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:. Unsteady or steady Laplace or Poisson two-or three-dimensional conduction equations or simplified one-dimensional equations for thin bodies. 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  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.
Heat Transfer Principles and Applications - 1st Edition
As shown  , 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 .
This approach has been applied  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.
A detailed review  of more than examples of conjugate modeling selected from a list of 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 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 two recent books.
August 1, Heat Transfer. August ; 3 : — This inverse heat conduction problem is a model of a situation where one wants to determine the surface temperature given measurements inside a heat-conducting body. The problem is ill-posed in the sense that the solution if it exists does not depend continuously on the data. In an earlier paper we showed that replacement of the time derivative by a difference stabilizes the problem. We discuss the numerical stability of this procedure, and we show that, in most cases, a usual explicit e.
Numerical examples are given. The approach of this paper is proposed as an alternative way of implementing space-marching methods for the sideways heat equation. Sign In or Create an Account.
Heat Transfer: A Problem Solving Approach
Sign In. Advanced Search. Article Navigation. Rework equation 1 to solve for P f , by dividing by volume. Method 6. The mass is the total mass of the two tanks because now both tanks are mix in this final state. The total mass is used because we are evaluating the final pressure in the final state. This is the state in which the gas is mixed together so the mass of the whole system needs to be considered.
Method 7. The volume V is the total amount of volume from both tanks for the same reason as the mass. Unfortunately, the volume of the tanks is not given so we need to solve for it. Since the initial pressure, temperature, and mass are given the initial volume of each tank can be calculated using the ideal gas equation shown in equation 1. This is where V 1 , P 1 , and T 1 denote the conditions in tank 1, and V 2 , P 2 , and T 2 denote the initial conditions in tank 2.
Reworking the ideal gas law to solve for V, by dividing by pressure:.
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Method 8. Simplify by removing common terms universal gas constant. Method 9. Method Apply the assumption that there is no work done on the system or change in kinetic or potential energy. This simplifies the equation above by setting work to zero. The initial internal energy is a summation of the internal energy in each tank at the beginning of the process. The general internal energy equation is shown below, where m is the total mass, and u T is the internal energy evaluated at temperature T.
Using the equations above we find the initial internal energy, where m1 is the mass in tank 1, m2 is the mass in tank 2, and T1 and T2 are the initial temperatures in tank one and tank two respectively. The final internal energy is found the same way as the initial internal energy. Use equation 16, yet use the total mass of the system. The final internal energy is as follows:. The law of specific heats allows for a simplification in the difference of the internal energies at two temperatures.
- Conjugate convective heat transfer.
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The use of a specific heat constant, c v , allows for the simplification of the difference of the internal energies at two states to just the temperatures at these states. This law applies to only ideal gases, and can be used due to our assumption of ideal gas. The relationship is seen below in equation Applying this to equation 22 we get. Convert the temperature from Celsius to Kelvin by adding to both initial temperatures.
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