Entrance Condition (Thermal Boundary Layer)

Entrance Condition – Thermal Boundary Layer

Introduction

Convection is a type of heat transfer mode responsible for the heat transfer between surfaces and moving fluids. Convection heat transfer is comprised of two different mechanisms, energy transfer due to the random molecular motion of particles as well as macroscopic/bulk fluid motion (referred to as advection). Given a situation where the flow temperature and the surface temperature differ, there exists an intermediate region where the temperature varies between these two regions

Boundary Layer Development

• • Random molecular motion (diffusion) contribution dominates where the fluid velocity is low. Where fluid velocity is zero (at the interface between the surface and fluid), this is the only mechanism for heat transfer present.

• Boundary layer grows in the downstream of the flow as a consequence of the macroscopic fluid motion carrying the heat generated upstream.

Relation to Velocity Boundary Layer

• May be larger, smaller or equal in size to the velocity boundary layer.

• The ratio between velocity and thermal boundary layer is related by the Prandtl number raised to a power of ‘n’.

Full Development Conditions

The existence of convection heat transfer between the surface and the fluid dictates that there must be a continuously changing with respect to x, hence full thermal development conditions cannot be reached the same way as described in the hydrodynamic situation where the rate of change of velocity reached zero at full development. Alternatively, the rate of change of temperature with respect to x cannot equal zero at any radius or value of x. Instead, thermal development is determined by the point where the dimensionless ratio becomes independent of x. Formally, this is expressed as the following condition:

The condition above can only be met under two different circumstances:

• • Uniform surface heat flux (q'' is constant). An example of this situation would be if uniform irradiation of the outer surface was achieved through external electrical heating.

  • Isothermal surface (T is constant throughout the surface). This condition would be achieved if a phase change was present (boiling/condensation).

It is important to note that both these conditions cannot be achieved at the same time. If heat flux is constant, surface temperature must change with x and vice versa. This condition allows inferences to be made that leads to multiple important conclusions. The evaluation of the derivative above (making the inference that the ratio must also be independent of x with respect to the radius) obtains the following.

The above illustrates how the rate of change of the dimensionless temperature ratio is not a function of displacement. Furthermore, using a substitution from Foriers law, the following may be derived

The equation above indicates that for a fully developed thermally flow, with constant thermodynamic properties (constant k), the local convection coefficient (h) can be considered to be constant. Note that due to the thin boundary length at the beginning of the flow, the local convection coefficient must be much higher as the transition region is much lower.

Additional Resources

The principle boundaries in place hindering the intuitive understanding of thermal boundary layers is as a result of the abstract nature of the concept. However, this concept becomes less abstract when shown visualisations of a thermal boundary layer as can be seen by the images below. (Click the images to see a helpful video)

NOTE. A lot of the information found above is summarised to be concise and is intended to be a revision or a supplement to learning. For a full description of the information above, a reliable source of which a lot of the content of this study tool is based (including figures) can be found in "Fundamentals of Heat and Mass Transfer - 7th edition".