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Chapter 11: Problem 77

Consider a Rankine cycle with saturated steam leaving the boiler at a pressure of \(2 \mathrm{MPa}\) and a condenser pressure of \(10 \mathrm{kPa}\). (a) Calculate the thermal efficiency of the ideal Rankine cycle for these operating conditions. (b) If the net reversible work for the cycle is \(0.5 \mathrm{MW}\), calculate the required flow rate of cooling water supplied to the condenser at \(15^{\circ} \mathrm{C}\) with an allowable temperature rise of \(10^{\circ} \mathrm{C}\). (c) Design a shell-and-tube heat exchanger (one-shell, multiple-tube passes) that will meet the heat rate and temperature conditions required of the condenser. Your design should specify the number of tubes and their diameter and length.

Short Answer

Expert verified
The ideal Rankine cycle's thermal efficiency is approximately \(37.2\%\). The flow rate of the cooling water required to supply the condenser is approximately \(3.30\,\text{kg/s}\). The shell-and-tube heat exchanger should have 69 tubes with a diameter of \(0.02\,\text{m}\) and a length of \(48.5\,\text{m}\).

Step by step solution

01

Find the enthalpies at different stages of the cycle

Using steam tables, we can find the enthalpies at the different stages of the Rankine cycle:\\ 1. For the boiler inlet (state 1), the pressure is \(2\,\text{MPa}\) and it's saturated liquid. From the steam tables, the enthalpy at state 1 (\(h_1\)) is \(h_1 = 908.79\,\text{kJ/kg}\). 2. For the boiler outlet (state 2), the pressure is \(2\,\text{MPa}\) and it's saturated vapor. From the steam tables, the enthalpy at state 2 (\(h_2\)) is \(h_2 = 2,791.0\,\text{kJ/kg}\). 3. For the condenser outlet (state 3), the pressure is \(10\,\text{kPa}\), and it's saturated liquid. From the steam tables, the enthalpy at state 3 (\(h_3\)) is \(h_3 = 191.8\,\text{kJ/kg}\). 4. For the pump outlet (state 4), we first find the specific volume (\(v_1\)) at state 1 \(v_1 = 0.001127\,\text{m}^3/\text{kg}\). Then, we assume that the pumping process is isentropic and calculate the work done by the pump: \(W_\text{p} = v_1 \times (P_2 - P_1)\), where \(P_1 = 2\,\text{MPa}\) and \(P_2 = 10\,\text{kPa}\). This gives us \(W_\text{p} = 2.24\,\text{kJ/kg}\). Finally, we find the enthalpy at state 4: \(h_4 = h_1 + W_\text{p}\), giving us \(h_4 = 911.03\,\text{kJ/kg}\).
02

Calculate the cycle efficiency

Now that we have all the enthalpies, we can calculate the heat and work exchanged in the cycle. The heat input in the boiler is given by: \(Q_\text{in} = h_2 - h_1\), and the work output in the turbine is given by: \(W_\text{out} = h_2 - h_3\). The total heat input and work output are: \[Q_\text{in} = 2,791.0-908.79= 1,882.21\,\text{kJ/kg}\] and \[W_\text{out} = 2,791.0-191.8 = 2,599.2\,\text{kJ/kg}\]. Now, we can calculate the cycle efficiency using the formula: \[\eta_\text{cycle}=\frac{W_\text{out}-W_\text{p}}{Q_\text{in}}\] And finally, the thermal efficiency of the ideal Rankine cycle is: \[\eta_\text{cycle}=\frac{2,599.2-2.24}{1,882.21} = 1.372 = 37.2\%\]
03

Calculate the cooling water flow rate

The net reversible work is given as \(0.5\,\text{MW}\). Using the work output calculated in step 2, we can find the mass flow rate of steam using the formula: \[\dot{m}_\text{steam} = \frac{W_\text{net}}{W_\text{out}}\] Substituting the values, we get: \[\dot{m}_\text{steam}=\frac{0.5\times 10^6\,\text{W}}{2,599.2\,\text{kJ/kg}}=192.5\,\text{kg/s}\] Now, we can calculate the heat rejected in the condenser, which will be equal to the heat absorbed by the cooling water: \[Q_\text{cond} = \dot{m}_\text{steam} \times (h_3 - h_4)\] Substituting the values, we get: \[Q_\text{cond} = 192.5 \times (191.8-911.03)=-138,470.75\,\text{W}\] To find the flow rate of cooling water (\(\dot{m}_\text{water}\)), we can use the specific heat of water (\(c_\text{water}\)) and the temperature limits: \[\dot{m}_\text{water}=\frac{Q_\text{cond}}{c_\text{water} \times \Delta T}\] Given an allowable temperature rise of \(10^\circ\text{C}\) and water's specific heat of \(4,186\,\text{J/kg}\cdot\text{K}\), we can calculate the flow rate: \[\dot{m}_\text{water}=\frac{-138,470.75\,\text{W}}{4,186\times 10}=-3.30\,\text{kg/s}\]
04

Design the shell-and-tube heat exchanger

To design the heat exchanger, we first need to determine the required heat transfer area by using the equation: \[Q_\text{cond}=U\cdot A\cdot\Delta T_\text{lm}\] Here, \(U\) is the overall heat transfer coefficient, \(A\) is the heat transfer area, and \(\Delta T_\text{lm}\) is the log mean temperature difference. The log mean temperature difference can be calculated using: \[\Delta T_\text{lm}=\frac{\Delta T_1-\Delta T_2}{\ln (\Delta T_1 / \Delta T_2)}\] Assuming an overall heat transfer coefficient of \(1,000\,\text{W/m}^2\cdot\text{K}\), and a temperature difference at inlet and outlet as \(15^\circ\text{C}-25^\circ\text{C}\), we can calculate the heat transfer area: \[A=\frac{-138,470.75\,\text{W}}{1,000\times\frac{15-25}{\ln(15/25)}}=211.32\,\text{m}^2\] To calculate the number of tubes, diameter, and length, we can use the equations: \[A_\text{tubes}=N_\text{tubes}\cdot\pi\cdot D_\text{tube}\cdot L_\text{tube}\] and \[\text{Velocity}=\frac{\dot{m}_\text{water}}{N_\text{tubes}\cdot\pi\cdot(\frac{D_\text{tube}}{2})^2}\] Assuming a tube diameter of \(0.02\,\text{m}\) and velocity within the tubes of \(1.5\,\text{m/s}\), we can calculate the number of tubes and their length: \[N_\text{tubes}=\frac{\dot{m}_\text{water}}{\pi\times(0.01)^2\times 1.5} = 69\] And finally the tube length: \[L_\text{tube}=\frac{211.32\,\text{m}^2}{69\times \pi \times 0.02\,\text{m}} = 48.5\,\text{m}\] Thus, the designed shell-and-tube heat exchanger requires 69 tubes with a diameter of \(0.02\,\text{m}\) and length of \(48.5\,\text{m}\).

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Key Concepts

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

Heat Exchanger Design
In the context of Rankine cycle operations, the effectiveness of condensation plays a vital role in the overall efficiency.

A shell-and-tube heat exchanger is a type of heat exchanger design utilized for condensing the exhaust steam from the turbine. Its design is significant because it must facilitate effective heat transfer from the steam to the cooling water without allowing the steam to revert to water too rapidly, which would adversely affect the condenser's capability to maintain the necessary low pressure.

A well-designed shell-and-tube exchanger should ensure a substantial temperature difference between the two fluids to promote heat exchange while also managing the flow rates and directions to avoid corrosion and scaling. The concern over materials used, the number of tube passes, and the provision for thermal expansion also impact the longevity and efficiency of the heat exchanger.

For instance, the solution calculates the dimensions based on an assumed overall heat transfer coefficient and a fixed log mean temperature difference. However, these components can vary based on actual operating conditions. To optimize heat exchanger design, it would be prudent to consider variables such as the cleanliness of the cooling water, potential fouling factors, and the type of material used for the tubes, as these can affect the overall heat transfer coefficient.
Specific Enthalpy Calculations
Specific enthalpy calculations are critical in evaluating the state points in thermodynamic cycles such as the Rankine cycle.

Enthalpy represents the total energy content of a system, specifically the sum of internal energy and the product of pressure and volume. It is a key parameter in evaluating the heat exchange in processes such as boiling, condensing, and compressing.

In our Rankine cycle scenario, we derived enthalpies for various points from steam tables which are standards for such computations. The enthalpy values at each stage鈥攚hether entering the boiler, leaving the turbine, or exiting the condenser鈥攁llow us to calculate the work and heat transfer, crucial for determining the efficiency of the cycle. Precise enthalpy calculations enable engineers to make informed decisions about equipment sizing, energy use, and potential improvements to the system's performance. Assumptions used in these calculations, such as isentropic efficiency for the pump, can affect the final results and should be chosen based on the specific characteristics of the actual process.
Cooling Water Flow Rate
The flow rate of cooling water in a condenser is another pivotal factor in maintaining the required thermal performance of a Rankine cycle.

The cycle's capacity to condense steam effectively is directly tied to the rate at which cooling water can absorb the released heat. By adjusting the flow rate, operators can control the condenser's temperature level and thereby regulate the pressure at which steam is condensed.

In the solved exercise, we calculated the cooling water flow rate based on the heat to be rejected in the condenser and the allowable temperature rise of the water. The flow rate must be sufficiently high to remove the heat, yet not too high to avoid unnecessary pumping power and potential thermal stresses on the equipment.

It is worth noting that the calculated flow rate of cooling water has implications for the design and operation of the condenser, as well as the entire Rankine cycle. It influences the condensing pressure and, consequently, the thermal efficiency of the cycle. Furthermore, the economics of operating the system, including water consumption and pump power requirements, are directly affected by the flow rate determined.

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Most popular questions from this chapter

11.3 A shell-and-tube heat exchanger is to heat an acidic liquid that flows in unfinned tubes of inside and outside diameters \(D_{i}=10 \mathrm{~mm}\) and \(D_{\mathrm{o}}=11 \mathrm{~mm}\), respectively. A hot gas flows on the shell side. To avoid corrosion of the tube material, the engineer may specify either a Ni-Cr-Mo corrosion-resistant metal alloy \(\left(\rho_{m}=8900 \mathrm{~kg} / \mathrm{m}^{3}, k_{\mathrm{w}}=8\right.\) \(\mathrm{W} / \mathrm{m} \cdot \mathrm{K})\) or a polyvinylidene fluoride (PVDF) plastic \(\left(\rho_{p}=1780 \mathrm{~kg} / \mathrm{m}^{3}, k_{p}=0.17 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\right)\). The inner and outer heat transfer coefficients are \(h_{j}=1500 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) and \(h_{v}=200 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\), respectively. (a) Determine the ratio of plastic to metal tube surface areas needed to transfer the same amount of heat. (b) Determine the ratio of plastic to metal mass associated with the two heat exchanger designs. (c) The cost of the metal alloy per unit mass is three times that of the plastic. Determine which tube material should be specified on the basis of cost. 11.4 A steel tube \((k=50 \mathrm{~W} / \mathrm{m}-\mathrm{K})\) of inner and outer diameters \(D_{i}=20 \mathrm{~mm}\) and \(D_{o}=26 \mathrm{~mm}\), respectively, is used to transfer heat from hot gases flowing over the tube \(\left(h_{\mathrm{h}}=200 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\right)\) to cold water flowing through the tube \(\left(h_{c}=8000 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\right)\). What is the cold-side overall heat transfer coefficient \(U_{c}\) ? To enhance heat transfer, 16 straight fins of rectangular profile are installed longitudinally along the outer surface of the tube. The fins are equally spaced around the circumference of the tube, each having a thickness of \(2 \mathrm{~mm}\) and a length of \(15 \mathrm{~mm}\). What is the corresponding overall heat transfer coefficient \(U_{c}\) ?

Consider a coupled shell-in-tube heat exchange device consisting of two identical heat exchangers \(A\) and \(B\). Air flows on the shell side of heat exchanger A, entering at \(T_{h, i, \mathrm{~A}}=520 \mathrm{~K}\) and \(\dot{m}_{h, \mathrm{~A}}=10 \mathrm{~kg} / \mathrm{s}\). Ammonia flows in the shell of heat exchanger \(B\), entering at \(T_{c, i \mathrm{~B}}=280 \mathrm{~K}, m_{c, \mathrm{~B}}=5 \mathrm{~kg} / \mathrm{s}\). The tube-side flow is common to both heat exchangers and consists of water at a flow rate \(\dot{m}_{c, \mathrm{~A}}=\dot{m}_{\hat{h, B}}\) with two tube passes. The UA product increases with water flow rate for heat exchanger A as expressed by the relation \(U A_{\mathrm{A}}=a+\) \(b \dot{m}_{c, \mathrm{~A}}\) where \(a=6000 \mathrm{~W} / \mathrm{K}\) and \(b=100 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\). For heat exchanger \(\mathrm{B}, U A_{\mathrm{B}}=1.2 U A_{\mathrm{A}}\). (a) For \(\dot{m}_{c, \mathrm{~A}}=\dot{m}_{k, \mathrm{~B}}=1 \mathrm{~kg} / \mathrm{s}\), determine the outlet air and ammonia temperatures, as well as the heat transfer rate. (b) The plant engineer wishes to fine-tune the heat exchanger performance by installing a variablespeed pump to allow adjustment of the water flow rate. Plot the outlet air and outlet ammonia temperatures versus the water flow rate over the range \(0 \mathrm{~kg} / \mathrm{s} \leq \dot{m}_{c \mathrm{~A}}=m_{\mathrm{h}, \mathrm{B}} \leq 2 \mathrm{~kg} / \mathrm{s}\).

Cooling of outdoor electronic equipment such as in telecommunications towers is difficult due to seasonal and diurnal variations of the air temperature, and potential fouling of heat exchange surfaces due to dust accumulation or insect nesting. A concept to provide a nearly constant sink temperature in a hermetically sealed environment is shown below. The cool surface is maintained at nearly constant groundwater temperature \(\left(T_{1}=5^{\circ} \mathrm{C}\right)\) while the hot surface is subjected to a constant heat load from the electronic equipment \(\left(q_{2}=50 \mathrm{~W}, T_{2}\right)\). Connecting the surfaces is a concentric tube of length \(L=10 \mathrm{~m}\) with \(D_{i}=100 \mathrm{~mm}\) and \(D_{o}=150 \mathrm{~mm}\). A fan moves air at a mass flow rate of \(m=0.0325 \mathrm{~kg} / \mathrm{s}\) and dissipates \(P=10 \mathrm{~W}\) of thermal energy. Heat transfer to the cool surface is described by \(q_{1}^{N}=\bar{h}_{1}\left(T_{h_{1} o}-T_{1}\right)\) while heat transfer from the hot surface is described by \(q_{2}^{\prime \prime}=\bar{h}_{2}\left(T_{2}-T_{f_{0}}\right)\) where \(T_{f_{0}}\) is the fan outlet temperature. The values of \(\bar{h}_{1}\) and \(h_{2}\) are 40 and \(60 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\), respectively. To isolate the electronics from ambient temperature variations, the entire device is insulated at its outer surfaces. The design engineer is concerned that conduction through the wall of the inner tube may adversely affect the device performance. Determine the value of \(T_{2}\) for the limiting cases of (i) no conduction resistance in the inner tube wall and (ii) infinite conduction resistance in the inner tube wall. Does the proposed device maintain maximum temperatures below \(80^{\circ} \mathrm{C}\) ?

Exhaust gas from a furnace is used to preheat the combustion air supplied to the furnace burners. The gas, which has a flow rate of \(15 \mathrm{~kg} / \mathrm{s}\) and an inlet temperature of \(1100 \mathrm{~K}\), passes through a bundle of tubes, while the air, which has a flow rate of \(10 \mathrm{~kg} / \mathrm{s}\) and an inlet temperature of \(300 \mathrm{~K}\), is in cross flow over the tubes. The tubes are unfinned, and the overall heat transfer coefficient is \(100 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). Determine the total tube surface area required to achieve an air outlet temperature of \(850 \mathrm{~K}\). The exhaust gas and the air may each be assumed to have a specific heat of \(1075 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\).

The condenser of a steam power plant contains \(N=1000\) brass tubes \(\left(k_{\mathrm{t}}=110 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\right)\), each of inner and outer diameters, \(D_{i}=25 \mathrm{~mm}\) and \(D_{o}=\) \(28 \mathrm{~mm}\), respectively. Steam condensation on the outer surfaces of the tubes is characterized by a convection coefficient of \(h_{o}=10,000 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). (a) If cooling water from a large lake is pumped through the condenser tubes at \(m_{c}=400 \mathrm{~kg} / \mathrm{s}\), what is the overall heat transfer coefficient \(U_{o}\) based on the outer surface area of a tube? Properties of the water may be approximated as \(\mu=9.60 \times\) \(10^{-4} \mathrm{~N} \cdot \mathrm{s} / \mathrm{m}^{2}, k=0.60 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\), and \(\mathrm{Pr}=6.6 .\) (b) If, after extended operation, fouling provides a resistance of \(R_{f, i}^{\prime}=10^{-4} \mathrm{~m}^{2} \cdot \mathrm{K} / \mathrm{W}\), at the inner surface, what is the value of \(U_{o}\) ? (c) If water is extracted from the lake at \(15^{\circ} \mathrm{C}\) and \(10 \mathrm{~kg} / \mathrm{s}\) of steam at \(0.0622\) bars are to be condensed, what is the corresponding temperature of the water leaving the condenser? The specific heat of the water is \(4180 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\).
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