Question

A single-pass cross-flow heat exchanger uses hot air (mixed) to heat water (unmixed), flowing with a mass flow rate of \(3 \mathrm{~kg} / \mathrm{s}\), from \(30^{\circ} \mathrm{C}\) to \(80^{\circ} \mathrm{C}\). The hot air enters and exits the heat exchanger at \(220^{\circ} \mathrm{C}\) and \(100^{\circ} \mathrm{C}\), respectively. If the overall heat transfer coefficient is \(200 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\), determine the required surface area.

Short answer

Answer: The required surface area of the single-pass cross-flow heat exchanger is approximately 30.86 m².

Step-by-step solution

Calculate the heat transfer rate (Q)

To find the heat transfer rate (Q), we can use the specific heat capacity of water (Cpw = 4.18 kJ/kg.K) and the mass flow rate (m) of water, as well as the temperature difference in water from the inlet to the outlet, as follows: Q = m * Cpw * (T_outlet_water - T_inlet_water) where m = 3 kg/s (mass flow rate of water) Cpw = 4.18 kJ/kg.K (specific heat capacity of water) T_outlet_water = 80°C T_inlet_water = 30°C Q = 3 kg/s * 4.18 kJ/kg.K * (80°C - 30°C) Converting kJ to J: Q = 3 kg/s * 4180 J/kg.K * (50 K) Q = 627000 W The heat transfer rate at which water is being heated is 627000 W.

Calculate the log-mean temperature difference (ΔTm)

Next, we need to find the log-mean temperature difference (ΔTm) which can be used to determine the surface area. The log-mean temperature difference can be determined by using the inlet and outlet temperatures of both the hot air and the water: ΔT₁ = T_inlet_hotair - T_outlet_water ΔT₂ = T_outlet_hotair - T_inlet_water To find ΔTm, use the formula: ΔTm = (ΔT₁ - ΔT₂) / ln(ΔT₁ / ΔT₂) where T_inlet_hotair = 220°C T_outlet_hotair = 100°C T_inlet_water = 30°C T_outlet_water = 80°C ΔT₁ = 220°C - 80°C = 140 K ΔT₂ = 100°C - 30°C = 70 K ΔTm = (140 K - 70 K) / ln(140 K / 70 K) ΔTm ≈ 101.46 K The log-mean temperature difference is approximately 101.46 K.

Calculate the required surface area (A)

Now that we have found the heat transfer rate (Q) and the log-mean temperature difference (ΔTm), we can determine the required surface area (A) of the heat exchanger using the formula: Q = U * A * ΔTm where U = 200 W/m².K (overall heat transfer coefficient) Rearranging the formula for A and substituting known values: A = Q / (U * ΔTm) A = 627000 W / (200 W/m².K * 101.46 K) A ≈ 30.86 m² The required surface area of the single-pass cross-flow heat exchanger is approximately 30.86 m².

Key concepts

Log-Mean Temperature Difference

The Log-Mean Temperature Difference (LMTD) is an essential concept when analyzing heat exchangers. It serves as the driving force for heat transfer in these systems.
The LMTD accounts for the temperature difference between two fluids at different points within the heat exchanger. This method provides a more accurate measure than simply using the average temperature difference. Given its accuracy, it is widely preferred in heat exchanger calculations.
To calculate the LMTD, you need the temperature differences at the entry and exit points of the fluids. Let's break this down:

• Calculate the temperature difference at the inlet and outlet of the hot and cold fluids.
• Use these differences to calculate the LMTD with the formula: \[ \Delta T_{m} = \frac{(\Delta T_{1} - \Delta T_{2})}{\ln(\Delta T_{1}/\Delta T_{2})} \] where \( \Delta T_{1} \) and \( \Delta T_{2} \) are the temperature differences at the start and end of the heat exchange process.

Understanding LMTD simplifies complex temperature variations and helps predict the efficiency of the heat exchanger effectively.

Heat Transfer Rate Calculation

The Heat Transfer Rate is pivotal in determining how efficient a heat exchanger is. This rate helps determine how much energy is being transferred between fluids.
It is quantified by the formula:

• First, identify the mass flow rate ( \( m \) ) and specific heat capacity (\( C_{p} \)) of the fluid being heated or cooled.
• Calculate the temperature difference of the fluid across the heat exchanger.
• The heat transfer rate ( \( Q \) ) is given by: \[ Q = m \cdot C_{p} \cdot \Delta T \] where \( \Delta T \) is the temperature change from inlet to outlet.

For example, in the given exercise, this translates to using the mass flow rate and specific heat capacity of water to calculate the required energy transfer. This rate tells us how much energy is being transferred at any given moment and can provide insights into whether the system meets the desired heat transfer performance.

Overall Heat Transfer Coefficient

The Overall Heat Transfer Coefficient (\( U \)) represents the heat transfer capability of the entire heat exchanger surface. It's a collective measure of how well the heat exchanger materials transfer heat between fluids.
Several factors can affect this coefficient:

• The type and thickness of the heat exchanger wall materials.
• The cleanliness or fouling of the heat exchange surfaces.
• The convection heat transfer coefficients on both sides of the exchanger.

Combine these to find the overall efficiency of heat transfer within the system using this coefficient.
The coefficient is key in the formula:
\[ Q = U \cdot A \cdot \Delta T_{m}\] Where:

• \( Q \) is the heat transfer rate, and
• \( A \) is the heat exchange surface area.

Efficient heat exchangers aim to maximize \( U \), as this leads to better heat transfer with less surface area, ultimately reducing costs and energy consumption.