Integral-heat-balance methods

Pleshanov (P4) extends the integral heat balance method to bodies symmetric in one, two, or three dimensions, using a quadratic polynomial for the approximate temperature function. Solutions are obtained in terms of modified Bessel functions which agree well with numerical finite-difference calculations. 

The mathematical model for the borehole grouting in the regions of permafrost is proposed and solved analytically by the approximate integral-heat-balance method. 

To calculate the amount of catalyst for a particular case, mass and heat balance have to be considered they can be described by two differential equations one gives the differential CO conversion for a differential mass of catalyst, and the other the associated differential temperature increase. As analytical integration is not possible, numerical methods have to be used for which today a number of computer programs are available with which the calculations can be performed on a powerful PC in the case of shift conversion. Thus the elaborate stepwise and graphical evaluation by hand [592], [609] is history. For the reaction rate r in these equations one of the kinetic expressions discussed above (for example, Eq. 83) together with the function of the temperature dependence of the rate constant has to be used. 

The basie equation for the development of methods for forecasting of distribution of temperature fields is the equation of heat balance in the integral form [8] ... 

Voller VR (1989) Development and application of a heat balance integral method for analysis of metallurgical solidification. Appl Math ModeU 13 3—11... 

The steady-state model consists of a set of coupled ordinary differential and algebraic equations. The simulation is obtained by integrating simultaneously the mass-balance equations for the gas and liquid phases in the axial direction of the reactor using a fourth-order Runge-Kutta method. The heat balance is used only for the simulation of the industrial reactors. The solid phase algebraic equations are solved between integration steps with the Newton-Raphson method. Physical properties and mass-transfer coefficients are also updated in every integration step. 

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. 

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. 

Unlike the radiant loss from an optically thin flame, conductive or convective losses never can be consistent exactly with the plane-flame assumption that has been employed in our development. Loss analyses must consider non-one-dimensional heat transfer and should also take flame shapes into account if high accuracy is to be achieved. This is difficult to accomplish by methods other than numerical integration of partial differential equations. Therefore, extinction formulas that in principle can be used with an accuracy as great as that of equation (21) for radiant loss are unavailable for convective or conductive loss. The most convenient approach in accounting for convective or conductive losses appears to be to employ equation (24) with L(7 ) estimated from an approximate analysis. The accuracy of the extinction prediction then depends mainly on the accuracy of the heat-loss estimate. Rough heat-loss estimates are readily obtained from overall balances. 

The chemical process constitutes the structural and motivational framework for the presentation of all of the text material. When we bring in concepts from physical chemistry—for example, vapor pressure, solubility, and heat capacity—we introduce them as quantities whose values are required to determine process variables or to perform material and energy balance calculations on a process. When we discuss computational techniques such as curve-fitting, rootfinding methods, and numerical integration, we present them on the same need-to-know basis in the context of process analysis. 

While the film and surface-renewal theories are based on a simplified physical model of the flow situation at the interface, the boundary layer methods couple the heat and mass transfer equation directly with the momentum balance. These theories thus result in anal3dical solutions that may be considered more accurate in comparison to the film or surface-renewal models. However, to be able to solve the governing equations analytically, only very idealized flow situations can be considered. Alternatively, more realistic functional forms of the local velocity, species concentration and temperature profiles can be postulated while the functions themselves are specified under certain constraints on integral conservation. Prom these integral relationships models for the shear stress (momentum transfer), the conductive heat flux (heat transfer) and the species diffusive flux (mass transfer) can be obtained. 

This method of writing the mass balance, like Eq. (3-10) for batch operation of a tank reactor, separates the extensive variables F and F and relates them to an integral dependent on the intensive conditions in the reaction mixture. It is worthwhile to note the similarity between Eq. (3-13) and the more familiar design equation for heat-transfer equipment based on an energy balance. This may be written... 

X = 0, Trx = Trx, inlet) are available at z = 0. This is known classically as a split boundary value problem, and it is characteristic of countercurrent flow heat exchangers. When numerical methods are required to integrate coupled mass and thermal energy balances subjected to split boundary conditions, it is necessary to do the following ... 

The final two-dimensional mathematical model thus consists of one partial parabolic differential mass balance equation (3.12) with boundary and initial conditions in (3.14) for each of the j reactions and one partial parabolic differential heat transfer equation (3.15) with boundary conditions in (3.17), (3.18) and initial conditions in (3.20). Simultaneously the pressure drop ordinary differential equation (3.7) and the differential equations for the temperature and pressure in each of the surrounding channels in (3.22) must be integrated. Catalyst effectiveness factors in the catalyst bed must be available in all axial and radial integration points using the methods in Section 3.4. 

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