Page 86 - Academic Press Encyclopedia of Physical Science and Technology 3rd Chemical Engineering

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Encyclopedia of Physical Science and Technology EN002E-49 May 17, 2001 20:13

50 Batch Processing

simultaneously,iftheyhappenatthesametime,orinseries states of the system; the terms δQ and δW are inexact
as the time for them to happen arrives. differentials since their values depend on the path followed
by the system during the change. Open systems exchange
both matter and energy with their surroundings. If δψ is
VIII. ANALYSIS
the resultant energy intake during the time dt due to heat
transfer and exchange of matter, the changes of energy
The importance of modeling batch processing systems
and matter of the system are
forces a review of the mathematical analysis needed to set
up and solve the models. The mathematical definition of dE = δψ + δW and dm/dt = 0 (4)
physical problems involves: (1) identification, (2) expres-
sion of the problem in mathematical language, (3) finding 1. Constraints
a solution, and (4) evaluating the solution. The completion
Problem definition requires specification of the initial state
of these steps in the order established determines whether
of the system and boundary conditions, which are mathe-
a solution can be attained. The problem must be identified
matical constraints describing the physical situation at the
before one spends time setting up equations; these and the
boundaries. These may be thermal energy, momentum,
initial and boundary conditions that define the problem
or other types of restrictions at the geometric boundaries.
must be well established before a solution is attempted;
The system is determined when one boundary condition
then a solution can be obtained and evaluated.
is known for each first partial derivative, two boundary
conditions for each second partial derivative, and so on.
A. Problem Identification
◦
In a plate heated from ambient temperature to 1200 F,
Problem identification is a human engineering problem; it the temperature distribution in the plate is determined by
2
arises from discussions with individuals working where the heat equation ∂T/∂t = α∇ T . The initial condition
the problem exists. The problem solver must listen to is T = 60 Fat t = 0, all over the plate. The boundary
◦
those individuals, use data to identify the problem, set up conditions indicate how heat is applied to the plate at
a model that best approximates it, fit the model to the data, the various edges: y = 0, 0 < x < a, ∂T/∂y = 0; y = b,
solve it, and compare the solution with the actual situation. 0 < x < a, ∂T/∂y = 0; x = 0, 0 < y < b, ∂T/∂x = 0;
Then, the problem solver must use the results and capital- x = a,0 < y < b, −k(∂T/∂x) = h(T − T A ). The first three
ize on any learning from the completed analysis. Problem conditions indicate that the plate is insulated while it is
identification leads to system definition, which is best ac- heated by convection along the fourth edge, x = a, from
complished by using a basis of thermodynamics. Physical an environment at temperature T A .
boundaries are set that separate the system from its sur-
roundings. A system may be isolated, closed, or open. Iso-
B. Mathematical Expression
lated systems do not exchange energy or matter with their
surroundings. The energy E and the mass m of the sys- Batch processing problems are described in terms of one
tem remain constant during the process: dE = 0; dm = 0. or more differential equations, sometimes combined with
We deal with systems that change with time; hence, any algebraic or integro-differential equations. The equations
changes in energy or mass must occur within the system. indicate changes in the system with time along its geom-
Consider a system composed of vapor and liquid; a third etry. The geometry is defined by means of a standard set
phase (solid), added at a given moment, reacts with the of space coordinates: x, y, z, Cartesian; r,θ, z, cylindrical;
liquid. The three-phase system is isolated, but there is en- r,θ,ϕ, spherical; or other type. System properties such as
ergy and mass exchange between the solid, liquid, and temperature, concentration, and velocity may change in
vapor. As the solid dissolves and reacts with the liquid, both time and space. Most real situations are described by
it produces heat, which vaporizes some of the liquid, but nonlinear equations, which cannot be solved by analytical
no energy or mass is lost outside the three phases. Closed means. The popularization of computers and the develop-
systems exchange energy, but not matter, with their sur- ment of numerical integrators, such as GEAR, EPISODE,
roundings. The total mass of the system is constant, as it DASSL, and others which can handle almost any situa-
is for an isolated system. The change in energy is given tion, has made possible the solution of many problems
by the law of energy conservation, defined in terms of ODEs both combined and not com-
bined with algebraic equations. A good number of cases
dE = δQ + δW (3)
that are set in terms of PDEs can be handled by trans-
The term dE in Eq. (3) is an exact differential; that is, its formation of the PDE to sets of simultaneous ODEs. The
value is independent of the path followed by the system limits of the solution are determined by the number of si-
during the change; it depends only on the initial and final multaneous equations generated, the size of the computer,