Abstract: In regulating the flow of cooling water sprayed upon a metallurgical product such as a steel slab in a machine for the continuous casting of such products which then undergo a straightening operation, the present, past and future speeds of the product are taken into account so as to compensate a projected change in the temperature of the product in the straightening stage, due to a planned or expected modification of the cooling conditions. These conditions are established by a regulating system responsive to changes in the speed of the product; the anticipated temperature change is compensated by substituting for the true speed, as a controlling parameter, a fictitious speed lying between the true speed and an advance image of an anticipated speed change whose effects upon the temperature are to be neutralized.