Application of Heat Transfer Data to Arc Characteristics

The chief experimental results of a study of the electric gradient $E$ (v/cm) and current density $I$ (amp./${\mathrm{cm}}^2$) in the arc in various gases and pressures can be correlated by means of conduction-convection heat loss data from solid bodies in fluids, and lead to an explanation of the variation of $E$ and $I$ with current $i$ and pressure $p$ which is in good agreement with the measurements. For this purpose we neglect radiation and use an equation of the Nusselt type $\left(\frac{W}{k\Delta T\pi }\right)=f\left(\frac{D^3{\rho }^{2}g\Delta T{\beta }_0}{{\eta }^2}\right)=f\left(x\right),$ where $W=\mathrm{watts}$ per unit length of arc column; $k=\mathrm{conductivity}$; $D=\mathrm{diameter}$; $\rho =\mathrm{density}$; ${\beta }_0=\mathrm{expansion}$ coefficient; and $\eta =\mathrm{viscosity}$. In the range of interest in arcs $f\left(x\right)=x^{\alpha }$, where $\alpha =0.1\mathrm{}$ is a satisfactory approximation. On the assumption that (1) the heat loss from the arc column is given by an equation of the Nusselt type and (2) the variation of arc temperature with $i$ is a small effect, the $E-i$ relation at constant $p$ is found to be $E=\mathrm{const}.\mathrm{}$ $i^{-n}$ where $n=\frac{(2-3\mathrm{}\alpha )}{(2+3\mathrm{}\alpha )}$. The exponent $n_{\exp }$ for nitrogen =0.6, while the $n_{\mathrm{calc}}=0.74\mathrm{}$. When $p$ varies, the arc temperature variation becomes important and the effect is included implicitly in the theory, leading to $E=\mathrm{const}.\mathrm{} p^m$, and $D=\mathrm{const}.\mathrm{} p^{-r}$. From heat transfer data and $N_2$ arc experiments we find that $m_{\exp }=0.32\mathrm{}$, $m_{\mathrm{calc}}=0.31\mathrm{}$, ${\gamma }_{\exp }=0.38\mathrm{}$, and ${\gamma }_{\mathrm{calc}}=0.28\mathrm{}$. For cases of forced convection we apply correlation data for forced convection cooling in the form $\frac{W}{\Delta \mathrm{Tk}\pi }=f\left(\frac{D{V}_{\rho }}{\eta }\right)={\left(\frac{D{V}_{\rho }}{\eta }\right)}^{\varphi }$, where $V$ is the fluid velocity. As above, this leads to the $E-i$ expression for an arc in forced convection $E=\mathrm{const}.\mathrm{} i^{-n^{\prime }}$ where $n^{\prime }=\frac{(2-\varphi )}{(2+\varphi )}$. The exponent ${n^{\prime }}_{\mathrm{calc}}=0.54\mathrm{}$, while ${n^{\prime }}_{\exp }$ is not known. For arcs in a variable gravity field $g$ (as in the experiments of Steenbeck), we find $E=\mathrm{const}.\mathrm{} g^q$ where $q=\frac{2\alpha }{(2+3\mathrm{}\alpha )}$. The $q_{\mathrm{calc}}=0.087\mathrm{}$, while $q_{\exp }$ is not known.