Heat exchangers are crucial in transferring thermal energy between fluids. The Log Mean Temperature Difference (LMTD) method is a key tool for analyzing their performance. It provides a more accurate average temperature difference than simpler methods.

The LMTD equation accounts for temperature changes along the exchanger's length. By using this method, engineers can calculate heat transfer rates and outlet temperatures for various flow arrangements, including parallel-flow, counter-flow, and multi-pass systems.

LMTD Equation for Heat Exchangers

Derivation of the LMTD Equation

• The LMTD equation is derived from the heat transfer rate equation, $Q = UA∆T$
  • $Q$ represents the heat transfer rate
  • $U$ represents the overall heat transfer coefficient
  • $A$ represents the heat transfer area
  • $∆T$ represents the temperature difference between the hot and cold fluids
• The LMTD equation is given by: $LMTD = (∆T1 - ∆T2) / ln(∆T1 / ∆T2)$
  • $∆T1$ represents the temperature difference between the hot and cold fluids at the inlet of the heat exchanger
  • $∆T2$ represents the temperature difference between the hot and cold fluids at the outlet of the heat exchanger
• The LMTD equation accounts for the logarithmic variation of the temperature difference along the length of the heat exchanger
• The LMTD provides a more accurate representation of the average temperature difference compared to the arithmetic mean temperature difference (AMTD)

Assumptions and Limitations of the LMTD Equation

• The LMTD equation assumes that the overall heat transfer coefficient ($U$) is constant throughout the heat exchanger
• The LMTD equation assumes that the specific heat capacities of the fluids are constant throughout the heat exchanger
• The LMTD equation assumes that there is no phase change (condensation or evaporation) of the fluids within the heat exchanger
• The LMTD equation does not account for the effects of axial conduction or heat losses to the surroundings

LMTD Method for Heat Transfer Analysis

Determining Heat Transfer Rate and Outlet Temperatures

• The LMTD method involves calculating the LMTD using the inlet and outlet temperatures of the hot and cold fluids
• The heat transfer rate equation, $Q = UA(LMTD)$, is used to determine the heat transfer rate
• The outlet temperatures of the hot and cold fluids can be determined using the energy balance equations:
  • For the hot fluid: $Qh = mh * Cph * (Th,in - Th,out)$
  • For the cold fluid: $Qc = mc * Cpc * (Tc,out - Tc,in)$
  • $m$ represents the mass flow rate
  • $Cp$ represents the specific heat capacity
  • Subscripts $h$ and $c$ refer to the hot and cold fluids, respectively

Applicability to Different Flow Arrangements

• The LMTD method is applicable to both parallel-flow and counter-flow heat exchangers
• For parallel-flow heat exchangers, the temperature differences ($∆T1$ and $∆T2$) are calculated using the inlet and outlet temperatures of the fluids on the same end of the heat exchanger
• For counter-flow heat exchangers, the temperature differences ($∆T1$ and $∆T2$) are calculated using the inlet and outlet temperatures of the fluids on opposite ends of the heat exchanger

Correction Factors for Heat Exchangers

Purpose and Definition of Correction Factors

• Correction factors ($F$) account for the deviation of the actual mean temperature difference from the LMTD in multi-pass and cross-flow heat exchangers
• The correction factor is a function of the heat exchanger geometry, flow arrangement, and the temperature effectiveness ($P$) and heat capacity rate ratio ($R$) of the fluids
• The corrected LMTD is given by: $LMTDcorrected = F LMTD$

Temperature Effectiveness and Heat Capacity Rate Ratio

• The temperature effectiveness ($P$) is defined as the ratio of the actual temperature change of one fluid to the maximum possible temperature change
• The heat capacity rate ratio ($R$) is the ratio of the smaller heat capacity rate to the larger heat capacity rate
  • The heat capacity rate is the product of the mass flow rate and specific heat capacity ($mCp$)
• Correction factor charts or correlations are used to determine the correction factor based on the heat exchanger configuration and the values of $P$ and $R$

LMTD Method for Flow Arrangements

Solving Heat Exchanger Problems

• The LMTD method can be applied to solve heat exchanger problems for various flow arrangements:
  • Parallel-flow heat exchangers
  • Counter-flow heat exchangers
  • Multi-pass heat exchangers (shell-and-tube)
  • Cross-flow heat exchangers with one or both fluids mixed or unmixed
• Given information typically includes:
  • Inlet temperatures of the hot and cold fluids
  • Mass flow rates of the hot and cold fluids
  • Specific heat capacities of the hot and cold fluids
  • Overall heat transfer coefficient
  • Heat transfer area
• Problem-solving approach:
  1. Calculate the LMTD using the appropriate temperature differences
  2. Determine the correction factor ($F$) for multi-pass and cross-flow heat exchangers
  3. Calculate the corrected LMTD ($LMTDcorrected = F LMTD$)
  4. Determine the heat transfer rate using the heat transfer rate equation ($Q = UA(LMTDcorrected)$)
  5. Calculate the outlet temperatures of the fluids using the energy balance equations

Examples of Heat Exchanger Problems

• Example 1: A counter-flow heat exchanger is used to cool oil (hot fluid) with water (cold fluid). Given the inlet temperatures, mass flow rates, specific heat capacities, overall heat transfer coefficient, and heat transfer area, determine the outlet temperatures of the oil and water.
• Example 2: A multi-pass shell-and-tube heat exchanger is used to heat water (cold fluid) with steam (hot fluid). Given the inlet temperatures, mass flow rates, specific heat capacities, overall heat transfer coefficient, and heat transfer area, determine the heat transfer rate and the outlet temperature of the water. Use the appropriate correction factor based on the heat exchanger configuration and the values of $P$ and $R$.