Suppose that a deep shaft were drilled in Earth's crust near...

Suppose that a deep shaft were drilled in Earth's crust near one of the poles, where the surface temperature is $-40^{\circ} \mathrm{C},$ to a depth where the temperature is $800^{\circ} \mathrm{C}$. (a) What is the theoretical limit to the efficiency of an engine operating between these temperatures? (b) If all the energy released as heat into the low-temperature reservoir were used to melt ice that was initially at $-40^{\circ} \mathrm{C},$ at what rate could liquid water at $0^{\circ} \mathrm{C}$ be produced by a $100 \mathrm{MW}$ power plant (treat it as an engine)? The specific heat of ice is $2220 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}$ water's heat of fusion is $333 \mathrm{~kJ} / \mathrm{kg}$. (Note that the engine can operate only between $0^{\circ} \mathrm{C}$ and $800^{\circ} \mathrm{C}$ in this case. Energy exhausted at $-40^{\circ} \mathrm{C}$ cannot warm anything above $-40^{\circ} \mathrm{C}$.)

Key Concepts

Carnot Efficiency

Carnot efficiency represents the theoretical maximum efficiency that a heat engine can achieve when operating between two thermal reservoirs. It is determined by the temperatures of the hot and cold reservoirs (expressed in absolute temperature scales such as Kelvin) and is calculated as 1 minus the ratio of the cold temperature to the hot temperature. This concept is central to understanding the limits of energy conversion devices and sets an upper bound on how much work can be extracted from a given thermal gradient.

Heat Engine Operation

A heat engine is a device that converts heat energy into work by exploiting the temperature difference between a high-temperature source and a lower-temperature sink. The performance of a heat engine is fundamentally limited by the second law of thermodynamics, and its theoretical maximum efficiency is described by the Carnot efficiency. Understanding how heat flows through such a system is key to analyzing energy conversion processes in various applications.

Temperature Scale Conversion

In thermodynamics, temperatures must be expressed in absolute units, typically Kelvin, to correctly calculate efficiencies and other performance metrics. Converting temperatures from Celsius (or other scales) to Kelvin is crucial because the Carnot efficiency formula depends on the absolute temperature values, ensuring that the relationships derived are physically meaningful in a universal context.

Specific Heat Capacity

Specific heat capacity is a property of a substance that defines the amount of energy required to raise the temperature of a unit mass of that substance by one degree (in a given unit of temperature). In the context of energy conversion and energy balance calculations, it is essential to determine how much energy is needed to change the temperature of materials, such as heating ice from below its melting point or warming water.

Latent Heat of Fusion

Latent heat of fusion refers to the amount of energy needed to convert a unit mass of a substance from the solid phase to the liquid phase at its melting point without changing its temperature. This concept is important in thermodynamics when analyzing phase change processes, where energy is absorbed or released during melting or freezing. It plays a significant role in energy management applications where phase transitions are involved.

Transcript

00:01 For this problem on the topic of entropy, we are to suppose that a deep shaft were to be drilled in the earth's crust near the poles where the temperature at surface is minus 40 degrees celsius to a depth with the temperatures 800 degrees celsius.

00:13 We want to find the limit to the efficiency of an engine operating between these temperatures.

00:19 And then if all the energy released as heat into the low temperature reservoir used to melt ice that was initially at minus 40 degrees celsius, we want to know the rate at which liquid water could be used.

00:30 Produced at 0 degrees celsius by a 100 megawatt power plant.

00:39 Now if th is the temperature of the high temperature reservoir and tl the temperature of the low temperature reservoir, then the maximum efficiency of the engine we know is epsilon defined by th minus tl all over th, which is the high temperature 800 plus the low temperature 40 calvin divided by 800 plus 200 plus 273 calvin which gives an efficiency for the engine of 0 .78 or 78 percent now for part b the efficiency calculated above, epsilon is defined by the magnitude of the work over the magnitude of qh.

01:50 So w is the work done by the engine and qh is the heat input.

01:54 And w here is positive.

01:56 So over a complete cycle, we have the heat input equal to the work done w plus the magnitude of ql, where qr is the heat output.

02:09 And so we get epsilon to be w divided by w plus the magnitude of the heat output ql.

02:24 And this means that the heat output ql has a magnitude of the work done w into 1 over epsilon minus 1.

02:39 And now epsilon is t h minus t l over t h where t h is the temperature of the high temperature heat reservoir and t l the temperature of the low temperature reservoir which means therefore that one minus the efficiency epsilon one over the efficiency epsilon minus one rather is equal to t l over t h minus t l and q l is equal to w .t .l.

03:14 Over th minus tl.

03:19 Now the heat output is used to melt ice at temperature ti, which is minus 40 degrees celsius...

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