91Ó°ÊÓ

A recuperator is a heat exchanger that heats the air used in a combustion process by extracting energy from the products of combustion (the flue gas). Consider using a single-pass, cross-flow heat exchanger as a recuperator. Eighty \((80)\) silicon carbide ceramic tubes \((k=20\) \(\mathrm{W} / \mathrm{m} \cdot \mathrm{K})\) of inner and outer diameters equal to 55 and \(80 \mathrm{~mm}\), respectively, and of length \(L=1.4 \mathrm{~m}\) are arranged as an aligned tube bank of longitudinal and transverse pitches \(S_{L}=100 \mathrm{~mm}\) and \(S_{T}=120 \mathrm{~mm}\), respectively. Cold air is in cross flow over the tube bank with upstream conditions of \(V=1 \mathrm{~m} / \mathrm{s}\) and \(T_{c i}=300 \mathrm{~K}\), while hot flue gases of inlet temperature \(T_{\mathrm{h}, \mathrm{I}}=1400 \mathrm{~K}\) pass through the tubes. The tube outer surface is clean, while the inner surface is characterized by a fouling factor of \(R_{f}^{N \prime}=2 \times 10^{-4} \mathrm{~m}^{2} \cdot \mathrm{K} / \mathrm{W}\). The air and flue gas flow rates are \(\dot{m}_{c}=1.0 \mathrm{~kg} / \mathrm{s}\) and \(m_{\mathrm{h}}=1.05 \mathrm{~kg} / \mathrm{s}\), respectively. As first approximations, (1) evaluate all required air properties at \(1 \mathrm{~atm}\) and \(300 \mathrm{~K},(2)\) assume the flue gas to have the properties of air at \(1 \mathrm{~atm}\) and \(1400 \mathrm{~K}\), and (3) assume the tube wall temperature to be at \(800 \mathrm{~K}\) for the purpose of treating the effect of variable properties on convection heat transfer. (a) If there is a \(1 \%\) fuel savings associated with each \(10^{\circ} \mathrm{C}\) increase in the temperature of the combustion air \(\left(T_{c o}\right)\) above \(300 \mathrm{~K}\), what is the percentage fuel savings for the prescribed conditions? (b) The performance of the recuperator is strongly influenced by the product of the overall heat transfer coefficient and the total surface area, UA. Compute and plot \(T_{c, \infty}\) and the percentage fuel savings as a function of UA for \(300 \leq U A \leq 600 \mathrm{~W} / \mathrm{K}\). Without changing the flow rates, what measures may be taken to increase \(U A^{*}\) ?

Step by step solution

01

The convective heat transfer coefficient, h (W / m² · K), can be determined using the following correlation for cross-flow over a cylinder: \[ h = 0.36 * (k / L) * Re_L^{0.6}\] In this correlation, k is the thermal conductivity of the fluid (W/m·K), and Re_L is the Reynolds number, which is given by: \[ Re_L = (\rho * V * L ) / \mu \] where \( \rho \) is the fluid density (kg/m³), V is the fluid velocity (m/s), L is the cylinder length (m), and \( \mu \) is the fluid dynamic viscosity (kg/m·s). #Step 2: Calculate overall heat transfer coefficient, U#

Using the formula for overall heat transfer coefficient, U (W/m²·K), in conjunction with the given fouling factor, R': \[ U = 1 / (1/h + R') \] #Step 3: Calculate total surface area, A#

02

The total surface area A (m²) of the silicon carbide tubes can be calculated using the formula: \[ A = N_t * (pi*D*L) \] where N_t is the number of tubes, D is the diameter of the tube, and L is the length of the tube. #Step 4: Calculate the Log Mean Temperature Difference (LMTD) #

For a cross-flow heat exchanger, the LMTD is given by: \[ LMTD = \frac{(T_{h,i} - T_{c,o}) - (T_{h,o} - T_{c,i})}{ln\left((T_{h,i}-T_{c,o})/(T_{h,o}-T_{c,i})\right) }\] where, \(T_{h,i}\) is the inlet temperature of the hot flue gases \(T_{h,o}\) is the outlet temperature of the hot flue gases \(T_{c,i}\) is the inlet temperature of the cold air \(T_{c,o}\) is the outlet temperature of the cold air #Step 5: Calculate energy balance and find T_c_o #

03

Using the energy balance, we can find the outlet temperature of the cold air: \[ \dot{Q} = \dot{m}_c * c_p_c * (T_{c,o} - T_{c,i}) = \dot{m}_h * c_p_h * (T_{h,i} - T_{h,o}) = U*A*LMTD\] #Step 6: Calculate percentage fuel savings for prescribed conditions#

Using the given relationship, we can find the percentage of fuel savings based on the increase in the combustion air temperature: \[ \frac{\Delta T_{c}}"10^{\circ}C" * 1\% = \% \text{fuel savings}\] #Step 7: Calculate the values of T_c_o and percentage fuel savings as a function of UA #

04

Create a range of UA values and compute the following quantities: T_c_o, \% fuel savings, and plot these against UA using the calculated results. #Step 8: Recommend measures for increasing UA#

To increase the value of UA without changing the flow rates, consider the following options: 1. Increase the number of tubes. 2. Increase the heat transfer surface area by making tubes longer or by increasing the number of fins. 3. Use a material with higher thermal conductivity for tubes to reduce the resistance to heat transfer. 4. Minimize fouling by periodic maintenance to keep the fouling factor as low as possible. 5. Improve the convective heat transfer coefficient by using inserts or modifying the flow channels.

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Key Concepts

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

Overall Heat Transfer Coefficient

The overall heat transfer coefficient, often represented by the symbol U, is a measure that demonstrates the ability of a heat exchanger to conduct heat from the hot side to the cold side. It encompasses the resistance to heat transfer in every component part of the heat exchanger system, such as the walls of the tubes, and any fouling that may occur on these surfaces. The formula for calculating this coefficient is:

\[\begin{equation}U = \frac{1}{\frac{1}{h} + R_f}\end{equation}\]
where h is the heat transfer coefficient for convection and R_f is the fouling factor. The latter is a correction factor that accounts for the degradation of performance over time due to the accumulation of unwanted materials on the heat transfer surfaces. Optimizing the overall heat transfer coefficient is crucial because it directly correlates to the heat exchanger’s efficiency. In the given problem, one can increase the value of U by using materials with higher thermal conductivity for the tubes, thus minimizing the resistance.To explore the impact on performance, changes in the U value could be graphed against the outlet temperature of the cold air and the percentage fuel savings. Thus, a practitioner can assess how different materials or designs affect the efficiency of heat exchange in practical scenarios.

Log Mean Temperature Difference (LMTD)

The Log Mean Temperature Difference, abbreviated as LMTD, is integral to the design and analysis of heat exchangers. It represents the driving force behind the heat exchange process and is used to calculate the total heat transfer when the overall heat transfer coefficient is known. The formula for LMTD in a cross-flow heat exchanger with unmixed streams is:

\[\begin{equation}LMTD = \frac{(T_{h,i} - T_{c,o}) - (T_{h,o} - T_{c,i})}{\ln\left(\frac{T_{h,i}-T_{c,o}}{T_{h,o}-T_{c,i}}\right)}\end{equation}\]
Here, T_h,i and T_h,o are the inlet and outlet temperatures of the hot fluid, respectively, while T_c,i and T_c,o are the inlet and outlet temperatures of the cold fluid, respectively.The significance of LMTD is that it provides a means to address the varying temperature difference along the length of the heat exchanger. In reality, this temperature difference is not constant due to the heat exchange process; thus, the LMTD serves to average the gradient to help estimate the total heat transfer. For students aiming to improve their understanding, concepts like LMTD are critical to grasp, as it directly determines the efficiency of a heat exchanger.

Fouling Factor

Fouling within a heat exchanger is a common issue that can severely impede the system's performance over time. Fouling happens when impurities, such as minerals, biological matter, or corrosion products, accumulate on the heat transfer surfaces. The fouling factor, denoted by R_f, quantifies the thermal resistance introduced by this layer of fouling material. It is essential to consider the fouling factor in the calculation of overall heat transfer coefficient by the equation:\[\begin{equation}U = \frac{1}{\frac{1}{h} + R_f}\end{equation}\]The fouling factor is pivotal when designing the heat exchanger as well as during maintenance planning. By anticipating the fouling factor, designers can ensure that a sufficient surface area is provided to permit effective heat exchange even as fouling develops. Additionally, maintenance procedures can be scheduled to clean and restore surfaces, hence keeping the fouling factor at a manageable level. Incorporating methods such as periodic cleaning or using anti-fouling coatings can help maintain the heat exchanger's efficiency close to its initial capabilities. Therefore, understanding and managing the fouling factor is crucial for sustaining the performance of heat exchanges in practical applications.