Momentum of fluid

Since in real fluids, some of the energy of fluid flow is typically converted into heat by viscous forces, it is convenient to generalize equation 9.7 so that it allows for dissipation. Consider the momentum of fluid flowing through the volume dT (= pv). Since its time rate of change is given by d pv)/dt = dp/dt)v -f p dv/dt), we can use equations 9.3 and 9.7 to rewrite this expression as follows ... 

Reynolds number. Experiments show that the velocity at which the transition from laminar to turbulent flow occurs depends on the physical properties of the fluid and the geometry of the flow. The nature of the flow is indicated by a dimensionless group known as the Reynolds number Re. The Reynolds number represents a ratio of inertial forces (rate of change of momentum of fluid elements) to viscous shear forces acting in a fluid. For flow in a pipe the Reynolds number is defined as... 

In order to evaluate mb, we further assume that the momentum of fluid in the axial direction is preserved, as proposed by Crowe and Riesebieter [22,24]. The momentum of gas is negligibly small compared to that of water, and hence, the following equation can be derived. 

If the speed, and thus the momentum, of fluid flowing in a conduit varies with time, a time-varying net force. 

If these assumptions are satisfied then the ideas developed earlier about the mean free path can be used to provide qualitative but useful estimates of the transport properties of a dilute gas. While many varied and complicated processes can take place in fluid systems, such as turbulent flow, pattern fonnation, and so on, the principles on which these flows are analysed are remarkably simple. The description of both simple and complicated flows m fluids is based on five hydrodynamic equations, die Navier-Stokes equations. These equations, in trim, are based upon the mechanical laws of conservation of particles, momentum and energy in a fluid, together with a set of phenomenological equations, such as Fourier s law of themial conduction and Newton s law of fluid friction. When these phenomenological laws are used in combination with the conservation equations, one obtains the Navier-Stokes equations. Our goal here is to derive the phenomenological laws from elementary mean free path considerations, and to obtain estimates of the associated transport coefficients. Flere we will consider themial conduction and viscous flow as examples. 

As velocity continues to rise, the thicknesses of the laminar sublayer and buffer layers decrease, almost in inverse proportion to the velocity. The shear stress becomes almost proportional to the momentum flux (pk ) and is only a modest function of fluid viscosity. Heat and mass transfer (qv) to the wall, which formerly were limited by diffusion throughout the pipe, now are limited mostly by the thin layers at the wall. Both the heat- and mass-transfer rates are increased by the onset of turbulence and continue to rise almost in proportion to the velocity. 

Fluid statics, discussed in Sec. 10 of the Handbook in reference to pressure measurement, is the branch of fluid mechanics in which the fluid velocity is either zero or is uniform and constant relative to an inertial reference frame. With velocity gradients equal to zero, the momentum equation reduces to a simple expression for the pressure field, Vp = pg. Letting z be directed vertically upward, so that g, = —g where g is the gravitational acceleration (9.806 mVs), the pressure field is given by... 

V/c is the ratio of fluid velocity to the speed of sound or aeoustie veloeity, c. The speed of sound is the propagation velocity of infinitesimal pressure disturbances and is derived from a momentum balance. The compression caused by the pressure wave is adiabatic and frictionless, and therefore isentropic. 

Jet pumps are a class of liquid-handling device that makes use of the momentum of one fluid to move another. 

The momentum equation is a mathematieal formulation of the law of eonservation of momentum. It states that the rate of ehange in linear momentum of a volume moving with a fluid is equal to the surfaee forees and body forees aeting on a fluid. Figure 3-2 shows the veloeity eomponents in a generalized turbomaehine. The veloeity veetors as shown are resolved into three mutually perpendieular eomponents the axial eomponent (FJ, the tangential eomponent (Fg), and the radial eomponent (F ). 

An equation for eonservation of momentum ean be derived by eonsidering Newton s seeond law, whieh states that the rate of ehange of momentum of a fluid partiele equals the sum of the forees on the partiele. That is. 

It is possible to determine the x-component of the momentum equation by setting the rate of change of x-momentum of the fluid particle equal to the total force in the x-direction on the element due to surface stresses plus the rate of increase of x-momentum due to sources, which gives ... 

Similarly, integrating the A component of Eq. (13.1) along a streamline shows that the angular momentum of the fluid remains constant throughout everv streamline as follows ... 

Conservation is a general concept widely used in chemical engineering systems analysis. Normally it relates to accounting for flows of heat, mass or momentum (mainly fluid flow) through control volumes within vessels and pipes. This leads to the formation of conservation equations, which, when coupled with the appropriate rate process (for heat, mass or momentum flux respectively), enables equipment (such as heat exchangers, absorbers and pipes etc.) to be sized and its performance in operation predicted. In analysing crystallization and other particulate systems, however, a further conservation equation is... 

Theoretical representation of the behaviour of a hydrocyclone requires adequate analysis of three distinct physical phenomenon taking place in these devices, viz. the understanding of fluid flow, its interactions with the dispersed solid phase and the quantification of shear induced attrition of crystals. Simplified analytical solutions to conservation of mass and momentum equations derived from the Navier-Stokes equation can be used to quantify fluid flow in the hydrocyclone. For dilute slurries, once bulk flow has been quantified in terms of spatial components of velocity, crystal motion can then be traced by balancing forces on the crystals themselves to map out their trajectories. The trajectories for different sizes can then be used to develop a separation efficiency curve, which quantifies performance of the vessel (Bloor and Ingham, 1987). In principle, population balances can be included for crystal attrition in the above description for developing a thorough mathematical model. 

We next consider the consequences of Newton s second law of motion i.e. the consequences of the fact that the rate of change of momentum of a fluid cell must equal the total force that is acting on it. 

By accelerating the gaseous combustion products through the exhaust nozzle, a thrust is imparted to the nozzle and motor case. This thrust is determined by the time rate-of-change of the total momentum of the bounded fluid, as indicated by the expression... 

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