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The present study is conducted to investigate the flow field and its impact on the thermal characteristics in the immediate surroundings of a rectangular flat plate, subjected to rotational oscillations in a quiescent fluid. The plate is heated with a constant heat flux on both faces during its rotational motion about one of its edges. The induced flow and thermal characteristics were simulated experimentally and computationally using the dynamic mesh method. The flow around a fabricated laboratory model was visualized with smoke particles and the surface temperature was recorded using J-types thermocouples. During the plate’s flapping cycles, computational and experimental results exhibit the presence of strong vortices near the free edges. When shed off, these vortices participate significantly in the enhancement of the cooling rate of the heated surface. The time dependent surface temperature distribution is symmetrical and characterized by a transient unsteady periodic variation which precedes a steady periodic phase. The results of current investigations present an interest in the cooling of portable electronic devices and shed light in the elephant’s metabolic heat regulation by the flapping of its pinnae.

Purpose – This paper aims to describe experience gained with a required six-credit year-long course, the Capstone Interdisciplinary Team Project, a key component of the Master of Science (MS) in Sustainability degree at the City College of New York. A common feature of sustainability problems is their interdisciplinary nature. Solutions to sustainability problems often require professionals with different training and backgrounds to work as a team. A sustainability curriculum should provide students with the skills needed to competently participate in an interdisciplinary team. Design/methodology/approach – Instructors drawn from different departments and divisions of the college developed a pool of sustainability-focused Capstone projects and acted as mentors to teams of students with diverse undergraduate backgrounds. Students attended workshops designed to provide training in teamwork, research and report preparation. An independent evaluator designed an evaluation protocol to assess the course’s impact both while it was implemented and after the course was concluded. Findings – Early experience with the program strongly indicates that the Capstone project requirement is an effective learning tool. However, identifying qualified mentors, developing suitable projects, assembling teams and administering the Capstone course are demanding tasks. Although students often experience difficulties in the early stages of their work, they ultimately express satisfaction and appreciation for the skills learned in the course. Practical implications – The inclusion of a capstone team project in a graduate sustainability curriculum is strongly recommended. Adopting such a course requires significant effort and sustained faculty engagement. Originality/value – Although there is considerable experience with undergraduate engineering Capstone course requirements, little is known about interdivisional capstone requirements at the level of master’s degree in Sustainability. This paper details new and relevant experience helpful to the implementation of such a requirement.

Analytic solutions for the gas and liquid velocity and temperature distribution are determined for steady state one-dimensional microchannel cylindrical Couette flow between a shaft and a concentric cylinder. The solution is based on the continuum model and takes into consideration the velocity slip and temperature jump in the gaseous phase defined by the Knudsen number range of 0.001 < Kn < 0.1. The two fluids are assumed immiscible. The gas layer is adjacent to the shaft which rotates with angular velocity xs and is thermally insulated. The outer cylinder rotates with angular velocity xo and is maintained at uniform temperature. The governing parameters are identified and the effects of the Knudsen number and accommodation coefficients on the velocity and temperature profiles, reduction in the overall temperature rise due to the gas layer, the Nusselt number and shear reduction are examined. It was found that the required torque to rotate the liquid in the annular space is significantly reduced by introducing a thin gas layer adjacent to the shaft. Also, reduction in shaft temperature is enhanced through a combination of high energy accommodation coefficient and low momentum accommodation coefficients. Results also indicate that the gas layer becomes more effective in reducing the shaft temperature when the housing angular velocity is much larger than the shaft angular velocity.

Reforming curricula has been likened to relocating graveyards. The implication, presumably, is that cur ricular reform should be done only when necessary, should be carried out respectfully, and demands a good deal of time and care. The creation of a new Sustainability in the Urban Environment program at City College of New York (CCNY)—a program that will graduate its first class in June 2011—certainly exhibited these characteristics. Given the dramatically increasing interest in sustainability and cleantech—in the academic world, the corporate world, the media world, and the planet in general—some sort of serious academic offering in sustainability seemed a virtual necessity. Of course it had to be done respectfully, given that much of the terrain of sustainability was at least partially covered, in differing ways, by existing programs and/or courses in environmental studies, earth sciences, environmental engineering, economics, management, law, and policy. We had to make clear that a new master’s degree program in Sustainability is not in any sense a replacement of existing disciplines, but rather a new focal point that draws heavily from the insights of these existing disciplines and juxtaposes these insights in ways that can be extremely useful in working toward solutions of real-world problems.

In this paper we consider steady laminar flow and heat transfer generated by two infinite parallel disks separated by a gas-filled micro-gap δ. One disk rotates with angular velocity Ω and the second with angular velocity s Ω . The analysis takes into consideration velocity slip, temperature jump, rarefaction and dissipation. A solution based on similarity transformation is obtained and used to examine the effects of the governing parameters on the velocity field, temperature distribution, disks’ torque and power, and Nusselt number. The solution requires numerical integration of the resulting coupled ordinary non-linear differential equations. An exact analytical solution is obtained for the limiting case of small Reynolds numbers.

The oscillatory rotational motion of the elephant pinna is considered a key mechanism in metabolic heat dissipation. Limited experimental investigations have revealed that the flapping of the elephant's pinna is responsible for surface heat transfer enhancement. The objective of the present experimental and computational work is to investigate the physics of the flow induced by the pinna's motion and its effects on the heat transfer. This was accomplished by designing, fabricating and testing two full-size laboratory models of elephant pinnae: one rigid and one flexible, both instrumented with small size thermocouples for time-dependent surface temperature measurements. A constant heat flux is applied to both sides of each model which is rotated about a fixed edge with a frequency of 2 rad/s in an infinite domain at ambient conditions. Of interest is the study of the impact of the flexural strength of the model's material on surface heat transfer. Additional computer simulations of the flow and thermal fields revealed a hooked-shape vortex tube around the free edges of the flapping pinna. This result is confirmed by the flow visualization with smoke particles. Both experimental and computational results exhibit local surface temperature profiles characterized by a transient unsteady periodic variation followed by a steady periodic phase. Flow visualization indicated significant interaction between the vortical structures shed off the edge and the flexible model's boundary layer. It has been found that the cooling of the flexible model is enhanced by 30%.

A new approximate analytic solution technique is developed for treating the heat exchange between two or more axially interacting vessels which are eccentrically embedded in a cylinder. The cylinder exchanges heat by convection with the environment and has a non-uniform surface temperature due to the vessel eccentricity. The flow in the vessels can be either countercurrent or unidirectional. The solution is constructed by superposition of a new solution for a single embedded vessel which exactly satisfies the boundary conditions when the thermal conductivity of the fluid is equal to that of the cylinder material and is a very good approximation when these conductivities are not equal. Comparison with an exact solution for a single embedded vessel, when the conductivities are not equal, shows that the approximate solution results in very minor errors for a wide range of the governing parameters. As an application of the new technique the two-vessel solution is used to examine the countercurrent heat exchange in a human arm.

There are many situations where analytic determination of the heat transfer coefficient h is difficult to obtain. As was shown in previous chapters, even after making many simplifying assumptions, analytical determination of h is generally not a simple mathematical problem. When complicating factors such as geometry, variable properties, turbulent flow, boiling, condensation, etc. are involved, the heat transfer coefficient is usually determined experimentally. This does not mean that each time there is a need for h for which there is no analytic solution we must conduct an experiment. Instead, we utilize the experimental results of other researchers. Experimental results are usually correlated and presented as dimensionless equations which are convenient to use. Such equations are known as correlation equations. They are extensively used in the solution of heat transfer problems and therefore deserve special attention.

Research on fluid flow and heat transfer in microchannels was partly driven by miniaturization of microelectronic devices. The need for efficient cooling methods for high heat flux components focused attention on the cooling features of microchannels. Microchannels are used in a variety of engineering and scientific applications. The inkjet printer is a classic example. Extensive use is found in medical applications and in micoelecro- mechanical systems (MEMS) such as micro heat exchangers, mixers, pumps, turbines, sensors and actuators.

This textbook presents the classical topics of conduction heat transfer and extends the coverage to include chapters on perturbation methods, heat transfer in living tissue, and microscale conduction. This makes the book unique among the many published textbook on conduction heat transfer. Other noteworthy features of the book are: The material is organized to provide students with the tools to model, analyze and solve a wide range of engineering applications involving conduction heat transfer. Mathematical techniques are presented in a clear and simplified fashion to be used as instruments in obtaining solutions. The simplicity of one-dimensional conduction is used to drill students in the role of boundary conditions and to explore a variety of physical conditions that are of practical interest. Examples are carefully selected to illustrate the application of principles and the construction of solutions. Students are trained to follow a systematic problem solving methodology with emphasis on thought process, logic, reasoning and verification. Solutions to all examples and end-of-chapter problems follow an orderly problems solving approach. (orig.)

There are many applications in which a material undergoes phase change such as in melting, freezing, casting, ablation, cryosurgery and soldering. Conduction with phase change is characterized by a moving interface separating two phases. Such problems are usually referred to as moving boundary or free boundary problems. They are inherently transient during interface motion. The motion and location of the interface are unknown a priori and thus must be determined as part of the solution. Since material properties change following phase transformation and since a discontinuity in temperature gradient exists at the interface, it follows that each phase must be assigned its own temperature function. Furthermore, changes in density give rise to motion of the liquid phase. If the effect of motion is significant, the heat equation of the liquid phase must include a convective term. However, in most problems this effect can be neglected.

Turbulent flow is a complicated physical phenomenon, and a daunting subject for students of engineering. Turbulent flow is disordered, with random and unsteady velocity fluctuations. This description by itself suggests that exact predictions cannot be determined, and that analysis of turbulent flows will not be easy. However intimidating turbulence may be, its importance is clear: turbulent flows are found in many industrial and natural processes, and affect quantities of great practical importance to engineers, like the local velocity distribution, drag force, and heat transfer. Moreover, a better understanding of this flow allows engineers to make design decisions that could either reduce the effects of turbulence or enhance them, in order to improve the performance of their devices.

The mathematical complexity of convection heat transfer is traced to the non-linearity of the Navier-Stokes equations of motion and the coupling of flow and thermal fields. The boundary layer concept, first introduced by Prandtl [1] in 1904, provides major simplifications. This concept is based on the notion that under special conditions certain terms in the governing equations are much smaller than others and therefore can be neglected without significantly affecting the accuracy of the solution. This raises two questions

There are various situations where it is desirable to obtain approximate analytic solutions. An obvious case is when an exact solution is not available or can not be easily obtained. Approximate solutions are also obtained when the form of the exact solution is not convenient to use. Examples include solutions that are too complex, implicit or require numerical integration. The integral method is used extensively in fluid flow, heat transfer and mass transfer. Because of the mathematical simplifications associated with this method, it can deal with such complicating factors as turbulent flow, temperature dependent properties and non-linearity.

Two-dimensional steady state conduction is governed by a second order partial differential equation. A solution must satisfy the differential equation and four boundary conditions. The method of separation of variables [1] will be used to construct solutions. The mathematical tools needed to apply this method will be outlined first. Examples will be presented to illustrate the application of this method to the solution of various problems.

We discussed laminar flow in channels in Chapter 6, and many of the features of turbulent flow are similar. We will begin this subject with the criteria for fully developed velocity and temperature profiles, and will focus most of our attention on analyzing fully developed flows. As in Chapter 6, our analysis is limited to general boundary conditions: (i) uniform surface temperature, and (ii) uniform surface heat flux.

In a moving fluid the three fundamental laws, conservation of mass, momentum, and energy, must be satisfied at every point in the domain. Thus the first step is to formulate (cast) the three laws in a form that satisfies this condition. This is accomplished by applying each law to a differential (infinitesimal) element. Following this approach, each law is described by a partial differential equation. Differential formulation of the three laws will be presented using rectangular coordinates. The corresponding forms in cylindrical and spherical coordinates will be stated without details.

Non-linearity in conduction problems arises when properties are temperature dependent or when boundary conditions are non-linear. Surface radiation and free convection are typical examples of non-linear boundary conditions. In phase change problems the interface energy equation is non-linear.

The determination of temperature distribution in blood perfused tissue is important in many medical therapies and physiological studies. Examples are found in cryosurgery, frost bite, hyperthermia, skin burns and body thermal regulation and response to environmental conditions and during thermal stress. The key to thermal modeling of blood perfused tissue is the formulation of an appropriate heat transfer equation. Such an equation must take into consideration three factors: (1) blood perfusion, (2) the vascular architecture, and (3) variation in thermal properties and blood flow rate. The problem is characterized by anisotropic blood flow in a complex network of branching arteries and veins with changing size and orientation. In addition, blood is exchanged between artery-vein pairs through capillary bleed-off along vessel walls, draining blood from arteries and adding it to veins. Energy is transported between neighboring vessels as well as between vessels and tissue. Thus, heat transfer takes place in a blood perfused inhomogeneous matrix undergoing metabolic heat production. The search for heat equations modeling this complex process began over half a century ago and remains an active topic among current investigators. Over the years, several bioheat transfer equations have been formulated. A brief description of some of these equations will be presented and their limitations, shortcomings and applicability outlined. Before these equations are presented, related aspects of the vascular circulation network and blood flow and temperature patterns are summarized.

The flow of fluid in porous media plays an important role in temperature distribution and heat transfer rate. The term transpiration cooling is used to describe heat removal associated with fluid flow through porous media. The idea is to provide intimate contact between the coolant and the material to be cooled. Thus heat transfer in porous media is a volumetric process rather than surface action.

Although the energy equation for constant properties is linear, the Navier- Stokes equations of motion are non-linear. Thus, in general, convection problems are non-linear since the temperature field depends on the flow field. Nevertheless, exact solutions to certain simplified cases can easily be constructed. One of the objectives of this chapter is to develop an appreciation for the physical significance of each term in the equations of continuity, Navier-Stokes and energy and to identify the conditions under which certain terms can be neglected. Simplification of the governing equations is critical to constructing solutions. The general procedure in solving convection problems is, whenever possible, to first determine the flow field and then the temperature field.

This paper examines the effects of rarefaction, dissipation, curvature, and accommodation coefficients on flow and heat transfer characteristics in rotating microdevices. The problem is modeled as a cylindrical Couette flow with a rotating shaft and stationary housing. The housing is maintained at uniform temperature while the rotating shaft is insulated. Thus, heat transfer is due to viscous dissipation only. An analytic solution is obtained for the temperature distribution in the gas filled concentric clearance between the rotating shaft and its stationary housing. The solution is valid in the slip flow and temperature jump domain defined by the Knudsen number range of 0.001

A three-dimensional approximate analytic solution is presented for the unequal countercurrent heat transfer between parallel paired vessels asymmetrically embedded in a long cylinder with surface convection. The analysis assumes that the velocity profile in the vessels is parabolic, the Peclet number is large compared to unity and the radius of the cylinder is several times larger than the distances from the center of the cylinder to the vessels. A perturbation method is employed by taking the reciprocal of the Peclet number as a small parameter. The new solution approach takes into consideration the axial thermal interaction and describes the three-dimensional asymmetric thermal field due to the vessels, the wall temperatures of which are nonuniform in the cross-sectional plane. Application of the model to whole limb heat transfer is discussed.

This paper examines the effects of rarefaction and dissipation on flow and heat transfer characteristics in rotating micro devices. The housing is assumed to be at uniform temperature while the rotating surface is insulated. Thus heat generation and transfer are due to viscous dissipation only. An analytic solution is obtained for the velocity and temperature distribution in the gas filled concentric clearance between a rotating shaft and its stationary housing. The solution is valid in the slip flow and temperature jump domain defined by the Knudsen number range of Kn < 0.1. The Nusselt number was found to depend on three parameters: the Knudsen number Kn, ratio of housing to shaft radius ro / ri , and Prandtl number-specific heat ratio group γ/(γ + 1) Pr. Results indicate that curvature and Knudsen number have significant effect on the Nusselt number. However, fluid temperature rise due to dissipation is negligible.Copyright © 2006 by ASME

Previous models of rat tail heat transfer have assumed that the tail is uniformly perfused along its length and have introduced questionable assumptions about the heat transfer role of the major axial arteries and the venous blood shunting between the superficial and deep veins. The recent experiments of Lemons and Wu (1994) have shown that (i) perfusion of the tail tip is more than tenfold higher than that in the tail base and (ii) the perfusion of the middle region of the tail increases eightfold during heat stress compared to threefold to fourfold in the base and tip. The authors' anatomical studies have shown that the lateral arteries are a series of radially arcading connections from the ventral artery and probably do not serve as major axial conduit vessels. These observations indicate that current views and models for the blood flow distribution and heat transfer in the major axial arteries and veins and in the rat tail cutaneous circulation need substantial revision. Based on these new experimental findings a new three-dimensional model is developed to determine the heat transfer function of the rat tail at different local and central temperatures. The predictions of the model show good agreement with the axial surface temperature distribution in the rat tail reported by Lemons and Wu. These results, when combined with the authors' anatomical studies, indicate that there is very little shunting of blood between the superficial lateral veins and the deep ventral vein as proposed by Raman et al. (1987). Although this model is based on the rat tail anatomy, it can be modified to treat the human limb and digit.

The microvascular organization and thermal equilibration of the primary and secondary arteries and veins that comprise the bleed off circulation to the muscle fibers from the parent countercurrent supply artery and veins are analyzed. The blood perfusion heat source term in the tissue energy equation is shown to be related to this vascular organization and to undergo a fundamental change in behavior as one proceeds from the more peripheral tissue, where the perfusion term is proportional to the Ta - Tv difference in the parent supply vessels, to the deeper tissue layers where the bleed off vessels themselves form a branching countercurrent system for each muscle tissue cylinder and the venous return temperature can vary between the local tissue temperature and Ta. The consequences of this change in behavior are examined for the Weinbaum-Jiji bioheat equation and a modified expression for the effective conductivity of perfused tissue is derived for countercurrent bleed off exchange.

The heat transfer in the boundary layer region of a circular free jet impinging on a flat solid surface with non-uniform wall temperature or wall heat flux is investigated analytically. The flow is laminar, incompressible and steady. The analysis begins with obtaining the solution to the problem with a step change in wall temperature or wall heat flux. The solution to the corresponding problem with arbitrary wall temperature or wall heat flux is obtained by superposition. This solution is then matched with that for the stagnation region obtained in the first part of this study so that the Nusselt number throughout the stagnation region and the boundary layer region is obtained. The results indicate that the Nusselt number for increasing wall temperature or wall heat flux can be considerably higher than that for constant wall temperature or wall heat flux outside the stagnation region. For the special case of constant wall temperature, the result is in good agreement with the integral solution ofChaudhury.

This paper deals with the inward solidification of liquid in an annular space which is initially not at the fusion temperature. The outer cylindrical surface is maintained at a subfreezing temperature while the inner cylindrical boundary is assumed to be either insulated or maintained at constant temperature. New perturbation solutions are obtained for the temperature distribution and the interface motion. The perturbation parameter e = C s (T f — T a ) L is the ratio of the sensible heat of the solid-phase to the latent heat of fusion. The non-uniformity of the long-time scale solutions is treated by constructing inner expansions in the short-time scale. The two solutions are matched using asymptotic theory. The solutions for the insulated case do not depart markedly, except on the short time scale, from the corresponding solutions for liquids which are initially at the fusion temperature T f . In contrast, the solutions for the isothermal inner boundary depart substantially from those with an initial temperature equal to T f . This is true even if the sensible heat is small compared to the latent heat of fusion. Similarly, curvature plays a minor role in the interface motion for the insulated case while its effect is dramatic in the isothermal case.

A new prototype model for whole limb heat transfer is proposed wherein the countercurrent heat exchange from the large central arteries and veins in the core of the limb is coupled to microvascular models for the surrounding muscle and the cutaneous tissue layers. The local microvascular temperature field in the muscle tissue is described by the bioheat equation of Weinbaum and Jiji [1]. The new model allows for an arbitrary axial variation of cross-sectional area and blood distribution between the muscle and cutaneous tissue, accounts for the blood flow to and heat loss from the hand and treats the venous return temperature and surface temperature distribution as unknowns that are determined as part of the solution to the overall boundary value problem. Representative solutions are presented for a wide range of environmental conditions for a limb in both the resting state and during exercise.

A theoretical and experimental study of the temperature characteristics of two-dimensional heated turbulent jets discharging parallel to and in the vicinity of a solid boundary is presented. The model is based on an integral formulation of the basic conservation laws and uses the flow field solutions of corresponding nonbuoyant jets. The theory takes into consideration the effects of free-stream motion and jet offset distance. Theoretical predictions of temperature were found to be in good agreement with experimental data.

A new simplified three-dimensional bioheat equation is derived to describe the effect of blood flow on blood-tissue heat transfer. In two recent theoretical and experimental studies [1, 2] the authors have demonstrated that the so-called isotropic blood perfusion term in the existing bioheat equation is negligible because of the microvascular organization, and that the primary mechanism for blood-tissue energy exchange is incomplete countercurrent exchange in the thermally significant microvessels. The new theory to describe this basic mechanism shows that the vascularization of tissue causes it to behave as an anisotropic heat transfer medium. A remarkably simple expression is derived for the tensor conductivity of the tissue as a function of the local vascular geometry and flow velocity in the thermally significant countercurrent vessels. It is also shown that directed as opposed to isotropic blood perfusion between the countercurrent vessels can have a significant influence on heat transfer in regions where the countercurrent vessels are under 70-μm diameter. The new bioheat equation also describes this mechanism.

An approximate fully three-dimensional numerical model for predicting the detailed flow and thermal characteristics of spray units is presented. The new model differs from all previous analyses of spray cooling systems in that it determines the local variation in the dry-bulb temperature, absolute humidity and the air streamlines throughout the flow field encompassing the spray umbrella and the effect of this local variation on drop cooling. The conservation equations to determine the local absolute humidity, velocity and dry-bulb temperature of the air-vapor phase are written in Lagrangian form where the droplets are treated as spatially varying sources of mass, momentum and energy. The analysis takes into consideration stable and unstable meteorological conditions, turbulent mixing in the atmospheric surface layer and nonuniform upwind and local air-velocity profiles. The numerical model is of intrinsic interest because it demonstrates the feasibility of computing a three-dimensional, two-phase flow field without the use of excessive computer time. Model predictions of droplet return temperature along the spray centerline were compared with data for the Spraco 1751 nozzle. Good agreement was observed.

In this paper the use of the boundary integral equation method (BIEM) for multidimensional problems with a moving phase-change interface is explored. The method is shown to be suited to heat transfer problems where the field equations are linear in each region and the boundary or interface matching conditions are both highly irregular and non-linear. For moving interface problems the BIE technique both reduces the dimensions of the problem by one, thus decreasing storage requirements, and directly solves for the unknown normal temperature gradient on each side of the interface for the determination of the instantaneous interface velocity. To illustrate the versatility of this technique the BIEM is applied to a previously unsolved problem: the melting/freezing around a pipe buried in a semi-infinite domain where the melting/freezing is initiated at the free surface and the medium is initially not at the phase-change temperature. For simplicity quasi-steady heat conduction is assumed in both the thawed and frozen zones. Solutions are presented for various values of the governing parameters.

In this paper the new bioheat equation derived in Weinbaum and Jiji [7] is applied to the three layer conceptual model of microvascular surface tissue organization proposed in [1]. A simplified one-dimensional quantitative model of peripheral tissue energy exchange is then developed for application in limb and whole body heat transfer studies. A representative vasculature is constructed for each layer and the enhancement in the local tensor conductivity of the tissue as a function of vascular geometry and blood flow is examined. Numerical solutions for the boundary value problem coupling the three layers are presented and these results used to study the thermal behavior of peripheral tissue for a wide variety of physiological conditions from supine resting state to maximum exercise.

This paper presents a quasi-steady state approximation solution for the three dimensional melting or freezing around a fluid carrying pipe which is buried in a semi-infinite phase change medium (PCM). The two-dimensional quasi-steady approximation solution method previously developed by the authors, the virtual free surface technique, has been extended to three dimensions where axial thermal interaction between the moving fluid and PCM is considered. Of particular interest herein is the motion of the phase change interface and the time variation of the axial temperature distribution in the fluid. Due to the singularities of the differential equations along the pipe surface, an axisymmetric analytic solution is provided for the region near the pipe wall. Solutions are presented for several representative dimensionless burial depths of the pipe and initial conditions. The computational time to predict the three dimensional interface location up to 10 years is of the order of several minutes on an IBM 4341 computer.

This paper presents an analytical solution to the two-dimensional free boundary problem of solidification of a liquid, initially at a uniform temperature and filling the quarter-space x, y > 0, subject to a constant wall temperature. The problem is the two-dimensional analog of Neumann’s freezing problem. The solution is characterized by similarity in the variables x/t1/2, y/t1/2 and is obtained by treating the heat of solidification as a moving heat source. A nonlinear, singular, integro-differential equation for the solid-liquid interface is thereby derived and used to establish superhyperbolas to approximate the interface position. Results are presented for a range of the two dimensionless parameters of the problem. The accuracy of the superhyperbolic representation of the interface position is determined by comparison with a finite-difference solution. Equations are given for the calculation of the temperature fields in the solid and liquid regions that are valid for all time (i.e., they are not necessarily short- or long-time solutions).

An analysis is made of the transient temperature behavior of a transpiration-cooled porous matrix entering a planetary atmosphere with constant velocity and negative entry angle. The analysis is based on one dimensional heat conduction in a porous plate subjected to a time dependent heat flux at one side and cooled internally by mass injection from a constant temperature reservoir at the opposite side. An exact closed-form solution is obtained and temperature charts are presented for a wide range of Fourier number and coolant flow parameters.

Two-dimensional solidification of forced viscous flow over a semi infinite flat plate is considered. The plate is initially maintained at a constant temperature above the freezing point of the liquid. At time t>0, heat is removed from the plate so that the liquid freezes. The continuity equation and the two-dimensional, nonsteady, momentum and energy equations in the liquid and solid phases are solved numerically. The growth of the solid layer as a function of time and position along the plate is determined.