Freezing Front Velocity Research Articles

Ice crystal formation in the carbon nanotube suspension: A modelling approach

Abstract Freeze-dried porous solid foams were prepared from carbon nanotube (CNT) suspensions via either contact freezing with a heat exchanger or immersion freezing into a cryo-bath. Microstructures in the freeze-dried foam cast by ice crystals formed during the freezing step. It was found that domains of the carbon nanotubes in the freeze-dried solid foams were fairly well interconnected, and the microstructures were largely influenced by the freezing condition and freezing method. A mathematical model was proposed to simulate the observed thermal history during the freezing step; then theoretical predictions of the microstructural features were attempted. The simulated thermal history was in good agreement with experimental result. The resulting mean ice crystal sizes L* could be estimated from the calculated freezing front velocity (R) and the temperature gradient in the frozen zone (G). Interestingly, it was found that a correlation based on the power law (L*∝R−0.2G−0.2) was applicable to the present system for both the contact and immersion freezing methods. Though the fundamental ice crystallization phenomena were essentially the same for both freezing methods, actual temperature distribution and movement of the freezing front through the whole freezing bulk could control the morphology of ice crystal in the frozen matrix. In other words, good control of thermal flow in the freezing system would greatly contribute to rational design of microstructures of the CNT foam.

Control of the freezing interface morphology in solidification processes in the presence of natural convection

AbstractThis paper presents a methodology for the solution of an inverse solidification design problem in the presence of natural convection. In particular, the boundary heat flux q0 in the fixed mold wall, δΩ0, is calculated such that a desired freezing front velocity and shape are obtained. As the front velocity together with the flux history qms on the solid side of the freezing front play a determinant role in the obtained cast structure, the potential applications of the proposed methods to the control of casting processes are enormous.The proposed technique consists of first solving a direct natural convection problem of the liquid phase in an a priori known shrinking cavity, ΩL(t), before solving an ill‐posed inverse design conduction problem in the solid phase in an a priori known growing region, ΩS(t). The direct convection problem is used to evaluate the flux qml in the liquid side of the freezing front. A front tracking deforming finite element technique is employed. The flux qml can be used together with the Stefan condition to provide the freezing interface flux qms in the solid side of the front. As such, two boundary conditions (flux qms and freezing temperature θm) are especified along the (known) freezing interface δΩI(t). The developed design technique uses the adjoint method to calculate in L2 the derivative of the cost functional, ∥θm – θ(x, t; q0)∥, that expresses the square error between the calculated temperature θ(x, t; q0) in the solid phase along δΩI(t) and the given melting temperature. The minimization of this cost functional is performed by the conjugate gradient method via the solutions of the direct, sensitivity and adjoint problems. A front tracking finite element technique is employed in this inverse analysis. Finally, an example is presented for the solidification of a superheated incompressible liquid aluminium, where the effects of natural convection in the moving interface shape are controlled with a proper adjustment of the cooling boundary conditions.

Control of the freezing interface motion in two‐dimensional solidification processes using the adjoint method

AbstractThe aim of this work is to calculate the optimum history of boundary cooling conditions that, in two‐dimensional conduction driven solidification processes, results in a desired history of the freezing interface location/motion. The freezing front velocity and heat flux on the solid side of the front, define the obtained solidification microstructure that can be selected such that desired macroscopic mechanical properties and soundness of the final cast product are achieved.The so‐called two‐dimensional inverse Stefan design problem is formulated as an infinite‐dimensional minimization problem. The adjoint method is developed in conjunction with the conjugate gradient method for the solution of this minimization problem. The sensitivity and adjoint equations are derived in a moving domain. The gradient of the cost functional is obtained by solving the adjoint equations backward in time. The sensitivity equations are solved forward in time to compute the optimal step size for the gradient method. Two‐dimensional numerical examples are analysed to demonstrate the performance of the present method.

On the solution of an ill‐posed design solidification problem using minimization techniques in finite‐ and infinite‐dimensional function spaces

AbstractThis paper provides a comparative study of two alternative methodologies for the solution of an inverse design solidification problem. It is the one‐dimensional solidification problem of calculating the boundary heat flux history that achieves a desired freezing front velocity and desired heat fluxes at the freezing front. The front velocity h(t) and flux history qmS(t) on the solid side of the front control the obtained cast structure. As such, the potential applications of the proposed methods to the control of casting processes are enormous.The first technique utilizes a finite‐dimensional approximation of the unknown boundary heat flux function q0(t). The second technique uses the adjoint method to calculate in L2 the derivative of the cost functional, ‖Tm – T(h(t), t;q0)‖, that expresses the square error between the calculated T(h(t), t; q0) and the given freezing front temperature Tm. Both steepest descent (SDM) and conjugate gradient methods (CGM) are examined. A front tracking FEM technique is used for the discretization of the state space. A detailed numerical analysis of the space and time discretization of the ‘parameter’ and state spaces, of the effect of the end condition of the adjoint problem and of other parameters in the solution are examined.

Control of the cryosurgical process in nonideal materials

A study of a controlled cryosurgical process is presented. This study is based on the energy equations describing the probe response and the phase change occurring in the medium. First-order nonlinear differential equations (state equations) are obtained by applying the integral-solution method. In order to obtain maximal cell destruction, it is desired to control a specific cooling rate at the solid-liquid interface. This cooling rate defines the desired trajectories of the state variables through the state equations. In order to satisfy the cooling rate condition on the freezing front, a closed-loop is designed to control the probe temperature program. A simple analysis of the system stability employed linearization at several points along the desired trajectories. Ranges of stability were obtained for a system containing a proportional-integral controller. It was demonstrated that these stability ranges depend mainly on the selected sampling time of the discrete control loop and that the phase-change process does not significantly affect the stability results. A complete study of the nonlinear equations was performed by a computer simulation program which enables the selection of the final values of the controller parameters, in order to minimize the error and to ensure stability. In addition, the simulation program gives information about the effects of the A/D and D/A converters accuracy on the performance of the control loop. An A/D converter accuracy of 12 bits was found necessary in order to reduce the oscillations in probe temperature to acceptable values. The simulation also yields a complete calculated temperature field in the tissue during the controlled process. From these simulated results it can be seen that oscillations of +/- 0.5 degrees C in the desired probe temperature do not significantly affect the desired cooling rate at the freezing front. An initial overshoot of 1.5 degrees C in the desired probe temperature was obtained both experimentally and theoretically from the simulation. When this initial overshoot occurs at the beginning of the freezing process, it causes an error in freezing front velocity and consequently in ice-front position. From the numerical simulation, it can be deduced that the cooling rate obtained at the front deviates from the desired value by approximately 1%. The probe-temperature error increases at two instants: a) during the super-cooling effect and the subsequent sudden crystallization, and b) when the probe temperature is below -80 degrees C and unstable boiling of the cooling medium causes oscillations.