Conservation, laws

Conservation of energy follows directly from Hamilton s equations. The differential change of the Hamiltonian along a trajectory is 

Since Hamilton s equations imply that pk = 0 if = 0, pk is a constant of motion if qt is such an ignorable coordinate. An ingenious choice of generalized coordinates can produce such constants and simplify the numerical or analytic task of integrating the equations of motion. 

The close connection between symmetry transformations and conservation laws was first noted by Jacobi, and later formulated as Noether s theorem invariance of the Lagrangian under a one-parameter transformation implies the existence of a conserved quantity associated with the generator of the transformation [304], The equations of motion imply that the time derivative of any function 3(p, q) is 

By Noether s theorem, invariance of the Lagrangian under an infinitesimal time displacement implies conservation of energy. This is consistent with the direct proof of energy conservation given above, when L and by implication H have no explicit time dependence. Define a continuous time displacement by the transformation t = t + oi(t ) whereat/(,) = a(t ) = 0. subject to a —0. Time intervals on the original and displaced trajectories are related by dt = (1 + a )dt or dt = (1 — a )dt. The transformed Lagrangian is 

Similarly for translational invariance, an infinitesimal coordinate translation is defined for a single particle by 

Tliree key conservation laws - mass, energy, and momentum this section. 

Tlie con sen at ion law for mass can be applied to any process or system. Tlie general form of tliis law is given by Eq. (4.5.1). 

A steady-state process is one in wliich there is no change in conditions (temperature, pressure, etc.) or rates of flow with time at any given point in die system. The accumulation term in Eq. (4.5.1) is dien zero. If diere is no cheniieid reaetion, the generation tenn is also zero. All other processes are unsteady state. 

In a batch process, a given quantity of reacttuits is plaeed in a container, and by ehemieal and/or physieal means, a eliange is made to oecur. At the end of die process, die container holds die product or products. In a continuous process, reactants are fed in an unending flow to a piece of equipment or to several pieces in series, and products are continuously removed from one or more points. A continuous process may or may not be steady state. 

As indicated previously, Eq. (4.5.1) may be applied to die total mass of each stream (referred to as an overall or total material balance) or to die individual eomponents of the streams (referred to as a componential or component material balance). Often the primary task in preparing i maleritil bahuice is to dei elop the qiumtitative relationsliips among the streams. 

The simplest demonstration of how symmetry fixes natural laws is by the effect of symmetries on the motion of non-relativistic classical particles. 

The relativistic, or Lorentz transformation (or boost) is a spatiotemporal transformation, which for relative motion along x reads 

The non-relativistic analogue, or Galilei transformation is the limiting form of the Lorentz boost as c — 00 vjc — 0), i.e. 

Assume a potential V that only depends on the coordinate x of the particle under one-dimensional translation. Since V does not depend on time the law that governs evolution of the system must remain constant in time. The total energy 

Thus the total energy does not change with time and therefore is conserved. The conservation of energy clearly is a consequence of the fact that the laws of Nature do not change with time, i.e. of their temporal homogeneity. 

There are several natural conservation laws. We have a natural feeling for some of these conservation laws. Distances in a distance table should follow certain conservation laws. Traveling from Aio B directly should not show a longer distance than traveling from A to B via C. But in practice, some of these distance tables show this unexpected property - due to errors. 

The equations of fluid motion can be derived from the physical principles of conservation of mass, momentum and energy. The standard form of such conservation equations is 

t) is the density of the quantity which is conserved, J(x, t) is the flux, i.e. the amount of that quantity crossing the unit surface at location x per unit of time. 5(x, t) is the production (or consumption) rate of the quantity A per unit of volume at location x. When 5 = 0, the balance between the terms in the left-hand-side indicates that the quantity A changes locally just because it is moving from place to place, so that it is globally conserved. 

The conservation of the mass of a fluid in a velocity field v(x, t) is expressed by Eq. (1.1) for the fluid density (mass per unit of volume) field p(x, t), with 5 = 0. The only process moving the fluid mass is transport by the velocity v, giving the advective flux J = pv. Thus, Eq. (1.1) becomes the continuity equation 

The second term can be written as a sum of two components v Vp that represents changes in the local density due to the flow bringing-in fluid of different density from elsewhere, and pV v representing compression or expansion of the fluid volume when the velocity field is convergent (V v 0) or divergent (V v 0). 

Under normal conditions most fluids are not compressed much in a flow. In general, if the typical flow velocity (U) is much smaller than the speed of sound (c) in the medium (cair 340 m/s, cwater 1500 m/s), i.e. the Mach number, Ma = U/c, is small, then the fluid is essentially incompressible. In this case the velocity field is a divergence-free (solenoidal) vector field 

This example also illustrates the use of the three basic concepts on which the analysis of more complex mass transfer problems is based namely, conservation laws, rate expressions, and equilibrium thermodynamics. The conservation of mass principle was implicitly employed to relate a measured rate of accumulation of sugar in the solution or decrease in undissolved sugar to the mass transfer rate ftom the crystals. The dependence of the rate expression for mass transfer on various variables (area, stirring, concentration, etc.) was explored experimentally. Phase-equilibrium thermodynamics was involved in setting limits to the final sugar concentration in solution as well as providing the value of the sugar concentration in solution at the solution-crystal interface. 

The various sections of this chapter develop and distinguish the conservation laws and various rate expressions for mass transfer. The laws of conservation of mass, energy, and momentum, which ate takm as universal principles, are formulated in both macroscopic and difletential forms in Section 2.2. 

In any particular region of space (Fig. 2.2-1) the macroscopic balances express the foct that the time rate of change of mass, species, momentum, or energy within the system is equad to the sum of the net flow across the boundaries of the region and the rate of generation within the region. 

FIGURE 2.2-f Mass balances on atbitraiy volume of space. 

in non-nuclear events, total mass is neither created nor destroyed, the conservation law for total mass in a system having a numter of discrete entiy and exit points may be written 

Expressions of the conservation of mass, a particular chemical species, momentum, and energy are fundamental principles which are used in the analysis and design of any separation device. It is appropriate to formulate these laws first without specific rate expressions so that a clear distinction between conservation laws and rate expressions is made. Some of these laws contain a source or generation term, for example, for a particular chemical species, so that the particular quantity is not actually conserved. A conservation law for entropy can also be formulated which contributes to a useful framework for a generalized transport theory. Such a discussion is beyond the scope of this chapter. The conservation expressions are first presented in their macroscopic forms, which are applicable to overall balances on energy, mass, and so on, within a system. However, such macroscopic formulations do not provide the information required to size equiprrwnt. Such analyses usually depend on a differential formulation of the conservation laws which permits consideration of spatial variations of composition, temperature, and so on within a system. 

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Conservation laws at a microscopic level of molecular interactions play an important role. In particular, energy as a conserved variable plays a central role in statistical mechanics. Another important concept for equilibrium systems is the law of detailed balance. Molecular motion can be viewed as a sequence of collisions, each of which is akin to a reaction. Most often it is the momentum, energy and angrilar momentum of each of the constituents that is changed during a collision if the molecular structure is altered, one has a chemical reaction. The law of detailed balance implies that, in equilibrium, the number of each reaction in the forward direction is the same as that in the reverse direction i.e. each microscopic reaction is in equilibrium. This is a consequence of the time reversal syimnetry of mechanics. 

The tliree conservation laws of mass, momentum and energy play a central role in the hydrodynamic description. For a one-component system, these are the only hydrodynamic variables. The mass density has an interesting feature in the associated continuity equation the mass current (flux) is the momentum density and thus itself is conserved, in the absence of external forces. The mass density p(r,0 satisfies a continuity equation which can be expressed in the fonn (see, for example, the book on fluid mechanics by Landau and Lifshitz, cited in the Furtlier Reading)... 

Hamiltonian, but in practice one often begins with a phenomenological set of equations. The set of macrovariables are chosen to include the order parameter and all otlier slow variables to which it couples. Such slow variables are typically obtained from the consideration of the conservation laws and broken synnnetries of the system. The remaining degrees of freedom are assumed to vary on a much faster timescale and enter the phenomenological description as random themial noise. The resulting coupled nonlinear stochastic differential equations for such a chosen relevant set of macrovariables are collectively referred to as the Langevin field theory description. 

The phenomenology of model B, where (j) is conserved, can also be outlined simply. Since (j) is conserved, it obeys a conservation law (continuity equation) ... 

Due to the conservation law, the diffiision field 5 j/ relaxes in a time much shorter than tlie time taken by significant interface motion. If the domain size is R(x), the difhision field relaxes over a time scale R Flowever a typical interface velocity is shown below to be R. Thus in time Tq, interfaces move a distanc of about one, much smaller compared to R. This implies that the difhision field 6vj is essentially always in equilibrium with tlie interfaces and, thus, obeys Laplace s equation... 

The hydrodynamical analogy now follows by comparing Eq. (B.6) to the conservation law for a classical fluid... 

Material and energy balances are based on the conservation law, Eq. (7-69). In the operation of liquid phase reactions at steady state, the input and output flow rates are constant so the holdup is fixed. The usual control of the discharge is on the liquid level in the tank. When the mixing is adequate, concentration and temperature are uniform, and the effluent has these same properties. The steady state material balance on a reacdant A is... 

Unsteady material and energy balances are formulated with the conservation law, Eq. (7-68). The sink term of a material balance is and the accumulation term is the time derivative of the content of reactant in the vessel, or 3(V C )/3t, where both and depend on the time. An unsteady condition in the sense used in this section always has an accumulation term. This sense of unsteadiness excludes the batch reactor where conditions do change with time but are taken account of in the sink term. Startup and shutdown periods of batch reactors, however, are classified as unsteady their equations are developed in the Batch Reactors subsection. For a semibatch operation in which some of the reactants are preloaded and the others are fed in gradually, equations are developed in Example 11, following. 

First, a mechanism is assumed whether completely mixed, plug flow, laminar, with dispersion, with bypass or recycle or dead space, steady or unsteady, ana so on. Then, for a differential element of space and/or time the elements of a conservation law. 

When (7.10)-(7.12) are combined with the expressions for mass and momentum conservation, we are then able to compare assumptions regarding and v with macroscale observations such as wave profiles, for example. The conservation laws are (in Lagrangian form Pq dX = p dx )... 

What is the conservation law for N = N+ + N when A1+ = N The latter situation is normally encountered in regions removed from surface sources of dislocations. 

In Eulerian coordinates x and t, the mass and momentum conservation laws and material constitutive equation are given by (u = = particle velocity,, = longitudinal stress, and p = material density)... 

J.B. Bell, P. Colella, and J.A. Trangenstein, Higher Order Godunov Methods for General Systems of Hyperbolic Conservation Laws, J. Comput. Phys. 82 (1989). 

I.L. Chern and P. Colella, A Conservative Front Tracking Method for Hyperbolic Conservation Laws, UCRL-97200, Lawrence Livermore National Laboratory, Livermore, CA, 1987. 

P. Colella, Multidimensional Upwind Methods for Hyperbolic Conservation Laws, LBL-17023, Lawrence Berkley Laboratory, Berkley, CA, 1984. 

J.B. Bell, C.N. Dawson, and G.R. Shubin, An Unsplit, Higher-Order Godunov Method for Scalar Conservation Laws in Multiple Dimensions, J. Comput. Phys. 74 No. 1 (1988). 

As required by the energy-conservation law reflected by the (5-function in (2.44), each intradoublet transition is accompanied by emission or absorption of a phonon with energy hAo. 

If chemical reactions occur only over the catalyst and none on the walls or in the homogeneous fluid stream in the recycle loop, then conservation laws require that the two balances should be equal. 

Steady-state operation is considered. In this case to satisfy conservation laws it will be assumed that the stream of a component that crosses a boundary inward, and does not come out, has been converted by chemical reaction. 

Ericksen, J. L., Conservation laws for liquid crystals, Trans Soc Rheol, 1961,5, 23 34. 

Using the conservation law, a similar expression can be derived for other connection pieces. The general formula structure is... 

The phase separation process at late times t is usually governed by a law of the type R t) oc f, where R t) is the characteristic domain size at time t, and n an exponent which depends on the universality class of the model and on the conservation laws in the dynamics. At the presence of amphiphiles, however, the situation is somewhat complicated by the fact that the amphiphiles aggregate at the interfaces and reduce the interfacial tension during the coarsening process, i.e., the interfacial tension depends on the time. This leads to a pronounced slowing down at late times. In order to quantify this effect, Laradji et al. [217,222] have proposed the scaling ansatz... 

The next step is the formulation of an equation of motion. We assume for this moment that h x) can only vary by surface diffusion, i.e., by peripheral diffusion of h along x. The classical conservation law holds that (5/5t)A + divy /, = 0. For the current the constitutive equation is, according to classical thermodynamics, j = n = 6F/6h = -V A,... 

Following a brief section on Units and Dimensions (Section 4.2), Sections 4.3 and 4.4 review some of the key physical and chemical properties, respectively. Three important conservation laws are presented in Section 4.5. Basic engineering principles are discussed in Section 4.6, to present a foundation for the theory underlying the proper design and operation of a chemical process. 

This chapter primarily serves as a review of process fundamentals such as units, dimensions, chemical and physical properties, conservation laws, and engineering principles. 

The Uiree basic conservation laws are mass, energy, and momentmn. 

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