Equations for the Mean Quantities

In the equations of motion and energy (16.39), the dependent variables n p, and 0 are random variables, making the equations virtually impossible to solve. We modify our goal 

Our objective is to determine equations for nj, 0, and p. To obtain these equations, we first substitute (16.40) into (16.39). We then average each term in the resulting equations with respect to time. The result, employing (16.41), is 

The good news is that these equations, now time-averaged, contain only smoothly time-varying average quantities, so that the difficulties associated with the stochastic nature of the original equations have been alleviated. However, there is also some bad news. We note the emergence of new dependent variables, m m) and u 0 for ij = 1,2,3. When the equations are written in the form of (16.45) and (16.46), we can see that p0 represents a new contribution to the total stress tensor and that poc ,m)0 is a new contribution to the heat flux vector. 

Equations (16.42) and (16.46) have, as dependent variables, ut,p, 0, u u -, and u -Q. We have thus 14 dependent variables (note that njwj = m-mJ, so there are six Reynolds stresses plus three u i) variables, three velocities, pressure, and temperature), but we have only five equations from (16.46). We need eight additional equations to close the system. We could attempt to write conservation equations for the new dependent variables. For example, we can derive such an equation for the variables w-wj by first subtracting (16.43) from the second equation in (16.39), leaving an equation for u r We then multiply by u - and average all terms. Although we can arrive at an equation for m m this way, we unfortunately have at the same time generated still more dependent variables This problem, 

Clearly, , (r) will depend on the averaging interval t. We need to choose r large enough so that an adequate number of fluctuations are included, but yet not so large that important macroscopic features of the flow would be masked. For example, if ri and tt are timescales associated with fluctuations and macroscopic changes in the flow, respectively, we would want t2 r Ti. 

What we seek, in principle, is a description of a turbulent flow at all points in space and time. Unfortunately, in the equations of motion and energy, the dependent variables m, p, and T are random variables, making the equations virtually impossible to solve. To proceed we decompose the velocities, temperature, and pressure into a mean and a fluctuating component. 

Thus a term of the form uju] can be written UjUj + u jUj, where the mean of the product of two fluctuations is not necessarily (and usually is not) equal to zero. If u ju j 0, m and u j are said to be correlated. 

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The result of the mixing-length idea used to derive the expressions (16.56) and (16.57) is that the turbulent momentum and energy fluxes are related to the gradients of the mean quantities. Substitution of these relations into (16.46) leads to closed equations for the mean quantities. Thus, except for the fact that KM and Kr vary with position and direction, these models for turbulent transport are analogous to those for molecular transport of momentum and energy. 

Reynolds [127] postulated that the Navier-Stokes equations are still valid for turbulent flows, but recognized that these equations could not be applied directly due to the complexity and irregularity of the fluid dynamic variables. A true description of these flows at all points in time and space was not feasible, and probably not very useful at the time. Instead, Reynolds proposed to develop equations governing the mean quantities that were actually measurable. 

Plan the solution. Recall that Plan, printed as you see it here, means to complete the first three steps in the problem-solving procedure (1) Write down what is Given. (2) Write down what is Wanted. (3) Decide how to solve the problem. If the given and wanted quantities are related by a Per expression, use dimensional analysis write the Per/Path. If the given and wanted quantities are related by an algebraic equation, use algebra by solving the equation for the Wanted quantity. Plan your solution now. 

The brackets symbolize fiinction of, not multiplication.) Smce there are only two parameters, and a, in this expression, the homogeneity assumption means that all four exponents a, p, y and S must be fiinctions of these two hence the inequalities in section A2.5.4.5(e) must be equalities. Equations for the various other thennodynamic quantities, in particular the singidar part of the heat capacity Cy and the isothemial compressibility Kp may be derived from this equation for p. The behaviour of these quantities as tire critical point is approached can be satisfied only if... 

When two substances react, they react in exact amounts. You can determine what amounts of the two reactants are needed to react completely with each other by means of mole ratios based on the balanced chemical equation for the reaction. In the laboratory, precise amounts of the reactants are rarely used in a reaction. Usually, there is an excess of one of the reactants. As soon as the other reactant is used up, the reaction stops. The reactant that is used up is called the limiting reactant. Based on the quantities of each reactant and the balanced chemical equation, you can predict which substance in a reaction is the limiting reactant. 

Equation for the estimation of the values of a measuring quantity from given values of a analytical quantity. The calibration function may be known a priori by natural laws or estimated experimentally by means of calibration samples. 

This equation is the basic relation for the mean residence time in a plug flow reactor with arbitrary reaction kinetics. Note that this expression differs from that for the space time (equation 8.2.9) by the inclusion of the term (1 + SAfA) and that this term appears inside the integral sign. The two quantities become identical only when 5a is zero (i.e., the fluid density is constant). The differences between the two characteristic times may be quite substantial, as we will see in Illustration 8.5. Of the two quantities, the reactor... 

The Reynolds-averaged approach is widely used for engineering calculations, and typically includes models such as Spalart-Allmaras, k-e and its variants, k-co, and the Reynolds stress model (RSM). The Boussinesq hypothesis, which assumes pt to be an isotropic scalar quantity, is used in the Spalart-Allmaras model, the k-s models, and the k-co models. The advantage of this approach is the relatively low computational cost associated with the computation of the turbulent viscosity, fit. For the Spalart-Allmaras model, one additional transport equation representing turbulent viscosity is solved. In the case of the k-e and k-co models, two additional transport equations for the turbulence kinetic energy, k, and either the turbulence dissipation rate, s, or the specific dissipation rate, co, are solved, and pt is computed as a function of k and either e or co. Alternatively, in the RSM approach, transport equations can be solved for each of the terms in the Reynolds stress tensor. An additional scale-determining equation (usually for s) is also required. This means that seven additional transport equations must be solved in 3D flows. 

The solution is complicated by the fact that many of the parameters in the design equations depend on the vessel dimensions (h or D or V). In addition to the design equations, we give expressions for these parameters in terms of D. The result is a set of nonlinear algebraic equations to be solved for the unknown quantities, including D. We solve these by means of the E-Z Solve software (file ex24-3.msp). 

To write down the differential equation for the rate of a chain reaction we employ the law of mass action, but the simple application of this involves the concentration of the transitorily formed activated molecules which propagate the chain, and as this concentration is unknown the equation written down would be useless unless some other relations are established by means of which the unknown quantities can be eliminated. These relations are provided by the condition that the chains shall be stable, or the Egerton and Gates, J. Inst. Petroleum, Tech., 1927, 13, 281. 

Using the kinetic theory of gases, the root mean square speed c was derived in Section 4.12. Using the Maxwell distribution of speeds, one can also calculate the mean speed and most probable speed of a collection of molecules. The equations for these two quantities are caverage = (HRT/-nM)m and cmost probable =... 

Physically this means that the temperature decrease due to incomplete combustion cannot exceed the characteristic quantity RT /E without full extinction or disruption of combustion also taking place. Substituting (16) into (14a) we obtain an equation for the critical reaction time at which extinction occurs ... 

It is instructive to compare the system of equations (3.46) and (3.47) with the system (3.37). One can see that both the radius of the tube and the positions of the particles in the Doi-Edwards model are, in fact, mean quantities from the point of view of a model of underlying stochastic motion described by equations (3.37). The intermediate length emerges at analysis of system (3.37) and can be expressed through the other parameters of the theory (see details in Chapter 5). The mean value of position of the particles can be also calculated to get a complete justification of the above model. The direct introduction of the mean quantities to describe dynamics of macromolecule led to an oversimplified, mechanistic model, which, nevertheless, allows one to make correct estimates of conformational relaxation times and coefficient of diffusion of a macromolecule in strongly entangled systems (see Sections 4.2.2 and 5.1.2). However, attempts to use this model to formulate the theory of viscoelasticity of entangled systems encounted some difficulties (for details, see Section 6.4, especially the footnote on p. 133) and were unsuccessful. 

The above equations for the pressure variation in Darcy flow involve the permeability, K. This quantity is mainly dependent on the porosity of the material, , i.e., the ratio of the pore volume. Vp, to the total volume, V, of the porous material, and on the mean diameter of the particles. The relation between these quantities depends on the shape of the particles and how they are arranged. If, for example, the particles... 

This means that we can write three independent equations for the x, y, z components of each vector quantity above. Each such equation will have the form of Eq. (VIII.10.1) and contain only quantities that involve the component chosen. 

It will be seen later (p. 230) that there does not appear to be any experimental method of evaluating the activity coefficient of a single ionic species, so that the Debye-Hiickel equations cannot be tested in the forms given above. It is possible, however, to derive very readily an expression for the mean activity coefficient, this being the quantity that is obtained experimentally. The mean activity coefficient f of an electrolyte is defined by an equation analogous to (30), and... 

The propagation of longitudinal acoustic waves in choked nozzles has been analyzed on the basis of the one-dimensional, time-dependent forms of equations (4-45) and (4-46) by introducing linearizations of the previously indicated type (for example, p = p(l + p )] for the stream wise velocity v as well—that is, v — v(l + i )—and by allowing the mean quantities p, p, and V to vary with the streamwise distance z through the nozzle, in a manner presumed known from a quasi-one-dimensional, steady-flow nozzle analysis [20]. The perturbation equations... 

This equation was first obtained by Clapeyron, a French engineer who continued the work of Carnot. He derived the equation for the evaporation of a liquid. In its general form it was established by Clausius, and is therefore called the Clausius-Clapeyron, or briefly the Clausius equation. By means of this equation we can calculate the change in pressure dp produced by an arbitrary change in temperature dT from the quantities L, 2, Vj, and T, which can all be determined by experiment. 

The emphasis in this section is to derive an equation of change for the average quantity iP)m- The mean value of the property function tp r, c, t) was defined by (2.55) in sec. 2.3.2. Notice that we are generally considering any property function ip of type of molecules that has position r and velocity c at time t. 

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