**The Integral ****Equations**

**The Integral****Equations**Many, if not most, of the quantities of interest in ﬂuid mechanics are integral quantities; they are found by integrating some property of interest over an area or a volume. Many times the property is essentially constant so the integration is easily performed but other times, the property varies over the area or volume, and the required integration may be quite difﬁcult. Some of the integral quantities of interest are: the rate of ﬂow through a pipe, the kinetic energy in the wind approaching a wind machine, the power generated by the blade of a turbine, and the drag on an airfoil. There are quantities that are not integral in nature, such as the minimum pressure on a body or the point of separation on an airfoil; such quantities will be considered in Chap. 5.

To perform an integration over an area or a volume, it is necessary that the integrand be known. The integrand must either be given or information must be avail- able so that it can be approximated with an acceptable degree of accuracy. There are numerous integrands where acceptable approximations cannot be made requiring the solutions of differential equations to provide the required relationships; external

ﬂow calculations, such as the lift and drag on an airfoil, often fall into this category. In this chapter, only those problems that involve integral quantities with integrands that are given or that can be approximated will be considered.

**4.1 ****System-to-Control-Volume**** ****T****r****ansformation**

The three basic laws that are of interest in ﬂuid mechanics are often referred to as the conservation of mass, energy, and momentum. The last two are more speciﬁcally called the ﬁrst law of thermodynamics and Newton’s second law. Each of these laws is expressed using a Lagrangian description of motion; they apply to a speciﬁed mass of the ﬂuid. They are stated as follows:

**Mass: **The mass of a system remains constant.

**Energy: **The rate of heat transfer to a system minus the work rate done by a system equals the rate of change of the energy *E *of the system.

**Momentum: **The resultant force acting on a system equals the rate of momentum change of the system.

Each of these laws applies to a collection of ﬂuid particles and the density, speciﬁc energy, and velocity can vary from point to point in the volume of interest. Using the material derivative and integration over the volume, the laws are now expressed in mathematical terms:

where the dot over *Q *and *W *signiﬁes a time rate and *e *is the speciﬁc energy included in the parentheses of Eq. (1.33). It is very difﬁcult to apply the above equations directly to a collection of ﬂuid particles as the ﬂuid moves along in a simple pipe ﬂow as well as in a more complicated ﬂow, such as ﬂow through a turbine. So, let’s convert these integrals that are expressed using a Lagrangian description to integrals expressed using a Eulerian description (see Sec. 3.1). This is a rather tedious**F****i****g****u****re 4.1 **The system and the ﬁxed control volume.

derivation but an important one. In this derivation, it is necessary to differentiate between two volumes: a *control volume *that, in this book is a ﬁxed volume in space, and a *system *that is a speciﬁed collection of ﬂuid particles. Figure 4.1 illustrates the difference between these two volumes. It represents a general ﬁxed volume in space through which a ﬂuid is ﬂowing; the volumes are shown at time *t *and at a slightly later time *t *+ Δ*t. *Let’s select the energy *E *= ∫sys *e*ρ*d V *with which to demonstrate the material derivative. We then write, assuming Δ

*t*to be a small quantity,

where we have simply added and subtracted *E*1 (*t *+ Δ*t*) in the last line. Note that the

ﬁrst ratio in the last line above refers to the control volume so that

where an ordinary derivative is used since we are no longer following a speciﬁed ﬂuid mass. Also, we have used “c.v.” to denote the control volume. The last ratio in

**F****i****g****u****re 4.2 **Differential volume elements from Fig. 4.1.

Eq. (4.4) results from ﬂuid ﬂowing into volume 3 and out of volume 1. Consider the differential volumes shown in Fig. 4.1 and displayed with more detail in Fig. 4.2.

Note that the area *A _{1}*+

*A*

_{3}completely surrounds the control volume so that

where “c.s.” is the control surface that surrounds the control volume. Substituting Eqs. (4.5) and (4.6) into Eq. (4.4) results in the *Reynolds transport theorem*, a system- to-control-volume transformation:

where, in general, *e *would represent the speciﬁc property of *E*. Note that we could have taken the limit as Δ*t *→ 0 to make the derivation more mathematically rigorous.

If we return to the energy equation of Eq. (4.2) we can now write it as

If we let *e *= 1 in Eq. (4.7) [see Eq. (4.1)], the conservation of mass results. It is

And ﬁnally, if we replace *e *in Eq. (4.7) with the vector **V **[see Eq. (4.3)], Newton’s second law results:

These three equations can be written in a slightly different form by recognizing that a *ﬁxed control volume *has been assumed. That means that the limits of the ﬁrst integral on the right-hand side of each equation are independent of time. Hence, the time derivative can be moved inside the integral if desired. Note that it would be written as a partial derivative should it be moved inside the integral since the integrand depends, in general, on *x, y, z, *and *t*. For example, the momentum equation would take the form

The following three sections will apply these integral forms of the basic laws to problems in which the integrands are given or in which they can be assumed.

**4.2 ****Continuity**** ****Equation**

The most general relationship for the conservation of mass using the Eulerian description that focuses on a ﬁxed volume was developed in the preceding section as Eq. (4.9). Since the limits on the volume integral do not depend on time, this can be written as

If the ﬂow of interest can be assumed to be a steady ﬂow so that time does not enter the above equation, the equation simpliﬁes to

Those ﬂows in which the density *r *is uniform over an area are of particular interest in our study of ﬂuids. Also, most applications have one entrance and one exit. For such a problem, the above equation can then be written as

where an over bar denotes an average over an area, i.e., *VA *= ∫*V**dA*. Note also that

at an entrance we use **n**ˆ ⋅ **V**1 = −*V*1 since the unit vector points out of the volume, and the velocity is into the volume. But at an exit, **n**ˆ ⋅ **V**2 = *V*2 since the two vectors are in the same direction.

For incompressible ﬂows in which the density does not change1 between the entrance and the exit, and the velocity is uniform over each area, the conservation of mass takes the simpliﬁed form:

We refer Eqs. (4.12) to (4.15) as the *continuity equation*. These equations are used most often to relate the velocities between sections.

The quantity *r**A**V *is the *mass ﬂux *and has units of kg/s. The quantity *VA *is the *ﬂow rate *(or *discharge*) and has units of m3/s. The mass ﬂux is usually used in a gas ﬂow and the discharge in a liquid ﬂow. They are deﬁned by

where *V *is the average velocity at a section of the ﬂow.

__EXAMPLE 4.1__

Water ﬂows in a 6-cm-diameter pipe with a ﬂow rate of 0.02 m3/s. The pipe is reduced in diameter to 2.8 cm. Calculate the maximum velocity in the pipe. Also calculate the mass ﬂux. Assume uniform velocity proﬁles.

__Solution__

The maximum velocity in the pipe will be where the diameter is the smallest. In the 2.8-cm diameter section we have

Water ﬂows into a volume that contains a sponge with a ﬂow rate of 0.02 m3/s. It exits the volume through two tubes, one 2 cm in diameter, and the other with a

mass ﬂux of 10 kg/s. If the velocity out of the 2-cm-diameter tube is 15 m/s, determine the rate at which the mass is changing inside the volume.

__Solution__

The continuity equation (4.9) is used. It is written in the form