Basic differential equation

Consider a fluid flowing steadily along a uniform pipe as depicted in Fig. 2.13 the fluid will be assumed to have a constant density so that the mean velocity u is constant. Let the fluid be carrying along the pipe a small amount of a tracer which has been injected at some point upstream as a pulse distributed uniformly over the cross-section the concentration C of the tracer is sufficiently small not to affect the density. Because the system is not in a steady state with respect to the tracer distribution, the concentration will vary with both z the position in the pipe and, at any fixed position, with time i.e. C is a function of both z and t but, at any given value of z and t, C is assumed to be uniform across that section of pipe. Consider a material balance on the tracer over an element of the pipe between z and (z + Sz), as shown in Fig. 2.13, in a time interval St. For convenience the pipe will be considered to have unit area of cross-section. The flux of tracer into and out of the element will be written in terms of the dispersion coefficient DL in accordance with equation 2.12. For completeness and for later application to reactors (see Section 2.3.7) the possibility of disappearance of the tracer by chemical reaction is also taken into account through a rate of reaction term 9L 

for unit cross-sectional area of pipe Inflow - Outflow 

This is the basic differential equation governing the transport of a dilute tracer substance along a pipe. Being a partial differential equation, its solution, which gives the concentration C as a function of z and /, will be very much dependent on the boundary conditions that apply to any particular case. 

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Recalling the basic differential equation for the elution curve given in chapter 2 is,... 

Assuming laminar flow through the filter chaimels, the basic differential equation of filtration is simply stated as follows ... 

This partial differential equation is most conveniently solved by the use of the Laplace transform of temperature with respect to time. As an illustration of the method of solution, the problem of the unidirectional flow of heat in a continuous medium will be considered. The basic differential equation for the X-direction is ... 

The basic differential equation for mass transfer accompanied by an nth order chemical reaction in a spherical particle is obtained by taking a material balance over a spherical shell of inner radius r and outer radius r + Sr, as shown in Figure 10.12. 

What are the general principles underlying the two-film, penetration and film-penetration theories for mass transfer across a phase boundary Give the basic differential equations which have to be solved for these theories with the appropriate boundary conditions. 

A soluble gas is absorbed into a liquid with which it undergoes a second-order irreversible reaction. The process reaches a steady-state with the surface concentration of reacting material remaining constant at (.2ij and the depth of penetration of the reactant being small compared with the depth of liquid which can be regarded as infinite in extent. Derive the basic differential equation for the process and from this derive an expression for the concentration and mass transfer rate (moles per unit area and unit time) as a function of depth below the surface. Assume that mass transfer is by molecular diffusion. 

Equation (9) is the basic differential equation that describes the rate of change of concentration of solute in the mobile phase in plate (p) with the volume flow of mobile phase through it. The integration of equation (9) will provide the equation for the elution curve of a solute for any plate in the column. A detailed integration of equation (9) will not be given here and the interested reader is again directed to reference (1) for further details. 

Combining Eqs. (A.l) and (A.5), we find the basic differential equation ofDebye-Htickel theory ... 

Although it is impossible to estimate accurately the amount of energy required in order to effect a size reduction of a given material, a number of empirical laws have been proposed. The two earliest laws are due to Kick17 and von Rittinger(8), and a third law due to Bond(910) has also been proposed. These three laws may all be derived from the basic differential equation ... 

In dimensionless form where z = ut x)IL and 6 - tl t = tu/L, the basic differential equation representing this dispersion model becomes... 

For this two-step process, the basic differential equations are those shown below (Equations 24-29). [The differential equations for the limiting case of the one-step reaction corresponding to Equation 20 may be obtained from Equations 24-29 by making the following identification ... 

The basic differential equation which describes the diffusion in this system is Eq. (51) written for diffusion in the z-direction with no chem-... 

In this line of reasoning, Eqn. (12.14) is the basic differential equation for ordering by component separation (the so-called spinodal decomposition) to be discussed in Section 12.3.2. 

Solution. Using Eq. 4.61 and the analogy between mass diffusion and thermal diffusion, the basic differential equation for the temperature distribution in graphite can be written... 

Prove that all of the results obtained in Exercise 14.5 for crystal growth (including the basic differential equation, its solution, and the expression for the sink efficiency) also hold for crystal evaporation. 

Solution. The basic differential equation is in a form that holds for both growth and evaporation. The diffusion boundary conditions are the same for the two cases, and therefore the solution is equally applicable. Finally, the same expression for the efficiency of the surface is obtained. (Note that the two changes of sign encountered in its derivation cancel.)... 

In most adsorption processes the adsorbent is contacted with fluid in a packed bed. An understanding of the dynamic behavior of such systems is therefore needed for rational process design and optimization. What is required is a mathematical model which allows the effluent concentration to be predicted for any defined change in the feed concentration or flow rate to the bed. The flow pattern can generally be represented adequately by the axial dispersed plug-flow model, according to which a mass balance for an element of the column yields, for the basic differential equation governing llie dynamic behavior,... 

From the appearance of the dispersion number DjuL in this dimensionless form of the basic differential equation of the plug-flow dispersion model it can be inferred that the dispersion number must be a significant characteristic parameter in any solution to the equation, as we have seen. 

As indicated in the problem statement the basic differential equations are ... 

Equations 3.30 and 3.31 are the basic differential equations for separation systems when solved, they show how concentration varies with time and distance, that is, the way in which concentration pulses move and evolve with the passage of time. 

This equation—which is in fact the basic differential equation dc/dt = D(d2c/dy2) applied to our assumed solution—must be valid at all points of space that is, the two sides must be equal for all values of space coordinate y. Therefore terms containing various powers of y can be grouped to form individual equations. First, the coefficients of y2 are equated... 

In the foregoing development we derived relations for the heat transfer from a rod or fin of uniform cross-sectional area protruding from a flat wall. In practical applications, fins may have varying cross-sectional areas and may be attached to circular surfaces. In either case the area must be considered as a variable in the derivation, and solution of the basic differential equation and the mathematical techniques become more tedious. We present only the results for these more complex situations. The reader is referred to Refs. 1 and 8 for details on the mathematical methods used to obtain the solutions. 

Three equations are basic to viscoelasticity (1) Newton s law of viscosity, a = ijy, (2) Hooke s law of elasticity. Equation 1.15, and (3) Newton s second law of motion, F = ma, where m is the mass and a is the acceleration. One can combine the three equations to obtain a basic differential equation. In linear viscoelasticity, the conditions are such that the contributions of the viscous, elastic, and the inertial elements are additive. The Maxwell model is ... 

In this chapter we will deal with steady-state and transient (or non steady-state) heat conduction in quiescent media, which occurs mostly in solid bodies. In the first section the basic differential equations for the temperature field will be derived, by combining the law of energy conservation with Fourier s law. The subsequent sections deal with steady-state and transient temperature fields with many practical applications as well as the numerical methods for solving heat conduction problems, which through the use of computers have been made easier to apply and more widespread. 

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