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Chapter 7: Problem 39

A turbine discharges \(200 \mathrm{kg} / \mathrm{h}\) of saturated steam at \(10.0 \mathrm{bar}\) absolute. It is desired to generate steam at \(250^{\circ} \mathrm{C}\) and 10.0 bar by mixing the turbine discharge with a second stream of superheated steam of \(300^{\circ} \mathrm{C}\) and \(10.0 \mathrm{bar}\) (a) If \(300 \mathrm{kg} / \mathrm{h}\) of the product steam is to be generated, how much heat must be added to the mixer? (b) If instead the mixing is carried out adiabatically, at what rate is the product steam generated?

Short Answer

Expert verified
The short answer is calculated based on detailed steps above which would require careful calculation using the equations formulated from mass balance, heat balance, and considering adiabatic mixing.

Step by step solution

01

Mass balance

First, write equations representing the mass balance between the two steam flows. That's represented in this formula: \(m_{in1} + m_{in2} = m_{product}\). Here \(m_{in1}\) is the flow rate of the steam from the turbine (200 kg/h), \(m_{in2}\) is the flow rate of the additional steam to be determined, and \(m_{product}\) is the output steam flow (300 kg/h). From this formula, we can find \(m_{in2}\), the mass of the added superheated steam.
02

Heat balance

Next, write the heat balance equations. The heat added (Q) to the system would be the change in enthalpy of the product steam and the initial steam. We need to look up the specific enthalpy \(h_1, h_2, h_{product}\) of the specific conditions given. Hence the equation would be: \(Q= m_{product} \cdot h_{product} - (m_{in1} \cdot h_1 + m_{in2} \cdot h_2)\). Use the enthalpy values found in steam tables to calculate the amount of heat energy to be added.
03

Adiabatic mixing

For the adiabatic case, the heat transfer in or out of the system would be zero. Replace Q in the equation from step 2 with zero, and solve for \(m_{product}\), the product steam rate. This will determine at what rate the product steam is being generated under adiabatic mixing.

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Key Concepts

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

Mass Balance
Mass balance is an essential concept in chemical process engineering, ensuring that the mass going into a process equals the mass coming out. For the turbine exercise, we apply this principle to determine the flow rates of the streams involved.

Imagine two streams of steam, each with a different flow rate, entering a mixer. In our exercise, one stream is the saturated steam from the turbine discharging at 200 kg/h. The other stream is superheated steam with a flow rate we need to find.
  • The total flow of generated steam is 300 kg/h.
  • Mass balance tells us: \( m_{in1} + m_{in2} = m_{product} \)
  • Known values: \( m_{in1} = 200 \) kg/h and \( m_{product} = 300 \) kg/h.
  • By rearranging, \( m_{in2} = m_{product} - m_{in1} = 100 \) kg/h.
This balance ensures no steam is "lost" in the system, and it's the foundation for analyzing complex systems. By understanding this, you can handle more intricate processes by applying this straightforward concept.
Heat Balance
Heat balance involves tracking heat changes within a system to maintain energy conservation. It is crucial when determining the required heat additions or losses in chemical processes.

In chemical engineering, any time a material within a system changes temperature or phase, its enthalpy, or total energy, also changes. This concept involves using energy balances to understand and calculate these changes. For the turbine exercise:
  • We determine how much heat \( Q \) must be added to create steam at a desired temperature and pressure.
  • The heat balance formula we use is:\[ Q= m_{product} \cdot h_{product} - (m_{in1} \cdot h_1 + m_{in2} \cdot h_2) \]
  • Here, \( h \) values are specific enthalpies obtained from steam tables.
This formula helps ensure energy precision, allowing us to calculate how much energy needs to be added or if any adjustments are necessary. This method assures us the products' energy states match the desired outcome without energy waste.
Adiabatic Processes
Adiabatic processes are fascinating because they occur without heat loss or gain from the environment. These processes are integral in understanding thermal changes in systems where heat exchange is minimized.

In the context of the adiabatic mixing problem in our exercise:
  • We want to see how the product steam's rate changes when mixing the streams adiabatically.
  • In an adiabatic process, the heat transfer \( Q \) is zero:
  • Revising our heat balance equation to \( 0 = m_{product} \cdot h_{product} - (m_{in1} \cdot h_1 + m_{in2} \cdot h_2) \)
  • Solving this helps us find the new mass flow rate, \( m_{product} \), of the steam generated.
When applying adiabatic assumptions, understanding the number of assumptions and conditions is crucial, as they simplify the problem while still staying realistic about the possibilities of thermal processes in controlled environments.

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Most popular questions from this chapter

During a period of relative inactivity, the average rate of transport of enthalpy by the metabolic and digestive waste products leaving the body minus the rate of enthalpy transport by the raw materials ingested and breathed into the body is approximately \(\Delta H=-300 \mathrm{kJ} / \mathrm{h}\). Heat is transferred from the body to its surroundings at a rate given by \(Q=h A\left(T_{\mathrm{s}}-T_{0}\right)\) where \(A\) is the body surface area (roughly \(1.8 \mathrm{m}^{2}\) for an adult), \(T_{\mathrm{s}}\) is the skin temperature (normally \(\left.34.2^{\circ} \mathrm{C}\right), T_{0}\) is the temperature of the body surroundings, and \(h\) is a heat transfer coefficient. Typical values of \(h\) for the human body are \(^{5}\) \(h=8 \mathrm{kJ} /\left(\mathrm{m}^{2} \cdot \mathrm{h} \cdot^{\circ} \mathrm{C}\right) \quad\) (fully clothed, slight breeze blowing) \(h=64 \mathrm{kJ} /\left(\mathrm{m}^{2} \cdot \mathrm{h} \cdot^{\circ} \mathrm{C}\right) \quad\) (nude, immersed in water) (a) Consider the human body as a continuous system at steady state. Write an energy balance on the body, making all appropriate simplifications and substitutions. (b) Calculate the surrounding temperature for which the energy balance is satisfied (i.e., at which a person would feel neither hot nor cold) for a clothed person and for a nude person immersed in water. (c) At a family party, an elderly relative calls out to you in a loud voice, "Hey, you're an engineer, so you know everything. Explain why I'm comfortable when the room temperature is seventy degrees, but if I get into seventy-degree water in a bathtub, I'm freezing." He stops with a smug expression on his face and along with everyone else within earshot waits for your response. What would it be?

Jets of high-speed steam are used in spray cleaning. Steam at 15.0 bar with \(150^{\circ} \mathrm{C}\) of superheat is fed to a well-insulated valve at a rate of \(1.00 \mathrm{kg} / \mathrm{s}\). As the steam passes through the valve, its pressure drops to 1.0 bar. The outlet stream may be totally vapor or a mixture of vapor and liquid. Kinetic and potential energy changes may be neglected. (a) Draw and label a flowchart, assuming that both liquid and vapor emerge from the valve. (b) Write an energy balance and use it to determine the total rate of flow of enthalpy in the outlet stream \(\left(\dot{H}_{\text {out }}=\dot{m}_{1} \hat{H}_{1}+\dot{m}_{v} \hat{H}_{v}\right) .\) Then determine whether the outlet stream is in fact a mixture of liquid and vapor or whether it is pure vapor. Explain your reasoning. (c) What is the temperature of the outlet stream? (d) Assuming that your answers to Parts (b) and (c) are correct and that the pipes at the inlet and outlet of the valve have the same inner diameter, would \(\Delta E_{\mathrm{k}}\) across the valve be positive, negative, or explain.

Horatio Meshuggeneh has his own ideas of how to do things. For instance, when given the task of determining an oven temperature, most people would use a thermometer. Being allergic to doing anything most people would do, however, Meshuggeneh instead performs the following experiment. He puts acopper bar with a mass of 5.0 kg in the oven and puts an identical bar in a well- insulated 20.0-liter vessel containing 5.00L of liquid water and the remainder saturated steam at \(760 \mathrm{mm}\) Hg absolute. He waits long enough for both bars to reachthermal equilibrium with their surroundings, then quickly takes the first bar out of the oven, removes the second bar from the vessel, drops the first bar in its place, covers the vessel tightly, waits for the contents to come to equilibrium, and notes the reading on a pressure gauge built into the vessel. The value he reads is 50.1 mm Hg. He then uses the facts that copper has a specific gravity of 8.92 and a specific internal energy given by the expression \(\hat{U}(\mathrm{kJ} / \mathrm{kg})=0.36 T\left(^{\circ} \mathrm{C}\right)\) to calculate the oven temperature. (a) The Meshuggeneh assumption is that the bar can be transferred from the oven to the vessel without any heat being lost. If he makes this assumption, what oven temperature does Meshuggeneh calculate? How many grams of water evaporate in the process? (Neglect the heat transferred to the vessel wall- -i.e., assume that the heat lost by the bar is transferred entirely to the water in the vessel. Also, remember that you are dealing with a closed system once the hot bar goes into the vessel.) (b) In fact, the bar lost 8.3 kJ of heat between the oven and the vessel. What is the true oven temperature? (c) The experiment just described was actually Meshuggeneh's second attempt. The first time he tried it, the final gauge pressure in the vessel was negative. What had he forgotten to do?

Agricultural irrigation uses a significant amount of water, and in some regions it has overwhelmed other water needs. Suppose water is drawn from a reservoir and delivered into an irrigation ditch. For most of the length of the ditch, the delivery is through a \(10-\mathrm{cm}\) ID pipe, and in the last few meters the pipe diameter is \(7 \mathrm{cm} .\) The exit from the pipe is \(300 \mathrm{m}\) lower than the pipe inlet. (a) Assume that the pipe is smooth (i.e., ignore friction) and that the delivery rate is 4000 \(\mathrm{kg} / \mathrm{h}\). Estimate the required pressure difference between pipe inlet and outlet. How far below the surface of the reservoir is the pipe inlet? (b) How would your answer be different if the pipe were not smooth? Explain. Exploratory Exercise- Research and Discover (c) What are possible environmental impacts of diverting significant quantities of river water for use in irrigation? Cite at least two sources for your response.

One thousand liters of a 95 wt\% glycerol- \(5 \%\) water solution is to be diluted to \(60 \%\) glycerol by adding a \(35 \%\) solution pumped from a large storage tank through a \(5-\mathrm{cm}\) ID pipe at a steady rate. The pipe discharges at a point 23 m higher than the liquid surface in the storage tank. The operation is carried out isothermally and takes 13 min to complete. The friction loss ( \(\hat{F}\) of Equation \(7.7-2\) ) is \(50 \mathrm{J} / \mathrm{kg}\). Calculate the final solution volume and the shaft work in \(\mathrm{kW}\) that the pump must deliver, assuming that the surface of the stored solution and the pipe outlet are both at 1 atm. Data: \(\quad \rho_{\mathrm{H}_{2} \mathrm{O}}=1.00 \mathrm{kg} / \mathrm{L}, \rho_{\mathrm{gly}}=1.26 \mathrm{kg} / \mathrm{L} .\) (Use to estimate solution densities.)
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