3. Matrix Methods and The Method of Synthesis
3.1. General
Application of the Rayleigh and Buckingham methods show obvious difficulty relating to complexity and direction. It will have been noticed that the solution arrived at, by either method, is dependent on how the initial equation is set up (eg 2.40, 2.65) or how the repeating variables are chosen. Among the infinite variety of correct solutions (available by compounding), there is little or no opportunity to direct the analysis to the ‘best’ or most convenient solution. This problem may be overcome by using Barr’s method of Synthesis (1969, 1971). Also, it will have been noticed that increasing the number of variables adds increasing complexity. While the Rayleigh and Buckingham methods both work for any number of variables and any number of dimensions, they both become excessively tedious as the number of variables increase. Matrix methods overcome this difficulty nicely.
3.2. Barr's method of Synthesis.
The method of synthesis was developed in an attempt to provide the analyst with the opportunity to apply guidance throughout the working of the method. This, it was hoped, would lead to more convenient solutions and would also open the analyst’s mind to the many alternative solutions that were available.
Consider, for example, a fluid system in which a velocity is dependent on viscous effects gravitational action and on one standard length. Thus,
(3.1)
Application of Rayleigh’s or Buckingham's method might then lead to a final standard solution such as
(3.2)
where υ = μ/ρ
However, any dimensionless equation could be considered to have arisen from a related dimensionally homogeneous equation. Thus, rearranging equation 3.2 to give ratios of lengths leads to
(3.3)
and this equation could have arisen from
(3.4)
However, an alternative solution to equation 3.2 (which can be obtained by compounding) is
(3.5)
which could have arisen from
(3.6)
Redundancy, and hence choice, is obtained by combining equations 3.4 and 3.6 to give
(3.7)
2 from
Depending on which two linear terms are chosen, equation 3.7 leads immediately to three different forms of dimensionally homogeneous equation, each of which is a valid development of equation 3.1. Considerable choice is then, again, available in developing the final dimensionless equation from the dimensionally homogeneous equation. This is so because if
(3.8)
is a dimensionally homogeneous equation then
(3.9a)
or
(3.9b)
or
(3.9c)
etc
The rules to be followed in developing equation 3.9 (the dimensionless equation) from equation 3.8 (the dimensionally homogeneous equation) are simple
1) Equation 3.9 must contain all terms of equation 3.8
2) The terms of equation 3.9 must be linked by the terms which appear in equation 3.8 (i.e. by A or B or C etc)
3) Equation 3.9 must contain one term less than equation 3.8
Application of these rules to equation 3.7 together with the choice implicit in the statement of equation 3.7, shows that there are nine possible solutions which can be written directly from equation 3.7. - and this is without any compounding.
Effectively then, the method of synthesis introduces a dimensionally homogeneous equation (eqn-3.7). between the statement of the problem (eqn-3.1) and the dimensionless answers (eqns-3.2 or 3.5). The method does not provide different answers from those which can be obtained by conventional methods. However, unlike conventional analyses, the method of synthesis gives the analyst direct access to a wide range of solutions. The built in redundancy increases with the complexity of the problem thus providing opportunities to guide the analysis to a convenient solution rather than a solution which is merely correct. Compounding can, of course, be used to transform an inconvenient solution to a convenient solution. However the literature suggests that such transformations are not always done. (An example is given at the end of this section).
The dimensionally homogeneous equation must include all relevant lengths of the system together with terms, called linear proportionalities, which are formed by combining the other relevant variables, two at a time to form combinations having the dimensions of length. For example V and g combine to V2/g. This is possible when only two dimensions are present (i.e. ). It is not possible, however, to combine only V and m (dynamic viscosity) because dimensions of mass, length and time are present in these two variables. Nevertheless, any valid non-dimensional parameter includes density, ρ, only in combination with other variables which include the mass dimension. Thus density is not relevant unless combined with another variable which has a mass dimension, and, for the purposes of forming linear proportionalities, density may be eliminated from the list of variables and included as required (Barr, 1969). Temperature may be dealt with in a similar way (Sharp, 1975). Because the dimensional equation is homogeneous in terms of the length dimension it is unnecessary to combine any variables which have the single dimension of length. These are simply added to the list of terms in the dimensionally homogeneous equation.
The method of forming linear proportionalities is purely dimensional and follows the inspectional procedure outlined in Chapter 2 for the development of p terms. For example, consider the combination of dynamic viscosity, m, with gravitational acceleration, g.
(3.10)
Density, r, may be used to eliminate the mass dimension giving
(3.11)
Gravitational acceleration is now combined to eliminate the time dimension:
(3.12)
Thus
(3.13)
Replacing by the kinematic viscosity,ν, gives n2/ 3/g1/ 3 as a linear proportionality.
If velocity, V, and viscosity, m, are to be combined, density would again be introduced, as in equation (3.11), to eliminate the mass dimension. Then, using velocity to eliminate the time dimension gives
(3.14)
and n/V is a linear proportionality.
A fairly complete list of linear proportionalities, relevant to fluid, thermal and structural systems is given in Tables 3.1 and 3.2.
As the number of variables increases, it is obvious that the choice available increases. With m variables to be combined two at a time, it is possible to form = (m-1) +(m-2) + (m-3) + …… + 1 linear proportionalities of which only (m-1) are required in any complete equation. The proviso must be made however that, in choosing the proportionalities needed to define the system, it is necessary to ensure that every variable is included at least once. This was not relevant in equation (3.7) because, with only three variables, it was not possible to choose two linear proportionalities without including V, g, and m.
As an example of the application of Barr’s method, consider again the heat transfer problem discussed in Chapter 2. Again the statement of the problem is
(3.15)
in which k = thermal conductivity, Cp = specific heat, f = function of, and other variables are as previously defined.
Linear proportionalities may be formed by combining the first five variables of Equation (3.15) in pairs, introducing r and DT as required to eliminate mass and temperature dimensions. Thus, from Tables 3.1 and 3..2.
(3.16)
four* from
where 4 * means any four may be chosen provided h, k, Cp, m, and g are each included at least once. One possible choice, leading to a conventional solution, is
(3.17)
term term term term term term
1 2 3 4 5 6
Further choice is now available in forming the non-dimensional equation. A correct arrangement is obtained by taking ratios as follows:
(3.18)
or
(3.19)
in which the first four terms may be recognized as forms of the Nusselt, Prandtl, local Grashof and expansion modulus numbers, respectively. Equation (3.19) provides a correct solution but is inconvenient for investigating k, m , Cp , b, x, or DT because each of these variables appears more than once in the equation. Thus the solution is satisfactory only if the study is restricted to determining the effects of varying h or g.
However, if the study was to concentrate on the effects of varying h, k, m , Cp, and b , a better solution (see Chapter 4) would be
(3.20)
and
(3.21)
Here x, g, DT, and r are used as the repeating variables but all other variables appear only once. Thus Eq. (3.21) provides a good arrangement for the study of these other variables and may be more convenient than the conventional solution of Eq. (3.19). Many other solutions to the problem can be obtained in the same manner. It is stressed that the advantage of the method lies in the choice available and the opportunity to exercise judgement in choosing the most appropriate proportionalities and the most appropriate ratios (Sharp, 1975, 1981)
In addition to providing a more flexible approach to dimensional reasoning, any proposed variant of dimensional analysis must be capable of quickly, and easily, generating the various standard non-dimensional numbers. Table (3.3) demonstrates that the method proposed here is eminently suitable for this purpose. A reasonably complete list of the standard numbers appropriate to fluid and thermal systems has been generated simply by taking ratios of linear proportionalities (obtained directly from Tables 3.1 and 3.2) or by taking the ratio of one proportionality to a relevant length.
Analysis of pipe flow provides a good example of the benefits of the method of synthesis. Flow in pipes may be charaterised by
V = φ ( d, k, Sg, ρ, μ ) (3.22)
Where V = velocity, d = diameter, k = internal equivalent roughness, Sg = reduced gravitational ecceleration, g = gravitational acceleration, S = rate of energy loss (energy gradient) μ = dynamic viscosity and ρ = density.
One solution is
(3.23)
and this is the basis of the well known Moody (or Stanton) diagram which appears in every basic textbook on Fluid Mechanics (and which, perhaps surprisingly, is still taught as a method for solving the basic pipe flow problems). Typically, however, equation (3.23) is written in terms of a friction factor λ (or f) defined by the D'Arcy Weisbach equation
(3.24)
where hf is the head loss over a length L. However,
(3.25)
or
(3.26)
Figure 3.1, and equation (3.23) on which it is based, have become well established because they show clearly the regions in which the flow is dominated by viscous or inertial effects together with the transition and smooth pipe zones. However, until the advent of personmal computers and advanced calculators equation 3.23 and Figure 3.1 were both highly inconvenient for practical purposes, requiring trial and error solutions for the most elementary problems.
The typical engineering problem is to determine the diameter of pipe,d, necessary to carry a given flow,Q, under specified conditions (pipe material, roughness, available head etc). Alternatively the problem could be to determine the discharge capable of being carried by a specified pipe size. In both cases a trial and error solution is required using Figure 3.1 because d appears three time in equation 3.33 ( or Figure 3.1) and V (Q = V π d2/4) appears twice. (Note that this is less of a problem now that computer and vcalculator solutions are easily available for the Colebrook White equation ( 1937, 1939) which specifies the mathematical function of equation 3.23). It is instructive, however, to examine the progression of ideas as a search was made for convenient answers to the most common problems.
The form of Figure 3.1 was first suggested by Stanton in 1914. Almost thirty years later, in 1943, Rouse suggested an improvement leading to a direct solution for velocity. Moody published his diagram in 1944. Then, in 1950, Powell produced a diagram giving a direct solution for diameter, discharge and energy gradient. Thirteen years after that, in 1963, Ackers published a series of charts giving essentially the same solution ( but in a more user friendly format). In addition Ackers included additional information for velocity. Presentation of this data was further developed by Barr and Smith (1967), Barr (1992 – 93), and Hydraulics Research (HRS 1990) in the closing years of the century.
All of these developments, which took almost fifty years to materialise, can be obtained directly (admittedly with hindsight) by the method of synthesis.
From equation 3.22 we can write
(3.27)
2 from
Also, recognising that equation (3.22) could have been written in terms of Q, the discharge, rather than V, the velocity,
(3.28)
2 from
All the solutions discussed earlier are apparent in these two equations. The Moody arrangement, based on equation (3.23), is obtained directly by selecting the first two linear proportionalities of equation (3.27) and using d as the repeating variable to form the dimensionless equation. Rouse's solution can be obtained from the first and third proportionlities of equation (3.27) to give
(3.29)
With d as the repeating variable this gives
(3.30)
which is Rouse's arrangement
Powell based his working on discharge rather than velocity. Thus from equation (3.28)
(3.31)
Now, using k as the repeating variable gives
(3.32)
which is Powell's solution.
Ackers also used k as the repeating variable but, with interest in Q or V and D, he based his solution on
(3.33)
and included Q/νk as an extra lining to his diagram.
Each of these various solutions can therefore be obtained directly from equations (3.27) and (3.28) simply be ensuring that the variables of interest appear as infrequently as possible. This means that proportionalities should be shosen carefully to avoid duplication in the dimensionally homogeneous equation. It also means that variables which are not of direct interest should be used as the repeating variables when the equation is made dimensionless. The long delay in progressing from equation (3.23) and Figure 3.1 to Ackers' solution and the current charts and tables tends to highlight the difficulties which are often exerienced in searching for a convenient solution. Compounding can, and should, be used but often it is not. The advantage of the method of synthesis is that it minimises the difficulties by providing a readily accessible range of solutions which are immediately apparent in the dimensionally homogeneous (length) equation.