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Concept# Mass flow rate

Résumé

In physics and engineering, mass flow rate is the mass of a substance which passes per unit of time. Its unit is kilogram per second in SI units, and slug per second or pound per second in US customary units. The common symbol is \dot{m} (ṁ, pronounced "m-dot"), although sometimes μ (Greek lowercase mu) is used.
Sometimes, mass flow rate is termed mass flux or mass current, see for example Schaum's Outline of Fluid Mechanics. In this article, the (more intuitive) definition is used.
Mass flow rate is defined by the limit:
\dot{m} = \lim_{\Delta t \to 0} \frac{\Delta m}{ \Delta t} = \frac{dm}{dt},
i.e., the flow of mass m through a surface per unit time t.
The overdot on the m is Newton's notation for a time derivative. Since mass is a scalar quantity, the mass flow rate (the time derivative of mass) is also a scalar quantity. The change in mass is the amount that flows after crossing the boundar

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The flow through fractured rock is of great importance for many civil engineering projects, for example, when considering the safe storage of nuclear waste or the stability and profitability of dams, tunnels and slopes. How the fluid actually flows through a fractured rockmass is still a matter of vivid discussion within the hydrogeologic community. Various approaches to calculate the flow through a fractured rockmass exist. Which approach is best depends on the rockmass properties and the size of the rockmass under investigation. This study presents a new model which assumes that groundwater flow will take place preferably at the intersection of fractures. Study of the scientific literature showed that flow through intersections of fractures exists and can play a major role in flow through a fractured rockmass. Fieldwork has been conducted at outcrops close to Granada, Spain. At these outcrops fossilized flow paths have been observed at the intersections of fractures. During this fieldwork, fracture data have been gathered, which could be used to reconstruct the fracture geometry. As part of this research a mathematical model has been developed to simulate flow along the intersections of fractures, neglecting flow within the fractures or through the rock matrix. To this end a computer program, CPA, developed for a different purpose, has been modified and completed in order to generate a stochastic tubular network of fracture intersections and calculate flow rates through this network. The new version of the CPA code has been applied to a number of problems to test the correctness of the model and investigate the effects of the different input parameters. The application part of this study can be roughly divided into: network generation, sensitivity analysis, and the comparison with a model, Joint-OKY, that assumes flow to take place through the fractures. The field data gathered during fieldwork in Spain have been used to confirm the geometric modelling capabilities of the CPA program. A comparison between the network model generated by CPA and the observations in the field showed good agreement. The use of an eigenvec- tor approach to represent fracture orientation distribution has proven to be a good and simple method. To better understand the influence of various input parameters in the CPA model, sensitivity analyses were performed on the models generated by the CPA code. The following parameters have been investigated: fracture density, size of the model area, fracture size and anisotropy of conductivity. Finally a comparison between the CPA code and the Joint-OKY code, which assumes flow to take place within the plane of the fractures, has been made. This comparison showed that a model assuming flow within the fractures is more conductive. Further studies are needed to better include the transport of contaminants in the current CPA program.

Improving the energy efficiency of cooling systems can contribute to reduce the emission of greenhouse gases. Currently, most microelectronic applications are air-cooled. Switching to two-phase cooling systems would decrease power consumption and allow for the reuse of the extracted heat. For this type of application, multi-microchannel evaporators are thought to be well adapted. However, such devices have not been tested for a wide range of operating conditions, such that their thermal response to the high non-uniform power map typically generated by microelectronics has not been studied. This research project aims at clarifying these gray areas by investigating the behavior of the two-phase flow of different refrigerants in silicon and copper multi-microchannel evaporators under uniform, non-uniform and transient heat fluxes operating conditions. The test elements use as a heat source a pseudo-chip able to mimic the behavior of a CPU. It is formed by 35 independent sub-heaters, each having its own temperature sensor, such that 35 temperature and 35 heat flux measurements can be made simultaneously. Careful measurements of each pressure drop component (inlet, microchannels and outlet) found in the micro-evaporators showed the importance of the inlet and outlet restriction pressure losses. The overall pressure drop levels found in the copper test section were low enough to possibly be driven by a thermosyphon system. The heat transfer coefficients measured for uniform heat flux conditions were very high and typically followed a V-shape curve. The first branch was associated to the slug flow regime and the second to the annular flow regime. By tracking the minimum level of heat transfer, a transition criteria between the regimes was established, which included the effect of heat flux on the transition. Then for each branch, a different prediction method was used to form the first flow pattern-based prediction method for two-phase heat transfer in microchannels. A non-uniform heat flux creates important temperature gradients in the evaporator, such that the data reduction procedure needs to be adapted to include heat spreading within the evaporator. To do so, a robust multi-dimensional thermal conduction scheme was developed. Once these effects were taken into consideration, the local heat transfer coefficients provided by two-phase flow were found to be the same for uniform and non-uniform heat fluxes, allowing the flow pattern-based method to be extended to non-uniform heat flux conditions. Lastly, with proper control of the mass flow, transient heat flux situations were well handled by the micro-evaporators.

The complexity of two-phase flow boiling on a tube bundle presents many challenges to the understanding of the physical phenomena taking place. It is important to quantify these numerous heat flow mechanisms in order to better describe the performance of tube bundles as a function of the operational conditions. In the present study, the bundle boiling facility at the Laboratory of Heat and Mass Transfer (LTCM) was modified to obtain high-speed videos to characterise the two-phase regimes and some bubble dynamics of the boiling process. It was then used to measure heat transfer on single tubes and in bundle boiling conditions. Pressure drop measurements were also made during adiabatic and diabatic bundle conditions. New enhanced boiling tubes from Wolverine Tube Inc. (Turbo-B5) and the Wieland-Werke AG (Gewa-B5) were investigated using R134a and R236fa as test fluids. The tests were carried out at saturation temperatures Tsat of 5°C and 15°C, mass flow rates from 4 to 35 kg/m2s and heat fluxes from 15 to 70 kW/m2, typical of actual operating conditions. The flow pattern investigation was conducted using visual observations from a borescope inserted in the middle of the bundle. Measurements of the light attenuation of a laser beam through the intertube two-phase flow and local pressure fluctuations with piezo-electric pressure transducers were also taken to further help in characterising the complex flow. Pressure drop measurements and data reduction procedures were revised and used to develop new, improved frictional pressure drop prediction methods for adiabatic and diabatic two-phase conditions. The physical phenomena governing the enhanced tube evaporation process and their effects on the performance of tube bundles were investigated and insight gained. A new method based on a theoretical analysis of thin film evaporation was used to propose a new correlating parameter. A large new database of local heat transfer coefficients were obtained and then utilised to generate improved prediction methods for pool boiling and bundle boiling, including a method for predicting the onset of dryout.