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Chapter 20: Problem 40

A Human Engine. You decide to use your body as a Carnot heat cngine. The operating gas is in a tube with one cnd in your mouth (where the temperature is \(37,0^{-} \mathrm{C}\) ) and the other end at the surface of your skin, at \(30.0^{\circ} \mathrm{C}\). (a) What is the maximum efficiency of such a heat engine? Would it be a very useful engine? (b) Suppose you want to use this human engine to lift a \(2.50 \mathrm{~kg}\) box from the floor to a tabletop \(1.20 \mathrm{~m}\) above the floor. How much must you increase the gravitational potential energy, and how much heat input is needed to accomplish this? (c) If your favorite candy bar has 350 food calories ( 1 food calorie \(=4186 \mathrm{~J}\) ) and \(80 \%\) of the food energy goes into heat, how many of these candy bars must you eat to lift the box in this way?

Short Answer

Expert verified
The maximum efficiency of the Carnot engine is approximately 0.023. This is not a very useful engine since the efficiency is quite low. The gravitational potential energy needed is 29.4 J. The heat input required to lift the box is 1278.3 J. Approximately 0.087 candy bars need to be eaten to provide the required energy.

Step by step solution

01

Calculate the efficiency of the Carnot engine

Using the formula for the efficiency of a Carnot engine: \[E = 1 - \frac{T_c}{T_h} \] where \(T_c\) is the temperature of the cold reservoir and \(T_h\) is the temperature of the hot reservoir. The temperatures must be in kelvin. Convert \(30.0^{\circ} C\) and \(37.0^{\circ} C\) to kelvin by adding 273. Then substitute \(T_c\) = 303 K and \(T_h\) = 310 K.
02

Calculate the gravitational potential energy

The formula for gravitational potential energy is given by: \[PE = mgh\] where \(m\) is the mass of the object, \(g\) is the acceleration due to gravity, and \(h\) is the height. The values in this case are \(m = 2.50 kg\), \(g = 9.8 m/s^2\) and \(h = 1.20 m\). Substituting these values into the formula gives the potential energy.
03

Calculate the heat input needed

To find the heat input required, divide the potential energy by the efficiency of the engine that was calculated in step 1.
04

Calculate the number of candy bars needed

To find the number of candy bars needed, first find the total energy in one candy bar in joules by multiplying the calories by \(4186 J\). Multiply this result by 0.8 because only 80% goes into heat. Then divide the heat input required (found in step 3) by the energy per candy bar to find the number of candies needed.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Efficiency of a Carnot Engine
The Carnot engine is known for having the maximum theoretical efficiency among heat engines. This efficiency measures how well the engine converts heat energy into work. It relies on the temperatures of the two thermal reservoirs it operates between: the hot reservoir (in this case, your mouth) and the cold reservoir (the surface of your skin).

To calculate efficiency, we use the formula:
  • \[ E = 1 - \frac{T_c}{T_h} \]where:
    • \(T_h\) is the temperature of the hot reservoir in Kelvin.
    • \(T_c\) is the temperature of the cold reservoir in Kelvin.
    Converting Celsius to Kelvin simply requires adding 273 to your Celsius measurement. In this problem, \(T_h = 310\, K\) and \(T_c = 303\, K\).
This results in a very small efficiency, indicating that as a human-engine, it isn't highly productive. Constructed engines have higher efficiencies partly because they often operate over larger temperature differences.
Gravitational Potential Energy
Gravitational potential energy (GPE) is the energy an object possesses because of its position in a gravitational field. To elevate an object means to increase its potential energy. The formula is straightforward:
  • \[ PE = mgh \]where:
    • \(m\) is the mass in kilograms.
    • \(g\) is the acceleration due to gravity, roughly \(9.8\, m/s^2\).
    • \(h\) is the height in meters.
For example, lifting a box weighing 2.50 kg to a height of 1.20 m requires calculating \(PE = 2.50 \times 9.8 \times 1.20\). Thus, the energy necessary for this task is a result of these three factors: mass, gravity, and height. Raising the mass's height increases its potential energy.
Heat Input Required
Heat input is crucial for understanding how much energy needs to be transferred into a system to perform work. After determining the energy required to lift the box, we find that the actual heat input must account for the inefficiency of the engine.

Since the engine's efficiency tells us the proportion of heat converted to work, we use:
  • \[ Q_{in} = \frac{PE}{E} \]where:
    • \(Q_{in}\) is the heat input.
    • \(PE\) is the potential energy calculated earlier.
    • \(E\) is the engine's efficiency calculated from the Carnot formula.
To find the number of candy bars needed, we consider the energy content of a candy bar since only a fraction (i.e., 80%) of it is usable as heat. Translating food calories into joules by recalling that 1 food calorie equals 4186 J helps determine total energy input per candy bar, thus allowing a comparison to the required heat input.

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Most popular questions from this chapter

Human Entropy. A person with skin of surface area \(1.85 \mathrm{~m}^{2}\) and temperature \(30.0^{\circ} \mathrm{C}\) is resting in an insulated room where the ambient air temperature is \(20.0^{\circ} \mathrm{C}\). Assume that this person gcts rid of excess heat by radiation only. By how much does the person change the entropy of the air in this room cach second? (Recall that the room radiates back into the person and that the emissivity of the skin is \(1.00 .\) )

A typical coal-fired power plant generates \(1000 \mathrm{MW}\) of usable power at an overall thermal cfficicncy of \(40 \%\). (a) What is the rate of heat input to the plant? (b) The plant burns anthracite coal, which has a heat of combustion of \(2.65 \times 10^{7} \mathrm{~J} / \mathrm{kg} .\) How much coal does the plant use per day, if it operates continuously? (c) At what rate is heat ejected into the cool reservoir, which is the nearby river? (d) The river is at \(18.0^{\circ} \mathrm{C}\) before it reaches the power plant and \(18.5^{\circ} \mathrm{C}\) after it has received the plant's waste heat. Calculate the river's flow rate, in cubic meters per second. (e) By how much does the river's entropy increase each second?

A Carnot engine operates between two heat reservoirs at temperatures \(T_{\mathrm{H}}\) and \(T_{\mathrm{C}} .\) An inventor proposes to increase the efficiency by running one engine between \(T_{\mathrm{H}}\) and an intermediate temperature \(T^{\prime}\) and a second engine between \(T^{\prime}\) and \(T_{C}\), using as input the heat expelled by the first engine. Compute the efficiency of this composite system. and compare it to that of the original engine.

For a refrigerator or air conditioner, the coefficient of performance \(K\) (often denoted as \(\mathrm{COP}\) ) is, as in Eq. (20.9) , the ratio of cooling output \(\left|Q_{\mathrm{C}}\right|\) to the required electrical energy input \(|W|,\) both in joules. The cocfficicnt of performance is also expressed as a ratio of powers, $$ K=\frac{\left|Q_{\mathrm{c}}\right| / t}{|W| / t} $$ where \(\left|Q_{\mathrm{C}}\right| / t\) is the cooling power and \(|W| / t\) is the electrical power input to the device, both in watts. The energy efficiency ratio (FER) is the same quantity expressed in units of Btu for \(\left|Q_{\mathrm{C}}\right|\) and \(\mathrm{W} \cdot \mathrm{h}\) for \(|W|\) (a) Derive a general relationship that expresses EER in terms of \(K\). (b) For a home air conditioner, FER is generally determined for a \(95^{\circ} \mathrm{F}\) outside temperature and an \(80^{\circ} \mathrm{F}\) return air temperature. Calculate EER for a Carnot device that operates between \(95^{\circ} \mathrm{F}\) and \(80 \mathrm{~F}\). (c) You have an air conditioner with an EER of 10.9 . Your home on average requires a total cooling output of \(\left|Q_{\mathrm{C}}\right|=1.9 \times 10^{10} \mathrm{~J}\) per year. If electricity costs you 15.3 cents per \(\mathrm{kW} \cdot \mathrm{h}\), how much do you spend per year, on average. to operate your air conditioner? (Assume that the unit's EER accurately represcnts the operation of your air conditioner. A seasonal energy efficiency ratio (SFER) is often used. The SFFR is calculated over a range of outside temperatures to get a more accurate seasonal average.) (d) You are considering replacing your air conditioner with a more efficient one with an EER of \(14.6 .\) Based on the EER, how much would that save you on electricity costs in an average year?

As a budding mechanical engineer, you are called upon to design a Carnot engine that has \(2.00 \mathrm{~mol}\) of a monatomic ideal gas as its working substance and operates from a high-temperature reservoir at \(500^{\circ} \mathrm{C}\). The engine is to lift a \(15.0 \mathrm{~kg}\) weight \(2.00 \mathrm{~m}\) per cycle, using \(500 \mathrm{~J}\) of heat input. The gas in the cngine chamber can have a minimum volume of \(5.00 \mathrm{~L}\) during the cycle. (a) Draw a \(p V\) -diagram for this cycle. Show in your diagram where heat enters and leaves the gas. (b) What must be the temperature of the cold reservoir? (c) What is the thermal efficicncy of the cngine? (d) How much heat cnergy does this engine waste per cycle? (e) What is the maximum pressure that the gas chamher will have to withstand?
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