12.2 Heat and Mass nansfer
371
In such a case, the properties are calculated using the temperature on the
current iteration, the temperature is updated, and the process is repeated.
Convergence is usually nearly as rapid as in the fixed-property case.
In some applications, heat conduction in a solid needs to be considered
along with convection in an adjacent fluid. Problems of this kind are called
conjugate heat transfer problems and need to be solved by iterating between
the equations describing the two types of heat transfer. Fully coupled methods
have also been suggested.
Radiation involving solid surfaces has little connection with fluid mechanics (except in problems with multiple active mechanisms of heat transfer). There are interesting problems (for example, flows in rocket nozzles and
combustors) in which both fluid mechanics and radiative heat transfer in the
gas are important. The combination also occurs in astrophysical applications
and in meteorology. We shall not deal with this type of problem here.
In laminar convective heat transfer, the dominant processes are advection (which we previously called convection!) in the streamwise direction and
conduction in the direction normal to the flow. When the flow is turbulent,
much of the role played by conduction in laminar flows is taken over by the
turbulence and is represented by a turbulence model; these models are discussed in Chapter 9. In either case, interest generally centers on exchange of
thermal energy with solid surfaces.
If the temperature differences are small (less than 5 K in water or 10
K in air), the variations of the fluid properties are not important and the
temperature behaves as a passive scalar. Problems of this sort can be treated
by methods described earlier in this book. Since the temperature is a passive
scalar in this case, it can be computed after the computation of the velocity
field has been completely converged, making the task much simpler. In the
case in which the flow is driven by density differences, the latter must be
taken into consideration. This can be done with the aid of the Boussinesq
approximation described below.
Another important special case is that of heat transfer occurring in flows
past bodies of smooth shape. In flows of this type, one can first compute
the potential flow around the body and then use the pressure distribution
obtained as input to a boundary layer code for the prediction of the heat
transfer. If the boundary layer does not separate from the body, it is possible
to compute these flows using the boundary-layer simplification of the NavierStokes equations (see, for example, Kays and Crawford, 1978, or Cebeci and
Bradshaw, 1984). The boundary-layer equations are parabolic and can be
solved in a matter of seconds (for the 2D case) or a minute or so (for the 3D
case) on a modern workstation or personal computer. Methods for computing
these flows have not been covered in detail in this work (but the general
principles are found in Chaps. 3 to 7); the interested reader can find them
in the works by Cebeci and Bradshaw (1984) and Patankar and Spalding
(1977).
371
In such a case, the properties are calculated using the temperature on the
current iteration, the temperature is updated, and the process is repeated.
Convergence is usually nearly as rapid as in the fixed-property case.
In some applications, heat conduction in a solid needs to be considered
along with convection in an adjacent fluid. Problems of this kind are called
conjugate heat transfer problems and need to be solved by iterating between
the equations describing the two types of heat transfer. Fully coupled methods
have also been suggested.
Radiation involving solid surfaces has little connection with fluid mechanics (except in problems with multiple active mechanisms of heat transfer). There are interesting problems (for example, flows in rocket nozzles and
combustors) in which both fluid mechanics and radiative heat transfer in the
gas are important. The combination also occurs in astrophysical applications
and in meteorology. We shall not deal with this type of problem here.
In laminar convective heat transfer, the dominant processes are advection (which we previously called convection!) in the streamwise direction and
conduction in the direction normal to the flow. When the flow is turbulent,
much of the role played by conduction in laminar flows is taken over by the
turbulence and is represented by a turbulence model; these models are discussed in Chapter 9. In either case, interest generally centers on exchange of
thermal energy with solid surfaces.
If the temperature differences are small (less than 5 K in water or 10
K in air), the variations of the fluid properties are not important and the
temperature behaves as a passive scalar. Problems of this sort can be treated
by methods described earlier in this book. Since the temperature is a passive
scalar in this case, it can be computed after the computation of the velocity
field has been completely converged, making the task much simpler. In the
case in which the flow is driven by density differences, the latter must be
taken into consideration. This can be done with the aid of the Boussinesq
approximation described below.
Another important special case is that of heat transfer occurring in flows
past bodies of smooth shape. In flows of this type, one can first compute
the potential flow around the body and then use the pressure distribution
obtained as input to a boundary layer code for the prediction of the heat
transfer. If the boundary layer does not separate from the body, it is possible
to compute these flows using the boundary-layer simplification of the NavierStokes equations (see, for example, Kays and Crawford, 1978, or Cebeci and
Bradshaw, 1984). The boundary-layer equations are parabolic and can be
solved in a matter of seconds (for the 2D case) or a minute or so (for the 3D
case) on a modern workstation or personal computer. Methods for computing
these flows have not been covered in detail in this work (but the general
principles are found in Chaps. 3 to 7); the interested reader can find them
in the works by Cebeci and Bradshaw (1984) and Patankar and Spalding
(1977).