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Engines

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Practice

practice problem 1

Read the following paragraph about environmental regulations for US coal-fired power plants.

More substantial emissions reductions could be achieved by installing supercritical and ultra-supercritical boilers (the latter operate at 760 °C or higher and pressures of around 35 megapascals). Ultra-supercritical units can achieve thermal efficiencies as high as 48%, far better than the 33% average efficiency of the current US coal fleet. Supercritical boilers are used for all large-capacity boiler operations in most European and Asian countries, according to the National Energy Technology Laboratory. More than 400 supercritical boiler plants are in operation worldwide. But US companies, which generally operate older plants, have been slower to adopt the technology.

David Kramer, 2013

Assume that coal-fired boilers in power plants in the US operate at the theoretical limit of efficiency.
  1. Determine the temperature of the exhaust steam from an ultra-supercritical boiler.
  2. Assuming the exhaust temperature you just calculated is typical, determine the temperature of the incoming steam for an average boiler.

solution

Answer it.

  1. 537 K or 264 °C
  2. 802 K or 529 °C

practice problem 2

0.40 moles of an ideal, monatomic gas runs through a four step cycle. All processes are either adiabatic or isochoric. The pressure and volume of the gas at the extreme points in the cycle are given in the first two data rows of the table below.
  1. Sketch the PV graph of this cycle.
  2. Determine the temperature at state A, B, C, and D.
  3. Calculate W, Q, and ΔU on the path A→B, B→C, C→D, D→A and for one complete cycle. (Include the algebraic sign with each value.)
A four step cycle (adiabatic & isochoric)
state A B C D  
P (Pa) 100,000 1,462,000 5,850,000 400,000  
V (m3) 0.010 0.002 0.002 0.010  
T (K)  
path A→B B→C C→D D→A ABCDA
description adiabatic isochoric adiabatic isochoric closed cycle
ΔU (J)  
Q (J)  
W (J)  
  1. Determine the…
    1. real efficiency
    2. ideal efficiency (Carnot efficiency)
    … of a heat engine running this cycle.

solution

  1. Answer
  2. Answer
  3. Answer it
A four step cycle (adiabatic & isochoric)
state A B C D  
P (Pa) 100,000 1,462,000 5,850,000 400,000  
V (m3) 0.010 0.002 0.002 0.010  
T (K) 300 880 3,520 1,200  
path A→B B→C C→D D→A ABCDA
description adiabatic isochoric adiabatic isochoric closed cycle
ΔU (J) +2,890 +13,160 −11,550 −4,500 0
Q (J) 0 +13,160 0 −4,500 +8,660
W (J) +2,890 0 −11,550 0 −8,660
  1. Answer
    1. it
    2. it

practice problem 3

Write something different.

solution

Answer it.

practice problem 4

Write something completely different.

solution

Answer it.