5.3 LMTD and ε-NTU methods for heat exchanger analysis

Heat exchangers are crucial in thermal systems. The LMTD method helps analyze simple exchangers, while the ε-NTU method tackles complex ones. Both use temperature differences and flow rates to determine heat transfer performance.

These methods are essential for designing and optimizing heat exchangers. LMTD is great for known temperatures, while ε-NTU shines when outlet temps are unknown. Understanding both helps engineers create efficient thermal systems.

Logarithmic Mean Temperature Difference

Definition and Equation

Top images from around the web for Definition and Equation

• 

  design and Analysis of a Heat Exchanger Network – Material Science Research India View original

  Is this image relevant?

• 

  Experimental Analysis of Performance of Heat Exchanger with Plate Fins and Parallel Flow of ... View original

  Is this image relevant?

• 

  design and Analysis of a Heat Exchanger Network – Material Science Research India View original

  Is this image relevant?

• 

  design and Analysis of a Heat Exchanger Network – Material Science Research India View original

  Is this image relevant?

• 

  Experimental Analysis of Performance of Heat Exchanger with Plate Fins and Parallel Flow of ... View original

  Is this image relevant?

1 of 3

Top images from around the web for Definition and Equation

• 

  design and Analysis of a Heat Exchanger Network – Material Science Research India View original

  Is this image relevant?

• 

  Experimental Analysis of Performance of Heat Exchanger with Plate Fins and Parallel Flow of ... View original

  Is this image relevant?

• 

  design and Analysis of a Heat Exchanger Network – Material Science Research India View original

  Is this image relevant?

• 

  design and Analysis of a Heat Exchanger Network – Material Science Research India View original

  Is this image relevant?

• 

  Experimental Analysis of Performance of Heat Exchanger with Plate Fins and Parallel Flow of ... View original

  Is this image relevant?

1 of 3

• The logarithmic mean temperature difference (LMTD) is a logarithmic average of the temperature difference between the hot and cold fluids at each end of the heat exchanger
• The LMTD equation is given by: $LMTD = (∆T₁ - ∆T₂) / ln(∆T₁/∆T₂)$, where $∆T₁$ and $∆T₂$ are the temperature differences between the hot and cold fluids at the two ends of the heat exchanger (inlet and outlet)
• The LMTD is a more accurate representation of the average temperature difference in a heat exchanger compared to the arithmetic mean temperature difference because it accounts for the logarithmic nature of the temperature profile

Derivation and Assumptions

• The LMTD equation is derived from the heat transfer rate equation, $Q = UA∆T$, where $Q$ is the heat transfer rate, $U$ is the overall heat transfer coefficient, $A$ is the heat transfer area, and $∆T$ is the temperature difference
• The derivation assumes constant specific heats for both fluids, no phase change occurring within the heat exchanger, and negligible heat loss to the surroundings
• These assumptions simplify the analysis and make the LMTD equation applicable to a wide range of heat exchanger problems

LMTD Method for Heat Exchangers

Determining Heat Transfer Rates and Outlet Temperatures

• The LMTD method is suitable for heat exchangers with known inlet and outlet temperatures and simple flow arrangements (parallel flow or counterflow)
• To determine the heat transfer rate, use the equation $Q = UA(LMTD)$, where $U$ is the overall heat transfer coefficient, $A$ is the heat transfer area, and $LMTD$ is the logarithmic mean temperature difference
• To find the outlet temperatures, use the energy balance equations for the hot and cold fluids: $Q = ṁ_h c_p,h (T_h,in - T_h,out) = ṁ_c c_p,c (T_c,out - T_c,in)$, where $ṁ$ is the mass flow rate, $c_p$ is the specific heat, and $T$ represents the temperatures
• Example: In a counterflow heat exchanger, hot water enters at 90°C and leaves at 60°C, while cold water enters at 20°C and leaves at 40°C. The LMTD can be calculated using these temperatures, and the heat transfer rate can be determined if the overall heat transfer coefficient and area are known

Correction Factors for Complex Flow Arrangements

• Correction factors ($F$) are used to modify the LMTD for heat exchangers with complex flow arrangements, such as cross-flow or shell-and-tube exchangers with multiple passes
• The corrected LMTD is given by: $LMTD_corrected = F × LMTD$, where $F$ is the correction factor, which depends on the flow arrangement and the temperature effectiveness ($P$ and $R$)
• $P$ and $R$ are dimensionless temperature ratios defined as: $P = (T_c,out - T_c,in) / (T_h,in - T_c,in)$ and $R = (T_h,in - T_h,out) / (T_c,out - T_c,in)$
• Correction factor charts or equations are available for various flow arrangements to determine the appropriate value of $F$ based on $P$ and $R$

Effectiveness and NTU in Heat Exchangers

Effectiveness (ε)

• Effectiveness ($ε$) is a dimensionless parameter that represents the ratio of the actual heat transfer rate to the maximum possible heat transfer rate in a heat exchanger
• The maximum possible heat transfer rate ($Q_max$) occurs when the fluid with the smaller heat capacity rate ($C_min$) undergoes the maximum temperature change, which is equal to the inlet temperature difference between the hot and cold fluids
• The effectiveness is given by: $ε = Q / Q_max$, where $Q$ is the actual heat transfer rate and $Q_max = C_min (T_h,in - T_c,in)$
• Example: If the actual heat transfer rate in a heat exchanger is 50 kW and the maximum possible heat transfer rate is 80 kW, the effectiveness would be $ε = 50 / 80 = 0.625$ or 62.5%

Number of Transfer Units (NTU)

• The number of transfer units (NTU) is a dimensionless parameter that represents the size of the heat exchanger relative to its heat transfer capacity
• NTU is defined as: $NTU = UA / C_min$, where $U$ is the overall heat transfer coefficient, $A$ is the heat transfer area, and $C_min$ is the smaller heat capacity rate of the two fluids
• The heat capacity rate ($C$) is the product of the mass flow rate and specific heat of a fluid: $C = ṁc_p$
• A higher NTU value indicates a larger heat exchanger or a higher overall heat transfer coefficient relative to the heat capacity rate, which generally results in a higher effectiveness

ε-NTU Method for Complex Heat Exchangers

Effectiveness Relations and Heat Capacity Rate Ratio

• The ε-NTU method is particularly useful when the outlet temperatures are unknown or the heat exchanger has a complex flow arrangement
• The effectiveness ($ε$) is related to NTU and the heat capacity rate ratio ($C_r$) through empirical relations that depend on the heat exchanger flow arrangement (parallel flow, counterflow, cross-flow, etc.)
• The heat capacity rate ratio is defined as: $C_r = C_min / C_max$, where $C_min$ and $C_max$ are the smaller and larger heat capacity rates of the two fluids, respectively
• Example: For a counterflow heat exchanger, the effectiveness relation is given by: $ε = (1 - exp[-NTU(1 - C_r)]) / (1 - C_r exp[-NTU(1 - C_r)])$ for $C_r < 1$, and $ε = NTU / (1 + NTU)$ for $C_r = 1$

Calculating Heat Transfer Rates and Outlet Temperatures

• Once the effectiveness is determined using the appropriate relation, the actual heat transfer rate can be calculated using: $Q = ε Q_max = ε C_min (T_h,in - T_c,in)$
• The outlet temperatures can then be found using the energy balance equations: $T_h,out = T_h,in - Q / (ṁ_h c_p,h)$ and $T_c,out = T_c,in + Q / (ṁ_c c_p,c)$
• The ε-NTU method assumes constant specific heats, no phase change, and negligible heat loss to the surroundings, similar to the LMTD method
• Example: In a cross-flow heat exchanger with both fluids unmixed, if the NTU is 2.5 and the heat capacity rate ratio is 0.8, the effectiveness can be determined using the appropriate relation. The heat transfer rate and outlet temperatures can then be calculated using the effectiveness, maximum heat transfer rate, and energy balance equations