Average Shear Zone Temperature

Equation(s) have to be developed for the purpose.One simple method is presented here. The cutting energy per unit time ,i.e. , PzVc gets used to cause primary shear and to overcome friction at the rake face as :

$P_{Z} \cdot V_{C}=P_{S} \cdot V_{S}+F \cdot V_{f}$

where, $\mathrm{V}_{\mathrm{S}}$ = slip velocity along the shear plane

and $\mathrm{V}_{\mathrm{f}}$ = average chip - velocity

So,$P_{Z} \cdot V_{s}=P_{S} \cdot V_{c}-F \cdot V_{f}$

Equating amount of heat received by the chip in one minute from shear zone and the heat contained by that chip,it appears.

$\frac{A q_{1}\left(P_{z} \cdot V_{c}-F \cdot V_{t}\right)}{J}-c_{v} a_{1} b_{1} \vee_{c}\left(\theta_{s}-\theta_{a}\right)$

where, A = fraction (of shear energy that is converted into heat)

$q_1 $= fraction(of heat that goes to the chip from the shear zone)

J = mechanical equivalent of heat of the chip / work

$C_v$ = volume specific heat of the chip

$\theta_{a}$ = ambient temperature

$\mathrm{a}_{1} \mathrm{b}_{1}$ = cross sectional area of uncut chip

$=\mathrm{ts}_{0}$

Therefore, $\theta_s = \frac{Aq_1 (P_z . V_c \ - \ F. V_f)}{Jts_o \ V_c} + \theta_a$

or, $\theta_s = \frac{Aq_1 (P_z \ - \ F / \zeta)}{Jts_o}$

Generally, A varies from 0.95 to 1.0 and q from 0.7 to 0.9 in machining like turning.

Average chip - Tool Interface Temperature

Using the two dimensions parameters, $Q_1 \ and \ Q_2$ and their simple relation (Buckingham)

$Q_1 = C_1 . Q_2^n$

where, $Q_1 = (\frac{v_v \theta_1}{E_c}) \ and Q_2 = (\frac{V_c c_v a_1}{\lambda})^0.5$

$E_c$ = specific cutting energy.

$c_v$ = volume specific heat.

$\lambda$ = thermal conductivity.

$c_1$ = a constant.

n = an index close to 0.25.

therefore, $\theta_i = c_1 E_c \sqrt{V_c a_1 / \lambda c_v}$

using equation 2.7.6 one can estimate the approximate value of average $\theta_1$ from the known other machining parameters.