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Isochoric process and isobaric process for constant Thermody

Isochoric process and isobaric process for constant Thermodynamics <?xml:namespace prefix = o ns = "urn:schemas-microsoft-com:office:office" />

Jegadeesan Singaram1 and P. Sangeetha2

1 Department of Mathematics, <?xml:namespace prefix = st1 ns = "urn:schemas-microsoft-com:office:smarttags" /> Thiagarajar College of Education, Affiliated to Periyar University, Salem, Tamil Nadu, India.

2Department of Mathematics, Jairam College of Arts and Science, Affiliated to Periyar University, Salem, Tamil Nadu, India.

Abstract

There are several specific types of thermodynamic processes that happen frequently enough (and in practical situations) that they are commonly treated in the study of thermodynamics. Each has a unique trait that identifies it, and which is useful in analyzing the energy and work changes related to the process. The laws of thermodynamics are applicable to any system, regardless of its complexity. They are sometimes helpful to model the system by considering only some of their aspects, and thus substantially simplifying the analysis. Here we introduce the concept of a simple system, which has only one mode of quasistatic work. In many applications this simplified concept describes fairly accurately real substances.

1. Introduction: Basic Processes in Simple Systems

In this part we apply the first law of thermodynamics to some basic processes in simple systems passing through equilibrium states.

The energy of a simple system is the internal energy and therefore the first law of thermodynamics, <?xml:namespace prefix = v ns = "urn:schemas-microsoft-com:vml" /> or (1)

It expresses the concept of conservation of energy in systems undergoing interactions.

Equation 1, is rewritten as

(2)

or in differential form

(3)

1.1. The Constant-Volume Process

A process in which the volume of the system remains constant during a change of state is called a constant-volume or isochoric process.

Consider a simple compressible system inside a rigid container undergoing a constant-volume change from states 1 to 2, shown by the vertical straight line on the p-V diagram in Figure 1a. The state of the system may be changed by several different processes.

Case a. The change of state is accomplished by a heat interaction as shown in Figure 1b. In this case W = 0 and the first law of thermodynamics, Equation 2, yields

(4)

where ΔU = U2U1 is the change in internal energy between states 1 and 2.

Figure 1 Constant-volume process.

Case b. The change of state is accomplished by a work interaction of a stirrer as shown in Figure 1b. In this case Q = 0 and the first law of thermodynamics, Equation 2, yields

(5)

We can also bring about the same change of state in the system by a process that involves both work and heat interactions in suitable amounts. For this case Equation 1 reads

Although the processes connecting points 1and 2 differ from each other by the interactions at the system boundary, they all result in the same changes in the internal energy. Heat and work interactions, however, are not the same for the processes between points 1 and 2, described earlier, because they are not properties.

The rate of variation of the internal energy with temperature is important in thermodynamics; hence, a new extensive property Cv is defined as

(6)

This property is the heat capacity at constant volume. The name is rather misleading, because it has nothing to do with the capacity of the system to store heat. Heat is not a property and cannot be stored. What is actually stored is the internal energy, and the change in the internal energy of the system can be achieved by either heat or work interactions or by a combination of the two. The term heat capacity is a remnant from the era of the caloric theory when heat was considered to be a property. The heat capacity per unit mass

(7)

is the specific heat capacity at constant volume or, in short, specific heat at constant volume. The units of heat capacity and specific heat are kJ/K and kJ/kg K, respectively, or in British units and Btu/lb°F.

The specific energy of a simple system is, in general, a function of two independent properties, say, u(T, v). It is obvious from the below Equation 8 that the specific heat at constant volume is also a function of the same independent properties, cv(T, v).

(8)

Now the internal energy, U, is an extensive property. The corresponding intensive prop­erty, the specific internal energy, u, is the internal energy per unit mass. An example of a functional relationship which holds for a simple system is

(9)

Differentiation of Equation. (9) results in

(10)

We now substitute Equation 8 into Equation 10 and obtain an alternative expression for the change in internal energy, which upon integration between states 1 and 2 yields

(11)

In the case of an isochoric process dv = 0, and Equation 12 simplifies to

(12)

or in terms of extensive properties

(13)

For the special case of cv = constant, we obtain

(14)

Example 1

A rigid vessel contains 0.3 kg of carbon monoxide (CO, cv = 0.7423 kJ/kg K) at 25°C. The vessel is heated to 150°C. Find the heat interaction of gas.

Solution

We use Equations 4 and 14 to calculate the heat interaction for this constant-volume process

1.2 The Constant-Pressure Process

A process in which the pressure of the system remains constant during a change of state is called a constant-pressure or isobaric process.

Consider a simple compressible system inside a piston-cylinder assembly. Properties of the system, such as energy, temperature, and volume, may be changed by several different processes. The pressure of the system, however, is kept constant and equal to the equivalent external pressure imposed by the piston. Figure 2a depicts the constant-pressure process on a p-V diagram.

Figure 2 Constant-pressure process.

The constant-pressure path from states 1 to 2 is shown by the horizontal line on the p-V diagram in Figure 2a. The state of the system may be changed by several different processes.

Case a. The change of state is accomplished by a heat interaction as shown in Figure 2b. This constant-pressure interaction results in the movement of the piston and a change of volume. The heat interaction is, therefore, accompanied by a work interaction on the piston. As the external equivalent pressure is equal to the pressure of the system p throughout this process, the work interaction is given by the product of the pressure and the change of volume. Thus

(15)

The first law of thermodynamics requires that

(16)

or

(17)

As the pressure is constant, Equation 17 may be rewritten as

(18)

Case b. As with the isochoric process, the same change of state may be obtained by the nonquasistatic work of the stirrer Ws (Figure 2b), instead of the heat interaction. The first law of thermodynamics requires now that

(19)

or

(20)

Note that two kinds of works are involved in this process, the work associated with the change of volume, which is quasistatic, and the work of the stirrer, which is not.

We can also bring about the same change of state in the system by a process that involves both stirrer work and heat interactions in suitable amounts such that

(21)

The combination of properties (U + pV) describes a new extensive property, with units of energy, which often appears in thermodynamics. This property is called enthalpy and is denoted by H.

(22)

The corresponding intensive property is the specific enthalpy, which is the enthalpy per unit mass.

(23)

Equation 21 may now be rewritten for this constant-pressure process as

(24)

The variation of enthalpy with temperature at constant pressure is also an extensive property called heat capacity at constant pressure and denoted by Cp.

(25)

The corresponding intensive property is the specific heat capacity at constant pressure called for short specific heat at constant pressure.

(26)

The specific enthalpy is a function of two independent properties, say, T and p.

(27)

Differentiating Equation 27 and using Equation 26 yields

(28)

which for an isobaric process, dp = 0, simplifies to

(29)

Hence, the change of enthalpy between two states at constant pressure is given by

(30)

The ratio of the specific heat at constant pressure to specific heat at constant volume is an intensive property denoted by k and generally it is a function of two independent properties.

(31)

Example 2

A coal-oil mixture (COM) is a fuel that can replace heating oil for certain applications. To prevent settling of the coal particles, mixing is required.

An open tank contains 5000 kg of COM at 22°C. The COM was stirred for 40 min by a 20 kW paddle mixer. During this operation the temperature of the mixture rose to 27.6°C. The specific heat of the COM is assumed to be constant at cp = 1.4 kJ/kg K.

1. Find the heat interaction of the COM with its surroundings.

2. What will be the temperature rise of the COM if the tank is perfectly insulated?

Solution

1. We use Equation 24 to calculate the heat interaction for this constant-pressure process.

H is calculated from Equation 30:

The work is

The work is negative because it is done on the system. The heat interaction is

The minus sign indicates that heat is removed from the system.

2. When perfectly insulated or in adiabatic process Q = 0; hence, from Equation 24

The process is isobaric (dp = 0) and the specific heat is constant. For this case Equation 31 becomes

Therefore

References

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