Jsmk wmk and t is the vector representing the

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J/s/m/K = W/m/K , and T is the vector representing the gradient of temperature. Recall that T is a vector pointing in the direction in which T rises most rapidly. Because of the minus sign, we see then that the thermal energy flows in the direction of most rapid temperature decrease. This law was developed by Joseph Fourier, who built an elegant and correct theory of a special case of non-equilibrium thermodynamics before the laws of equilibrium thermodynamics were formulated, let alone fully understood. Fourier is depicted in Fig. 4.15. Figure 4.15: Jean Baptiste Joseph Fourier (1768-1830), French physicist and math- ematician who developed a correct theory of heat conduction. Image from history/Biographies/Fourier.html . In one dimension, we get q = k dT dx . (4.111) 1 J. B. J. Fourier, 1822, Th´ eorie Analytique de la Chaleur , Chez Firmin Didot, Paris. 1878 English translation. CC BY-NC-ND. 2011, J. M. Powers.
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98 CHAPTER 4. WORK AND HEAT If we multiply by the local cross-sectional area, we find ˙ Q = q A , and ˙ Q = k A dT dx ∼ − k A T hot T cold L . (4.112) Here ˙ Q has units J/s or W (Watts). convection . This is actually a version of conduction, albeit enhanced by fluid flow. For some systems, convective effects are well modeled by Newton’s law of cooling 2 3 : q = h ( T hot T cold ) , (4.113) ˙ Q = q A = h A ( T hot T cold ) . (4.114) Here h is a constant with units W/m 2 /K . thermal radiation . Physically this is due to remote effects. The earth is heated by the sun via radiation effects, not conductive energy diffusion. For some systems, the radiative heat transfer rate is well modeled by q = σ ( T 4 hot T 4 cold ) , (4.115) ˙ Q = q A = σA ( T 4 hot T 4 cold ) . (4.116) Here σ is the Stefan-Boltzmann constant, σ = 5 . 67 × 10 8 W/m 2 /K 4 . We adopt the traditional engineering sign convention for heat: + heat enters the system, - heat leaves the system. The sign convention again is motivated by nineteenth century steam engines. Heat had to be added into a system to get work out of the system. Since engineers were and are concerned with this problem, this convention is taken. We define a special kind of process in which Q = 0 as Adiabatic : a type of process for which there is no heat transfer. The word “adiabatic” was first used by Rankine. 4 It is from the Greek , αδι ´ αβατoς : not to be passed through; in detail, , α (not) + δι ´ α (through) + βατ ´ (passable). An image of Rankine’s text containing the first use of the word is shown in Fig. 4.16. 2 Anonymous, 1701, “Scala graduum caloris,” Philosophical Transactions , 270: 824-829; often attributed to I. Newton. 3 J. A. Ruffner, 1963, “Reinterpretation of the genius of Newton’s ‘law of cooling,’ ” Archive for History of Exact Sciences , 2(2):138-152. 4 W. J. M. Rankine, 1859, A Manual of the Steam Engine and Other Prime Movers , Griffin, London, p. 302.
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