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Heated air required for a food-drying process is generated by passing ambient air at \(20^{\circ} \mathrm{C}\) through long, circular tubes \((D=50 \mathrm{~mm}, L=5 \mathrm{~m})\) housed in a steam condenser. Saturated steam at atmospheric pressure condenses on the outer surface of the tubes, maintaining a uniform surface temperature of \(100^{\circ} \mathrm{C}\). (a) If an airflow rate of \(0.01 \mathrm{~kg} / \mathrm{s}\) is maintained in each tube, determine the air outlet temperature \(T_{m, o}\) and the total heat rate \(q\) for the tube. (b) The air outlet temperature may be controlled by adjusting the tube mass flow rate. Compute and plot \(T_{m \rho}\) as a function of \(\dot{m}\) for \(0.005 \leq \dot{m} \leq\) \(0.050 \mathrm{~kg} / \mathrm{s}\). If a particular drying process requires approximately \(1 \mathrm{~kg} / \mathrm{s}\) of air at \(75^{\circ} \mathrm{C}\), what design and operating conditions should be prescribed for the air heater, subject to the constraint that the tube diameter and length be fixed at \(50 \mathrm{~mm}\) and \(5 \mathrm{~m}\), respectively?

Step by step solution

01

(Step 1: Calculate the outlet temperature of the air)

To find the air outlet temperature, we can use the following energy balance equation: \[(T_{m,o} - T_{m,i}) = UAT_{lm} / C_p \dot{m}\] Where: \(T_{m,o}\) = outlet temperature of the air (°C) \(T_{m,i}\) = inlet temperature of the air (°C) \(U\) = overall heat transfer coefficient (W/m²·K) \(A\) = heat transfer area (m²) \(T_{lm}\) = logarithmic mean temperature difference (°C) \(C_p\) = specific heat capacity of air (J/kg·K) \(\dot{m}\) = mass flow rate of air (kg/s) For this problem, \(T_{m,i} = 20 °C\) and \(\dot{m} = 0.01 \ \text{kg/s}\). We first find the value of \(T_{lm}\) using the equation: \[T_{lm} = \frac{(T_s - T_{m,o})-(T_s - T_{m,i})}{\ln [(T_s - T_{m,o})/(T_s – T_{m,i})]}\] Where \(T_s = 100 °C\) is the surface temperature of the tube. Rearrange the equation to find the outlet temperature \(T_{m,o}\) and solve using an iterative method like the Newton-Raphson method.

02

(Step 2: Find the total heat transfer rate)

Once we have the outlet temperature of the air, we can find the total heat transfer rate using the following formula: \[q = C_p \dot{m} (T_{m,o} - T_{m,i})\]

03

(Step 3: Outlet temperature as a function of mass flow rate)

Next, we need to compute the air outlet temperature \(T_{m \rho}\) for different mass flow rates \(\dot{m}\) in the range of 0.005 kg/s to 0.050 kg/s. For each mass flow rate, follow step 1 to find the outlet temperature and plot the results.

04

(Step 4: Determine design and operating conditions for the air heater)

We are given the constraints on the tube diameter and length, and we need to find the design and operating conditions for the air heater to provide approximately 1 kg/s of air at 75 °C. To accomplish this, we can adjust the mass flow rate of the air and tube properties to achieve the desired air outlet temperature and flow rate. Iterate over different mass flow rates and tube configurations until you find a combination that satisfies the requirements.

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Key Concepts

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

Energy Balance

Energy balance is a fundamental concept in heat transfer that involves equating the heat gained or lost by a system to the heat provided to or extracted from it. The principle of energy balance is based on the law of conservation of energy. In the context of our exercise, the heat balance equation is used to determine the outlet temperature of air in a heating process.

The energy balance equation for the air passing through the steam-heated tubes is given by:\[ (T_{m,o} - T_{m,i}) = \frac{UA T_{lm}}{C_p \dot{m}} \]

• \( U \): Overall heat transfer coefficient, which measures the heat transfer per unit area and temperature difference.
• \( A \): Surface area over which the heat transfer takes place.
• \( T_{lm} \): Logarithmic mean temperature difference, a key concept in heat exchanger design.
• \( C_p \): Specific heat capacity of the air, showing how much energy is required to change the temperature of a unit mass by one degree.
• \( \dot{m} \): Mass flow rate of the air.

This equation helps in predicting how the incoming air's temperature will change as it absorbs heat from the steam-condensed tube. To solve for the outlet temperature \( T_{m,o} \), often an iterative method is employed due to the integral nature of certain parameters like \( T_{lm} \).

Logarithmic Mean Temperature Difference

The Logarithmic Mean Temperature Difference (LMTD) is crucial when analyzing heat exchangers because it accounts for varying temperature differences across the exchanger's length. It provides a true average of the temperature difference between two flows, ensuring accurate calculations.

The formula for LMTD is:\[ T_{lm} = \frac{(T_s - T_{m,o}) - (T_s - T_{m,i})}{\ln \left(\frac{T_s - T_{m,o}}{T_s - T_{m,i}}\right)} \]In this equation:

• \(T_s\) denotes the surface temperature of the heat exchange interface, here 100°C.
• \(T_{m,o}\) and \(T_{m,i}\) are outlet and inlet temperatures of the air, respectively.

Understanding LMTD helps in recognizing that the temperature difference isn't uniform across the tube; it changes from the inlet to the outlet. LMTD accounts for this change by smoothing out these temperature variations into a singular comparable value. This concept is important for optimizing heat exchanger performance and ensuring the predicted values are closely aligned with actual behavior.

Mass Flow Rate

Mass flow rate refers to the quantity of mass passing through a section per unit of time, usually expressed in kilograms per second (kg/s). It plays a critical role in determining how much energy can be transported in a heat exchanger.

In the given problem, adjusting the mass flow rate directly influences the air's outlet temperature. As mass flow rate affects the dwell time of air inside the heated tubes, it indirectly governs the extent to which the air will absorb heat.
The formula relating mass flow rate and heat transfer rate is:\[ q = C_p \dot{m} (T_{m,o} - T_{m,i}) \]Here:

• \(q\) is the total heat transfer rate.
• \(C_p\) is the specific heat capacity of air.
• \(\dot{m}\) is the mass flow rate of the air.
• \(T_{m,o}\), \(T_{m,i}\) are the outlet and inlet temperatures of the air, respectively.

Understanding mass flow rate is essential to achieve the desired outlet temperature. Higher flow rates might lead to lower temperatures as there is less time for heat absorption, whereas lower flow rates might result in more heat being absorbed, increasing the outlet temperature. Thus, carefully adjusting the mass flow rate can fine-tune the thermal output to desired specifications.