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Chapter 8: Problem 65

Saturated propane vapor at \(2.00 \times 10^{2}\) psia is fed to a well- insulated heat exchanger at a rate of \(3.00 \times 10^{3} \mathrm{SCFH}\) (standard cubic feet per hour). The propane leaves the exchanger as a saturated liquid (i.e., a liquid at its boiling point) at the same pressure. Cooling water enters the exchanger at \(70^{\circ} \mathrm{F},\) flowing cocurrently (in the same direction) with the propane. The temperature difference between the outlet streams (liquid propane and water) is \(15^{\circ} \mathrm{F}\). (a) What is the outlet temperature of the water stream? (Use the Antoine equation.) Is the outlet water temperature less than or greater than the outlet propane temperature? Briefly explain. (b) Estimate the rate (Btu/h) at which heat must be transferred from the propane to the water in the heat exchanger and the required flow rate \(\left(1 \mathrm{b}_{\mathrm{m}} / \mathrm{h}\right)\) of the water. (You will need to write two separate energy balances.) Assume the heat capacity of liquid water is constant at \(1.00 \mathrm{Btu} /\left(\mathrm{lb}_{\mathrm{m}} \cdot^{\circ} \mathrm{F}\right)\) and neglect heat losses to the outside and the effects of pressure on the heat of vaporization of propane.

Short Answer

Expert verified
The outlet temperature of the water stream is \(85^{\circ} F\), which is greater than the outlet propane temperature indicating heat transfer from propane to water. The heat transfer rate and water flow rate need to be calculated step by step as per the solution.

Step by step solution

01

Determine the boiling point of propane

Using the Antoine equation and the given pressure, the boiling point of propane is obtained. The Antoine equation relates the vapour pressure and temperature for a pure substance. The constants for propane (found in thermodynamics book) can be used to solve the equation. This temperature is the outlet temperature for the propane as it's a saturated liquid.
02

Determine the outlet temperature of the water stream

The outlet temperature of the water stream is determined by adding the temperature difference between the outlet streams to the temperature at which the water enters the exchanger. Therefore, outlet water temperature = inlet water temperature + temperature difference = \(70^{\circ} F + 15^{\circ} F = 85^{\circ} F\).
03

Compare the outlet water and propane temperatures

The outlet water temperature is compared to the outlet propane temperature. If the water temperature is higher it means that heat transfer has occurred from the propane to the water.
04

Determine the heat transfer rate

The heat transfer rate is determined by the equation: Q = m * Cp * (T_out - T_in), where m is the mass flow rate of propane, Cp is the heat capacity of propane, T_out is the outlet temperature, and T_in is the inlet temperature. The mass flow rate of propane is given in SCFH and should be converted to lbm/h using conversion factors. Assume the Cp of propane to be near that of saturated liquid and can be found in thermodynamics book. The outlet and inlet temperatures are in \(^{\circ} F\).
05

Estimate the required water flow rate

A separate energy balance is written for the cooling water. Assuming its heat capacity is constant at 1 Btu / (lbm * \(^{\circ} F\)), the flow rate is the unknown. The formula used is: m_water = Q / (Cp_water * (T_out - T_in)).

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Key Concepts

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

Heat Transfer
Heat transfer is a fundamental concept in thermodynamics that involves the movement of thermal energy from one substance to another. In this exercise, heat is transferred from propane to water in a heat exchanger.
Propane enters as saturated vapor and leaves as saturated liquid, implying it has released its latent heat to the water.
  • Heat transfer can occur through conduction, convection, or radiation.
  • In our context, convection is most common as propane and water flow along the heat exchanger.
To calculate heat transfer, you'll often use equations involving the mass flow rate, specific heat capacities, and temperature differences. A well-understood concept here ensures accurate control and optimization in systems like HVAC or energy generation.
Energy Balance
An energy balance involves setting up equations to account for energy entering and exiting a system, ensuring conservation of energy according to the first law of thermodynamics.
For the heat exchanger in this problem, you need to consider separate energy balances for propane and water.
  • For propane, you'll calculate the energy it releases when condensing from vapor to liquid.
  • For the water, you'll calculate the energy it gains as it absorbs the heat from propane.
The energy lost by propane must equal the energy gained by water, considering the heat exchanger is well-insulated and there are no losses to the surroundings.
Phase Change
Phase change refers to the transition of a substance from one phase to another, like from gas to liquid, which happens to propane in this scenario.
When propane undergoes phase change, it goes from vapor to liquid as it cools at constant pressure.
  • During this process, propane releases latent heat, which is absorbed by the cooling water.
  • This phase change occurs at a specific temperature known as the boiling point, which stays constant for a pure substance at a given pressure.
Understanding phase change is crucial, as it involves large amounts of energy transfer, indicating systems like heat exchangers are optimized for energy efficiency.
Antoine Equation
The Antoine equation is crucial in determining the relationship between vapor pressure and temperature for substances like propane.
This equation helps determine the boiling point at a given pressure, a key part of solving the problem.
  • The Antoine equation is expressed as: \( \log_{10}(P) = A - \frac{B}{C+T} \), where \( P \) is the vapor pressure, \( T \) the temperature, and \( A, B, C \) are substance-specific constants.
  • For propane, these constants can be found in thermodynamics tables.
Using this equation helps you accurately calculate temperatures for phase changes, ensuring precise energy management in thermal systems.

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Most popular questions from this chapter

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 mixture of \(n\) -hexane vapor and air leaves a solvent recovery unit and flows through a \(70-\mathrm{cm}\) diameter duct at a velocity of \(3.00 \mathrm{m} / \mathrm{s}\). At a sampling point in the duct the temperature is \(40^{\circ} \mathrm{C}\), the pressure is \(850 \mathrm{mm}\) Hg, and the dew point of the sampled gas is \(25^{\circ} \mathrm{C}\). The gas is fed to a condenser in which it is cooled at constant pressure, condensing \(70 \%\) of the hexane in the feed. (a) Perform a degree-of-freedom analysis to show that enough information is available to calculate the required condenser outlet temperature \(\left(^{\circ} \mathrm{C}\right)\) and cooling rate \((\mathrm{kW})\) (b) Perform the calculations. (c) If the feed duct diameter were \(35 \mathrm{cm}\) for the same molar flow rate of the feed gas, what would be the average gas velocity (volumetric flow rate divided by cross-sectional area)? (d) Suppose you wanted to increase the percentage condensation of hexane for the same feed stream. Which three condenser operating variables might you change, and in which direction?

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.

A natural gas containing 95 mole \(\%\) methane and the balance ethane is burned with \(20.0 \%\) excess air. The stack gas, which contains no unburned hydrocarbons or carbon monoxide, leaves the furnace at \(900^{\circ} \mathrm{C}\) and \(1.2 \mathrm{atm}\) and passes through a heat exchanger. The air on its way to the furnace also passes through the heat exchanger, entering it at \(20^{\circ} \mathrm{C}\) and leaving it at \(245^{\circ} \mathrm{C}\). (a) Taking as a basis \(100 \mathrm{mol} / \mathrm{s}\) of the natural gas fed to the furnace, calculate the required molar flow rate of air, the molar flow rate and composition of the stack gas, the required rate of heat transfer in the preheater, \(\dot{Q}\) (write an energy balance on the air), and the temperature at which the stack gas leaves the preheater (write an energy balance on the stack gas). Note: The problem statement does not give you the fuel feed temperature. Make a reasonable assumption, and state why your final results should be nearly independent of what you assume. (b) What would \(\dot{Q}\) be if the actual feed rate of the natural gas were 350 SCMH [standard cubic meters per hour, \(\left.\mathrm{m}^{3}(\mathrm{STP}) / \mathrm{h}\right] ?\) Scale up the flowchart of Part (a) rather than repeating the entire calculation.

A sheet of cellulose acetate film containing 5.00 wt\% liquid acetone enters an adiabatic dryer where \(90 \%\) of the acetone evaporates into a stream of dry air flowing over the film. The film enters the dryer at \(T_{\mathrm{f} 1}=35^{\circ} \mathrm{C}\) and leaves at \(T_{\mathrm{f} 2}\left(^{\circ} \mathrm{C}\right) .\) The air enters the dryer at \(T_{\mathrm{al}}\left(^{\circ} \mathrm{C}\right)\) and 1.01 atm and exits the dryer at \(T_{\mathrm{a} 2}=49^{\circ} \mathrm{C}\) and 1 atm with a relative saturation of \(40 \% . C_{p}\) may be taken to be \(1.33 \mathrm{kJ} /\left(\mathrm{kg} \cdot^{\circ} \mathrm{C}\right)\) for dry film and \(0.129 \mathrm{kJ} /\left(\mathrm{mol} \cdot^{\circ} \mathrm{C}\right)\) for liquid acetone. Make a reasonable assumption regarding the heat capacity of dry air. The heat of vaporization of acetone may be considered independent of temperature. Take a basis of \(100 \mathrm{kg}\) film fed to the dryer for the requested calculations. (a) Estimate the feed ratio [liters dry air (STP)/kg dry film]. (b) Derive an expression for \(T_{\mathrm{al}}\) in terms of the film temperature change, \(\left(T_{\mathrm{f} 2}-35\right),\) and use it to answer Parts (c) and (d). (c) Calculate the film temperature change if the inlet air temperature is \(120^{\circ} \mathrm{C}\). (d) Calculate the required value of \(T_{\mathrm{al}}\) if the film temperature falls to \(34^{\circ} \mathrm{C},\) and the value if it rises to \(36^{\circ} \mathrm{C}.\) (e) If you solved Parts (c) and (d) correctly, you found that even though the air temperature is consistently higher than the film temperature in the dryer, so that heat is always transferred from the air to the film, the film temperature can drop from the inlet to the outlet. How is this possible?
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