## 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 (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 (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:

**Special Case – Thermal Symmetry**

Another 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:

**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**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**, *h***_{c}**, 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**. 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:

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.

**Heat Transfer:**

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