Question

A cross-flow heat exchanger consists of 80 thinwalled tubes of \(3-\mathrm{cm}\) diameter located in a duct of \(1 \mathrm{~m} \times 1 \mathrm{~m}\) cross section. There are no fins attached to the tubes. Cold water \(\left(c_{p}=4180 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\right)\) enters the tubes at \(18^{\circ} \mathrm{C}\) with an average velocity of \(3 \mathrm{~m} / \mathrm{s}\), while hot air \(\left(c_{p}=1010 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\right)\) enters the channel at \(130^{\circ} \mathrm{C}\) and \(105 \mathrm{kPa}\) at an average velocity of \(12 \mathrm{~m} / \mathrm{s}\). If the overall heat transfer coefficient is \(130 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\), determine the outlet temperatures of both fluids and the rate of heat transfer.

Short answer

Determine the outlet temperatures of both fluids (cold water and hot air) and the rate of heat transfer in a cross-flow heat exchanger given the fluid properties, mass flow rates, and heat transfer coefficients. Follow the steps below to solve the problem: Step 1: Calculate the mass flow rates of cold water and hot air using the formula, \(\dot{m} = \rho AV\), and the given fluid properties. Step 2: Apply the energy conservation equation for the heat exchanger, which states that the rate of heat transfer from hot air to cold water equals the difference in energy content between the inlet and outlet. Step 3: Apply the heat transfer formula to find the outlet temperatures, which states that the rate of heat transfer equals the product of the overall heat transfer coefficient \(U\), heat transfer surface area \(A_H\), and the logarithmic mean temperature difference \(\Delta T_{LM}\). Step 4: Calculate the rate of heat transfer using the energy conservation equation from Step 2 with the outlet temperatures of both fluids.

Step-by-step solution

Calculate mass flow rates of fluids

We need to calculate the mass flow rates of both cold water and hot air. We will use the formula: \(\dot{m} = \rho AV\), where \(\rho\) is the fluid density, \(A\) is the cross-sectional area, and \(V\) is the fluid velocity. Although we do not have the density of the fluids directly, we do know that for an ideal gas, \(PV = mRT\), so \(\rho = \frac{m}{V} = \frac{P}{RT}\). We'll assume water is incompressible, so its density remains constant at 1000 kg/m³. For cold water: \(\dot{m} = \rho AV\) For hot air: \(\rho = \frac{P}{RT}\) \(\dot{m} = \rho AV\) Calculating the mass flow rate for each fluid is the first step.

Apply the energy conservation equation

We need to apply the energy conservation equation for the heat exchanger, which states that the rate of heat transfer from hot air to cold water equals the difference in energy content between the inlet and outlet. \(q = \dot{m}_c c_{p_c}(T_{out_c} - T_{in_c}) = \dot{m}_h c_{p_h}(T_{in_h} - T_{out_h})\) Where \(q\) is the rate of heat transfer, \(\dot{m}_c\) and \(\dot{m}_h\) are the mass flow rates of cold water and hot air respectively, \(c_{p_c}\) and \(c_{p_h}\) are the specific heat capacities of cold water and hot air, and \(T_{in_c}\), \(T_{in_h}\), \(T_{out_c}\), and \(T_{out_h}\) are the inlet and outlet temperatures of cold water and hot air.

Apply the heat transfer formula

We will now apply the heat transfer formula to find the outlet temperatures, which states that the rate of heat transfer equals the product of the overall heat transfer coefficient \(U\), heat transfer surface area \(A_H\) and the logarithmic mean temperature difference \(\Delta T_{LM}\): \(q = UA_H \Delta T_{LM}\) Where, \(\Delta T_{LM} = \frac{(\Delta T_1 - \Delta T_2)}{\ln(\Delta T_1 / \Delta T_2)}\) \(\Delta T_1 = T_{h1} - T_{c1}\) \(\Delta T_2 = T_{h2} - T_{c2}\) Applying the formula will help us find the outlet temperatures of both fluids.

Calculate the rate of heat transfer

With the outlet temperatures of both fluids, we can calculate the rate of heat transfer using the energy conservation equation from Step 2: \(q = \dot{m}_c c_{p_c}(T_{out_c} - T_{in_c}) = \dot{m}_h c_{p_h}(T_{in_h} - T_{out_h})\) #get_template-Solution#

Key concepts

Mass Flow Rate Calculation

To determine the mass flow rates of the fluids in a heat exchanger, we'll need to start with the fundamental equation of mass flow rate: \(\dot{m} = \rho AV\), where:

• \(\dot{m}\) is the mass flow rate,
• \(\rho\) is the density of the fluid,
• \(A\) is the cross-sectional area the fluid flows through, and
• \(V\) is the velocity of the fluid.

This equation helps us relate the physical characteristics of the fluid flow to the rate at which mass is moving through the system.
For incompressible fluids like cold water, the density \(\rho\) remains constant, so substituting a typical value for water, such as \(1000\ \text{kg/m}^3\), is a practical approach.
For gases like hot air, the situation is a bit different. Since air is compressible, its density can be found using the ideal gas law rearranged as \(\rho = \frac{P}{RT}\). Here:

• \(P\) is the pressure,
• \(R\) is the specific gas constant, and
• \(T\) is the absolute temperature in Kelvin.

Calculating these mass flow rates provides the foundation for understanding how much of each fluid is involved in the heat exchange process.

Energy Conservation in Heat Exchanger

The energy conservation principle in heat exchangers is built around the notion that energy lost by the hot fluid equals the energy gained by the cold fluid if no heat is lost to the surroundings. This is an expression of the conservation of energy law. In mathematical terms, it can be represented as:\[q = \dot{m}_c c_{p_c}(T_{out_c} - T_{in_c}) = \dot{m}_h c_{p_h}(T_{in_h} - T_{out_h})\]Where:

• \(q\) is the rate of heat transfer,
• \(\dot{m}_c\) and \(\dot{m}_h\) are the mass flow rates of the cold and hot fluids, respectively,
• \(c_{p_c}\) and \(c_{p_h}\) are their specific heat capacities,
• \(T_{out_c}\), \(T_{in_c}\), \(T_{in_h}\), \(T_{out_h}\) are the outlet and inlet temperatures of the cold and hot fluids.

This step is critical as it directly ties the thermodynamic properties of the fluids to their flow characteristics and temperatures, offering insights into how efficiently the heat exchanger operates.

Heat Transfer Coefficient

The heat transfer coefficient (\(U\)) plays a crucial role in determining how effectively heat is transferred between hot and cold fluids in a heat exchanger. It is affected by the material of the heat exchanger, the fluid properties, and the flow dynamics. The formula connecting heat transfer with \(U\) is given as:\[q = UA_H \Delta T_{LM}\]Where:

• \(U\) represents the overall heat transfer coefficient,
• \(A_H\) is the area available for heat exchange,
• \(\Delta T_{LM}\) is the logarithmic mean temperature difference (LMTD) across the exchanger.

The LMTD accounts for temperature changes along the length of the heat exchanger and is calculated using:\[\Delta T_{LM} = \frac{(\Delta T_1 - \Delta T_2)}{\ln(\Delta T_1 / \Delta T_2)}\]With:

• \(\Delta T_1 = T_{h1} - T_{c1}\)
• \(\Delta T_2 = T_{h2} - T_{c2}\)

Understanding the heat transfer coefficient and LMTD is essential for evaluating the performance and efficiency of the exchanger.

Outlet Temperature Calculation

Determining the outlet temperatures of the fluids in a heat exchanger involves using both the heat transfer and energy conservation principles. The governing formula looks like this:\[q = \dot{m}_c c_{p_c}(T_{out_c} - T_{in_c}) = \dot{m}_h c_{p_h}(T_{in_h} - T_{out_h})\]Start by calculating the total heat transfer \(q\) using known parameters and rearranging the equation to solve for unknown outlet temperatures.
Generally, the procedure includes:

• Knowing or deriving \(q\) from heat transfer principles using \(q = UA_H \Delta T_{LM}\).
• Using mass flow rate and heat capacity data to rearrange the conservation equation for the correct fluid outlet temperature.

These calculations allow engineers to deduce the thermal output conditions of the system, aiding in designing more efficient thermal management solutions.