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Pressure and Temperature

Objectives The relationships between the various types of pressures. The thermodynamic meaning of temperature.

Systems that use thermodynamic processes for their operation can be described by certain physical characteristics. Any such characteristic of a system is called a property. In its broadest engineering context, a property can refer to any physical aspect of a system such as length, density, velocity, modulus of elasticity, viscosity, etc. In thermodynamics, a property usually refers to a characteristic that relates directly to the energy of the system. Two of the most important properties in thermodynamics are pressure andtemperature.

1 Pressure

When a fluid (liquid or gas) is confined by a solid boundary, the fluid exerts a force on the boundary. The direction of the force is normal (perpendicular) to the boundary. From a microscopic point of view, the force is the result of a change in momentum experienced by the fluid molecules as they collide with the solid surface. Molecules collide with the surface in many directions, but the overall effect of the collisions is a net force that is normal to the surface. Pressure is defined as the normal force exerted by a fluid per unit area. Thus, the formula for pressure is

 

where P is pressure, (N/m²), F is the normal force, (N) and A is area, (m²). For a liquid at rest, pressure increases with depth as a consequence of the weight of the liquid. For example, the pressure exerted by sea water on the hull of a submarine at a depth of 200 m is greater than the pressure exerted by seawater on a scuba diver at a depth of 10 m. The pressure in a tank containing a gas is essentially constant, however, because the weight of the gas is usually negligible compared to the force required to compress the gas. For example, consider a gas enclosed in a piston-cylinder device, as illustrated in Figure 4. A force, F, is applied to the piston, compressing the gas in the cylinder. A constant pressure, whose magnitude is given by Equation 6-1, acts on all interior surfaces of the enclosure. If the force, F, increases, the pressure, P, increases accordingly.

. An enclosed gas exerts a pressure on the walls of its container.

The unit of pressure in the SI system is N/m², which is defined as the pascal (Pa). Hence, 1 Pa=1 N/m². The pascal is a very small unit of pressure, so it is customary to use the standard SI multiples kPa (kilopascal) and MPa (megapascal), which stand for 10³ Pa and 10⁶ Pa, respectively. Pressure is sometimes expressed in terms of the bar (1 bar=10⁵ Pa). The most commonly used unit of pressure in the English system is pound-force per square inch (lb_f/in²), typically written in shorthand notation as psi.

When doing calculations involving pressure, care must be taken to specify the reference on which the pressure is based. Pressure referenced to a perfect vacuum is called absolute pressure, P_abs. Atmospheric pressure, P_atm, is the pressure exerted by the atmosphere at a specified location. At sea level, atmospheric pressure is P_atm=101,325 Pa=14.696 psi. At higher elevations, the atmospheric pressure is lower due to decreasing air density. The pressure that uses atmospheric pressure as the reference is called gauge pressure, P_gauge. Gauge pressure is the difference between the absolute pressure and the local atmospheric pressure. Most pressure-measuring instruments, such as an automobile tire gauge, measure gauge pressure. A pressure below atmospheric pressure is called vacuum pressure, P_vac. Vacuum pressure is measured by vacuum gauges that indicate the difference between the local atmospheric pressure and absolute pressure. Gauge, absolute, and vacuum pressures are all positive quantities and are related to one another by the relations

 

 

The majority of thermodynamic equations and data tables use absolute pressure. Sometimes, the letter “a” is used to specify absolute pressure and the letter “g” is used to specify gauge pressure. For example, absolute, and gauge pressures are sometimes written in English units as psia and psig, respectively.

2 Temperature

Our physiological sense of temperature tells us how hot or how cold something is, but does not provide a quantitative definition of temperature for engineering use. A scientific definition, based on microscopic considerations, is that temperature is a measure of atomic and molecular kinetic energy of a substance. Thus, at a temperature of absolute zero, all translational, rotational, and vibrational motions of atoms and molecules cease. A practical engineering definition is that temperature, or more specifically a temperature difference, is an indicator of heat transfer. As illustrated in Figure 5, heat flows from a region of high temperature to a region of low temperature. This engineering definition of temperature is consistent with our common experiences. For example, a hot beverage will gradually cool until it reaches the temperature of the surroundings. Conversely, a cold beverage will eventually warm until it reaches the temperature of the surroundings. In either case, when the beverage attains the temperature of the surroundings, heat transfer stops and the beverage and surroundings are said to be in thermal equilibrium because their temperatures are equal. We can therefore state that when any two bodies have the same temperature, the bodies are in thermal equilibrium.

. Heat is transferred from a high-temperature region to a low-temperature region.

The zeroth law of thermodynamics states that if two bodies are in thermal equilibrium with a third body, they are also in thermal equilibrium with each other. This law is analogous to the arithmetic axiom which states that if A=C and B=C, then A=B. The zeroth law, as obvious as it sounds, cannot be derived from the first or second laws of thermodynamics. The zeroth law of thermodynamics is the underlying physical basis for a key element of thermodynamics, temperature measurement. By the zeroth law of thermodynamics, if body A and body B are in thermal equilibrium with body C, then body A and body B are in thermal equilibrium with each other. By letting body C be a thermometer, the zeroth law of thermodynamics infers that bodies A and B are in thermal equilibrium if their temperatures, as measured by the thermometer, are equal. The interesting aspect of the zeroth law is that bodies A and B do not even have to be in physical contact with each other. They only have to have the same temperature to be in thermal equilibrium.

Like length, mass, time, electrical current, luminous intensity, and amount of substance, temperature is a base dimension. As a base dimension, temperature is predicated on a measurable physical standard. Temperature scales enable engineers to make temperature measurements on a common basis. International temperature scales have been adopted that are based on fixed reproducible thermodynamic states of matter. The ice point and boiling point of water at 1 atmosphere pressure are defined as 0°C and 100°C, respectively, on the Celsius temperature scale. On the Fahrenheit temperature scale, these points have the values 32°F and 212°F, respectively. The Kelvin and Rankine temperature scales are absolute temperature scales that have 0 K and 0°R as their lowest possible temperature values. Thus, we say that absolute zero temperature refers to either 0 K or 0°R. By convention, the degree symbol “°” is used for the Celsius, Fahrenheit, and Rankine temperature scales but not the Kelvin scale. A comparison of these four temperature scales is shown in Figure 6.

Because engineers use four different temperature scales in the analysis of thermodynamic systems, it is important to know how to convert from one temperature scale to another. The Kelvin scale is related to the Celsius scale by the formula

 

. The Celsius, Kelvin, Fahrenheit and Rankine temperature scales.

and the Rankine scale is related to the Fahrenheit scale by the formula

 

In the majority of practical applications, temperature precision beyond the decimal point is not required, so the constants in Eqs. (6-4) and (6-5) are typically rounded to 273 and 460, respectively. The Rankine and Kelvin scales are related by the formula

 

and the Fahrenheit and Celsius scales are related by the formula

 

Equation (6-4)), Equation (6-5), Equation (6-6), and Equation (6-7) are used to convert one temperature value or measurement to another. Using Figure 6, the validity of these relations can be readily checked by converting the boiling point and ice point of water from one temperature scale to the other three scales.

We mentioned earlier in this section that temperature difference is an indicator of heat transfer. When calculating a temperature difference, it is important to note that the size of the temperature divisions for the Kelvin and Celsius scales are equal and that the size of the temperature divisions for the Rankine and Fahrenheit scales are equal. In other words, increasing the temperature of a substance by 1 K is the same as increasing the temperature by 1°C. Similarly, increasing the temperature of a substance by 1°R is the same as increasing the temperature by 1°F. Thus, we write the relations for temperature differences as

 

 

where the Greek symbol, , refers to a difference or change. When doing thermodynamics calculations involving temperature differences in the SI system, it does not matter whether K or °C is used. Similarly, when doing thermodynamics calculations involving temperature differences in the English system, it does not matter whether °R or °F is used. In analysis work, care must be taken to distinguish between a single temperature value, T, and a temperature difference, T. If the thermodynamic relation is of the form x=yT, it does not matter whether T is expresses in K or °C. If the thermodynamic relation is of the form x=yT, however, the temperature T must be expressed in K. The same rules apply for the corresponding English temperature units, °R and °F.

EXAMPLE

The atmospheric pressure in Denver, Colorado (elevation 1 mile) is approximately 83.4 kPa. If we were to inflate the tire of an automobile in Denver to a gauge pressure of 35 psi, what is the absolute pressure in units of kPa?   Solution In order to work in a consistent set of units, we convert the gauge pressure to units of kPa. Solving for absolute pressure from Equation 6-2, we have

EXAMPLE

Steam in a boiler has a temperature of 300°C. What is this temperature in units of K, °R, and °F? If the temperature drops to 225 °C, what is the temperature change in units of K, °R, and °F?   Solution Using Equation 6-4, the temperature in K is Now that the temperature in K is known, we use Equation 6-6 to find the temperature in °R. Using Equation 6-7, the temperature in °F is The temperature change is The Rankine and Kelvin temperature scales are related through Equation 6-6. Because these temperature scales are absolute scales, we can write Equation 6-6 in terms of temperature differences as Hence,

 

Practice!

1. A pressure gauge on the discharge side of an air compressor reads 260 kPa. What is the absolute pressure at this point in units of psi if the local atmospheric pressure is 95 kPa?
2. A force of 1.2 kN is applied to the piston of a cylinder, compressing the gas within the cylinder. The piston has a radius of 4 cm. If the local atmospheric pressure is 100 kPa, what is the pressure inside the cylinder?
3. A vacuum gauge connected to a tank reads 30 kPa. If the local atmospheric pressure is 13.5 psi, what is the absolute pressure in units of psi?
4. A boiler at sea level contains superheated steam at 0.4 MPa absolute pressure and 300°C. Find the gauge pressure in the boiler and the steam temperature in units of K, °R, and °F.
5. A hard-boiled egg removed from a pot of boiling water at 96°C is placed in a 40°F refrigerator to cool. Find the temperature of the egg in units of K, °C, and °R after the egg has attained thermal equilibrium with the refrigerator. What is the temperature change of the egg in units of °F, °C and K?