91Ó°ÊÓ

It is common practice to recover waste heat from an oilor gas-fired furnace by using the exhaust gases to preheat the combustion air. A device commonly used for this purpose consists of a concentric pipe arrangement for which the exhaust gases are passed through the inner pipe, while the cooler combustion air flows through an annular passage around the pipe. Consider conditions for which there is a uniform heat transfer rate per unit length, \(q_{i}^{\prime}=1.25 \times 10^{5} \mathrm{~W} / \mathrm{m}\), from the exhaust gases to the pipe inner surface, while air flows through the annular passage at a rate of \(\dot{m}_{a}=2.1 \mathrm{~kg} / \mathrm{s}\). The thin- walled inner pipe is of diameter \(D_{i}=2 \mathrm{~m}\), while the outer pipe, which is well insulated from the surroundings, is of diameter \(D_{o}=2.05 \mathrm{~m}\). The air properties may be taken to be \(c_{p}=1030 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}, \mu=270 \times 10^{-7} \mathrm{~N} \cdot \mathrm{s} / \mathrm{m}^{2}, k=0.041\) \(\mathrm{W} / \mathrm{m} \cdot \mathrm{K}\), and \(P r=0.68\). (a) If air enters at \(T_{a, 1}=300 \mathrm{~K}\) and \(L=7 \mathrm{~m}\), what is the air outlet temperature \(T_{a, 2}\) ? (b) If the airflow is fully developed throughout the annular region, what is the temperature of the inner pipe at the inlet \(\left(T_{s, i, 1}\right)\) and outlet \(\left(T_{s, i, 2}\right)\) sections of the device? What is the outer surface temperature \(T_{s, \theta, 1}\) at the inlet?

Step by step solution

01

Find the air outlet temperature (Ta2)

Using the heat transfer rate per unit length and the air mass flow rate, we can find the temperature increase in the air across the length of the pipe. Then, we add this temperature increase to the initial air temperature to find the air outlet temperature. Calculate the total heat transfer rate from the exhaust gases to the annular air passage: \(Q = q_{i}^{\prime} \times L\) Now we need to find the temperature increase (∆T) using the mass flow rate, specific heat capacity, and the heat transfer rate: \(\Delta T = \frac{Q}{\dot{m}_{a} \times c_{p}}\) Finally, calculate the air outlet temperature Ta2 using the initial air temperature (Ta1) and the temperature increase: \(T_{a,2} = T_{a,1} + \Delta T\)

02

Determine the temperature of the inner pipe at the inlet (Ts,i,1) and outlet (Ts,i,2)

We can find the temperature of the inner pipe at the inlet and outlet by using the heat transfer rate and the air properties. Use the given uniform heat transfer rate per unit length to find the total heat transfer rate at the inlet and outlet sections: \(q_{i,1}^{\prime} = q_{i,2}^{\prime} = q_{i}^{\prime}\) Then, determine the convection heat transfer coefficient (h) at the inlet and outlet using Nusselt number and thermal conductivity (k) : \(h_{1} = h_{2} = \frac{q_{i,1}^{\prime}}{T_{s,i,1} - T_{a,1}} = \frac{q_{i,2}^{\prime}}{T_{s,i,2} - T_{a,2}}\) Now, we have two equations with two unknowns, Ts,i,1 and Ts,i,2. Solve for both temperatures using these equations.

03

Find the outer surface temperature at the inlet (Ts,θ,1)

To find the outer surface temperature at the inlet, we need to use the temperature distribution for fully developed flow in the annular passage. The temperature distribution for fully developed flow in an annular passage is given by: \(T(r) = T_{s,i,1} + k \left(\frac{1}{r} - \frac{1}{r_{o}}\right)\left(\frac{T_{s,i,1} - T_{a,1}}{h}\right)\) By substituting the values of inner pipe temperature at the inlet Ts,i,1, and air inlet temperature Ta1, we can find the temperature at the outer surface (r = ro) of the annular passage, Ts,θ,1.

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Key Concepts

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

Concentric Pipe Arrangement

The concentric pipe arrangement is a common setup in heat exchanger design. Imagine two pipes, one inside the other. The inner pipe carries a fluid, such as exhaust gases, and the outer pipe carries a different fluid, like combustion air. In this setup, heat is transferred from one fluid to the other.
This type of arrangement is particularly useful when you want to recover waste heat. For example, hot exhaust gases can preheat cooler air flowing through the outer pipe. This makes fuel combustion more efficient.

• Inner Pipe: Carries the hot fluid.
• Outer Pipe: Carries the cooler fluid.
• Annular Space: The gap between the pipes where the cooler fluid flows.

The concentric arrangement ensures maximum surface area contact between the hot and cooler fluids. This contact enhances the overall heat transfer efficiency.

Heat Transfer Rate

The heat transfer rate indicates how much heat energy is being transferred over time. In heat exchangers, this is a critical factor to ensure efficiency. In the exercise, the heat transfer rate per unit length was given as \( q_{i}^{\prime} = 1.25 \times 10^{5} \mathrm{~W/m} \).
This value tells us how much heat is transferred from the exhaust gases to the air flowing through the annular space.

• Calculation of Total Heat Transfer: To find the total heat transferred, multiply the heat transfer rate per unit length by the total pipe length \( L \). This gives the total heat energy that flows from the exhaust gases to the air.
• Importance: The heat transfer rate determines how effective the heat exchanger is in transferring energy.

Understanding this helps in designing systems that are energy-efficient and meet the desired heating or cooling requirements.

Thermal Conductivity

Thermal conductivity \( k \) measures a material's ability to conduct heat. It is a crucial property in heat exchanger design as it influences heat transfer efficiency. In this scenario, the thermal conductivity of the air was \( k = 0.041 \, \mathrm{W/m \cdot K} \).
Higher thermal conductivity means better heat conduction. Materials such as metals often have high thermal conductivity, making them excellent for heat exchangers.

• Influence on Heat Exchanger: A higher \( k \) value ensures more effective heat transfer across materials, enhancing system performance.
• Role in Calculations: Thermal conductivity is used in conjunction with the Nusselt number and the convection heat transfer coefficient to evaluate overall heat transfer.

Understanding thermal conductivity helps engineers choose appropriate materials for different parts of the heat exchanger to optimize performance.

Nusselt Number

The Nusselt number \( Nu \) is a dimensionless quantity that provides insight into the convection heat transfer occurring in a fluid. It relates the convective to conductive heat transfer across a boundary.
For heat exchangers, the Nusselt number helps determine the efficiency of heat transfer from one fluid to another. In general, a high Nusselt number indicates efficient convection compared to conduction.

• Importance in Heat Exchangers: The Nusselt number is used to calculate the convection heat transfer coefficient \( h \), which in turn defines how well heat is transferred between the pipe surface and the fluid.
• Relation to Thermal Conductivity \( k \): The Nusselt number is derived from the formula \( Nu = \frac{hL}{k} \), where \( h \) is the heat transfer coefficient and \( L \) is the characteristic length.

Understanding and calculating the Nusselt number is key to improving heat exchanger design, ensuring optimal thermal performance.