In thermodynamics, the Clausius inequality is the name given to the inequality: [1]

where the circle integral symbol means that we integrate the surface of the working substance over the entire cycle, dQ is an inexact differential amount of heat, often signified as đQ or δQ, added to the body of substance, T is the absolute temperature of the surface of the substance, the value of the integral equal zero for a reversible or ideal cycle, and is less than zero for an irreversible cycle. [2] The equation is named after German physicist Rudolf Clausius.

Etymology
The expression, according to the 1875 views of Clausius, is a mathematical way of stating that the sum of uncompensated transformations in a cyclical process must always be positive; and is thus an amendment of the proof of the second main principle (second law) for variations of the body that are irreversible. [3]

At the turn of the 20th century the expression “Clausius’ theorem” was common. One definitive 1904 version of Clausius’ theorem, from Irish mathematician William Orr, was: “if a body or system of bodies undergoes any irreversible cycle of operations in which, in addition to interchanges of heat which may take place between different parts of the system, it receives heat from (or gives heat to) any external bodies, then for the cycle (using the symbols H for heat and θ for temperature)", the quantity:

is negative, where “δH denotes the small quantity of heat absorbed by any small portion of the system when at the absolute temperature θ, whether this heat be supplied by conduction or by radiation, and whether by external bodies or by any other part of the system, the double sign of the integration being used to indicate that the integration is to be performed over the whole mass as throughout the cycle.”

Proof
Clausius' theorem (c.1854) can be proved on the basis of Carnot’s theorem (1824), which states that: “no engine operating between two heat reservoirs can be more efficient than a Carnot engine operating between the same reservoirs”. In equation form, no heat engine can be more efficient η than the efficiency ηrev of Carnot engine:

whose working substance can theoretically be reversed, with no net utilization of heat internally.Starting for his argument, with some substitution into the this expression, one can thus derive the expression for the Clausius theorem. [5]

References
1. Perrot, Pierre. (1998). A to Z of Thermodynamics. Oxford University Press.
2. (a) Dincer, Ibrahim, Rosen, Marc A. (2007). Exergy (section 1.3.7.: The Clausius inequality, pgs. 10-12). Elsevier.
(b) Cengel, Yunus A., and Turner, Robert A. (2004). Fundamentals of Thermal-Fluid Sciences (pg. 274). McGraw-Hill.
3. Clausius, Rudolf. (1879). The Mechanical Theory of Heat (ch. 10: On Non-Reversible Processes, pgs. 213-34). London: Macmillan & Co.
4. (a) Orr, William McFadden. (1904). “On Clausius’ Theorem for Irreversible Cycles, and on the Increase of Entropy” (section: Clausius’ Theorem: an Enunciation, pg. 519), Philosophical Magazine, Vol. 8, pgs. 509-27. Taylor & Francis.
(b) William McFadden Orr – Wikipedia.
5. Adkins, Clement J. (1983). Equilibrium Thermodynamics (section 5.1: Clausius’ theorem, pgs. 68-71). Cambridge University Press.

External links
● Clausius inequality – Wikipedia.
● Clausius theorem and inequality – HyperPhysics.