Phase change

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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}}
Eq. 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 }
Eq. 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). Figure 1.3 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
Eq. 3

The total volume of the liquid and vapor

V = {m_\ell }{v_\ell } + {m_v}{v_v}
Eq. 4

remains constant during the phase change process. Thus

\frac{{d{m_\ell }}}{{dt}}{v_\ell } + \frac{{d{m_v}}}{{dt}}{v_v} = 0
Eq. 5

Combining eqs. (3) and (5) yields

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

The first law of thermodynamics for the control volume (which will be presented in detail in Chapter 3) is

\frac{{d{E_{cv}}}}{{dt}} = q - \dot m{h_v}
Eq. 8

where the internal energy of the control volume is

{E_{cv}} = {m_\ell }{e_\ell } + {m_v}{e_v}
Eq. 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)
Eq. 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]
Eq. 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 }}
Eq. 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.