Conjugate convective heat transfer

Last updated

The contemporary conjugate convective heat transfer model was developed after computers came into wide use in order to substitute the empirical relation of proportionality of heat flux to temperature difference with heat transfer coefficient which was the only tool in theoretical heat convection since the times of Newton. This model, based on a strictly mathematically stated problem, describes the heat transfer between a body and a fluid flowing over or inside it as a result of the interaction of two objects. The physical processes and solutions of the governing equations are considered separately for each object in two subdomains. Matching conditions for these solutions at the interface provide the distributions of temperature and heat flux along the body–flow interface, eliminating the need for a heat transfer coefficient. Moreover, it may be calculated using these data.

Contents

History

The problem of heat transfer in the presence of liquid flowing around the body was first formulated and solved as a coupled problem by Theodore L. Perelman in 1961, [1] who also coined the term conjugate problem of heat transfer. Later T. L. Perelman, in collaboration with A.V. Luikov, [2] developed this approach further. At that time, many other researchers [3] [4] [5] [6] [7] [8] started to solve simple problems using different approaches and joining the solutions for body and fluid on their interface. A review of early conjugate solutions may be found in the book by Dorfman. [9]

Conjugate problem formulation

The conjugate convective heat transfer problem is governed by the set of equations consisting in conformity with physical pattern of two separate systems for body and fluid domains which incorporate the following equations:

Body domain

Unsteady or steady (Laplace or Poisson) two-or three-dimensional conduction equations or simplified one-dimensional equations for thin bodies

Fluid domain

Initial, boundary and conjugate conditions

Methods of conjugation body-fluid separation solutions

Numerical methods

One simple way to realize conjugation is to apply the iterations. The idea of this approach is that each solution for the body or for the fluid produces a boundary condition for other components of the system. The process starts by assuming that one of conjugate conditions exists on the interface. Then, one solves the problem for body or for fluid applying the guessing boundary condition and uses the result as a boundary condition for solving a set of equations for another component, and so on. If this process converges, the desired accuracy may be achieved. However, the rate of convergence highly depends on the first guessing condition, and there is no way to find a proper one, except through trial and error.

Another numerical conjugate procedure is grounded on the simultaneous solution of a large set of governing equations for both subdomains and conjugate conditions. Patankar [10] proposed a method and software for such solutions using one generalized expression for continuously computing the velocities and temperature fields through the whole problem domain while satisfying the conjugate boundary conditions.

Analytical reducing to conduction problem

As shown, [9] the well-known Duhamel's integral for heat flux on a plate with arbitrary variable temperature is a sum of series of consequent temperature derivatives. This series in fact is a general boundary condition which becomes a condition of the third kind in the first approximation. Each of those two expressions in the form of Duhamel's integral or in a series of derivatives reduces a conjugate problem to the solution of only the conduction equation for the body at given conjugate conditions. An example of an early conjugate problem solution using Duhamel's integral has been performed. [8] This approach has been applied [9] both in integral and in series forms and is generalized for laminar and turbulent flows with pressure gradient, for flows at wide range of Prandtl and Reynolds numbers, for compressible flow, for power-law non-Newtonian fluids, for flows with unsteady temperature variations and some other more specific cases.

Applications

Starting from simple examples in the 1960s, the conjugate heat transfer methods have become a more powerful tool for modeling and investigating nature phenomena and engineering systems in different areas ranging from aerospace and nuclear reactors to thermal goods treatment and food processing, from complex procedures in medicine to atmosphere/ocean thermal interaction in meteorology, and from relatively simple units to multistage, nonlinear processes. A detailed review [9] of more than 100 examples of conjugate modeling selected from a list of 200 early and modern publications shows that conjugate methods is now used extensively in a wide range of applications. That also is confirmed by numerous results published after this book appearance (2009) that one may see, for example, at the Web of Science. The applications in specific areas of conjugate heat transfer at periodic boundary conditions [11] and in exchanger ducts [12] are considered in two recent books.

Related Research Articles

<span class="mw-page-title-main">Fluid dynamics</span> Aspects of fluid mechanics involving flow

In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids—liquids and gases. It has several subdisciplines, including aerodynamics and hydrodynamics. Fluid dynamics has a wide range of applications, including calculating forces and moments on aircraft, determining the mass flow rate of petroleum through pipelines, predicting weather patterns, understanding nebulae in interstellar space and modelling fission weapon detonation.

Convection Fluid flow that occurs due to heterogeneous fluid properties and body forces.

Convection is single or multiphase fluid flow that occurs spontaneously due to the combined effects of material property heterogeneity and body forces on a fluid, most commonly density and gravity. When the cause of the convection is unspecified, convection due to the effects of thermal expansion and buoyancy can be assumed. Convection may also take place in soft solids or mixtures where particles can flow.

In fluid dynamics, the Nusselt number (Nu) is the ratio of convective to conductive heat transfer at a boundary in a fluid. Convection includes both advection and diffusion (conduction). The conductive component is measured under the same conditions as the convective but for a hypothetically motionless fluid. It is a dimensionless number, closely related to the fluid's Rayleigh number.

The Grashof number (Gr) is a dimensionless number in fluid dynamics and heat transfer which approximates the ratio of the buoyancy to viscous force acting on a fluid. It frequently arises in the study of situations involving natural convection and is analogous to the Reynolds number. It's believed to be named after Franz Grashof. Though this grouping of terms had already been in use, it wasn't named until around 1921, 28 years after Franz Grashof's death. It's not very clear why the grouping was named after him.

The process by which heat is transferred from the hotter end to the colder end of object is known as conduction.

Newton's law of cooling states that the rate of heat loss of a body is directly proportional to the difference in the temperatures between the body and its environment. The law is frequently qualified to include the condition that the temperature difference is small and the nature of heat transfer mechanism remains the same. As such, it is equivalent to a statement that the heat transfer coefficient, which mediates between heat losses and temperature differences, is a constant. This condition is generally met in heat conduction as the thermal conductivity of most materials is only weakly dependent on temperature. In convective heat transfer, Newton's Law is followed for forced air or pumped fluid cooling, where the properties of the fluid do not vary strongly with temperature, but it is only approximately true for buoyancy-driven convection, where the velocity of the flow increases with temperature difference. Finally, in the case of heat transfer by thermal radiation, Newton's law of cooling holds only for very small temperature differences.

The Biot number (Bi) is a dimensionless quantity used in heat transfer calculations. It is named after the eighteenth century French physicist Jean-Baptiste Biot (1774–1862), and gives a simple index of the ratio of the thermal resistances inside of a body and at the surface of a body. This ratio determines whether or not the temperatures inside a body will vary significantly in space, while the body heats or cools over time, from a thermal gradient applied to its surface.

Heat transfer Transport of thermal energy in physical systems

Heat transfer is a discipline of thermal engineering that concerns the generation, use, conversion, and exchange of thermal energy (heat) between physical systems. Heat transfer is classified into various mechanisms, such as thermal conduction, thermal convection, thermal radiation, and transfer of energy by phase changes. Engineers also consider the transfer of mass of differing chemical species, either cold or hot, to achieve heat transfer. While these mechanisms have distinct characteristics, they often occur simultaneously in the same system.

Boundary layer Layer of fluid in the immediate vicinity of a bounding surface

In physics and fluid mechanics, a boundary layer is the thin layer of fluid in the immediate vicinity of a bounding surface formed by the fluid flowing along the surface. The fluid's interaction with the wall induces a no-slip boundary condition. The flow velocity then monotonically increases above the surface until it returns to the bulk flow velocity. The thin layer consisting of fluid whose velocity has not yet returned to the bulk flow velocity is called the velocity boundary layer.

Lumped-element model

The lumped-element model simplifies the description of the behaviour of spatially distributed physical systems, such as electrical circuits, into a topology consisting of discrete entities that approximate the behaviour of the distributed system under certain assumptions. It is useful in electrical systems, mechanical multibody systems, heat transfer, acoustics, etc. This may be contrasted to distributed parameter systems or models in which the behaviour is distributed spatially and cannot be considered as localized into discrete entities.

In mathematics and its applications, particularly to phase transitions in matter, a Stefan problem is a particular kind of boundary value problem for a system of partial differential equations (PDE), in which the boundary between the phases can move with time. The classical Stefan problem aims to describe the evolution of the boundary between two phases of a material undergoing a phase change, for example the melting of a solid, such as ice to water. This is accomplished by solving heat equations in both regions, subject to given boundary and initial conditions. At the interface between the phases the temperature is set to the phase change temperature. To close the mathematical system a further equation, the Stefan condition, is required. This is an energy balance which defines the position of the moving interface. Note that this evolving boundary is an unknown (hyper-)surface; hence, Stefan problems are examples of free boundary problems.

The heat transfer coefficient or film coefficient, or film effectiveness, in thermodynamics and in mechanics is the proportionality constant between the heat flux and the thermodynamic driving force for the flow of heat :

Thermal hydraulics is the study of hydraulic flow in thermal fluids. The area can be mainly divided into three parts: thermodynamics, fluid mechanics, and heat transfer, but they are often closely linked to each other. A common example is steam generation in power plants and the associated energy transfer to mechanical motion and the change of states of the water while undergoing this process. Thermal-hydraulic analysis can determine important parameters for reactor design such as plant efficiency and coolability of the system.

Falling film evaporator

A falling film evaporator is an industrial device to concentrate solutions, especially with heat sensitive components. The evaporator is a special type of heat exchanger.

Convection (heat transfer) Heat transfer due to combined effects of advection and diffusion

Convection is the transfer of heat from one place to another due to the movement of fluid. Although often discussed as a distinct method of heat transfer, convective heat transfer involves the combined processes of conduction and advection. Convection is usually the dominant form of heat transfer in liquids and gases.

A nanofluid is a fluid containing nanometer-sized particles, called nanoparticles. These fluids are engineered colloidal suspensions of nanoparticles in a base fluid. The nanoparticles used in nanofluids are typically made of metals, oxides, carbides, or carbon nanotubes. Common base fluids include water, ethylene glycol and oil.

In engineering, physics and chemistry, the study of transport phenomena concerns the exchange of mass, energy, charge, momentum and angular momentum between observed and studied systems. While it draws from fields as diverse as continuum mechanics and thermodynamics, it places a heavy emphasis on the commonalities between the topics covered. Mass, momentum, and heat transport all share a very similar mathematical framework, and the parallels between them are exploited in the study of transport phenomena to draw deep mathematical connections that often provide very useful tools in the analysis of one field that are directly derived from the others.

Within thermodynamics, the thermal entrance lengthLeh is used to outline the boundary between the fully developed heat flow and the non-fully developed heat flow in a heated/cooled pipe flow of a fluid.

CFD stands for computational fluid dynamics. As per this technique, the governing differential equations of a flow system or thermal system are known in the form of Navier–Stokes equations, thermal energy equation and species equation with an appropriate equation of state. In the past few years, CFD has been playing an increasingly important role in building design, following its continuing development for over a quarter of a century. The information provided by CFD can be used to analyse the impact of building exhausts to the environment, to predict smoke and fire risks in buildings, to quantify indoor environment quality, and to design natural ventilation systems.

In fluid dynamics, the entrance length is the distance a flow travels after entering a pipe before the flow becomes fully developed. Entrance length refers to the length of the entry region, the area following the pipe entrance where effects originating from the interior wall of the pipe propagate into the flow as an expanding boundary layer. When the boundary layer expands to fill the entire pipe, the developing flow becomes a fully developed flow, where flow characteristics no longer change with increased distance along the pipe. Many different entrance lengths exist to describe a variety of flow conditions. Hydrodynamic entrance length describes the formation of a velocity profile caused by viscous forces propagating from the pipe wall. Thermal entrance length describes the formation of a temperature profile. Awareness of entrance length may be necessary for the effective placement of instrumentation, such as fluid flow meters.

References

  1. ↑ Perelman, T. L. (1961). "On conjugated problems of heat transfer". International Journal of Heat and Mass Transfer . 3 (4): 293–303. doi:10.1016/0017-9310(61)90044-8.
  2. ↑ Luikov, A. V.; Perelman, T. L.; Levitin, R. S.; Gdalevich, L. B. (1971). "Heat transfer from a plate in a compressible gas flow". International Journal of Heat and Mass Transfer . 13 (8): 1261–1270. doi:10.1016/0017-9310(70)90067-0.
  3. ↑ Siegel, R.; Perlmutter, M. (1963). "Laminar Heat Transfer in a Channel with Unsteady Flow and Wall Heating Varying with Position and Time". Journal of Heat Transfer. 85 (4): 358–365. doi:10.1115/1.3686125.
  4. ↑ Chambre, P. L. (1964). "Theoretical analysis of the transient heat transfer into fluid flowing over a flat plate containing internal source". In Johnson, H. A. (ed.). Heat Transfer, Thermodynamics and Education. New York: McGraw-Hill. pp. 59–69.
  5. ↑ Soliman, M.; Johnson, H. A. (1967). "Transient Heat Transfer for Turbulent Flow over a Flat Plate of Appreciable Thermal Capacity and Containing Time-Dependent Heat Source". Journal of Heat Transfer. 89 (4): 362–370. doi:10.1115/1.3614398.
  6. ↑ Sparrow, E. M.; De Farias, F. N. (1968). "Unsteady heat transfer in ducts with time varying inlet temperature and participating walls". International Journal of Heat and Mass Transfer . 11 (5): 837–853. doi:10.1016/0017-9310(68)90128-2.
  7. ↑ Dorfman, A. S. (1970). "Heat transfer from liquid to liquid in a flow past two sides of a plate". High Temperature. 8: 515–520.
  8. 1 2 Viskanta, R.; Abrams, M. (1971). "Thermal interaction of two streams in boundary layer flow separated by a plate". International Journal of Heat and Mass Transfer . 14 (9): 1311–1321. doi:10.1016/0017-9310(71)90180-3.
  9. 1 2 3 4 Dorfman, A. S. (2009). Conjugate Problems in Convective Heat Transfer. Boca Raton: CRC Press.
  10. ↑ Patankar, S. V., 1980, Numerical Heat Transfer and Fluid Flow. Taylor & Francis.
  11. ↑ Zudin, Y. B., 2011, Theory of Periodic Conjugate Heat Transfer, Springer
  12. ↑ Zhang, Li-Zhi, 2013, Conjugate Heat and Mass Transfer in Heat Mass Exchanger Ducts, Academic Press inc.
This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.