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Chapter 8: Problem 5

Estimate the specific enthalpy of steam (kJ/kg) at \(100^{\circ} \mathrm{C}\) and 1 atm relative to steam at \(350^{\circ} \mathrm{C}\) and 100 bar using: (a) The steam tables. (b) Table B.2 or APEx and assuming ideal-gas behavior. What is the physical significance of the difference between the values of \(\hat{H}\) calculated by the two methods?

Short Answer

Expert verified
The specific enthalpy of steam differs when calculated using steam tables versus the ideal gas equation due to the assumptions inherent in the ideal gas law which becomes less accurate at higher temperatures and pressures. The discrepancy between the values indicates the extent to which these assumptions are invalid under the given conditions.

Step by step solution

01

Using steam tables

First, refer to the steam tables to find the specific enthalpy values directly at the given conditions. To do this,\n\nFor steam at \(100^{\circ} \mathrm{C}\) and 1 atm, read the specific enthalpy, \( \hat{H}_{100C,1atm} \), directly from the steam tables. \n\nNext, for steam at \(350^{\circ} \mathrm{C}\) and 100 bar, read the specific enthalpy, \( \hat{H}_{350C,100bar} \), directly from the steam tables.\n\nThe difference between the two can then be calculated as follows: \( \Delta\hat{H}_{1} = \hat{H}_{100C,1atm} - \hat{H}_{350C,100bar} \).
02

Using Table B.2 or APEx

Next, calculate the specific enthalpy assuming ideal gas behavior using an appropriate equation of state. This typically involves calculating specific enthalpy as \( \hat{H} = C_{p}T \) where \(C_{p}\) is the heat capacity at constant pressure and T is temperature in Kelvin. The heat capacities can be found in Table B.2 or through APEx software.\n\nFind the specific enthalpy at the two conditions and then calculate the difference as follows: \( \Delta\hat{H}_{2} = \hat{H}_{100C,1atm} - \hat{H}_{350C,100bar} \).
03

Analyzing the results

Once the specific enthalpies are calculated using both methods, discuss the difference between the values and the physical significance of this difference.\n\nWhy might there be a discrepancy? Remember, steam tables are more accurate because they account for the specific behavior of water. By contrast, the ideal gas assumption, while easier to calculate, is less accurate because it assumes that the gas behaves ideally, which is not often the case in real life, especially for steam at high temperatures and pressures. Remember also to consider if the calculated values are reasonable, given known values for the specific enthalpy of steam.

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Key Concepts

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

Understanding Specific Enthalpy
Specific enthalpy is a crucial concept in thermodynamics, especially when dealing with energy transfer in processes involving gases and vapors. It represents the total energy content of a substance for a given mass and is expressed in units like kJ/kg.
Specific enthalpy combines two forms of energy:
  • Internal energy (the energy stored within the molecules)
  • Flow work (the energy required to push the fluid into or out of a control volume)
The formula for specific enthalpy is often expressed as:\[ \hat{H} = U + PV \]Where:
  • \(U\) is the specific internal energy
  • \(P\) is pressure
  • \(V\) is specific volume
Knowing specific enthalpy helps engineers and scientists calculate the efficient transfer of energy in open systems like turbines and engines.
Reading Steam Tables
Steam tables are essential tools for thermodynamic calculations involving water and steam. They provide the specific enthalpy values at various temperatures and pressures, which are not always straightforward to calculate from first principles.
  • Steam tables give data for both the saturated and superheated states of water and steam.
  • They help avoid complexities in calculating thermodynamic properties and ensure precision.
To use a steam table effectively:
  1. Identify the state of steam (saturated or superheated)
  2. Locate the corresponding temperature and pressure in the table
  3. Read off the specific enthalpy values provided
Steam tables are more reliable for calculating properties in applications because they are derived from experimental data specific to water and steam under various states.
Exploring Ideal Gas Behavior
The ideal gas law is a simplified model used to describe the behavior of gases. An ideal gas is an imaginary gas that perfectly follows the gas law equation:\[ PV = nRT \]Though this model is often idealized, it can provide reasonable approximations for real gases under certain conditions.
  • It assumes no interactions between gas particles (no attractions or repulsions).
  • It presumes that the volume of individual gas molecules is negligible compared to the volume of the container.
While this approach simplifies calculations, it's important to note:
  • Real gases deviating significantly from ideal behavior at high pressures and low temperatures.
  • For steam, especially at high pressures and temperatures, such deviations can lead to inaccuracies in enthalpy calculations.
Thus, care must be taken when applying ideal gas principles, particularly for substances like steam.
The Process of Enthalpy Calculation
Calculating enthalpy, whether through steam tables or ideal gas equations, involves an understanding of both empirical data and mathematical approximations.
  • Using steam tables gives immediate results based on conducted experiments.
  • Ideal gas calculations involve integrating specific heat capacity over a temperature range:
The formula for enthalpy at constant pressure is:\[ \hat{H} = C_p(T_2 - T_1) \]Where:
  • \(C_p\) is the heat capacity at constant pressure
  • \(T_2\) and \(T_1\) are the final and initial temperatures, respectively
By comparing these methods, one gains insight into the practical applications of thermodynamics and the trade-offs between accuracy and simplicity.
Explaining Heat Capacity
Heat capacity is a material-specific property that signifies the amount of heat required to change the temperature of an object by a given amount. It is an essential factor in calculating specific enthalpy for thermodynamic systems.
  • At constant pressure, it is denoted as \(C_p\) and at constant volume as \(C_v\).
  • The heat capacity at constant pressure is pertinent in many enthalpy calculations as it appears frequently in the ideal gas formula for enthalpy.
Knowing the heat capacity helps determine how much energy is needed to affect the temperature, and assists in comprehending the thermal properties of gases and vapors during processes.Understanding how heat capacity changes with temperature and pressure helps refine thermodynamic calculations and develop better models of energy exchange.

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Most popular questions from this chapter

The heat required to raise the temperature of \(m\) (kg) of a liquid from \(T_{1}\) to \(T_{2}\) at constant pressure is $$ Q=\Delta H=m \int_{T_{1}}^{T_{2}} C_{p}(T) d T $$ In high school and in first-year college physics courses, the formula is usually given as $$ Q=m C_{p} \Delta T=m C_{p}\left(T_{2}-T_{1}\right) $$ (a) What assumption about \(C_{p}\) is required to go from Equation 1 to Equation \(2 ?\) (b) The heat capacity \(\left(C_{p}\right)\) of liquid \(n\) -hexane is measured in a bomb calorimeter. A small reaction flask (the bomb) is placed in a well- insulated vessel containing \(2.00 \mathrm{L}\) of liquid \(n-\mathrm{C}_{6} \mathrm{H}_{14}\) at \(T=300 \mathrm{K} .\) A combustion reaction known to release \(16.73 \mathrm{kJ}\) of heat takes place in the bomb, and the subsequent temperature rise of the system contents is measured and found to be \(3.10 \mathrm{K}\). In a separate experiment, it is found that \(6.14 \mathrm{kJ}\) of heat is required to raise the temperature of everything in the system except the hexane by \(3.10 \mathrm{K}\). Use these data to estimate \(C_{p}[\mathrm{kJ} /(\mathrm{mol} \cdot \mathrm{K})]\) for liquid \(n\) -hexane at \(T \approx 300 \mathrm{K},\) assuming that the condition required for the validity of Equation 2 is satisfied. Compare your result with a tabulated value.

Saturated steam at \(300^{\circ} \mathrm{C}\) is used to heat a countercurrently flowing stream of methanol vapor from \(65^{\circ} \mathrm{C}\) to \(260^{\circ} \mathrm{C}\) in an adiabatic heat exchanger. The flow rate of the methanol is 6500 standard liters per minute, and the steam condenses and leaves the heat exchanger as liquid water at \(90^{\circ} \mathrm{C}.\) (a) Calculate the required flow rate of the entering steam in \(\mathrm{m}^{3} / \mathrm{min}\). (b) Calculate the rate of heat transfer from the water to the methanol ( \(\mathrm{kW}\) ). (c) Suppose the outlet temperature of the methanol is measured and found to be \(240^{\circ} \mathrm{C}\) instead of the specified value of \(260^{\circ} \mathrm{C}\). List five possible realistic explanations for the \(20^{\circ} \mathrm{C}\) difference. 7 An adiabatic heat exchanger is one for which no heat is exchanged with the surroundings. All of the heat lost by the hot stream is transferred to the cold stream.

The brakes on an automobile act by forcing brake pads, which have a metal support and a lining, to press against a disk (rotor) attached to the wheel. Friction between the pads and the disk causes the car to slow or stop. Each wheel has an iron brake disk with a mass of \(15 \mathrm{lb}_{\mathrm{m}}\) and two brake pads, each having a mass of \(11 \mathrm{b}_{\mathrm{m}}\). (a) Suppose an automobile is moving at 55 miles per hour when the driver suddenly applies the brakes and brings the car to a rapid halt. Take the heat capacity of the disk and brake pads to be \(0.12 \mathrm{Btu} /\left(\mathrm{lb}_{\mathrm{m}} \cdot^{\circ} \mathrm{F}\right)\) and assume that the car stops so rapidly that heat transfer from the disk and pads has been insignificant. Estimate the final temperature of the disk and pads if the car is (i) a Toyota Camry, which has a mass of about \(3200 \mathrm{Ib}_{\mathrm{m}},\) or (ii) a Cadillac Escalade, which has a mass of about \(5.900 \mathrm{lb}_{\mathrm{m}}.\) (b) Why are the linings on brake pads no longer made of asbestos? Your answer should provide information on specific issues or concerns caused by the use of asbestos.

Propane is to be burned with \(25.0 \%\) excess air. Before entering the furnace, the air is preheated from \(32^{\circ} \mathrm{F}\) to \(575^{\circ} \mathrm{F}\) (a) At what rate (B tu/h) must heat be transferred to the air if the feed rate of propane is \(1.35 \times 10^{5}\) SCFH (ft \(^{3} / \mathrm{h}\) at \(\mathrm{STP}\) )? (b) The stack gas leaves the furnace at \(855^{\circ} \mathrm{F}\). How is the air likely to be preheated?

A liquid stream containing 50.0 mole \(\%\) benzene and the balance toluene at \(25^{\circ} \mathrm{C}\) is fed to a continuous single-stage evaporator at a rate of \(1320 \mathrm{mol} / \mathrm{s}\). The liquid and vapor streams leaving the evaporator are both at \(95.0^{\circ} \mathrm{C}\). The liquid contains 42.5 mole \(\%\) benzene and the vapor contains 73.5 mole\% benzene. (a) Calculate the heating requirement for this process in \(\mathrm{kW}\). (b) Using Raoult's law (Section 6.4b) to describe the equilibrium between the vapor and liquid outlet streams, determine whether or not the given benzene analyses are consistent with each other. If they are, calculate the pressure (torr) at which the evaporator must be operating; if they are not, give several possible explanations for the inconsistency.
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