Boundary and Initial Conditions

Boundary and Initial Conditions

Boundary and Initial Conditions

As for another differential equation, the solution is given by boundary and initial conditions. With regard to the boundary conditions, there are several common possibilities that are simply expressed in mathematical form.

Because the heat equation is second order in the spatial coordinates, to describe a heat transfer problem completely, two boundary conditions must be given for each direction of the coordinate system along which heat transfer is significant. Therefore, we need to specify four boundary conditions for two-dimensional problems, and six boundary conditions for three-dimensional problems.

Four kinds of boundary conditions commonly encountered in heat transfer are summarized in following section:

 
Dirichlet Boundary Condition
dirichlet boundary condition - type IIn mathematics, the Dirichlet (or first-type) boundary condition is a type of boundary condition, named after a German mathematician Peter Gustav Lejeune Dirichlet (1805–1859). When imposed on an ordinary or a partial differential equation, the condition specifies the values in which the derivative of a solution is applied within the boundary of the domain.

In heat transfer problems, this condition corresponds to a given fixed surface temperature. The Dirichlet boundary condition is closely approximated, for example, when the surface is in contact with a melting solid or a boiling liquid. In both cases, there is heat transfer at the surface, while the surface remains at the temperature of the phase change process.

Neumann Boundary Condition
Neumann boundary condition - type IIIn mathematics, the Neumann (or second-type) boundary condition is a type of boundary condition, named after a German mathematician Carl Neumann (1832–1925). When imposed on an ordinary or a partial differential equation, it specifies the values that a solution needs to take on along the boundary of the domain.

In heat transfer problems, the Neumann condition corresponds to a given rate of change of temperature. In other words, this condition assumes that the heat flux at the surface of the material is known. The heat flux in the positive x-direction anywhere in the medium, including the boundaries, can be expressed by Fourier’s law of heat conduction.

Special Case – Adiabatic Boundary – Perfectly Insulated Boundary

A special case of this condition corresponds to the perfectly insulated surface for which (∂T/∂x = 0). Heat transfer through a properly insulated surface can be taken to be zero since adequate insulation reduces heat transfer through a surface to negligible levels. Mathematically, this boundary condition can be expressed as:

Neumann boundary condition - adiabatic

Special Case – Thermal Symmetry

Neumann boundary condition - thermal symmetryAnother very important case, that can be used for solving heat transfer problems involving fuel rods, is the thermal symmetry. For example, the two surfaces of a large hot plate of thickness L suspended vertically in air will be subjected to the same thermal conditions, and thus the temperature distribution will be symmetrical (i.e. in one half of the plate will be the same temperature profile as that in the other half). As a result, there must be a maximum in in the center line of the plate and the center line can be viewed as an insulated surface (∂T/∂x = 0). The thermal condition at this plane of symmetry can be expressed as:

Neumann boundary condition - thermal symmetry

Convection Boundary Condition
Newton boundary conditionIn heat transfer problems, the convection boundary condition, known also as the Newton boundary condition, corresponds to the existence of convection heating (or cooling) at the surface and is obtained from the surface energy balance. Convection boundary condition is probably the most common boundary condition encountered in practice since most heat transfer surfaces are exposed to a convective environment at specified parameters.

In other words, this condition assumes that the heat conduction at the surface of the material is equal to the heat convection at the surface in the same direction. Since the boundary cannot store energy, the net heat entering the surface from convective side must leave the surface from the conduction side.

Similarly, the radiation boundary condition can be constructed and used.

Interface Boundary Condition
interface boundary conditionIn heat transfer problems, the interface boundary condition can be used when the material is made up of layers of different materials. Solution of a heat transfer problem in such a medium requires the solution of the heat transfer problem in each layer and one must specify an interface condition at each interface. The interface boundary conditions at an interface are based on the two following requirements:
  • two bodies in contact must have the same temperature at the area of contact (i.e. an ideal contact without contact resistance)
  • an interface cannot store any energy, and therefore the heat conduction at the surface of the first material is equal to the heat conduction at the surface of the second material

Noteworthy, when components are bolted or otherwise pressed together, a knowledge of the thermal performance of such joints are also needed. In these composite systems, the temperature drop across the interface between materials may be appreciable. This temperature drop is characterized by the thermal contact conductance coefficient, hc, which is a property indicating the thermal conductivity, or ability to conduct heat, between two bodies in contact.

The interface boundary condition can be mathematically expressed in the way depicted in the figure.

 
Initial Condition
If the situation is time dependent (transient heat conduction), we have to specify also the initial condition. Since the heat equation is first order in time, only one condition must be specified. In rectangular coordinates, the initial condition can be the initial temperature field specified in the general form as:

Initial Condition - equation

where the function f(x, y, z) represents the temperature field inside the material at time t = 0. Note that under steady conditions, the heat conduction equation does not involve any time derivatives (∂T/∂t = 0), and thus we do not need to specify an initial condition.

 
References:
Heat Transfer:
  1. Fundamentals of Heat and Mass Transfer, 7th Edition. Theodore L. Bergman, Adrienne S. Lavine, Frank P. Incropera. John Wiley & Sons, Incorporated, 2011. ISBN: 9781118137253.
  2. Heat and Mass Transfer. Yunus A. Cengel. McGraw-Hill Education, 2011. ISBN: 9780071077866.
  3. Fundamentals of Heat and Mass Transfer. C. P. Kothandaraman. New Age International, 2006, ISBN: 9788122417722.
  4. U.S. Department of Energy, Thermodynamics, Heat Transfer and Fluid Flow. DOE Fundamentals Handbook, Volume 2 of 3. May 2016.

Nuclear and Reactor Physics:

  1. J. R. Lamarsh, Introduction to Nuclear Reactor Theory, 2nd ed., Addison-Wesley, Reading, MA (1983).
  2. J. R. Lamarsh, A. J. Baratta, Introduction to Nuclear Engineering, 3d ed., Prentice-Hall, 2001, ISBN: 0-201-82498-1.
  3. W. M. Stacey, Nuclear Reactor Physics, John Wiley & Sons, 2001, ISBN: 0- 471-39127-1.
  4. Glasstone, Sesonske. Nuclear Reactor Engineering: Reactor Systems Engineering, Springer; 4th edition, 1994, ISBN: 978-0412985317
  5. W.S.C. Williams. Nuclear and Particle Physics. Clarendon Press; 1 edition, 1991, ISBN: 978-0198520467
  6. G.R.Keepin. Physics of Nuclear Kinetics. Addison-Wesley Pub. Co; 1st edition, 1965
  7. Robert Reed Burn, Introduction to Nuclear Reactor Operation, 1988.
  8. U.S. Department of Energy, Nuclear Physics and Reactor Theory. DOE Fundamentals Handbook, Volume 1 and 2. January 1993.
  9. Paul Reuss, Neutron Physics. EDP Sciences, 2008. ISBN: 978-2759800414.

Advanced Reactor Physics:

  1. K. O. Ott, W. A. Bezella, Introductory Nuclear Reactor Statics, American Nuclear Society, Revised edition (1989), 1989, ISBN: 0-894-48033-2.
  2. K. O. Ott, R. J. Neuhold, Introductory Nuclear Reactor Dynamics, American Nuclear Society, 1985, ISBN: 0-894-48029-4.
  3. D. L. Hetrick, Dynamics of Nuclear Reactors, American Nuclear Society, 1993, ISBN: 0-894-48453-2. 
  4. E. E. Lewis, W. F. Miller, Computational Methods of Neutron Transport, American Nuclear Society, 1993, ISBN: 0-894-48452-4.

See above:

Heat Equation