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## Elementary Thermodynamics - Phase Changes

Phase changes are very important in thermodynamics, principally because energy transfer (heat loss or heat gain) must occur for a substance to change states, often with negligible change in temperature. To cite an example, consider the case of water (a liquid) turning into steam (a vapor) at atmospheric pressure. At sea level, this phase change will occur at a temperature of 100 degrees Celsius, or 212 degrees Fahrenheit. The amount of energy required to increase the temperature of water up to its boiling point is a simple function of the sample’s mass and its original temperature. For instance, a sample of water 70 grams in mass starting at 24 degrees Celsius will require 5320 calories of heat to reach the boiling point:

However, actually boiling the 70 grams of water into 70 grams of steam (both at 100 degrees Celsius) requires a comparatively enormous input of heat: 37,734 calories – over seven times as much heat to turn the water to steam than required to warm the water to its boiling point. Furthermore, this additional input of 37,734 calories does not increase the temperature of the water one bit: the resulting steam is still at (only) 100 degrees Celsius. If further heat is added to the 70 gram steam sample, its temperature will rise, albeit at a rate proportional to the value of steam’s specific heat (0.48 calories per gram degree Celsius, or BTU per pound degree Fahrenheit).

What we see here is a fundamentally different phenomenon than we did with specific heat. Here, we are looking at the thermal energy required to transition a substance from one phase to another, not to change its temperature. We call this quantity latent heat rather than specific heat, because no temperature change is manifest_{1}. Conversely, if we allow the steam to condense back into liquid water, it must release the same 37,734 calories of heat energy we invested in it turning the water into steam before it may cool at all below the boiling point (100 degrees Celsius).

As with specific heat, there is a formula relating mass, latent heat, and heat exchange:

Q = mL

Where,

Q = Heat of transition required to completely change the phase of a sample (metric calories or British BTU)

m = Mass of sample (metric grams or British pounds)

L = Latent heat of substance

Each substance has its own set of latent heat values, one_{2} for each phase-to-phase transition. Water, for example, exhibits a latent heat of vaporization (boiling/condensing) of 539.1 calories per gram, or 970.3 BTU per pound. Water also exhibits a latent heat of fusion (melting/freezing) of 79.7 calories per gram, or 143.5 BTU per pound. Both figures are enormous compared to water’s specific heat value of 1 calorie per gram-degree Celsius (or 1 BTU per pound-degree Fahrenheit_{3}): it takes only one calorie of heat to warm one gram of water one degree Celsius, but it takes 539.1 calories of heat to boil that same gram of water into one gram of steam, and 79.7 calories of heat to melt one gram of ice into one gram of water. The lesson here is simple: phase changes involve huge amounts of heat.

A table showing various latent heats of vaporization (all at room temperature, 70 degrees Fahrenheit) for common industrial fluids appears here, contrasted against their specific heat values (as liquids). In each case you will note how much larger L is than c:

Fluid @ 70 | ^{O}F Lvaporization, BTU/lb |
Lvaporization, cal/g | cliquid |

Water | 970.3 | 539.1 | 1 |

Ammonia | 508.6 | 282.6 | 1.1 |

Carbon dioxide | 63.7 | 35.4 | 0.66 |

Butane | 157.5 | 87.5 | 0.56 |

Propane | 149.5 | 83.06 | 0.6 |

One of the most important, and also non-intuitive, consequences of latent heat is the relative stability of temperature during the phase-change process. Referencing the table of latent heats of vaporization, we see how much more heat is needed to boil a liquid into a vapor than is needed to warm that same liquid one degree of temperature. During the process of boiling, all heat input to the liquid goes into the task of phase change (latent heat) and none of it goes into increased temperature. In fact, until all the liquid has been vaporized, the liquid’s temperature cannot rise above its boiling point! The requirement of heat input to vaporize a liquid forces temperature to stabilize (not rise further) until all the liquid has evaporated from the sample.

If we take a sample of ice and add heat to it over time until it melts, warms, boils, and then becomes steam, we will notice a temperature profile that looks something like this:

The flat areas of the graph during the melting and boiling periods represents times where the sample’s temperature does not change at all, but where all heat input goes into the work of changing the sample’s phase. Only where we see the curve rising does the temperature change.

To use our liquid-filled vessel analogy again, it is as if at some point along the vessel’s height there is a pipe connection leading to a large, relatively flat expansion chamber, so that the vessel “acts” as if its area were much larger at one point, requiring much more fluid volume (heat) to change height (temperature):

Liquid poured into this vessel will fill it at a rate proportional to the volume added and inversely proportional to the vessel’s cross-sectional area at the current liquid height. As soon as the liquid level reaches the expansion chamber, a great deal more liquid must be added to cause level to increase, since this chamber must fill before the liquid level may rise above it. Once that happens, the liquid level rises at a different rate with addition introduced volume, because now the phase is different (with a different specific heat value).

Remember that the filling of a vessel with liquid is merely an analogy for heat and temperature, intended to provide an easily visualized process mimicking another process not so easily visualized. The important concept to realize with latent heat and phase change is that it constitutes a discontinuity in the temperature/heat function of any given substance.

A vivid demonstration of this phenomenon is to take a paper_{4} cup filled with water and place it in the middle of a roaring fire_{5}. “Common sense” might tell you the paper will burn through with the fire’s heat, so that the water runs out of the cup through the burn-hole. This does not happen, however. Instead, the water in the cup will rise in temperature until it boils, and there it will maintain that temperature no matter how hot the fire burns. The boiling point of water happens to be substantially below the burning point of paper, and so the boiling water keeps the paper cup too cool to burn. As a result, the paper cup remains intact so long as water remains in the cup. The rim of the cup above the water line will burn up because the steam does not have the same temperature-stabilizing effect as the water, leaving a rimless cup that grows shorter as the water boils away.

The point at which a pure substances changes phase not only relates to temperature, but to pressure as well. We may speak casually about the boiling point of water being 100 degrees Celsius (212 degrees Fahrenheit), but that is only if we assume the water and steam are at atmospheric pressure (at sea level). If we reduce the ambient air pressure

_{6}, water will boil at a lesser temperature. Anyone familiar with cooking at high altitudes knows you must generally cook for longer periods of time at altitude, because the decreased boiling temperature of water is not as effective for cooking. Conversely, anyone familiar with pressure cooking (where the cooking takes place inside a vessel pressurized by steam) knows how comparatively little cooking time is required because the pressure raises water’s boiling temperature. In either of these scenarios, where pressure influences boiling temperature, the latent heat of water acts to hold the boiling water’s temperature stable until all the water has boiled away. The only difference is the temperature at which the water begins to boil (or when the steam begins to condense).

Many industrial processes use boiling liquids to convectively transfer heat from one object (or fluid) to another. In such applications, it is important to know how much heat will be carried by a specific quantity of the vapor as it condenses into liquid over a specified temperature drop. The quantity of enthalpy (heat content) used for rating the heat-carrying capacity of liquids applies to condensing vapors as well. Enthalpy is the amount of heat lost by a unit mass (one gram metric, or one pound British) of the fluid as it cools from a given temperature all the way down to the freezing point of water (0 degrees Celsius, or 32 degrees Fahrenheit). When the fluid’s initial state is vapor, and it condenses into liquid as it cools down to the reference temperature (32 ^{o}F), the heat content (enthalpy) is not just a function of specific heat, but also of latent heat. Steam at atmospheric pressure and 212 degrees Fahrenheit (the boiling point of water) has an enthalpy of about 1150 BTU per pound. 970 BTU is released due to the phase change from vapor to liquid, while the rest is due to the water’s specific heat of (approximately) one as it cools from 212 degrees Fahrenheit to 32 degrees Fahrenheit (approximately 180 BTU released).

If the vapor in question is at a temperature greater than its boiling point at that pressure, the vapor is said to be superheated. The enthalpy of superheated vapor comes from three different heat-loss mechanisms:

• Cooling the vapor down to its condensing temperature (specific heat of vapor)

• Phase-changing from vapor to liquid (latent heat of phase change)

• Cooling the liquid down to the reference temperature (specific heat of liquid)

Using steam as the example once more, a sample of superheated steam at 500 degrees Fahrenheit and atmospheric pressure (boiling point = 212 degrees Fahrenheit) has an enthalpy of approximately 1287 BTU per pound. We may calculate the heat lost by one pound of this superheated steam as it cools from 500 ^{o}F to 32 ^{o}F in each of the three steps previously described. Here, we will assume a specific heat for steam of 0.476, a specific heat for water of 1, and a latent heat of vaporization for water of 970:

Heat loss mechanism | Formula | Quantity |

Cooling vapor | Q = mcΔT | (1)(0.476)(500-212) = 137 BTU |

Phase change | Q = mL | (1)(970) = 970 BTU |

Cooling liquid | Q = mcΔT | (1)(1)(212-32) = 180 BTU |

TOTAL | 1287 BTU |

Enthalpy values are very useful in steam engineering to predict the amount of thermal energy delivered to a load given the steam’s initial temperature, its final (cooled) temperature, and the mass flow rate. Although the definition of enthalpy – where we calculate heat value by supposing the vapor cools all the way down to the freezing point of water – might seem a bit strange and impractical (how often does steam lose so much heat to a process that it reaches freezing temperature?), it is not difficult to shift the enthalpy value to reflect a more practical final temperature. Since we know the specific heat of liquid water is very nearly one, all we have to do is offset the enthalpy value by the amount that the final temperature differs from freezing in order to calculate how much heat the steam will lose (per pound) to its load.

For example, suppose we were to use the same 500 ^{o}F superheated steam used in the previous example to heat a flow of oil through a heat exchanger, with the steam condensing to water and then cooling down to 170 degrees Fahrenheit as it delivers heat to the flowing oil. Here, the steam’s enthalpy value of 1287 BTU per pound may simply be offset by 138 (170 degrees minus 32 degrees) to calculate how much heat (per pound) this steam will deliver to the oil: 1287 - 138 = 1149 BTU per pound. If we require a heat transfer rate of 45,000 BTU per hour to the flowing oil, the steam flow rate will have to be just over 39 pounds per hour.

_{1}The word “latent” refers to something with potential that is not yet realized. Here, heat exchange takes place without there being any realized change in temperature.

_{2}Latent heat of vaporization also varies with pressure, as different amounts of heat are required to vaporize a liquid depending on the pressure that liquid is subject to. Generally, increased pressure (increased boiling temperature) results in less latent heat of vaporization.

_{3}The reason specific heat values are identical between metric and British units, while latent heat values are not, is because latent heat does not involve temperature change, and therefore there is one less unit conversion taking place between metric and British when translating latent heats. Specific heat in both metric and British units is defined in such a way that the three different units for heat, mass, and temperature all cancel each other out. With latent heat, we are only dealing with mass and heat, and so we have a proportional conversion of 59 or 95 left over, just the same as if we were converting between degrees Celsius and Fahrenheit alone.

_{4}Styrofoam and plastic cups work as well, but paper exhibits the furthest separation between the boiling point of water and the burning point of the cup material, and it is usually thin enough to ensure good heat transfer from the outside (impinging flame) to the inside (water).

_{5}This is a lot of fun to do while camping!

_{6}This may be done in a vacuum jar, or by traveling to a region of high altitude where the ambient air pressure is less than at sea level.

**Go Back to Lessons in Instrumentation Table of Contents**

written by Edward, May 01, 2017

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