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How does the second law of thermodynamics explain why heat engines produce waste heat?

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Engraved portrait of Rudolf Julius Emanuel Clausius, a bearded man in formal attire with sideburns, facing forward. — thermodynamics Rudolf Clausius (1822 - 1888) is considered one of the central founders of the science of thermodynamics.

The second law of thermodynamics dictates that heat engines cannot convert all absorbed heat into work, necessitating the production of waste heat.

A heat engine operates by absorbing heat ( $Q_{1}$ ) from a high-temperature reservoir and exhausting a portion of that heat ( $Q_{2}$ ) to a lower-temperature environment to complete its cycle. According to the second law, the net entropy change of the system and its reservoirs must be greater than or equal to zero ( $Δ S \geq 0$ ). If an engine were to convert 100% of the absorbed heat into work ( $Q_{2} = 0$ ), the resulting entropy change would be negative, which is a violation of this fundamental law.123

Consequently, to satisfy the requirement that $Δ S \geq 0$ , a heat engine must exhaust a minimum amount of heat to the environment. This unavoidable discharge of energy is known as waste heat. While design improvements can reduce losses from factors like friction, no amount of engineering can eliminate waste heat entirely, as the limit on efficiency is a fundamental constraint of nature.124