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# 物理代写|热力学代写Thermodynamics代考|ENME485 Joule Heating: Kirchhoff’s Principle

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## 物理代写|热力学代写Thermodynamics代考|Joule Heating: Kirchhoff’s Principle

The contribution of Joule heating to the internal entropy production density (we refer to Chap. 3 of $[6]$ and to $[8,17,18]$ is:
$$\sigma=\frac{\mathbf{j}_{e l} \cdot(\mathbf{E}+\mathbf{v} \wedge \mathbf{B})}{T}$$

The quantity $\mathbf{E}+\mathbf{v} \wedge \mathbf{B}$ is the electric field in the reference frame which moves at velocity $\mathbf{v}$. In steady state, this field is $=-\nabla \phi_{e l}, \phi_{e l}$ electrostatic potential. Moreover, let us introduce the electric charge density $\rho_{e l}$; the electric charge balance reads, therefore
$$\nabla \cdot \mathbf{j}{e l}+\frac{\partial \rho{e l}}{\partial t}=0$$
Finally, the relationship between $\mathbf{j}{e l}$ and $\mathbf{E}+\mathbf{v} \wedge \mathbf{B}$ (‘Ohm’s law’) is often linear: $$\mathbf{j}{e l}=\sigma_{\Omega}(\mathbf{E}+\mathbf{v} \wedge \mathbf{B})$$
where $\sigma_{\Omega}$ is the electrical conductivity. ${ }^{12}$ It follows that the Joule heating power density is $T \sigma=\frac{\left|\mathbf{j}e\right|^2}{\sigma{\Omega}}$. Joule heating is an irreversible process, then $\sigma>0$ and $\sigma_{\Omega}>0$.

If the electric charge carriers (usually, electrons) are acted upon by a local electric field $\mathbf{E}+\mathbf{v} \wedge \mathbf{B}$, then in steady state their acceleration is damped by collisions at frequency, say, $\nu_{\text {coll }}$. If $\mathbf{v}{e l}, m{e l}, n_{e l}$ and $q_{e l}$ are the velocity, the mass, the particle density and the electric charge of charge carriers, respectively, then the definition of $\mathbf{j}{e l}$ and compensation between the accelerating and the damping force ${ }^{13}$ lead to: $$\mathbf{j}{e l}=n_{e l} q_{e l} \mathbf{v}{e l} \quad ; \quad q{e l}(\mathbf{E}+\mathbf{v} \wedge \mathbf{B})=m_{e l} \nu_{c o l l} \mathbf{v}{e l}$$ Dot multiplication of both sides of the latter equation by $\frac{n{e l} q_{e l}}{\nu_{c o l l}}$ leads to:
$$\sigma_{\Omega}=\frac{n_{e l} q_{e l}^2}{m_{e l} \nu_{\text {coll }}}$$
As for LNET, two cases are possible.

## 物理代写|热力学代写Thermodynamics代考|Electric Arc

In particular, radiative transport may flatten $\nabla T$ in the bulk of radiation-cooled, freeflowing electric arcs. By ‘free-flowing’ we mean that the arc is in contact with no solid walls but the electrodes, so that the exchange of matter between the arc bulk and the external world is reduced. If the external world supplies the arc with a constant electric current $i_{\text {arc }}$ and a voltage drop $v_{\text {arc }}$ is obtained, then Kirchhoff’s principle reduces to minimization of Joule power $i_{\text {arc }} \cdot v$ at constant $i_{\text {arc }}$, i.e. to minimization of $v_{\text {arc }}$ at constant $i_{a r c}$. Such minimization has been proposed by Steenbeck in order to explain experimental observations [22]. A discussion of the relationship between Steenbeck’s result and MinEP is due to Peters [23]. Our discussion shows that $v_{\text {arc }}$ is quite insensitive to arc temperature. An independent investigation of SF6 arcs confirms this result-see Fig. 8 and Sect. V of [24]. For further discussion, see Sects. $5.6 .5$ and $6.2 .8$.

## 物理代写|热力学代写Thermodynamics代考|Joule Heating: Kirchhoff’s Principle

$$\sigma=\frac{\mathbf{j}{e l} \cdot(\mathbf{E}+\mathbf{v} \wedge \mathbf{B})}{T}$$ 数量 $\mathbf{E}+\mathbf{v} \wedge \mathbf{B}$ 是参考系中以速度运动的电场 $\mathbf{v}$. 在稳定状态下，该场为 $=-\nabla \phi{e l}, \phi_{e l}$ 静电势。此外，让我们介绍电荷密度 $\rho_{e l}$; 电荷平䔤卖数，因此
$$\nabla \cdot \mathbf{j} e l+\frac{\partial \rho e l}{\partial t}=0$$

$$\mathbf{j} e l=\sigma_{\Omega}(\mathbf{E}+\mathbf{v} \wedge \mathbf{B})$$

$$\mathbf{j} e l=n_{e l} q_{e l} \mathbf{v e l} \quad ; \quad q e l(\mathbf{E}+\mathbf{v} \wedge \mathbf{B})=m_{e l} \nu_{\text {coll }} \mathbf{v} e l$$

$$\sigma_{\Omega}=\frac{n_{e l} q_{e l}^2}{m_{e l} \nu_{\mathrm{coll}}}$$

## MATLAB代写

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