4. Gas, vapor, heat, and mass

气体、蒸气、热量与质量

4.1 A pressure law is a thermodynamic assumption

压力定律是一项热力学假设

For fixed-mass ideal gas, an isothermal process gives $p_gV=\mathrm{constant}$, whereas a reversible adiabatic process gives $p_gV^\gamma=\mathrm{constant}$, with $\gamma=c_p/c_v$. A polytropic exponent $\kappa$ interpolates phenomenologically between some behaviors, but it is not automatically a material constant. Thermal gradients, finite-rate exchange, and phase change can make a single exponent inadequate. [S413, S409]

对固定质量理想气体，等温过程满足$p_gV=\mathrm{constant}$，可逆绝热过程则满足$p_gV^\gamma=\mathrm{constant}$，其中$\gamma=c_p/c_v$。多方指数$\kappa$可在某些情况下作为不同过程之间的现象学近似，但它并不自动成为材料常数。温度梯度、有限速率热交换和相变，都可能使单一指数失效。[S413, S409]

Under the ideal-gas, fixed-mass polytropic assumption only, $T_g/T_{g0}=(R_0/R)^{3(\kappa-1)}$. This formula is useful for understanding why compression heats gas. It is not permission to extrapolate to molecular radii and announce a measured collapse temperature. At extreme compression, the thermal model, equation of state, vapor chemistry, and liquid compressibility all deserve scrutiny.

仅在理想气体、固定质量、多方过程假设下，才有$T_g/T_{g0}=(R_0/R)^{3(\kappa-1)}$。这个公式有助于理解压缩为何使气体升温，却不能据此外推到分子尺度半径，并宣称得到了实验塌缩温度。极端压缩时，热模型、状态方程、蒸气化学和液体可压缩性都需要重新检查。

4.2 Vapor pressure and vapor mass are different closures

蒸气压与蒸气质量是不同闭合量

Setting $p_v=p_{\mathrm{sat}}(T_\infty)$ approximates a vapor component that remains near saturation at an effectively fixed interfacial temperature. It does not mean the vapor mass is constant. During growth, evaporation may add mass; during collapse, condensation may remove it. Conversely, assigning fixed vapor mass and compressing it as ideal gas is a different model. Never impose both assumptions on the same vapor component without reconciling them. [S419, S409]

令$p_v=p_{\mathrm{sat}}(T_\infty)$，相当于近似认为蒸气组分在有效恒定的界面温度下保持接近饱和状态。这不意味着蒸气质量恒定。膨胀时，蒸发可能增加质量；塌缩时，凝结可能移除质量。相反，将蒸气质量固定并按理想气体压缩，是另一个模型。不能不加协调地对同一蒸气组分同时施加这两种假设。[S419, S409]

A lumped gas energy model might start from $d(m_gu_g)/dt=-p_g\,dV/dt+\dot Q$ for a noncondensable fixed mass. An open vapor component adds enthalpy carried by interfacial mass transfer. A detailed model also solves temperature fields and couples saturation pressure to interface temperature. The choice is dictated by the needed observable, not by a preference for a more complicated equation.

对于固定质量不凝性气体，集中参数能量模型可从$d(m_gu_g)/dt=-p_g\,dV/dt+\dot Q$出发。开放的蒸气组分还需加入界面质量输运携带的焓。更详细的模型还要求解温度场，并将饱和蒸气压与界面温度耦合。模型选择应由目标可观测量决定，而不是因为更复杂的方程看起来更高级。

4.3 The Stefan condition and interface kinematics

Stefan条件与界面运动学

Let the unit normal point from the bubble into the liquid, and let $j>0$ denote mass flux from liquid to bubble, measured per unit interface area. Mass conservation gives $j=\rho_L(\dot R-u_L)$ and, for a pure vapor interior, $j=\rho_V(\dot R-u_V)$. Thus the familiar kinematic condition $u_L=\dot R$ is an approximation when $j/\rho_L$ is negligible.

令单位法向由气泡指向液体，并令$j>0$表示单位界面面积上从液体进入气泡的质量通量。质量守恒给出$j=\rho_L(\dot R-u_L)$；对于纯蒸气内部，还有$j=\rho_V(\dot R-u_V)$。因此，常用运动学条件$u_L=\dot R$实际上依赖于$j/\rho_L$可以忽略的近似。

Define $q_{L\to i}$ and $q_{V\to i}$ as conductive heat fluxes directed toward the interface from the liquid and vapor sides. Neglecting interfacial energy storage and kinetic corrections, the latent-heat balance is

将$q_{L\to i}$和$q_{V\to i}$分别定义为从液体侧和蒸气侧流向界面的导热通量。忽略界面储能与动能修正，潜热平衡为

The diagram locates the physical system; its key lists the symbols in the formula. Read the definition in the surrounding derivation when a letter has different roles.

图示标出物理体系，图例列出公式中的符号。若某字母有多种作用，应以邻近推导中的定义为准。

This sign convention makes the physics transparent: positive evaporation requires heat delivered to the interface. If you instead define heat flux along a common outward normal, one of the signs changes. Many apparent disagreements between published Stefan conditions are merely different normal conventions.

这一符号约定使物理意义十分清楚：正的蒸发通量需要热量输入界面。若改为沿同一个外法向定义热流，其中一项的符号会改变。文献中某些看似矛盾的Stefan条件，实际上只是法向约定不同。

4.4 Inertia-limited versus heat-limited growth

惯性控制与热控制的增长

With a sustained positive driving pressure and negligible losses, inertial growth has a characteristic speed of order $\sqrt{\Delta p/\rho}$, so radius can increase approximately linearly over a suitable interval. In a diffusion-controlled thermal regime, the available thermal penetration depth grows like $\sqrt{\alpha t}$; bubble growth can inherit a square-root time dependence under additional assumptions. These are different limits, and a single experiment can move between them. Laser-heated microabsorbers and delayed plasmonic nucleation provide examples where the heating history cannot be discarded. [S414, S415]

在持续正驱动压差且损失可忽略时，惯性增长的特征速度为$\sqrt{\Delta p/\rho}$量级，因此半径在适当时间段内可以近似线性增加。在热扩散控制的情形中，热渗透深度按$\sqrt{\alpha t}$增长；在附加假设下，气泡增长也可能表现为时间的平方根关系。这是两个不同极限，同一实验也可能在二者之间转换。激光加热微吸收体和延迟等离激元成核表明，加热历程不能被随意忽略。[S414, S415]

A Jakob-type number compares sensible heat available in superheated liquid with latent heat demand. Different conventions use different density factors. Always state the definition. For a simple mass-specific comparison, $Ja=c_{p,L}(T_\infty-T_{\mathrm{sat}})/L_v$; other bubble-growth formulations multiply this by $\rho_L/\rho_V$. Numbers quoted under different definitions must not be compared directly.

Jakob类无量纲数比较过热液体可提供的显热与相变所需潜热。不同定义可能包含不同密度因子，使用时必须写出具体定义。例如，简单的单位质量比较为$Ja=c_{p,L}(T_\infty-T_{\mathrm{sat}})/L_v$；其他气泡增长理论还会乘以$\rho_L/\rho_V$。不同定义下的数值不能直接比较。

4.5 Dissolved gas and shells

溶解气体与壳层

A dissolved gas concentration field obeys an advection–diffusion equation in the liquid. At the interface, an equilibrium approximation can use a clearly defined Henry relation, such as $C_s=H_{cp}p_g$ with $H_{cp}$ in concentration-per-pressure units. Other Henry constants are reciprocals or use mole fractions, so importing an unlabeled constant is dangerous. Diffusive gas exchange can control long-lived bubbles even when the first inertial collapse is much faster.

液体中的溶解气体浓度场满足对流—扩散方程。界面平衡近似可以采用明确定义的Henry关系，例如$C_s=H_{cp}p_g$，其中$H_{cp}$的单位为浓度除以压力。其他Henry常数可能是其倒数，或采用摩尔分数定义，因此直接导入未说明定义的常数很危险。即使第一次惯性塌缩极快，扩散气体交换仍可能主导长寿命气泡的后续行为。

For a coated bubble, write a general additional shell traction $\Pi_s(R,\dot R,\text{history})$ in the interfacial balance before choosing a constitutive law. A shell may buckle, rupture, or have surface viscosity. Simply assigning a different constant $\sigma$ cannot represent every shell effect. The formulation dependence in your perfluorocarbon papers is a reason to preserve this distinction. [S387, S349]

对于包覆气泡，应先在界面平衡中加入一般壳层牵引$\Pi_s(R,\dot R,\text{history})$，再选择具体本构关系。壳层可能发生屈曲、破裂或具有表面黏性。单纯更换一个恒定$\sigma$，不能表示所有壳层效应。你提供的全氟碳资料中存在明显配方依赖性，因此更应保留这一物理区别。[S387, S349]