Flow, adiabatic forced

The first term on the right-hand side of Eq. (14.113) comes from the inertial forces. Because of the pressure drop the density of gas decreases in the di rection of the flow and therefore, on the basis of mass balance of gas flow, the velocity v increases along the flow. If the pipe is isolated, then the flow can be treated as adiabatic, which on the basis of energy balance implies that along the flow we have... 

For adiabatic, steady-state, and developed gas-liquid two-phase flow in a smooth pipe, assuming immiscible and incompressible phases, the essential variables are pu, pG, Pl, Pg, cr, dh, g, 9, Uls, and Uas, where subscripts L and G represent liquid and gas (or vapor), respectively, p is the density, p is the viscosity, cr is the surface tension, dh is the channel hydraulic diameter, 9 is the channel angle of inclination with respect to the gravity force, or the contact angle, g is the acceleration due to gravity, and Uls and Ugs are the liquid and gas superficial velocities, respectively. The independent dimensionless parameters can be chosen as Ap/pu (where Ap = Pl-Pg), and... 

The first attempt to analyze the flow pattern of an adiabatic gas-liquid two-phase flow in terms of the dominant physical forces acting on the system was made by Quandt (1965). The criteria for prediction of major flow patterns were developed... 

Pattern transition in horizontal adiabatic flow. An accurate analysis of pattern transitions on the basis of prevailing force(s) with flows in horizontal channels was performed and reported by Taitel and Dukler (1976b). In addition to the Froude and Weber numbers, other dimensionless groups used are... 

The scope of coverage includes internal flows of Newtonian and non-Newtonian incompressible fluids, adiabatic and isothermal compressible flows (up to sonic or choking conditions), two-phase (gas-liquid, solid-liquid, and gas-solid) flows, external flows (e.g., drag), and flow in porous media. Applications include dimensional analysis and scale-up, piping systems with fittings for Newtonian and non-Newtonian fluids (for unknown driving force, unknown flow rate, unknown diameter, or most economical diameter), compressible pipe flows up to choked flow, flow measurement and control, pumps, compressors, fluid-particle separation methods (e.g.,... 

In some cases, it will be possible to consider the system as isolated (i.e., not interacting with the surroundings). In order to be isolated, the boundaries of a system must be impermeable to mass and energy. Such boundaries cannot allow any interaction with external mechanical or electrical forces. For example, if there is an external pressure, the walls of the system must be rigid so that they cannot be moved by the pressure. In addition, the system must also be adiabatic (i.e., not allowing any energy to flow through the walls in the absence of such forces). 

A very useful equation to deal with phenomena associated with the flow of fluids is the Bernoulli equation. It can be used to analyse fluid flow along a streamline from a point 1 to a point 2 assuming that the flow is steady, the process is adiabatic and that frictional forces between the fluid and the tube are negligible. Various forms of the equation appear in textbooks on fluid mechanics and physics. A statement in differential form can be obtained ... 

All authors, for instance, consider the jacket oil at constant temperature. This assumption, equivalent to that of infinite oil flow rate, makes it impossible to correctly compute the overall heat transfer coefficient and the thermal driving force. Since heat exchange plays an important role in the conduction of industrial reactors, where more than one third of the polymerization heat is removed through the external cooling oil (only very low conversion reactors can be assumed adiabatic, as claimed by Chen et al.), this limitation cannot be accepted. 

It seems reasonable to assume that similar temperature profiles will also exist when viscous dissipation is important. Attention will first be given to the adiabatic wall case. If the wall is adiabatic and viscous dissipation is neglected, then the solution to the energy equation will be T = Ti everywhere in die flow. However, when viscous dissipation effects are important, the work done by the viscous forces leads to a rise in fluid temperature in the fluid. This temperature will be related to the kinetic energy of the fluid in the freestream flow, i.e., will be related to u /2cp. For this reason, the similarity profiles in the adiabatic wall case when viscous dissipation is important are assumed to have the form ... 

It will be seen that the heat transfer rate is a maximum when is about 50°. At this angle, the flow along both the heated wall and the adiabatic top wall is acted upon by buoyancy forces that are near parallel to the wall as shown in Fig. 10.32. It will also be noted that the highest magnitude of center occurs near this value of . 

These studies have implications for the stability of cratons and diamond exploration. The steep gradients focus flow and adiabatic decompression melting along cratonic boundaries where exten-sional body forces arising from the density contrast between normal asthenosphere and hot plume material will be the largest, indirectly focusing tectonism. This in turn further shields the cratonic cores, although lateral erosion of... 

Here aB is the heat transfer coefficient for nucleate boiling from section 4.2.6, ac is that by forced, single phase flow, sections 3.7.4, page 338, and 3.9.3, page 384, L is the adiabatic mixing temperature of the liquid. 

For systems where the adiabatic temperature rise is low (as is the case considered here) the thermal spikes introduced by the flow reversals do not dramatically affect the reactor performance. However, the concentration of feed streams to such treatment reactors can fluctuate to a high level which can result in a high temperature thermal spike developing within the reactor. Pinjala, Chen, and Luss characterized this dynamic response and showed that reactor runaway could occur within the single-pass reactor. Their work is directly applicable to the RFR as the forced oscillations in the gas flow direction can result in a thermal spike formation at the beginning of each half cycle. Thus, there is a need to understand thermal stability within these systems. Further complicating the matter is the fact that the temperature spikes are very narrow and are thus difficult to detect using thermocouples or other sensors imbedded within the reactor. 

On the other hand, for slow reactions, adiabatic and isothermal calorimeters are used and in the case of very small heat effects, heat-flow micro-calorimeters are suitable. Heat effects of thermodynamic processes lower than 1J are advantageously measured by the micro-calorimeter proposed by Tian (1923) or its modifications. For temperature measurement of the calorimetric vessel and the cover, thermoelectric batteries of thermocouples are used. At exothermic processes, the electromotive force of one battery is proportional to the heat flow between the vessel and the cover. The second battery enables us to compensate the heat evolved in the calorimetric vessel using the Peltier s effect. The endothermic heat effect is compensated using Joule heat. Calvet and Prat (1955, 1958) then improved the Tian s calorimeter, introducing the differential method of measurement using two calorimetric cells, which enabled direct determination of the reaction heat. 

In the foregoing discussion we have seen how the case Ma 0 with an adiabatic wall is an example of incompressible flow. In other instances there is significant heat transfer through the wall. In this case we can isolate the flow situation by imagining that the wall is held at some fixed temperature Tw that is different from Tq. The non-dimensional scale for the temperature is redefined, so we need to redo the analysis of the resulting dimensionless equations. The problem now has a characteristic temperature scale, To — Tw, which is a driving force for the conduction of heat from the wall into the fluid. Since we expect that all temperatures will lie between these two values, the proper non-dimensional temperature is T =. The temperature... 

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