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Why q = h{ΔT}ΔT Has Been the de facto Law of Convection Heat Transfer for More Than 100 Years

Eugene F. Adiutori

Ventuno Press, 1094 Sixth Lane N., Naples, FL 34102

Abstract: The purpose of engineering laws is to describe parametric behavior. Experiments indicate that h is a constant or a variable. The law of convection heat transfer cannot be q = hΔT because it states that h is always a constant. Since sometime near the beginning of the 20th century, the equation q = h{ΔT}ΔT has been the de facto law of convection heat transfer because it states that h is a constant or a variable. q = h{ΔT}ΔT is NOT dimensionally homogeneous if h is a variable because the dimension of q is not equal to the dimension of h{ΔT}ΔT. q = hΔT is not acceptable because it states that h is always a constant. q = h{ΔT}ΔT is not acceptable because it is not dimensionally homogeneous if h is a variable. A new law is required that always describes behavior, and is always dimensionally homogeneous.

Keywords: Convection heat transfer, de facto law of convection heat transfer, Fourier, law of convection heat transfer, nonlinear heat transfer behavior

Received October 19, 2023; Revised November 21, 2023; Accepted November 27, 2023

Citation: Eugene F. Adiutori, Why q = h{ΔT}ΔT Has Been the de facto Law of Convection Heat Transfer for More Than 100 Years. American Journal of Mechanical Engineering, 11 (4). 141-142. January 2023. doi:10.12691/ajme-11-4-3

1. Introduction

The first engineering law, q = hΔT, and the methodology Fourier [1] used to create it.

Until 1822, there were no engineering laws because scientists and engineers agreed that equations cannot describe how parameters are related because parameter dimensions cannot be multiplied or divided. That is why Newton’s [2] second law of motion is not f = ma. It is a α f.

Fourier’s treatise [1] published in 1822 presents the first engineering law, Eq. (1), and describes the methodology used to create it.

[picture] | (1)

Fourier’s methodology is based on the following revolutionary and unproven tenets that are still engineering tenets:

• Dimensions can be assigned to constants in equations.

• Parameter dimensions can be multiplied and divided.

• Equations must be dimensionally homogeneous.

Fourier performed a heat transfer experiment and concluded that, if a warm solid body is cooled by the steady-state forced convection of ambient air, heat flux equals a constant times the boundary layer temperature difference. To the constant in the equation, Fourier assigned the symbol h and the dimension of q/ΔT, resulting in Eq. (1), the proportional law of heat transfer from a warm solid body to the steady-state flow of ambient air.

2. Fourier’s Warning

Fourier warned that Eq. (1) is a proportional equation, and therefore it applies only if q is proportional to ΔT. He stressed that Eq. (1) does not apply if heat transfer is by natural convection because the coolant flow rate would not be steady-state, q would not be proportional to ΔT, and h would not be a constant. For most of the nineteenth century, Fourier’s warning was heeded, and Eq. (1) was the proportional law of convection heat transfer from a warm solid body to the steady-state flow of ambient air.

Fourier’s warning has been ignored since sometime near the beginning of the twentieth century.

Sometime near the beginning of the twentieth century, the heat transfer community decided to ignore Fourier’s warning, and use proportional Eq. (1) even if q is not proportional to ΔT, as in natural convection, condensation, and boiling.

The law that should have replaced Eq. (1).

When it was decided to use Eq. (1) even if q is not proportional to ΔT, Eq. (1) should have been replaced by Eq. (2) because it correctly states that q may or may not be proportional to ΔT, and h may be a constant or a variable.

[picture] | (2)

Note that if h is a variable, Eq. (2) is not dimensionally homogeneous because the dimension of q is not equal to the dimension of h{ΔT}ΔT.

Why Eq. (2) has been the de facto law of convection heat transfer.

Eq. (2) has been the de facto law of convection heat transfer since sometime near the beginning of the twentieth century because it underlies conventional heat transfer analyses of problems that concern nonlinear thermal behavior.

Note that if heat transfer is by natural convection, condensation, or boiling, h correlations generally indicate that q is a nonlinear function of ΔT, and h is a variable.

In problems that concern natural convection, condensation, or boiling, the value of q should be determined by first calculating the value of h using a given value of ΔT and an h correlation that states h is a variable, then calculating q from de facto Eq. (2).

The value of q cannot rationally be determined from Eq. (1) because, based on conventional symbolism, Eq. (1) is not an equation if h is a variable. It is a definition of h in the inappropriate form of an equation, and it defines h to be a symbol for q/ΔT.

The problem with Eq. (2).

The problem with Eq. (2) is that it violates the engineering tenet that parametric equations must be dimensionally homogeneous.

3. Conclusions

• Eq. (1) is unacceptable because it indicates that q is always proportional to ΔT.

• Equation (2) has been the de facto law of convection heat transfer for more than 100 years.

• Eq. (2) is unacceptable because it is not dimensionally homogeneous if h is a variable.

• A new law is required that always describes behavior, and is always dimensionally homogeneous.

4. Symbols

a acceleration

f force

h symbol for q/ΔT

m mass

q heat flux

ΔT boundary layer temperature difference

References

[1]   | Fourier, J., 1822, The Analytical Theory of Heat, 1955 Dover edition of 1878 English translation.
[2]   | Newton, I., The Principia, 1726, 3rd edition, translation by Cohen, I. B., and Whitman, A. M.,1999, p. 460, University of California Press.