Question

The Newcomen engine was the first working steam engine with a piston; it was constructed by Thomas Newcomen through incremental improvements between 1705 and 1711 and was ultimately widely used. A large cylinder (e.g. radius 1 metre, height several metres) was closed by a piston attached to one end of a heavy balanced beam. As the beam rocked to and fro, the other end provided work, for example to drive a water pump. The working principle of the Newcomen engine is as follows. A coal fire heats a quantity of water in a boiler. During the down-stroke on the pump, which is the up-stroke on the piston, steam fills the cylinder, but no useful work is done because the steam pressure is roughly $1 \mathrm{~atm}$; the weight of the pump rod provides the force required to lift the piston. At the top of the piston stroke, a valve to the boiler is closed, and another valve to a supply of cold water under pressure is briefly opened. Some cold water sprays into the chamber. Since this water is much colder than the steam, heat flows from the latter to the former, and the steam condenses. The pressure in the chamber therefore drops considerably, 'sucking' the piston downwards-which is to say, the atmospheric pressure above the piston pushes it down. This is why such engines are called 'atmospheric' engines. This is the power stroke. When the piston reaches the bottom, the valve to the boiler is re-opened and the incoming steam pushes the cold water out of the chamber. (i) If the cylinder radius is $1 \mathrm{~m}$, and the working stroke of the piston is $3 \mathrm{~m}$, calculate the mass of steam required to fill the chamber at $100^{\circ} \mathrm{C}, 100 \mathrm{kPa}$. Determine the heat input required to produce this amount of steam from water initially at $15^{\circ} \mathrm{C}$. (ii) Determine the amount of water at $15^{\circ} \mathrm{C}$ that will suffice to lower the pressure to $50 \mathrm{kPa}$. (iii) Find the work output per cycle, and hence the average power output if the machine executes 15 cycles per minute. How does this compare to the heat input in part (i)?

Name: Problem 7, Chapter 11: Thermodynamics: A complete undergraduate course
Uploaded: 2020-07-08T19:00:08-08:00
Duration: 12 min 21 s
Channel: Sanjeev Kumar

    The Newcomen engine was the first working steam engine with a piston; it was constructed by Thomas Newcomen through incremental improvements between 1705 and 1711 and was ultimately widely used. A large cylinder (e.g. radius 1 metre, height several metres) was closed by a piston attached to one end of a heavy balanced beam. As the beam rocked to and fro, the other end provided work, for example to drive a water pump. The working principle of the Newcomen engine is as follows. A coal fire heats a quantity of water in a boiler. During the down-stroke on the pump, which is the up-stroke on the piston, steam fills the cylinder, but no useful work is done because the steam pressure is roughly $1 \mathrm{~atm}$; the weight of the pump rod provides the force required to lift the piston. At the top of the piston stroke, a valve to the boiler is closed, and another valve to a supply of cold water under pressure is briefly opened. Some cold water sprays into the chamber. Since this water is much colder than the steam, heat flows from the latter to the former, and the steam condenses. The pressure in the chamber therefore drops considerably, 'sucking' the piston downwards-which is to say, the atmospheric pressure above the piston pushes it down. This is why such engines are called 'atmospheric' engines. This is the power stroke. When the piston reaches the bottom, the valve to the boiler is re-opened and the incoming steam pushes the cold water out of the chamber.
(i) If the cylinder radius is $1 \mathrm{~m}$, and the working stroke of the piston is $3 \mathrm{~m}$, calculate the mass of steam required to fill the chamber at $100^{\circ} \mathrm{C}, 100 \mathrm{kPa}$. Determine the heat input required to produce this amount of steam from water initially at $15^{\circ} \mathrm{C}$.
(ii) Determine the amount of water at $15^{\circ} \mathrm{C}$ that will suffice to lower the pressure to $50 \mathrm{kPa}$.
(iii) Find the work output per cycle, and hence the average power output if the machine executes 15 cycles per minute. How does this compare to the heat input in part (i)?

Thermodynamics: A complete undergraduate course

Andrew M. Steane 1st Edition

Chapter 11, Problem 7

Instant Answer

Solved by Expert Sanjeev Kumar

07/08/2020

Step 1

- The volume $ V $ of a cylinder is given by the formula $ V = \pi r^2 h $. - Given the radius $ r = 1 \, \text{m} $ and the height (working stroke) $ h = 3 \, \text{m} $, we have: \[ V = \pi (1)^2 (3) = 3\pi \, \text{m}^3 \] Show more…

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The Newcomen engine was the first working steam engine with a piston; it was constructed by Thomas Newcomen through incremental improvements between 1705 and 1711 and was ultimately widely used. A large cylinder (e.g. radius 1 metre, height several metres) was closed by a piston attached to one end of a heavy balanced beam. As the beam rocked to and fro, the other end provided work, for example to drive a water pump. The working principle of the Newcomen engine is as follows. A coal fire heats a quantity of water in a boiler. During the down-stroke on the pump, which is the up-stroke on the piston, steam fills the cylinder, but no useful work is done because the steam pressure is roughly $1 \mathrm{~atm}$; the weight of the pump rod provides the force required to lift the piston. At the top of the piston stroke, a valve to the boiler is closed, and another valve to a supply of cold water under pressure is briefly opened. Some cold water sprays into the chamber. Since this water is much colder than the steam, heat flows from the latter to the former, and the steam condenses. The pressure in the chamber therefore drops considerably, 'sucking' the piston downwards-which is to say, the atmospheric pressure above the piston pushes it down. This is why such engines are called 'atmospheric' engines. This is the power stroke. When the piston reaches the bottom, the valve to the boiler is re-opened and the incoming steam pushes the cold water out of the chamber. (i) If the cylinder radius is $1 \mathrm{~m}$, and the working stroke of the piston is $3 \mathrm{~m}$, calculate the mass of steam required to fill the chamber at $100^{\circ} \mathrm{C}, 100 \mathrm{kPa}$. Determine the heat input required to produce this amount of steam from water initially at $15^{\circ} \mathrm{C}$. (ii) Determine the amount of water at $15^{\circ} \mathrm{C}$ that will suffice to lower the pressure to $50 \mathrm{kPa}$. (iii) Find the work output per cycle, and hence the average power output if the machine executes 15 cycles per minute. How does this compare to the heat input in part (i)?

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Key Concepts

Work Output and Efficiency in Thermodynamic Cycles

This involves calculating the work done per cycle by the engine and comparing it to the heat input. It is a fundamental thermodynamic analysis that assesses the efficiency of energy conversion processes in engines, bridging the gap between theoretical energy inputs and practical mechanical outputs.

Phase Change and Condensation Impact on Pressure

This concept explains how the condensation of steam into water within the cylinder reduces the pressure, leading to a pressure differential that enables the atmospheric pressure to push the piston. Analyzing the thermodynamic implications of phase change is vital for understanding the power stroke in steam engines.

Thermal Energy Calculations (Sensible and Latent Heat)

Understanding how much energy is required to heat water and convert it into steam involves both sensible heat (energy needed to raise the temperature of water) and latent heat (energy needed for the phase change from liquid to vapor). This concept is essential for quantifying the total heat energy input in steam generation processes.

Ideal Gas Law and Thermodynamic States

This concept involves applying the ideal gas law to analyze the state of steam under various conditions of pressure and temperature. It is crucial for determining properties such as mass, volume, and density of steam, which are necessary for calculating both the energy input and the performance of the engine during different phases of its cycle.

Piston-Cylinder Mechanism

The piston-cylinder mechanism is a fundamental component in many heat engines, where a piston moves inside a cylinder to convert pressure into mechanical motion. It involves the understanding of geometry, forces, and motion, which are key to calculating the work output and mechanical performance of the engine.

Atmospheric Steam Engine

This concept covers the basic operating principle of early steam engines where atmospheric pressure is utilized to move a piston. In these engines, steam is used to create a pressure differential, and then its condensation in the cylinder causes a drop in pressure that allows atmospheric pressure to push the piston, converting thermal energy into mechanical work.

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Transcript

00:02 There is a six -phase cycle of gasoline internal combustion engine.

00:22 So here these six path of the cycle is given as path oa is is is obaric, where from o2a the pressure remains constant, path a from a point to b is as adiatic path b to c is is isochoric path c to d ischoric path c to d is a the last one completing the cycle from a to o is isobaric.

01:47 So we are also given that e is equal to w upon q1.

01:58 So take it as equation 1.

02:04 W is given as q1 minus q2, take it as equation 2, q1 is equal to, q1 is equal.

02:14 To n c b at constant volume at tc temperature difference minus c to b c b and q2 so take it as a equation 3 q2 is equal to n c v t d minus t a so this is equation 4 so for part a of of the question we know that the efficiency of a heat engine is given by equation one above so now we will see as e is equal to q1 minus q2 upon q1 so here e will be as e is equal to 1 minus q2 upon q1 so simply putting the value of w from 2 into 1 so we get e is equal to q2 upon q1 and take it this as equation 5 where now putting the value of q1 and q2 from 3 to 4 into equation 5 now this e would come as 1 minus n cv t d minus t a upon ncv tc minus t v tc minus t v so e will be as 1 minus t d d minus t a upon tc minus d v so we take it as a part a now calculating for the part v so we are given as va, the volume at a and volume at d is equal.

04:40 So take it this relationship at 6 and given that vv is equal to vc, that volume at v point and volume at c point is again same.

04:51 So this is again 7.

04:54 Now we know that from second we can say this same.

05:09 Stroke av so this is stroke av is a v is a v is adiatic and from sixth the stroke of cd as cd is again adiatic hence for part v we know that the path ab and the cd are adiatic process so here we can compare as t a v a row minus 1 is equal to t v v v v rho minus 1 so this is equation 9 t c vc row minus 1 is equal to td vd row minus 1 so this is equation 10 now, substrating equation 9 from 10.

06:38 So, subtracting equation from 9 from 10, so we will get as tc, vc, r0 minus 1, minus tv, vv, r0 minus 1 is equal to t d v d row minus 1 minus t a va va rau minus 1 here we know that v a is equal to v d d and v v v c so we are can here put this values in this equation to make it simplify as tc v v rho minus 1 minus t v v v rho minus 1 is equal to t d v a row minus 1 minus ta a v a rae a row minus 1 rau minus 1 so now simplifying this equation in this order we get as vv row minus 1 as common factor as dc minus t v v v v.

08:23 V...

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