Thermodynamics, Chapter 2

METBD 330: Thermodynamics

Chapter 2:

PURE SUBSTANCE: Fixed chemical composition, throughout H₂O, N₂, CO₂, Air (even a mixture of ice and water is pure)

COMPRESSED LIQUID: NOT about to vaporize

(Sub-cooed liquid) e.g., water at 20^oC and 1 atmosphere

SATURATED LIQUID: about to vaporize

e.g., water at 100^oC and 1 atmosphere

SATURATED VAPOR: about to condense

e.g., water vapor (steam) at 100^oC and 1 atm.

SUPERHEATED VAPOR: NOT about to condense

e.g., water vapor (steam) at >100^oC and 1 atm.

Tv Diagram for Heating H₂O at Constant Pressure (Figure 2-11):

Fig2_11.gif (4835 bytes)

Pressure Cooker example: the boiling temperature varies with pressure

Saturation Temperature: the boiling temperature at a given pressure

Saturation Pressure: the pressure at which boiling occurs at a given T

Liquid-Vapor Saturation Curve for Water (Figure 2-12):

Fig2_12.gif (4163 bytes)

T-v Diagrams: useful in studying and understanding phase change processes.

For water (Figure 2-13):

Fig2_13.gif (5808 bytes)

CRITICAL POINT:

the saturated liquid and saturated vapor states are identical

No saturated mixture exists - the substance changes directly from the liquid to vapor states.

LOOK at Table A-1:

for H₂O:

P_CR = 22.09 MPa

T_CR = 374.14 ^oC (or 647.3 ^oK)

v_CR = 0.003155 m³/kg (or .0568 m³/kmol)

MORE T-v DIAGRAMS (Figure 2-15):

Fig2_15.gif (5438 bytes)

P-v DIAGRAMS (Figure 2-16): Note: Constant Temperature lines go downward.

Fig2_16.gif (5377 bytes)

P-T DIAGRAM (Figure 2-22): Shows the TRIPLE POINT

The ONLY point where all 3 phases (solid, liquid, and vapor) can exist in equilibrium

The triple point for H₂O:

T = 0.01^oC
P = 0.6113 kPa
(0.6% of 1 atm)

ENTHALPY, H

another property of pure substances (like internal energy, U)

U is a function ONLY of the temperature of the substance

H is formed by combining the internal energy, U, and work done in the form, PV. PV = (N/m²)(m³) = N-m = Joules

on a per unit mass basis: h = u + Pv (specific enthalpy)

H is a function of BOTH temperature and pressure.

PROPERTY TABLES

used to look up the properties for various substances at many temperatures and pressures

LOOK at Tables A-4 and A-5 for saturated water (in SI units, Appendix 2 has English units)

Table A-4 is property data, tabulated vs. Temperature

Table A-5 is property data, tabulated vs. Pressure

v_f = specific volume of the saturated LIQUID

v_g = specific volume of the saturated VAPOR

v_fg = the difference between the specific volume of the saturated vapor and saturated liquid = (v_g - v_f)

LOOK in Table A-4 ……

find Temperature of 100^oC in the left column
the saturation pressure at this temperature is 0.10133 MPa (or 101.33 kPa)
v_f = 0.001044 m³/kg (saturated LIQUID)
v_g = 1.6729 m³/kg (saturated VAPOR)
Calculate: v_fg = v_g - v_f = 1.6729 - 0.001044 m³/kg

Also, in this table:

u_f, u_fg, and u_g, the specific internal energy data (kJ/kg)

h_f, h_fg, and h_g, the specific enthalpy data (kJ/kg)

h_fg is the "heat of vaporization", or the energy required to completely vaporize from the saturated liquid state to the saturated vapor state.

s_f, s_fg, and s_g, the specific entropy data (kJ/kg^oK)

SATURATED LIQUID-VAPOR MIXTURES:

Find the properties of a mixture using the QUALITY.

QUALITY defines the proportions of the liquid and vapor phases in the mixture.

QUALITY is defined:

QUALITY, x, is never used to describe compressed liquid or superheated vapor !!

Quality may be expressed as a percentage: from 0% to 100%, where 0% is a saturated liquid and 100% is a saturated vapor.

is treated as

m_t v_avg = m_f v_f + m_gv_g
m_f = m_t - m_g

<-- dividing by the total mass and
applying the definition of Quality, x

v_avg = (1 - x) v_f + xv_g = v_f - x v_f + x v_g

= v_f + x (-v_f + v_g)

v_avg = v_f + x v_fg

Also, x = (v_avg - v_f) /v_fg
(a common way to evaluate the quality)

Quality is used to find other properties of saturated liquid-vapor mixtures (Fig. 2-32):

u_avg = u_f + x u_fgand,

x = (u_avg - u_f )/u_fg

h_avg = h_f + x h_fgand,

x = (h_avg - h_f )/h_fg

ALWAYS check:

h_f <= h_avg <= h_g

EXAMPLE:

GIVEN: 0.05 kg of water at 25^oC in a container of 1.0 m³ volume.

FIND: the phase description, pressure, and quality (if appropriate).

WHY might the quality be inappropriate ???

Procedure:

Find the proper table: Water + given T, use Table A-4

Look up v_f = 0.001003 m³/kg v_g = 43.36 m³/kg

Calculate v (= v_avg) = V/m = 1.0 m³/0.05 kg = 20 m³/kg

Since v_f < v_avg < v_g the phase is saturated liquid-vapor mixture

Since this is a saturated state: P = P_sat@25C = 3.169 kPa

Finally, x = (v_avg - v_f)/(v_g - v_f) = (20 - 0.001003)/(43.36 - 0.001003) = 0.461 = 46.1%

SUPERHEATED VAPOR:

no longer any liquid, NOT about to condense.

Fig2_40.gif (5703 bytes)

characteristics:

at a given P, T > T_SAT or

at a given T, P < P_SAT or

at either a given P or T, v > v_g or h > h_g or u > u_g

Evaluate the properties of superheated water in Table A-6

Table A-6 has a different format than saturated data tables (A-4 and A-5), because P and T are now independent i.e., for a given pressure, a series of temperatures are tabulated

EXAMPLE 2-7

Water at P = 0.5 MPa, h = 2890 kJ/kg. Find T.

Check Table A-5 (given P) finding h_g = 2748.7 kJ/kg
since h > h_g, this is a superheated vapor !
From Table A-6 at P = 0.5 MPa, find two rows that bracket the given h

Temperature (^oC)	Enthalpy (kJ/kg)	h of 2890 kJ/kg is between these values
200	2855.4
250	2960.7

Make your own Interpolation Table:

Temperature (^oC)	Enthalpy (kJ/kg)
200	2855.4
T	2890
250	2960.7

COMPRESSED LIQUID:

no vapor, all liquid, NOT about to vaporize

Fig2_4b.gif (5602 bytes)

characteristics:

at a given P, T < T_SAT or

at a given T, P > P_SAT or

at either a given P or T, v < v_f or h < h_f or u < u_f

Usually we approximate compressed liquid behavior (Figure 2-33) evaluated at the given TEMPERATURE (don’t use the pressure)

v = v_f@T

h = h_f@T

u = u_f@T

Before we leave this section, look at Tables A-8, A-9, A-10.

This data is for Refrigerant 134a, commonly used in air conditioning systems. You have the same tables: Saturated liquid-vapor mixtures (both a temperature and pressure table), and the superheated vapor table.

Problems with R-134a can be treated using the tables just like problems with water or steam.

IDEAL GASES:

The Ideal Gas Equation is: P v = R T

Alternative to using Tables

Defines the state of a gas by relating T, P, and v

R is the gas constant for the specific gas being analyzed

T is the absolute temperature

P is the absolute pressure

P v = R T is valid for gases with low density (r). Low density is found when pressure is low and/or temperature is high.

Test for Ideal Gas behavior (p. 62):

The gas is:

at any temperature,

if P_R << 1

IDEAL

(P_R = P/P_CR)

at high temperatures,

if T_R > 2

IDEAL

unless P_R >> 1

(T_R = T/T_CR)

The gas constant, R = R_u/M

where: R_u = the universal gas constant

R_u = 8.314 kJ/kmole-K (for all gases)

M = the molecular weight of the gas (kg/kmole ) Table A-1

P V = N R_u T

uses N, the number of moles of the gas molecules

also, since V = m v, P V = m R T where m is the mass

UNITS:

P v = R T gives (kPa) (m³/kg) = (kJ/kg-^oK) (^oK)

Notice: 1 kJ = 1 kPa-m³

For changes of state (with a fixed mass of Ideal gas):

P₁ v₁ = R T₁ and P₂ v₂ = R T₂

(P₁ v₁)/T₁ = R and (P₂ v₂)/T₂ = R

(P₁ v₁)/T₁ = (P₂ v₂)/T₂

Can you use the Ideal Gas Law for water vapor ????

Almost NEVER, only at very low pressures, < 10 kPa

NON-IDEAL Gas Behavior: Compressibility Factor

P v = Z R T

where Z = compressibility factor for gases which are not quite IDEAL

Z is the ratio of the ACTUAL specific volume to the IDEAL specific volume of the gas

Z = (v_ACTUAL)/(v_IDEAL)

Z is found in the charts in Figure 2-40 and Appendix A-13, using the quantities:

P_R (Reduced Pressure) = P/P_CRITICAL

T_R (Reduced Temperature) = T/T_CRITICAL

v_R ("Reduced" Specific Vol.)

= v_ACTUAL P_CR/ (R T_CR)

The Compressibility charts can be used for ALL GASES.

EXAMPLE: Air at 164^oK, 10.17 MPa. What is Z ??

Table A-1: Find T_CR and P_CR for AIR …

T_CR = 132.5^oK and P_CR = 3.77 MPa

T_R = T / T_CR = 164^oK / 132.5^oK = 1.24

P_R = P / P_CR = 10.17 MPa / 3.77 MPa = 2.7

Use Figure 2-40, READ Z from the graph.

Also notice, in Fig A-13, you can also read v_R from the graph

OTHER EQUATIONS OF STATE

(have more terms - applicable in larger range of T_R and P_R, see Fig. 2-48)

Van der Waals (2 constants): (P + a/v²)(v - b) = R T

Beattie-Bridgeman (5 constants)

Benedict-Webb-Rubin (8 constants)

Strobridge (16 constants)

Virial (power series form)

m_t v_avg = m_f v_f + m_gv_g	m_f = m_t - m_g
	<-- dividing by the total mass and applying the definition of Quality, x
v_avg = (1 - x) v_f + xv_g	= v_f - x v_f + x v_g
	= v_f + x (-v_f + v_g)
	v_avg = v_f + x v_fg
Also, x = (v_avg - v_f) /v_fg	(a common way to evaluate the quality)