PROPERTY RELATIONS - THERMODYNAMICS, MODELING, AND

THERMODYNAMICS, MODELING, AND

2.3 PROPERTY RELATIONS

To apply the mass, energy, and entropy balances to a system of interest re- quires knowledge of the properties of the system and how the properties are related. This section reviews property relations for simple compressible systems, which include a wide range of industrially important gases and liquids.

2.3.1

The T Equations. Internal energy, enthalpy, and entropy are calculated using data on that are more readily measured. The calculations use the T equations and relations derived from them.

To establish the equations consider a closed system undergoing an internally reversible process in the absence of overall system motion and the effect of gravity. An energy balance in differential form is

Basic Relations for Pure Substances

From Equation 2.1 T so

2.3 PROPERTY RELATIONS 65

T -

A pure, simple compressible system is one in which the only significant energy transfer by work in an internally reversible process is associated with volume change and is given by ⁼ Accordingly, the first T equation for pure, simple compressible systems takes the form

Since H =

+

⁼

+

V dp. Combining this with Equation the second T equation follows

+

V d p

Using the same approach, a further relation can be derived from the Gibbs function, defined as G = - TS. The result is

V d p -

The last three equations can be expressed on a per unit mass (or a per mole) basis:

dh = T

+

v -

(2.33 b)

Relations. Pressure, specific volume, and temperature are relatively easily measured and considerable data have been accumulated for industrially important gases and liquids. These data can be represented in the form p = T ) , called an equation of state. Equations of state can be expressed in tabular, graphical, and analytical forms [ 1,

The graph of a function p = T ) is a surface in three-dimensional space.

Figure 2.2 shows the relationship for water, a substance that expands upon freezing. Figure shows the projection of the surface onto the pressure-temperature plane. The projection onto the plane is shown in Figure Figure 2.2 has three regions labeled solid, liquid, and vapor where the substance exists only in a single phase. Between the single-phase regions lie two-phase regions, where two phases coexist in equilibrium. The line separating a single-phase region from a two-phase region is called a

line. Any state represented by a point on a saturation line is a saturation state.

Thus, the line separating the liquid phase and the two-phase liquid-vapor

THERMODYNAMICS, MODELING, AND DESIGN ANALYSIS

Temperature

Specific volume

Figure 2.2 Pressure-specific volume-temperature surface and projections for water.

region is the saturated liquid line. The state denoted by f i n Figure is a saturated liquid state. The saturated vapor line separates the vapor region and the two-phase liquid-vapor region. The state denoted by g in Figure is a saturated vapor state. The saturated liquid line and the saturated vapor line meet at a point, called the critical point. At the critical point, the pressure is called the critical pressure and the temperature is called the critical tem- perature

2.3 RELATIONS 67

When a phase change occurs during constant pressure heating or cooling, the temperature remains constant as long as both phases are present. Accord- ingly, in the two-phase liquid-vapor region, a line of constant pressure is also a line of constant temperature. For a specified pressure, the corresponding temperature is called the saturation temperature. For a specified temperature, the corresponding pressure is called the saturation pressure. The region to the right of the saturated vapor line is often referred to as the superheated vapor region because the vapor exists at a temperature greater than the saturation temperature for its pressure. The region to the left of the saturated liquid line is also known as the compressed liquid region because the liquid is at a pressure higher than the saturation pressure for its temperature.

When a mixture of liquid and vapor coexist in equilibrium, the liquid phase is at the saturated liquid state and the vapor is at the corresponding saturated vapor state. The total volume of any such mixture is V =

+

V,, where f internal energy, enthalpy, and entropy.

Property Tables for Gases and Liquids. Values for specific internal energy, enthalpy, and entropy are calculated for gases and liquids using the T equations and relations developed from them, and specific heat data, and other thermodynamic data. The results of such calculations, supplemented by data taken from direct physical measurement or determined from these measurements, are presented in tabular form and, increasingly, as computer software. The form of the tables and the way they are used is assumed to be familiar.

The calculated specific internal energy, enthalpy, and entropy data are determined relative to arbitrary datums and the datums used vary from substance to substance. For example, in Reference the internal energy of saturated liquid water at is set to zero, but for refrigerants the saturated liquid enthalpy is zero at -40°C (-40°F for tables in English units). When calculations are being performed that involve only differences in a particular specific property, the datum cancels. Care must be taken, however, when there are changes in chemical composition during the process. The approach to be

68 THERMODYNAMICS, MODELING, AND DESIGN ANALYSIS

followed when composition changes due to chemical reaction is considered in Section 2.4.

Specific Heats. The properties c, and are defined as partial derivatives of the functions and respectively

(2.37)

(2.38) where the subscripts and p denote, respectively, the variables held fixed in the differentiation process. These properties are known commonly as

heats. The property k is simply their ratio

(2.39)

Incompressible Liquid Model. Reference to property data shows that for the liquid phase of water there are regions where the variation in the specific volume is slight and the dependence of internal energy on pressure (at fixed temperature) is weak. This behavior is also exhibited by other substances. As a model, it is often convenient to assume in such regions that the specific volume (density) is constant and the internal energy depends only on temperature. A liquid modeled in this way is said to be incompressible.

Since the incompressible liquid regards only on temperature, the specific heat c, is also alone:

internal to depend a function of temperature

Although specific volume is constant, enthalpy varies with both temperature and pressure as shown by

Differentiating Equation 2.40 with respect to temperature, holding pressure fixed gives = c,. That is, for an incompressible substance the two specific heats are equal. The common specific heat is commonly shown simply as c.

For an incompressible substance Equation reduces to du = T ds.

Then, with = the change in specific entropy is

2.3 PROPERTY RELATIONS 69

As = (2.41)

As an alternative to the incompressible liquid model when saturated liquid data are available, the following equations can be used to estimate property values at liquid states

where the subscript f denotes the saturated liquid state at the temperature T and is the corresponding saturation pressure.

Ideal-Gas Model. For vapor and liquid states, the general pattern of the p - u-T relation can be conveniently shown in terms of the compressibility factor Z, defined by Z = where is the universal gas constant (Section 2.1).

Using the compressibility factor, generalized charts can be developed from which reasonable approximations to the p-u-T behavior of many substances can be obtained. In one form of generalized charts the compressibility factor Z is plotted versus the reduced pressure and reduced temperature de- fined as = and = where p , and are, respectively, the critical pressure and critical temperature. Generalized compressibility charts suitable for engineering calculations, together with illustrations of their use, are available from the literature 1, 2,

By inspection of a generalized compressibility chart, it can be concluded that when is small, and for many states when is large, the value of the compressibility factor Z is closely 1. That is, for pressures that are low relative to p,, and for many states with temperatures high relative to the compressibility factor approaches a value of Within the indicated limits, it may be assumed with resonable accuracy that Z ⁼1. That is,

= (2.43)

It can be shown that the internal energy and enthalpy for any gas whose equation of state is exactly given by Equation 2.43 depends only on temperature

These considerations lead to the introduction of an ideal-gas model for each real gas. For example, the ideal-gas model of nitrogen obeys the equation of state = and its internal energy is a function of temperature alone.

The ideal-gas model of oxygen also obeys = and its internal energy

70 THERMODYNAMICS, MODELING, AND DESIGN ANALYSIS

is also a function of temperature alone, but the function differs from that for nitrogen since each gas has a unique internal structure. The real gas approaches the ideal-gas model in the limit of low reduced pressure. At other states the actual behavior may deviate substantially from the predictions of the model.

Dividing the ideal-gas equation of state by the molecular weight the equation is placed on a unit mass basis

p v = RT

where R , defined as is the specific gas constant. Other common use are

Since = for an ideal gas, the ideal-gas enthalpy and internal energy are related by = Differentiating with respect to temperature

With Equations 2.37 and 2.38, this gives

+

That is, the two ideal-gas specific heats depend on temperature alone, and their difference is the gas constant. Since k =

, R

= -

k - 1

Specific heat data can be obtained by direct measurement. When extrapolated to zero pressure, ideal-gas model specific heats result. Ideal-gas specific heats also can be obtained from theory based on molecular models of matter using spectroscopic measurements. Ideal-gas specific heat functions in both tabular and equation form are available in the literature for a number of substances Since = and = for an ideal gas, expressions for internal energy and enthalpy can be obtained by integration:

2.3 PROPERTY RELATIONS 71

+

(2.45)

where T‘ is an arbitrary reference temperature.

The first T equation, T =

+

^p can be used to determine the entropy change of an ideal gas between two states. Using ⁼ and

T V

Integrating between states 1 and 2

Similarly, from the second T equation (Equation follows

- = -

PI Equation 2.47 can be rewritten using defined by

(2.47)

(2.48) as

- = - - ^(2.49)

Equations 2.45 and 2.48 provide the basis for a simple tabular display.

That is, by specifying the reference temperature T’ the quantities can be tabulated versus temperature. Tabulations for several gases based on the specification ⁼ ⁼ ⁼0 at T‘ ⁼0 K are available in the literature In Section 2.4, is identified as the absolute entropy at tem- perature T and a reference pressure

72 THERMODYNAMICS, MODELING, AND DESIGN ANALYSIS

When the temperature interval is relatively small, the ideal-gas specific heats are nearly constant. It is often convenient in such instances to assume them to be constant, usually their arithmetic average over the interval. The expressions for changes in internal energy, enthalpy, and entropy of an ideal gas then appear as

- -

- = -

+

- - P z

- = - - R -

Ideal-Gas Mixtures. Consider a phase consisting of a mixture of N gases for which the Dalton mixture model applies: Each gas is uninfluenced by the presence of the others, each can be treated as an ideal gas, and each acts as if it exists separately at the volume and temperature of the mixture. Writing the ideal-gas equation of state for the mixture as a whole and for any com- ponent k

where p is the mixture pressure and is the partial pressure of component k. The partial pressure is the pressure gas k would exert if moles oc- cupied the mixture volume alone at the mixture temperature.

Forming the ratio of these two equations gives

(2.51) n

where defined as is the mole fraction of component k. Since

= 1, we have p = The pressure of a mixture of ideal gases is equal to the sum of the partial pressures of the gases. This is known as Dalton’s model additive partial pressures.

The internal energy, enthalpy, and entropy of the mixture can be determined as the sum of the respective properties of the component gases, provided that the contribution from each gas is evaluated at the condition at which the gas exists in the mixture. Thus

2.3 PROPERTY RELATIONS

N N

u

⁼ or ⁼

I I

N N

H ⁼ or ⁼

I I

N N

= or ⁼

I I

(2.52 b)

Since the internal energy and enthalpy of an ideal gas depend only on temperature, the and terms appearing in these equations are evaluated at the temperature of the mixture. Entropy is a function of two independent properties. Accordingly, the terms are evaluated either at the temperature and volume of the mixture or at the mixture temperature and the partial pressure

of the component. In the latter case

(2.53)

Differentiation of ⁼ and ⁼ with respect to temperature results, respectively, in expressions for the two specific heats and for the mixture in terms of the corresponding specific heats of the components :

I N

where = and =

molecular weights of the components as

The molecular weight of the mixture is determined in terms of the

M ⁼

(2.55)

Inserting the expressions for H and given by Equations and into the Gibbs function G = H - TS results in

THERMODYNAMICS, MODELING, AND DESIGN ANALYSIS

(2.56) where the molar specific Gibbs function of component k is =

The Gibbs function of component k can be expressed alternatively by integrating Equation at fixed temperature from an arbitrarily selected reference pressure to pressure = as follows:

For an ideal gas, = thus

(2.57)

In this section we consider some important terms, defini- tions, and principles of the study of systems consisting of dry air and water. Moist air refers to a mixture of dry air and water vapor in which the dry air is treated as if it were a pure component. Ideal-gas mixture principles are assumed to apply to moist air. In particular, the Dalton model is applicable, and so the mixture pressure p is the sum of the partial pressures and of the dry air and water vapor, respectively. Saturated air is a mixture of dry air and saturated water vapor. For saturated air, the partial pressure of the water vapor equals the saturation pressure of water corresponding to the mixture temperature T.

The composition of a given moist air sample can be described in terms of the humidity ratio humidity) w, defined as the ratio of the mass of the water vapor to the mass of dry air:

m..

which can be expressed alternatively as

P v

w 0.622 -

P - ^{P v} ^(2.58^b)

2.3 PROPERTY RELATIONS

The makeup of moist air also can be described in terms of the relative hu- midity defined as the ratio of the water vapor mole fraction in a given moist air sample to the water vapor mole fraction in a saturated moist air sample at the same temperature T and pressure p . The relative humidity can be expressed as

(2.59)

For moist air, the values of H and can be found by adding the contribution of each component at the condition at which it exists in the mixture. Thus, the enthalpy is

H ⁼

+

⁼

+

H ma

_ -

_(2.60)

where h, and h, are each evaluated at the mixture temperature. When using steam table data for the water vapor, is calculated from = To evaluate the entropy of moist air, the contribution of each component in the mixture is determined at the mixture temperature and the partial pressure of the component:

(2.61)

When using steam table data, is calculated from p , ) ⁼ R This equation is obtained by applying Equation 2.47 on a mass basis between the saturated vapor state at temperature T and the vapor state at and using Equation 2.59.

When a sample of moist air is cooled at constant pressure, the temperature at which the sample becomes saturated is called the dew point

Cooling below the dew point temperature results in the condensation of some of the water vapor initially present. When cooled to a final equilibrium state at a temperature below the dew point temperature, the original sample would consist of a gas phase of dry air and saturated water vapor in equilibrium with a liquid water phase. Further discussion including a numerical illustration is provided in Section 2.5.

Several important parameters of moist air are represented graphically by psychrometric charts. For moist air at a pressure of 1 atm, Reference 1 gives

76 THERMODYNAMICS, MODELING, AND DESIGN ANALYSIS

detailed charts in SI and English units. Familiarity with such charts is as- sumed.

2.3.2 Multicomponent Systems

In this section we introduce some general aspects of the properties of multicomponent systems consisting of nonreacting mixtures in a single phase.

Elaboration is provided elsewhere [ The special case of ideal-gas mixtures is considered in the previous section.

Partial Molal Properties. Any extensive thermodynamic property X of a single-phase, single-component system is a function of two independent intensive properties and the size of the system. Selecting temperature and pressure as the independent properties and the number of moles as the measure of size, we have X = p , n). For a single-phase, multicomponent system consisting of components, the extensive property X must then be a function of temperature, pressure, and the number of moles of each component present in the mixture, X = p , As X is mathematically homoge- neous of degree I in the the function is expressible as

where the partial molal property is, by definition,

(2.62)

(2.63) The subscript denotes that all are held fixed during differentiation. The partial molal property is a of the mixture and not simply a property of the component. depends general on temperature, pressure, and mixture composition: n,, Partial molal properties are intensive properties of the mixture.

Selecting the extensive property X in Equation 2.62 to be volume, internal energy, enthalpy, entropy, and the Gibbs function, respectively, gives

(2.64)

N N

I I

- - - -

where and denote the partial molal volume, internal energy, enthalpy, entropy, and Gibbs function, respectively.

2.3 PROPERTY RELATIONS

Chemical Potential. Because of its importance in the study of multicomponent systems, the partial molal Gibbs function of the kth component is given a special name and symbol. It is called the chemical potential and symbolized by

(2.65)

Like temperature and pressure, the chemical potential is an intensive prop- erty. The chemical potential is a measure of the escaping tendency of a sub- stance: Any substance will try to move from the phase having the higher chemical potential for that substance to the phase having a lower chemical potential. A necessary condition for phase equilibrium is that the chemical potential of each component has the same value in every phase.

Using Equation 2.65, together with the expression for the Gibbs function given in Equation 2.64

(2.66)

For a single-component system, Equation 2.66 reduces to G ⁼ That is, for a single component the chemical potential equals the molar Gibbs function. Comparing Equations 2.56 and 2.66, we conclude = That is, the chemical potential of component k in an ideal-gas mixture is equal to its Gibbs function per mole of gas k, evaluated at the mixture temperature and the partial pressure of the kth gas of the mixture.

With G = H - TS and H =

+

Equation 2.66 can be expressed as

u

⁼^TS^-

+

(2.67)

Using Equation 2.67, the following can be derived:

(2.68)

When the mixture composition is constant, Equation 2.68 reduces to Equation

78 THERMODYNAMICS, MODELING, AND DESIGN ANALYSIS

In document Bejan (Page 79-93)