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CONJUGATE CONVECTIVE HEAT TRANSFER

The contemporary conjugate heat transfer model was developed after computer came to use in order to substitute the empirical relation of proportionality heat flux to temperature difference with heat transfer coefficient which was only one tool in theoretical heat convection since time of Newton. This model based on strictly mathematical stated problem describes the heat transfer between body and flowing over or inside it as a result of interaction of two objects. The physical processes and solutions of governing equations are considered separately for each object in two subdomains. Following conjugation of such obtained results gives the distributions of temperature and heat flux along the body/flow interface, and there is no need in heat transfer coefficient. Moreover, it may be calculated using these data.

CONTENTS

History

Conjugate problem formulation

Methods of conjugation of body/fluid separation solutions

Applications

History: The term “Conjugate problem” was coined by the group of sciences leading by academic A. V. Luikov in former Soviet Union [1,2] who among the others in the sixties of the last century began to investigate heat transfer as a coupled problem. At that time, many other researches (for example, [3-8]) 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 [9].

Conjugate problem formulation: 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

a)For laminar flow

Navier-Stokes and energy equations or simplified equations: boundary layer for large and creeping for small Reynolds numbers,respectively.

b)For turbulent flow

Reynolds average Navier-Stokes and energy equations or boundary layer equations for large Reynolds numbers

Initial,boundary and conjugate conditions

a) Conditions specifying the spatial distributions of variables for dynamic and thermal equations at initial time                        b) No-slip condition on the body and other usual conditions for dynamic equations c) Conditions of the first or the second kind specifying temperature or heat flux distribution on the domain boundaries    d) Conjugate conditions on the body/fluid interface providing continuity of the thermal fields by specifying the equalities of temperatures and heat fluxes of a body and a flow at the vicinity of interface:T(+)=T(-),q(+)=q(-).

Methods of conjugation body-fluid separation solutions

a)Numerical methods

One simple way to realize conjugation is to apply the iterations. Idea of such approach is that each solution for the body or for the fluid produces a boundary condition for other component of 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 the 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 solution using one generalized expression for continuous computing the velocities and temperatures fields through whole problem domain with satisfying the conjugate boundary conditions.

b) Analytical reducing to conduction problem.

As shown in [9], the well-known Duhamel 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 integral or in series of derivatives reduces a conjugate problem to the solution of only conduction equation for the body at given conjugate conditions. Paper [8] is an example of early conjugate problem solution using Duhamel integral. In [9] this approach applying both integral and in series forms 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 1960s of the last century, the conjugate heat transfer methods become now a powerful tool for modeling and investigating nature phenomena and engineering systems in different areas from aerospace and nuclear reactors to thermal goods treatment and food processing, from complex procedures in medicine to atmosphere/ocean thermal interaction in meteorology, from relatively simple units to multistage, nonlinear processes. A detailed review of more than 100 examples of conjugate modeling selected from a list of 200 early and modern publications considered in book [9] 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 and in exchanger ducts are considered in books [11] and [12], respectively.

REFERENCES

[1] Perelman, T. L., On conjugated problems of heat transfer, Int. J. Heat Mass Transfer 3: 293-303, 1961.

[2] Luikov, A. V., Perelman, T. L., Levitin, R. S., and Gdalevich, L. B., Heat transfer from a plate in a compressible gas flow, Int. J. Heat Mass Transfer 13: 1261-1270, 1971

[3] Siegel, R., and Perlmutter, M., Laminar heat transfer in a channel with unsteady flow and heating varying with position and time, ASME J. Heat Transfer 85: 358-365, 1963

[4] Chambre, P. L., Theoretical analysis of the transient heat transfer into fluid flowing over a flat plate containing internal source, in Heat Transfer, Thermodynamics and Education, edited by H. A. Johnson, pp59-69, McGraw-Hill, New York,1964

[5] Soliman, M., and Johnson, H. A., Transient heat transfer from turbulent flow over a flat of appreciable thermal capacity and containing time-dependent heat source, ASME J. Heat Transfer 89: 362-370, 1967

[6] Sparrow, E. M., and De Farias, F. N., Unsteady heat transfer in ducts with time varying inlet temperature and participating walls, Int. J. Heat Mass Transfer 11: 837-853, 1968

[7] Dorfman, A. S., Heat transfer from liquid to liquid in a flow past two sides of a plate, High Temperature 8: 515-520, 1970

[8] Viskanta, R., and Abrams, M., Thermal interaction of two streams in boundary layer Flow separated by a plate, Int. J. Heat Mass Transfer 14: 1311-1321, 1971

[9] Dorfman, A. S., 2009, Conjugate Problems in Convective Heat Transfer CRC Press Taylor & Francis, Boca Raton

[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. Navigation menu

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Welcome!
Hello, Abram Dorfman, and welcome to Wikipedia! Thank you for your contributions. I hope you like the place and decide to stay. Here are a few links to pages you might find helpful:


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Re: Conjugate convective heat transfer
Hello Abram Dorfman I have added projects Physics, to your article. You may wish to join them, check their to-do, and meet new people with interest in these topics. ( To reply click "edit" next to this section, and add your reply at the end. ) Cheers, --Gryllida (talk) 02:29, 13 February 2019 (UTC)