Abstract: Considering that the thermal-conductivity coefficient is a linear function of temperature, a temperature distribution equations is derived from the 1-D heat conduction equation to characterize the concave/convex form of a quadratic curve. A conclusion is thus drawn that the concave/convex form of the quadratic curve depends on the changing tendency of the temperature-dependent thermal-conductivity coefficient. Then, the transition relation between the surface temperature and mean temperature is provided to relation solve the problem that the result using the conventional quadratic equation in concave form to calculate the temperature distribution in thickness direction with approximate fitting technique is different from that using the same equation in convex form for thick hot-rolled strips during cooling control after rolling. Such a transition relation has been applied to the model of online coiling temperature control and proved available to improve its accuracy.