Mass balances unsteady differential

Differential and Integral Balances. Two types of material balances, differential and integral, are applied in analyzing chemical processes. The differential mass balance is valid at any instant in time, with each term representing a rate (i.e., mass per unit time). A general differential material balance may be written on any material involved in any transient process, including semibatch and unsteady-state continuous flow processes ... 

Note that since there are two independent variables of both length and time, the defining equation is written in terms of the partial differentials, 3C/dt and 3C/dZ, whereas at steady state only one independent variable, length, is involved and the ordinary derivative function is used. In reality the above diffusion equation results from a combination of an unsteady-state mass balance, based on a small differential element of solid length dZ, combined with Pick s Law of diffusion. 

The TDE moisture module (of the model) is formulated from three equations (1) the water mass balance equation, (2) the water momentum, (3) the Darcy equation, and (4) other equations such as the surface tension of potential energy equation. The resulting differential equation system describes moisture movement in the soil and is written in a one dimensional, vertical, unsteady, isotropic formulation as ... 

An unsteady state mass balance over the typical reactor element lying between x and x + 5x gives rise to the partial differential equation... 

In this section we develop a dynamic model from the same basis and assumptions as the steady-state model developed earlier. The model will include the necessarily unsteady-state dynamic terms, giving a set of initial value differential equations that describe the dynamic behavior of the system. Both the heat and coke capacitances are taken into consideration, while the vapor phase capacitances in both the dense and bubble phase are assumed negligible and therefore the corresponding mass-balance equations are assumed to be at pseudosteady state. This last assumption will be relaxed in the next subsection where the chemisorption capacities of gas oil and gasoline on the surface of the catalyst will be accounted for, albeit in a simple manner. In addition, the heat and mass capacities of the bubble phases are assumed to be negligible and thus the bubble phases of both the reactor and regenerator are assumed to be in a pseudosteady state. Based on these assumptions, the dynamics of the system are controlled by the thermal and coke dynamics in the dense phases of the reactor and of the regenerator. 

An unsteady-state mass balance for solids in a differential element with a small perturbation in a system otherwise in equilibrium yields... 

Consider a differential length, dz, of cylindrical conduit as shown in Figure 4.7. Fluid with a uniform velocity u in the axial direction is passing through the differential volume Adz, where A is the cross section of the tube. At a time t = 0, a step input of tracer is introduced uniformly across the cross section at the entrance to the tube at a concentration Cq. We write the general unsteady-state mass balance for the differential volume in the normal fashion ... 

The coefficient of dNi/dt in the summation of (6-4) is defined as the partial molar internal energy of species i, because differentiation with respect to mole numbers of component i is performed at constant T, p and mole numbers of all other species in the mixture. The unsteady-state mass balance for species i in a batch reactor,... 

The parameters n and X(T) which characterize the rate law are evaluated via the differential method of reaction-rate data analysis based on the unsteady-state mass balance ... 

Now, it is instructive to re-analyze the unsteady-state macroscopic mass balance on an isolated solid pellet of pure A with no chemical reaction. The rate of output due to interphase mass transfer from the solid particle to the liquid solution is expressed as the product of a liquid-phase mass transfer coefficient c, liquids a Concentration driving force (Ca, — Ca), and the surface area of one spherical pellet, 4nR. The unsteady-state mass balance on the solid yields an ordinary differential equation for the time dependence of the radius of the peUet. For example,... 

For the following holdup simulation a combination of an empirical holdup correlation (Table 4.3, Lange [33]) with the total mass balance (Eq. (4.1)) is used. This leads to a partial differential equation that describes the liquid holdup oscillation under unsteady-state-operation conditions. The simulation based on this equation combination shows a strongly asymmetrical decay of the liquid pulses (see Fig. 4.8). 

Derive the differential equation for unsteady state for diffusion and reaction for this system. [ Hint First make a mass balance for A for a Az length of tube as follows rate of input (diffusion) + rate of generation (heterogeneous) = rate of output (diffusion) + rate of accumulation.]... 

The extension to a Level IV or unsteady-state model is obvious, the mass balance equations taking the form of differential equations, i.e. ... 

Examples of linear first-order differential equations occur frequently in chemical engineering practice through unsteady state mass balances or first-order chemical reaction problems. 

Presented below is a brief recap of the definition of linear equations in the context of differential equations. Following the recap are examples of unsteady mass balances, which lead to linear first-order problems. Also presented are examples involving chemical reactions that can be treated as linear first-order problems. 

For unsteady-state mass balance inside the floe, considering that all of the reactions producing ethanol and consuming sugar is taking place inside the floe, we obtain the following parabolic partial differential equations ... 

Effective axial transport properties can be determined using an adiabatic reactor. Steady state mass and heat balances result in second-order ordinary differential equations when the axial dispersion is taken into consideration, solutions of which can readily be obtained. Based on these solutions and temperature or concentration measurements, the effective transport properties can be calculated in a manner similar to the procedures used for the radial transport properties. As indicated earlier, a transient experiment can also be used for the determination. Here, experimental and analytical procedures are illustrated for the determination of the effective axial transport property for mass. An unsteady state mass balance for an adiabatic reactor can be written as ... 

The alternative viewpoint involves changing the differential volume on which we make the mass balance. In the foregoing problem, we chose a volume fixed in space, a volume through which liquid was flowing. This volume accumulated no solute, so its use led to a steady-state differential equation. Alternatively, we can choose a differential volume floating along with the fluid at a speed v ax- The use of this volume leads to an unsteady-state differential equation like Eq. 2.3-5. Which viewpoint is correct ... 

As the problems that we discuss in this chapter become more and more complex, the development of the differential equations becomes more and more tedious. Such tedium can be avoided by using the generalized mass balances developed in this section. These mass balances automatically include both steady- and unsteady-state situations. They imply the usual variety of coordinate systems, and they reflect the vectorial nature of mass fluxes. They are excellent weapons. 

Writing unsteady-state component balances for each liquid phase results in the following pair of partial differential equations which are linked by the mass transfer rate and equilibrium relationships... 

A dynamic model of a CSTR can be derived based on unsteady mass and energy balance (see Chapters 4 and 8). The model contains a few nonlinear differential equations, being amenable to analytic or numerical investigation. When n -order reaction is considered, the mass and heat balance equations can be written in the following dimensionless form ... 

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