__Introduction__

All the physical quantities are given by a few fundamental quantities or their combinations. The units of such fundamental quantities are called base units, combinations of them being called derived units. The system in which length, mass and time is adopted as the basic quantities, and from which the units of other quantities are derived, is called the absolute system of units.

__Dimension:__

A dimension is the measure by which a physical variable is expressed quantitatively.

__Unit:__

A unit is a particular way of attaching a number to the quantitative dimension.

Measured physical properties have a basic dimension in which they are measured. There may be many units that are used to measure this dimension. This is best shown by example. The thickness of an object has the dimension of length. Length can be measured in a wide range of units including inches, feet, yards, meters, kilometers, micrometers, Angstrom units, furlongs, fathoms, light-years and many more. The thickness of an object cannot be measured in kilograms, however. That is because the kilogram is a unit used to measure quantities that have the fundamental dimension of mass.

The choice of the units to use in a particular measurement is a matter of convention or convenience. Whatever units you use, however, must correspond to the correct dimension for the physical quantity.

Systems of units usually start by making arbitrary definitions of a unit for fundamental dimensions. Typically these fundamental dimensions are mass, length, time, electric charge and temperature. Once these units are selected for the fundamental dimensions the units for other physical quantities can be determined from the physical relations among quantities having the fundamental units. For example velocity is found as distance divided by time. Thus the dimensions of velocity must be length/time. Similarly the dimensions of acceleration, found as velocity divided by time, must be length/time^{2}, and the dimensions of force can be found from Newton’s second law: force equals mass times acceleration. This gives the dimensions of force as the dimensions of mass times the dimensions of acceleration or (mass) times (length) divided by (time). The symbols M, L, and T are usually used to represent dimensions of mass, length, and time, respectively.

We can continue in this fashion. Pressure is force per unit area; this must have dimensions of (force) / (length)^{2} = (mass) (length) / [(time)^{2}(length)^{2}] = (mass) / [(time)^{2}(length)]. Work is the product of force time’s distance. Thus the dimensions of work must be the dimensions of force times the dimensions of distance. This means that the dimensions of work are (mass)(length)^{2} /(time)^{2}. Because work is a form of energy we should expect that any kind of energy should have the same dimensions. We can check this by recalling the formula for kinetic energy: mV^{2}/2. This has the dimensions of mass times the dimensions of velocity squared. But, this is just (mass) times (length/time)^{2}, which is the result that we just found for work. Similarly, potential energy, mgz, has dimensions of mass time’s dimensions of acceleration time’s dimensions of length. This gives (mass) times [(length)/(time)^{2}] times (length). Again the same result: the dimensions of energy are (mass) (length)^{2} /(time)^{2}. Finally, power which is energy divided by time must have dimensions of (mass) (length)^{2} /(time)^{3}.

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__SI units __

In the SI system of units the first three fundamental dimensions are mass, length and time. The units used to measure these dimensions (and their abbreviations) are defined to be the kilogram (kg), the meter (m) and the second (s). With these definitions the units for velocity and acceleration become meters per second (m/s) and meters per second per second (or meters per second2, m/s2), respectively.

The kilogram is defined as the mass of a reference material located at the International Bureau of Weights and Measures. The meter is defined as the length of a path traveled by light in a vacuum during a time interval of 1/299,792,458 of a second. (This is equivalent to defining the speed of light as 299,792,458 m/s.) The second is defined in terms of the wavelength of a certain emission from a cesium atom and the kelvin (temperature) is defined as 1/273.16 of the triple point temperature of water. The ampere is defined as the basic unit of electrical current.

As mentioned above the force units must be the same as the mass unit’s times the acceleration units. Thus the SI units of force are kg•m/s2. These units are given a special name, the newton (N). Note that we use a capital N when we spell the last name of Sir Isaac Newton, but a lower case n when we spell the name of the SI force unit, the newton. However, we use an upper case N when we abbreviate one newton as 1 N. This is the common practice in SI unit naming conventions.

This definition of the newton can be written as a unit-conversion factor equation:

1 N = 1 kg•m•s-2

Similarly the units for pressure, energy and power, which can be written in terms of fundamental units, are also given separate names for convenience. Table 1 gives a summary of the various units in the SI system of units.

__Prefixes __

The wide scale of units in engineering applications is handled by using prefixes that account for multiplicative factors on the SI units. Thus a kilopascal (abbreviation kPa) means 1000 Pascals. The following prefixes, and their associated factors and abbreviations are commonly used with SI units.

Note that in areas and volumes we speak of square kilometers or cubic millimeters. We apply the prefix to the length measurement. So parallelepiped that has sides of 7 mm, 12 mm, and 25 mm has a volume of 2,100 mm3. This is a volume of 2.1×10-6 m3. The kilogram is an unusual SI unit. Although it is a basic unit, it has the kilo prefix, because it was originally defined as 1000 grams. For purposes of naming mass units, the gram = 10-3 kilograms provides the base name. Thus we can speak of megagrams (instead of kiliokiilograms), micrograms (instead of nanokilograms), and the like rather than using a prefix in front of kilograms.

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__Engineering units __

In the engineering system of units the word pound is used to define both a unit of force and a unit of mass. To distinguish between the two we will speak of a pound-force (lbf) and a pound-mass (lbm). The pound-mass is defined in terms of the kilogram as a mass unit. One pound mass equals 0.4535924 kilogram. The basic unit of length, the foot, is defined to be exactly 0.3048 meters. The time unit, the second, is the same as in SI units.

If we were to define a force unit in the same way as defined in SI units we would create a name for the mass times acceleration units of 1 lbm•ft•s-2. In fact there is a name for this quantity. It is called a poundal; a name found only in obscure physics texts. The usual unit for force, the pound-force, is defined as the force that is generated when one pound mass is weighed in a standard gravitational field** of 32.174 ft•s-2. Thus we define

1 lb_{f} = 32.174 lb_{m}•ft•s^{-2}

Because this unit conversion factor is based on a standard gravitational acceleration, it is a fixed number, which is always the same regardless of the local gravitational acceleration. The weight of a mass is the product of the mass times the local gravitational acceleration. At a point where the local gravitational acceleration, g = 32.11 ft/ s^{2}, the weight of a 100 pound mass would be given as

W = (100 lbm ) ( 32.11 ft/ s2.) = 3211 lbm• ft/ s2

We can convert these unwieldy units to a more conventional one by using the unit conversion factor for pound force. A convenient way to use unit conversion factors is to write them as a ratio equal to unity. Thus we would write:

1 = 32.174 lbm•ft /(lbf•s^{2})

We can multiply anything by 1 and not change the result; so we can multiply our answer of 3211 lbm• ft/ s^{2}, found above, by the unit conversion factor written in this form to obtain

Note that we can regard the abbreviations for the units as algebraic symbols to be cancelled. When we do this we wind up with the correct set of units. In a similar manner we could compute the kinetic energy of a 2000 pound mass moving at a velocity of 100 ft/s as follows:

Again we see that the use of the unit conversion factor and the canceling of the unit abbreviations as algebraic quantities give us the result in the units we desire. Note that an answer of 107 lbm • ft/s2 would have the correct dimensions, but the units are not the conventional ones. The technique outlined above for algebraic cancellation of units is a helpful tool for ensuring that you have correct units when you are working with new equations or new systems of units.

Another approach to mass and force units is to start by taking the pound force as a defined term, equal to 4.4482216 N. Then we can define the mass unit as the mass that requires a force of 1 lb_{f} to accelerate it at a rate of 1 ft/s^{2}. This mass is called the slug, and the relationship between slug and pound force is similar to the one between the kilogram and the newton.

1 lb_{f} = 1 slug•ft•s^{-2}

By comparing the relationship of the pound mass and the slug with the pound force, we obtain the following relationship between the two mass units.

1 slug = 32.174 lb_{m}^{}

The English system of units has no formal name for the energy unit of ft•lbf.There is a commonly used energy unit, the British thermal unit or Btu[1], which is defined in terms of ft•lbf as follows:

1 Btu = 778.169 ft-lbf = 1.055056 kJ

In addition to the usual SI unit of watts for power, the engineering system of units sometimes uses the unit of horsepower (hp). The unit conversion factors for power are:

1 hp = 550 ft•lbf / s = 2544.433 Btu / hr = 0.7456999 kW 1 kW = 3412.14 Btu / hr

__Temperature Units__

As noted above, the fundamental unit of temperature in the SI system is defined as the kelvin that is 1/273.16 of the temperature of the triple point of water. The abbreviation for kelvins is the letter K, without the degree sign. This is known as an absolute temperature since the zero on the Kelvin scale is the known to be the absolute zero of temperature; no temperature lower than this is possible. Conventionally, the relative temperature measurement, degrees Celsius (^{o}C) is used. This is defined as follows:

^{o}C = K – 273.15

In engineering units the relative temperature unit is the degree Fahrenheit, (^{o}F) that is related to the degree Celsius by the following equations.

^{o}C = (^{o}F – 32)/1.8 ^{o}F = 1.8(^{o}C) + 32

The engineering absolute temperature unit is rankine. The increment on the rankine scale is 1.8 times the increment on the kelvin scale. Since both Kelvin and Rankine are absolute temperatures they are related by the simple equation shown below. The relation below between rankine and degrees Fahrenheit can be derived from previous temperature relations.

R = 1.8 K = ^{o}F + 459.69

In several applications the important variable is the temperature difference. Since the kelvin (or rankine) temperatures are found by adding a constant to the degrees Celsius (or Fahrenheit), taking the difference of two temperatures in absolute or relative units will give the same result. For example, the difference between 100^{o}C and 200^{o}C is 100^{o}C. If both temperatures are converted to kelvins, the difference is between 373.15 K and 473.15 K, which is 100 K. The following general rules always hold: the numerical value of the temperature difference between the same temperatures expressed in ^{o}C or K is the same. Similarly, the numerical value of the temperature difference between the same temperatures expressed in ^{o}F or R is the same.

**Other variables **

In thermodynamics the **heat capacity**, c, (sometimes called the specific heat) represents the amount of heat that is transferred per unit mass per unit temperature difference. The usual units for heat capacity are J/kg·K in SI units and Btu/lb_{m}·R in engineering units. The heat capacity depends on the type of process and is usually subscripted as c_{p} for a constant pressure process and c_{v} for a constant volume process.

Surface thermodynamics and fluid mechanics use the **surface tension**, s, associated with the force generated by a surface. Units for surface tension are N/m or lb_{f}/m.

The viscosity, m, is an important variable in fluid mechanics and convective heat transfer. The shear stress in a fluid is proportional to velocity gradients and the proportionality constant is called the viscosity. The dimensions for viscosity are (force)(time)/(length)^{2} which is the same as (mass)/(length)/(time). Typical units are N·s/m = kg/m·s in SI units or lb_{f}·s/ft^{2} = slug/ft·s = 32.174 lb_{m}/ft·s in engineering units.

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__Properties of fluids:__

Fluids are divided into liquids and gases. A liquid is hard to compress and as in the ancient saying ‘Water takes the shape of the vessel containing it, it changes its shape according to the shape of its container with an upper free surface. Gas on the other hand is easy to compress, and fully expands to fill its container. There is thus no free surface.

Consequently, an important characteristic of a fluid from the viewpoint of fluid mechanics is its compressibility. Another characteristic is its viscosity. Whereas a solid shows its elasticity in tension, compression or shearing stress, a fluid does so only for compression. In other words, a fluid increases its pressure against compression, trying to retain its original volume. This characteristic is called compressibility. Furthermore, a fluid shows resistance whenever two layers slide over each other. This characteristic is called viscosity.

__Density, Specific gravity and specific volume:__

The mass per unit volume of material is called the density, which is generally expressed by the symbol ρ. The density of a gas changes according to the pressure, but that of a liquid may be considered unchangeable in general. The units of density are kg/m3 (SI). The density of water at 4°C and 1 atm (101 325 Pa, standard atmospheric pressure; see Section 3.1.1) is 1000 kg/m3. The ratio of the density of a material ρ to the density of water ρ , is called the specific gravity, which is expressed by the symbol s:

__Viscosity:__

As shown in Fig. 2.1, suppose that liquid fills the space between two parallel plates of area A each and gap h, the lower plate is fixed, and force F is needed to move the upper plate in parallel at velocity U. Whenever Uh/v < 1500 (v = µ/ρ: kinematic viscosity), laminar flow is maintained, and a linear velocity distribution, as shown in the figure, is obtained. Such a parallel flow of uniform velocity gradient is called the Couette flow.

In this case, the force per unit area necessary for moving the plate, i.e. the shearing stress (Pa), is proportional to U and inversely proportional to h. Using a proportional constant ρ, it can be expressed as follows:

The proportional constant ρ is called the viscosity, the coefficient of viscosity or the dynamic viscosity.

Fig. 2.1 Couette flow

Fig. 2.2 Flow between parallel plates

Such a flow where the velocity u in the x direction changes in the y direction is called shear flow. Figure 2.1 shows the case where the fluid in the gap is not flowing. However, the velocity distribution in the case where the fluid is flowing is as shown in Fig. 2.2. Extending eqn (2.4) to such a flow, the shear stress z on the section dy, distance y from the solid wall, is given by the following equation:

This relation was found by Newton through experiment, and is called Newton’s law of viscosity.

Fig. 2.3 Change in viscosity of air and of water under 1 atm

In the case of gases, increased temperature makes the molecular movement more vigorous and increases molecular mixing so that the viscosity increases. In the case of a liquid, as its temperature increases molecules separate from each other, decreasing the attraction between them, and so the viscosity decreases. The relation between the temperature and the viscosity is thus reversed for gas and for liquid. Figure 2.3 shows the change with temperature of the viscosity of air and of water.

The units of viscosity are Pa s (Pascal second) in SI, and g/(cm s) in the CGS absolute system of units. lg/(cm s) in the absolute system of units is called 1 P (poise) (since Poiseuille’s law, stated in Section 6.3.2, is utilized for measuring the viscosity, the unit is named after him), while its 1/100th part is 1 CP (centipoise). Thus

The value v obtained by dividing viscosity µ by density ρ is called the kinematic viscosity or the coefficient of kinematic viscosity:

Since the effect of viscosity on the movement of fluid is expressed by v, the name of kinematic viscosity is given. The unit is m2/s regardless of the system of units. In the CGS system of units 1 cm2/s is called 1 St (stokes) (since Stokes’ equation, to be stated in Section 9.3.3, is utilized

Table 2.3 Viscosity and kinematic viscosity of water and air at standard atmospheric pressure

for measuring the viscosity, it is named after him), while its 1 / 100th part is 1 cSt (centistokes).

Thus

The viscosity ρ and the kinematic viscosity µ of water and air under standard atmospheric pressure are given in Table 2.3.

The kinematic viscosity v of oil is approximately 30-100 cSt. Viscosity sensitivity to temperature

Fig. 2.4 Rheological diagram

is expressed by the viscosity index VI, a non-dimensional number. A VI of 100 is assigned to the least temperature sensitive oil and 0 to the most sensitive. With additives, the VI can exceed

100. While oil is used under high pressure in many cases, the viscosity of oil is apt to increase somewhat as the pressure increases.

For water, oil or air, the shearing stress z is proportional to the velocity gradient du/dy. Such fluids are called Newtonian fluids. On the other hand, liquid which is not subject to Newton’s law of viscosity, such as a liquid pulp, a high-molecular-weight solution or asphalt, is called a

non-Newtonian fluid. These fluids are further classified as shown in Fig. 2.4 by the relationship between the shearing stress and the velocity gradient, i.e. a rheological diagram. Their mechanical behavior is minutely treated by rheology, the science allied to the deformation and

flow of a substance.

__Compressibility:__

As shown in Fig. 2.9, assume that fluid of volume V at pressure p decreased its volume by AV due to the further increase in pressure by Δρ. In this case, since the cubic dilatation of the fluid is ΔV/V, the bulk modulus K is expressed by the following equation:

Its reciprocal β

is called the compressibility, whose value directly indicates how compressible the fluid is. For water of normal temperature/pressure K = 2.06 x lo9 Pa, and for air K = 1.4 x los Pa assuming adiabatic change. In the case of water, B = 4.85 x lo-” l/Pa, and shrinks only by approximately 0.005% even if the atmospheric pressure is increased by 1 atm.

Putting ρ as the fluid density and M as the mass, since ρV = M = constant, assume an increase in density Δρ whenever the volume has decreased by ΔV, and

The bulk modulus K is closely related to the velocity a of a pressure wave propagating in a liquid, which is given by the following equation (see Section 13.2)

**Fig. 2.9 **Measuring of bulk modulus of fluid

__Vapour Pressure:__

Liquids exhibit a free surface in the container whereas vapours and gases fill the full volume. Liquid molecules have higher cohesive forces and are bound to each other. In the gaseous state the binding forces are minimal. Molecules constantly escape out of a liquid surface and an equal number constantly enter the surface when there is no energy addition. The number of molecules escaping from the surface or re-entering will depend upon the temperature.

Under equilibrium conditions these molecules above the free surface exert a certain pressure. This pressure is known as vapour pressure corresponding to the temperature.

As the temperature increases, more molecules will leave and re-enter the surface and so the vapour pressure increases with temperature. All liquids exhibit this phenomenon. Sublimating solids also exhibit this phenomenon. The vapour pressure is also known as saturation pressure corresponding to the temperature. The temperature corresponding to the pressure is known as saturation temperature.

If liquid is in contact with vapour both will be at the same temperature and under this condition these phases will be in equilibrium unless energy transaction takes place. The vapour pressure data for water and refrigerants are available in tabular form. The vapour pressure increases with the temperature. For all liquids there exists a pressure above which there is no observable difference between the two phases. This pressure is known as critical pressure. Liquid will begin to boil if the pressure falls to the level of vapour pressure corresponding to that temperature. Such boiling leads to the phenomenon known as cavitation in pumps and turbines. In pumps it is usually at the suction side and in turbines it is usually at the exit end.

__Gas Laws:__

**Boyle’s law**

At constant temperature, the product of an ideal gas’s pressure and volume is always constant.

where,

P_{1} is pressure

V_{1} is volume

P_{2} is pressure

V_{2} is volume

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**Charles’s Law:**

For an ideal gas at constant pressure, the volume is directly proportional to its temperature.

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where,

P_{1} is pressure

T_{1} is temperature

P_{2} is pressure

T_{2} is temperature

__Gay-Lussac’s law or pressure law):__

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It states that the pressure exerted on the sides of a container by an ideal gas of fixed volume is proportional to its temperature.

__Avogadro’s law:__

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It states that the volume occupied by an ideal gas is proportional to the number of moles (or molecules) present in the container.

__Ideal gas law:__

It shows the relationship between the pressure, volume, and temperature for a fixed mass of gas.

Where,

P is pressure

V is volume

n is the number of moles

R is the universal gas constant

T is temperature

__Surface tension:__

The surface of a liquid is apt to shrink, and its free surface is in such a state where each section pulls another as if an elastic film is being stretched. The tensile strength per unit length of assumed section on the free surface is called the surface tension.

Surface tensions of various kinds of liquid are given in below table.

Table -Surface tension of liquid (20°C)

As shown in Fig 2.5, a dewdrop appearing on a plant leaf is spherical in shape. This is also because of the tendency to shrink due to surface tension. Consequently its internal pressure is higher than its peripheral pressure. Putting d as the diameter of the liquid drop, T as the surface tension, and p as the increase in internal pressure, the following equation is obtained owing to the balance of forces as shown in fig.

Fig. 2.5 A dewdrop on a taro leaf

__Capillarity:__

Whenever a fine tube is pushed through the free surface of a liquid, the liquid rises up or falls in the tube as shown in Fig. 2.7 owing to the relation between the surface tension and the adhesive force between the liquid and the solid. This phenomenon is called capillarity. As shown in Fig. 2.8, d is the diameter of the tube, q the contact angle of the liquid to the wall, r the density of liquid, and h the mean height of the liquid surface. The following equation is obtained owing to the balance between the adhesive force of liquid stuck to the wall, trying to pull the liquid up the tube by the surface tension, and the weight of liquid in the tube:

. 2.6 Balance between the pressure increase within a liquid drop and the surface tension

Fig. 2.7 Change of liquid surface due to capillarity

Whenever water or alcohol is in direct contact with a glass tube in air under normal temperature, q~0.In the case of mercury, q = 130″-150″. In the case where a glass tube is placed in liquid,

(in mm). Whenever pressure is measured using a liquid column, it is necessary to pay attention to the capillarity correction.

__Concepts of system and control volume:__

A system is defined as a quantity of matter or a region in space chosen for study. The mass or region outside the system is called the surroundings.

The real or imaginary surface that separates the system from its surroundings is called the boundary. These terms are illustrated in Fig.

The boundary of a system can be fixed or movable. Note that the boundary is the contact surface shared by both the system and the surroundings. Mathematically speaking, the boundary has zero thickness, and thus it can neither contain any mass nor occupy any volume in space. Systems may be considered to be closed or open, depending on whether a fixed mass or a fixed volume in space is chosen for study. A closed system (also known as a control mass) consists of a fixed amount of mass, and no mass can cross its boundary. That is, no mass can enter or leave a closed system, as shown in Fig.

But energy, in the form of heat or work, can cross the boundary; and the volume of a closed system does not have to be fixed. If, as a special case, even energy is not allowed to cross the boundary, that system is called an isolated system.

Consider the piston-cylinder device shown in Fig.

Let us say that we would like to find out what happens to the enclosed gas when it is heated. Since we are focusing our attention on the gas, it is our system. The inner surfaces of the piston and the cylinder form the boundary, and since no mass is crossing this boundary, it is a closed system. Notice that energy may cross the boundary, and part of the boundary (the inner surface of the piston, in this case) may move. Everything outside the gas, including the piston and the cylinder, is the surroundings.

An open system, or a control volume, as it is often called, is a properly selected region in space. It usually encloses a device that involves mass flow such as a compressor, turbine, or nozzle. Flow through these devices is best studied by selecting the region within the device as the control volume. Both mass and energy can cross the boundary of a control volume.

A large number of engineering problems involve mass flow in and out of a system and, therefore, are modeled as control volumes. A water heater, a car radiator, a turbine, and a compressor all involve mass flow and should be analyzed as control volumes (open systems) instead of as control masses (closed systems). In general, any arbitrary region in space can be selected as a control volume. There are no concrete rules for the selection of control volumes, but the proper choice certainly makes the analysis much easier. If we were to analyze the flow of air through a nozzle, for example, a good choice for the control volume would be the region within the nozzle.

The boundaries of a control volume are called a control surface, and they can be real or imaginary. In the case of a nozzle, the inner surface of the nozzle forms the real part of the boundary, and the entrance and exit areas form the imaginary part, since there are no physical surfaces there, as shown in fig a.

A control volume can be fixed in size and shape, as in the case of a nozzle, or it may involve a moving boundary, as shown in Fig b

Most control volumes, however, have fixed boundaries and thus do not involve any moving boundaries. A control volume can also involve heat and work interactions just as a closed

system, in addition to mass interaction. As an example of an open system, consider the water heater shown in Fig.

Fig: An open system (a control volume) with one inlet and one exit.

Let us say that we would like to determine how much heat we must transfer to the water in the tank in order to supply a steady stream of hot water. Since hot water will leave the tank and be replaced by cold water, it is not convenient to choose a fixed mass as our system for the analysis. Instead, we can concentrate our attention on the volume formed by the interior surfaces of the tank and consider the hot and cold water streams as mass leaving and entering the control volume. The interior surfaces of the tank form the control surface for this case, and mass are crossing the control surface at two locations.

In an engineering analysis, the system under study must be defined carefully. In most cases, the system investigated is quite simple and obvious, and defining the system may seem like a tedious and unnecessary task. In other cases, however, the system under study may be rather involved, and a proper choice of the system may greatly simplify the analysis.

__Application of control volume to continuity equation:__

Physical principle: Mass is conserved

The governing flow equation which results from the application of this physical principle to any one of the four models of the flow shown in Fig. 2.2a and b is called the continuity equation. Moreover, in this section we will carry out in detail the application of this physical principle using all four of the flow models illustrated in Fig. 2.2a and b; in this way we hope to dispel any mystery surrounding the derivation of the governing flow equation. That is, we will derive the continuity equation four different ways, obtaining in a direct fashion four different forms of the equation. Then, by indirect manipulation of these four different forms, we will show that they are all really the same equation. In addition, we will invoke the idea of conservation versus non conservation forms, helping to elucidate the meaning of those words. Let us proceed.

**2.5.1 Model of the Finite Control Volume Fixed in Space**

Consider the flow model shown at the left of Fig. 2.2a, namely, a control volume of arbitrary shape and of finite size. The volume is fixed in space. The surface that bounds this control volume is called the control surface, as labeled in Fig. 2.2a. The fluid moves through the fixed control volume, flowing across the control surface. This flow model is shown in more detail in Fig. 2.5. At a point on the control surface in Fig. 2.5, the flow velocity is V and the vector elemental surface area (as defined in Sec. 2.4) is dS. Also let dV be an elemental volume inside the finite control volume. Applied to this control volume, our hdamental physical principle that mass is conserved means

B=C

where B and C are just convenient symbols for the left and right sides, respectively,

FIG. 2.5 Finite control volume fixed in space.

of Eq. (2.15a). First, let us obtain an expression for B in terms of the quantities shown in Fig. 2.5. The mass flow of a moving fluid across any fixed surface (say, in lulograms per second or slugs per second) is equal to the product of (density) x (area of surface) x (component of velocity perpendicular to the surface). Hence the elemental mass flow across the area dS is

Examining Fig. 2.5, note that by convention, dS always points in a direction out of

the control volume. Hence, when V also points out of the control volume (as shown

in Fig. 2.5), the product pV . dS is positive. Moreover, when V points out of the

control volume, the mass flow is physically leaving the control volume; i.e., it is an

outflow. Hence, a positive pV dS denotes an outflow. In turn, when V points into

the control volume, pV dS is negative. Moreover, when V points inward, the

mass flow is physically entering the control volume; i.e., it is an inflow. Hence, a

negative pV dS denotes an inflow. The net mass flow out of the entire control

volume through the control surface S is the summation over S of the elemental mass

flow expressed in Eq. (2.16). In the limit, this becomes a surface integral, which is

physically the left sides of Eqs. (2.15a) and (2.15b); that is,

Now consider the right sides of Eqs. (2.15a) and (2.15b). The mass contained within

the elemental volume d V is p d-k”. The total mass inside the control volume is

therefore

The time rate of increase of mass inside V is then

Time rate of mass increase =

The physical principle that mass is conserved, when applied to the fixed element in

Fig. 2.7, can be expressed in words as follows: the net mass flow out of the element

must equal the time rate of decrease of mass inside the element. Denoting the mass

decrease by a negative quantity, this statement can be expressed in terms of Eqs.

In Eq. (2.24), the term in brackets is simply V . (pV). Thus, Eq. (2.24) becomes

Equation (2.25) is a partial differential equation form oj’the continuity equation. It

was derived on the basis of an i~zfinitesimally small element fixed in space. The

injinitestimally small aspect of the element is why the equation is obtained directly

in partial differential equation form. The fact that the element wasJixed in space

leads to the specific differential form given by Eq. (2.25), which is called the

consewation.form. As stated earlier, the forms of the governing flow equations that

are directly obtained from a flow model which is fixed in space are, by definition,

called the conservation form.

Equation (2.25) is displayed in box III in Fig. 2.6. It is the form that most

directly stems fiom the model of an infinitesimally small element fixed in space. On

the other hand, it can also be obtained by indirect manipulation fiom either of the

integral equations displayed in boxes 1 and 11, as will be shown in Sec. 2.2.5.