Momentum of a flowing fluid

Momentum flow Into a tank (a) Discrete mass(b) Flowing liquid 

When each mass is brought to rest its momentum is destroyed and a corresponding impulse is thereby imposed on the tank. As the input of a succession of masses increases towards a steady stream, the impulses merge into a steady force. This is also the case with the stream of liquid the fluid s momentum is destroyed at a constant rate and by Newton s second law there must be a force acting on the fluid equal to the rate of change of its momentum. If there is no accumulation of momentum within the tank, the jet s momentum must be destroyed at the same rate as it flows into the tank. The rate of change of momentum of the jet can be expressed as 

Consequently, a force equal to -Mvx is required to retard the jet, ie a force of magnitude Mvx acting in the negative x-direction. By Newton s third law of motion, there must be a reaction of equal magnitude acting on the tank in the positive x-direction. 

Similarly, if a jet of liquid were to issue from the tank with a velocity component vx and mass flow rate Af, there would be a reaction -Mvx acting on the tank. 

Although the fluid flows continuously through the section, the change of momentum is the same as if the fluid were brought to rest in the section then ejected from it. Consequently, Newton s second law of motion can be written as 

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In the preceding examples, cases in which there is a change in the momentum of a flowing fluid have been considered and the reactions on the pipe-work due solely to changes of fluid momentum have been determined. Sometimes it is required to make calculations of all forces acting on a piece of equipment as a result of the presence of the fluid and its flow through the equipment this is illustrated in Example 1.6. 

A second aim has been to make the book more nearly self-contained and to this end a substantial introductory chapter has been written. In addition to the material provided in the first edition, the principles of continuity, momentum of a flowing fluid, and stresses in fluids are discussed. There is also an elementary treatment of turbulence. 

The mass elements of a flowing fluid transfer momentum. This is understood to be the product of the mass and velocity. A mass element dM, which flows at a velocity of transports a momentum w dM = w gdV. The total momentum / transported in a fluid of volume V(/,) is therefore... 

MACROSCOPIC MOMENTUM BALANCE. A momentum balance, similar to the overall mass balance, can be written for the control volume shown in Fig. 4.3, assuming that flow is steady and unidirectional in the x direction. The sum of all forces acting on the fluid in the x direction, by the momentum principle, equals the increase in the time rate of momentum of the flowing fluid. That is to say, the sum of forces acting in the x direction equals the difference between the momentum leaving with the fluid per unit time and that brought in per unit time by the fluid, or... 

In practice, the loss term AF is usually not deterrnined by detailed examination of the flow field. Instead, the momentum and mass balances are employed to determine the pressure and velocity changes these are substituted into the mechanical energy equation and AFis deterrnined by difference. Eor the sudden expansion of a turbulent fluid depicted in Eigure 21b, which deflvers no work to the surroundings, appHcation of equations 49, 60, and 68 yields... 

It is necessary to be able to calculate the energy and momentum of a fluid at various positions in a flow system. It will be seen that energy occurs in a number of forms and that some of these are influenced by the motion of the fluid. In the first part of this chapter the thermodynamic properties of fluids will be discussed. It will then be seen how the thermodynamic relations are modified if the fluid is in motion. In later chapters, the effects of frictional forces will be considered, and the principal methods of measuring flow will be described. 

Derive the momentum equation for the flow of a viscous fluid over a small plane surface. 

Derive the momentum equation for the flow of a viscous fluid over a small plane surface. Show that the velocity profile in the neighbourhood of the surface may be expressed as a sine function which satisfies the boundary conditions at the surface and at the outer edge of the boundary layer. 

Show how the Hagen-Poiseuille equation for the steady laminar flow of a Newtonian fluid in a uniform cylindrical tube can be derived starting from the general microscopic equations of motion (e.g., the continuity and momentum equations). 

If the relative velocity is sufficiently low, the fluid streamlines can follow the contour of the body almost completely all the way around (this is called creeping flow). For this case, the microscopic momentum balance equations in spherical coordinates for the two-dimensional flow [vr(r, 0), v0(r, 0)] of a Newtonian fluid were solved by Stokes for the distribution of pressure and the local stress components. These equations can then be integrated over the surface of the sphere to determine the total drag acting on the sphere, two-thirds of which results from viscous drag and one-third from the non-uniform pressure distribution (refered to as form drag). The result can be expressed in dimensionless form as a theoretical expression for the drag coefficient ... 

If the velocity had the uniform value u, the momentum flow rate would be mfpu2. Thus for laminar flow of a Newtonian fluid in a pipe the momentum flow rate is greater by a factor of 4/3 than it would be if the same fluid with the same mass flow rate had a uniform velocity. This difference is analogous to the different values of a in Bernoulli s equation (equation 1.14). 

Because the quantitative analysis of transport processes in terms of the microscopic description of turbulence is difficult, Kdrmdn suggested (K2) the use of a macroscopic quantity called eddy viscosity to describe the momentum transport in turbulent flow. This quantity, which is dimensionally and physically analogous to kinematic viscosity in the laminar motion of a Newtonian fluid, is defined by... 

During recent years experimental work continued actively upon the macroscopic aspects of thermal transfer. Much work has been done with fluidized beds. Jakob (D5, J2) made some progress in an attempt to correlate the thermal transport to fluidized beds with transfer to plane surfaces. This contribution supplements work by Bartholomew (B3) and Wamsley (Wl) upon fluidized beds and by Schuler (S10) upon transport in fixed-bed reactors. The influence of thermal convection upon laminar boundary layers and their transition to turbulent boundary layers was considered by Merk and Prins (M5). Monaghan (M7) made available a useful approach to the estimation of thermal transport associated with the supersonic flow of a compressible fluid. Monaghan s approximation of Crocco s more general solution (C9) of the momentum and thermal transport in laminar compressible boundary flow permits a rather satisfactory evaluation of the transport from supersonic compressible flow without the need for a detailed iterative solution of the boundary transport for each specific situation. None of these references bears directly on the problem of turbulence in thermal transport and for that reason they have not been treated in detail. 

Fluid motion may be described by applying Newton s second law to a particle. The momentum flow of a substance pvv is equal to the product of the mass flow pv and the barycentric velocity. Newton s second law of motion states that the change in the momentum of a body is equal to the resultant of all forces, mass force F and surface force a, acting on that body. If F, is the force exerted per unit mass of component i, we have... 

Consider turbulent flow in a horizontal pipe, and the upward eddy motion of fluid particles in a layer of lower velocity to an adjacent layer of higher velocity through a differential area ri4 as a result of the velocity fluctuation v, as shown in Fig. 6-21. The mass flow rate of the fluid panicles rising through dA is pu dA, arid its net effect on the layer above dA is a reduction in its average flow velocity because of nioraentum transfer to the fluid particles with lower average flow velocity. This momentum transfer causes the horizontal velocity of the fluid particles to increase by and thus its momentum in the horizontal direction to increase at a rate of pv dA)u, which must be equal to the decrease in the momentum of the upper fluid layer. 

Por the same initial conditions, one can expect the laminar thermal and momentum boundary layers on a fiat plate to have the same thickness when the Prandtl number of the flowing fluid is... 

In addition to these impediments to rheological measurements, some complex fluids exhibit wall slip, yield, or a material instability, so that the actual fluid deformation fails to comply with the intended one. A material instability is distinguished from a hydrodynamic instability in that the former can in principle be predicted from the constitutive relationship for the material alone, while prediction of a flow instability requires a mathematical analysis that involves not only the constitutive equation, but also the equations of motion (i.e., momentum and mass conservation). 

By assuming incompressible flow, the conservation of momentum and mass for a flowing fluid can be described by the Navier-Stokes equation,... 

For integrating Eq. (4-9), vji= ei Er) should be known as a function of and operating variables. However, the momentum diffusivity is the only term we know, with essentially no systematic data for In the case of free turbulence of a homogeneous fluid, the diffusivity of a scalar quantity like heat and mass is estimated to be about two times that of momentum (S4) and the two diffusivities are not far apart for turbulent pipe flow (S8). However, such a relation is not available yet for gas-liquid bubble flow in bubble columns. Generally the local radial mass diffusivity may be expressed by a, with a being a numerical coefficient of order unity. 

Not all of the balance equations are independent of one another, thus the set of equation used to solve particular problems is not solely a matter of convenience. In chemical reactor modeling it is important to recall that all chemical species mass balance equations or all chemical element conservation equations are not independent of the total mass conservation equation. In a similar manner, the angular momentum and linear momentum constraints are not independent for flow of a simple fluid . 

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