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This post was last edited by wiseboy on 2012-1-11 10:26. If one side is a heat source at a constant temperature of T, and the fluid on the other side is heated from temperature t1 to t2, then the logarithmic mean temperature difference is: Δtm = ln((t2 – t1)) / ln(1). Theorem 1: If t2 = t1 (= t), that is, the cold end also has a uniform temperature, then Δtm = T – t. Proof: Δtm = lim {(t2 – t1)/ln}. Mathematically, for a 0/0 form of limit, L’Hôpital’s rule is applied by taking derivatives of the numerator and denominator with respect to t1. As t1 approaches t2, it becomes T – t2 = T – t. For example, in a steam-heated tank where the liquid is at temperature T, and if the tank is stirred, then the temperature t throughout the tank is uniform at all times; in this case, Δtm = T – t. It should be noted that if t1 = t2 = t, using the logarithmic mean temperature formula to calculate Δtm will yield an incorrect result; instead, Δtm must be calculated as T – t. 2. Wiseboy’s inference 1: A heat source at a constant temperature of T is used to heat the fluid on the other side from temperature t1 to t2; if the difference between t1 and t2 is not large, then Δtm ≈ T – (t1 + t2)/2. The mathematical proof is omitted here. 3. Wiseboy’s inference 2: A heat source at a constant temperature of T is used to heat the fluid on the other side from temperature t1 to t2. If T >> t1 and T >> t2, then Δtm ≈ T – (t1 + t2)/2; the mathematical proof is omitted here. The significance of Inferences 1 and 2 is that they transform logarithmic calculations into subtraction operations, which reduces the difficulty of analytically solving complex heat transfer models, especially in finding analytical solutions to differential equations and integral equations. Note: The logarithmic solution method is meaningless. These conclusions, which are based on rigorous mathematics, are also used in Vivid Software’s heat transfer calculation software. Statement: Inferences 1 and 2 are the work of Wiseboy; if reproducing them, please indicate “HaiChuan Forum Wiseboy’s inferences””