Evaluation of the transient thermal state of thick-walled pressure elements of power engineering machinery and equipment is addressed in Sect. 6.1. Methods of evaluation of the transient thermal statfe of cylindrical elements are discussed in Sect. 6.1.1. First, simple and quick methods for calculating thermal stresses are presented, but the least accurate. The first of these methods is a determination of thermal stresses based on the characteristic difference in temperature inside the element. In the second method, a quasi-steady temperature field is assumed in the pressure component. A more complex and accurate power series method is then presented. When using this method, the time derivatives of the measured temperature should be accurately determined, such as using moving digital filters. Methods based on the FEM in combination with the superposition method or Duhamel's integral can be used to determine thermal stresses in the pressure components of complex shapes. However, this approach can only be applied to linear problems. The optimum heating and cooling of thick-walled pressure elements is outlined in Sect. 6.4. Optimum heating and cooling of the component, assuming a quasi-steady temperature field, is analysed at first. Then, the optimum heating and cooling of a pressure component with fluid at a saturation temperature is presented. The determination of optimum changes in the fluid temperature during the heating and cooling of the boiler drum is analysed in detail in Sect. 6.4.