Constant density assumption

Strehlow (1975) achieved a solution by conducting a mass balance over the flow field. Such a balance can be drawn up under the assumptions of similarity and a constant density between shock and flame. The assumption of constant density violates the momentum-conservation equation, and is a drastic simplification. The maximum overpressure is, therefore, substantially underestimated over the entire flame speed range. An additional drawback is that the relationship of overpressure to flame speed is not produced in the form of a tractable analytical expression, but must be found by trial and error. 

Here in Chapter 1 we make the additional assumptions that the fluid has constant density, that the cross-sectional area of the tube is constant, and that the walls of the tube are impenetrable (i.e., no transpiration through the walls), but these assumptions are not required in the general definition of piston flow. In the general case, it is possible for u, temperature, and pressure to vary as a function of z. The axis of the tube need not be straight. Helically coiled tubes sometimes approximate piston flow more closely than straight tubes. Reactors with square or triangular cross sections are occasionally used. However, in most of this book, we will assume that PFRs are circular tubes of length L and constant radius R. 

Example 1.4 Determine the reactor design equations for the various elementary reactions in a piston flow reactor. Assume constant temperature, constant density, and constant reactor cross section. (Whether or not all these assumptions are needed will be explored in subsequent chapters.)... 

The feed is charged all at once to a batch reactor, and the products are removed together, with the mass in the system being held constant during the reaction step. Such reactors usually operate at nearly constant volume. The reason for this is that most batch reactors are liquid-phase reactors, and liquid densities tend to be insensitive to composition. The ideal batch reactor considered so far is perfectly mixed, isothermal, and operates at constant density. We now relax the assumption of constant density but retain the other simplifying assumptions of perfect mixing and isothermal operation. 

Chapter 2 developed a methodology for treating multiple and complex reactions in batch reactors. The methodology is now applied to piston flow reactors. Chapter 3 also generalizes the design equations for piston flow beyond the simple case of constant density and constant velocity. The key assumption of piston flow remains intact there must be complete mixing in the direction perpendicular to flow and no mixing in the direction of flow. The fluid density and reactor cross section are allowed to vary. The pressure drop in the reactor is calculated. Transpiration is briefly considered. Scaleup and scaledown techniques for tubular reactors are developed in some detail. 

Compare the results of the simulation to the analytical solution which can also be calculated with ISIM. For the constant density assumption, simply set b = 0. 

Consider the gas-phase decomposition of ethane (A) to ethylene at 750°C and 101 kPa (assume both constant) in a PFR. If the reaction is first-order with kA = 0.534s-1(Fro-ment and Bischoff, 1990, p. 351), and r is 1 s, calculate /a- For comparison, repeat the calculation on the assumption that density is constant. (In both cases, assume the reaction is irreversible.)... 

The assumption of constant density leads to a result 15% greater than that from equation (C). Desirable as this may be, it is an overestimate of the performance of the PFR based on the other assumptions made. 

Because of the small change in moles on reaction, and the dilution with N2(inert), together with an anticipated (but not calculated) relatively small (-AZ ), the assumption of constant density is a reasonable one. It can be calculated that for every mole of A entering, the total moles entering is 10.57 and the total leaving is 10.92, an increase of only 3.3%. The actual molar feed rates of A (FAo) and total gas (Fto) are ... 

We again assume constant-density flow so that U is solenoidal. The first equality requires this assumption. 

With the assumption that the reaction is irreversible, bimolecular, and of constant density, the rate expressions are given by... 

Further, we want to be able to work problems with numerical solutions. This will require simplifying assumptions wherever possible so that the equations we need to solve are not too messy. This wiU require that fluids are at constant density so that we can use concentrations in molesA olume. This is a good approximation for hquid solutions but not for gases, where a reaction produces a changing number of moles, and temperatures and... 

In deriving these equations we have made many assumptions to keep them simple. We have assumed constant density so that we can use concentration as the composition variable. We have also assumed that the parameters in these systems are independent of temperature and composition. Thus parameters such as AAr, pCp, and JJ are considered to be constants, even though we know they all depend at least weakly on temperature. To be exact, we would have to find the heat of reaction, heat capacity, and heat transfer coefficient as functions of temperature and composition, and for the PFTR insert them within the integrals we must solve for temperature and composition. However, in most situations these variations are small, and the equations written will give good approximations to actual performance. 

A first study refers to liquid water [77]. The signals AS q,x) and A5[r,r] were measured using time-resolved X-ray diffraction techniques with 100 ps resolution. Laser pulses at 266 and 400 nm were employed. Only short times x were considered, where thermal expansion was assumed to be negligible and the density p to be independent of x. To prove this assumption, the authors compared their values of AS q, x) to the values of AS q) obtained from isochoric (i.e., p = const) temperature differential data [78-80]. Their argument is based on the fact that liquid H2O shows a density maximum at 4 °C. Pairs of temperatures Ti, T2 thus exist for which the density p is the same constant density conditions can thus be created in this unusual way. The experiment confirmed the existence of the acoustic horizon (Fig. 8). 

If one assumes that the resin has a constant density and thermal conductivity, an energy equation for the IP process can be obtained by simplifying Equation 5.41. First by using the mass balance equation (i.e., V (Ur) — 0) one of the convective heat transfer terms in the resin phase can be neglected. Another term can be neglected by setting V (Uf ) = 0. This assumption is justified since in the IP process der/dt = 0. 

Under assumption of constant density of states the total spin is related with by S = N/2 = vl... 

In Eqs. (4.1) to (4.6), k is the reaction rate constant. Fk and zk are the molar flow rate and reactant mole fraction in stream k. Note that the assumption of constant density implies that all streams have the same total molar concentration, which is... 

We shall further consider the interaction between two flat plates immersed into an aqueous electrolyte solution with an emphasis on the dielectric colloid material. These dielectric plates bear the surface charge of a constant density. All these assumptions simplify the boundary conditions (33)-(36) sufficiently, yielding... 

The fundamental resonant frequency (/0) shifts when a thin film is deposited on the surface of the quartz crystal. Under the assumption that the density and the shear modulus of the film are the same as those of quartz and that the film is uniform (constant density and thickness) and covering the acoustically active area of the whole... 

In special cases an analytical expression can be derived for Eqn. 7.4. The assumption of constant density allows us to write ... 

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