Lumped systems

Fig. 39.14. (a) Catenary compartmental model representing a reservoir (r), absorption (a) and plasma (p) compartments and the elimination (e) pool. The contents X, Xa, Xp and X,. are functions of time t. (b) The same catenary model is represented in the form of a flow diagram using the Laplace transforms Xr, Xa and Xp in the j-domain. The nodes of the flow diagram represent the compartments, the boxes contain the transfer functions between compartments [1 ]. (c) Flow diagram of the lumped system consisting of the reservoir (r), and the absorption (a) and plasma (p) compartments. The lumped transfer function is the product of all the transfer functions in the individual links. 

The geometry of the tubes allows the heat transfer being considered one dimensional, and each tube to be a lumped system in front of the ambient air. This two conditions are fulfilled when Bi < 0.1 (Biot number Bi = a /(/(2a ), where R is the radius of the sample, X its thermal conductivity and a the heat transfer coefficient between the tube and the environment). Once the temperature-time curves of the PCM and the reference substance are obtained (Figure 160), the data can be used to determine the thermophysical properties of the PCM. 

We will discuss in this book only deterministic systems that can be described by ordinary or partial differential equations. Most of the emphasis will be on lumped systems (with one independent variable, time, described by ordinary differential equations). Both English and SI units will be used. You need to be familiar with both. 

We have used an ordinary derivative since t is the only independent variable in this lumped system. The units of this component continuity equation are moles of A per unit time. The left-hand side of the equation is the dynamic term. The first two terms on the right-hand side are the convective terms. The last term is the generation term. 

When we make a balance to obtain a differential equation, we are invoking a natural law, the conservation of matter in our case. If the net flux of any conserved quantity into a lumped system over its boundaries is F, the rate of generation within the system is G, and the amount contained in it is H, then the balance gives... 

This is a partial differential equation, as we should expect from a plug-flow tubular reactor with a single reaction. We note in passing that the solution requires the specification of an initial distribution and a boundary, or feed, value. These are both functions (the first of z because t = 0 the second of t because z = 0) in the distributed system. Of the corresponding quantities, c0 and cin, in the lumped system, the latter is embodied in the ordinary differential equation itself and the former is the initial value. 

Chapter 3 tries to give students the essential tools to solve lumped systems that are governed by scalar equations. It starts with the simplest continuous-start reactor, a CSTR in the adiabatic case. The first section should be studied carefully since it represents the basis of what follows. Our students should write their own codes by studying and eventually rewriting the codes that are given in the book. These personal codes should be run and tested before the codes on the CD are actually used to solve the unsolved problems in the book. Section 3.2 treats the nonadiabatic case. 

The current section has covered numerical techniques and MATLAB codes for investigating the static bifurcation behavior of nonadiabatic lumped systems. 

For each component i the material-balance equation (6.1) is correct regardless of whether the system is lumped or distributed. However, when turning the material-balance equations (6.1) into design equations given in (6.4) below, the situation differs for lumped and distributed systems. The design equations for a lumped system such as depicted in Figure 6.1 is given by... 

The Design Equations (Steady-State Models) for Isothermal Heterogeneous Lumped Systems... 

The above six equations (6.44) to (6.49) are the design equations for this two-phase system when both phases are lumped systems. 

As explained in detail for homogeneous systems in Chapter 4, a distributed system includes variations in the space direction. Therefore for a distributed system we can not use the overall rate of reaction, i.e., the rate of reaction per unit volume multiplied by the total volume, nor the overall rate of mass transfer, i.e., the mass transfer per unit area multiplied by the area of mass transfer. For lumped systems the area of mass transfer is treated as the multiple of the area per unit volume of the process multiplied by the volume of the process for the design equations. 

In this section we develop the heat-balance design equations for heterogeneous systems. Based on the previous sections it is clear how to use the heat-balance and heat-balance design equations that were developed earlier for homogeneous systems, as well as the principles that were used to develop the mass-balance and mass-balance design equations for heterogeneous systems for our purpose. We will start with lumped systems. 

A key aspect of modeling is to derive the appropriate momentum, mass, or energy conservation equations for the reactor. These balances may be used in lumped systems or derived over a differential volume within the reactor and then integrated over the reactor volume. Mass conservation equations have the following general form ... 

Crilcha for Lumped System Analysis 219 Some Remarks on Heat Transfer in Lumped Systems 221... 

We start this chapter with the analysis of lumped systems in which the temperature of a body varies with time but remains uniform throughout at any time. Then we consider the variation of temperature with time as well as position for one-dimensional heat conduction problems such as those associated with a large plane wall, a long cylinder, a sphere, and a semi infinite medium using transient temperature charts and analytical solutions. Finally, we consider transient heat conduction in multidimensional systems by utilizing the product solution. 

A small copper ball can be modeled as a lumped system, but a roast beef camiot. 

In heat transfer analysts, some bodies are observed to behave like a lump whose interior temperature remains essentially uniform at all times during a heat transfer process. The temperature of such bodies can be taken to be a function of time only, 7(r). Heat transfer analysis that utilizes this idealization is known as lumped system analysis, which provides great simplification in certain classes of heat transfer problems without much sacrifice from accuracy. 

Now let tis go to the other extreme and consider a large roast in an oven. If you have done any roasting, you must liave noticed that the temperature distribution within the roast is not even close to being unifonn. You can easily verify this by taking the roast outbefote it is completely done and cutting it in half. You will see that the outer parts of the roast are well done while the center part is barely warm. Thus, lumped system analysis is not applicable in this case. Before presenting a criterion about applicability of lumped system analysis, we develop the formulation associated with it. 

Consider a body of arbitrary shape of mass m, volume J, surface area density p, and specific heat initially at a uniform temperature 7) (Fig. 4-2). At time t = 0, the body is placed into a medium at temperature T., and heat transfer takes place between the body and its environment, with a heat transfer coefficient h. For the sake of discussion, we assume that 7) > 7), but the analysis is equally valid for the opposite case. We assume lumped system analysis to be applicable, so lhat the temperature remains uniform wilhin the body at all times and changes with time only, T T t). 

The lumped system analysis certainly provides great convenience in heat transfer analysis, and naturally we would like to know when it is appropriate... 

The temperature of a lumped system approaches the environment temperature as time gets larger. 

Lumped system analysis assumes a uniform temperature dislribulion throughout the body, which is the case only when the thermal resistance of the body to heat conduction (the conduction resixtance) is zero. Thus, lumped system analysis is exact when Bi = 0 and approximate when Bi > 0. Of course, the smaller the Bi number, the more accurate the lumped system analysis. Then the question we must answer is, How much accuracy are we willing to sacrifice for the convenience of the lumped system analysis ... 

The first step in ilie application of lumped system analysis is the calculation of the Biot number, and the assessment of the applicability of this approach. One may still wish to use Inmped system analysis even when the criterion Bi < 0.1 is not satisfied, if high accuracy is not a major concern. 

Note that the Biot number is tlic ratio of the convection at the surface to conduction within the body, and this number should be as small as possible for lumped system analysis to be applicable. Therefore, small bodies with high tlieniial conductivity are good candidates for lumped system analysis, especially when they are in a medium that is a poor conductor of heat (such as air or another gas) and motionless. Thus, Ihe hot small copper ball placed in quiescent air, discussed eailier, is most likely to satisfy the criterion for lumped system analysis (Fig. 4-6). 

Small bodies with high thermal conductivities and low convection coefficients arc most likely to satisfy the criterion for lumped system analysis. 

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