Constant volume incompressibility

The ratios in parentheses express the constant volume rate per unit filter area. Hence, Equation 24 is the relationship between time i and pressure drop Ap. For incompressible cakes, rg is constant and independent of pressure. For compressible cakes, the relationship between time and pressure at constant-rate filtration is ... 

If density (p) is constant, the fluid is referred to as isochoric (i.e., a given mass occupies a constant volume), although the somewhat more restrictive term incompressible is commonly used for this property (liquids are normally considered to be incompressible or isochoric fluids). If gravity (g) is also constant, the only variables in Eq. (4-5) are pressure and elevation, which can then be integrated between any two points (1 and 2) in a given fluid to give... 

As can be expected, the angular scattering intensity distribution has become anisotropic. From the data obtained, two values of the average correlation distance between neighboring nodules can be calculated, parallel and perpendicular to the deformation direction respectively. Although the accuracy of the data is not outstanding, it can be seen that d increases with the macroscopic extension ratio, Ax, whereas dx decreases only slightly (Fig. 10). It is not possible to ascertain experimentally whether dL is proportional to AJ1/2 as it should be if the deformation proceeded at constant volume L e., if the network could be considered incompressible. 

Liquids are virtually incompressible and fluid and maintain a constant volume. 

Volume variations with conversion are large for constant-pressure gas-phase reactions with change in mole number. Here, as a rule, operation at constant volume poses no difficulties. Liquid-phase reactions may also entail volume contraction or expansion. However, these are not related to changes in mole number and can be predicted only if information on partial molar volumes is at hand. Because liquids are essentially incompressible, even at elevated temperature, it is unsafe to conduct liquid-phase reactions without a gas cap in a closed reactor. Some variation of liquid-phase volume with conversion therefore is apt to occur. Fortunately, the variation at constant temperature is usually so small that it can be neglected in the evaluation or accounted for by a minor correction. 

The constant-volume and constant-pressure specific heats are identical for incompressible substances (Fig. 1-10). Therefore, for solids and liquids the subscripts on c and Cp can be dropped and both. specific heals can be represented by a single symbol, c. That is, Cp = tv = c- This result could also be deduced from the physical definitions of constant-volume and constant-pressure specific heats. Specific heats of several common gases, liquids, and solids are given in the Appendix. 

If w e consider the constant-volume reactor with incompressible fluid (a = 0,Cv = Cp), Equation 6.16 reduces to Equation 6.15 as it should because Equation 6.15 is valid for any reactor operation with an incompressible fluid. We also notice that, in the constant-pressure case, the same energy balance applies for any fluid mixture (ideal gas, incompressible fluid, etc.), and that this balance is the same as the balance for an incompressible fluid in a constant-volume reactor. Although the same final balances are obtained for these two cases, the physical situations they describe are completely different. 

An easily compressible substance has a larger value of k, so larger volume fluctuations occur at a given pressure than in a more incompressible substance. Conversely, in a constant volume simulation a less compressible substance shows larger fluctuations in the pressure. The isothermal compressibility is the pressure analogue of the heat capacity, which is related to the energy fluctuations. 

Therefore, the free energy is a function of the temperature alone. We emphasize that for the isothermal incompressible body the heat capacity at constant volume is not accessible. When we change the temperature, which is needed to measure the C then we would also change the volume, due to Eq. (4.46). An exception is the special case dy(r)/dr = 0 that occurs in liquid water at the density maximum. Therefore, C is not defined in general however, Cp is readily accessible. 

If the extension occurs at constant volume, (the material is incompressible)... 

Here v represents the Poisson ratio, defined as the ratio between the linear contraction and the elongation in the axis of stretching. In the case of constant volume (an incompressible body like rubber), i> = 0.5 and therefore E = 3G for rigid materials p < 0.3. There is also a direct test for tear strength, mainly in the case of thin films, similar to those used in the paper industry. 

Another well-developed mean-field model, known as the free association model, was developed by Dudowicz, Douglas and Freed and is based on the mean-field Flory-Huggins incompressible lattice model. ° Douglas and co-workers incorporated two temperature-independent parameters - polymerization enthalpy AHp and entropy ASp -along with parameters describing the flexibility of the polymer and a solvent-monomer interaction parameter (x). The model allows the calculation of various temperature-dependent properties, such as the number-average DP, constant volume specific heat (Q,), and osmotic pressure, and predias similar temperature-dependent behavior with the van der Schoot treatment. 

These are the most striking features of smectic textures [19]. Smectic layers of constant thickness (incompressible, modulus B— oo) form surfaces called Dupin cyclides. We have seen some of them, which have the form of tori including disclinations, see Fig. 4.7b. Such cyclides can fill any volume of a liquid crystal by cones of different size. An example is afocal-conic pair, namely, two cones with a common base. The common base is an ellipse with apices at A and C and foci at O and O , see Fig. 8.30a. The hyperbola B-B passes through focus O. The focus of... 

Equation (2.8) indicates that in thermodynamics, the internal energy is used as a function of state to characterize the system at constant volume, and also when no work is being performed on or by the system. But the majority of real processes, especially for polymers, take place at constant pressure, because solids and liquids (the only physical states for polymers) are virtually incompressible. For such processes (i.e., those taking place at constant pressure), Gibbs introduced a new function of state, enthalpy H... 

If Poisson s ratio is v 0.5, the change in volume is zero or the material is incompressible. Here it is important to note that Poisson s ratio for metals and many other materials in the linear elastic range is approximately 0.33 (i.e., V 1/3). However, near and beyond the yield point, Poisson s ratio is approximately 0.5 (i.e., V 1/2). That is, when materials yield, neck or flow, they do so at constant volume. 

Equation (3.15) is ihe generalized relationship between the residence time distribution, E, and the internal age distribution, I, in the transient state for an arbitrary flow system. For an incompressible fluid (given a constant volume). 

Solids and liquid have no distinction between constant-pressure or constant-volume specific heats, and a single specific heat value is appropriate. The specific heat of most incompressible substances varies only slightly with temperature and can generally be considered constant over a limited temperature range of interest to fuel cell studies. For example, the mass specific heat of liquid water at 25°C is 4.179 kJ/kg K, while at 1(X)°C, the specific heat of the liquid phase is 4.218 kJ/kg K, only about 1% higher. 

In 1873, van der Waals [2] first used these ideas to account for the deviation of real gases from the ideal gas law P V= RT in which P, Tand T are the pressure, molar volume and temperature of the gas and R is the gas constant. Fie argried that the incompressible molecules occupied a volume b leaving only the volume V- b free for the molecules to move in. Fie further argried that the attractive forces between the molecules reduced the pressure they exerted on the container by a/V thus the pressure appropriate for the gas law isP + a/V rather than P. These ideas led him to the van der Waals equation of state ... 

This expression shows the relationship between filtration time and filtrate volume. The equation is applicable to both incompressible or compressible calces, because at constant AP the values and x are constant. For constant AP, an increase in the filtrate volume results in a reduction in the filtration rate. If we assume a definite filtering apparatus and set up a constant temperature and filtration pressure, then the values of Rf, r , fi and AP will be constant. We now take note of the well-known filtration constants K and C, which are derived from the above expressions ... 

If the pilot reactor is turbulent and closely approximates piston flow, the larger unit will as well. In isothermal piston flow, reactor performance is determined by the feed composition, feed temperature, and the mean residence time in the reactor. Even when piston flow is a poor approximation, these parameters are rarely, if ever, varied in the scaleup of a tubular reactor. The scaleup factor for throughput is S. To keep t constant, the inventory of mass in the system must also scale as S. When the fluid is incompressible, the volume scales with S. The general case allows the number of tubes, the tube radius, and the tube length to be changed upon scaleup ... 

Depart from Geometric Similarity. Adding length to a tubular reactor while keeping the diameter constant allows both volume and external area to scale as S if the liquid is incompressible. Scaling in this manner gives poor results for gas-phase reactions. The quantitative aspects of such scaleups are discussed... 

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