Fig. 2.1 Balance of fluxes in a linear system in .NET

Creator gs1 datamatrix barcode in .NET Fig. 2.1 Balance of fluxes in a linear system
Fig. 2.1 Balance of fluxes in a linear system
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The tube contains the thermodynamic quantity in an amount M (amount of a substance, thermal energy, etc.), which has a density (concentration, energy density, etc.) ip(x) at each point in the tube defined by the relationship
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where V is the volume of the system. It thus holds for M that M = A I ydx (2.2.2) Jo The change in amount M in the whole tube per unit time is given by the relationship
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Consider a section of the tube with length Ax; inside this section and in its vicinity, J is a continuous function of x. If an amount J(x)AAt flows into this section at point x in time At and an amount J(x + Ax)A At at point x + Ax, then its change in volume A Ax is given as
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dxAt = -A[J(x dt
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+ AJC) - J(x)]At
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Dividing both sides of t h e equation by A Ax At a n d taking t h e limit yields the continuity relation
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T h e general case of a three-dimensional body enclosed by a surface S will b e treated using vector analysis. T h e time change in t h e a m o u n t M is given by the relationship
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= -<j)JdS
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The normal vector dS points out of the body and J > 0 for a quantity flowing out of the body; thus the right-hand side of Eq. (2.2.2) must be negative in sign. Obviously, 3M (dip = dV dt J dt In view of the Gauss-Ostrogradsky theorem (2.2.7) V
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substitution into Eq. (2.2.6) gives \^-dV
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(2.2.8)
= - fdivJdK
(2.2.9)
and differentiation with respect to V yields the continuity equation -^=-divJ (2.2.10) dt If in the system the thermodynamic quantity is formed (e.g. by chemical reaction) at a rate of pi units of quantity per unit volume and unit time, then Eq. (2.2.10) must be completed to yield -^- = - d i v J + pi (2.2.11)
For thermodynamic quantities other than the amount of substance, the quantity ip can be expressed by the equation dM dMdw (2.2.12) where m is the amount of the quantity M per unit of mass (e.g. energy per kilogram), w denotes mass and p is the density of the substance present in the system. Equation (2.2.11) then takes the form ^ /i (2.2.13)
References De Groot, S. R., and P. Mazur, see page 81. Denbigh, K. G., see page 81. Ibl, N., Fundamentals of transport phenomena in electrolytic systems, CTE, 6, 1 (1983). Levich, V. G., Physico-Chemical Hydrodynamics, Prentice-Hall, Englewood Cliffs, 1962.
Production of Entropy, the Driving Forces of Transport Phenomena
If the system considered is closed and thermally insulated, then its entropy increases during a transport phenomenon
(2.3.1)
If the system is neither closed nor thermally insulated, then the change in the entropy with time consists of two quantities: of the time change in the entropy as a result of processes occurring within the system St and of entropy changes in the surroundings, caused by transfer of the entropy from the system in the reversible process S"e
-- +
dt dt dt If we consider a system of infinitesimal dimensions, then the entropy balance can be described directly by Eq. (2.2.13):
(2.3.2) }
p^=-divJ5 + a (2.3.3) at where s is the entropy of the system, related to 1 kg of substance, Js is the entropy flux and o is the rate of entropy production per unit volume. The first term on the right-hand side of this equation corresponds to dSJdt and the second to dSJdt in Eq. (2.3.2). The rate of entropy formation for a single transport process (Eq. 2.1.1) is given as
o = ~3X = ^LX2 (2.3.4)
(unit J m ' V 1 ^ 1 ) . X is the absolute value of the vector of the driving force X. Generally,
o = ^ 2 J,X; = ^ 2 2 UXiX*