Solicitud de Financiamiento
5. SOLICITUD DE FINANCIAMIENTO
5.1. Análisis de las brechas financieras - VIH
One useful involves the amount of any quantity per unit denoted as
. Changing variables from b to the left-hand side of Table 2-1
becomes
According to the first bracketed term in Eq. (2.3-6) is exactly zero. Thus, (C) of Table 2-1 is equivalent to
This is employed, for example, in Chapter to obtain a conservation equation for An especially useful form of conservation equation applies to fluids at constant density. In this case, it follows from that
.
(b v) = v V b b(V . v) = v Vb. Using this to introduce the material derivative, (C) of Table 2-1 becomes- (constant
Dt
This result is employed in Sections 2.4 and 2.6 to obtain the most common forms of the energy and species conservation equations. If the flux is of the form f =
(Chapter 1) and if the is constant, then becomes Db
- = (constant p and
Dt
This is the conservation equation for a "constant-property fluid."
2.4 CONSERVATION O F ENERGY: THERMAL EFFECTS
In many problems which require an energy equation, thermal effects are more important
than
mechanical ones. This is usually true when there are large variations, when heats of reaction or latent heats are involved. An approximate equation for conservation of energy that is suitable for such applications is obtained from8) by setting f =q, and Thus, assuming that and are constant, we have
DT
.
+
(constant p andThe source term represents the rate of energy input from external power sources, unit volume. The most common example is resistive heating due to the passage of
electric current.
If the fluid density is constant, the continuity equation simplifies to
v = (constant
The density of any pure fluid is related to the pressure and temperature by an equation of state of the form T). Liquids are virtually incompressible, so that the effects of pressure on p can be ignored. Although gases are not incompressible in the
sense, the effect of pressure on density in gas flows is often negligible. As dis-
cussed in (1968, pp. gases tend to behave as if they are incompress- ible if the velocity is much smaller than the speed of sound. In that the Mach number (Ma) is the ratio of the flow velocity to the speed of sound, this criterion for applying (2.3-2) may be stated as (For dry air at O°C, the speed of sound is 331 For both liquids and gases, spatial variations in density due to temperature may be quite important, in that they cause buoyancy-driven flows convection). However, as discussed in Chapter 12, even in such flows Eq. (2.3-2) is usually an excellent ap- proximation. For these reasons, this form of the continuity equation is employed when solving all problems involving fluid mechanics or convective heat and transfer in this book.
Material Derivative
Using conservation of mass, we can write the general conservation equations for scalar quantities in other These are expressed most easily in terms of the differential operator,
is called the derivative (or substantial derivative).
The material derivative has a specific physical interpretation: It is rate of change as seen by an observer moving with the fluid. This interpretation can be under- stood by considering the rate of change of a scalar perceived by an observer
moving at any velocity u, written as (dbldt),. At any fixed position r, the time depen- dence of b will cause a variation equal to over a small time At. An additional change is caused by movement of the observer, provided that there are spatial variations in b. In particular, a small displacement described by a vector Ar, tangent to causes a change in b equal to (see Section A.6). Given that = for intervals, the resulting change in b over the time
A
t is Adding the contributions to the change in b and letting the rate of change seen by the moving observer isFor an observer moving with the and
as indicated above.
Effects 39
Alternative Conservation Equations
One useful involves the amount of any quantity per unit denoted as
. Changing variables from b to the left-hand side of Table 2-1
becomes
According to the first bracketed term in Eq. (2.3-6) is exactly zero. Thus, (C) of Table 2-1 is equivalent to
This is employed, for example, in Chapter to obtain a conservation equation for An especially useful form of conservation equation applies to fluids at constant density. In this case, it follows from that
.
(b v) = v V b b(V . v) = v Vb. Using this to introduce the material derivative, (C) of Table 2-1 becomes- (constant
Dt
This result is employed in Sections 2.4 and 2.6 to obtain the most common forms of the energy and species conservation equations. If the flux is of the form f =
(Chapter 1) and if the is constant, then becomes Db
- = (constant p and
Dt
This is the conservation equation for a "constant-property fluid."
2.4 CONSERVATION O F ENERGY: THERMAL EFFECTS
In many problems which require an energy equation, thermal effects are more important
than
mechanical ones. This is usually true when there are large variations, when heats of reaction or latent heats are involved. An approximate equation for conservation of energy that is suitable for such applications is obtained from8) by setting f =q, and Thus, assuming that and are constant, we have
DT
.
+
(constant p andThe source term represents the rate of energy input from external power sources, unit volume. The most common example is resistive heating due to the passage of
electric current.
Assuming that k is constant, (2.4-1) and (2.4-2) are combined to give
This energy equation is the one used most frequently in this book. Notice that it is of same form as Equation (2.4-3) is given in rectangular, cylindrical, and spherical coordinates in Table
kinds of mechanical effects are ignored in Eqs. (2.4-1) and (2.4-3), both of would appear as additional source terms. One is viscous dissipation, which is the conversion of energy to heat due to the internal friction in a fluid. This is usually negligible, with the exception of certain high-speed flows (with large velocity gradients) or flows of extremely viscous fluids. The other term not included, which involves pres- sure variations, is related to the work done in compressing a fluid. It is zero for fluids which are incompressible; specifically, it vanishes if at constant pressure is zero. Accordingly, this term is ordinarily negligible for liquids. For gases, it will be negligible if where A P and are the magnitudes of the pressure and temperature variations. For air at ambient conditions, a
change of 8 is energetically equivalent to a pressure change of 1 Pa 0.1 Thus, this effect is negligible also for many applications involving gases.
In Chapter 9 there is a rigorous derivation of conservation of energy for a pure fluid, including the mechanical terms just discussed. A general treatment of energy con- servation in a mixture is given in Chapter
TABLE 2-2
Approximate Forms of the Energy Conservation Equation in Rectangular, Cylindrical, and Spherical Coordinates (Thermal Only)'
Rectangular: z, t )
Cylindrical:
Spherical: T = 8, t )
that and k are and certain effects are negligible.
equations in Section 9.6 for pure fluids and in 1.2 for
Heat Transfer at Interfaces 4 1
2.5
HEAT TRANSFER
ATINTERFACES
In this section we derive the conditions satisfied by the temperature and the heat at boundaries between pure materials when mass transfer is absent). Either material may be a fluid or a solid, but the two phases are understood to be mutually insoluble or immiscible. These interfacial conditions involving temperatures and temperature gradi- ents form the basis for the boundary conditions used most commonly with
The interfacial conditions for simultaneous heat and mass transfer are discussed in Chapter 11.