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Module 2: Characteristics of Gases - Heat Capacity and Enthalphy
Practice Problems
Objective
- Determine the change in enthalpy of a gas stream when it changes temperature.
The heat capacity of a gas is the amount of heat required to change the temperature of a unit-mass of gas one temperature degree. Common units of heat capacity are provided below.
Actually, the numerical values for specific heat are the same regardless of which system of units is used. The same is true for molar heat capacity. For example, the molar heat capacity expressed in Btus per pound mole degree Fahrenheit is identical to the heat capacity expressed in calories per gram mole degree Celsius. This can be demonstrated by substituting the appropriate correction factors and deriving one set of dimensional units from another.
Example Problem 1.
Similarity of Heat Capacity Data in Cgs and American Engineering Units
The heat capacity of water vapor at 400°F is 8.3 cal/[(gm mole)(°C)]. What is the heat capacity expressed in Btu/[(lb mole)(°F)]?
Solution:
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Equation 1 can be represented in symbols as follows:
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The heat capacities of common gaseous compounds are different due to the differences in the ways the molecules take up energy in vibrational, rotational, and translational forms. The heat capacities also vary as a function of gas temperature. These compound-by-compound differences and the influence of temperature are illustrated in Figure 1.
The heat capacity data shown in Figure 1 provide a convenient way to visualize the differences between compounds such as carbon dioxide and water vapor and other common gaseous compounds. It is a measure of the quantity of energy needed to increase the temperatures of a specific quantity of material by one degree.
Enthalpy represents the total quantity of internal energy, such as heat, measured for a specific quantity of material at a given temperature. Enthalpy data are often represented in units of energy (e.g. Btu, kcal, joule, etc.). The enthalpy content change is often expressed in Btu/unit mass (Btu/lbm) or Btu/unit gas volume (Btu/SCF). The change in enthalpy of the total quantity of material present in a system is expressed in units of Btu/unit time (Btu/min). The symbols, H and 
H, denote enthalpy and the change in enthalpy, respectively.
Table 2 provides enthalpy data for a number of common gases. For example, one pound of ambient air at 500°F and kept at standard pressure is given an enthalpy value of 106.7 Btu. One pound of water vapor at 500°F and kept at standard pressure is given an enthalpy value of 154.3 Btu. If one pound of air at 500°F were added to one pound of water vapor at 500°F, the total enthalpy of the two-pound gas mixture would be 261 Btu/lbm (106.7 Btu/lbm for air and 154.3 Btu/lbm for water vapor).
It is often necessary to use enthalpy data for a temperature value that is between the values listed in the left column of Table 2. The enthalpy value at a specific temperature can be calculated by interpolating the data in the Table above. The general equation used for interpolating the data is shown below.
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Example Problem 2.
Interpolation of Data in Table 2
Calculate the enthalpy of oxygen at 800°F as follows.
Solution:
To calculate the change in enthalpy of a volume of gas as it changes temperature from Temperature 1 (T1) to Temperature 2 (T2), simply multiply the mass of the gas volume by the difference in enthalpies of this specific gas mixture at the two temperatures. Equations 4 and 5 can be used to calculate enthalpy changes of a gas volume as its temperature changes.
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Enthalpy data is useful in a number of different situations in the field of air pollution control. They can be used in energy balances to evaluate temperature changes in an air pollution control system. Enthalpy data are often used in combustion calculations to determine the amount of heat that must be added to a gas stream in order to achieve the desired operating temperature (see Example Problem 3). Enthalpy data can also be used to estimate both the heating and cooling requirements for hot gas streams entering air pollution control systems (see Example Problem 4).
Converting Volumetric Flow Rates to Mass and Molar Rates
When making heat capacity and enthalpy calculations, you will often have to convert volumetric flow rates (i.e. ACFM, SCFM, m3/min, or Nm3/min) to molar flow rates (e.g. lb mole/min) or mass flow rates (e.g. lbm/min). These conversions are necessary due to the units of measure for heat capacity (per molar basis) and enthalpy (per mass basis).
Introducing units of time to the ideal gas law shows the relationship between volumetric and molar flow rates. Start with the following rearranged version of the ideal gas law:
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This is identical to the following equation that states that the number of moles per unit time (molar flow rate) is equivalent to the gas volume per unit time multiplied by (P/RT). The gas volume per minute (V/min) is simply the gas flow rate, which is expressed in SCFM when standard conditions of temperature and pressure are applied.
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The relationships provided below are derived from ideal gas law calculations for the volume of one mole of gas at standard temperature and pressure. (See Example Problems 1 and 2 in the Ideal Gas Law Lesson.)
For American Engineering Systems:
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For Cgs System:
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Example Problem 3.
Gas Stream Preheating Requirements
How much heat must be added to a 1,000 SCFM gas stream to raise the temperature of a VOC-laden air stream from 100°F to 700°F to allow for oxidation of the organic compounds in a catalytic incinerator? Use an enthalpy content of 9.6 Btu/lbm for the VOC-laden air at 100°F and an enthalpy content of 156.7 Btu/lbm for the VOC-laden air at 700°F. Assume the molecular weight of air is 29 lbm/lb mole.
Solution:
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Convert the gas flow rate data to the mass flow rate, lbm/min.
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Calculate the change in enthalpy needed, H.
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Calculate the total amount of heat required.
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Example Problem 4.
Gas Stream Cooling Requirements
A 3,000 SCFM gas stream containing 45% CO2 and 55% N2 must be cooled from 1,000°F to 100°F prior to entering a carbon adsorber to lower concentrations of organic compound contaminants. How much heat must be removed in the condenser to achieve this temperature reduction? Use the enthalpy data in Table 2.
Solution:
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Convert the gas flow rate data to lbm/min for each major constituent.
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Calculate the change in enthalpy (H) corresponding to lowering the temperature of the gas from 1,000°F to 100°F. For enthalpy data, see Table 2.
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Practice Problems
Heat Capacity and Enthalpy
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Instructions:
- Complete the Practice Problems before proceeding to the next lesson. Click on the button below.