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Negative head loss coefficients in branched junctions, has been controversialfor long. Herwig et al. showed, that the cause is a diffusive shear workexchange. Based on their work, a new junction internal model is described,while the conventional head loss is named external. The latter is obtainedexperimentally, while the first cannot. This fact seems to push back reaching apractical solution. Conventionally two head `loss' coefficients are required.However the internal model needs three: two `pure' head loss coefficients and awork coefficient. Based on previous works, the paper shows that the missingequation comes from the Minimum Energy Dissipation Principle (MinEDP). Thecharacteristic differential equation of a T-junction is discovered, whichrelates the two `pure' head loss coefficients. A particular case is presentedto illustrate how to obtain the internal model from the external one by usingit. The results are applied to empirical and numerical data from the samebranched junction; Zhu's measurements (external) and Herwig's CFD computations(internal), respectively. Finally, an example illustrates how powerful this newmethod is to analyse a new, recently published, exhaust return duct.

Characteristic differential equation of a T-junction: diffusive shear work exchange from its head loss coefficients