Steady State Heat Transfer Process Research Articles

A Variational Formulation for Fins with Nonzero Contact Thermal Resistance at the Base

This paper considers the steady-state heat transfer process in a fin with a Robin boundary condition at the base (instead of the usual Dirichlet boundary condition at the base). Robin boundary condition models the effect of the thermal resistance between the base of the fin and the surface on which the fin is placed. This work presents an equivalent minimum principle, represented by a convex and coercive functional, ensuring the solution’s existence and uniqueness. In order to illustrate the use of the proposed functional for reaching approximations, the heat-transfer process in a trapezoidal fin considering a piecewise linear approximation is simulated. The Appendix presents a case in which the exact solution in a closed form has been achieved.

Experimental study of steam-air mixture thermal stratification during the dropping of containment pressure

The passive containment cooling system (PCCS) as a passive technology has been applied in Generation Ⅲ nuclear power plants. Experiments were conducted in a scale model to analyze the thermal stratification in containment during the containment pressure dropping. Results indicate that the obvious thermal stratification occurs in the containment during the transient-state heat transfer process but not during the steady-state heat transfer process. For the transient-state process, the steam-air mixture temperature increases by 2 °C for each 100 mm increase in the elevation direction of the pressure tank. The nonuniform distribution of air in containment causes thermal stratification, which is more pronounced for lower air mass fraction and higher initial containment pressure. The thermal stratification effect on the condensation heat transfer process is assessed via diffusion layer heat transfer and film heat transfer to evaluate its effect. The diffusion layer heat transfer coefficient (HTC) is strongly affected by thermal stratification, whereas no effect is observed on the film HTC. This leads to the total condensation HTC at the top of the heat exchanger is approximately 3–5 times than that at the bottom in the first 100 s of the containment pressure dropping process. To improve the overall heat transfer performance of PCCS, approaches that enhance the heat transfer capacity at the bottom of the heat exchanger should be considered for prioritization, especially the methods of steam diffusion heat transfer capacity.

Development of a two-dimensional mathematical model of flow boiling heat transfer in micro- and minichannels

The paper concerns flow boiling heat transfer in micro- and minichannels. In the mathematical model, the steady state heat transfer process in a single asymmetrically heated minichannel was considered. Calculations with the use of Trefftz functions were based on the data from own experiments. The temperature of the heater and the refrigerant were assumed to satisfy the Laplace equations and the energy equation respectively. The problem was solved by the Trefftz method using two sets of Trefftz functions. The known heater and refrigerant temperature distributions were used to determine the heat transfer coefficient at the heater – refrigerant contact. To verify the proposed mathematical model, data from experiments were applied to calculations. The essential part of the experimental stand was the test section which comprises a minichannel heat sink. The heated element for Fluorinert FC-72 flowing along minichannels was a thin foil. The temperature of its outer side was measured using infrared thermography. Thermocouples and pressure transducers installed at the inlet and outlet of the test section monitored fluid temperature and pressure. Mass flow rate, the current supplied to the heater and the voltage drop were also recorded. The resulting graphs presented thermograms of measured temperature on outer surface of the heater, temperature distributions of fluid temperature and local values of the heat transfer coefficient.

A Variational Principle for the Steady-State Heat Transfer Process in a Rigid Continuous Mixture

The subject of this paper is the steady-state heat transfer process in a rigid mixture with N continuous constituents, each of them representing a given continuous body. A continuous mixture consists of a convenient representation for bodies composed by several different materials or phases, in which the actual interfaces do not allow an adequate Classical Continuum Mechanics approach, once that the boundary conditions make the mathematical description of the problem unfeasible (as for instance in reinforced concrete, polymer strengthened concrete, and porous media). The phenomenon is mathematically described by a set of N partial differential equations coupled by temperature-dependent terms that play the role of internal energy sources. These internal energy sources arise because, at each spatial point, there are different temperatures, each one associated with one constituent of the mixture. The coupling among the partial differential equations arises from the thermal interchange among continua in a thermal nonequilibrium context (different temperature levels). In this work, it is presented a functional whose minimization is equivalent to the solution of the original steady-state problem (variational principle). The features of this functional give rise to proofs of solution existence and solution uniqueness. It is remarkable that, with the functional to be proposed here, instead of solving a system of N coupled partial differential equations, we need to look only for the minimum of a single functional.

Heat Transfer Coefficient Determination during FC-72 Flow in a Minichannel Heat Sink Using the Trefftz Functions and ADINA Software

This work focuses on subcooled boiling heat transfer during flow in a minichannel heat sink with three or five minichannels of 1 mm depth. The heated element for FC-72 flowing along the minichannels was a thin foil of which temperature on the outer surface was measured due to the infrared thermography. The test section was oriented vertically or horizontally. A steady state heat transfer process and a laminar, incompressible flow of the fluid in a central minichannel were assumed. The heat transfer problem was described by the energy equations with an appropriate system of boundary conditions. Several mathematical methods were applied to solve the heat transfer problem with the Robin condition to determine the local heat transfer coefficients at the fluid/heated foil interface. Besides the 1D approach as a simple analytical method, a more sophisticated 2D approach was proposed with solutions by the Trefftz functions and ADINA software. Finite element method (FEM) calculations were conducted to find the temperature field in the flowing fluid and in the heated wall. The results were illustrated by graphs of local heated foil temperature and transfer coefficients as a function of the distance from the minichannel inlet. Temperature distributions in the heater and the fluid obtained from the FEM computations carried out by ADINA software were also shown. Similar values of the heat transfer coefficient were obtained in both the FEM calculations and the 1D approach. Example boiling curves indicating nucleation hysteresis are shown and discussed.

Numerical Solution of Axisymmetric Inverse Heat Conduction Problem by the Trefftz Method

In this paper, the issue of flow boiling heat transfer in an annular minigap was discussed. The main aim of the paper was determining the boiling heat transfer coefficient at the HFE-649 fluid–heater contact during flow along an annular minigap. The essential element of the experimental stand was a test section vertically oriented with the minigap 2 mm wide. Thermocouples were used to measure the temperature of the heater and fluid at the inlet and the outlet to the minigap. The mathematical model assumed that the fluid flow was laminar and the steady–state heat transfer process was axisymmetric. The temperatures of the heated surface and of the flowing fluid were assumed to fulfill energy equations with adequate boundary conditions. The problem was solved by the Trefftz method. The local heat transfer coefficients at the fluid–test surface interface were calculated due to the third kind boundary condition at the saturated boiling. Graphs were used to illustrate: the measurement of the heater surface temperature, 2D temperature distributions in the pipe and fluid, and the heat transfer coefficient as a function of the distance from the minigap inlet. The measurement uncertainties and accuracy of the heat transfer coefficient determination were estimated.

Thermal performance of non-load-bearing cold-formed steel walls under different design fire conditions

Most previous fire experiments of cold-formed steel (CFS) walls focused on the ISO 834 or the E119 fire curve. Although the fire performance of CFS walls under ISO 834 or E119 curve can be theoretically converted to that under other fire curves, the corresponding time equivalency methods have not been experimentally examined. This paper presents an experimental investigation of non-load-bearing CFS walls under four different fire conditions. It is proved by the experiments that the maximum temperature of the design fire curve is one of the most important indexes for representing the severity of fire exposure. Especially, the maximum temperature of 680 °C for external fire was not severe to the present non-load-bearing CFS walls. Two specimens were subjected to 285 min of external fire but demonstrated an almost steady-state heat transfer process. Meanwhile, the hydrocarbon fire and realistic design fire demonstrated much more severe damage. It is also proved for the first time that neither the equal area method nor the energy-based method was suitable for the fire-resistance time equivalency of CFS walls. Based on post fire observation, significant local crushing of studs was identified, especially after the fall off of fire-side sheathing. Additionally, it was found that replacing the fire-side base-layer gypsum plasterboard with low-bulk-density calcium-silicate board extends the insulation and structure performance of CFS walls under fire conditions. Moreover, a simplified temperature distribution model and the critical temperature on the fall off of fire-side gypsum plasterboard were recommended for the numerical modeling of CFS walls.

Thermal Conductivity Measurements of Novel Porous Copper Fiber Sintered Sheet

In this work, a porous copper fiber sintered sheet (PCFSS) is fabricated by using a low-temperature solid-phase sintering method. Copper fiber is a raw material that is produced by a cutting method with a multi-tooth tool. A novel reference plate method (RPM) with a steady state heat transfer process is introduced to measure the thermal conductivity of PCFSS. The study involves experimentally investigating the effects of reference plate materials as well as the porosity and porosity gradients of PCFSS on thermal conductivity. The findings indicate that reasonable measurement results of thermal conductivity of PCFSS are obtained when 304 stainless steel is selected as a reference plate material when compared with that in the case of a red copper plate. The thermal conductivity increases with a decrease in the porosities of PCFSS in the approximate range of 70%–90%. With respect to the approximate measuring temperature range of 34°C~58°C, the thermal conductivities of PCFSSs with 70%, 80% and 90% porosities correspond to 25.35 W/(m·°C), 15.01 W/(m·°C) and 11.24 W/(m·°C), respectively. The thermal conductivity of PCFSS with 70%-80%-90% gradient porosity corresponds to 13.43 W/(m·°C), and this value is between 80% and 90% porosity of PCFSS.

NUMERICAL STUDY OF HEAT TRANSFER IN EXTENDED SURFACESWITH MUTUAL RADIATION BETWEEN PARALLEL FINS

The present work shows the influence of the mutual heat transfer on the effectiveness of finned surfaces. Numerical simulations are carried out through a sequence of linear problems, possessing an equivalent minimum principle, that has as its limit the solution of the original problem. The original nonlinear problem is regarded as the limit (which always exists) of a sequence of linear problems like the classical conduction-convection ones. In this work the nonlinear conduction-radiation heat transfer process is considered and simulated by means of a finite difference linear scheme. Such a limit is reached in an easy way by means of standard procedures, allowing the employment of more realistic hypotheses, like some nonlinear boundary conditions, since the mathematical complexities are not a constraint for simulating the elliptic partial differential equation. This work accounts for the the steady state heat transfer process in rigid fins which experiences convective and radiative heat exchange. Some typical results are shown in order to illustrate the methodology. Results have shown both the relevance of the radiation and the importance of the thermal interaction between the fins, so that there is an effective and realistic thermal mapping. Neglecting the thermal interaction can lead to errors of up to 20 percent.

Optimization of the one-dimensional transient heat conduction problems using extended entransy analyses

Heat transfer optimization principle is critically important for further explaining the underlying mechanisms and guiding practical designs of heat transfer processes. Recently, the entransy theory has been successfully used to optimize various steady-state heat transfer processes. Nevertheless, it is still an open question whether this theory can be utilized in transient cases. Here, we examined the applicability of the entransy analyses on the one-dimensional transient heat conduction process. It was found that the entransy dissipation rate can neither derive the transient governing equation nor correspond to the optimal result in the transient optimization problem. Therefore, an extended entransy dissipation rate was defined as the convolution integral of heat flux and negative temperature gradient. The total extended entransy dissipation rate over the time and space domain can correspond to the optimal result of the transient optimization problem. Additionally, Fourier transform was used to convert the transient problem from the time domain into the frequency domain, and the total entransy dissipation rate in the frequency domain will give a convenient optimization criterion that the temperature gradient field should be spatially uniform to reach the shortest characteristic time. Also, the inverse Fourier transform of the entransy dissipation rate in the frequency will be the extended entransy dissipation rate in the time domain. Finally, these findings were used to optimize a practical transient heat conduction problem for a solid thermal energy storage unit.

Perturbation Solutions for Hagen-Poiseuille Flow and Heat Transfer of Third-Grade Fluid with Temperature-Dependent Viscosities and Internal Heat Generation

Regular perturbation technique is applied to analyze the fluid flow and heat transfer in a pipe containing third-grade fluid with temperature-dependent viscosities and heat generation under slip and no slip conditions. The obtained approximate solutions were used to investigate the effects of slip on the heat transfer characteristics of the laminar flow in a pipe under Reynolds’s and Vogel’s temperature-dependent viscosities. Also, the effects of parameters such as variable viscosity, non-Newtonian parameter, viscous dissipation, and pressure gradient at various values were established. The results of this work were compared with the numerical results found in literature and good agreements were established. The results can be used to advance the analysis and study of the behavior of third-grade fluid flow and steady state heat transfer processes such as those found in coal slurries, polymer solutions, textiles, ceramics, catalytic reactors, and oil recovery applications.

Mathematical modelling of the steady state heat transfer processes in the convectional elements of passive solar heating systems

This paper presents the developed mathematical models of the steady state heat transfer processes in the convectional elements of passive solar heating systems as well as the numerical results of these processes’ simulation. The influence of the adjustable radiant barrier installation on the solar heat gain through the convectional elements of a building’s solar passive heating system has also been analyzed.

Steady State Heat Transfer Within a Nanoscale Spatial Domain

In this paper, we study the steady state heat transfer process within a spatial domain of the transporting medium whose length is of the same order as the distance traveled by thermal waves. In this study, the thermal conductivity is defined as a function of a spatial variable. This is achieved by analyzing an effective thermal diffusivity that is used to match the transient temperature behavior in the case of heat wave propagation by the result obtained from the Fourier theory. Then, combining the defined size-dependent thermal conductivity with Fourier’s law allows us to study the behavior of the heat flux at nanoscale and predict that a decrease of the size of the transporting medium leads to an increase of the heat transfer coefficient which reaches its finite maximal value, contrary to the infinite value predicted by the classical theory. The upper limit value of the heat transfer coefficient is proportional to the ratio of the bulk value of the thermal conductivity to the characteristic length of thermal waves in the transporting medium.