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title: LMTD for Condensers and Evaporators
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# LMTD for Condensers and Evaporators
Unit 4, Lecture 33

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## LMTD for Condensers and Evaporators

- Course: Heat and Mass Transfer (DI05019071)
- Unit 4: Heat Exchanger
- Lecture 33: Condensers, Evaporators, and Cross-flow

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Welcome back. Today we look at special cases of heat exchangers where phase change occurs.
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## Lecture Agenda

- 1. Concept of Phase Change in Exchangers
- 2. Temperature Profiles in Condensers
- 3. Temperature Profiles in Evaporators
- 4. LMTD for Phase Change
- 5. Cross-flow and Multipass Exchangers
- 6. The Correction Factor (F)

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Our focus will be on fluids that maintain a constant temperature while exchanging heat, and then we will tackle complex flow geometries.
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## Phase Change in Heat Exchangers

- In many industrial processes, one fluid undergoes a phase change (boiling or condensation).
- During phase change at a constant pressure, the temperature of the fluid remains constant.
- Specific heat capacity (cp) is effectively infinite during phase change.
- Therefore, heat capacity rate (C = m cp) approaches infinity.

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Because the temperature doesn't change, the fluid acts like an infinite heat sink or source. This simplifies our LMTD analysis.
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## Temperature Distribution in Condensers

- Hot fluid (vapor) condenses into liquid at a constant temperature (Th).
- Cold fluid temperature (Tc) increases as it absorbs the latent heat.
- Th,in = Th,out = Th (constant line on the graph).
- Tc rises from Tc,in to Tc,out.

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Notice that the hot fluid line is perfectly horizontal. The cold fluid approaches this temperature but cannot exceed it.
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## Temperature Distribution in Evaporators

- Cold fluid (liquid) boils into vapor at a constant temperature (Tc).
- Hot fluid temperature (Th) decreases as it provides the latent heat.
- Tc,in = Tc,out = Tc (constant line on the graph).
- Th drops from Th,in to Th,out.

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Similar to the condenser, but here the cold fluid temperature remains constant.
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## LMTD for Condensers and Evaporators

- Because one temperature is constant, ΔT1 and ΔT2 are the same regardless of flow direction.
- Parallel flow and counter flow arrangements give the exact same LMTD.
- Condenser: ΔT1 = Th - Tc,in ; ΔT2 = Th - Tc,out
- Evaporator: ΔT1 = Th,in - Tc ; ΔT2 = Th,out - Tc
- LMTD = (ΔT1 - ΔT2) / ln(ΔT1 / ΔT2)

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You do not need to specify parallel or counter flow for a condenser or evaporator. The LMTD will be identical in both cases.
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## Beyond Parallel and Counter Flow

- Pure parallel or counter flow is often impractical for large surface areas.
- Industry uses Cross-flow (fluids flow perpendicular to each other).
- Industry also uses Multipass shell-and-tube exchangers (e.g., 1 shell pass, 2 tube passes).
- The standard LMTD formula assumes pure parallel or counter flow.

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To fit more area in a smaller volume, we cross the flows or loop them back and forth. Our basic LMTD formula needs an adjustment.
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## The Correction Factor (F)

- For cross-flow and multipass exchangers, we modify the LMTD equation.
- q = U A F LMTD_cf
- Where LMTD_cf is the LMTD calculated assuming pure counter flow.
- F is a correction factor (0 < F ≤ 1).
- If F < 0.75, the heat exchanger design is generally considered uneconomical.

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We always calculate the counter flow LMTD, and then multiply by F. The factor F represents the deviation from ideal counter flow.
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## How to Find F

- F is found using published charts (Bowman, Mueller, and Nagle charts).
- The charts depend on two dimensionless temperature ratios: P and R.
- P (Capacity ratio) = (t2 - t1) / (T1 - t1)
- R (Temperature ratio) = (T1 - T2) / (t2 - t1)
- (Where T = shell side temps, t = tube side temps)

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You will use these charts extensively in your exams and engineering practice. Calculate P and R, locate the intersection, and read F.
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## Correction Factor for Phase Change

- What happens to the Correction Factor (F) in a condenser or evaporator?
- Since one fluid temperature is constant, R = 0 or R = infinity.
- Looking at the F-charts, for these values, F = 1.0 exactly.
- Conclusion: For phase change, q = U A LMTD (no correction needed for cross/multipass).

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This is a great simplification. If you have a condenser, even if it's cross-flow, F is always 1.
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## Steps for LMTD Method

- 1. Calculate heat duty (q) if temperatures and mass flows are known.
- 2. Determine unknown inlet/outlet temperatures using energy balance.
- 3. Calculate LMTD for counter flow arrangement.
- 4. Find P and R, then read F from the appropriate chart (if cross/multipass).
- 5. Calculate Area A = q / (U F LMTD_cf).

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This is the standard algorithm for sizing a heat exchanger when all inlet and outlet temperatures are known or can be found.
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## Limitations of the LMTD Method

- LMTD is great for sizing problems (finding Area A when temps are known).
- LMTD is difficult for performance problems (finding outlet temps when A is known).
- If outlet temps are unknown, LMTD requires tedious trial-and-error iterations.
- To solve this, we will introduce the Effectiveness-NTU method in upcoming lectures.

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If I give you a heat exchanger and ask you what the outlet temperatures will be, LMTD becomes a mathematical headache. We need a better tool.
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## Summary & Next Lecture

- Flow direction doesn't matter for condensers and evaporators.
- Cross-flow and multipass designs require a correction factor F.
- F = 1 for any phase change process.
- LMTD method requires iteration if outlet temperatures are unknown.
- Next Lecture: Overall Heat Transfer Coefficient and Fouling Factors.

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Review the F-charts from your textbook. Next time, we will dive into how we actually determine the U-value used in these equations.
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