Fenômentos de Transporte
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Fenômentos de Transporte

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just two. Many of these 
entities will be handled by the simple accounting procedure which we will now 
describe - in particular, those relating to mass, energy, and momentum. Our 
accounting procedure (the entity balance) must be applied to a system or 
control volume. This accounting procedure will lead us in a natural way to 
the mathematical model describing a process. 
By system or control volume we mean some unambiguously defined region 
of space. For example, if we fill a balloon with air and release it, We might take 
as our system the space interior to the balloon. This would provide a perfectly 
acceptable system even though it moves about and changes size in a peculiar 
way as it flies about the room. Although acceptable, i t might or might not be 
the most convenient system, and one of our problems in many situations is 
the selection of the most convenient system. 
One sometimes uses as a system a specific quantity of matter, all parts of 
which remain in proximity; for example, a "clump" of fluid. This is consistent 
with the concept of a system since this matter unambiguously defines a region 
in space. The idea of a system can be illustrated by considering fluid flow in a 
pipe. We could choose as our system the region bounded by two pl'anes normal 
An entity is a thing which has reality and distinctness of being, either in fact or for 
thought. (Webster's Third New Internutionul Dictionary of the English Lunguuge, G 
& C Merriam Company, Springfield, MA, 1969.) 
Chapter I : Essentials 13 
to the pipe axis and the inner wall of the pipe. We could equally well use the 
region between the two planes and the outer wall of the pipe. Or, to take yet a 
third system, we could select a certain mass of fluid and follow the mass as it 
moves in space. 
The entity balance used for our accounting system is 
(1.2- 1) Input + Generation = Output + Accumulation 
By input we mean that which crosses the system boundary from outside to 
inside in time At; by output, the converse. (There is no real need to define the 
second term, output. One could instead speak of positive and negative inputs - 
it is simply traditiond.l0 ) 
By accumulation we mean the result of subtracting that which was in the 
system at the beginning of some time interval from that which was there at 
the end of the interval. As opposed to the input/output case, we do not define a 
term called "depletion" (although we could); we speak instead of a "negative 
By generation we mean that which appears within the system without 
either being present initially or being transferred in across the boundary. It 
materializes, somewhat as the ghost of Hamlet's father, but in a far more 
predictable fashion. Similarly to the case of accumulation, we do not refer to 
"consumption, It but rather to "negative accumulation. " 
You will note that the above is full of "that which"; we have carefully 
avoided saying just what it is for which we are accounting. This is deliberate, 
and is done to stress the generality of the procedure, which, as we shall see 
below, is applicable to people and money as well as to mass, energy, and 
Note that our entity balance is equally valid when applied to rates.ll By 
considering smaller and smaller time intervals we obtain 
l 0 As Tevye says in Fiddler on the Roof before singing "Tradition": "You may ask, 
'How did this tradition start?' I'll tell you - I don't know! But it's a tradition. Because 
of our traditions, everyone knows who he is and what God expects him to do." Even 
engineering and science are bound hy tradition in their methods of communication. 
The definition of a instantaneous rate for an entity is 
1 At -+ 0 I At ,im change in enti ty in At 
I4 Chapter I : Essentials 
1 Input + Generation - Output + Accumulation At - A l + O li-[ At 
so the entity balance applies to rates as well as to amounts. 
Dle 1.2-1 An entity balance 
The entity balance can be applied to very general sorts of quantities so long 
as they are quantifiable, either on a discrete scale (e.g., particles) or a continuous 
scale (e.g., energy). When applied to discrete entities it is frequently called a 
population balance. One such entity, of course, is people, although 
automobiles, tornadoes, buildings, trees, bolts, marriages, etc., can also be 
described using the entity balance approach. 
If the entity balance is applied to people with a political unit (e.g., a city, 
county, state, etc.) as the system, the input term is calculable from immigration 
statistics and the output from emigration statistics. The accumulation term is 
calculable from census figures (the differences in the number within the system 
over some prescribed time interval). The generation term is made up of births 
(positive) and deaths (negative). 
For example, suppose that a political unit (the system) had a population of 
l,OOO,OOO pe~ple at the beginning of the year (t = 0). At the end of the year (t + 
At) suppose the population was 1,010,OOO. Thus the accumulation over a year 
(At) was 
1,010,Ooo - 1,Ooo,Ooo = 10,000 (1.2-4) 
Chapter I : Essentials I5 
During this time suppose that 18,000 people died and 30,000 were born, 
giving a net generation of 
30,000 - 18,000 = 12,000 people (1.2-5) 
If 40,000 people immigrated (input), how many emigrated (output)? 
Application of the entity balance shows 
40,000 + 12,000 = Output + 10,OOO (1.2-6) 
Solving, we see that 42,000 people emigmted. 
An edifying exercise is to check published statistics on population, 
birthldeath, and immigratiodemigration for consistency by using the entity 
balance. Such a calculation performed on the world population reveals a large 
accumulation term, often referred to as the "population explosion." 
1.2.1 Conserved quantities 
It is interesting that when our equation is applied to some entities, there is 
never any generation. Quantities which do not exhibit generation - that is, 
quantities which are neither created nor destroyed - we term conserved 
quantities. For these quantities the entity balance contains only three terms 
Input = Output + Accumulation (1.2.1-1) 
This balance can also be written in terms of rates 
Input Rate = Output Rate + Accumulation Rate (1.2.1-2) 
For example, if I apply the entity balance to the amount in my checking 
account as the system, I never find money spontaneously appearing. (If 1 didn't 
put it in, it isn't there to remove - the first great law of personal finance. We 
note in passing that an embezzler may be regarded as either an output or a 
negative accumulation term but the total amount of money is still conserved.) 
I6 Chapter I : Essentials 
Quantities which are conserved under some assumptions or approximations 
may not be conserved under other assumptions. Banks and the government can 
create and destroy money, but this requires a model of a different level, much as 
including nuclear reactions in our energy balance models would destroy the 
assumption of conservation of energy. 
We know that matter and energy are not conserved in processes where 
nuclear reactions are taking place or where things move at speeds approaching 
the speed of light. Matter and energy can, instead, be transmuted into one 
another, and only their totality is conserved. 
In the course of this book, we will assume that total mass and total 
energy are conserved quantities, as they are for all practical purposes in the 
processes we will consider. On the other hand, momentum (which can be created 
or destroyed by applied forces), mass of an individual species (which can be 
created or destroyed by chemical reaction), and mechanical energy and thermal 
energy (which can each be transformed into the other) are not generally conserved 
in our processes. 
Another difference in applying