Module 3. Dimensional analysis and heat transfer

Lesson 11

SIMILARITY OF HEAT-TRANSFER PROCESSES

11.1  Introduction

One of the major means facilitating the solution of heat transfer problem is theory of similarity which is essentially the theory of experimentation.

11.2  Conditions of Unambiguity

Differential equations express things in a most general way because they are based on the general laws of physics. There is an infinite number of heat transfer processes expressed by the equation mentioned, but these processes at the same time differ from each other by certain peculiarities. To limit the problem, isolate the process considered from the infinite number and determine it unambiguously, i.e. to give it a comprehensive mathematical expression, the system of differential equations must be supplemented with a mathematical expression of all individual peculiarities which are defined as conditions of unambiguity.

The condition of unambiguity consists of

1) Geometric conditions which characterise the shape and size of the body in which the process occurs;

2) Physical conditions which characterise the physical properties of the environment and body;

3) Boundary conditions which characterize the peculiarities of the process taking place at the boundaries of the body;

4) Time condition which characterise the peculiarities of the process in terms of time.

The condition of unambiguity may be given in numerical values in the form of functional dependence or in the form of a differential equation. By way of illustration, let us consider heat transfer (heating and cooling) in a flow inside the tube. In this case the following conditions of unambiguity may be given

1. Tube, round, smooth, d in diameter and l long.

2. The working body, i.e. the heat transfer agent, is water, which is incompressible, and its physical parameters are λ (t), c (t), μ (t) and γ (t). If the problem permits neglecting the dependence of the physical parameters on temperature, then the physical properties of water are given in the numerical values of λ, c, μ and γ. If the heat transfer agent is a compressible fluid (gas), then the equation of state must be written for the fluid involved.

3. Fluid inlet temperature is , and that at the surface of the tube t_w. Inlet velocity is w and at the wall w=0.

4. For steady-state processes the conditions characterizing the process in time are immaterial.

Thus, the mathematical expression of a heat-transfer process consists of

1) Equation of heat-transfer

2) Equation of conduction

3) Equation of flow

4) Continuity equation

5) Condition of unambiguity

The application of mathematical analysis to problem in convection heat transfer is in most cases confined to formulation of the problem, i.e. to the composition of the differential equations and statement of the conditions of unambiguity. These equations, however, can be solved only in certain particular cases, provided a number of simplifying assumptions are made; such solutions exist. For example, in solving the problem of heat transfer in a fluid flowing inside a tube, the following simplification are assumed: the tube is considered absolutely smooth, of round cross section; the fluid is incompressible; the motion is steady and laminar, and velocity distribution is parabolic; the fluid inlet temperature is constant; the physical parameters of the fluid are constant and do not depend on temperature. The solution obtained poorly agrees with experiment, because the enumerated assumptions do not satisfy the actual conditions in which the considered process of heat transfer develops. That is why theoretical methods are still of no decisive importance in heat-transfer studies.

11.3  Fundamentals of  Theory of  Similarity

Theory of similarity is the doctorine of similarity of different phenomenon. Concept of similarity is first encountered in geometry. As we know for geometrically similar triangles there is

Where, l₁’, l₂’, l₃’ are some linear dimension of one geometrical figure, and l₁”, l₂”, l₃” are corresponding linear dimension of other figure, but similar to first where c is similarity constant. It is possible to solve a number of practical problems if condition of similarity is known.

Therefore, concept of similarity can be applied to any physical process.

For example

· Similarity of two streams of fluid is called kinematic similarity.

· Similarity of forces giving origin to similar flows is called dynamic similarity.

· Similarity of temperature and heat flows is called heat similarity.

To utilize these concepts, it is necessary to know condition of similarity of considered processes.

1. Concept of similarity is applicable only to physical phenomenon of one and same kind equal qualitatively and described analytically by equations which are similar both in form and content.

Those phenomenon whose analytical equations are similar in form but not in content such phenomenon are defined as Analogous.

For example: Analogy exists between phenomenon of heat conduction and diffusivity.

2. Another necessary pre-requisite for similarity of physical phenomenon is geometrical similarity, which means that similar physical phenomenon should occur in a geometrically similar system.

3. In analyzing similar phenomenon one may compare only homogenous values and only at corresponding space points and corresponding moments of time. Homogenous values are those values which are expressed in equal units.

4. Finally, two physical phenomena are similar if all the values characterizing the phenomena considered are similar.

This means that at corresponding space point and corresponding moment of time any value ∅’ of first phenomenon is proportional to ∅” of second phenomenon in such a way that

∅"= C_P ∅’

where, C_P = similarity constant or factor of similarity transformation.

Factor of similarity transformation depends neither on space-coordinates nor time. Many physical values such as velocity and temperature, as well as physical properties say ρ₀, λ, μ etc. may be different at different points of flow. The similarity of processes requires all these values to be similar in entire space of systems considered. Further, thermal similarity of two streams of fluid requires the streams to be limited by walls of geometrically similar configuration and other physical properties characterizing phenomenon to be similar over entire surface.

By analogy, we can write

In this case of complex processes determined by many physical properties, the similarity constants of these properties are related in a definite way and cannot be chosen arbitrarily. A more profound study reveals that, apart from the constancy of the ratios of similar additional conditions. We shall expound these conditions by considering the following particular examples.

Consider the following case of fluid flow. By definition, velocity w is the ratio of the path l travelled by particle in a period of time τ to this period of time, i.e.

Applying this formula to the corresponding particles of two similar streams of fluid which have travelled similar distances, we have:

Dividing the two equations term-wise, we get

Proceeding from the definition of similarity we may write the following relationships for our case:

Replacing the ratios in equation by their similarity constants from equation we get:

That is exactly sought-for condition of similarity which limits the arbitrary choice of the similarity constant c_w, c_t, and c_τ.

This condition may be presented in a more convenient form. If the constant c_φ are replaced by their values from the relationship and if all terms indexed (‘) are grouped in the left side of the equation and the terms indexed (“) in the right side, we get

This illustrates the main property of similar system. The criteria of similarity are dimensionless terms composed of values characterizing the phenomenon. The main property of the criteria of similarity is their zero dimensionality, which makes it possible to verify the proper composition of the terms.

It is customary to name criteria of similarity or dimensionless terms after the scientists engaged in respective fields of science, and denote them by the first two letters of their names, for instance, Ne (Newton), Re (Reynolds), Eu (Euler), Nu (Nusselt) or simply by capital K.