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title: Effectiveness of Heat Exchanger and NTU Method
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# Effectiveness of Heat Exchanger and NTU Method
Unit 4, Lecture 35

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## Effectiveness of Heat Exchanger and NTU Method

- Course: Heat and Mass Transfer (DI05019071)
- Unit 4: Heat Exchanger
- Lecture 35: Effectiveness-NTU Method (Part 1)

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Welcome. Today we introduce a new analytical method that avoids the tedious iterations required by LMTD for performance problems.
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## Lecture Agenda

- 1. Why not use LMTD?
- 2. Heat Capacity Rate (C)
- 3. Maximum Possible Heat Transfer (q_max)
- 4. Heat Exchanger Effectiveness (ε)
- 5. The Number of Transfer Units (NTU)
- 6. Capacity Ratio (c)

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We will build the foundation for the Effectiveness-NTU method by defining several new dimensionless parameters.
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## The Limitation of LMTD

- LMTD is excellent for 'Sizing' problems:
- - You know inlet and outlet temps, you find LMTD, then find Area.
- LMTD is terrible for 'Performance' problems:
- - You have a specific heat exchanger (Area known).
- - You know inlet temps, but NOT outlet temps.
- - Finding outlet temps requires guessing them, calculating LMTD, and iterating until things match.

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Iterative solutions are time-consuming and prone to error without software. In 1955, Kays and London formalized the NTU method to bypass this.
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## Heat Capacity Rate (C)

- First, we define the Heat Capacity Rate, C.
- C = mass flow rate (m) * specific heat (cp)
- Ch = m_h * cp_h (Hot fluid capacity rate)
- Cc = m_c * cp_c (Cold fluid capacity rate)
- Units: W/K or J/s·K

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Capacity rate tells us how much energy is required to change a fluid's temperature by 1 degree per second.
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## Identifying C_min and C_max

- Calculate both Ch and Cc.
- The smaller value is designated as C_min.
- The larger value is designated as C_max.
- The fluid with C_min will experience the LARGER temperature change (ΔT) in the heat exchanger.
- Since q = C * ΔT, if C is small, ΔT must be large for the same q.

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Identifying C_min is the most critical first step in the NTU method.
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## Maximum Possible Heat Transfer (q_max)

- What is the theoretical maximum heat transfer possible?
- It happens in an infinitely long counter-flow heat exchanger.
- The fluid with C_min will reach the inlet temperature of the other fluid.
- q_max = C_min * (Th,in - Tc,in)
- The max temperature difference is (Th,in - Tc,in).

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Only the fluid with C_min can achieve the maximum theoretical temperature change. If C_max tried to do it, energy balance would be violated.
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## Defining Effectiveness (ε)

- Effectiveness (ε) is the ratio of actual heat transfer to maximum possible heat transfer.
- ε = q_actual / q_max
- Effectiveness is dimensionless and 0 ≤ ε ≤ 1.
- q_actual = ε * C_min * (Th,in - Tc,in)

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If we know the effectiveness of a heat exchanger, finding the actual heat transfer becomes trivial. We don't even need the outlet temperatures to find it!
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## Calculating Actual Heat Transfer

- Once q_actual is found via effectiveness, outlet temps are easy:
- q_actual = Ch * (Th,in - Th,out)
- Th,out = Th,in - (q_actual / Ch)
- q_actual = Cc * (Tc,out - Tc,in)
- Tc,out = Tc,in + (q_actual / Cc)

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Notice how we bypass LMTD completely. If we have ε, we find q, and then we find the outlet temperatures with basic algebra.
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## How to find Effectiveness?

- Effectiveness depends on the geometry of the heat exchanger and two dimensionless groups.
- ε = function(NTU, c, flow arrangement)
- We need to define NTU and c to use the formulas and charts.

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So the goal shifts from finding LMTD to finding effectiveness. For that, we need two new variables.
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## Number of Transfer Units (NTU)

- NTU is a dimensionless parameter indicating the 'size' or heat transfer capability.
- NTU = (U * A) / C_min
- Higher NTU means a larger heat exchanger (more Area or better U).
- Practically, for NTU > 3, the size of the heat exchanger grows massively for very small gains in effectiveness.

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NTU is proportional to Area. If NTU is small, the exchanger is small. If NTU is huge, the exchanger is massive.
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## Capacity Ratio (c)

- Capacity ratio 'c' is the ratio of the minimum to maximum heat capacity rates.
- c = C_min / C_max
- Since C_min ≤ C_max, the ratio c must be: 0 ≤ c ≤ 1.
- If c = 0, one fluid is undergoing phase change (C_max = infinity).
- If c = 1, both fluids have the same capacity rate (balanced flow).

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This ratio tells us how the thermal mass of the two fluids compares. It completes our set of parameters.
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## Effectiveness Formulas Preview

- Parallel Flow: ε = [1 - exp(-NTU*(1+c))] / (1+c)
- Counter Flow (c<1): ε = [1 - exp(-NTU*(1-c))] / [1 - c*exp(-NTU*(1-c))]
- Counter Flow (c=1): ε = NTU / (1 + NTU)
- These are standard formulas provided in data books.

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You don't need to memorize these; they are in your heat transfer data book. You just need to know how to calculate NTU and c, then plug them in.
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## Summary & Next Lecture

- LMTD requires iteration for performance problems.
- q_max occurs when the C_min fluid reaches the opposite inlet temperature.
- Effectiveness ε = q_actual / q_max.
- ε depends on NTU (UA/Cmin) and c (Cmin/Cmax).
- Next Lecture: Using Effectiveness charts and solving numerical problems.

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Review these new definitions carefully. C_min, NTU, and effectiveness form the core of modern heat exchanger analysis.
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