Thermal crown of the work rolls is known as one the effective parameters changing the unloaded roll gap profile and hence the thickness of products. Estimating the thermal crown of the work roller and other factors that cause to create the roll crown in addition to other effects such as bending , wear and initial crown will determine the loaded roll gap profile and hence the strip profile. This thesis presents the mathematical formula for the developed thermal crown in flat rolling of strip. Considering the heat transfer to the roller at the contact point with the work-piece and also the heat generated due to the plastic work and radiation and the heat lost from splashing water via nozzle and the transmission of heat to the environment, the temperature of the work roll at any point can be predicted dynamically at any instance. The differential equation for the conservation of thermal energy for incompressible constant-property substance in 3D situation is the main equation considered to be solved in this problem. The equation was simplified considering the work rolls conditions such as not moving in radial direction, roll shifting does not affect the thermal energy traort and the heat conduction in tangential direction. The main element of energy traort in this direction is considered to be convection. Hence some of the terms are omitted making the main equation as simple as possible. To complete the formulation of the model, the boundary conditions are considered as follows: The arc of contact at the interface of the roll and plate is considered to be a circular arc, the radius of which is calculated using the Hitchcock formula. The roll shifting is also included in the thermal model by considering different unloaded part for the rolls, where in general, not equal. The heat transfer coefficients between the rollers and the plate were chosen according the relevant references. Dissipated heat due to friction was also included in the model. In the present model, according to data that taken from Mobarakeh Steel Complex, the friction factor considered to be 0.255. The interface between backup and work rolls is treated similar to the roll-plate arc of contact, except that the frictional heating is assumed to be negligible. The temperature gradient is assumed zero at the roll centerline. Hence the Heat flux at the interface of water jet and the roll surface may be computed. The remaining surface of the work roll is either covered partially by water or is in contact with the surrounding air. The differential equation of the work roll is solved by the control-volume-based finite- difference method. The mesh, which is nonuniform in r and uniform in z direction, is generated by an algebraic method. A typical control volume is considered and the differential equation for the work roll, is integrated over the control volume with respect to space and time coordinates. Substitution of integrated terms in the differential equation and simplifying it can be transform to a simple algebraic equation. Solution of these equations provides temperature of the roll from which thermal expansion and thermal crown can be obtained. According to the mathematical model and the computer program, it is shown that the work roll temperature and hence the thermal crown can accurately be predicted. The results of this program with those of the other available resources are in good agreements. This program fast enough so that it can be utilized for online predictions of thermal crown in a hot strip mill. The heat transfer in hot rolling is considered to be one of the effective parameters in strip quality. Variations in roll temperature can cause obvious defect on strip crown and flatness. Some of the published works cannot be utilized in hot rolling of strip due to their disability of such model to control the roll gap on line. In this thesis, an attempt was made to simplify the mathematical model for reducing the computational time so that it can be used online. Key words Thermal Crown, Hot Rolling, Online Prediction, Index Unit