Unsteady lumped

Mathematical models can also be classified according to the mathematical foundation the model is built on. Thus we have transport phenomena-bas A models (including most of the models presented in this text), empirical models (based on experimental correlations), and population-based models, such as the previously mentioned residence time distribution models. Models can be further classified as steady or unsteady, lumped parameter or distributed parameter (implying no variation or variation with spatial coordinates, respectively), and linear or nonlinear. 

Inserting A and into Eqs. (3.41) and (3.43) gives, after some rearrangement, the unsteady lumped temperature of the ball and that of the bath,... 

Before proceeding to unsteady distributed problems, we next consider a class of unsteady lumped problems periodically depending on time. These problems (lumped or distributed) find many practical applications. 

Having completed our discussion of unsteady lumped problems, we proceed now to unsteady distributed problems. 

Clearly, a DC amplifier may also be used as a differentiator. However, whenever possible this operation should be avoided because of noise. So far, we have seen the use of the DC amplifier as a multiplier, a sign changer, an adder, and an integrator.16 This background is sufficient for solving unsteady lumped problems, which are illustrated next. ... 

Reconsider Ex. 3.4. We wish to simultaneously determine the unsteady lumped ball and bath temperatures. 

In the case of C02, system was broken down into muscle and nonmuscle tissue compartments with unsteady lumped analysis being applicable in each. For 02, only one tissue compartment was employed. 

Figures 9.17-9.19 clearly show that, as the Biot number approaches zero, the temperature becomes uniform within the solid, and the lumped capacity method may be used for calculating the unsteady-state heating of the particles, as discussed in section (2). The charts are applicable for Fourier numbers greater than about 0.2.

What type of model would you use to represent the process shown in the figure Lumped or distributed Steady state or unsteady state Linear or nonlinear ... 

Lumped models (usually called lumped parameter models , which is wrong terminology since the state variables are lumped, not the input variables or parameters) described by transcendental equations for the steady state and ODEs for the unsteady state. 

In polymer processing, the mathematical models are by and large deterministic (as are the processes), generally transport based, either steady (continuous process, except when dynamic models for control purposes are needed) or unsteady (cyclic process), linear generally only to a first approximation, and distributed parameter (although when the process is broken into small, finite elements, locally lumped-parameter models are used). 

Measurements of the rate of deposition of particles, suspended in a moving phase, onto a surface also change dramatically with ionic strength (Marshall and Kitchener, 1966 Hull and Kitchener, 1969 Fitzpatrick and Spiel-man, 1973 Clint et al., 1973). This indicates that repulsive double-layer forces are also of importance to the transport rates of particulate solutes. When the interactions act over distances that are small compared to the diffusion boundary-layer thickness, the rate of transport can be computed (Ruckenstein and Prieve, 1973 Spiel-man and Friedlander, 1974) by lumping the interactions into a boundary condition on the usual convective-diffusion equation. This takes die form of an irreversible, first-order reaction on tlie surface. A similar analysis has also been performed for the case of unsteady deposition from stagnant suspensions (Ruckenstein and Prieve, 1975). 

What is meant by a lumped capacity What are the physical assumptions necessary for a lumped-capacity unsteady-state analysis to apply ... 

Due to the high thermal conductivity of sphere, the conductive resistance within the sphere can be neglected in comparison to the convective resistance at its surface. Accordingly, this unsteady state heat transfer situation could be analyzed as a lumped system. 

In this chapter wefhave already classified unsteady problems with respect to their dependence on space (as lumped or distributed) and have so far studied the lumped problems. Now we may also classify these problems with respect to their dependence on time (as transient or periodic). Consequently... 

In Section 3.2 we focused on the unsteady solution and its steady part for periodic lumped problems. We learned then the practical importance of steady periodic solutions and, in terms of the method of complex temperature, an easy way of obtaining only the steady part of periodic solutions. In this section we apply the method of complex... 

For a lumped detector active junction (bismuth and tellurium layers of thickness 5-1 and 82, respectively, and surface area. 4), neglecting the conductive loss to the Mylar as well as the convective and radiative losses to the ambient, determine (a) the steady temperature, (b) the unsteady temperature, and (c) the time constant of the thermopile. 

These are systems where the state variables describing the system are lumped in space (invariant in all space dimensions). The simplest chemical reaction engineering example is thp perfectly mixed continuous stirred tank reactor. These systems are described at steady state by algebraic equations while in the unsteady state they are described by initial value ordinary differential equations where time is the independent variable. 

Various simplified models can be used with varying degrees of accuracy for the simulation of the transient behaviour of non-porous catalyst pellets. The most suitable unsteady state model for this problem is that with infinite thermal conductivity. This simplified model is quite accurate for metal and metal oxide catalysts. In this model, equation (5.45) disappears and the model becomes strictly lumped parameter described only by ordinary initial value differential equations. 

Grodins, Gray, Schroeder, Norins, and Jones (28) Unsteady state lumped alveolar compartment with continuous ventilation equilibrium between gas in lungs and arterial blood. Lumped tissue compartment with equilibrium between tissue and venous blood. 

Unsteady state lumped compartment made up of arterial blood and the functional residual capacity of lung. 

Unsteady state lumped lung compartment with continuous ventilation with equilibrium between alveolar gas and arterial blood. 

Batch reactors can be considered unsteady-state lumped systems with no input and no output, or it can be considered a distributed system in time (not in space) with the initial conditions as the feed conditions and the final conditions as the exit conditions. 

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