Mathematical Analogies Among Mass, Heat, and Momentum Transfer

1 Mathematical Analogies Among Mass, Heat, and Momentum Transfer 

To explore these analogies, we remember that the diffusion of mass and the conduction of heat obey very similar equations. In particular, diffusion in one dimension is described by the following form of Pick s law  

Although we have not discussed momentum transport in this book, we should mention that this process is also described within the same framework. The basic law is due to Newton  

As a student, I found this conventional analogy confusing. Sure, Eqs. 21.1-1,21.1-3, and 20.1 -5 all say that a flux varies with a first derivative. Sure, Eqs. 21.1 -2,21.1 -4, and 21.1 -6 all have an error function in them. But D, kr, and n do not have the same physical dimensions. Moreover, D appears in both Eq. 21.1-1 and Eq. 21.1-2. In contrast, hr appears in Eq. 21.1-3, but it must be replaced by the thermal diffusivity a in Eq. 21.1 -4. The viscosity n in Eq. 21.1-5 is replaced by the kinematic viscosity u in Eq. 21.1-6. These changes confused me, and initially they undercut any value that these analogies might have. 

The source of my confusion stemmed from the ways in which the basic laws are written. In Tick s law (Eq. 21.1-1), the mass flux is proportional to the gradient of mass per volume, or the molar flux varies with the gradient in moles per volume. To be analogous, the energy flux q should be proportional to the gradient of the energy per volume (pCpT). In other words, Eq. 21.1-3 should be rewritten as 

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