91Ó°ÊÓ

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 \(4180 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\).

Step by step solution

01

Compute the inner heat transfer coefficient (h_i)

Using the Sieder-Tate correlation, we can estimate the inner heat transfer coefficient (h_i) as: \(h_{i}=0.023\left(\frac{4 m_{c}}{\pi D_{i}}\right)^{0.8} \mathrm{Pr}^{0.3}\frac{k}{D_{i}}\) We can plug in values for mc, Di, and Pr.

02

Compute the overall heat transfer coefficient without fouling (U_o)

Now, we can compute U_o using the resistance network as follows: \(\frac{1}{U_{o}}=R_{i}+R_{m}+R_{o}\) Where, \(R_{i}=\frac{1}{h_{i} A_{i}}\) \(R_{m}=\frac{\ln (D_{o} / D_{i})}{2 \pi k_{t} L}\) \(R_{o}=\frac{1}{h_{o} A_{o}}\) In which Ai and Ao are the inner and outer tube surface areas, respectively. Calculate each resistance term, then find U_o accordingly. #b) Calculating the Overall Heat Transfer Coefficient (U_o) with fouling#

03

Recalculate overall heat transfer coefficient (U_o) with fouling:

Now, we need to account for the added resistance due to fouling on the inner surface (R_f,i). The new overall heat transfer coefficient will be: \(\frac{1}{U_{o}'}=R_{i}+R_{f, i}+R_{m}+R_{o}\) Compute U_o' as described earlier, including the fouling resistance R_f,i. #c) Calculating the temperature of the water leaving the condenser#

04

Apply energy balance to the condenser:

Assuming no heat lost to the surroundings, we can write the energy balance equation as: \(m_{w} C_{p,w}(T_{out} - T_{in}) = m_{s} h_{fg}\) Where: mw = mass flow rate of the cooling water, Cp,w = specific heat of water, Tin = inlet temperature of the cooling water, Tout = outlet temperature of the cooling water, ms = mass flow rate of the steam, hfg = latent heat of vaporization. We are given water mass flow rate, inlet temperature, steam mass flow rate, and can find the latent heat of vaporization using steam tables for the given pressure. Now, solve for Tout.

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Key Concepts

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

Understanding Condenser Heat Transfer

The transfer of heat in a condenser is an essential aspect of thermal management in systems such as power plants and refrigeration cycles. In the context of a steam power plant, the condenser serves as a device to convert the exhaust steam from the turbine into liquid water by removing heat from the steam.

For an effective heat transfer process, we consider various coefficients that characterize how well heat is transferred from the steam to the cooling water. An important coefficient is the convection coefficient \( h_o \), which describes the heat transfer capability of the steam at the condenser outer surface. This coefficient greatly influences the overall heat transfer coefficient \( U_o \), which is a measure of the total heat transfer resistance including the condenser tube material and the characteristics of the inner fluid flow.

The efficiency of condensers is often limited by fouling, which affects the heat transfer and can be accounted for by adding fouling resistance into the calculation of \( U_o \). This process requires a detailed and multidimensional analysis of each component's effect on the overall efficiency of the heat transfer. The final goal is to ensure steam is effectively condensed while minimizing any losses in energy transfer.

The Impact of Fouling Resistance

Over time, the inner surfaces of the condenser tubes can accumulate deposits, a phenomenon known as fouling. This layer of unwanted material adds thermal resistance to the heat transfer process, described as fouling resistance \( R_{f, i} \). Fouling can drastically reduce the efficiency of a condenser by forming an insulating layer that impedes the flow of heat.

Fouling resistance increases the overall resistance to heat transfer and thus lowers the overall heat transfer coefficient \( U_o \). When calculating the impact of fouling, we add this resistance to the existing network of thermal resistances within the condenser. The greater the fouling resistance, the more significant the effect on the overall performance of the condenser, necessitating regular maintenance or cleaning to maintain heat transfer efficiency.

Energy Balance in Condensers

An energy balance in a condenser accounts for the energy transferred from the steam to the cooling water within the system. In an ideal scenario, the energy lost by the steam, which condenses as it loses heat, equals the energy gained by the cooling water, as it absorbs heat.

The energy balance can be mathematically expressed using the conservation of energy principle. The mass flow rate of the cooling water \( m_w \), its specific heat capacity \( C_{p,w} \), and the difference between its outlet and inlet temperatures \( T_{out} - T_{in} \), should equal the mass flow rate of the steam \( m_s \) multiplied by its latent heat of vaporization \( h_{fg} \).

This balance allows us to predict the exit temperature of the cooling water, \( T_{out} \), based on the amount of steam being condensed. If the amount of condensed steam increases, we expect the exit temperature of the cooling water to rise accordingly, assuming the specific heat and mass flow rate remain constant. Regular monitoring and adjustments based on energy balance are crucial for maintaining the desired performance levels of the condenser.

Applying the Sieder-Tate Correlation

The Sieder-Tate correlation is an empirical relationship used to calculate the internal heat transfer coefficient \( h_i \) for turbulent flow inside tubes. This correlation takes into account fluid properties, tube dimensions, and flow characteristics to provide an estimate of the heat transfer coefficient.

According to the correlation, \( h_i \) is proportional to \(0.023\left(\frac{4 m_c}{\pi D_i}\right)^{0.8} \Pr^{0.3}\frac{k}{D_i}\), where \( m_c \) is the mass flow rate of the cooling water, \( D_i \) is the inner diameter of the tube, \( \Pr \) is the Prandtl number, and \( k \) is the thermal conductivity of the fluid. By plugging in these values, we can calculate \( h_i \) and subsequently determine the overall heat transfer coefficient \( U_o \) for the condenser.

The Sieder-Tate correlation is particularly useful when precise measurements of the heat transfer coefficient are not available, allowing engineers and students alike to estimate the performance of heat exchange equipment with reasonable accuracy. However, it’s important to note that this correlation is valid under certain conditions, such as fully developed turbulent flow and assuming that the physical properties of the fluid are constant across the tube.