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PRINCIPLES OF THERMODYNAMIC PHASE A Teaching Approach for Engineers HERMAN BIEBER Esso Research and Engineering Co., Linden, New Jersey ERNEST J. HENLEY1 Stevens Institute of Technology, Hoboken, New Jersey
T m m o n m m , n c diagrams are generally variations on only three distinct types. These three types stem from the fact that properties may be extensive or intensive. Diagram T v ~ Ie ~ ; ~I1e
Type I11
Abscissa Intensive Intensive Extensive Extensive
Ordinate Intensive Extensive Intensive Extensive
Intensive properties are independent of the amount of mass in the system under consideration. Examples are temperature, pressure, Young's modulus, etc. Extensive properties depend upon the size of the system. For example, volume and internal energy are meaningless unless an amount such as one pound or one mole of the system in question is specified.
THE PRFSSURE TEMPERATURE DIAGRAM The P-T plot is a frequently used thermodynamic graph, and it is perhaps the commonest example of
a Type I diagram. Figure 1 depicts the familiar P-T diagram for the pure substance, H20,schematically. The coordinates are chosen so as to include all the common regions (about 0.01 to 3500 psia and -100 to +400°C.). This hounded area in the P-T "plane" will include all the common states of water. The solid lines are equilibrium lines and thus represent phase boundaries. The dashed lines depict several simple processes and serve to help explain the diagram. Lines and axes of the insert2 have been distorted for clarity and are not t o scale. Consider the constant pressure melting of ice a t O°C. and atmospheric pressure. Starting below O°C. (point 7) on Figure 1 (insert), one has subcooled ice (since it is not in equilibrium with liquid water). It will, however, he in equilibrium with water (point 9) when the temperature reaches 0°C. (point 8). The word saturated is applicable t o phases in equilibrium. During the process (7 8), sensible heat has been added to the system t o raise its temperature. Adding more heat a t point 8 results in melting of the ice (8 + 91, but no immediate additional rise in temperature occurs. This is due to the constraint of the phase rule. The temperature in a one-component two-phase system is invariant if the pressure is fixed. Latent heat (of fusion) is being added to the system to change its state. When all the ice has melted, the temperature rise continues along path (9 + 10). This represents the locus of all subcooled liquid states a t one atmosphere pressure. At point 10 the water is saturated. Again, the heating process changes from one of addmg sensible heat t o one of adding latent heat (of vaporization). Again, the temperature remains fixed (at 100°C.) until all the water has evaporated (10 11). Finally, the temperature of the steam is raised along the path (11 -+ 12). Here the vapor above its boiling point is termed superheated. Note that one cannot tell during any point of the phase changes (8 -+ 9 and 10 11) hoxv much of the change has occurred. This is true for any Type I diaeram or alone anv intensive urouertv axis. The
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Written while a t Columbia University, New York, N. Y. The figurehas been simplified for there are ~ctuallllysix forms of ice. Further data relating to the special peculiarities of the water system can be found in any advanced text on physical chemistry. I
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whole process of fusion or vaporization takes place a t a point. The diagram only states a t what temperature this occurs if P is known, or vice versa. There is a unique temperature for each substance known as a critical temperature, above which it is impossible to condense a vapor to the liquid state. More precisely, the critical state is that state of temperature and pressure a t which the properties of the two coexisting fluid phases become so nearly alike that they no longer can coexist. This occurs a t the critical point (point 4), the abrupt end of the liquid vapor equilibrium line. The corresponding ordinate is termed the critical pressure. States with coordinates (P, T ) such that T > T4 are often referred t o as the "fluid" region since the terms liquid and vapor have no meaning there. The only other point of special interest on the P-T plane is the triple point (point 1, 2, 3). This is unique for each system since for a11 the phases to be in equilibrium there are no degrees of freedom left. Again, one cannot tell from the P-T diagram how much ice, liquid water, and &earn, respectively, are in equilibrium a t the triple point. It only relates the temperature and pressure that must exist. At pressures below the triple point it is possible t o pass directly from the solid to the vapor phase as by path 13 14 15 16. At points 14 and 15 the phases are saturated and sublimation is occurring. Process 5 + 17 + 6 represents a constant temperature compression starting with subcooled ice. This causes melting by the process of regelation (at point 17), a phenomenon familiar in ice skating. Ice melts under the runners and the water acts as a lubricant. As soon as the pressure under the runners is released, the water refreezes by a reverse process analogous to (6 17 5). Similarly, constant temperature compression of vapor between the triple point and the critical temperature will cause condensation to the liquid state a t some pressure between the triple point and the critical pressure (path 19 + 30 + 31 18).
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THE PRESSURE-ENTHALPY DIAGRAM
The P-h diagram is of Type I1 since one now has an extensive property along one of the axes. Prrssurevolume and temperature-enthalpy plots are other common diagrams in this category; all have similar shapes and identical characteristics. Figure 2 is a sketch of the P-h diagram for H20. The P scale has been distorted so as to cover the wide range of pressure between triple and critical points, but the enthalpy axis is approximately to scale. To understand how this is related to the first type of diagram (Figure I), imagine that the equilibrium lines in the P-T plot have been "stretched-out" in the X direction. This creates three new two-phase regions, solid-vapor, solid-liquid, and liquid-vapor. The first two are parallelogram-shaped bands, the latter is dome-shaped because the vapor-liquid line has a discrete terminus a t the critical point (point 4). These regions come into being because the value of an extensive property (as enthalpy) for one pound of saturated ice is not the same as the value of this property for one pound of saturated water a t the same temperature and pressure. Carrying the analogy further, it must be obvious that VOLUME 35, NO. 12, DECEMBER, 1958
the triple point is now the line along the base of the vapor-liquid dome, i.e., line 1-2-3. The abscissa1values a t points 1,2, and 3 represent the enthalpy in Btu of one pound of saturated ice, liquid water, and steam, respectively, a t the triple point. On the other hand the critical point is still a point (4 a t the top of the vapor liquid dome). This is because the distinguishing differences between liquid and vapor vanish a t the critical point, so that: One can use Type I1diagrams to examine what happens during a phase change and follow the amount quantitatively as well as qualitatively. On the P-h diagram the constant pressure heating of ice initially below O°C. and a t 1 atmosphere may now be repeated. Starting with the subcooled ice (point 7), it is heated to the saturation point 8 and begins to melt. At point 9 fusion is complete. The "ice water" is now heated to the boiling point, 100°C. (point lo), where vaporization commences. Finally, after complete evaporation (point l l ) , the resultant steam is superheated to the final temperature, point 12. Likewise the process, 13 14 15 16, is analogous to the sublimation process in Figure 1. Consider point B in the solid-vapor region. How much ice has evaporated? The applicability of the lever rule is apparent. The weight fraction of ice is BC/AC, the weight fraction of vapor is AB/AC, and the weight ratio of ice to vapor is BCIAB. Of course these statements apply equally well to mole fractions for this system. The lever rule is tantamount to solving the energy balance:
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where the h's refer to enthalpies of one pound of material and x is the weight fraction of vapor. It is of course impossible to solve for the mass ratios along the triple point (line 1-2-3) as this involves three unknowns. The other curved lines (TI T2,. . . . . . . . . .T6) in Figure 2 are isotherms. The slopes of the constant. temperature lines in any single phase region can he deduced from the observations that the heat content
w
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la0
k
Figure 2.
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P-h Diavam for Water
617
direction, the single and two-phase regions are distorted hut do not alter their general shape. The triple point constitutes the only real difference. The line (1-2-3) in Figure 2 is resolved into a triangle, the comers of which represent the pure saturated phases. Otherwise, the vapor liquid area is still a dome, and the critical point is still a point (point 4). This is depicted in Figure 3. Again the axes of Figure 3 h a w been distorted somewhat to aid illustration. Note the isotherms (T, and TB). I n the single phase region they are nearly perpendicular to the E axis, since the internal energy function is relatively intensitive to pressure and volume changes as compared to temperature chanqes. In other words, AE =
CdT
and
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for watn
(enthalpy) of any substance rises as its temperature increases a t constant P, i.e., (b h/bT), is positive. Similarly, the temperature will rise as the pressure increases a t constant h, i.e., (bT/bP)* is positive. Hence by the calculus
w, = CS/(CS
and the isotherm slope, (bP/bh),, must he negative. The isotherms become horizontal lines in any two phase region because of phase rule constraints. Note that the right-hand part of the dome overhangs the base so that it is possible to have two different saturated phases (points 20 and 21) with the same enthalpy. This trait is not characteristic of Type I1 diagrams hut is a peculiarity of the enthalpy function. As one increases the pressure of a saturated vapor (such as point 20) at constant enthalpy, it is sometimes possible for the changes in E and the product Pv to he = (E of equal and opposite signs, i.e., (E P u ) ~ . On a P v diagram, the vapor-liquid dome would have no overhang.
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I n the two-phase region, the isotherms rise steeply, since there is a large absorption of energy and an expansion of volume (at constant pressure). The triple point constitutes the only case where an isotherm is ever an area. Process (7 8 9 + 10 + 11 12) represents the same heating operation as (7 8 9 10 11 -+ 12) in Figure 2. It, then, gives a qualitative picture of the shape of isobars (constant pressure lines) on the E-v plane. As before, tie lines can he employed to estimate the amount of phases in equilibrium in a twwphase region. I n the triangular region 1-2-3, it can he shown that the relative amount of each pure phase present is proportional to the length of the perpendicular drawn from a net composition point such as point C to the side of the triangle opposite the vertex in question. For example, a t C, the weight fraction vapor is:
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Figure 3. E-0 Di-am
(z)?
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+ CT + C R )
The weight fraction liquid is: w, = CT/(CS
+ CT + C R )
The weight fraction solid is: I n many cases, it will he iust as easy to obtain this information by solving the following system of equations simultaneously.
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THE ENERGY-VOLUME DIAGRAM
Typical of the third class of diagrams, one can choose the E-u diagram. ( E is internal energy; u is specific volume.) Now both axes represent extensive properties. The shape of the resulting plot is, however, not much different from the previous example (Fig. 2). If one should "stretch" this figure out in the Y-
Again these relationships apply equally well on a molal basis for a one component system. It is hoped that the foregoing treatment will help the student to understand the principles involved in thermodynamic diagrams. Such an understanding is essential, particularly to engineering students who must use the diagram to solve problems involving the interconversion of heat and work.
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