《惠崇春江晚景二首》蘇軾

竹外桃花三兩枝，春江水暖鴨先知。
蔞蒿滿地蘆芽短，正是河豚欲上時。

兩兩歸鴻欲破群，依依還似北歸人。
遙知朔漠多風雪，更待江南半月春。

所謂人間『冷』『暖』，識得一個『熱』字也難！？

Heat

In thermodynamics, heat is energy transferred from one system to another as a result of thermal interactions.^[1] The amount of heat transferred in any process can be defined as the total amount of transferred energy excluding any macroscopic work that was done and any transfer of part of the object itself.^[2]^[3]^[4]^[5]^[6] When two systems with different temperatures are put in contact, heat flows spontaneously from the hotter to the colder system. Transfer of energy as heat can occur through direct contact, through a barrier that is impermeable to matter (as in conduction), by radiation between separated bodies, by way of an intermediate fluid (as in convective circulation), or by a combination of these.^[7]^[8]^[9] By contrast to work, heat involves the stochastic (random) motion of particles (such as atoms or molecules) that is equally distributed among all degrees of freedom, while work is confined to one or more specific degrees of freedom such as those of the center of mass.

Like thermodynamic work, heat is a property of a process, not a property of a system. Energy exchanged as heat (a process function) changes the internal energy (a state function) of each system by equal and opposite amounts. This is to be distinguished from the common conception of heat as a property of high-temperature systems.

Although heat flows spontaneously from a hotter body to a cooler one, it is possible to construct a heat pump or refrigeration system that does work to increase the difference in temperature between two systems. In contrast, a heat engine reduces an existing temperature difference to do work on another system.

As a form of energy, the SI unit of heat is the joule (J). The conventional symbol used to represent the amount of heat exchanged in a thermodynamic process is $Q$ . Heat is measured by its effect on the states of interacting bodies, for example, by the amount of ice melted or a change in temperature.^[10] The quantification of heat via the temperature change of a body is called calorimetry.

眾『理』尋它千百遍，是否已然了斷？！

Macroscopic view

According to Planck, there are three main conceptual approaches to heat.^[46] One is the microscopic or kinetic theory approach. The other two are macroscopic approaches. One is the approach through the law of conservation of energy taken as prior to thermodynamics, with a mechanical analysis of processes, for example in the work of Helmholtz. This mechanical view is taken in this article as currently customary for thermodynamic theory. The other macroscopic approach is the thermodynamic one, which admits heat as a primitive concept, which contributes, by scientific induction^[47] to knowledge of the law of conservation of energy. This view is widely taken as the practical one, quantity of heat being measured by calorimetry.

Bailyn also distinguishes the two macroscopic approaches as the mechanical and the thermodynamic.^[48] The thermodynamic view was taken by the founders of thermodynamics in the nineteenth century. It regards quantity of energy transferred as heat as a primitive concept coherent with a primitive concept of temperature, measured primarily by calorimetry. A calorimeter is a body in the surroundings of the system, with its own temperature and internal energy; when it is connected to the system by a path for heat transfer, changes in it measure heat transfer. The mechanical view was pioneered by Helmholtz and developed and used in the twentieth century, largely through the influence of Max Born.^[49] It regards quantity of heat transferred as heat as a derived concept, defined for closed systems as quantity of heat transferred by mechanisms other than work transfer, the latter being regarded as primitive for thermodynamics, defined by macroscopic mechanics. According to Born, the transfer of internal energy between open systems that accompanies transfer of matter “cannot be reduced to mechanics”.^[50] It follows that there is no well-founded definition of quantities of energy transferred as heat or as work associated with transfer of matter.

Nevertheless, for the thermodynamical description of non-equilibrium processes, it is desired to consider the effect of a temperature gradient established by the surroundings across the system of interest when there is no physical barrier or wall between system and surroundings, that is to say, when they are open with respect to one another. The impossibility of a mechanical definition in terms of work for this circumstance does not alter the physical fact that a temperature gradient causes a diffusive flux of internal energy, a process that, in the thermodynamic view, might be proposed as a candidate concept for transfer of energy as heat.

In this circumstance, it may be expected that there may also be active other drivers of diffusive flux of internal energy, such as gradient of chemical potential which drives transfer of matter, and gradient of electric potential which drives electric current and iontophoresis; such effects usually interact with diffusive flux of internal energy driven by temperature gradient, and such interactions are known as cross-effects.^[51]

If cross-effects that result in diffusive transfer of internal energy were also labeled as heat transfers, they would sometimes violate the rule that pure heat transfer occurs only down a temperature gradient, never up one. They would also contradict the principle that all heat transfer is of one and the same kind, a principle founded on the idea of heat conduction between closed systems. One might to try to think narrowly of heat flux driven purely by temperature gradient as a conceptual component of diffusive internal energy flux, in the thermodynamic view, the concept resting specifically on careful calculations based on detailed knowledge of the processes and being indirectly assessed. In these circumstances, if perchance it happens that no transfer of matter is actualized, and there are no cross-effects, then the thermodynamic concept and the mechanical concept coincide, as if one were dealing with closed systems. But when there is transfer of matter, the exact laws by which temperature gradient drives diffusive flux of internal energy, rather than being exactly knowable, mostly need to be assumed, and in many cases are practically unverifiable. Consequently, when there is transfer of matter, the calculation of the pure ‘heat flux’ component of the diffusive flux of internal energy rests on practically unverifiable assumptions.^[52]^{[quotations 1]}^[53] This is a reason to think of heat as a specialized concept that relates primarily and precisely to closed systems, and applicable only in a very restricted way to open systems.

In many writings in this context, the term “heat flux” is used when what is meant is therefore more accurately called diffusive flux of internal energy; such usage of the term “heat flux” is a residue of older and now obsolete language usage that allowed that a body may have a “heat content”.^[54]

Microscopic view

In the kinetic theory, heat is explained in terms of the microscopic motions and interactions of constituent particles, such as electrons, atoms, and molecules.^[55] The immediate meaning of the kinetic energy of the constituent particles is not as heat. It is as a component of internal energy. In microscopic terms, heat is a transfer quantity, and is described by a transport theory, not as steadily localized kinetic energy of particles. Heat transfer arises from temperature gradients or differences, through the diffuse exchange of microscopic kinetic and potential particle energy, by particle collisions and other interactions. An early and vague expression of this was made by Francis Bacon.^[56]^[57] Precise and detailed versions of it were developed in the nineteenth century.^[58]

In statistical mechanics, for a closed system (no transfer of matter), heat is the energy transfer associated with a disordered, microscopic action on the system, associated with jumps in occupation numbers of the energy levels of the system, without change in the values of the energy levels themselves.^[59] It is possible for macroscopic thermodynamic work to alter the occupation numbers without change in the values of the system energy levels themselves, but what distinguishes transfer as heat is that the transfer is entirely due to disordered, microscopic action, including radiative transfer. A mathematical definition can be formulated for small increments of quasi-static adiabatic work in terms of the statistical distribution of an ensemble of microstates.

Calorimetry

Quantity of heat transferred can be measured by calorimetry, or determined through calculations based on other quantities.

Calorimetry is the empirical basis of the idea of quantity of heat transferred in a process. The transferred heat is measured by changes in a body of known properties, for example, temperature rise, change in volume or length, or phase change, such as melting of ice.^[60]^[61]

A calculation of quantity of heat transferred can rely on a hypothetical quantity of energy transferred as adiabatic work and on the first law of thermodynamics. Such calculation is the primary approach of many theoretical studies of quantity of heat transferred.^[30]^[62]^[63]

Engineering

The discipline of heat transfer, typically considered an aspect of mechanical engineering and chemical engineering, deals with specific applied methods by which thermal energy in a system is generated, or converted, or transferred to another system. Although the definition of heat implicitly means the transfer of energy, the term heat transfer encompasses this traditional usage in many engineering disciplines and laymen language.

Heat transfer is generally described as including the mechanisms of heat conduction, heat convection, thermal radiation, but may include mass transfer and heat in processes of phase changes.

Convection may be described as the combined effects of conduction and fluid flow. From the thermodynamic point of view, heat flows into a fluid by diffusion to increase its energy, the fluid then transfers (advects) this increased internal energy (not heat) from one location to another, and this is then followed by a second thermal interaction which transfers heat to a second body or system, again by diffusion. This entire process is often regarded as an additional mechanism of heat transfer, although technically, “heat transfer” and thus heating and cooling occurs only on either end of such a conductive flow, but not as a result of flow. Thus, conduction can be said to “transfer” heat only as a net result of the process, but may not do so at every time within the complicated convective process.

A red-hot iron rod from which heat transfer to the surrounding environment will be primarily through radiation.

想那『焦耳』墓誌銘猶在，

一八八九年十月十一日，焦耳在塞爾的家中逝世，被埋葬在該市的布魯克蘭公墓。在他的墓碑上刻有數字『772.55』，這是他在一八七八年的關鍵測量中得到的熱功當量值。墓碑上還刻有約翰福音的一段話，『趁著白日，我們必須做那差我來者的工；黑夜將到，就沒有人能做工了。【9：4】』

─── 摘自《【SONIC Π】電聲學之電路學《一》中》

測量『熱功當量』從早到晚︰

Electrical resistance and conductance

The electrical resistance of an electrical conductor is a measure of the difficulty to pass an electric current through that conductor. The inverse quantity is electrical conductance, and is the ease with which an electric current passes. Electrical resistance shares some conceptual parallels with the notion of mechanical friction. The SI unit of electrical resistance is the ohm (Ω), while electrical conductance is measured in siemens (S).

An object of uniform cross section has a resistance proportional to its resistivity and length and inversely proportional to its cross-sectional area. All materials show some resistance, except for superconductors, which have a resistance of zero.

The resistance (R) of an object is defined as the ratio of voltage across it (V) to current through it (I), while the conductance (G) is the inverse:

$\displaystyle R={V \over I},\qquad G={I \over V}={\frac {1}{R}}$

For a wide variety of materials and conditions, V and I are directly proportional to each other, and therefore R and G are constant (although they can depend on other factors like temperature or strain). This proportionality is called Ohm’s law, and materials that satisfy it are called ohmic materials.

In other cases, such as a transformer, diode or battery, V and I are not directly proportional. The ratio V/I is sometimes still useful, and is referred to as a “chordal resistance” or “static resistance”,^[1]^[2] since it corresponds to the inverse slope of a chord between the origin and an I–V curve. In other situations, the derivative $\displaystyle {\frac {dV}{dI}}\,\!$ may be most useful; this is called the “differential resistance”.

……

Energy dissipation and Joule heating

Resistors (and other elements with resistance) oppose the flow of electric current; therefore, electrical energy is required to push current through the resistance. This electrical energy is dissipated, heating the resistor in the process. This is called Joule heating (after James Prescott Joule), also called ohmic heating or resistive heating.

The dissipation of electrical energy is often undesired, particularly in the case of transmission losses in power lines. High voltage transmission helps reduce the losses by reducing the current for a given power.

On the other hand, Joule heating is sometimes useful, for example in electric stoves and other electric heaters (also called resistive heaters). As another example, incandescent lamps rely on Joule heating: the filament is heated to such a high temperature that it glows “white hot” with thermal radiation (also called incandescence).

The formula for Joule heating is:

$\displaystyle P=I^{2}R$

where P is the power (energy per unit time) converted from electrical energy to thermal energy, R is the resistance, and I is the current through the resistor.

Running current through a material with high resistance creates heat, in a phenomenon called Joule heating. In this picture, a cartridge heater, warmed by Joule heating, is glowing red hot.

───

Joule heating

Joule heating, also known as Ohmic heating and resistive heating, is the process by which the passage of an electric current through a conductor produces heat.

Joule’s first law, also known as the Joule–Lenz law,^[1] states that the power of heating generated by an electrical conductor is proportional to the product of its resistance and the square of the current:

$\displaystyle P\propto I^{2}\cdot R$

Joule heating affects the whole electric conductor, unlike the Peltier effect which transfers heat from one electrical junction to another.

A coiled heating element from an electric toaster, showing red to yellowincandescence

History

James Prescott Joule first published in December 1840, an abstract in the Proceedings of the Royal Society, suggesting that heat could be generated by an electrical current. Joule immersed a length of wire in a fixed mass of water and measured the temperature rise due to a known current flowing through the wire for a 30 minute period. By varying the current and the length of the wire he deduced that the heat produced was proportional to the square of the current multiplied by the electrical resistance of the immersed wire.^[2]

In 1841 and 1842, subsequent experiments showed that the amount of heat generated was proportional to the chemical energy used in the voltaic pile that generated the current. This led Joule to reject the caloric theory (at that time the dominant theory) in favor of the mechanical theory of heat (according to which heat is another form of energy).^[2]

Resistive heating was independently studied by Heinrich Lenz in 1842.^[1]

The SI unit of energy was subsequently named the joule and given the symbol J. The commonly known unit of power, the watt, is equivalent to one joule per second.

其與『德汝德』實在無緣★☆

舉例來說『鈉』 $Na$ 很容易形成一價的『鈉離子』，就說它的 $Z_c = 1$ ，如此 $D_{eNa} = 6.02 \times {10}^{23} atoms/mole \frac {1 e / atom \cdot 0.968 \times {10}^6 g / m^3}{22.98 g/mole} = 2.54 \times {10}^{28} e / m^3$ ，這樣一克的鈉，體積大約一立方公分，就有『數量級』為 ${10}^{22}$ 個『自由電子』。

假使將它看成『自由電子氣體』，再利用奧地利物理學家路德維希‧愛德華‧波茲曼 Ludwig Eduard Boltzmann 所發展的古典氣體『運動理論』Kinetic theory 來探討這些『自由電子』，就如同理想氣體一樣，在『熱平衡』時，一個『自由電子』的『熱速度』 $v_{thermal}$ 可以用 $\frac {1} {2} m \cdot \overline{{v_{thermal}}^2} = \frac {3} {2} k_B T$ 來計算，此處 $k_B$ 是波茲曼常數 $k_b = 1.3806488(13) \times 10^{-23} \mbox{ JK}^{-1}$ ， $T$ 是『絕對溫標』。那麼室溫下 ${25}^{\circ} C = {298.16}^{\circ} K$ 的一個『自由電子』的『熱速度』大約是 $v_{thermal} = \sqrt{\frac {3 k_B T}{m}} = 1.16 \times {10}^5 m/s$ 。

統計力學拓荒者

這個速度一秒大於百公里，不可謂之不大，假使用『費米氣體』的量子統計力學來講，更要大上個十倍，不過由於它在『各方向』的『均等性』，因此統計上來說『淨電流』的貢獻為『零』。也就是說 $\langle \vec{v}_{thermal} \rangle = 0$ 。

那麼德汝德是如何看待這些『碰撞』作用的呢？或者說他做了哪些『假設』的呢？這點正是探討一個『物理模型』的『合理性』與『適切性』的重要之處。依據現今的說法，德汝德假設了︰

一、如果沒有外部的『電磁場』作用，『自由電子』將會作『直線運動』，彼此間的『電磁作用力』可以被忽略。這意味著是一種『獨立電子』的假設，它處於一個由『正離子』與『其他電子』所構成的『平均的環境』 ── 因此淨作用為零 ──，統計上來講這一般認為是『合宜的』。

二、『電子』和『正離子』之間的『碰撞』是『即時』的，統計上無關之『隨機事件』，所以總體來說這沒有任何『淨貢獻』，雖然有不同的學者『批評』它的『合宜性』。然而如果從『散射事件』來看，這也許只是說某些『物質屬性』之『均向性』的另一種說法罷了。

三、假設了『平均碰撞時間』 $\tau$ 的『存在』，所以我們可以說很小的一段時距 $\delta t$ 發生『碰撞』的『機會』是 $\frac {\delta t}{\tau}$ ，而且這個『機率』和一個『自由電子』的『位置』與『動量』無關。這正像是『丟一根』長度為 $\delta t$ 的『針』投到一個以 $\tau$ 為『格子線』板子上，問『針』掉到『線上』的『機率』大小如何，通常被認為是很好的『近似』。

四、『碰撞』後的『熱電子』應該保有該處『熱平衡』的速度。這是一個作用『鄰近原則』的假設，一般從『物理因果』上講，以為應是『正確的』。

那麼我們如何推導『自由電子』受到一個外在時變的『力場』 $\vec{F}(t)$ 中之『平均動量』方程式的呢？假使在 $t$ 時刻，一個『自由電子』的『動量』是 $\vec{p}(t)$ ，到了 $t + dt$ 時刻它的動量 $\vec{p}(t+dt)$ 可以這樣考慮，如果說這個『自由電子』發生了『碰撞』，按造『假設三』它的『碰撞』機率是 $P_c = \frac{dt}{\tau}$ ，再依據『假設二』，它的淨『平均動量』貢獻將會是『零』， $\langle {\vec{p}}_c(t+dt) \rangle = 0$ 。如果說此時這個『自由電子』沒有發生『碰撞』，於是按造『牛頓第二運動定律』 $\langle {\vec{p}}_{nc}(t+dt) \rangle = \langle \vec{p}(t) \rangle + \vec{F}(t)dt$ ，這個不發生『碰撞』的機率 $P_{nc}$ 是 $1 - P_c = 1 - \frac{dt}{\tau}$ ，因此

$\langle \vec{p}(t+dt) \rangle = P_c \cdot \langle {\vec{p}}_c(t+dt) \rangle + P_{nc} \cdot \langle {\vec{p}}_{nc}(t+dt) \rangle$

$= \left( 1 - \frac{dt}{\tau} \right) \left( \langle \vec{p}(t) \rangle + \vec{F}(t)dt \right)$ ，所以可得

$\frac{d \langle \vec{p}(t) \rangle}{dt} = \frac{\langle \vec{p}(t+dt) \rangle - \langle \vec{p}(t) \rangle}{dt} = - \frac{\langle \vec{p}(t) \rangle}{\tau} + \vec{F}(t)$ 。

這就是德汝德模型之電子的運動方程式。首先我們考慮一些典型的『時變力場』 $\vec{F}(t)$ 情況︰

一、沒有外力存在 $\vec{F}(t) = 0$ 時， $\langle \vec{p}(t) \rangle = \langle \vec{p}(0) \rangle \cdot e^{- \frac {t}{\tau}}$ ，這說明了『弛豫時間』 Relaxation Time $\tau$ 的物理意義，每經過 $\tau$ 時距，『平均動量』以 $e^{-1}$ 為比率『衰減』。事實上，電子的運動方程式中的 $- \frac{\langle \vec{p}(t) \rangle}{\tau}$ 項就是一種『阻力』的啊！

二、常量不隨時變的外力 $\vec{F}$ 時， $\langle \vec{p}(t) \rangle = \langle \vec{p}(0) \rangle \cdot e^{- \frac {t}{\tau}} + \vec{F} \cdot \tau$ 。當 $t >> \tau$ 時，『暫態解』可以被忽略，這時 $\langle \vec{p}(t) \rangle = \vec{F} \cdot \tau$ 。假使將此應用於『導體』中的電子在一個『均勻恆定的電場』 $\vec{E}$ 情況下，這時 $\langle \vec{p}(t) \rangle = e \cdot \vec{E} \cdot \tau$ ，由於

$\langle \vec{p}(t) \rangle = m \cdot \langle \vec{v}(t) \rangle$ ，

$\vec{J} = D_e \cdot e \langle \vec{v}(t) \rangle$

，於是就得到了 $\vec{J} = \left( \frac{D_e e^2 \tau}{m} \right) \vec{E}$ ，也就是說電流密度 $\vec{J}$ 與電場 $\vec{E}$ 成正比，這就是『歐姆定律』的『微觀表述』。通常人們將『電場』作用下的電子『平均速度』稱作『漂移速度』 Drift Velocity，那麼這個『漂移速度』有多大的呢？如果考慮一根直徑一公釐的銅線，因為銅的密度是每立方公分 8.94 克，它的莫爾原子量是 63.546 克，所以假設每個銅原子貢獻一個自由電子，那麼一立方公尺的銅，就有 $8.5 * 10^{28}$ 個自由電子。假使這根銅線上流過 3 A 安培的電流，『漂移速度』可以用下式來計算

$v = {I \over nAq}$

$v = \frac{3}{\left(8.5 \times 10^{28}\right) \left(7.85 \times 10^{-7}\right) \left(-1.6 \times 10^{-19}\right)}$

$v = -0.00028$

上式中，I 是電流量，n 是電流密度，A 是銅線的截面積，q 是電子的電荷量，因次分析的結果是︰

$v = \dfrac{\text{A}}{\dfrac{\text{electron}}{\text{m}^3}{\cdot}\text{m}^2\cdot\dfrac{\text{C}}{\text{electron}}} = \dfrac{\text{C}}{\text{s}{\cdot}\dfrac{1}{\text{m}}{\cdot}\text{C}} = \dfrac{\text{m}}{\text{s}}$

，由此可知『電子』在電場中的『漂移速度』如果和『熱速度』作比較其實是非常的小。那麼『電流』的『速度』到底有多快的呢？『電子』果真會從『電力公司』長途跑到『你家裡』的嗎？？

─── 摘自《【SONIC Π】電聲學補充《三》上》

FreeSandal

每日彙整: 2018-07-08

STEM 隨筆︰古典力學︰轉子【五】《電路學》三【電阻】I