/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 98 An air-cooled steam condenser is... [FREE SOLUTION] | 91影视

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Chapter 7: Problem 98

An air-cooled steam condenser is operated with air in cross flow over a square, in-line array of 400 tubes \(\left(N_{L}=N_{T}=20\right.\) ), with an outside tube diameter of \(20 \mathrm{~mm}\) and longitudinal and transverse pitches of \(S_{L}=60 \mathrm{~mm}\) and \(S_{T}=30 \mathrm{~mm}\), respectively. Saturated steam at a pressure of \(2.455\) bars enters the tubes, and a uniform tube outer surface temperature of \(T_{s}=390 \mathrm{~K}\) may be assumed to be maintained as condensation occurs within the tubes. (a) If the temperature and velocity of the air upstream of the array are \(T_{i}=300 \mathrm{~K}\) and \(V=4 \mathrm{~m} / \mathrm{s}\), what is the temperature \(T_{o}\) of the air that leaves the array? As a first approximation, evaluate the properties of air at \(300 \mathrm{~K}\). (b) If the tubes are \(2 \mathrm{~m}\) long, what is the total heat transfer rate for the array? What is the rate at which steam is condensed in \(\mathrm{kg} / \mathrm{s}\) ? (c) Assess the effect of increasing \(N_{L}\) by a factor of 2 , while reducing \(S_{L}\) to \(30 \mathrm{~mm}\). For this configuration, explore the effect of changes in the air velocity.

Short Answer

Expert verified
In this air-cooled steam condenser problem, we can determine the outlet air temperature using the effectiveness-NTU relation, given by \(T_o = T_s - (T_s - T_i) \cdot e^{(- NTU)}\). Using the given parameters and the appropriate formulas, we can calculate the total heat transfer rate \(Q = \dot{m}_a c_p (T_o - T_i)\) and the rate of steam condensation \(\dot{m}_s = \frac{Q}{h_{fg}}\). To assess the effect of changing the parameters and air velocities, repeat these calculations with the modified parameters.

Step by step solution

01

Given parameters

We are given the following parameters: - Number of tubes: \(N_L = N_T = 20\) - Outside tube diameter: \(D_o = 20 \, \text{mm}\) - Longitudinal pitch: \(S_L = 60 \, \text{mm}\) - Transverse pitch: \(S_T = 30 \, \text{mm}\) - Steam pressure: \(P = 2.455 \, \text{bars}\) - Tube surface temperature: \(T_s = 390 \, \text{K}\) - Initial air temperature: \(T_i = 300 \, \text{K}\) - Air velocity: \(V = 4 \, \text{m/s}\) (a) We need to find the temperature \(T_o\) of the air leaving the array.
02

Calculate heat transfer coefficient

For air properties, at \(T_i = 300 \, \text{K}\), using a steam table, we have: - Specific heat capacity: \(c_p = 1005 \, \text{J/kgK}\) - Air viscosity: \(\mu = 1.846 \times 10^{-5} \, \text{kg/ms}\) - Air thermal conductivity: \(k_a = 0.02624 \, \text{W/mK}\) - Air density: \(\rho_a = 1.177 \, \text{kg/m}^3\) Calculate the Reynolds number based on air properties: \(Re = \frac{\rho_a V D_o}{\mu}\) Calculate the Nusselt number based on Reynolds number: \(Nu = \frac{2}{3} + \frac{0.4 Re^{3/4} Pr_{a}^{1/3}}{\left(1 + \left(\frac{0.05 Re}{Pr_{a}}\right)^{1/2}\right)^{4/1}}\) Where \(Pr_a\) is the Prandtl number of air, calculated as: \(Pr_a = \frac{c_p \mu}{k_a}\) Calculate the heat transfer coefficient (using Nusselt number): \(h_a = \frac{k_a}{D_o} Nu\)
03

Determine outlet air temperature

We can use an effectiveness-NTU relation to find the outlet air temperature, i.e., the temperature of air leaving the array: \(T_o = T_s - (T_s - T_i) \cdot e^{(- NTU)}\) Where \(NTU = \frac{h_a A_t}{\dot{m}_a c_p} = \frac{h_a N_L N_T \pi D_o L}{\dot{m}_a c_p}\) Here, \(A_t\) is the total tube outer surface area, and \(\dot{m}_a\) is the air mass flow rate. We can calculate the air mass flow rate as: \(\dot{m}_a = \rho_a V (N_L S_T)\) After calculating \(T_o\), we can proceed to part (b) of the problem. (b) We need to find the total heat transfer rate and the rate at which steam is condensed.
04

Calculate heat transfer rate

The total heat transfer rate for the array can be calculated using the formula: \(Q = \dot{m}_a c_p (T_o - T_i)\)
05

Calculate rate of steam condensation

The rate at which steam is condensed can be calculated using: \(\dot{m}_s = \frac{Q}{h_{fg}}\) Where \(h_{fg}\) is the enthalpy of vaporization at the given pressure. We can obtain this value from a steam table. (c) Finally, we need to assess the effect of doubling \(N_L\) while reducing \(S_L\) to \(30 \, \text{mm}\), as well as the effect of changing the air velocity. Repeat steps 2-5 with the new parameters and air velocities to assess their impact on the performance of the steam condenser.

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Key Concepts

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

Steam Condenser
Steam condensers play a crucial role in power plants and various thermal systems, where they are primarily used to efficiently convert steam into liquid water. This conversion helps in conserving water and maximizing thermal efficiency. By reducing steam back to water, the steam cycle can continue without the need for constant re-boiling, thus saving energy and optimizing system performance.
In a typical setup, the steam passes through a series of tubes where heat is transferred from the steam to a cooling medium, usually air or water. The `air-cooled steam condenser` of our concern uses air in cross-flow over the tubes to achieve the condensation process. This specific exchange of heat is critical because it affects not only the efficiency of the condenser itself but also the overall performance of the entire system it's part of.
Nusselt Number
The Nusselt Number ( Nu ) is a dimensionless number that represents the ratio of convective to conductive heat transfer across a boundary. It is a crucial parameter in analyzing heat transfer in flows over surfaces, as it helps ascertain the effectiveness of the heat transfer process. For a steam condenser, understanding Nu helps gauge how well the air transfers heat from the tube surface.
In our exercise, calculating the Nu requires first determining the Reynolds Number, because it profiles the flow condition, which is then used alongside other factors in a Nusselt correlation formula. Essentially, a higher Nusselt number indicates a more efficient heat transfer process, necessary for the design and optimization of heat exchangers, including our air-cooled steam condenser.
Reynolds Number
The Reynolds Number ( Re ) is another pivotal dimensionless parameter in fluid mechanics, used to predict flow patterns in different fluid flow situations. It is determined by the ratio of inertial forces to viscous forces within a fluid, helping differentiate between laminar and turbulent flow regimes. In the context of our steam condenser setup, Re is calculated using air properties such as density, velocity, and viscosity, in conjunction with the tube diameter.
Understanding whether the flow is laminar or turbulent is crucial. Turbulent flows tend to have higher rates of heat transfer. This behavior helps govern how we adjust conditions to improve heat exchanger performance. Changes in tube arrangements or flow velocity might aim to achieve a desirable Re that optimizes the system鈥檚 heat transfer capabilities.
Thermal Conductivity
Thermal conductivity ( k ) measures a material's ability to conduct heat. It's expressed in watts per meter kelvin ( W/mK ). In the context of the air-cooled steam condenser, the thermal conductivity of air ( k_a ) is vital in determining how effectively heat can be transferred from the steam to the air.
The air's thermal conductivity influences the calculation of Nu and, directly, the heat transfer coefficient ( h_a ), highlighting why k is integral to the understanding heat transfer mechanisms. For condenser designs, selecting materials with suitable thermal conductivity can have profound effects on performance, as materials that effectively transfer heat can enhance the overall efficiency of the condenser process. Ultimately, precise knowledge of thermal conductivity supports improved design and operation strategies.

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Most popular questions from this chapter

In an extrusion process, copper wire emerges from the extruder at a velocity \(V_{e}\) and is cooled by convection heat transfer to air in cross flow over the wire, as well as by radiation to the surroundings. (a) By applying conservation of energy to a differential control surface of length \(d x\), which either moves with the wire or is stationary and through which the wire passes, derive a differential equation that governs the temperature distribution, \(T(x)\), along the wire. In your derivation, the effect of axial conduction along the wire may be neglected. Express your result in terms of the velocity, diameter, and properties of the wire \(\left(V_{e}, D, \rho, c_{p}, \varepsilon\right)\), the convection coefficient associated with the cross flow \((\bar{h})\), and the environmental temperatures \(\left(T_{w}, T_{\text {sur }}\right)\). (b) Neglecting radiation, obtain a closed form solution to the foregoing equation. For \(V_{c}=0.2 \mathrm{~m} / \mathrm{s}\), \(D=5 \mathrm{~mm}, V=5 \mathrm{~m} / \mathrm{s}, T_{\infty}=25^{\circ} \mathrm{C}\), and an initial wire temperature of \(T_{i}=600^{\circ} \mathrm{C}\), compute the temperature \(T_{o}\) of the wire at \(x=L=5 \mathrm{~m}\). The density and specific heat of the copper are \(\rho=8900 \mathrm{~kg} / \mathrm{m}^{3}\) and \(c_{p}=400 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\), while properties of the air may be taken to be \(k=0.037 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \nu=3 \times\) \(10^{-5} \mathrm{~m}^{2} / \mathrm{s}\), and \(P r=0.69\). (c) Accounting for the effects of radiation, with \(\varepsilon=0.55\) and \(T_{\text {sur }}=25^{\circ} \mathrm{C}\), numerically integrate the differential equation derived in part (a) to determine the temperature of the wire at \(L=5 \mathrm{~m}\). Explore the effects of \(V_{e}\) and \(\varepsilon\) on the temperature distribution along the wire.

The cylindrical chamber of a pebble bed nuclear reactor is of length \(L=10 \mathrm{~m}\), and diameter \(D=3 \mathrm{~m}\). The chamber is filled with spherical uranium oxide pellets of core diameter \(D_{p}=50 \mathrm{~mm}\). Each pellet generates thermal energy in its core at a rate of \(\dot{E}_{g}\) and is coated with a layer of non-heat-generating graphite, which is of uniform thickness \(\delta=5 \mathrm{~mm}\), to form a pebble. The uranium oxide and graphite each have a thermal conductivity of \(2 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\). The packed bed has a porosity of \(\varepsilon=0.4\). Pressurized helium at 40 bars is used to absorb the thermal energy from the pebbles. The helium enters the packed bed at \(T_{i}=450^{\circ} \mathrm{C}\) with a velocity of \(3.2 \mathrm{~m} / \mathrm{s}\). The properties of the helium may be assumed to be \(c_{p}=5193 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\), \(k=0.3355 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \rho=2.1676 \mathrm{~kg} / \mathrm{m}^{3}, \mu=4.214 \times\) \(10^{-5} \mathrm{~kg} / \mathrm{s} \cdot \mathrm{m}, \operatorname{Pr}=0.654\). (a) For a desired overall thermal energy transfer rate of \(q=125 \mathrm{MW}\), determine the mean outlet temperature of the helium leaving the bed, \(T_{o}\), and the amount of thermal energy generated by each pellet, \(\dot{E}_{g^{*}}\) (b) The amount of energy generated by the fuel decreases if a maximum operating temperature of approximately \(2100^{\circ} \mathrm{C}\) is exceeded. Determine the maximum internal temperature of the hottest pellet in the packed bed. For Reynolds numbers in the range \(4000 \leq R e_{D} \leq 10,000\), Equation \(7.81\) may be replaced by \(\varepsilon \bar{j}_{H}=2.876 R e_{D}^{-1}+0.3023 R e_{D}^{-0.35}\).

Worldwide, over a billion solder balls must be manufactured daily for assembling electronics packages. The uniform droplet spray method uses a piezoelectric device to vibrate a shaft in a pot of molten solder that, in turn, ejects small droplets of solder through a precision-machined nozzle. As they traverse a collection chamber, the droplets cool and solidify. The collection chamber is flooded with an inert gas such as nitrogen to prevent oxidation of the solder ball surfaces. (a) Molten solder droplets of diameter \(130 \mu \mathrm{m}\) are ejected at a velocity of \(2 \mathrm{~m} / \mathrm{s}\) at an initial temperature of \(225^{\circ} \mathrm{C}\) into gaseous nitrogen that is at \(30^{\circ} \mathrm{C}\) and slightly above atmospheric pressure. Determine the terminal velocity of the particles and the distance the particles have traveled when they become completely solidified. The solder properties are \(\rho=8230 \mathrm{~kg} / \mathrm{m}^{3}\), \(c=240 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}, k=38 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, h_{s f}=42 \mathrm{~kJ} / \mathrm{kg}\). The solder's melting temperature is \(183^{\circ} \mathrm{C}\). (b) The piezoelectric device oscillates at \(1.8 \mathrm{kHz}\), producing 1800 particles per second. Determine the separation distance between the particles as they traverse the nitrogen gas and the pot volume needed in order to produce the solder balls continuously for one week.

Consider a liquid metal \((\operatorname{Pr} \leqslant 1)\), with free stream conditions \(u_{\infty}\) and \(T_{\infty}\), in parallel flow over an isothermal flat plate at \(T_{s}\). Assuming that \(u=u_{\infty}\) throughout the thermal boundary layer, write the corresponding form of the boundary layer energy equation. Applying appropriate initial \((x=0)\) and boundary conditions, solve this equation for the boundary layer temperature field, \(T(x, y)\). Use the result to obtain an expression for the local Nusselt number \(\mathrm{Nu}_{x}\). Hint: This problem is analogous to one-dimensional heat transfer in a semiinfinite medium with a sudden change in surface temperature.

Consider the velocity boundary layer profile for flow over a flat plate to be of the form \(u=C_{1}+C_{2} y\). Applying appropriate boundary conditions, obtain an expression for the velocity profile in terms of the boundary layer thickness \(\delta\) and the free stream velocity \(u_{\infty}\). Using the integral form of the boundary layer momentum equation (Appendix G), obtain expressions for the boundary layer thickness and the local friction coefficient, expressing your result in terms of the local Reynolds number. Compare your results with those obtained from the exact solution (Section 7.2.1) and the integral solution with a cubic profile (Appendix \(G\) ).
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