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The analytical study of heat conduction in an anharmonic chain is considered here. We investigate an one-dimensional system (directly related to the Frenkel-Kontorova model) with anharmonic cosine on-site potential and harmonic interparticle interaction. We start with a stochastic thermal reservoir connected to each site of the system, and analyse the behaviour of the conductivity in the steady state with all the heat baths as we turn off the interior reservoirs, i.e., as we keep the heat baths at the boundaries only. For a weak interparticle potential and small anharmonicity, in a perturbative computation, we derive an analytic expression for the heat conductivity which indicates that the Fourier's law holds only when each site is connected to a heat bath. To show the trustworthiness of our perturbative computation, we also derive an expression for the conductivity by starting from the exact solution of the linear part of the dynamics and compare with the result which comes from the previous perturbative analysis.

It is known that the presence of trapped air within water distribution pipes can lead to irregular operation or even damage to the distribution systems and their components. The presence of trapped air may occur while the pipes are being filled with water, or while the pumping systems are in operation. The formation of large air pockets can produce the water hammer phenomenon, the instability and the loss of pressure in the water distribution networks. As a result, it can overload the pumps, increase the consumption of electricity, and damage the pumping system. In order to avoid its formation, all of the trapped air should be removed through "air trap valves". In Brazil, manufacturers frequently have unreliable sizing charts, which cause malfunctioning of the "air trap valves". The result of these malfunctions causes accidents of substantial damage. The construction of a test facility will provide a foundation of technical information that will be used to help make decisions when designing a system of pipelines where "air trap valves" are used. To achieve this, all of the valve characteristics (geometric, mechanic, hydraulic and dynamic) should be determined. This paper aims to describe and analyze the experimental apparatus and test procedure to be used to test "air trap valves". The experimental apparatus and test facility will be located at the University of Campinas, Brazil at the College of Civil Engineering, Architecture, and Urbanism in the Hydraulics and Fluid Mechanics laboratory. The experimental apparatus will be comprised of various components (pumps, steel pipes, butterfly valves to control the discharge, flow meter and reservoirs) and instrumentation (pressure transducers, anemometer and proximity sensor). It should be emphasized that all theoretical and experimental procedures should be defined while taking into consideration flow parameters and fluid properties that influence the tests.

We address the problem of the existence of thermal rectification in inhomogeneous (in particular, graded) chains. Searching for analytical results, we investigate the symmetry properties of the thermal conductivity of the quantum inhomogeneous harmonic chain of oscillators with self-consistent reservoirs, an analytically treatable effective anharmonic model (the inner reservoirs mimic the anharmonic on-site potentials). Considering the linear response regime, i.e., for the system submitted to a small gradient of temperature, we show that, even with quantum effects in the conductivity, there is no thermal rectification. Moreover, as a secondary result, we analyze an old expression derived for the thermal conductivity of pure harmonic chains (i.e., a chain with baths at the boundaries only), and prove that there is no thermal rectification in such inhomogeneous systems, as suggested by numerical simulations in previous works.