Copyright © 1998, 1999, 2000 by Prentice Hall, Inc.
NOT FOR DISTRIBUTION

Work and Heat

Objectives How to determine various forms of work. The difference between heat and temperature.

Work, like energy, is a word that is commonly used in our everyday language and a word that has many meanings. As a student, you know that studying engineering is a lot of “work.” When we participate in sports or exercise at the gym, we get a “work out.” A person who travels to a place of employment goes to “work,” and when a mechanical device stops functioning, we say that it doesn't “work.” While various day-to-day usages of this term are thrown about quite casually, engineering defines “work” precisely, with no ambiguity. Work is defined as a form of energy that is transferred across the boundary of a system. A system is a quantity of matter or a region in space chosen for study, and the boundary of a system is a real or imaginary surface that separates the system from the surroundings. For example, propane in a fuel tank is a thermodynamic system, and the boundary of the system is the inside surface of the tank wall. Besides work, there is a second form of energy that can be transferred across the boundary of a system. The second form of energy is heat. Heat is a special kind of energy transfer that is easily recognizable and differentiated from work. Heat is defined as the form of energy that is transferred across the boundary of a system by virtue of a temperature difference. A system with both work and heat crossing the boundary is illustrated in Figure 9. Depending on the nature of the interactions of the system with the surroundings, work and heat can be transferred across the boundary in either direction. The only requirement for heat transfer is a temperature difference between the system and the surroundings. If there is no temperature difference between the system and the surroundings, heat cannot be transferred, so the only form of energy transfer is work. Because work and heat are forms of energy, both quantities have the same units. Work and heat have units of J in the SI system and Btu in the English system. The most commonly used symbols for work and heat in thermodynamics are W and Q, respectively.

. Energy in the form of work or heat can be transferred across the boundary of a system.

Now that work and heat have been defined in general terms, let us examine these forms of energy transfer more closely. In thermodynamics, work is usually categorized as mechanical work or nonmechanical work. The nonmechanical forms of work include electrical, magnetic, and electrical polarization work. Mechanical forms of work are generally the most important, so we will consider these in some detail.

1 Mechanical Work

There are several types of mechanical work. From basic physics, the work, W, done by a force, F, acting through a displacement, s, in the same direction of the force is given by the relation

 

Equation 6-14 is valid only if the force is constant. If the force is not constant; i.e., if the force is a function of displacement, the work is obtained by integration. Thus, Equation 6-14 becomes

 

where the limits 1 and 2 denote the initial and final positions of the displacement, respectively. Equation 6-15 is a general mathematical definition from which equations for the various types of mechanical work are derived. Consider, for example, a vehicle that climbs a rough hill, as shown in Figure 10. As the vehicle climbs the hill, it encounters two forces that tend to oppose its motion. Gravity exerts a downward force on the vehicle that retards its upward motion, and friction between the wheels and the rough surface retards its motion along the surface. The vehicle does work against these two forces, and the magnitude of that work is found by integrating the total force from position s₁ to s₂, which is graphically interpreted as the area under the force-displacement curve. The various types of mechanical work are now considered.

. As a vehicle climbs a hill, gravitational and friction forces act on it.

Gravitational Work

Gravitational work is defined as the work done on an object by a gravitational force. In a gravitational field, the force acting on a body is the weightof the body, and is given by

 

where m is mass (kg) and g is the local gravitational acceleration (m/s²). Consider a vehicle that climbs a hill from elevation, z₁, to a higher elevation, z₂, as shown in Figure 11. Substituting Equation 6-16 into Equation 6-15 and integrating, we obtain the gravitational work, W_g, as

 

Note that the displacement in Equation 6-17 is in terms of elevation,z, because work is defined as a force acting through a distance in the same direction of the force. Gravity acts in the vertical direction, so Equation 6-17 is written in terms of a vertical distance (elevation) and not a horizontal distance. The gravitational work for the vehicle in Figure 11 is negative because the direction of the displacement (upward) is opposite to the direction of the gravitational force (downward). If the vehicle descends the hill, the gravitational work is positive because the desplacement is in the same direction as the force. Note also that gravitational work is equivalent to a change in potential energy because PE=mgz.

. Gravitational work is done on a body as it changes elevation.

Acceleration Work

Acceleration work is the work associated with a change in velocity of a system. Newton's second law states that the force acting on a body equals the product of the body's mass and acceleration. But acceleration, a, is the time derivative of velocity, v, so Newton's second law may be written as

 

Velocity is the time derivative of displacement,

 

so the differential displacement, ds, in Equation 6-15 is ds=v dt. Thus, acceleration work, W_a, is

 

As shown in Figure 12, a vehicle traveling along a horizontal road increases its velocity from 10 mi/h to 65 mi/h. In doing so, the vehicle does acceleration work because its velocity changes. We note that the acceleration work is equivalent to a change in kinetic energy because .

. A body does acceleration work as its velocity changes.

Boundary Work

Boundary work is the work associated with the movement of a solid boundary. The most common instance of boundary work is the compression or expansion of a gas within a piston-cylinder device, as illustrated in Figure 13. A force, F, is applied to the piston, compressing the gas within the cylinder. Because the cylinder is a closed vessel, the pressure increases as the gas volume decreases. As the gas volume decreases from V₁ to V₂, the pressure increases along a path that depends on certain physical characteristics of the compression process. Pressure is defined as a force divided by area, so the force causing the compression is given by the relation

 

where A is the surface area of the face of the piston. A differential change in volume, dV, is the product of the piston's differential displacement,ds, and the surface area of the piston, A. Hence, dV=A ds, and the boundary work, W_b, becomes

 

Because the product P dV appears in the definition, boundary work is sometimes referred to as “P dV” work. As indicated in Figure 13, the magnitude of the boundary work is the area under the pressure-volume curve. In order to evaluate the integral in Equation 6-22, we would have to know the functional relationship between pressure, P, and volume, V. This relationship may be an analytical expression for P as a function of V or a graph that shows the variation of P with V.

. Boundary work is performed by a piston as it compresses a gas.

Shaft Work

Shaft work is energy transfer by a rotating shaft. Numerous engineering systems transfer energy by means of a rotating shaft. The drive shaft of an automobile, for example, transfers energy from the transmission to the axle. Energy is transferred from a boat motor to the propeller by a shaft. Even the mixing blades of a food blender perform shaft work on the food. As a shaft rotates, a constant torque is usually applied to the shaft that tends to retard its rotation. As illustrated in Figure 14, the torque, , is produced by a force, F, acting through a moment arm, r, according to the relation

 

The force acts through a distance,s, equal to the circumference times the number of revolutions of the shaft, n. Thus,

 

Upon substituting Equations 6-23 and (6-24) into Equation 6-14, the shaft work, W_sh, becomes

 

. Work is produced by a rotating shaft.

Spring Work

Spring work is the work done in deforming a spring. A force is required to compress or stretch a spring, so work is done. From elementary physics, we know that the force required to deform a linear elastic spring is proportional to the deformation. This principle is known as Hooke's law, and is expressed as

 

where F is force, x is displacement (change in spring length) andk is the spring constant. Substituting Equation 6-26 into Equation 6-15 and noting that ds=dx, the spring work, W_sp, becomes

 

As indicated in Figure 15, the initial and final spring displacements arex₁ and x₂, respectively, as measured from the rest (undeformed) position of the spring.

. Work is done by stretching or compressing a spring.

2 Heat

Heat is the transfer of energy across the boundary of a system by virtue of a temperature difference. In order for heat transfer to occur, there must be a temperature difference between the system and the surroundings. The transfer or flow of heat is not the flow of a material substance, as in the case of the flow of a fluid such as air or water. Rather, there is an exchange of internal energy across the system boundary by atomic or molecular motion or by electromagnetic waves. Heat transfer can occur by three distinct mechanisms: conduction, convection, and radiation. Conduction is the transfer of internal energy in solids and fluids at rest. The actual mechanism of conduction involves kinetic energy exchange between molecules in contact or, in the case of metals, movement of free electrons. Convection is the mechanism by which internal energy is transferred to or from a fluid near a solid surface. Convection is basically conduction at the solid surface with the added complexity of energy transfer by moving fluid molecules. Radiation is the mechanism by which energy is transferred by electromagnetic waves. Unlike conduction and convection, radiation does not require a medium. A familiar example of radiation is the thermal energy that we receive from the sun across the vacuum of space. Regardless of the heat transfer mechanism involved, the direction of heat transfer is always from a high temperature region to a low temperature region.

Heat transfer occurs all around us. As a familiar example, consider the hot beverage shown in Figure 16. Heat is transferred from the beverage to the surroundings by all three heat transfer mechanisms. A portion of the energy is transferred by convection from the liquid to the solid cup wall where the heat is subsequently conducted through the cup wall. That energy is then transferred to the surroundings by convection and radiation. The portion of the energy conducted into the bottom portion of the cup is transferred directly into the table top by conduction. The remaining energy is transferred from the surface of the liquid directly to the surroundings by convection and radiation.

. A hot beverage resting on a table transfers thermal energy to the surroundings by conduction, convection and radiation.

In the following example, we use the general analysis procedure of: (1) problem statement, (2) diagram, (3) assumptions, (4) governing equations, (5) calculations, (6) solution check and (7) discussion.

EXAMPLE

Problem Statement A 1200-kg automobile accelerates up a hill, increasing its speed from 5 mi/h to 45 mi/h, along a straight 100-m stretch of road. If the hill makes an angle of 6° with respect to the horizontal, find the total work.   Diagram The diagram for this problem is shown in Figure 17. . Example.   Assumptions . Neglect friction between wheels and road. Neglect aerodynamic friction. Mass of automobile is constant.   Governing Equations Two forms of work, gravitational and acceleration, are involved as the automobile ascends the hill, so we have two governing equations:   Calculations The quantities in the problem statement are given in a mixed set of units, so we first convert the units of all quantities to SI units. Converting the velocities, we obtain The vertical position, z2, of the automobile when it attains a speed of 45 mi/h is Assigning the position of the ground as z1=0 m, the gravitational work is The acceleration work is The total work is the sum of the gravitational and acceleration work.   Solution Check No errors are found.   Discussion Even though the gravitational work is negative, the total work is positive because the acceleration work is larger in magnitude. We must remember that gravitational work is the work done by gravity on the body and not the work done by the body in overcoming gravity. The work done by the automobile's engine in order to climb the hill is 123 kJ, but the gravitational work is 123 k.   Application: Boundary Work During a Constant Pressure Process In some thermodynamic systems, boundary work is performed while the pressure remains constant. A common example is the heating of a gas contained in a piston-cylinder device, as illustrated in figure -a18. As heat is transferred to the gas within the cylinder, the internal energy of the gas increases, as exhibited by an increase in the gas temperature, and the piston moves up. If we assume that the piston-cylinder device is frictionless, the pressure of the gas remains constant, however, but boundary work is still done because the piston moves. Suppose that the frictionless piston-cylinder device shown in figure -18a contains 2.5 L of nitrogen at 120 kPa. Heat is then transferred to the nitrogen until the volume is 4 L. Find the boundary work done by the nitrogen during this process. . A constant pressure process. Boundary work, Wb, is given by the relation where P is pressure and V is volume. Because the process occurs at constant pressure,P can be brought outside the integral, giving the relation The initial and final volumes of the nitrogen are Thus, the boundary work is The boundary work calculated here is the work done by the nitrogen on the piston, not the work done on the nitrogen by the piston. Figure -18b shows the process path for the constant pressure process that occurs in the piston-cylinder device. The boundary work of 180 J is the shaded area under the process path.

 

Practice!

1. As a 2500-kg truck climbs a hill, it changes speed from 20 mi/h to 50 mi/h along a straight 1600-ft section of road. If the hill is inclined at an angle of 8 ° with respect to the horizontal, find the total work.
2. A 95-slug automobile changes speed from 55 mi/h to 30 mi/h while climbing a 3° hill. If the change in speed occurs over a 1355-ft straight section of road, find the total work.
3. A shaft rotating at 1200 rpm (revolutions per minute) experiences a constant torque of 60 N?m. How much work does the shaft perform in one hour?
4. The pressure inside a frictionless piston-cylinder device varies according to the function P=abV where a and b are constants and V is volume. The initial and final volumes for the process are 1 m³ and 0.1 m³, respectively. If a=500 Pa and b=2000 Pa/m³, find the boundary work.
5. A linear elastic spring is compressed 3.5 cm from its at-rest position. The spring is then compressed an additional 7.5 cm. If the spring constant is 2600 N/cm, find the work done in compressing the spring.
6. A frictionless piston-cylinder device has a diameter of 10 cm. As the gas inside the cylinder is heated, the piston moves a distance of 16 cm. If the gas pressure is maintained at 120 kPa, how much work is done?