What’s wrong with the mathematics of modern engineering?

 

In modern engineering, problems are generally solved with the variables combined.  This is mathematically unsound because it is generally easier to solve problems if the variables are separated.

 

 

Solving problems with the variables combined vs separated

Equations (1) and (2) are identical. 

 

                        (y/x) = 3 + (7/x)                                                                       (1)

 

                        y = 3x + 7                                                                                (2)

 

The only difference between Eqs. (1) and (2) is that x and y are combined in Eq (1), and separated in Eq (2).

 

Because x and y are combined in (y/x), Eq. (1) cannot be solved directly for x given y.  It must be solved iteratively, or by trial-and-error.   Because x and y are separated in Eq. (2), it can be solved directly for x given y, and for y given x. 

 

For example, if Eq. (2) is used to determine the value of x at y = 12, the problem is solved simply and directly, and the solution is obviously x = 5/3. 

 

But if Eq. (1) is used to determine the value of x at y = 12 without first separating x and y, the problem cannot be solved in a simple and direct manner because we do not know the value of (y/x) or the value of (7/x).  It must be solved iteratively, or by trial-and-error.

 

Similarly, if a chart is in the combined variable form y/x vs x, it cannot be read directly to determine x given y.   It must be read iteratively, or by trial-and-error. 

 

But if a chart is in the separated variable form y vs x, it can be read directly to determine x given y, and to determine y given x.

 

 

“Heat transfer coefficient” combines the variables q and DT

“Heat transfer coefficient” is used in modern engineering.  It is defined by Eq. (3):

 

q = h DT                                                                                              (3)

 

where q is heat flux, h is heat transfer coefficient, and DT is temperature difference.

 

In modern engineering, Eq. (3) is said to be “the defining equation for h”.  (The phrase “defining equation” is an oxymoron because equations describe behavior, whereas definitions do not describe behavior.) 

 

It was appropriate for Fourier to write Eq. (3) because he intended it to be a global description of the proportional behavior he had experimentally observed. 

 

But in modern engineering, Eq. (3) does not describe behavior, and therefore it should not be written in the form of an equation.  It should be written in the form of a definition—ie it should be written in the form

 

h ≡ q/ΔT

 

in order to correctly indicate that it does not describe behavior.  It combines the variables q and DT in the ratio q/DT, and assigns the symbol “h” to this ratio.

 

 

A simple example

Nu contains h (ie q/DT) and Ra contains DT.  Therefore heat transfer charts in the form Nu vs Ra must be read iteratively or by trial-and-error if q is given, and DT is to be determined.  (Nu is Nusselt number, and Ra is Rayleigh number.  Charts in the form Nu vs Ra are often used to correlate free convection heat transfer data.) 

 

On the other hand, if a Nu vs Ra chart is first transformed so that q and DT are separated (thereby eliminating h), the transformed chart can be read directly to determine DT given q, and conversely. 

 

Similarly, correlations in the form Nu{Ra} must be solved iteratively or by trial-and-error if q is given, and DT is to be determined.  However, if the correlation is first transformed so that q and DT are separated (thereby eliminating h), the resultant correlation can be solved directly to determine DT given q, and conversely.

 

(The transformation of Nu vs Ra charts and correlations is described in An Improved Form for Natural Convection Heat Transfer Correlations by Eugene F. Adiutori, presented at The ASME-ZSIS International Thermal Science Seminar II held in Bled, Slovenia, June 13-16, 2004.)

 

 

“Electrical resistance” combines the variables E and I

“Electrical resistance” is used in modern engineering.  It is defined by Eq. (4):

 

E = IR                                                                                      (4)

 

where E is electromotive force, I is electric current, and R is electrical resistance.  Eq. (4) combines the variables E and I in the ratio E/I, and assigns the symbol R to this ratio.

 

 (It is interesting to note that Eq. (4) is generally referred to as “Ohm’s law”.  However, the law expressed by Ohm in his 1827 treatise “The Galvanic Circuit Investigated Mathematically”, was not the homogeneous Eq. (4).  It was the inhomogeneous equation

 

                        E = IL                                                                                      (5)

 

in which L is the length of an equivalent copper wire of standard diameter.)

 

 

“Material modulus” combines the variables s and e.

“Material modulus” is used in modern engineering:  It is defined by Eq. (6)

 

s = Ee                                                                                     (6)

 

where s is stress, e is strain, and E is material modulus.  Equation (6) combines the variables s and e in the ratio s/e, and assigns the symbol E to this ratio.

 

 

Why modern engineering is mathematically unsound

In modern engineering, problems are generally solved with the variables combined.  This is mathematically unsound because it is generally easier to solve problems if the variables are separated.