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title: LMTD for parallel and counter flow exchanger, condenser and evaporator
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# LMTD for parallel and counter flow exchanger, condenser and evaporator
Unit 4, Lecture 29

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## Log Mean Temperature Difference (LMTD)

- Course: Heat and Mass Transfer (DI05019071)
- Unit 4: Heat Exchanger
- Lecture 29: LMTD for Parallel, Counter Flow, Condenser & Evaporator

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Welcome. Today we derive and apply the Log Mean Temperature Difference, the standard method for sizing heat exchangers.
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## Lecture Agenda

- 1. The Concept of LMTD
- 2. Derivation of LMTD for Parallel Flow
- 3. Derivation of LMTD for Counter Flow
- 4. Comparing Parallel vs. Counter Flow
- 5. LMTD for Condensers
- 6. LMTD for Evaporators

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We will rigorously define the LMTD for both major flow types and see how it simplifies for phase change devices.
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## The Concept of LMTD

- From Newton's Law of Cooling, total heat transfer Q = U * A * delta_T_m.
- delta_T_m must account for the non-linear temperature profiles of the fluids.
- By integrating the local heat transfer equations along the length, we obtain a logarithmic average.
- This is called the Log Mean Temperature Difference (LMTD).

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Because the temperature difference between the fluids decays exponentially, the correct average to use is logarithmic, not arithmetic.
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## Derivation Setup: Parallel Flow

- Consider a differential area dA in a parallel flow exchanger.
- Hot fluid temperature changes by dT_h (negative).
- Cold fluid temperature changes by dT_c (positive).
- Local temperature difference delta_T = T_h - T_c.

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We begin by analyzing a tiny slice of the heat exchanger where the heat transfer is dQ.
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## LMTD Parallel Flow Equation

- Integrating d(delta_T) / delta_T from inlet to outlet yields:
- LMTD = (delta_T_1 - delta_T_2) / ln(delta_T_1 / delta_T_2)
- Where for Parallel Flow:
- delta_T_1 = T_hi - T_ci (Inlet temperature difference)
- delta_T_2 = T_ho - T_co (Outlet temperature difference)

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The integration results in this elegant logarithmic formula. Note carefully how delta_T_1 and delta_T_2 are defined for parallel flow.
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## Derivation Setup: Counter Flow

- For counter flow, fluids move in opposite directions.
- Hot fluid enters at one end, cold fluid enters at the other.
- The derivation steps are similar, but the temperature boundary conditions change.

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If we reverse the direction of the cold fluid, we must adjust our boundary conditions for the integration.
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## LMTD Counter Flow Equation

- The LMTD formula remains the same:
- LMTD = (delta_T_1 - delta_T_2) / ln(delta_T_1 / delta_T_2)
- However, the definitions change for Counter Flow:
- delta_T_1 = T_hi - T_co (Hot inlet minus Cold outlet)
- delta_T_2 = T_ho - T_ci (Hot outlet minus Cold inlet)

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The formula looks identical, but the physical meaning of the temperature differences at ends 1 and 2 has completely changed.
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## Comparing Parallel vs. Counter Flow

- For the same inlet and outlet temperatures, LMTD_counter > LMTD_parallel.
- Since Q = U * A * LMTD, a larger LMTD means a smaller required surface area (A) for the same Q and U.
- Therefore, counter flow heat exchangers are more compact and economical.

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This mathematical result proves why counter flow is preferred in industry: it requires less surface area to transfer the same amount of heat.
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## Condensers in Heat Exchangers

- In a condenser, the hot fluid (vapor) condenses at a constant temperature (T_h = constant).
- The cold fluid absorbs heat and its temperature rises.
- Since T_h is constant, parallel and counter flow configurations yield the same temperature profile.

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When the hot fluid condenses, its temperature stays flat. In this case, flow direction doesn't matter mathematically.
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## Evaporators in Heat Exchangers

- In an evaporator, the cold fluid (liquid) boils at a constant temperature (T_c = constant).
- The hot fluid rejects heat and its temperature drops.
- Similar to condensers, the flow direction (parallel vs counter) does not affect the LMTD.

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Evaporators are the mirror image. The cold fluid stays at a constant boiling temperature.
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## LMTD for Phase Change

- For condensers and evaporators, use the standard LMTD formula.
- delta_T_1 and delta_T_2 are simply the temperature differences at the two ends.
- LMTD_parallel = LMTD_counter for any phase change exchanger where one temperature is constant.

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You don't need to worry about flow arrangement when designing a condenser or evaporator; the LMTD is the same either way.
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## Summary

- LMTD is the exact mean temperature difference for heat exchanger analysis.
- Counter flow LMTD is greater than parallel flow LMTD for the same conditions.
- Phase change devices have identical LMTD regardless of flow arrangement.

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To summarize, LMTD allows us to relate terminal temperatures to heat exchanger size, with counter flow offering the highest efficiency.
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## Next Lecture Preview

- Topic: Overall Heat Transfer Coefficient
- - Thermal resistance network in heat exchangers
- - Definition of U based on inner and outer areas
- - Fouling factors and their impact on performance

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Next time, we will explore the 'U' term in our equation, the overall heat transfer coefficient, and how scaling and dirt reduce efficiency.
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