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Fluid Properties

Objectives About density, specific weight,and specific gravity of fluids. The concept of compressibility. Learn how viscosity affects shear forces in fluids.

A property is a physical characteristic or attribute of a substance. Matter in either state, solid or fluid, may be characterized in terms of properties. For example, Young's modulus is a property of solids that relates stress to strain. Density is a property of solids and fluids that provides a measure of mass contained in a unit volume. In this section, we examine some of the more commonly used fluid properties. Specifically, we will discuss 1) density, specific weight, and specific gravity, 2) bulk modulus, and 3) viscosity.

1 Density, Specific Weight, and Specific Gravity

A fluid is a continuous medium; i.e., a substance that is continuously distributed throughout a region in space. Because a fluid is a continuous medium, it would be rather awkward to analyze the fluid as a single entity with a total mass, m,total weight, W, or total volume, V. It is more convenient to analyze the fluid in terms of the mass of fluid contained in a specified volume. Density is defined as mass per unit volume. Density is a property that applies to solids as well as fluids. The mathematical definition for density, , is

 

The most commonly used units for density are kg/m³ in the SI system and slug/ft³ in the English system. Values for density can vary widely for different fluids. For example, the densities of water and air at 4°C and 1 atm pressure are approximately 1000 kg/m³ (1.94 slug/ft³) and 1.27 kg/m³ (0.00246 slug/ft³), respectively. Densities of liquids are higher than those of gases because the intermolecular spacing is smaller. Physical properties vary with temperature and pressure to some extent. For liquids, density does not vary significantly with changes in temperature and pressure, but the densities of gases are strongly influenced by changes in temperature and pressure.

A fluid property that is similar to density is specific weight. Specific weight is defined as weight per unit volume. The mathematical definition for specific weight, , is

 

The most commonly used units for specific weight are N/m³ in the SI system and lb_f/ft³ in the English system. Note that the unit for specific weight in the English system is not lb_m/ft³. The unit lb_m is a unit of mass, not a unit of weight. A quick inspection of Equation 7-1 and Equation 7-2 reveals that specific weight is essentially the same property as density with mass replaced by weight. A formula that relates density, , and specific weight, , may be obtained by noting that the weight of a unit volume of fluid is W=mg, where g is the local gravitational acceleration. Substituting the relation for weight, W, into Equation 7-2 and combining the result with Equation 7-1, we obtain the relation

 

Using the standard value of gravitational acceleration, g=9.807 m/s², water at 4°C has a specific weight of

Doing the same calculation in English units, noting that the standard value of gravitational acceleration is g=32.174 ft/s², water at 4°C (39.2°F) has a specific weight of

The rationale for finding the density and specific weight of water at 4°C in the foregoing discussion is that 4°C is a reference temperature on which specific gravity is based. Specific gravity is defined as the ratio of the density of a fluid to the density of water at a reference temperature. Typically, the reference temperature is taken as 4°C because the density of water is maximum (about 1000 kg/m³) at this temperature. The mathematical definition for specific gravity, sg, is

 

Because specific gravity is a ratio of two properties with the same units, it is a dimensionless quantity. Furthermore, the value of sg does not depend on the system of units used. For example, the density of mercury at 20°C is 13,550 kg/m³ (26.29 slug/ft³). Using SI units, the specific gravity of mercury is

Using English units, we obtain the same value.

Specific gravity may also be defined as the ratio of the specific weight of a fluid to the specific weight of water at a reference temperature. This definition, which is derived by combining Equation 7-4 and Equation 7-3, is expressed as

 

It does not matter whether Equation 7-4 or Equation 7-5 is used to find sg because both relations yield the same value. The definitions given by Equation 7-4 and Equation 7-5 apply regardless of the temperature at which the specific gravity is being determined. In other words, the reference temperature for water is always 4°C, but the density and specific weight of the fluid being considered are based on the temperature specified in the problem. Table 7-1 summarizes the reference values used in the definitions of specific gravity.

Density and Specific Weight of Water at 4°C.

	P
SI	1000 kg/m3	9810 N/m3
English	1.94 slug/ft3	62.4 lbf/ft3

2 Bulk Modulus

An important consideration in the analysis of fluids is the degree to which a given mass of fluid changes its volume (and therefore its density) when there is a change in pressure. Stated another way, how compressible is the fluid? Compressibility refers to the change in volume, V, of a fluid subjected to a change in pressure, P. The property used to characterize compressibility is the bulk modulus, K, defined by the relation

 

where P is the change in pressure, V is the change in volume, and V is the volume before the pressure change occurs. The negative sign is used in Equation 7-6 because an increase in pressure causes a decrease in volume, thereby assigning a negative sign to the quantity V. The negative signs on P and V cancel, leaving a positive bulk modulus, K, which is always a positive quantity. Because the ratio V/V is dimensionless, the bulk modulus has units of pressure. Typical units used for K are MPa and psi in the SI and English systems, respectively. A large value of K means that the fluid is relatively incompressible; i.e., it takes a large change in pressure to produce a small change in volume. Equation 7-6 applies for liquids only. Compared to liquids, gases are considered compressible fluids, and the formula for bulk modulus depends on certain thermodynamic considerations. Only liquids will be considered here. Liquids are generally considered incompressible fluids because they compress very little when subjected to a large change in pressure. Hence, the value of K for liquids is typically large. For example, the bulk modulus for water at 20°C is K=2.24 GPa. For mercury at 20°C, K=28.5 GPa. A list of bulk modulus values for some common liquids is given in Table 7-2.

Bulk Modulus for Common Liquids at 20°C

Compressibility is an important consideration in the analysis and design of hydraulic systems. Hydraulic systems are used to transmit and amplify forces by pressurizing a fluid in a cylinder. A tube or hose connects the fluid in the cylinder with a mechanical actuator. The hydraulic fluid completely fills the cylinder, connecting line, and actuator so that when a force is applied to the fluid in the cylinder, the fluid is pressurized with equal pressure everywhere in the system. A relatively low force applied to the fluid in the cylinder can produce a large actuator force because the cross-sectional area over which the pressure is applied is much larger in the actuator than in the cylinder. Thus, the force applied at the cylinder is amplified at the actuator. Hydraulic systems are used in a variety of applications such as heavy construction equipment, manufacturing processes, and transportation systems. The brake system in your automobile is a hydraulic system. When you press the brake pedal, the brake fluid in the system is pressurized, causing the brake mechanism in the wheels to transmit friction forces to the wheels thereby slowing the vehicle. Brake fluids must have high bulk modulus values for the brake system to function properly. If the value of the bulk modulus of the brake fluid is too low, a large change in pressure will produce a large change in volume which will cause the brake pedal to bottom out on the floor of the automobile rather than activating the brake mechanism in the wheels. In principle, this is what happens when air becomes trapped inside the brake system. Brake fluid is incompressible, but air is compressible, so the brakes do not function. As an engineering student, you will understand the underlying engineering principles on which this hazardous situation is based. (See Figure 4.)

. An engineering student explains a brake system failure. (Art by Kathryn Hagen)

3 Viscosity

The fluid properties density, specific weight, and specific gravity are measures of the “heaviness” of a fluid, but these properties do not completely characterize a fluid. Two different fluids, water and oil for example, have similar densities but exhibit distinctly different flow behavior. Water flows readily when poured from a container, whereas oil, which is a “thicker” fluid, flows more slowly. Clearly, an additional fluid property is required to adequately describe the flow behavior of fluids. Viscosity may be qualitatively defined as the property of a fluid that signifies the ease with which the fluid flows under specified conditions.

To investigate viscosity further, consider the hypothetical experiment depicted in Figure 5. Two parallel plates, one stationary and the other moving with a constant velocity, u, enclose a fluid. We observe in this experiment that the fluid in contact with both plates “sticks” to the plates. Hence, the fluid in contact with the bottom plate has a zero velocity, and the fluid in contact with the top plate has a velocity, u. The velocity of the fluid changes linearly from zero at the bottom plate to u at the top plate, giving rise to a velocity gradient in the fluid. This velocity gradient is expressed as a derivative, du/dy, where y is the coordinate measured from the bottom plate. Because a velocity gradient exists in the fluid, adjacent parallel “layers” of fluid at slightly different y values have slightly different velocities. This means that adjacent layers of fluid slide over each other in the same direction as the velocity, u. As adjacent layers of fluid slide across each other, they exert a shear stress, , in the fluid. Our experiment reveals that the shear stress, , is proportional to the velocity gradient, du/dy,which is the slope of the function u(y). Thus,

 

This result indicates that for common fluids such as water, oil, and air, the proportionality in Equation 7-7 may be replaced by the equality

 

where the constant of proportionality, , is called the dynamic viscosity. Equation 7-8 is known as Newton's law of viscosity, and fluids that conform to this law are referred to as Newtonian fluids. Common liquids such as water, oil, glycerin, and gasoline are Newtonian fluids as are common gases such as air, nitrogen, hydrogen and argon. The value of the dynamic viscosity depends on the fluid. Liquids have higher viscosities than gases, and some liquids are more viscous than others. For example, oil, glycerin and other gooey liquids have higher viscosities than water, gasoline, and alcohol. The viscosities of gases do not vary significantly from one gas to another, however.

. A velocity gradient is established in a fluid between a stationary and a moving plate.

Shear stress has the same units as pressure. In the SI system of units, shear stress is expressed in N/m², which is defined as a pascal (Pa). In the English system, shear stress is usually expressed in lb_f/ft² or lb_f/in² (psi). Velocity gradient has units of s¹, so a quick inspection of Equation 7-8 shows that dynamic viscosity, , has units of Pa·s in the SI system. The units of Pa·s may be broken down into their base units of kg/m·s. The units for are lb_f·s/ft² or slug/ft·s in the English system.

Consider once again the configuration illustrated in Figure 5. As the fluid flows between the plates, shear forces caused by viscosity are resisted by inertia forces in the fluid. Inertia forces are forces that tend to maintain a state of rest or motion in all matter, as stated by Newton's first law. A second viscosity property that denotes the ratio of viscous forces to inertia forces in a fluid is kinematic viscosity. Kinematic viscosity, , is defined as the ratio of dynamic viscosity to the density of the fluid. Thus,

 

In the SI system of units, kinematic viscosity is expressed in m²/s, and in the English system it is expressed in ft²/s. Because the ratio of dynamic viscosity to density often appears in the analysis of fluid systems, kinematic viscosity may be the preferred viscosity property.

Viscosity, like all physical properties, is a function of temperature. For liquids, dynamic viscosity decreases dramatically with increasing temperature. For gases, however, dynamic viscosity increases, but only slightly, with increasing temperature. The kinematic viscosity of liquids behaves essentially the same as dynamic viscosity because liquid densities change little with temperature. The kinematic viscosities of gases, however, increase drastically with increasing temperature because gas densities decrease sharply with increasing temperature.

EXAMPLE

A graduated cylinder containing 100 mL of alcohol has a combined mass of 280 g. If the mass of the cylinder is 200 g, what is the density, specific weight, and specific gravity of the alcohol?   Solution The combined mass of the cylinder and alcohol is 280 g. By subtraction, the mass of the alcohol is Converting 100 mL to m3, we obtain The density of the alcohol is The weight of the alcohol is so the specific weight is The specific gravity of the alcohol is

EXAMPLE

Find the change in pressure required to decrease the volume of water at 20°C by 1 percent.   Solution From Table 7-2, the bulk modulus of water at 20°C is K=2.24 GPa. A 1 percent decrease in volume denotes that V/V=0.01. Rearranging Equation 7-6 and solving for P, we obtain

EXAMPLE

Two parallel plates, spaced 3 mm apart, enclose a fluid. One plate is stationary, while the other plate moves parallel to the stationary plate with a constant velocity of 10 m/s. Both plates measure 60 cm×80 cm. If a 12-N force is required to sustain the velocity of the moving plate, what is the dynamic viscosity of the fluid?   Solution The velocity varies from zero at the stationary plate to 10 m/s at the moving plate, and the spacing between the plates is 0.003 m. The velocity gradient in Newton's law of viscosity may be expressed in terms of differential quantities as The shear stress is found by dividing the force by the area of the plates. Thus, Rearranging Equation 7-8, and solving for dynamic viscosity, , we obtain

 

Practice!

1. A cylindrical container with a height and diameter of 16 cm and 10 cm, respectively, contains 1.1 kg of liquid. If the liquid fills the container, find the density, specific weight, and specific gravity of the liquid.
2. A swimming pool measuring 30 ft×18 ft×8 ft is to be filled using a water truck with a capacity of 5500 gallons. How many trips does the water truck have to make to fill the pool? If the density of the water is 1.93 slug/ft³, what is the mass and weight of the water in the pool after it has been filled?
3. A cylinder containing benzene at 20°C has a piston that compresses the fluid from 0 to 37 MPa. Find the percent change in the volume of the benzene.
4. Hydraulic fluid is compressed by a piston in a cylinder producing a change in pressure of 40 MPa. Before the piston is activated, the hydraulic fluid fills a 20-cm length of the cylinder. If the axial displacement of the piston is 6.5 mm, what is the bulk modulus of the hydraulic fluid?
5. Glycerin at 20°C (=1260 kg/m³, =1.48 Pa·s) occupies a 1.6-mm space between two square parallel plates. One plate remains stationary while the other plate moves with a constant velocity of 8 m/s. If the both plates measure 1 m on a side, what force must be exerted on the moving plate to sustain its motion? What is the kinematic viscosity of the glycerin?