Understanding Heat Exchanger Effectiveness and NTU in Gas Turbines

In a gas turbine system, heat exchangers are often used to recover heat from the hot exhaust gases. This process, known as regeneration or recuperation, helps improve the overall thermal efficiency of the turbine. Two important parameters used to analyze heat exchanger performance are Effectiveness ($\epsilon$) and the Number of Transfer Units (NTU).

What is Heat Exchanger Effectiveness?

Effectiveness ($\epsilon$) is a measure of the actual heat transfer rate compared to the maximum possible heat transfer rate. It is defined as:

\(\epsilon = \frac{\text{Actual heat transfer rate}}{\text{Maximum possible heat transfer rate}}\)

The maximum possible heat transfer rate occurs in a counter-flow heat exchanger of infinite length, where the outlet temperature of the cold fluid reaches the inlet temperature of the hot fluid, or vice versa, limited by the fluid with the minimum heat capacity rate.

What is the Number of Transfer Units (NTU)?

The Number of Transfer Units (NTU) is a dimensionless parameter that indicates the size of the heat exchanger. It is defined as:

\(NTU = \frac{UA}{C_{min}}\)

Where:

• \(U\) is the overall heat transfer coefficient.
• \(A\) is the heat transfer surface area.
• \(C_{min}\) is the minimum heat capacity rate (\(C = \dot{m}c_p\), where \(\dot{m}\) is mass flow rate and \(c_p\) is specific heat at constant pressure) between the two fluids.

NTU is related to effectiveness and the capacity rate ratio (\(C_r = C_{min}/C_{max}\)). The exact relationship depends on the type of heat exchanger (e.g., parallel flow, counter flow, cross flow) and whether the fluids are mixed or unmixed.

Calculating NTU for a Gas Turbine Heat Exchanger

The question provides the effectiveness (\(\epsilon = 0.8\)) of a compact heat exchanger used in a gas turbine to heat compressed air with hot exhaust gases. We need to find the NTU. The relationship between effectiveness and NTU depends on the heat exchanger configuration and the capacity rate ratio, \(C_r\).

Gas turbine recuperators typically involve two gases (air and exhaust gas) with similar mass flow rates and specific heats, meaning the capacity rate ratio \(C_r = C_{min}/C_{max}\) is often close to 1. Also, counter-flow heat exchangers are generally used for high effectiveness. For a counter-flow heat exchanger with a capacity rate ratio \(C_r = 1\), the relationship between effectiveness and NTU is given by:

\(\epsilon = \frac{NTU}{1 + NTU}\)

Given the effectiveness \(\epsilon = 0.8\), we can substitute this value into the formula:

\(0.8 = \frac{NTU}{1 + NTU}\)

Now, we solve for NTU:

\(0.8 \times (1 + NTU) = NTU\)

\(0.8 + 0.8 \times NTU = NTU\)

\(0.8 = NTU - 0.8 \times NTU\)

\(0.8 = NTU \times (1 - 0.8)\)

\(0.8 = NTU \times 0.2\)

\(NTU = \frac{0.8}{0.2}\)

\(NTU = 4\)

Thus, for a counter-flow heat exchanger with a capacity rate ratio of 1 and an effectiveness of 0.8, the Number of Transfer Units (NTU) is 4.

Based on this calculation, the value of NTU is 4.