4.3: Injani za joto

Malengo ya kujifunza

Mwishoni mwa sehemu hii, utaweza:

• Eleza kazi na vipengele vya inji ya joto
• Eleza ufanisi wa inji
• Tumia ufanisi wa inji kwa mzunguko uliopewa wa gesi bora

Injini ya joto ni kifaa kinachotumiwa kutolea joto kutoka chanzo halafu kukibadilisha kuwa kazi ya mitambo ambayo hutumiwa kwa kila aina ya maombi. Kwa mfano, inji ya mvuke kwenye treni ya zamani inaweza kuzalisha kazi inayohitajika kwa kuendesha treni. Maswali kadhaa yanajitokeza kutoka kwa ujenzi na matumizi ya inji za joto. Kwa mfano, ni asilimia gani ya juu ya joto iliyotolewa ambayo inaweza kutumika kufanya kazi? Hii inageuka kuwa swali ambalo linaweza tu kujibiwa kupitia sheria ya pili ya thermodynamics.

Sheria ya pili ya thermodynamics inaweza kutajwa rasmi kwa njia kadhaa. Taarifa moja iliyotolewa hadi sasa ni kuhusu mwelekeo wa mtiririko wa joto la kawaida, unaojulikana kama taarifa ya Clausius. Taarifa nyingine kadhaa zinategemea inji za joto. Wakati wowote tunazingatia inji za joto na vifaa vinavyohusishwa kama vile friji na pampu za joto, hatutumii mkataba wa kawaida wa ishara kwa joto na kazi. Kwa urahisi, sisi kudhani kwamba alama\(Q_h, \, Q_c\), na W kuwakilisha tu kiasi cha joto kuhamishwa na kazi mikononi, bila kujali watoaji au kupokea ni. Ikiwa joto linaingia au kuacha mfumo na kazi imefanywa au kwa mfumo huonyeshwa na ishara sahihi mbele ya alama na kwa maelekezo ya mishale katika michoro.

Kielelezo\(\PageIndex{1}\): A schematic representation of a heat engine. Energy flows from the hot reservoir to the cold reservoir while doing work.

It turns out that we need more than one heat source/sink to construct a heat engine. We will come back to this point later in the chapter, when we compare different statements of the second law of thermodynamics. For the moment, we assume that a heat engine is constructed between a heat source (high-temperature reservoir or hot reservoir) and a heat sink (low-temperature reservoir or cold reservoir), represented schematically in Figure \(\PageIndex{1}\). The engine absorbs heat \(Q_h\) from a heat source (hot reservoir) of Kelvin temperature \(T_h\), uses some of that energy to produce useful work W, and then discards the remaining energy as heat \(Q_c\), into a heat sink (cold reservoir) of Kelvin temperature \(T_c\). Power plants and internal combustion engines are examples of heat engines. Power plants use steam produced at high temperature to drive electric generators, while exhausting heat to the atmosphere or a nearby body of water in the role of the heat sink. In an internal combustion engine, a hot gas-air mixture is used to push a piston, and heat is exhausted to the nearby atmosphere in a similar manner.

Figure \(\PageIndex{2}\): The heat exhausted from a nuclear power plant goes to the cooling towers, where it is released into the atmosphere.

Actual heat engines have many different designs. Examples include internal combustion engines, such as those used in most cars today, and external combustion engines, such as the steam engines used in old steam-engine trains. Figure \(\PageIndex{2}\) shows a photo of a nuclear power plant in operation. The atmosphere around the reactors acts as the cold reservoir, and the heat generated from the nuclear reaction provides the heat from the hot reservoir.

Heat engines operate by carrying a working substance through a cycle. In a steam power plant, the working substance is water, which starts as a liquid, becomes vaporized, is then used to drive a turbine, and is finally condensed back into the liquid state. As is the case for all working substances in cyclic processes, once the water returns to its initial state, it repeats the same sequence.

For now, we assume that the cycles of heat engines are reversible, so there is no energy loss to friction or other irreversible effects. Suppose that the engine of Figure \(\PageIndex{1}\) goes through one complete cycle and that \(Q_h\), \(Q_c\), and W represent the heats exchanged and the work done for that cycle. Since the initial and final states of the system are the same, \(\Delta E_{int} = 0\) for the cycle. We therefore have from the first law of thermodynamics,

\[\begin{align} W &= Q - \Delta E_{int} \\[4pt] &= (Q_h - Q_c) - 0, \label{eq1} \end{align} \]

so that

\[W = Q_h - Q_c.\label{eq2} \]

The most important measure of a heat engine is its efficiency (e), which is simply “what we get out” divided by “what we put in” during each cycle, as defined by

\[e = \dfrac{W_{out}}{Q_{in}}. \label{eq3}\]

With a heat engine working between two heat reservoirs, we get out \(W\) and put in \(Q_h\), so the efficiency of the engine is

\[\begin{align} e &= \dfrac{W}{Q_h} \\[4pt] &= 1 - \dfrac{Q_c}{Q_h}. \label{eq4} \end{align} \]

Here, we used Equation \ref{eq2}, in the final step of this expression for the efficiency.

Example \(\PageIndex{1}\): A Lawn Mower

A lawn mower is rated to have an efficiency of \(25\%\) and an average power of 3.00 kW. What are

1. the average work and
2. the minimum heat discharge into the air by the lawn mower in one minute of use?

Strategy

From the average power—that is, the rate of work production—we can figure out the work done in the given elapsed time. Then, from the efficiency given, we can figure out the minimum heat discharge \(Q_c = Q_h(1 - e)\) with \(Q_h = Q_c + W\).

Solution

1. The average work delivered by the lawn mower is \[\begin{align} W &= P\Delta t \\[4pt] &= 3.00 \times 10^3 \times 60 \times 1.00 \, J \\[4pt] &= 180 \, kJ.\end{align} \]
2. The minimum heat discharged into the air is given by \[\begin{align} Q_c &= Q_h(1 - e) \\[4pt] &= (Q_c + W)(1 - e), \end{align} \] which leads to \[\begin{align} Q_c &= W(1/e - 1) \\[4pt] &= 180 \times (1/0.25 - 1)kJ = 540 \, kJ. \end{align} \]

Significance

As the efficiency rises, the minimum heat discharged falls. This helps our environment and atmosphere by not having as much waste heat expelled.