Heat Exchanger LMTD Calculation for Counter Flow System

This problem involves calculating the Log Mean Temperature Difference (LMTD) for a counter-flow heat exchanger. We are given the fluid properties, mass flow rates, inlet temperatures, and the heat exchanger's effectiveness. Understanding these parameters is crucial for heat exchanger design and analysis, especially when determining the required heat transfer area.

Key Concepts in Heat Exchangers

• Heat Exchanger: A thermal device designed to transfer heat efficiently between two or more fluids that are at different temperatures and often separated by a solid wall.
• Counter Flow: A configuration in heat exchangers where the hot and cold fluids flow in opposite directions. This arrangement typically results in a more uniform temperature difference along the heat exchanger's length and often achieves higher heat transfer rates compared to parallel flow.
• Effectiveness (\(\epsilon\)): A dimensionless performance parameter for heat exchangers. It is defined as the ratio of the actual heat transfer rate to the maximum possible heat transfer rate that could be achieved in an infinitely large heat exchanger.
• Log Mean Temperature Difference (LMTD): A logarithmic average of the temperature differences between the hot and cold fluids at the two ends of the heat exchanger. The LMTD is used in the fundamental heat exchanger design equation: \(Q = U A \text{ LMTD}\), where \(Q\) is the heat transfer rate, \(U\) is the overall heat transfer coefficient, and \(A\) is the heat transfer surface area.

Heat Exchanger Parameters: Given Data

We are provided with the following information for the counter flow heat exchanger:

• Hot Fluid (Water):
  • Specific Heat Capacity, \(C_{p,h}\) = 4.18 kJ/kg.K
  • Inlet Temperature, \(T_{h,in}\) = 80°C
  • Mass Flow Rate, \(\dot{m}_h\) = 0.5 kg/s
• Cold Fluid (Air):
  • Specific Heat Capacity, \(C_{p,c}\) = 1 kJ/kg.K
  • Inlet Temperature, \(T_{c,in}\) = 30°C
  • Mass Flow Rate, \(\dot{m}_c\) = 2.09 kg/s
• Heat Exchanger Effectiveness, \(\epsilon\) = 0.8
• Type of Flow: Counter flow

LMTD Calculation: Step-by-Step Process

1. Heat Capacity Rates Determination

The heat capacity rate (\(C\)) for each fluid is calculated as the product of its mass flow rate and specific heat capacity. This value represents the rate at which a fluid can absorb or release heat per unit temperature change.

• For Hot Fluid (Water):
• \(C_h = \dot{m}_h \times C_{p,h}\)
• \(C_h = 0.5 \text{ kg/s} \times 4.18 \text{ kJ/kg.K}\)
• \(C_h = 2.09 \text{ kJ/s.K}\)
• For Cold Fluid (Air):
• \(C_c = \dot{m}_c \times C_{p,c}\)
• \(C_c = 2.09 \text{ kg/s} \times 1 \text{ kJ/kg.K}\)
• \(C_c = 2.09 \text{ kJ/s.K}\)

In this problem, we observe that the heat capacity rates for both fluids are equal: \(C_h = C_c = 2.09 \text{ kJ/s.K}\).

2. Minimum Heat Capacity Rate Identification

The minimum heat capacity rate (\(C_{min}\)) is the smaller of the two calculated heat capacity rates. This is a crucial value as it limits the maximum possible heat transfer in the heat exchanger.

• Since \(C_h = C_c\), then \(C_{min} = 2.09 \text{ kJ/s.K}\).

3. Maximum Possible Heat Transfer Calculation

The maximum possible heat transfer (\(Q_{max}\)) occurs if the fluid with the minimum heat capacity rate were to experience the largest possible temperature change, which is the difference between the inlet temperatures of the hot and cold fluids.

• Formula: \(Q_{max} = C_{min} \times (T_{h,in} - T_{c,in})\)
• \(Q_{max} = 2.09 \text{ kJ/s.K} \times (80^\circ\text{C} - 30^\circ\text{C})\)
• \(Q_{max} = 2.09 \text{ kJ/s.K} \times 50^\circ\text{C}\)
• \(Q_{max} = 104.5 \text{ kW}\)

4. Actual Heat Transfer Determination using Effectiveness

The actual heat transfer (\(Q_{actual}\)) that occurs in the heat exchanger is calculated using its given effectiveness and the maximum possible heat transfer.

• Formula: \(\epsilon = \frac{Q_{actual}}{Q_{max}}\)
• Therefore, \(Q_{actual} = \epsilon \times Q_{max}\)
• \(Q_{actual} = 0.8 \times 104.5 \text{ kW}\)
• \(Q_{actual} = 83.6 \text{ kW}\)

5. Outlet Temperatures Calculation

Using the actual heat transfer rate, we can determine the outlet temperatures of both the hot water and cold air. The heat transferred away from the hot fluid equals the heat absorbed by the cold fluid.

• For Hot Fluid (Water):
• \(Q_{actual} = C_h \times (T_{h,in} - T_{h,out})\)
• \(83.6 \text{ kW} = 2.09 \text{ kJ/s.K} \times (80^\circ\text{C} - T_{h,out})\)
• \((80^\circ\text{C} - T_{h,out}) = \frac{83.6}{2.09} = 40^\circ\text{C}\)
• \(T_{h,out} = 80^\circ\text{C} - 40^\circ\text{C} = 40^\circ\text{C}\)

• For Cold Fluid (Air):
• \(Q_{actual} = C_c \times (T_{c,out} - T_{c,in})\)
• \(83.6 \text{ kW} = 2.09 \text{ kJ/s.K} \times (T_{c,out} - 30^\circ\text{C})\)
• \((T_{c,out} - 30^\circ\text{C}) = \frac{83.6}{2.09} = 40^\circ\text{C}\)
• \(T_{c,out} = 30^\circ\text{C} + 40^\circ\text{C} = 70^\circ\text{C}\)

6. Temperature Differences at Ends for Counter Flow

For a counter-flow heat exchanger, the temperature differences at the two ends (inlet and outlet) are defined as:

• \(\Delta T_1 = T_{h,in} - T_{c,out}\) (Temperature difference at one end)
• \(\Delta T_1 = 80^\circ\text{C} - 70^\circ\text{C} = 10^\circ\text{C}\)

• \(\Delta T_2 = T_{h,out} - T_{c,in}\) (Temperature difference at the other end)
• \(\Delta T_2 = 40^\circ\text{C} - 30^\circ\text{C} = 10^\circ\text{C}\)

Notice that in this specific case, the temperature differences at both ends are equal: \(\Delta T_1 = \Delta T_2 = 10^\circ\text{C}\).

7. Log Mean Temperature Difference (LMTD) Calculation

The general formula for LMTD for a counter-flow heat exchanger is:

However, when the temperature differences at both ends are equal (\(\Delta T_1 = \Delta T_2\)), the LMTD is simply equal to this common temperature difference. This special condition arises when the heat capacity rates of the hot and cold fluids are identical (\(C_h = C_c\)), as seen in this problem.

Since \(\Delta T_1 = \Delta T_2 = 10^\circ\text{C}\), the LMTD is:

LMTD Calculation Conclusion

Based on the detailed calculations, the Log Mean Temperature Difference (LMTD) for the given counter-flow heat exchanger operating with water and air is \(10^\circ\text{C}\).