Until 1822, engineers and s cientists agreed that equations cannot describe how parameters are related because parameter dimensions cannot be multiplied or divided . In 1822, Fourier claimed (without proof ) that equations can rationally describe how parameter s are related because parameter dimensions can be multiplied or divided, and dimension s can be assigned to numbers . Fourier’s unproven claim s are the only reason that, since 1822 , equations have been used to describe how engineering parameters are related . However , for more than 70 years , it has been widely agreed that dimensions must not be assigned to numbers . Because p arameters such as h and E were created by assigning dimensions to numbers , they are irrational , and equations in which they appear should be abandoned . The proposed paradigm shift requires that parameter symbols represent only numerical values , and results in engineering laws that are analogs of y = f{x} . The new laws state that the numerical value of parameter y is a function of the numerical value of parameter x, and the function may be proportional, linear, or nonlinear. Because parameter symbols represent only numerical value, all proportions and equations are dimensionally homogeneous because they are inherently dimensionless. If an equation is quantitative , the dimension units that underlie parameter symbol s must be specified in an accompanying nomenclature . The proposed paradigm shift result s in a rational engineering science that is much easier to learn and apply because irrational parameters such as h and E are abandoned . They are not replaced because they are not necessary. Key words: Dimensional homogeneity, d imensions, ir rational equations, irrational laws, Newton’s laws, paradigm shift, parameters , proportions, rational equations, rational laws. 1. Introduction Until 1822, it was generally agreed that equations cannot describe how parameters are related because parameter dimension s cannot be multiplied or divided . In 1822, Fourier [1] claimed that equations can rationally describe how parameters are related because: • Parameter dimensions can be multiplied or divided . • Dimension s can rationally be assigned to numbers . These revolutionary claims made it possible for the first time in history to create dimensionally homogeneous equations that quantitatively describe how parameters are related. 2 However, s ince s ometime before 1951, it has been widely agreed that dimensions must not be assigned to numbers. Therefore all conventional engineering laws and equations that include parameters created by assigning dimensions to numbers (such as h and E) are irrational , and should be replaced by the rational laws and equations that result from the paradigm shift . The proposed paradigm shift requires that parameter symbols in equations represent only numerical values , and results in laws that are analogs of Eq. (1) in which symbols represent only the numerical values of parameters. y = f{x} (1) These laws make it much simpler to learn and apply engineering science because all parameters (such as h and E) created by assigning dimensions to numbers are abandoned . They are not replaced because they are not necessary . 2. Dimensional homogeneity until 1822. Until 1822, scientists and engineers such as Galileo and Newton agreed that: • Parameter symbols in proportions and equations represent numerical value and dimension. • Parameters cannot be multiplied or divided because parameter dimensions cannot be multiplied or divided.1 • Because equations generally require that parameter s be multiplied or divided, and because parameter dimension s cannot be multiplied or divided, equations cannot describe how parameters are related. • Proportions need not be dimensionally homogeneous. • Equations must be dimensionally homogeneous. Because proportions need not be dimensionally homogeneous, and because proportions that relate two parameters do not require that parameter s be multiplied or divided, proportions are generally used instead of equations . That is why Hooke’s [ 2] law is Proportion (1) instead of an equation , Newton’s [ 3] law of cooling2 is Proportion ( 2) instead of Eq. (3) , and Newton’s [4] second law of motion is Proportion (4) instead of Eq. (5). 1 However, a dimension can be divided by the same dimension. For example, meters can be divided by meters, and seconds can be divided by seconds, but meters cannot be divided by seconds. In the qualitative equations that result , all terms are dimensionless ratios. This methodology was used by Galileo . 2 American heat transfer texts generally refer to Eq. (3) as “Newton’s law of cooling” , and claim that Newton created h. However, Eq. (3) cannot be Newton’s law of cooling because cooling is a transient phenomenon , and Eq. (3) is a steady -state equation . Also because Eq. (3) requires that h and T be multiplied , whereas in Newton’s time, it was irrational to multiply parameters. Newton could not have created h because he could not rationally have multiplied h times another parameter. 3    (1) (dT body/dt)  (Tair – Tbody) (2) q = hT (3) a  f (4) f = ma (5) 3. Fourier’s heat transfer experiment, and the proportion and equation that resulted. Fourier performed a heat transfer experiment in which a warm, solid body is cooled by the steady -state forced convection of ambient air. Fourier concluded that Proportion (6) and Eq. (7) correlate th e data. q  T (6) q = cT (7) Newton and his colleagues would have been satisfied by Proportion (6), but it did not satisfy Fourier because he wanted an equation , and it had to be dimensionally homogeneous. Equation (7) did not satisfy Fourier because it is not dimensionally homogeneous. 4. Fourier ’s revolutionary and unproven view of dimensional homogeneity that enabled him to transform inhomogeneous Eq. (7) to homogeneous Eq. (8). Fourier recognized that Eq. (7) could be transformed to a dimensionally homogeneous equation only if it were rational to assign dimensions to number c in Eq. (7), and rational to multiply and divide parameter dimensions. Fourier [1] describes his revolutionary and unproven view of dimensional homogeneity in the following: . . . every undetermined magnitude or constant has one dimension proper to itself, and the terms of one and the same equation could not be compared, if they had not the same exponent of dimension. . . this consideration is derived from primary notions on quantities; for which reason, in geometry and mechanics, it is the equivalent of the fundamental lemmas which the Greeks have left us without proof . It is important to note that, in Fourier’s nearly 500 page treatise, The Analytical Theory of Heat [1], he made no effort to prove that his view of dimensional homogeneity is rational. He did not include the Greek lemmas, he did not cite a reference where the Greek lemmas could be found, and he did not include his own proof. Fourier’s revolutionary view of dimensional homogeneity includes the following unproven claims: 4 • Parameter dimensions can be multiplied or divided. • Dimension s can be assigned to numbers . In accordance with his revolutionary and unproven view of dimensional homogeneity, Fourier assigned the symbol h and the dimension of q/T to number c in Eq. (7), then multiplied h and T, resulting in dimensionally homogeneous Eq. (8). q = hT (8) 5. The definition of h. American heat transfer texts generally do not define h. Nomenclatures in heat transfer texts generally state only that h is “heat transfer coefficient ”. However, rearranging Eq. (8) results in Eq. (9). h = q/T (9) Equation s (8) and (9) state that h and q/T are identical and interchangeable . They also state that h is a symbol for the dimensional group q/T. Combining Eqs. (8) and (9) results in Eq. (10). Note that Eqs. (8) and (10) are identical because h and q/T are identical and interchangeable . q = (q/T)T (10) The nomenclature in every conventional heat transfer text should state “ h is a symbol for the dimensional group q/T—i.e. h and q/T are identical and interchangeable ”. 6. What Eq. (8) meant in most of the nineteenth century. In most of the nineteenth century, Eq. (8) was a proportional equation , and h was a proportionality constant . Fourier warned that Eq. (8) applies only if a solid, warm body is cooled by the steady -state forced convection of ambient air . He emphasized that Eq. (8 ) does not apply if a solid, warm body is cooled by the natural convection of ambient air because the coolant flow rate would vary, and consequently the relationship between q and T would not be proportional. 7. What Eq. (8) has meant since sometime near the end of the nineteenth century. Sometime near the end of the nineteenth century, the heat transfer community decided to ignore Fourier’s warning that Eq. (8) applies only if the heat transfer behavior is proportional. It decided to apply Eq. (8) even if the relationship between q and T is nonlinear . 5 When Eq. (8) is applied to nonlinear heat transfer phenomena, it is not an equation because a proportional equation cannot describe nonlinear behavior. Even though Eq. (8) is a proportional equation, it must now be interpreted to mean that the relationship between q and T may be proportional, linear, or nonlinear, and h may be a constant or a variable . 8. The equation that should have replaced Eq. (8) when it began to be applied to nonlinear phenomena . When the decision was made to apply Eq. (8) to nonlinear phenomena, Eq. (8) should have been abandoned because it obviously cannot describe nonlinear behavior. Equation (8) should have been replaced by Eq. ( 11) because it correctly states that the relationship between q and T may be proportional, linear, or nonlinear , and h may be a constant or a variable . q = h{T}T (11) Note that Eqs. (11) and (11a) are identical . They both state that q is a function of T, and the function may be proportional, linear, or nonlinear. q = f{T} (11a) However, Eq. (11 a) could not rationally have replaced Eq. (8) because , based on conventional parameter symbolism, Eq. (11a) is not dimensionally homogeneous. 9. Why conventional engineering laws are mathematically undesirable . Substituting q/T for h in Eq. ( 11) results in Eq. (1 2). q = (q/T){T}T (12) Equation (12) is a rigorously correct expression of the modern law of convective heat transfer. Note that Eq. (1 2) is an analog of Eq. (1 3), and q/T (i.e. h) is an analog of (y/x){x}. y = (y/x) {x}x (13) In mathematics, if y is a nonlinear function of x, Eq. (1 3) is never used because (y/x){x} is a third variable , and it greatly complicates problem solutions. Equation (1 4) is always used because it always has only two variables, and therefore it always allows nonlinear problems to be solved in the simplest possible way —ie with y and x separated rather than combined in an analog of (y/x){x} . y = f{x} (14) Laws such as Eqs. (8), (10), (11), and (12) are mathematically undesirable because , if q is a nonlinear function of T, they include three variables ( q, q/T, and T) to describe the relationship between two variables ( q and T). And similarly for all proportional engineering laws that are applied to nonlinear phenomena. 6 10. Proof that Fourier was wrong . Dimensions cannot rationally be assigned to numbers. Langhaar [5] explains why dimensions cannot rationally be assigned to numbers: Dimensions must not be assigned to numbers, for then any equation could be regarded as dimensionally homogeneous . 11. Proof that Fourier was wrong . Equations cannot describe how parameters are related because p arameter dimensions cannot rationally be multiplied or divided . Conventional engineering laws and equations are based in part on Fourier’s unproven claim that parameters can be multiplied or divided because parameter dimensions can be multiplied or divided. Fourier was wrong . As demonstrated by the following, parameter dimensions cannot rationally be multiplied or divided. “Multiply four times seven ” means “add seven four times”. Therefore “multiply meters times kilograms” must mean “add kilograms meters times”. Because “add kilograms meters times” has no meaning , dimensions cannot be multiplied. “Divide forty by five” means “how many fives are in forty ”. Therefore “divide meters by seconds” must mean “how many seconds are in meters”. Because “how many seconds are in meters” has no meaning , dimensions cannot be divided. If meters could be divided by seconds, it would be possible to determine how many seconds are in meters. 12. Irrational views in conventional engineering . The following are irrational views in conventional engineering. • Hooke’s law, Proportion (15), and Young’s law, Eq. (16), are identical . They both state that stress equals a constant times strain. Therefore it is irrational to require Young’s law to be dimensionally homogeneous, and not require Hooke’s law to be dimensionally homogeneous.    (15)  = E elastic (16) • A chart of q vs T is a picture of Eq. (1 7). q = f{T} (17) It is irrational to reject Eq. (1 7) because it is not dimensionally homogeneous, and to accept a chart of Eq. (1 7) that is not dimensionally homogeneous. • Charts are dimensionless because they describe how the numerical value of parameter group y is related to the numerical value of parameter group x. If a chart is to be quantitative , the dimension units that underlie paramete r groups x and y must be specified on the chart, or in an accompanying nomenclature. 7 Because charts are pictures of equations, and because charts are dimensionless , it is irrational to not have equations in which parameter symbols are dimension less. The above irrational views have no place in the engineering science that results from the paradigm shift . 13. What e quations can rationally describe about how parameters are related. Equations can rationally describe how the numerical values of parameters are related because equations are inherently dimensionally homogeneous if parameter symbols represent only numerical value . If a dimensionless equation is quantitative , the dimension units that underlie parameter symbols (that are not in dimensionless groups) must be specified in an accompanying nomenclature . 14. The proposed paradigm shift. The proposed paradigm shift requires th at parameter symbols in proportions and equations represent only numerical values . 15. The engineering science that results from the proposed paradigm shift. The engineering science that results from the proposed paradigm shift i s described by : • All proportions and equations are dimensionless because all parameter symbols represent only numerical value. • All proportions and equations are dimensionally homogeneous because they are dimensionless. • If an equation is quantitative , the dimension units that underlie parame ter symbols are specified in an accompanying nomenclature (excep t for symbols in dimens ionless groups). • All engineering laws are replaced by analogs of Eq. ( 18) which states that the numerical value of parameter y is a function of the numerical value of parameter x, and the function may be proportional, linear, or nonlinear. y = f{x} (18) • There are no parameters that were created by assigning dimensions to numbers. 16. How to transform conventional texts to texts based on the proposed paradigm shift . To transform conventional engineering texts to texts based on the proposed paradigm shift: • Replace laws with analogs of y = f{x} . • In equations that include analog s of (y/x), replace analog s with y/x, then separate x and y. 8 For example, to tra nsform Eq. ( 19) to a paradigm shift equation, replace h and kwall/twall with q/T, then separate q and T, resulting in Eq. ( 20). U = (1/h 1 + twall/kwall + 1/h 2)-1 (19) Ttotal = T1{q} + Twall{q} + T2{q} (20) It is important to note that Eqs. ( 19) and ( 20) are identical . They mean exactly the same thing . Equation ( 19) is written in the opaque language of conventional engineering. Equation (2 0) is written in the transparent language of the proposed paradigm shift. (Convection heat transfer correlations are generally in the form T{q} because that is the form required by Eq. (20.)) Textbooks for o ther branches of engineering are transformed to paradigm shift texts in the same way heat transfer texts are transformed —i.e. by replacing conventional laws with laws that are analogs of Eq. (14), and transforming equations by separating x and y. 17. How data are correlated in conventional engineering, and in engineering based on the proposed paradigm shift. Experimenters cannot obtain h data or E data because there is no such thing as h or E. They are symbol s for the dimensional groups q/T and /. In conventional engineering, experimenters obtain q data and T data, and use it to determine q/T values and (q/T){T} correlations —ie to determine h values and h{T} correlations . In engineering based on the proposed paradigm shift, experimenters obtain q data and T data, and use it to determine T{q} correlations . And similarly for other engineering branches . 18. Correlation transformation s and experiments . It is important to n ote that the proposed paradigm shift does not require that experiments that resulted in current correlations be repeated . It requires merely that current correlations be transformed analytically as described in Section 15, or that the data that resulted in current correlations be used to determine correlations that are analogs of y = f{x} . 20. Conclusions Conventional engineering science works well when applied to problems that concern proportional behavior because it is founded on laws that are proportional equations , and the coefficients in the laws ( such as h and E) are proportionality constants . It does not work well when applied to problems that concern nonlinear behavior because the coefficients in the laws (such as h and E) are extraneous variables , and they greatly complicate problem solutions. Engineering science should be founded on the proposed paradigm shift because it results in laws that work well with all forms of behavior —proportional, linear, and nonlinear. 9 Nomenclature a acceleration c arbitrary constant E modulus F force h q/T k q/(dT/dx) m mass q heat flux T temperature t time or wall thickness x arbitrary variable y arbitrary variable  strain  stress