The Isothermal Condition

If the compression cycle approaches the isothermal condition, pV = constant, as is the case when several stages with intercoolers are used, a simple approximation of the power is obtained from the following formula ... 

We shall now prove that P, for fixed values of 7r and the temperature, is definite for a given solution. For this purpose we have first of all to show that the dilution or concentration of the solution can be effected isothermally and reversibly. If the above apparatus is constructed of some good conductor of heat, placed in a large constant-temperature reservoir, and if all processes are carried out very slowly, the isothermal condition is satisfied. Further, suppose the end pistons fixed, and then apply to the septum an additional small pressure SP towards the solution. There will be a slight motion of the septum, through a small volume SV, and work... 

Note that the isothermal condition can be considered a special case of the isentropic condition for k = 1. The constant in Eq. (8-19) or (8-20) can be evaluated from known conditions at some point in the system (e.g., Px and 7)). Using Eq. (8-19) to eliminate the density from Eq. (8-15) and evaluating the integral leads to... 

In the case of adiabatic flow we use Eqs. (9-1) and (9-3) to eliminate density and temperature from Eq. (9-15). This can be called the locally isentropic approach, because the friction loss is still included in the energy balance. Actual flow conditions are often somewhere between isothermal and adiabatic, in which case the flow behavior can be described by the isentropic equations, with the isentropic constant k replaced by a polytropic constant (or isentropic exponent ) y, where 1 < y < k, as is done for compressors. (The isothermal condition corresponds to y= 1, whereas truly isentropic flow corresponds to y = k.) This same approach can be used for some non-ideal gases by using a variable isentropic exponent for k (e.g., for steam, see Fig. C-l). 

As a consequence the heating rate must not only be linear but also adapted to the type of reaction which is to be expected. In certain cases stepwise heating may be chosen, since this better approaches the isothermal conditions. Figure 31 shows the effect of the heating rate on the shape of the TG-curve in the case of palygorskite, a complex layer silicate28. Only the low heating rate (0.5 °C/min) allowed to resolve the individual dehydration steps. 

If the calorimeter could respond instantaneously to the heat effects associated with the addition of titrant, then the measured curve would coincide with the dashed lines in figure 11.5. The deviation of the data from this ideal behavior corresponds to periods in which the isothermal condition is not observed. When necessary, however, it is possible to use deconvolution techniques to generate the input function represented by the dashed line from the observed experimental curve. 

So far, the isothermal condition was examined. However, a temperature gradient can be developed in the gas film surrounding the catalyst pellet. The rate of heat transfer through the film is given by... 

The gas is now reversibly compressed under the isothermal conditions (tc) of the low-temperature reservoir, with energy change... 

The approximation in the preceding equation is to circumvent any conflict with Eq. (5.100), which defines the intrinsic average of velocity. This approximate relation becomes an equivalent one under the isothermal condition of phase k with the expansion work estimated by... 

For comparison to the isothermal conditions in the well, an adiabatic calculation with the same well geometry was executed. This time, all uncoated faces were set to be adiabatic while the coated channel faces obeyed the above-given boundary conditions. Figure 3.61 shows that there is a tremendous temperature increase if the exothermal reaction heat is not allowed to leave the control volume. 

When only the feed side and permeate side mass balance equations are considered under the isothermal condition, the resulting equations arc a set of first-order ordinary differential equations. Furthermore, a co-current purge stream renders the set of equations an initial value problem and well established procedures such as the... 

To understand the mechanism of simultaneous heat and mass transfer, consider the evaporation of water from a swimming pool into aiir. Let us assume that the water and the air are initially at the same temperature. If the air is saturated (a relative humidity of 0 = 100 percent), there will be no heat or mass transfer as long as the isothermal conditions remain. But if the air is not... 

We start the analysis of CSTRs by considering isothermal operations with single chemical reactions. Isothermal CSTRs are defined as those where Bom = 0in-Since we do not have to determine the reactor temperature, we have to solve only the design equations. The energy balance equation provides tbe heating (or cooling) load necessary to maintain the isothermal conditions. Also, for isothermal operations, the individual reaction rates depend only on the species concentrations, and, when the reactor temperature is taken as the reference temperature, T=To, and Eq. 8.1.5 reduces to... 

For a steady flow of an incompressible fluid in a uniform pipe, the only property that varies along the pipe is pressure. However, for a compressible fluid when the pressure varies (i.e., drops), the density also drops, which means that the velocity must increase for a given mass flow. The kinetic energy thus increases, which results in a decrease in the internal energy and the temperature. This process is usually described as adiabatic, or locally isentropic, with the effect of friction loss included separately. A limiting case is the isothermal condition, although special means are usually required to achieve constant temperature. Under isothermal conditions for an ideal gas. 

Although a three-dimensional method of representation is necessary in order to give a complete picture of the isothermal conditions in a quaternary system, it is very convenient, for practical purposes, to make use of a plane diagram. Such a diagram can be obtained by projecting the curves in Fig. 143 perpendicularly on the base of the tetrahedron. The vertex of the tetrahedron (D) then comes to lie at the centre D of the triangle ABC (Fig. 144), and the edges AD, BD, and CD find their projections in the three medians AD, BD, and CD. 

Under the isothermal condition imposed in this example, gypsum precipitates only at the boundary a = 0 in both constant porosity and variable porosity cases. Significant changes of porosity occur at this location in the variable porosity case. Figure 5 shows the accumulation of gypsum at this location at times up to 10 seconds for the two cases. The results are qualitatively similar to the results of the previous example. In the case of constant porosity the volume of precipitated gypsum exceeds available pore space at times greater than about... 

The relationships developed for the isothermal conditions for the turbulent free jet can be extrapolated to the chemically reacting flows by using the concept of the equivalent nozzle diameter doech- This is defined as the diameter of the nozzle through which the entrained fluid of density p would expand having the same mass flow rate and momentum as that of the nozzle jet initially considered (Table 31.10). 

Using the combination of the models that describe the regenerated and coked catalyst, an accurate prediction of the performance of the catalyst mixture c28 can be calculated, as shown in Figure 6. So, at the isothermal conditions of this work, the mixture of coked and regenerated catalyst behaves as a physical mixture without the presence of non-linear blending effects. 

Figure 10-17. Plasma-catalytic conversion degree of ethanol as a function of specific energy input. Curve 1 corresponds to an experiment with only thermal energy input point 2 corresponds to an experiment with the microwave discharge dashed lines correspond to the isothermal conditions.

The calorimetry techniques have been used for many purposes in the cure of rubber, including to measure the heat of the overall cure reaction, but also for evaluating the kinetics of the heat evolved from the overall reaction. The only question at first was what way to use the calorimetry In the early eighties, it seemed to a majority of the authors that the isothermal condition would be preferable, but finally the operation in scanning mode was preferred. 

The value of the cure enthalpy of the overall cure reaction is obtained in the same way as previously done by using the isothermal condition process. The area measured between the two curves shown in Figure 3.5 is the basis for the measure of the cure enthalpy. 

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