Single tube model

A special case of the two tube model is the single tube or uniform tube model where the cross sectional area is constant along the entire length of the tube. This is a reasonable approximation of the vocal tract when producing the schwa vowel. If.4i =. 42, then the refection coefficient ri = 0 and hence Equation 11.19 simplifies to 

Recall that D is the length of the vocal tract in units of the normalised distance that we used to simplify the equation for discrete time analysis. By use of Equation 11.9, we can determine a 

The numerator of the transfer function tells us that the output is delayed by a factor which is a function of the length of the tube. All these zeros lie at 0, and can be ignored for purposes of determining the frequency response of the transfer function. The poles are evenly spaced on the unit circle at intervals of n/5. The first is at 7i/10 = 0.314, which when converted to a real frequency value with Equation 10.29, gives 500Hz. Subsequent resonances occur every lOOOHz after that, i.e. 1500Hz, 2500Hz and so on. 

Recall that the losses in the system come fi om the wave exiting the vocal tract at the glottis and lips, and the degree to which this is done is controlled by the special reflection coefficients Ai and ro- Let us first note, that since these are multiplied, it doesn t affect the model as to where the losses actually occur, and in fact in some formulations the losses are taken to occur only at the glottis or only at the lips. Secondly, recall from Section 11.3.3, that the special case of complete closure at the glottis would produce a reflection coefficient of 1, and the special condition of complete openness at the lips would produce a reflection coefficient of -1. This is a situation where there are no losses at all in the system (unrealistic of course, for the simple fact that no sound would escape and hence a listener would never hear the speech.) For these special cases the product is riro is 1, which means that the denominator term is 

It can be shown that when this is factorised, all the poles are spaced as before, but lie on the unit circle. This is of course exactly what we would expect from a lossless case. 

---

In today s competitive climate, investigators cannot spend much time on the clarification of the kinetics for a new process. At Union Carbide Corporation in the 1970s the study to replace the old and not very efficient butyraldehyde hydrogenation was done in three months. In another three months a kinetic model was developed and simultaneously tested in an existing single tube in a pilot-plant (Cropley et al,1984). Seldom is a completely new process studied for which no similar example exists in the industry. 

The Ball and Wire model is identical to the Wire model, exeept that atom positions are represented by small spheres. This makes it possible to identify all atom locations in all molecules. The Tube model is identical to the Wire model, except that bonds, whether single, double or triple, are represented by single colored tubes. The tubes are useful because they better eonvey the three-dimensional shape of a molecule. The Ball and Spoke model is a variation on the Ibbe model atom positions are represented by colored spheres, making it possible to see all atom locations in all molecules. 

The intestinal permeability may be determined from the rate of drug appearance in mesenteric blood (i.e. dM/dt) at steady state, using Eq. 2.12. Estimating the term C[ en will again depend on the flow dynamics of the model chosen. The most commonly used experimental procedure is the single-pass perfusion (i.e. parallel tube model) and the luminal concentration can be estimated using the logarithmic mean of inlet and outlet concentrations (i.e. ). 

Fig. 2. The tube model replaces the many-chain system left) with an effective constraint on each single chain right). The tube permits diffusion of chains along their own contours only...

Several models have been proposed in the literature for the steady-state and transient simulation of various SOFC concepts (for example, see [9, 13-17]). Here, a onedimensional (1-D), steady state model of an IP-SOFC single tube and bundle has been developed [18], under the hypothesis reported below ... 

Therefore, an attempt was made to determine the kinetic reaction scheme and effective heat transfer as well as kinetic parameters from a limited number of experimental results in a single-tube reactor of industrial dimensions with side-stream analysis. The data evaluation was performed with a pseudohomo-geneous two-dimensional continuum model without axial dispersion. The model was tested for its suitability for prediction. 

Table 2. Model conditions and assumptions used in the single tube example...

A constricted tube model is used to analyze viscous and capillary effects associated with foam flow in porous media. The foam moves through the pore structure as single layer, continuous bubble trains. Capillary resistance stems from the drainage and imbibition surfaces as well as the internal lamellae structure. 

Here X ,ax is the single chain limiting extension ratio in the isotropic, unoriented polymer with the same entanglement weight. A complication is that the process of orientation above Tg may result in a loss of entanglement constraint, effectively increasing and One of the ways this loss can take place in the current versions of the tube model is by tube relaxation whereby the process of tube... 

Despite these complications, there are now numerous evidences that the tube model is basically con-ect. The signatory mark that the chain is trapped in a tube is that the chain ends relax first, and the center of the chain remains unrelaxed until relaxation is almost over. Evidence that this occurs has been obtained in experiments with chains whose ends are labeled, either chemically or isotopically (Ylitalo et al. 1990 Russell et al. 1993). These studies show that the rate of relaxation of the chain ends is distinctively faster than the middle of the chain, in quantitative agreement with reptation theory. The special role of chain ends is also shown indirectly in studies of the relaxation of star polymers. Stars are polymers in which several branches radiate from a single branch point. The arms of the star cannot reptate because they are anchored at the branch point (de Gennes 1975). Relaxation must thus occur by the slower process of primitive-path fluctuations, which is found to slow down exponentially with increasing arm molecular weight, in agreement with predictions (Pearson and Helfand 1984). 

Single-chain models, such as the Doi Edwards reptation model [Eq. (9.21)] or the Doi tube length fluctuation model, assume a linear contribution to... 

We shall develop next a single-channel model that captures the key features of a catalytic combustor. The catalytic materials are deposited on the walls of a monolithic structure comprising a bundle of identical parallel tubes. The combustor includes a fuel distributor providing a uniform fuel/air composition and temperature over the cross section of the combustor. Natural gas, typically >98% methane, is the fuel of choice for gas turbines. Therefore, we will neglect reactions of minor components and treat the system as a methane combustion reactor. The fuel/air mixture is lean, typically 1/25 molar, which corresponds to an adiabatic temperature rise of about 950°C and to a maximum outlet temperature of 1300°C for typical compressor discharge temperatures ( 350°C). Oxygen is present in large stoichiometric excess and thus only methane mass balances are needed to solve this problem. 

These expressions demonstrate that the normalized mean residence time and variance of the normalized residence time distribution increase with increased values of the axial dispersion number Dj. In the limit of = 0, the signal is convected and behavior corresponding to the parallel tube model is approximated the normalized residence time fi= 1 and Act = 0. For very large values of D, the behavior corresponds to a single well-mixed compartment. 

Assuming a pseudohomogeneous two-dimensional reactor model with plug flow of fluid and constant properties, calculate the axial concentration and temperature profiles at several radial positions along the axis of a single tube. Use the following property/parameter values (Doraiswamy, 2001) ... 

In order to estimate the accuracy of the quasi-one-dimensional results, a two-dimensional model was employed to study the single-tube PDE dynamics. The work also provides a basis for multitube PDE analysis. 

Film condensation in tube bundles (more commonly used in shell-and-tube heat exchangers) characterize more complex physical conditions compared to condensation on a single tube. The gravity-controlled and surface-shear-stress-influenced condensate films must be modeled in different ways to accommodate combined influences of condensate drain to lower tubes (i.e., condensate inundation) and shear effects. Such a correlation, the fourth correlation from the top of Table 17.24, was proposed by Kern and modified by Butterworth [81]. 

---