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Carnot Cycle And The Efficiency Of A Perfect Heat Engine

(a) Derive the equation for the efficiency of a perfect heat engine in terms of the temperatures of the primary heat reservoir and the secondary heat reservoir using the Carnot cycle.

(b) Determine the efficiency of a steam engine (heat engine) operated reversibly between a primary reservoir and a secondary reservoir at 35^{o}C.

**The strings**:

S_{7}P_{3}A_{32} (Force - Push)
**The math**:

Pj Problem of Interest is of type *force* (force-push). Work problems are generally of type *force*. It is then left to determine whether the problem is of type *pull* or *push*. Usually a heat engine absorbs heat from the primary reservoir and releases (push) the heat to the secondary reservoir. It is in this sense that the primary problem of interest is of type *force-push*.

(a) In 1824 A.D. Sardi Carnot (1796-1832) presented a cycle of changes in the state of a gas interacting with its environment. This cycle of changes is now known as the Carnot Cycle (figure 14.28).

The Carnot cycle consists of four stages:

(1) An isothermal expansion of the gas from volume V_{1} to volume V_{2} in which work is done on the environment and heat absorbed from the environment at temperature T_{hot}

(2) Expansion of the gas from volume V_{2} to volume V_{3} at a lower temperature T_{cold} without heat transfer (adiabatic expansion).

(3) Isothermal compression of gas at volume V_{3} to volume V_{4}.

(4) Adiabatic compression of gas from volume V_{3} to its original volume, V_{1}.

The Carnot Circle assumes that all processes are reversible so that net entropy change in the universe is zero. Processes (2) and (4) (adiabatic processes) have no heat transfer (q = 0), so no change in entropy of the environment. Processes (1) and (3) involve the transfer of heat, so there is change in the entropy of the environment.

Let heat transfer during processes (1) and (3) be q_{hot} and q_{cold} respectively, then:

entropy change in the environment due to process (1):

= -q_{hot}/T_{hot} (heat is absorbed from the environment)

entropy change in the environment due to process (3):

= q_{cold}/T_{cold} (heat is transfered to the environment)

Since net entropy change in the universe is zero, we have:

-q_{hot}/T_{hot} + q_{cold}/T_{cold} = 0

So, q_{hot}/T_{hot} = q_{cold}/T_{cold} ------(5)

by the conservation of energy, we have:

work done by system = w = q_{hot} - q_{cold}

So, w/q_{hot} = (T_{hot} - T_{cold})/T_{hot}

So efficiency of a perfect heat engine (reversible heat engine);

= (T_{hot} - T_{cold})/T_{hot}

where T_{hot} is temperature of primary reservoir and T_{cold} is temperature of secondary reservoir.

(b) Temperature of primary reservoir = 100^{o}C = 373^{o}K

Temperature of secondary reservoir = 35^{o}C = 308^{o}K

So, efficiency of steam engine = (373-308)/373 = 0.174 or 17.4%.

Efficiency is less if engine is not perfect.

Math

The *point* **.** is a mathematical abstraction. It has negligible size and a great sense of position. Consequently, it is front and center in abstract existential reasoning.

Derivation Of The Area Of A Circle, A Sector Of A Circle And A Circular Ring

Derivation Of The Area Of A Trapezoid, A Rectangle And A Triangle

Derivation Of The Area Of An Ellipse

Derivation Of Volume Of A Cylinder

Derivation Of Volume Of A Sphere

Derivation Of Volume Of A Cone

Derivation Of Volume Of A Torus

Derivation Of Volume Of A Paraboloid

Volume Obtained By Revolving The Curve y = x^{2} About The X Axis

Single Variable Functions

Absolute Value Functions

Conics

Real Numbers

Vector Spaces

Equation Of The Ascent Path Of An Airplane

Calculating Capacity Of A Video Adapter Board Memory

Probability Density Functions

Boolean Algebra - Logic Functions

Ordinary Differential Equations (ODEs)

Infinite Sequences And Series

Introduction To Group Theory

Advanced Calculus - Partial Derivatives

Advanced Calculus - General Charateristics Of Partial Differential Equations

Advanced Calculus - Jacobians

Advanced Calculus - Solving PDEs By The Method Of Separation Of Variables

Advanced Calculus - Fourier Series

Advanced Calculus - Multiple Integrals

Production Schedule That Maximizes Profit Given Constraint Equation

Separation Of Variables As Solution Method For Homogeneous Heat Flow Equation

Newton And Fourier Cooling Laws Applied To Heat Flow Boundary Conditions

Fourier Series

Derivation Of Heat Equation For A One-Dimensional Heat Flow

Homogenizing-Non-Homogeneous-Time-Varying-IBVP-Boundary-Condition

The Universe is composed of *matter* and *radiant energy*. *Matter* is any kind of *mass-energy* that moves with velocities less than the velocity of light. *Radiant energy* is any kind of *mass-energy* that moves with the velocity of light.

Periodic Table

Composition And Structure Of Matter

How Matter Gets Composed

How Matter Gets Composed (2)

Molecular Structure Of Matter

Molecular Shapes: Bond Length, Bond Angle

Molecular Shapes: Valence Shell Electron Pair Repulsion

Molecular Shapes: Orbital Hybridization

Molecular Shapes: Sigma Bonds Pi Bonds

Molecular Shapes: Non ABn Molecules

Molecular Orbital Theory

More Pj Problem Strings