Phase change

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Under normal conditions at the macroscopic level, there are three phases (or states) of matter: solid phase, liquid phase, and gaseous phase. Plasma is sometimes called the fourth phase of matter. In the description of matter, phase indicates how particles group together to form a substance. The structure of a substance can vary from compactly-arranged particles to highly-dispersed ones. In a solid, the particles are close together in a fixed pattern, while in a liquid the particles are almost as close together as in a solid but are not held in any fixed pattern. In a gas, the particles are also not held in any fixed pattern, but the average distance between particles is large. Both liquids and gases are called fluids. A gas that is capable of conducting electricity is called plasma. Gases do not normally conduct electricity, but when they are heated to high temperatures or collide intensely with each other, they form electrically-charged particles called ions. These ions give the plasma the ability to conduct an electrical current. Because of the high temperatures that prevail in the sun and other stars, their constituent matter exists almost entirely in the plasma phase.

Based on the phases that are involved in the system, phase change problems can be classified as: (1) solid-liquid phase change (melting and solidification), (2) solid-vapor phase change (sublimation and deposition), and (3) liquid-vapor phase change (boiling/evaporation and condensation). Melting and sublimation are also referred to as fluidification because both liquid and vapor are regarded as fluids.

Liquid-vapor change in rigid tank with relief valve
Liquid-vapor change in rigid tank with relief valve.

When a process involves a change in phase, the heat absorbed or released can be expressed as the product of the mass quantity and the latent heat of the material. For liquid-vapor phase change, such as vaporization or condensation, the heat transferred can be expressed as

q = \dot m{h_{\ell v}} \qquad\qquad(1)

where m is the mass of material changing phase per unit time. The latent heat of vaporization, {h_{\ell v}}, is the difference between the enthalpy of vapor and of liquid, i.e.,

{h_{\ell v}} = {h_v} - {h_\ell }\qquad\qquad(2)

For a pure substance, liquid-vapor phase change always occurs at the saturation temperature. Since the pressure for the saturated liquid-vapor mixture is a function of temperature only, liquid-vapor phase change usually occurs at a constant pressure. As will become evident, eq. 1 needs to be modified for a liquid-vapor phase change process occurring at a constant pressure (Bejan, 1993). The figure on the right shows a rigid tank filled with a mixture of liquid and vapor at equilibrium. To maintain a constant pressure during evaporation, a relief valve at the top of the tank is opened; the mass flow rate of vapor through the valve is m. At time t, the masses of the liquid and vapor are, respectively, {m_\ell} and mv. The change in mass of the liquid and of the vapor satisfies

\frac{{d{m_\ell }}}{{dt}} + \frac{{d{m_v}}}{{dt}} =  - \dot m \qquad\qquad(3)

The total volume of the liquid and vapor

V = {m_\ell }{v_\ell } + {m_v}{v_v}\qquad\qquad(4)

remains constant during the phase change process. Thus

\frac{{d{m_\ell }}}{{dt}}{v_\ell } + \frac{{d{m_v}}}{{dt}}{v_v} = 0 \qquad\qquad(5)

Combining eqs. (3) and (5) yields

\frac{{d{m_\ell }}}{{dt}} =  - \dot m\left( {\frac{{{v_v}}}{{{v_v} - {v_\ell }}}} \right) \qquad\qquad(6)
\frac{{d{m_v}}}{{dt}} = \dot m\left( {\frac{{{v_\ell }}}{{{v_v} - {v_\ell }}}} \right) \qquad\qquad(7)

The first law of thermodynamics for the control volume is

\frac{{d{E_{cv}}}}{{dt}} = q - \dot m{h_v} \qquad\qquad(8)

where the internal energy of the control volume is

{E_{cv}} = {m_\ell }{e_\ell } + {m_v}{e_v} \qquad\qquad(9)

Differentiating eq. (9) and considering eqs. (6) and (7), one obtains

\frac{{d{E_{cv}}}}{{dt}} =  - \dot m\left( {\frac{{{v_v}{e_\ell } - {v_\ell }{e_v}}}{{{v_v} - {v_\ell }}}} \right) \qquad\qquad(10)

Substituting eq. (10) into eq. (8) and considering eq. (2) yields

q = \dot m\left[ {{h_{\ell v}} + \left( {{h_\ell } - \frac{{{v_v}{e_\ell } - {v_\ell }{e_v}}}{{{v_v} - {v_\ell }}}} \right)} \right] \qquad\qquad(11)

where the terms in the parentheses are the correction on latent heat required for this specific process. This correction is necessary in order to adjust the internal energy and maintain constant pressure and temperature in the phase change process. It should be pointed out, however, that this correction is usually very insignificant except near the critical point. Therefore, eq. 1 is usually valid for constant-pressure liquid-vapor phase change processes. During melting and solidification processes, heat transfer can be expressed as

q = \dot m{h_{s\ell }} \qquad\qquad(12)

where {h_{s\ell }} is the latent heat of fusion. Since the change of volume during solid-liquid phase change is insignificant, a correction similar to that in eq. (12) usually is not necessary.


Bejan, A., 1993, Heat Transfer, 2nd edition, John Wiley & Sons, New York.

Faghri, A., and Zhang, Y., 2006, Transport Phenomena in Multiphase Systems, Elsevier, Burlington, MA

Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.

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