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title: LMTD for Parallel and Counter Flow Exchangers
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# LMTD for Parallel and Counter Flow Exchangers
Unit 4, Lecture 32

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## LMTD for Parallel and Counter Flow Exchangers

- Course: Heat and Mass Transfer (DI05019071)
- Unit 4: Heat Exchanger
- Lecture 32: LMTD for Parallel and Counter Flow Exchangers

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Welcome to Unit 4. Today we'll learn how to analyze the temperature differences in heat exchangers using LMTD.
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## Lecture Agenda

- 1. Introduction to Mean Temperature Difference
- 2. Temperature Distribution in Parallel Flow
- 3. Derivation of LMTD for Parallel Flow
- 4. Temperature Distribution in Counter Flow
- 5. Derivation of LMTD for Counter Flow
- 6. Comparison of Flow Types

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Here is our roadmap. We will derive the Logarithmic Mean Temperature Difference for both parallel and counter flow.
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## Introduction to LMTD

- In a heat exchanger, temperatures of both fluids change as they flow.
- The temperature difference ΔT between hot and cold fluids varies along the length.
- Newton's Law of Cooling: q = U A ΔT_mean
- We need an appropriate average ΔT, which is the Logarithmic Mean Temperature Difference (LMTD).

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Since ΔT is not constant, we cannot simply use the inlet or outlet temperature differences. We need a proper mean.
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## Why Not Arithmetic Mean?

- Arithmetic Mean Temperature Difference (AMTD) = (ΔT1 + ΔT2) / 2
- AMTD is only accurate if the temperature profiles are linear.
- In reality, heat transfer rate decreases as ΔT decreases, leading to exponential temperature profiles.
- Therefore, AMTD overestimates the actual mean temperature difference.

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An arithmetic mean assumes a linear change, but temperatures approach each other exponentially, meaning LMTD is required for accurate analysis.
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## Temperature Distribution in Parallel Flow

- Both hot and cold fluids enter at the same end and flow in the same direction.
- Hot fluid temperature (Th) drops; cold fluid temperature (Tc) rises.
- ΔT is maximum at the inlet (ΔT1) and minimum at the outlet (ΔT2).
- ΔT1 = Th,in - Tc,in
- ΔT2 = Th,out - Tc,out

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Notice how the temperatures approach each other but can never cross in a parallel flow configuration.
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## Derivation of LMTD for Parallel Flow (Part 1)

- Consider a small area element dA.
- Heat transfer: dq = U dA (Th - Tc) = U dA ΔT
- Energy balance for hot fluid: dq = - m_h cp_h dTh = - Ch dTh
- Energy balance for cold fluid: dq = m_c cp_c dTc = Cc dTc
- d(ΔT) = dTh - dTc = -dq(1/Ch + 1/Cc)

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We start with a differential energy balance on a small segment of the heat exchanger.
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## Derivation of LMTD for Parallel Flow (Part 2)

- Substitute dq = U dA ΔT into the equation:
- d(ΔT) = - U dA ΔT (1/Ch + 1/Cc)
- Rearranging: d(ΔT)/ΔT = - U (1/Ch + 1/Cc) dA
- Integrating from inlet (1) to outlet (2):
- ln(ΔT2 / ΔT1) = - U A (1/Ch + 1/Cc)

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By integrating along the length of the exchanger, we get the natural logarithm of the temperature differences.
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## Final LMTD Formula for Parallel Flow

- From overall energy balance: 1/Ch = (Th,in - Th,out)/q and 1/Cc = (Tc,out - Tc,in)/q
- Substitute these back:
- ln(ΔT2 / ΔT1) = - U A / q * (ΔT1 - ΔT2)
- Rearranging for q: q = U A [(ΔT1 - ΔT2) / ln(ΔT1 / ΔT2)]
- Thus, LMTD = (ΔT1 - ΔT2) / ln(ΔT1 / ΔT2) = (ΔT2 - ΔT1) / ln(ΔT2 / ΔT1)

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This is the final LMTD expression for parallel flow. Note that it works whether you use (1-2) or (2-1) as long as you are consistent.
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## Temperature Distribution in Counter Flow

- Fluids enter at opposite ends and flow in opposite directions.
- Inlet 1: Hot fluid in, Cold fluid out. ΔT1 = Th,in - Tc,out
- Inlet 2: Hot fluid out, Cold fluid in. ΔT2 = Th,out - Tc,in
- Temperature difference ΔT is more uniform along the length.

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In counter flow, the cold fluid exits where the hot fluid enters. The cold fluid can actually exit at a higher temperature than the hot fluid's exit temperature.
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## Derivation of LMTD for Counter Flow

- Energy balances are similar, but dTc is negative along positive x.
- dq = - Ch dTh = - Cc dTc (since cold fluid flows opposite)
- d(ΔT) = dTh - dTc = -dq(1/Ch - 1/Cc)
- Integrating gives: ln(ΔT2 / ΔT1) = - U A (1/Ch - 1/Cc)

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The derivation is nearly identical, except the sign for the cold fluid capacity rate changes due to the flow direction.
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## Final LMTD Formula for Counter Flow

- Substituting the capacity rates as before yields the exact same mathematical form:
- LMTD = (ΔT1 - ΔT2) / ln(ΔT1 / ΔT2)
- IMPORTANT: The definitions of ΔT1 and ΔT2 are different from parallel flow!
- Parallel: ΔT1 = Th,in - Tc,in ; ΔT2 = Th,out - Tc,out
- Counter: ΔT1 = Th,in - Tc,out ; ΔT2 = Th,out - Tc,in

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The formula looks identical! But be extremely careful: the definitions of the temperature differences at each end are physically different.
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## Comparison: Parallel vs Counter Flow

- For the same inlet and outlet temperatures, LMTD_counter > LMTD_parallel.
- Higher LMTD means a smaller required Area (A) for the same heat transfer (q).
- Counter flow is more efficient and requires a smaller heat exchanger.
- Counter flow allows Tc,out > Th,out, which is impossible in parallel flow.

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Counter flow is almost always preferred in industry because it maximizes the average temperature difference, leading to smaller, cheaper heat exchangers.
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## Summary & Next Lecture

- LMTD is the appropriate average temperature difference for heat exchangers.
- The formula is LMTD = (ΔT1 - ΔT2) / ln(ΔT1 / ΔT2).
- ΔT definitions depend heavily on the flow arrangement.
- Counter flow is thermally more efficient than parallel flow.
- Next Lecture: LMTD for Condensers and Evaporators, and Cross-flow arrangements.

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Review these derivations. Next time, we will see what happens when one of the fluids changes phase, like in a boiler or condenser.
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