Steady-state mass balance method

Steady-State Mass Balance Method In theory, the Ki a in an apparatus that is operating continuously under steady-state conditions could be evaluated from the flow rates and the concentrations of the gas and liquid streams entering and leaving, and the known rate of mass transfer (e.g., the oxygen consumption rate of microbes in the case of a fermentor). However, such a method is not practical, except when the apparatus is fairly large and highly accurate instruments such as flow meters and oxygen sensors (or gas analyzers) are available. 

Metabolic flux analysis is one of the most powerful analytical and experimental tools used for physiological characterisation of cell metabolism. In its most basic form, the method is essentially based on the conservation principles used for macrochemical and biological systems applied to the internal environment of cellular systems. The fundamental equation of MFA considers the steady-state mass balances around all intracellular metabolic intermediates such that... 

The main assumptions of the method are not explicitly declared. For example, Rogers does not discuss the problems with respect to the conflict between the transient behaviour of a batch process and the implicit steady-state assumption of the mass balance. However, the mathematical model behind the mass-balance method is quite clear. The work does not include any uncertainty propagation analysis of the mass-balance method. 

Stable-Isotope Mass-Balance Method. The equations presented in this section apply to groundwater-lake systems that are at hydrological and isotopic steady states. Equations that describe isotopic mass balances for non-steady-state systems and forms that pertain to the estimation of evaporation from lakes have been presented by other authors (13, 14). 

Reliable estimates of relative humidity are critical for use of the isotope mass-balance method however, humidity data are difficult to interpret and commonly not available for a specific study area. The index-lake method provides a means for checking the accuracy of these data and the validity of their use in isotope hydrology. The equation that describes the steady-state isotopic composition of a lake (35) is... 

In constrast to the situation described above, the major disadvantages of using CSTR s for kinetic studies stem from the fact that they can be difficult to construct and operate and they consume considerably larger supplies of raw materials. This last factor can be particularly troublesome for academic laboratories. On the other hand, a CSTR system which is operated at steady state provides a method for direct measurement of reaction rate, particle nucleation etc. without data differentiation. The required mass balance measurements yield accurate values for the various rates. 

The utihty stream gets started at operating temperature and flow rate. In the following experiments, the utihty stream is heated so as to initiate the reaction. The main and secondary process tines are fed with water at room temperature and with the same flow rate as one of the experiments. Once steady state is reached, operating parameters are recorded. Process tines are then fed with the reactants, hydrogen peroxide and sodium thiosulfate. At steady state, operating parameters are recorded, and a sample of a known mass of reactor products is introduced in the Dewar vessel. Temperature in the Dewar vessel is recorded until equilibrium is reached, that is, until the reaction ends. This calorimetric method is aimed at calculating the conversion rate at the product outlet and thus the conversion rate in the reactor. The latter is also determined by thermal balances between process inlet and outlet of the reactor. Finally, the reactor is rinsed with water. This procedure is repeated for each experiment... 

The steady-state calculation method for surface water is based on the mass balance schematically shown in Figure 9. 

However, it is possible to directly or indirectly measure the mass flux (mass flow) of conversion gas. Several authors have measured the mass loss of the fuel bed as function of primary air velocities and biofuel [12,33,38,53] by means of a balance. Most of them have used the over-fired batch conversion concept. They utilise the relationship illustrated by Eq. 16 (formulised in amounts instead of flows) above and the assumption that no ash is entrained. As a consequence, the mass loss of the batch bed with time equals the conversion gas. In the simple three-step model [3], an assumption of steady state is made, which is not relevant for batch studies. If it is practically possible, the method of using a balance to measure the conversion gas rate is especially appropriate for transient processes, that is, batch processes. 

The process inputs are defined as the heat input, the product flow rate and the fines flow rate. The steady state operating point is Pj =120 kW, Q =.215 1/s and Q =.8 1/s. The process outputs are defined as the thlrd moment m (t), the (mass based) mean crystal size L Q(tK relative volume of crystals vr (t) in the size range (r.-lO m. In determining the responses of the nonlinear model the method of lines is chosen to transform the partial differential equation in a set of (nonlinear) ordinary differential equations. The time responses are then obtained by using a standard numerical integration technique for sets of coupled ordinary differential equations. It was found that discretization of the population balance with 1001 grid points in the size range 0. to 5 10 m results in very accurate solutions of the crystallizer model. 

For the calculation of kLa two methods based on the liquid and gas phase mass balances (equations 3-17 and 3-18) are possible. For steady state and the case with reaction only in the liquid phase one obtains ... 

The steady-state heat and mass balance equations of the different models were numerically integrated using a fourth-order Runge-Kutta-Gill method for the one-dimensional models, while the Crank-Nicholson finite differences method was used to solve the two-dimensional models. 

This type of solution method is possible for reactions where deactivation is slow, and a pseudo steady-state assumption can be made when solving the mass balance equations. Thus, these equations are applicable to reactions where the activity loss is first-order in both the poison and the active sites, and where deactivation is slow compared to the main reaction. A similar type of approach was taken by Johnson et al. (5), for oxygen consumption and carbon content during catalyst regeneration and by Bohart and Adams (6), for chlorine consumption and absorbence capacity of charcoal. 

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