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Chapter 7: Problem 21

Air is heated from \(25^{\circ} \mathrm{C}\) to \(140^{\circ} \mathrm{C}\) prior to entering a combustion furnace. The change in specific enthalpy associated with this transition is \(3349 \mathrm{J} / \mathrm{mol}\). The flow rate of air at the heater outlet is \(1.65 \mathrm{m}^{3} / \mathrm{min}\) and the air pressure at this point is \(122 \mathrm{kPa}\) absolute. (a) Calculate the heat requirement in \(\mathrm{kW}\), assuming ideal-gas behavior and that kinetic and potential energy changes from the heater inlet to the outlet are negligible. (b) Would the value of \(\Delta \dot{E}_{k}\) [which was neglected in Part (a)] be positive or negative, or would you need more information to be able to tell? If the latter, what additional information would be needed?

Short Answer

Expert verified
(a) The heat requirement calculated is in kW. (b) The value of \(\Delta \dot{E}_{k}\) would be zero. If not, the additional information needed would be the change in flow speed and the change in height between the inlet and outlet.

Step by step solution

01

Calculating the Molar FlowRate

The ideal gas equation of state says that \( P V = n R T \), where P is the pressure, V is the volumetric flow rate, n is the molar flow rate, R is the universal gas constant, and T is the temperature. With rearrangement and by knowing all the properties, we can solve for n, the molar flow rate. So \( n = P V / R T \). We use \( P= 122 kPa = 122000 Pa \), \( V = 1.65 m^{3}/min = 1.65/60 m^{3}/s \), \(R = 8314.5 J/(K.mol)\), and \(T = 140+273=413 K\).
02

Computing the Heat requirement

The change in enthalpy, which is the heat requirement, under constant pressure can be given as \( \dot{Q} = \dot{n} \Delta h \) where \( \Delta h = 3349 J/mol \) and \( \dot{n} \) is the molar flow rate that we calculated in step 1. This calculation gives the heat requirement in Joules per second, which is Watts.
03

Converting the Heat requirement to kW

By definition, 1 kilowatt = 1000 Watt. So to convert the heat requirement from Watts to Kilowatts, we divide the calculated value in Watts by 1000.
04

Analyzing kinetic energy change

Since the kinetic and potential energy changes are neglected, it implies that the velocity and height of the air at the inlet and outlet remain the same, hence \(\Delta \dot{E}_{k}\) would be zero.
05

Determining the additional information needed

If \(\Delta \dot{E}_{k}\) needs to be considered, we'd need information about velocity changes and height changes from the heater inlet to the outlet.

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Key Concepts

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

Specific Enthalpy
Specific enthalpy is a concept in thermodynamics that helps us understand the energy changes in a system. It is the amount of enthalpy (thermal energy) per unit mass. Enthalpy itself is the sum of the internal energy and the product of pressure and volume. In simpler terms, it reflects the total heat content of a system.

When air is heated from one temperature to another, like from \(25^{\circ} \mathrm{C}\) to \(140^{\circ} \mathrm{C}\) in this problem, the specific enthalpy of the air changes. This change represents how much energy is required to achieve this temperature increase.

In this particular exercise, the specific enthalpy change is given as \(3349 \, \mathrm{J/mol}\), meaning each mole of air needs this amount of energy to reach the higher temperature.
Ideal Gas Behavior
Ideal gas behavior is a simplifying assumption in thermodynamics where gases are considered to have no intermolecular forces and occupy no volume. This assumption makes calculations easier because it allows us to use the ideal gas law: \(PV = nRT\). Where:
  • \(P\) is the pressure of the gas
  • \(V\) is the volume
  • \(n\) is the number of moles
  • \(R\) is the universal gas constant
  • \(T\) is the temperature in Kelvin
This concept is relevant here because we estimate the behavior of air as an ideal gas, which simplifies the calculation of molar flow rate needed to determine the heat requirement later.

By applying the ideal gas law, one can deduce the necessary flow characteristics, such as the number of moles of gas passing through a point in a given time, by rearranging the equation to find \(n\).
Molar Flow Rate
The molar flow rate is a measure of how many moles of a substance pass a given point per unit time. It is essential for calculating the total amount of energy associated with gases in processes like heating.

Using the ideal gas law, the molar flow rate can be calculated by rearranging the equation to solve for \(n\), which gives \(n = \frac{PV}{RT}\).

In this exercise, we know:
  • Pressure, \(P = 122,000 \, \mathrm{Pa}\)
  • Volume flow rate, \(V = \frac{1.65}{60} \, \mathrm{m}^3/\mathrm{s}\)
  • Universal gas constant, \(R = 8314.5 \, \mathrm{J}/(\mathrm{K} \cdot \mathrm{mol})\)
  • Temperature, \(T = 413 \, \mathrm{K}\)
Plugging these values in gives us the molar flow rate, which helps in calculating the total heat required.
Heat Requirement Calculation
Calculating the heat requirement involves determining how much energy is needed to raise the temperature of the gas from an initial to a final state. This is found by using the relation \(\dot{Q} = \dot{n} \Delta h\). Here:
  • \(\dot{Q}\) is the heat transfer rate in Joules per second or Watts
  • \(\dot{n}\) is the molar flow rate
  • \(\Delta h\) is the change in specific enthalpy
For our problem, since we have already calculated \(\dot{n}\) using the ideal gas law and we know \(\Delta h = 3349 \, \mathrm{J/mol}\), we can substitute these values to find the energy rate requirement.

Finally, to convert this heat requirement from Watts to kilowatts, we divide by 1000. This provides an understandable measure of energy requirement often used in practical applications.

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Most popular questions from this chapter

A fuel oil is burned with air in a boiler furnace. The combustion produces \(813 \mathrm{kW}\) of thermal energy, of which \(65 \%\) is transferred as heat to boiler tubes that pass through the furnace. The combustion products pass from the furnace to a stack at \(550^{\circ} \mathrm{C}\). Water enters the boiler tubes as a liquid at \(30^{\circ} \mathrm{C}\) and leaves the tubes as saturated steam at 20 bar absolute. (a) Calculate the rate ( \(\mathrm{kg} / \mathrm{h}\) ) at which steam is produced. (b) Use the steam tables to estimate the volumetric flow rate of the steam produced. (c) Repeat the calculation of Part (b), only assume ideal-gas behavior instead of using the steam tables. Would you have more confidence in the estimate of Part (b) or Part (c)? Explain. (d) What happened to the \(35 \%\) of the thermal energy released by the combustion that did not go to produce the steam?

A certain gasoline engine has an efficiency of \(30 \% ;\) that is, it converts into useful work \(30 \%\) of the heat generated by burning a fuel. (a) If the congine consumes \(0.80 \mathrm{L}\) /h of a gasoline with a heating value of \(3.25 \times 10^{4} \mathrm{kJ} / \mathrm{L}\), how much power does it provide? Express the answer both in \(\mathrm{kW}\) and horsepower. (b) Suppose the fuel is changed to include \(10 \%\) ethanol by volume. The heating value of ethanol is approximately \(2.34 \times 10^{4} \mathrm{kJ} / \mathrm{L}\) and volumes of gasoline and ethanol may be assumed additive. At what rate ( \((\text { / } h\) ) does the fuel mixture have to be consumed to produce the same power as gasoline?

Methane enters a 3 -cm ID pipe at \(30^{\circ} \mathrm{C}\) and 10 bar with an average velocity of \(5.00 \mathrm{m} / \mathrm{s}\) and emerges at a point 200 m lower than the inlet at \(30^{\circ} \mathrm{C}\) and 9 bar. (a) Without doing any calculations, predict the signs ( \(+\) or \(-\) ) of \(\Delta \dot{E}_{\mathrm{k}}\) and \(\Delta \dot{E}_{\mathrm{p}},\) where \(\Delta\) signifies (outlet - inlet). Briefly explain your reasoning. (b) Calculate \(\Delta \dot{E}_{\mathrm{k}}\) and \(\Delta \dot{E}_{\mathrm{p}}(\mathrm{W}),\) assuming that the methane behaves as an ideal gas. (c) If you determine that \(\Delta \dot{E}_{\mathrm{k}} \neq-\Delta \dot{E}_{\mathrm{p}},\) explain how that result is possible.

Energy may be produced from solid waste in two ways: (1) generate methane from anaerobic decomposition of the waste and burn it (landfill-gas-to-energy, or LFGTE) or(2) burn the waste directly (waste-to-energy, or WTE). The heat generated by either method can be used to produce steam, which impinges on a turbine rotor connected to a generator to produce electricity. LFGTE produces about 215 k Wh electricity/ton of waste, and WTE produces roughly 600 kWh/ton of waste. The average output of a large power plant is 1 GW, which is enough to supply the annual residential energy consumption of a city of roughly 800,000 people. (a) The current rate of municipal solid-waste generation in the United States is approximately 413 million tons per year. If all of it were used for energy recovery, how many \(1 \mathrm{GW}\) power plants could LFGTE supply? How many if WTE is used? A useful source of information regarding LFGTE is the U.S. EPA Landfill Methane Outreach Program, http://www.epa.gov//mop/; the Waste-to-Energy Research and Technology Council at Columbia University provides useful information on WTE, http://www.seas.columbia.edu/earth/wtert/; and information on natural gas can be obtained from the U.S. Energy Information Administration, http:// www.eia.doe.gov/oil_gas/natural_gas/info_glance/natural_gas.html.

Liquid water at 60 bar and \(250^{\circ} \mathrm{C}\) passes through an adiabatic expansion valve, emerging at a pressure \(P_{\mathrm{f}}\) and temperature \(T_{\mathrm{f}} .\) If \(P_{\mathrm{f}}\) is low enough, some of the liquid evaporates. (a) If \(P_{\mathrm{f}}=1.0\) bar, determine the temperature of the final mixture \(\left(T_{\mathrm{f}}\right)\) and the fraction of the liquid feed that evaporates \(\left(y_{\mathrm{v}}\right)\) by writing an energy balance about the valve and neglecting \(\Delta \dot{E}_{\mathrm{k}}\) (b) If you took \(\Delta \dot{E}_{\mathrm{k}}\) into account in Part (a), how would the calculated outlet temperature compare with the value you determined? What about the calculated value of \(y_{\mathrm{v}} ?\) Explain. (c) What is the value of \(P_{\mathrm{f}}\) above which no evaporation would occur? (d) Sketch the shapes of plots of \(T_{\mathrm{f}}\) versus \(P_{\mathrm{f}}\) and \(y_{\mathrm{v}}\) versus \(P_{\mathrm{f}}\) for 1 bar \(\leq P_{\mathrm{f}} \leq 60\) bar. Briefly explain your reasoning.
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