PNG 520
Phase Relations in Reservoir Engineering

Heat Capacities

PrintPrint

The constant volume heat capacity is defined by:

C v ( U T ) V This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
(19.1)
 

To see the physical significance of the constant volume heat capacity, let us consider a 1 lbmol of gas within a rigid-wall (constant volume) container. Heat is added to the system through the walls of the container and the gas temperature rises. It is evident that the temperature rise ( ΔT This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers. ) is proportional to the amount of heat added,

QΔT This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
(19.2)
 

Introducing a constant of proportionality “cv”,

Q= c v ΔT This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
(19.3)
 

In our experiment, no work was done because the boundaries (walls) of the system remained unchanged. Applying the first law of thermodynamics to this closed system, we have:

ΔU= c v ΔT This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
(19.4)
 

Therefore, for infinitesimal changes,

C v = ( U T ) V This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
(19.5)
 

As we have seen, constant volume heat capacity is the amount of heat required to raise the temperature of a gas by one degree while retaining its volume.

Let us now consider the same 1 lbmol of gas confined in a piston-cylinder equipment (i.e., a system with non-rigid walls or boundaries). When heat is added to the system, the gas temperature rises and the gas expands so that the pressure in the system remains the same at any time. The piston displaces a volume Δ This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers. V and the gas increases its temperature in Δ This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers. T degrees. Again, the temperature rise ( Δ This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers. T) is proportional to the amount of heat added, and the new constant of proportionality we use here is “cp”,

Q= c P ΔT This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
(19.6)
 

This time, some work was done because the boundaries (walls) of the system changed from their original position. Applying the first law of thermodynamics to this closed system, we have that:

ΔU=QW This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
(19.7)
 

If the pressure remained the same both inside and outside the container, the system made some work against the surroundings in the amount of W= P Δ V This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers. . Introducing (19.7) into (19.6),

ΔU+PΔV= c P ΔT This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
(19.8)
 

The left hand side of this equation represents the definition of enthalpy change ( ΔH This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers. ) for a constant-pressure process. Therefore:

ΔH= c P ΔT This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
(19.9)
 

Finally, for infinitesimal changes,

c P = ( H T ) V This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
(19.10)
 

The function “cp” is called the constant pressure heat capacity. The constant pressure heat capacity is the amount of heat required to raise the temperature of a gas by one degree while retaining its pressure.

The units of both heat capacities are (Btu/lbmol-°F) and (cal/gr-°C). Their values are never equal to each other, not even for ideal gases. In fact, the ratio “cp/cv” of a gas is known as “k” — the heat capacity ratio — and it is never equal to unity. This ratio is frequently used in gas-dynamics studies.

k= c p c v This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
(19.11)
 

Heat capacities can be calculated using equations of state. For instance, Peng and Robinson (1976) presented an expression for the departure enthalpy of a fluid mixture, shown below:

H ¨ =H H · =RT(Z1)+ T d (aα) m dT (aα) m 2 2 b m ln( Z+( 2 +1)B Z( 2 1)B ) This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
(19.12)
 

The value of the enthalpy of the fluid (H) is obtained by adding up this enthalpy of departure (shown above) to the ideal gas enthalpy (H*). Ideal enthalpies are sole functions of temperature. For hydrocarbons, Passut and Danner (1972) developed correlations for ideal gas properties such as enthalpy, heat capacity and entropy as a function of temperature. Therefore, an analytical relationship for “cp” can be derived taking the derivative of (19.12), as shown below:

C P = C P · +R( T ( Z T ) P +Z1 )+ T d (aα) m dT (aα) m 2 2 b m [ ( Z T ) P +2.414 ( B T ) P Z+2.414B ( Z T ) P 0.414 ( B T ) P Z0.414B ]+ T d 2 ( aα ) m d T 2 2 2 b m ln( Z+2.414B Z414B ) This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
(19.13)
 

where:

C P · = ( H · T ) P This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers. = ideal gas CP,

also found in the work of Passut and Danner (1972).

The second derivative of ( aα ) m This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers. with respect to temperature can be calculated through the expression:

d 2 d T 2 ( aα ) m = 0.45724 R 2 2 T i j c i c j ( 1 k ij ) [ f( w j )( α i 0.5 T ci P ci 0.5 ) ( T cj P cj ) 0.5 ψ i +f( w i )( α j 0.5 T cj P cj 0.5 ) ( T ci P ci ) 0.5 ψ j ] This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
(19.14a)
 

where,

ψ i = f( w i ) 2 T ci T α i 1 2T This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
(19.14b)
 

For the evaluation of expression (19.13), the derivative of the compressibility factor with respect to temperature is also required. Using the cubic version of Peng-Robinson EOS, this derivative can be written as:

( Z T ) P =( ( Ω 2 T ) P Z 2 + ( Ω 3 T ) P Z+ ( Ω 4 T ) P 3 Z 2 +2 Ω 2 Z+ Ω 3 ) This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
(19.15)
 

where,

( Ω 2 T ) P = ( B T ) P ( Ω 3 T ) P = ( A T ) P 6B ( B T ) P 2 ( B T ) P ( Ω 4 T ) P =[ A ( B T ) P +B ( A T ) P 2B ( B T ) P 3 B 2 ( B T ) P ] ( A T ) P = A ( aα ) m d ( aα ) m dT 2 A T ( B T ) P = B T This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.

“cp” and “cv” values are thermodynamically related. It can be proven that this relationship is controlled by the P-V-T behavior of the substances through the relationship:

c p c v =T ( V T ) P ( P T ) V This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
(19.16)
 

For ideal gases, PV=nRT This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers. and Equation (18.28) collapses to:

c p · c v · =R This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
(19.17)