of the plate, hence it is unnecessary to use shear correction factors. The plates subjected to the two types of thermal loadings, namely; uniform temperature rise and nonlinear temperature change across the thickness. Numerical results for a variety of FG plates with simply supported edges are given and compared with the available results.

The purpose of this paper is to investigate the effects of thermal radiation and viscous dissipation on steady natural convection flow of a viscous incompressible fluid along a uniformly moving infinite vertical porous plate with Newtonian heating in the presence of transverse magnetic field. The governing non-linear boundary layer equations are solved by using homotopy analysis method (HAM). The effects of various system parameters on velocity and temperature fields are discussed graphically, and the numerical values for skin friction and Nusselt number are presented in tabular form.

The problem is formulated using the Boussinesq approximation under the effects of thermal radiation and transverse magnetic field. The resulting coupled system of non-linear differential equations is solved using HAM with appropriate boundary conditions for Newtonian heating of the plate. HAM is a powerful method which provides rapidly converging series solution for the velocity and temperature fields. The effects of Prandtl number, Grashof number, suction parameter, magnetic field parameter, radiation parameter and Eckert number on the fluid velocity, temperature, skin friction and Nusselt number have been investigated.

The HAM solution has been successfully applied to find the converging series solution for velocity and temperature fields in terms of pertinent system parameters. Comparison of the exact solution results agree well with the HAM solution results in the absence of Eckert number and this indicates that the HAM solutions are accurate. It is found that the velocity and temperature profiles decreases with the increase of thermal radiation and suction parameters. An increase in the magnetic field parameter leads to a rise in the fluid temperature and fall in the fluid velocity.

The present analysis is limited to steady state laminar natural convection flow only. Unsteady natural- /mixed-convection laminar flow in the presence of thermal radiation, chemical reaction and transverse magnetic field will be investigated in a future work.

The study provides very useful information for heat transfer engineers to understand the heat transfer rate when the moving vertical porous surface temperature is not known a prior. The present results have immediate relevance in the design of nuclear reactors where vertical moving porous plates are using as control rods.

The present research work is relatively original and illustrates the effects of thermal radiation, viscous dissipation and transverse magnetic field on natural convection flow past a uniformly moving infinite vertical porous plate with Newtonian heating.

This paper aims to study the thermal and thermo-hydraulic performances of ferro-nanofluid flow in a three-dimensional trapezoidal microchannel heat sink (TMCHS) under uniform heat flux and magnetic fields.

To investigate the effect of direction of Lorentz force the magnetic field has been applied: transversely in the x direction (Case I);transversely in the y direction (Case II); and parallel in the z direction (Case III). The three-dimensional governing equations with the associated boundary conditions for ferro-nanofluid flow and heat transfer have been solved by using an element-based finite volume method. The coupled algorithm has been used to solve the velocity and pressure fields. The convergence is reached when the accuracy of solutions attains 10–6 for the continuity and momentum equations and 10–9 for the energy equation.

According to thermal indicators the Case III has the best performance, but according to performance evaluation criterion (PEC) the Case II is the best. The simulation results show by increasing the Hartmann number from 0 to 12, there is an increase for PEC between 845.01% and 2997.39%, for thermal resistance between 155.91% and 262.35% and ratio of the maximum electronic chip temperature difference to heat flux between 155.16% and 289.59%. Also, the best thermo-hydraulic performance occurs at Hartmann number of 12, pressure drop of 10 kPa and volume fraction of 2%.

The embedded electronic chip on the base plate generates heat flux of 60 kW/m2. Simulations have been performed for ferro-nanofluid with volume fractions of 1%, 2% and 3%, pressure drops of 10, 20 and 30 kPa and Hartmann numbers of 0, 3, 6, 9 and 12.

The authors obtained interesting results, which can be used as a design tool for magnetohydrodynamics micro pumps, microelectronic devices, micro heat exchanger and micro scale cooling systems.

Review of the literature indicated that there has been no study on the effects of magnetic field on thermal and thermo-hydraulic performances of ferro-nanofluid flow in a TMCHS, so far. In this three dimensional study, flow of ferro-nanofluid through a trapezoidal heat sink with five trapezoidal microchannels has been considered. In all of previous studies, in which the effect of magnetic field has been investigated, the magnetic field has been applied only in one direction. So as another innovation of the present research, the effect of applying magnetic field direction (transverse and parallel) on thermo-hydraulic behavior of TMCHS is investigated.

The purpose of this study is to address one-dimensional, unsteady heat conduction in a large plane wall exchanging heat convection with a nearby fluid under “small time” conditions.

The Transversal Method of Lines (TMOL) was used to reformulate the unsteady, one-dimensional heat conduction equation in the space coordinate and time into a transformed “quasi-steady”, one-dimensional heat conduction equation in the space coordinate housing the time as an embedded parameter. The resulting ordinary differential equation of second order with heat convection boundary conditions is solved analytically with the method of undetermined coefficients.

Semi-analytical TMOL dimensionless temperature profiles of compact form with/without regressed terms are obtained for the whole spectrum of Biot number (0 < Bi < ∞) in the “small time” sub-domain. In addition, a new “large time” sub-domain is redefined, that is, setting a smaller critical dimensionless time or critical Fourier number τ_cr = 0.18.

The computed dimensionless center, surface and mean temperature profiles in the large plane wall accounting for all Biot number (0 < Bi < ∞) in the “small time” sub-domain τ < τ_cr = 0.18 exhibit excellent quality while carrying reasonable relative errors for engineering applications. The exemplary level of accuracy indicates that the traditional evaluation of the center, surface and mean temperatures with the standard infinite series retaining a large number of terms is no longer necessary.

To study quantitatively the effects of combined temperature dependent thermodynamics and transport fluid properties on the heat transfer rate, heat function fields and profiles in a fluid filled square enclosure.

Navier‐Stokes equations in two‐dimensions, which are the flow governing equations, were transformed into stream function and vorticity transport equations. These equations together with the energy and heat function equations were cast into their non‐dimensional forms. Numerical solutions of the resulting equations were done by the use of finite‐difference method.

The numerical investigations conducted covered the Rayleigh and Prandtl numbers in the range 10³≤Ra≤10⁶ and 0.01≤Pr≤450, respectively, and expansion parameter ε=(T_h−T_c)/T_R in the range 0.05≤ε≤1. Results show that Boussinesq‐approximation is not sufficient to simulate natural convective flow when the difference between T_h and T_c is high and close to the reference state temperature. The effects of the other fluids properties other than density can be disregarded in computation without significant loss of accuracy. Combined fluid properties have very strong effects on the heat transfer, heat function fields and profiles.

The results of this study will serve as baseline information to designers of heat transfer or process equipment in which fluid at very high temperature occurs.

The purpose of this paper is to analyse and improve the temperature uniformity of aluminium billets heated by superconducting DC induction heaters.

A 3D electromagnetic model coupled with a heat transfer model is developed to calculate the heating process of the billets which are rotated in uniform transverse DC magnetic field. A laboratory-scale DC induction heater prototype has been built to validate the model. The results from simulation and measurement have a good agreement. The model is used to investigate the factors affecting the temperature uniformity of aluminium billets.

The results from simulation show that lower rotation speeds always mean better temperature uniformity along the radial direction, due to the increase in power penetration. However, the situation is very different for the temperature distribution along the axial direction. When the rotation speed is low, the temperature at the ends is lower than other parts. The situation reverses as the rotation speeds increase. This phenomenon is referred to as the “ending effect” in this paper.

Because of the ending effect, a lower rotation speed does not always result in better overall temperature uniformity, especially for billets of smaller sizes.

There is an optimal rotation speed that yields the best overall temperature uniformity. Lower rotation speeds are not always preferred. The results and numerical model developed are very useful in the design of a superconducting DC induction heater.

The temperature uniformity of aluminium billets heated by DC induction heaters is investigated and optimized.

The purpose of this paper is to examine the effects of Darcian drag force and radiation-conduction on unsteady two-dimensional magnetohydrodynamic flow of viscous, electrically conducting and Newtonian fluid over a vertical plate adjacent to a Darcian regime in presence of thermal radiation and transversal magnetic field. A well-tested, numerically stable Crank-Nicolson finite-difference procedure is employed for the conservation equations. Excellent agreement is obtained for numerical solutions with previously published work.

In this investigation, an efficient, accurate, extensively validated and unconditionally stable finite-difference scheme based on the Crank-Nicolson model is developed to solve the governing coupled, non-linear partial differential equations. The accuracy and effectiveness of the method are demonstrated.

Different numerical results are obtained and presented graphically to explain the effect of various physical parameters on the velocity and temperature profiles, local, as well as average, skin friction and Nusselt number. It is found that, with a rise in Darcian drag force, flow velocity and temperature are reduced, but increased for all times. Both average and local skin frictions are reduced considerably with an increase in Darcian drag force, but reversed behavior is observed for the local Nusselt number. Increasing the thermal radiation effects accelerated the flow velocity as well as the fluid temperature and wall local skin friction in a saturated porous medium, but effectively reduced the local Nusselt number and average Nusselt number at the wall. Comparison with previously published works in the limits shows excellent agreement.

The analysis is valid for unsteady, two-dimensional laminar flow of an optically thick no-gray gas, electrically conducting, and Newtonian fluid past an isothermal vertical surface adjacent to the Darcian regime with variable surface temperature. An extension to three-dimensional flow case is left for future work.

Practical interest of such study includes applications in electromagnetic lubrication, boundary cooling, bio-physical systems and in many branches of engineering and science. It is well known that the effect of thermal radiation is important in space technology and high temperature processes. Thermal radiation also plays an important role in controlling heat transfer process in polymer processing industry.

The purpose of this paper is to investigate the use of medical‐grade polymethylmethacrylate (PMMA) in fused deposition modeling (FDM) to fabricate porous customized freeform structures for several applications including craniofacial reconstruction and orthopaedic spacers. It also aims to examine the effects of different fabrication conditions on porosity and mechanical properties of PMMA samples.

The building parameters and procedures to properly and consistently extrude PMMA filament in FDM for building 3D structures were determined. Two experiments were performed that examined the effects of different fabrication conditions, including tip wipe frequency, layer orientation, and air gap (AG) (or distance between filament edges) on the mechanical properties and porosity of the fabricated structures. The samples were characterized through optical micrographs, and measurements of weight and dimensions of the samples were used to calculate porosity. The yield strength, strain, and modulus of elasticity of the samples were determined through compressive testing.

Results show that both the tip wipe frequency (one wipe every layer or one wipe every ten layers) and layer orientation (transverse or axial with respect to the applied compressive load) used to fabricate the scaffolds have effects on the mechanical properties and resulting porosity. The samples fabricate in the transverse orientation with the high tip wipe frequency have a larger compressive strength and modulus than the lower tip wipe frequency samples (compressive strength: 16±0.97 vs 13±0.71 MPa, modulus: 370±14 vs 313±29 MPa, for the high vs low tip wipe frequency, respectively). Also, the samples fabricated in the transverse orientation have a larger compressive strength and modulus than the ones fabricated in the axial orientation (compressive strength: 16±0.97 vs 13±0.83 MPa, modulus: 370±14 vs 281±22 MPa; for samples fabricated with one tip wipe per layer in the transverse and axial orientations, respectively). In general, the stiffness and yield strength decreased when the porosity increased (compressive strength: 12±0.71 to 7±0.95 MPa, Modulus: 248±10 to 165±16 MPa, for samples with a porosity ranging from 55 to 70 percent). As a demonstration, FDM is successfully used to fabricate patient‐specific, 3D PMMA implants with varying densities, including cranial defect repair and femur models.

This paper demonstrates that customized, 3D, biocompatible PMMA structures with varying porosities can be designed and directly fabricated using FDM. By enabling the use of PMMA in FDM, medical implants such as custom craniofacial implants can be directly fabricated from medical imaging data improving the current state of PMMA use in medicine.