Sink term

The analysis of the heat exchanger network first identifies sources of heat (termed hot streams) and sinks (termed cold streams) from the material and energy balance. Consider first a very simple problem with just one hot stream (heat source) and one cold stream (heat sink). The initial temperature (termed supply temperature), final temperature (termed target temperature), and enthalpy change of both streams are given in Table 6.1. 

The latter contribute to the fluxes in time-varying conditions and provide source or sink terms in the presence of chemical reaction, but they have no influence on steady state diffusion or flow measurements in a non-reactive sys cem. 

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. 

The third term on the left side of the equation has significance in reactive systems only. It is used with a positive sign when material is produced as a net result of all chemical reactions a negative sign must precede this term if material is consumed by chemical reactions. The former situation corresponds to a source and the latter to a sink for the material under consideration. Since the total mass of reactants always equals the total mass of products in a chemical reaction, it is clear that the reaction (source/sink) term (R should appear explicitly in the equation for component material balances only. The overall material balance, which is equivalent to the algebraic sum of all of the component balance equations, will not contain any (R term. 

T if and only if there are initially no particles in either the east or west directions to site (i,j) and there arc particles to the north and south, and a sink term —... 

Therefore if the carbon substrate is present at sufficiently high concentration anywhere in the rhizosphere (i.e., p p, ax), the microbial biomass will increase exponentially. Most models have considered the microbes to be immobile and so Eq. (33) can be solved independently for each position in the rhizosphere provided the substrate concentration is known. This, in turn, is simulated by treating substrate-carbon as the diffusing solute in Eq. (32). The substrate consumption by microorganisms is considered as a sink term in the diffusion equation, Eq. (8). 

The TDE solute module is formulated with one equation describing pollutant mass balance of the species in a representative soil volume dV = dxdydz. The solute module is frequently known as the dispersive, convective differential mass transport equation, in porous media, because of the wide employment of this equation, that may also contain an adsorptive, a decay and a source or sink term. The one dimensional formulation of the module is ... 

From the pollutant and biological cycles the processes of advection, diffusion, volatilization (diffusion at the soil-air interface), adsorption or desorption (equilibrium), and degradation or decay, which are also the most important chemical processes in the soil zone. All other processes can be lumped together under the source or sink term of equation (3). 

The motions of the individual fluid parcels may be overlooked in favor of a more global, or Eulerian, description. In the case of single-phase systems, convective transport equations for scalar quantities are widely used for calculating the spatial distributions in species concentrations and/or temperature. Chemical reactions may be taken into account in these scalar transport equations by means of source or sink terms comprising chemical rate expressions. The pertinent transport equations run as... 

Note that the particle diffusion term is ignored, just like particle dispersion due to SGS motions (this was found justified in a separate simulation). The shape of the sink term in the right-hand term of this equation is due to Von Smoluchowski (1917) while the local value of the agglomeration kernel /i0 is assumed to depend on the local 3-D shear rate according to a proposition due to Mumtaz et al. (1997). 

So eq. (11.47) can be viewed as a diffusion equation in the spatial coordinates of the electrons with a diffusion coefficient D equal to j. The source and sink term S is related to the potential energy V. In regions of space where V is attractive (negative) the concentration of diffusing material (here the wavefunction) will accumulate and it will decrease where V is positive. It turns out that if we start from an initial trial wavefunction and propagate it forward in time using eq. (11.47),... 

Duhamel s principle can be extended to cases of surface conditions being functions of both time and space variables and to variable source and sink terms as well (Zwillinger, 1989). 

The distance dependence is characterized by the parameter re, which is in the range 0.5-2 A. The diffusion equation (B4.1.2) must be modified by adding a distance-dependent sink term k(r)... 

Finally, as described in Box 4.1 of Chapter 4, an exact numerical solution of the diffusion equation (based on Fick s second law with an added sink term that falls off as r-6) was calculated by Butler and Pilling (1979). These authors showed that, even for high values of Ro ( 60 A), large errors are made when using the Forster equation for diffusion coefficients > 10 s cm2 s 1. Equation (9.34) proposed by Gosele et al. provides an excellent approximation. 

The Gaussian plume foimulations, however, use closed-form solutions of the turbulent version of Equation 5-1 subject to simplifying assumptions. Although these are not treated further here, their description is included for comparative purposes. The assumptions are reflection of species off the ground (that is, zero flux at the ground), constant value of vertical diffusion coefficient, and large distance from the source compared with lateral dimensions. This Gaussian solution to Equation 5-1 is obtained under the assumption that chemical transformation source and sink terms are all zero. In some cases, an exponential decay factor is applied for reactions that obey first-order kinetics. A typical solution (with the time-decay factor) is ... 

In the above, D rn is the water diffusion coefficient through the membrane phase only. Note also that the water fluxes through the membrane phase, via electro-osmotic drag and molecular diffusion, represent a source/sink term for the gas mixture mass in the anode and cathode, respectively. 

Chapter 4 Mass, Heat, and Momentum Transport Analogies. The transport of mass, heat, and momentum is modeled with analogous transport equations, except for the source and sink terms. Another difference between these equations is the magnitude of the diffusive transport coefficients. The similarities and differences between the transport of mass, heat, and momentum and the solution of the transport equations will be investigated in this chapter. 

Table 2.1 Common source and sink terms used in the diffusion equation...

A mass balance on one compound in our box is based on the principle that whatever comes in must do one of three things (1) be accumulated in the box, (2) flux out of another side, or (3) react in the source/sink terms. If it seems simple, it is. 

We will introduce the product rule through demonstrating its use in an example problem. The product rule can be used to expand a solution without source and sink terms to the unsteady, one-dimensional diffusion equation to two and three dimensions. It does not work as well in developing solutions to all problems and therefore is more of a technique rather than a rule. Once again, the final test of any solution is (1) it must solve the governing equation(s) and (2) it must satisfy the boundary conditions. 

Note that the reaction rate coefficients are not divided by R, because radioactive decay will occur whether or not the elements are adsorbed to the sediments. With the exception of the source/sink terms, these equations look like that provided in Example 2.2. The solution will be simply given here with the reader required to apply a technique similar to Example 2.2 that shows that the solution is correct. 

A linear PDF, such as the diffusion equation, can only have first-order and zero-order source or sink rates. But what if the source or sink term is of higher order An example would be the generalized reaction... 

EXAMPLE 2.9 Degradation of 1,3-butadiene resulting from a spill (linearized source-sink terms)... 

In Chapter 2, we used the control volume technique represented by equation (2.1) to transport mass into and out of our control volume. Inside of the control volume, there were source and sink rates that acted to increase or reduce the mass of the compound. Anything left after these flux and source/sink terms had to stay in the control volume, and was counted as accumulation of the compound. 

It is apparent that the source and sink terms can be different between mass, heat, and momentum transport. There is another significant difference, however, related to the magnitude of the diffusion coefficient for mass, heat, and momentum. 

We will not consider any of the source or sink terms for acrolein. 

EXAMPLE 6.4 Development of the Streeter-Phelps (1925) equation for DO sag below a point BOD source in a river (plug flow with a first-order and zero-order source/sink terms)... 

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