Working with Reaction EquationsA gas mixture with the

A vessel contains a mixture of 60% \(\mathrm{O}_{2}\) and 40% CO on a mass basis. Determine the percent excess or percent deficiency of oxygen, as appropriate.

A gas turbine burns octane (\(\mathrm{C}_{8} \mathrm{H}_{18}\)) completely with 400% of theoretical air. Determine the amount of \(\mathrm{N}_{2}\) in the products, in kmol per kmol of fuel.

Ten grams of propane (\(\mathrm{C}_{3} \mathrm{H}_{8}\)) burns with just enough oxygen (\(\mathrm{O}_{2}\)) for complete combustion. Determine the amount of oxygen required and the amount of combustion products formed, each in grams.

Using the result of Problem 13.111(b), determine the total specific flow exergy at locations 1, 2, and 3 and the rate of exergy destruction, each in Btu/min, for the device of Problem 12.102. Let the environment be a mixture of dry air and water vapor at \(95^{\circ} \mathrm{F}\), 1 atm with \(y_{\mathrm{v}}^{\mathrm{e}}=0.022, y_{\mathrm{a}}^{\mathrm{e}}=0.978\). Also, let \(c_{p \mathrm{a}}=0.24 \mathrm{Btu} / \mathrm{lb} \cdot{ }^{\circ} \mathrm{R} \text { and } c_{p \mathrm{v}}=0.44 \mathrm{Btu} / \mathrm{lb} \cdot{ }^{\circ} \mathrm{R}\)

One hundred kmol of butane (\(\mathrm{C}_{4} \mathrm{H}_{10}\)) together with 4000 kmol of air enter a furnace per unit of time. Carbon dioxide, carbon monoxide, and unburned fuel appear in the products of combustion exiting the furnace. Determine the percent excess or percent deficiency of air, whichever is appropriate.

Ethane (\(\mathrm{C}_{2} \mathrm{H}_{6}\)) burns completely with the theoretical amount of air. Determine the air–fuel ratio on a (a) molar basis, (b) mass basis.

Butane (\(\mathrm{C}_{4} \mathrm{H}_{10}\)) burns completely with air. The equivalence ratio is 0.9. Determine (a) the balanced reaction equation. (b) the percent excess air.

A fuel mixture with the molar analysis 40% \(\mathrm{CH}_{3} \mathrm{OH}\), 50% \(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}\), and 10% \(\mathrm{N}_{2}\) burns completely with 33% excess air. Determine (a) the balanced reaction equation. (b) the air–fuel ratio, both on a molar and mass basis.

A natural gas mixture having a molar analysis 60% \(\mathrm{CH}_{4}\), 30% \(\mathrm{C}_{2} \mathrm{H}_{6}\), 10% \(\mathrm{N}_{2}\) is supplied to a furnace like the one shown in Fig. P13.8, where it burns completely with 20% excess air. Determine (a) the balanced reaction equation. (b) the air–fuel ratio, both on a molar and a mass basis.

Propane (\(\mathrm{C}_{3} \mathrm{H}_{8}\)) is burned with air. For each case, obtain the balanced reaction equation for complete combustion (a) with the theoretical amount of air. (b) with 20% excess air. (c) with 20% excess air, but only 90% of the propane being consumed in the reaction.

A gas mixture with the molar analysis 25% \(\mathrm{H}_{2}\), 25% CO, 50% \(\mathrm{O}_{2}\) reacts to form products consisting of \(\mathrm{CO}_{2}\), \(\mathrm{H}_{2} \mathrm{O}\), and \(\mathrm{O}_{2}\) only. Determine the amount of each product, in kg per kg of mixture.

A natural gas with the molar analysis 78% \(\mathrm{CH}_{4}\), 13% \(\mathrm{C}_{2} \mathrm{H}_{6}\), 6% \(\mathrm{C}_{3} \mathrm{H}_{8}\), 1.7% \(\mathrm{C}_{4} \mathrm{H}_{10}\), 1.3% \(\mathrm{N}_{2}\) burns completely with 40% excess air in a reactor operating at steady state. If the molar flow rate of the fuel is 0.5 kmol/h, determine the molar flow rate of the air, in kmol/h.

A natural gas fuel mixture has the molar analysis shown below. Determine the molar analysis of the products for complete combustion with 70% excess dry air.

Coal with the mass analysis 77.54% C, 4.28% H, 1.46% S, 7.72% O, 1.34% N, 7.66% noncombustible ash burns completely with 120% of theoretical air. Determine (a) the balanced reaction equation. (b) the amount of \(\mathrm{SO}_{2}\) produced, in kg per kg of coal.

A sample of dried feedlot manure is being tested for use as a fuel. The mass analysis of the sample is 42.7% carbon, 5.5% hydrogen (H), 31.3% oxygen (O), 2.4% nitrogen (N), 0.3% sulfur, and 17.8% noncombustible ash. The sample is burned completely with 120% of theoretical air. Determine (a) the balanced reaction equation. (b) the air–fuel ratio on a mass basis.

A sample of dried Appanoose County coal has a mass analysis of 71.1% carbon, 5.1% hydrogen (H), 9.0% oxygen (O), 1.4% nitrogen (\(\mathrm{N}_{2}\)), 5.8% sulfur, and the rest noncombustible ash. For complete combustion with the theoretical amount of air, determine (a) the amount of \(\mathrm{SO}_{2}\) produced, in kg per kg of coal. (b) the air–fuel ratio on a mass basis.

A coal sample has a mass analysis of 80.4% carbon, 3.9% hydrogen (H), 5.0% oxygen (O), 1.1% nitrogen (N), 1.1% sulfur, and the rest is noncombustible ash. For complete combustion with 120% of the theoretical amount of air, determine the air–fuel ratio on a mass basis.

Octane (\(\mathrm{C}_{8} \mathrm{H}_{18}\)) burns completely with 120% of theoretical air. Determine (a) the air–fuel ratio on a molar and mass basis. (b) the dew point temperature of the combustion products, in \({ }^{\circ} \mathrm{C}\), when cooled at 1 atm.

Butane (\(\mathrm{C}_{4} \mathrm{H}_{10}\)) burns completely with 150% of theoretical air. If the combustion products are cooled at 1 atm to temperature T, plot the amount of water vapor condensed, in kmol per kmol of fuel, versus T ranging from 20 to \(60^{\circ} \mathrm{C}\).

Ethylene (\(\mathrm{C}_{2} \mathrm{H}_{4}\)) burns completely with air and the combustion products are cooled to temperature T at 1 atm. The air–fuel ratio on a mass basis is AF. (a) Determine for AF = 15 and \(T=70^{\circ} \mathrm{F}\), the percent excess air and the amount of water vapor condensed, in lb per lbmol of fuel. (b) Plot the amount of water vapor condensed, in lb per lbmol of fuel, versus T ranging from 70 to \(100^{\circ} \mathrm{F}\), for AF = 15, 20, 25, 30.

The gas driven off when low-grade coal is burned with insufficient air for complete combustion is known as producer gas. A particular producer gas has the following volumetric analysis: 3.8% \(\mathrm{CH}_{4}\), 0.1% \(\mathrm{C}_{2} \mathrm{H}_{6}\), 4.8% \(\mathrm{CO}_{2}), 11.7% \(\mathrm{H}_{2}\), 0.6% \(\mathrm{O}_{2}\), 23.2% CO, and the remainder \(\mathrm{N}_{2}\), Determine, for complete combustion with the theoretical amount of air (a) the molar analysis of the dry products of combustion. (b) the amount of water vapor condensed, in lbmol/lbmol of producer gas, if the products are cooled to \(70^{\circ} \mathrm{F}\) at a constant pressure of 1 atm.

Acetylene (\(\mathrm{C}_{2} \mathrm{H}_{2}\)) enters a torch and burns completely with 110% of theoretical air entering at \(74^{\circ} \mathrm{F}\), 1 atm, 50% relative humidity. Obtain the balanced reaction equation, and determine the dew point temperature of the products, in \({ }^{\circ} \mathrm{F}\), at 1 atm.

A gaseous fuel mixture with a specified molar analysis burns completely with moist air to form gaseous products as shown in Fig. P13.20. Determine the dew point temperature of the products, in \({ }^{\circ} \mathrm{C}\).

Butane (\(\mathrm{C}_{4} \mathrm{H}_{10}\)) burns completely with 160% of theoretical air at \(20^{\circ} \mathrm{C}\), 1 atm, and 90% relative humidity. Determine (a) the balanced reaction equation. (b) the dew point temperature, in \({ }^{\circ} \mathrm{C}\), of the products, when cooled at 1 atm.

Ethane (\(\mathrm{C}_{2} \mathrm{H}_{6}\)) enters a furnace and burns completely with 130% of theoretical air entering at \(25^{\circ} \mathrm{C}\), 85 kPa, 50% relative humidity. Determine (a) the balanced reaction equation. (b) the dew point temperature of the combustion products, in \({ }^{\circ} \mathrm{C}\), at 85 kPa.

Propane (\(\mathrm{C}_{3} \mathrm{H}_{8}\)) burns completely with the theoretical amount of air at \(60^{\circ} \mathrm{F}\), 1 atm, 90% relative humidity. Determine (a) the balanced reaction equation. (b) the dew point temperature of the combustion products at 1 atm. (c) the amount of water condensed, in lbmol per lbmol of fuel, if the combustion products are cooled to \(75^{\circ} \mathrm{F}\), at 1 atm.

A liquid fuel mixture that is 40% octane (\(\mathrm{C}_{8} \mathrm{H}_{18}\)) and 60% decane (\(\mathrm{C}_{10} \mathrm{H}_{22}\)) by mass is burned completely with 10% excess air at \(25^{\circ} \mathrm{C}\), 1 atm, 80% relative humidity. (a) Determine the equivalent hydrocarbon composition, \(\mathrm{C}_{a} \mathrm{H}_{b}\), of a fuel that would have the same carbon–hydrogen ratio on a mass basis as the fuel mixture. (b) If the combustion products are cooled to \(25^{\circ} \mathrm{C}\) at a pressure of 1 atm, determine the amount of water vapor that condenses, in kg per kg of fuel mixture.

Hydrogen (\(\mathrm{H}_{2}\)) enters a combustion chamber with a mass flow rate of 5 lb/h and burns with air entering at \(85^{\circ} \mathrm{F}\), 1 atm with a volumetric flow rate of \(75 \mathrm{ft}^{3} / \mathrm{min}\). Determine the percent of theoretical air used.

Octane (\(\mathrm{C}_{8} \mathrm{H}_{18}\)) is burned with 20% excess air, yielding \(\mathrm{CO}_{2}\), CO, \(\mathrm{O}_{2}\), \(\mathrm{H}_{2} \mathrmOH}\), and \(\mathrm{N}_{2}\) only. If 5% of the dry products (molar basis) is \(\mathrm{O}_{2}\), determine (a) the balanced reaction equation. (b) the analysis of the products on a dry molar basis.

Methyl alcohol (\(\mathrm{CH}_{3} \mathrm{OH}\)) burns with 200% theoretical air, yielding \(\mathrm{CO}_{2}\), \(\mathrm{H}_{2} \mathrm{O}\), \(\mathrm{O}_{2}\), and \(\mathrm{N}_{2}\). Determine the (a) balanced reaction equation. (b) air–fuel ratio on a mass basis. (c) molar analysis of the products.

The components of the exhaust gas of a spark-ignition engine using a fuel mixture represented as \(\mathrm{C}_{8} \mathrm{H}_{17}\) have a dry molar analysis of 8.7% \(\mathrm{CO}_{2}\), 8.9% CO, 0.3% \(\mathrm{O}_{2}\), 3.7% \(\mathrm{H}_{2}\), 0.3% \(\mathrm{CH}_{4}\), and 78.1% \(\mathrm{N}_{2}\). Determine the equivalence ratio.

Hexane (\(\mathrm{C}_{6} \mathrm{H}_{14}\)) burns with dry air to give products with the dry molar analysis 8.5% \(\mathrm{CO}_{2}\), 5.2% CO, 3% \(\mathrm{O}_{2}\), 83.3% \(\mathrm{N}_{2}\), Determine (a) the balanced reaction equation. (b) the percent of theoretical air. (c) the dew point temperature, in \({ }^{\circ} \mathrm{C}\), of the products at 1 atm.

The combustion of a hydrocarbon fuel, represented as \(\mathrm{C}_{a} \mathrm{H}_{b}\), results in products with the dry molar analysis 11% \(\mathrm{CO}_{2}\), 0.5% CO, 2% \(\mathrm{CH}_{4}\), 1.5% \(\mathrm{H}_{2}\), 6% \(\mathrm{O}_{2}\), and 79% \(\mathrm{N}_{2}\). Determine the air–fuel ratio on (a) a molar basis, (b) a mass basis

Decane (\(\mathrm{C}_{10} \mathrm{H}_{22}\)) burns completely in dry air. The air– fuel ratio on a mass basis is 33. Determine the (a) analysis of the products on a dry molar basis. (b) percent of theoretical air.

Butane (\(\mathrm{C}_{4} \mathrm{H}_{10}\)) burns with air, giving products having the dry molar analysis 11.0% \(\mathrm{CO}_{2}\), 1.0% CO, 3.5% \(\mathrm{O}_{2}\), 84.5% \(\mathrm{N}_{2}\). Determine (a) the percent theoretical air. (b) the dew point temperature of the combustion products, in \({ }^{\circ} \mathrm{C}\), at 1 bar.

A natural gas with the volumetric analysis 97.3% \(\mathrm{CH}_{4}\), 2.3% \(\mathrm{CO}_{2}\), 0.4% \(\mathrm{N}_{2}\) is burned with air in a furnace to give products having a dry molar analysis of 9.20% \(\mathrm{CO}_{2}\), 3.84% \(\mathrm{O}_{2}\), 0.64% CO, and the remainder \(\mathrm{N}_{2}\). Determine (a) the percent theoretical air. (b) the dew point temperature, in \({ }^{\circ} \mathrm{F}\), of the combustion products at 1 atm.

A fuel oil having an analysis on a mass basis of 85.7% C, 14.2% H, 0.1% inert matter burns with air to give products with a dry molar analysis of 12.29% \(\mathrm{CO}_{2}\), 3.76% \(\mathrm{O}_{2}\), 83.95% \(\mathrm{N}_{2}\). Determine the air–fuel ratio on a mass basis.

Ethyl alcohol (\(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}\)) burns with air. The product gas is analyzed and the laboratory report gives only the following percentages on a dry molar basis: 6.9% \(\mathrm{CO}_{2}\), 1.4% CO, 0.5% (\(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}\)). Assuming the remaining components consist of \(\mathrm{O}_{2}\) and \(\mathrm{N}_{2}\), determine (a) the percentages of \(\mathrm{O}_{2}\) and \(\mathrm{N}_{2}\) in the dry molar analysis. (b) the percent excess air.

A fuel oil with the mass analysis 87% C, 11% H, 1.4% S, 0.6% inert matter burns with 120% of theoretical air. The hydrogen and sulfur are completely oxidized, but 95% of the carbon is oxidized to \(\mathrm{CO}_{2}\) and the remainder to CO. (a) Determine the balanced reaction equation. (b) For the CO and \(\mathrm{SO}_{2}\), determine the amount, in kmol per 106 kmol of combustion products (that is, the amount in parts per million).

Pentane (\(\mathrm{C}_{5} \mathrm{H}_{12}\)) burns with air so that a fraction x of the carbon is converted to \(\mathrm{CO}_{2}\). The remaining carbon appears as CO. There is no free \(\mathrm{O}_{2}\) in the products. Develop plots of the air–fuel ratio and the percent of theoretical air versus x, for x ranging from zero to unity.

For each of the following mixtures, determine the equivalence ratio and indicate if the mixture is lean or rich: (a) 1 kmol of butane (\(\mathrm{C}_{4} \mathrm{H}_{10}\)) and 32 kmol of air. (b) 1 lb of propane (\(\mathrm{C}_{3} \mathrm{H}_{8}\)) and 14.5 lb of air.

Methyl alcohol (\(\mathrm{CH}_{3} \mathrm{OH}\)) burns in dry air according to the reaction \(\begin{aligned} \mathrm{CH}_{3} \mathrm{OH} & +3.3\left(\mathrm{O}_{2}+3.76 \mathrm{~N}_{2}\right) \rightarrow \mathrm{CO}_{2}+2 \mathrm{H}_{2} \mathrm{O} \\ & +1.8 \mathrm{O}_{2}+12.408 \mathrm{~N}_{2} \end{aligned}\) Determine the (a) air–fuel ratio on a mass basis. (b) equivalence ratio. (c) percent excess air.

percent excess air. 13.42 Ethyl alcohol (\(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}\)) burns in dry air according to the reaction \(\begin{aligned} \mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH} & +2.16\left(\mathrm{O}_{2}+3.76 \mathrm{~N}_{2}\right) \rightarrow 0.32 \mathrm{CO}_{2}+1.68 \mathrm{CO} \\ & +3 \mathrm{H}_{2} \mathrm{O}+8.1216 \mathrm{~N}_{2} \end{aligned}\) Determine the (a) air–fuel ratio on a mass basis. (b) equivalence ratio. (c) percent theoretical air.

Octane (\(\mathrm{C}_{8} \mathrm{H}_{18}\)) enters an engine and burns with air to give products with the dry molar analysis of \(\mathrm{CO}_{2}\), 10.5%; CO, 5.8%; \(\mathrm{CH}_{4}\), 0.9%; \(\mathrm{H}_{2}\), 2.6%; \(\mathrm{O}_{2}\), 0.3%; \(\mathrm{N}_{2}\), 79.9%. Determine the equivalence ratio.

Methane (\(\mathrm{CH}_{4}\)) burns with air to form products consisting of \(\mathrm{CO}_{2}\), CO, \(\mathrm{H}_{2} \mathrm{O}\), and \(\mathrm{N}_{2}\) only. If the equivalence ratio is 1.25, determine the balanced reaction equation.

Liquid octane (\(\mathrm{C}_{8} \mathrm{H}_{18}\)) at \(77^{\circ} \mathrm{F}\), 1 atm enters a combustion chamber operating at steady state and burns completely with 50% excess dry air entering at \(120^{\circ} \mathrm{F}\), 1 atm. The products exit at \(1060^{\circ} \mathrm{F}\), 1 atm. Determine the rate of heat transfer between the combustion chamber and its surroundings, in Btu per lbmol of fuel entering. Kinetic and potential energy effects are negligible.

Propane (\(\mathrm{C}_{3} \mathrm{H}_{8}\)) at 298 K, 1 atm, enters a combustion chamber operating at steady state with a molar flow rate of 0.7 kmol/s and burns completely with 200% of theoretical air entering at 298 K, 1 atm. Kinetic and potential energy effects are negligible. If the combustion products exit at 560 K, 1 atm, determine the rate of heat transfer for the combustion chamber, in kW. Repeat for an exit temperature of 298 K.

Methane (\(\mathrm{CH}_{4}\)) at \(25^{\circ} \mathrm{C}\), 1 atm enters a furnace operating at steady state and burns completely with 140% of theoretical air entering at 400 K, 1 atm. The products of combustion exit at 700 K, 1 atm. Kinetic and potential energy effects are negligible. If the rate of heat transfer from the furnace to the surroundings is 400 kW, determine the mass flow rate of methane, in kg/s.

Methane gas (\(\mathrm{CH}_{4}\)) at \(25^{\circ} \mathrm{C}\), 1 atm enters a steam generator operating at steady state. The methane burns completely with 140% of theoretical air entering at \(127^{\circ} \mathrm{C}\), 1 atm. Products of combustion exit at \(427^{\circ} \mathrm{C}\), 1 atm. In a separate stream, saturated liquid water enters at 8 MPa and exits as superheated vapor at \(480^{\circ} \mathrm{C}\) with a negligible pressure drop. If the vapor mass flow rate is 3.7 3 105 kg/h, determine the volumetric flow rate of the methane, in \(\mathrm{m}^{3} / \mathrm{h}\).

Liquid ethanol (\(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}\)) at \(77^{\circ} \mathrm{F}\), 1 atm enters a combustion chamber operating at steady state and burns completely with dry air entering at \(340^{\circ} \mathrm{F}\), 1 atm. The fuel flow rate is 50 lb/s, and the equivalence ratio is 0.8. Products of combustion exit at \(2000^{\circ} \mathrm{F}\), 1 atm. Ignoring kinetic and potential energy effects, determine (a) the air–fuel ratio on a mass basis. (b) the rate of heat transfer, in Btu/s.

Octane gas (\(\mathrm{C}_{8} \mathrm{H}_{18}\)) at \(25^{\circ} \mathrm{C}\), 1 atm enters a combustion chamber operating at steady state and burns with 120% theoretical air entering at \(25^{\circ} \mathrm{C}\), 1 atm. The combustion products exit at 1200 K and include only \(\mathrm{CO}_{2}\), \(\mathrm{H}_{2} \mathrm{O}\), \(\mathrm{O}_{2}\), and \(\mathrm{N}_{2}\). If the rate of heat transfer from the combustion chamber to the surroundings is 2500 kW, determine the mass flow rate of the fuel, in kg/s.

Liquid propane (\(\mathrm{C}_{3} \mathrm{H}_{8}\)) at \(25^{\circ} \mathrm{C}\), 1 atm, enters a wellinsulated reactor operating at steady state. Air enters at the same temperature and pressure. For liquid propane, \(\bar{h}_{\mathrm{f}}^{\circ}=-118,900 \mathrm{~kJ} / \mathrm{kmol}\). Determine the temperature of the combustion products, in K, for complete combustion with (a) the theoretical amount of air. (b) 300% of theoretical air.

The energy required to vaporize the working fluid passing through the boiler of a simple vapor power plant is provided by the complete combustion of methane with 110% of theoretical air. The fuel and air enter in separate streams at \(25^{\circ} \mathrm{C}\), 1 atm. Products of combustion exit the stack at \(150^{\circ} \mathrm{C}\), 1 atm. Plot the mass flow rate of fuel required, in kg/h per MW of power developed by the plant versus the plant thermal efficiency, \(\eta\). Consider \(\eta\) in the range 30–40%. Kinetic and potential energy effects are negligible.

Methane (\(\mathrm{CH}_{4}\)) at \(25^{\circ} \mathrm{C}\), enters the combustor of a simple open gas turbine power plant and burns completely with 400% of theoretical air entering the compressor at \(25^{\circ} \mathrm{C}\), 1 atm. Products of combustion exit the turbine at \(577^{\circ} \mathrm{C}\), 1 atm. The rate of heat transfer from the gas turbine is estimated as 10% of the net power developed. Determine the net power output, in MW, if the fuel mass flow rate is 1200 kg/h. Kinetic and potential energy effects are negligible.

Octane gas \(\mathrm{C}_{8} \mathrm{H}_{18}\) at \(25^{\circ} \mathrm{C}\) enters a jet engine and burns completely with 300% of theoretical air entering at \(25^{\circ} \mathrm{C}\), 1 atm with a volumetric flow rate of \(42 \mathrm{m}^{3} / \mathrm{s}\). Products of combustion exit at 990 K, 1 atm. If the fuel and air enter with negligible velocities, determine the thrust produced by the engine in kN.

One lbmol of octane gas (\(\mathrm{C}_{8} \mathrm{H}_{18}\)) reacts with the theoretical amount of air in a closed, rigid tank. Initially, the reactants are at \(77^{\circ} \mathrm{F}\), 1 atm. After complete combustion, the pressure in the tank is 3.98 atm. Determine the heat transfer, in Btu.

A closed, rigid vessel initially contains a gaseous mixture at \(25^{\circ} \mathrm{C}\), 1 atm with the molar analysis of 20% ethane (\(\mathrm{C}_{2} \mathrm{H}_{6}\)), 80% oxygen (\(\mathrm{O}_{2}\)). The initial mixture contains one kmol of ethane. Complete combustion occurs, and the products are cooled to \(25^{\circ} \mathrm{C}\). Determine the heat transfer, in kJ, and the final pressure, in atm.

A closed, rigid vessel initially contains a gaseous mixture of 1 kmol of pentane (\(\mathrm{C}_{5} \mathrm{H}_{12}\)) and 150% of theoretical air at \(25^{\circ} \mathrm{C}\), 1 atm. If the mixture burns completely, determine the heat transfer from the vessel, in kJ, and the final pressure, in atm, for a final temperature of 800 K.

Figure P13.55 provides data for a boiler and air preheater operating at steady state. Methane (\(\mathrm{CH}_{4}\)) entering the boiler at \(25^{\circ} \mathrm{C}\), 1 atm is burned completely with 170% of theoretical air. Ignoring stray heat transfer and kinetic and potential energy effects, determine the temperature, in \({ }^{\circ} \mathrm{C}\), of the combustion air entering the boiler from the preheater.

Calculate the enthalpy of combustion of gaseous pentane (\(\mathrm{C}_{5} \mathrm{H}_{12}\)), in kJ per kmol of fuel, at \(25^{\circ} \mathrm{C}\) with water vapor in the products.

Plot the enthalpy of combustion for gaseous propane (\(\mathrm{C}_{3} \mathrm{H}_{8}\)), in Btu per lbmol of fuel, at 1 atm versus temperature in the interval 77 to \(500^{\circ} \mathrm{F}\). Assume water vapor in the products. For propane, let \(c_{p}=0.41 \mathrm{Btu} / \mathrm{lb} \cdot{ }^{\circ} \mathrm{R}\).

For the producer gas of Prob. 13.21, determine the enthalpy of combustion, in Btu per lbmol of mixture, at \(77^{\circ} \mathrm{F}\), 1 atm, assuming water vapor in the products.

Plot the enthalpy of combustion for gaseous methane (\(\mathrm{CH}_{4}\)), in Btu per lbmol of fuel, at 1 atm versus temperature in the interval from 537 to \(1800^{\circ} \mathrm{R}\). Assume water vapor in the products. For methane, let\(\bar{c}_{p}=4.52+7.37(T / 1000) \text { Btu/lbmol } \cdot{ }^{\circ} \mathrm{R}\), where T is in \({ }^{\circ} \mathrm{R}\).

For a natural gas with a molar analysis of 86.5% \(\mathrm{CH}_{4}\), 8% \(\mathrm{C}_{2} \mathrm{H}_{6}\), 2% \(\mathrm{C}_{3} \mathrm{H}_{8}\), 3.5% \(\mathrm{N}_{2}\), determine the lower heating value, in kJ per kmol of fuel and in kJ per kg of fuel, at \(25^{\circ} \mathrm{C}\), 1 atm.

Determine the lower heating value, in kJ per kmol of fuel and in kJ per kg of fuel, at \(25^{\circ} \mathrm{C}\), 1 atm for (a) gaseous ethane (\(\mathrm{C}_{2} \mathrm{H}_{6}\)). (b) liquid ethanol (\(\mathrm{C}_{2} \mathrm{H}_{5}\mathrm{OH}\)). (c) gaseous propane (\(\mathrm{C}_{3} \mathrm{H}_{8}\)). (d) liquid octane (\(\mathrm{C}_{8} \mathrm{H}_{18}\)).

For each of the following fuels, plot the adiabatic flame temperature, in K, versus percent excess air for complete combustion in a combustor operating at steady state. The reactants enter at \(25^{\circ} \mathrm{C}\), 1 atm. (a) carbon. (b) hydrogen (\(\mathrm{H}_{2}\)). (c) liquid octane (\(\mathrm{C}_{8} \mathrm{H}_{18}\)).

Liquid octane (\(\mathrm{C}_{8} \mathrm{H}_{18}\)) at (\25^{\circ} \mathrm{C}\), 1 atm enters an insulated reactor operating at steady state and burns with 90% of theoretical air at \(25^{\circ} \mathrm{C}\), 1 atm to form products consisting of \(\mathrm{CO}_{2}\), CO, \(\mathrm{H}_{2} \mathrm{O}\), and \(\mathrm{N}_{2}\) only. Determine the temperature of the exiting products, in K. Compare with the results of Example 13.8 and comment.

Propane gas (\(\mathrm{C}_{3} \mathrm{H}_{8}\)) at \(25^{\circ} \mathrm{C}\), 1 atm enters an insulated reactor operating at steady state and burns completely with air entering at \(25^{\circ} \mathrm{C}\), 1 atm. Plot the adiabatic flame temperature versus percent of theoretical air ranging from 100 to 400%. Why does the adiabatic flame temperature vary with increasing combustion air?

Methane (\(\mathrm{CH}_{4}\)) at 258C, 1 atm enters an insulated reactor operating at steady state and burns with the theoretical amount of air entering at 258C, 1 atm. The products contain \(\mathrm{CO}_{2}\), CO, \(\mathrm{H}_{2} \mathrm{O}\), \\mathrm{O}_{2}\), and \(\mathrm{N}_{2}\), and exit at 2260 K. Determine the fractions of the entering carbon in the fuel that burn to \(\mathrm{CO}_{2}\) and CO, respectively.

Hydrogen (\(\mathrm{H}_{2}\)) at \(77^{\circ} \mathrm{F}\), 1 atm enters an insulated reactor operating at steady state and burns completely with x% of theoretical air entering at \(77^{\circ} \mathrm{F}\) 1 atm. Plot the adiabatic flame temperature for x ranging from 100 to 400%.

Ethane (\(\mathrm{C}_{2} \mathrm{H}_{6}\)) gas at \(77^{\circ} \mathrm{F}\), 1 atm enters a well-insulated reactor operating at steady state and burns completely with air entering at \(240^{\circ} \mathrm{F}\), 1 atm. Determine the temperature of the products, in \({ }^{\circ} \mathrm{F}\). Neglect kinetic and potential energy effects.

Liquid methanol (\(\mathrm{CH}_{3} \mathrm{OH}\)) at \(25^{\circ} \mathrm{C}\), 1 atm enters an insulated reactor operating at steady state and burns completely with air entering at \(100^{\circ} \mathrm{C}\), 1 atm. If the combustion products exit at \(1256^{\circ} \mathrm{C}\), determine the percent excess air used. Neglect kinetic and potential energy effects.

Methane (\(\mathrm{CH}_{4}\)) at \(77^{\circ} \mathrm{F}\) enters the combustor of a gas turbine power plant operating at steady state and burns completely with air entering at \(400^{\circ} \mathrm{F}\). The temperature of the products of combustion flowing from the combustor to the turbine depends on the percent excess air for combustion. Plot the percent excess air versus combustion product temperatures ranging from 1400 to \(1800^{\circ} \mathrm{F}\). There is no significant heat transfer between the combustor and its surroundings, and kinetic and potential energy effects can be ignored.

A mixture of gaseous octane (\(\mathrm{C}_{8} \mathrm{H}_{18}\)) and 200% of theoretical air, initially at \(25^{\circ} \mathrm{C}\), 1 atm, reacts completely in a rigid vessel. (a) If the vessel were well-insulated, determine the temperature, in \({ }^{\circ} \mathrm{C}\), and the pressure, in atm, of the combustion products. (b) If the combustion products were cooled at constant volume to \(25^{\circ} \mathrm{C}\), determine the final pressure, in atm, and the heat transfer, in kJ per kmol of fuel.

Methane gas (\(\mathrm{CH}_{4}\)) reacts completely with the theoretical amount of oxygen (\(\mathrm{O}_{2}\)) in a piston-cylinder assembly. Initially, the mixture is at \(77^{\circ} \mathrm{F}\), 1 atm. If the process occurs at constant pressure and the final volume is 1.9 times the initial volume, determine the work and the heat transfer, each in Btu per lbmol of fuel.

Carbon monoxide (CO) at \(\25^{\circ} \mathrm{C}\), 1 atm enters an insulated reactor operating at steady state and reacts completely with the theoretical amount of air entering in a separate stream at \(25^{\circ} \mathrm{C}\), 1 atm. The products of combustion exit as a mixture at 1 atm. For the reactor, determine the rate of entropy production, in kJ/K per kmol of CO entering. Neglect kinetic and potential energy effects.

A \(5 \times 10^{-3} \mathrm{~kg}\) sample of liquid benzene (\(\mathrm{C}_{6} \mathrm{H}_{6}\)) together with 20% excess air, initially at \(25^{\circ} \mathrm{C}\) and 1 atm, reacts completely in a rigid, insulated vessel. Determine the temperature, in \({ }^{\circ} \mathrm{C}\), and the pressure, in atm, of the combustion products.

Methane (\(\mathrm{CH}_{4}\)) at \(\77^{\circ} \mathrm{F}\), 1 atm enters an insulated reactor operating at steady state and burns completely with air entering in a separate stream at \(77^{\circ} \mathrm{F}\), 1 atm. The products of combustion exit as a mixture at 1 atm. For the reactor, determine the rate of entropy production, in \(\mathrm{Btu} /{ }^{\circ} \mathrm{R}\) per lbmol of methane entering, for combustion with (a) the theoretical amount of air. (b) 200% of theoretical air. Neglect kinetic and potential energy effects.

A gaseous mixture of butane (\(\mathrm{C}_{4} \mathrm{H}_{10}\)) and 80% excess air at \(25^{\circ} \mathrm{C}\), 3 atm enters a reactor. Complete combustion occurs, and the products exit as a mixture at 1200 K, 3 atm. Coolant enters an outer jacket as a saturated liquid and saturated vapor exits at essentially the same pressure. No significant heat transfer occurs from the outer surface of the jacket, and kinetic and potential energy effects are negligible. Determine for the jacketed reactor (a) the mass flow rate of the coolant, in kg per kmol of fuel. (b) the rate of entropy production, in kJ/K per kmol of fuel. (c) the rate of exergy destruction, in kJ per kmol of fuel, for \(T_{0}=25^{\circ} \mathrm{C}\). Consider each of two coolants: water at 1 bar and ammonia at 10 bar.

Liquid ethanol (\(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}\)) at \(25^{\circ} \mathrm{C}\), 1 atm enters a reactor operating at steady state and burns completely with 130% of theoretical air entering in a separate stream at \(25^{\circ} \mathrm{C}\), 1 atm. Combustion products exit at \(227^{\circ} \mathrm{C}\), 1 atm. Heat transfer from the reactor takes place at an average surface temperature \(T_{b}\). For \(T_{b}\) ranging from 25 to \(200^{\circ} \mathrm{C}\), determine the rate of exergy destruction within the reactor, in kJ per kmol of fuel. Kinetic and potential energy effects are negligible. Let \(T_{0}=25^{\circ} \mathrm{C}\).

Determine the change in the Gibbs function, in kJ per kmol of methane, at \(\25^{\circ} \mathrm{C}\), 1 atm for \(\mathrm{CH}_{4}+2 \mathrm{O}_{2} \rightarrow \mathrm{CO}_{2}+2 \mathrm{H}_{2} \mathrm{O}\), using (a) Gibbs function of formation data. (b) enthalpy of formation data, together with absolute entropy data.

A gaseous mixture of ethane (\(\mathrm{C}_{2} \mathrm{H}_{6}\)) and the theoretical amount of air at \(25^{\circ} \mathrm{C}\), 1 atm enters a reactor operating at steady state and burns completely. Combustion products exit at \(627^{\circ} \mathrm{C}\), 1 atm. Heat transfer from the reactor takes place at an average surface temperature \(T_{b}\). For \(T_{b}\) ranging from 25 to \(600^{\circ} \mathrm{C}\), determine the rate of exergy destruction within the reactor, in kJ per kmol of fuel. Kinetic and potential energy effects are negligible. Let \(T_{0}=25^{\circ} \mathrm{C}\).

Determine the change in the Gibbs function, in Btu per lbmol of hydrogen, at \(77^{\circ} \mathrm{F}\), 1 atm for \(\mathrm{H}_{2}+\frac{1}{2} \mathrm{O}_{2} \rightarrow \mathrm{H}_{2} \mathrm{O}(\mathrm{g})\), using (a) Gibbs function of formation data. (b) enthalpy of formation data, together with absolute entropy data.

Separate streams of hydrogen (\(\mathrm{H}_{2}\)) and oxygen (\(\mathrm{O}_{2}\)) at \(25^{\circ} \mathrm{C}\), 1 atm enter a fuel cell operating at steady state, and liquid water exits at \(25^{\circ} \mathrm{C}\), 1 atm. The hydrogen flow rate is \(2 \times 10^{-4} \mathrm{kmol} / \mathrm{s}\). If the fuel cell operates isothermally at \(25 ^{\circ} \mathrm{C}\), determine the maximum theoretical power it can develop and the accompanying rate of heat transfer, each in kW. Kinetic and potential energy effects are negligible.

Streams of methane (\(\mathrm{CH}_{4}\)) and oxygen (\(\mathrm{O}_{2}\)), each at \(\25^{\circ} \mathrm{C}), 1 atm, enter a fuel cell operating at steady state. Streams of carbon dioxide and water exit separately at \(25^{\circ} \mathrm{C}\), 1 atm. If the fuel cell operates isothermally at \(25^{\circ} \mathrm{C}\), 1 atm, determine the maximum theoretical work that it can develop, in kJ per kmol of methane. Ignore kinetic and potential energy effects.

An inventor has developed a device that at steady state takes in liquid water at \(25^{\circ} \mathrm{C}\), 1 atm with a mass flow rate of 4 kg/h and produces separate streams of hydrogen (\(\mathrm{H}_{2}\)) and oxygen (\(\mathrm{O}_{2}\)), each at \(25^{\circ} \mathrm{C}\), 1 atm. The inventor claims that the device requires an electrical power input of 14.6 kW when operating isothermally at \(25^{\circ} \mathrm{C}\). Heat transfer with the surroundings occurs, but kinetic and potential energy effects can be ignored. Evaluate the inventor’s claim.

Showing all important steps, derive (a) Eqs. 13.41a, b (b) Eqs. 13.44 a, b.

Justify the use of Eq. 13.36 for liquid methanol, \(\mathrm{CH}_{3} \mathrm{OH}\), and liquid ethanol, \(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}\), and apply it to evaluate the chemical exergy, in kJ/kmol, of each substance relative to the environment of Prob. 13.91. Compare with the respective standard chemical exergy values from Table A-26 (Model II).

The accompanying table shows an environment consisting of a gas phase and a condensed water phase. The gas phase forms an ideal gas mixture. (a) Show that the chemical exergy of the hydrocarbon \(\mathrm{C}_{a} \mathrm{H}_{b}\) can be determined as \(\begin{aligned} \overline{\mathrm{e}}^{\mathrm{ch}}= & {\left[\bar{g}_{\mathrm{F}}+\left(\mathrm{a}+\frac{\mathrm{b}}{4}\right) \bar{g}_{\mathrm{O}_{2}}-\mathrm{a}_{\mathrm{CO}_{2}}\right.} \\ & \left.-\frac{\mathrm{b}}{2} \bar{g}_{\mathrm{H}_{2} \mathrm{O}(\mathrm{1})}\right]+\bar{R} T_{0} \ln \left[\frac{\left(y_{\mathrm{O}_{2}}^{\mathrm{e}}\right)^{\mathrm{a}+\mathrm{b} / 4}}{\left(y_{\mathrm{CO}_{2}}^{\mathrm{e}}\right)^{\mathrm{a}}}\right] \end{aligned}\) (b) Using the result of part (a), repeat parts (a) through (c) of Problem 13.91.

Applying Eq. 13.36 for (a) carbon, (b) hydrogen (\(\\mathrm{H}_{2}\)), (c) methane, (d) carbon monoxide, (e) nitrogen (\(\mathrm{N}_{2}\)), (f) oxygen (\(\mathrm{O}_{2}\)), and (g) carbon dioxide, determine the chemical exergy, in kJ/kg, relative to the following environment in which the gas phase obeys the ideal gas model:

Evaluate the total specific flow exergy of an equimolar mixture of oxygen (\(\mathrm{O}_{2}\)) and nitrogen (\(\mathrm{N}_{2}\)), in kJ/kg, at \(227^{\circ} \mathrm{C}\), 1 atm. Neglect the effects of motion and gravity. Perform calculations (a) relative to the environment of Problem 13.91. (b) using data from Table A-26 (Model II).

Using data from Tables A-25 and A-26, together with Eq. 13.44b, determine the standard molar chemical exergy, in kJ/kmol, of propane \(\\mathrm{C}_{3} \mathrm{H}_{8})(g). Compare this value with the standard chemical exergy from Table A-26 (Model II).

A mixture of methane gas (\(\mathrm{CH}_{4}\)) and 150% of theoretical air enters a combustion chamber at \(\77^{\circ} \mathrm{F}\), 1 atm. Determine the total specific flow exergy of the entering mixture, in Btu per lbmol of methane. Ignore the effects of motion and gravity. Perform calculations (a) relative to the environment of Problem 13.91. (b) using data from Table A-26 (Model II).

A mixture having an analysis on a molar basis of 85% dry air, 15% CO enters a device at \(\125^{\circ} \mathrm{C}), 2.1 atm, and a velocity of 250 m/s. If the mass flow rate is 1.0 kg/s, determine the rate exergy enters, in MW. Neglect the effect of gravity. Perform calculations (a) relative to the environment of Problem 13.91. (b) using data from Table A-26 (Model II).

Carbon at \(25^{\circ} \mathrm{C}\), 1 atm enters an insulated reactor operating at steady state and reacts completely with the theoretical amount of air entering separately at \(25^{\circ} \mathrm{C}\), 1 atm. For the reactor, (a) determine the rate of exergy destruction, in kJ per kmol of carbon, and (b) evaluate an exergetic efficiency. Perform calculations relative to the environment of Problem 13.91. Neglect the effects of motion and gravity.

The following flow rates in lb/h are reported for the exiting syngas (synthesis gas) stream in a certain process for producing syngas from bituminous coal: If the syngas stream is at 77°F, 1 atm, determine the rate at which exergy exits, in Btu/h. Perform calculations relative to the environment of Problem 13.91. Neglect the effects of motion and gravity.

Carbon monoxide (CO) at \(\25^{\circ} \mathrm{C}), 1 atm enters an insulated reactor operating at steady state and reacts completely with the theoretical amount of air entering in a separate stream at \(25^{\circ} \mathrm{C}\), 1 atm. The products exit as a mixture at 1 atm. Determine in kJ per kmol of CO (a) the exergy entering with the carbon monoxide. (b) the exergy exiting with the products. (c) the rate of exergy destruction. Also, evaluate an exergetic efficiency for the reactor. Perform calculations relative to the environment of Problem 13.91. Neglect the effects of motion and gravity.

Propane gas (\(\mathrm{C}_{3} \mathrm{H}_{8}\)) at \(25^{\circ} \mathrm{C}\), 1 atm and a volumetric flow rate of \(0.03 \mathrm{m}^{3} / \mathrm{min}\) enters a furnace operating at steady state and burns completely with 200% of theoretical air entering at \(25^{\circ} \mathrm{C}\), 1 atm. The furnace provides energy by heat transfer at \(227^{\circ} \mathrm{C}\) for an industrial process and combustion products at \(227^{\circ} \mathrm{C}\), 1 atm for cogeneration of hot water. For the furnace, perform a full exergy accounting, in kJ/min, of the exergy supplied by the fuel. Use standard chemical exergies from Table A-26 (Model II), as required, and ignore the effects of motion and gravity.

Figure P13.104 shows a coal gasification reactor making use of the carbon–steam process. The energy required for the endothermic reaction is supplied by an electrical resistor. The reactor operates at steady state, with no stray heat transfers and negligible effects of motion and gravity. Evaluate in Btu per lbmol of carbon entering (a) the required electrical input. (b) the exergy entering with the carbon. (c) the exergy entering with the steam. (d) the exergy exiting with the product gas. (e) the exergy destruction within the reactor. Perform calculations relative to the environment of Problem 13.91

Acetylene gas (\(\\mathrm{C}_{2} \mathrm{H}_{2}\)) at \(77^{\circ} \mathrm{F}\), 1 atm enters an insulated reactor operating at steady state and burns completely with 180% of theoretical air, entering in a separate stream at \(77^{\circ} \mathrm{F}\), 1 atm. The products exit as a mixture at 1 atm. Determine in Btu per lbmol of fuel (a) the exergy of the fuel entering the reactor. (b) the exergy exiting with the products. (c) the rate of exergy destruction. Also, evaluate an exergetic efficiency for the reactor. Perform calculations relative to the environment of Problem 13.91. Neglect the effects of motion and gravity.

Evaluate the total specific flow exergy of water vapor, in kJ/kg, at \(200^{\circ} \mathrm{C}\), 1 bar. Neglect the effects of motion and gravity. Perform calculations (a) relative to the environment of Problem 13.91. (b) using data from Table A-26 for each of the models.

Liquid octane (\(\mathrm{C}_{8} \mathrm{H}_{18}\)) at \(25^{\circ} \mathrm{C}\), 1 atm and a mass flow rate of 0.57 kg/h enters an internal combustion engine operating at steady state. The fuel burns with air entering the engine in a separate stream at \(25^{\circ} \mathrm{C}\), 1 atm. Combustion products exit at 670 K, 1 atm with a dry molar analysis of 11.4% \(\mathrm{CO}_{2}\), 2.9% CO, 1.6% \(\mathrm{O}_{2}\), and 84.1% \(\mathrm{N}_{2}\). If the engine develops power at the rate of 3 kW, determine (a) the rate of heat transfer from the engine, in kW. (b) an exergetic efficiency for the engine. Use the environment of Problem 13.91 and neglect the effects of motion and gravity.

Figure P13.108 shows a simple vapor power plant. The fuel is methane that enters at \(77^{\circ} \mathrm{F}\), 1 atm and burns completely with 200% theoretical air entering at \(77^{\circ} \mathrm{F}\), 1 atm. Steam exits the steam generator at \(900^{\circ} \mathrm{F}\), \(500 \mathrm{lbf} / \mathrm{in.}^{2}\) The vapor expands through the turbine and exits at \(1 \mathrm{lbf} / \mathrm{in.}^{2}\) , and a quality of 97%. At the condenser exit, the pressure is \(1 \mathrm{lbf} / \mathrm{in.}^{2}\) and the water is a saturated liquid. The plant operates at steady state with no stray heat transfers from any plant component. Pump work and the effects of motion and gravity are negligible. Determine (a) the balanced reaction equation. (b) the vapor mass flow rate, in lb per lbmol of fuel. (c) the cooling water mass flow rate, in lb per lbmol of fuel. (d) each of the following, expressed as a percent of the exergy entering the steam generator with the fuel, (i) the exergy exiting with the stack gases, (ii) the exergy destroyed in the steam generator, (iii) the power developed by the turbine, (iv) the exergy destroyed in the turbine, (v) the exergy exiting with the cooling water, (vi) the exergy destroyed in the condenser. Base exergy values on the environment of Problem 13.91.

For each of the following, use the result of Problem 13.111(a) to determine the total specific flow exergy, in kJ/kg, relative to an environment consisting of moist air at \(20^{\circ} \mathrm{C}\), 1 atm, \(\phi=100 \%\) (a) moist air at \(20^{\circ} \mathrm{C}\), 1 atm, \(\phi=90 \%\). (b) moist air at \(20^{\circ} \mathrm{C}\), 1 atm, \(\phi=50 \%\). (c) moist air at \(20^{\circ} \mathrm{C}\), 1 atm, \(\phi=10 \%\).

For psychrometric applications such as those considered in Chap. 12, the environment often can be modeled simply as an ideal gas mixture of water vapor and dry air at temperature \(T_{0}\) and pressure \(p_{0}\). The composition of the environment is defined by the dry air and water vapor mole fractions \(y_{\mathrm{a}}^{\mathrm{e}}, y_{\mathrm{v}}^{\mathrm{e}}\), respectively. (a) Show that relative to such an environment the total specific flow exergy of a moist air stream at temperature T and pressure p with dry air and water vapor mole fractions \(y_{\mathrm{a}}\) and \(y_{\mathrm{v}}\), respectively, can be expressed on a molar basis as \(\begin{aligned} \overline{\mathrm{e}}_{\mathrm{f}}= & T_{0}\left\{( y _ { \mathrm { a } } \overline { c } _ { p \mathrm { a } } + y _ { \mathrm { v } } \overline { c } _ { p \mathrm { v } } ) \left[\left(\frac{T}{T_{0}}\right)\right.\right. \\ & \left.\left.-1-\ln \left(\frac{T}{T_{0}}\right)\right]+\bar{R} \ln \left(\frac{p}{p_{0}}\right)\right\} \\ & +\bar{R} T_{0}\left[y_{\mathrm{a}} \ln \left(\frac{y_{\mathrm{a}}}{y_{\mathrm{a}}^{\mathrm{e}}}\right)+y_{\mathrm{v}} \ln \left(\frac{y_{\mathrm{v}}}{y_{\mathrm{v}}^{\mathrm{e}}}\right)\right] \end{aligned}\) where \(\bar{c}_{p \mathrm{a}}\) and \(\bar{c}_{p \mathrm{v}}\) denote the molar specific heats of dry air and water vapor, respectively. Neglect the effects of motion and gravity. (b) Express the result of part (a) on a per unit mass of dry air basis as \(\begin{aligned} \mathrm{e}_{\mathrm{f}}= & T_{0}\left\{\left(c_{p \mathrm{a}}+\omega c_{p \mathrm{v}}\right)\left[\frac{T}{T_{0}}-1-\ln \left(\frac{T}{T_{0}}\right)\right]+(1+\widetilde{\omega}) R_{\mathrm{a}} \ln \left(p / p_{0}\right)\right\} \\& +R_{\mathrm{a}} T_{0}\left\{(1+\widetilde{\omega}) \ln \left(\frac{1+\widetilde{\omega}^{\mathrm{e}}}{1+\widetilde{\omega}}\right)+\widetilde{\omega} \ln \left(\frac{\widetilde{\omega}}{\widetilde{\omega}^{\mathrm{e}}}\right)\right\} \end{aligned}\) where \(R_{\mathrm{a}}=\bar{R} / M_{\mathrm{a}} \text { and } \widetilde{\omega}=\omega M_{\mathrm{a}} / M_{\mathrm{v}}=y_{\mathrm{v}} / y_{\mathrm{a}}\).

Evaluate the total specific flow exergy of nitrogen (\(\mathrm{N}_{2}\)), in Btu/lb, at \(200^{\circ} \mathrm{F}\), 4 atm. Neglect the effects of motion and gravity. Perform calculations (a) relative to the environment of Problem 13.91. (b) using data from Table A-26 (Model II).