114 CHAPTER 4  Fundamentals of convection

FIGURE 4.6

The conservation of momentum in x-direction for an infinitesimal two-dimensional control volume.

In summary, the equation physically says that the inertial forces on the fluid are balanced by the stresses and body forces acting upon it.

This equation, unfortunately, is still unusable in correct form because of some unknown terms—X, σ, and τ, which are not in terms of the variables we already know. X is the body force per unit volume and usually does not pose a problem because it is a property we specify based on the nature of the body force. The real challenge before us is to relate σxx and τxy to u and v or their derivatives so that u,v remain the only unknowns of the equation. This relation depends on the nature of the fluid’s molecular arrangement and is hence called a constitutive relationship.

The most common type of fluid is called a Newtonian fluid (water, oil, etc., are examples). This is a fluid in which the stress is proportional to rate of deformation or strain. For the normal stress, this is mathematically given by

NOTE: The appearance of 2/3 in the above equation is referred to as the Stokes’ hypothesis and is not always accurate for compressible flow. For incompressible

Substituting for the normal and shear stresses in Eq. (4.21) and rearranging, the X momentum equation becomes

116 CHAPTER 4  Fundamentals of convection

Physically, the first multiplicative term in the RHS of the above equation represents the mass and the second term represents the specific energy. Now, by following an exercise similar to conduction, we obtain for advection

We now move on to the source terms Esource. These sources are internal sources of heat generation; work done by the body forces; and normal and shear stresses.

As earlier, the internal sources of heat generation per unit volume are denoted by qv. The work done by the body forces is given by X V = Xu + Yv

The work done by the normal and shear stresses is given by Wnet , x + Wnet , y where

Substituting all the individual terms in the balance equation, we obtain

the energy equation in the conduction chapter as well.

•Xu + Yv is the work done by the body forces.

•∂∂x (σ xxu) + ∂∂y (σ yyv) is the work done by the normal forces.

•∂∂y (τ xxu) + ∂∂x (τ xyv) is the work done by the shear forces. Eq. (4.38) is not yet usable. We need to

•Apply a constitutive law for σ and τ. We will assume the Newtonian fluid with Stokes’ hypothesis.

•Make any required assumptions about the thermal conductivity. We will assume that k is a constant.

•Make assumptions about compressibility. We will assume that the flow is incom- pressible.

•For an incompressible flow, we can also apply cp = cv = c

•Incorporate the fact that the flow field also satisfies the continuity and momentum equations and make any further simplifications of terms.

On making all the above assumptions, and after a couple of pages of tedious algebra, it is possible to reduce the energy equation to the following froma

aThe tedious derivation is available in many heat transfer textbooks and has been left here as an exercise to the motivated reader.

118 CHAPTER 4  Fundamentals of convection

The terms in the above equation have the following physical meanings:

•The left-hand side represents the total rate of change of internal energy of a fluid element following the fluid. It is a combination of the local rate of change and the advective flux of the internal energy.

•k 2T is the energy flux due to conduction.

•qv is the volumetric heat generation term.

•µ is the heat dissipation due to viscous forces.

In summary, the energy equation shows that the internal energy of a fluid element changes due to a combination of conduction, heat generation, or viscous dissipation. This is not a trivial statement. Note, for instance, that pressure or gravitational work does not play a direct role in internal energy change.

4.4.5  Summary of equations

For a two-dimensional, incompressible flow of constant property fluid the governing equations are,

These are four equations in four unknowns—namely u, v, T, and P; qv is known in the problem a priori. Hence, the problem satisfies closure in the mathematical sense. Given the appropriate initial and boundary conditions, this is self-sufficient to determine the solutions to any convection problem.

4.5  Summary

This chapter covers the concepts of the heat transfer coefficient, the full equations of fluid motion, and the physical significance of the terms. Despite having the full equations, a complete picture of convection is impossible without an understanding of boundary layer theory. We will be looking at boundary layer theory along with its application to forced convection in the next chapter.