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title: Overall Heat Transfer Coefficient
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# Overall Heat Transfer Coefficient
Unit 4, Lecture 34

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## Overall Heat Transfer Coefficient

- Course: Heat and Mass Transfer (DI05019071)
- Unit 4: Heat Exchanger
- Lecture 34: Overall Heat Transfer Coefficient and Fouling

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Welcome. We have been using 'U' in our LMTD equations. Today, we dissect what U actually represents and how to calculate it.
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## Lecture Agenda

- 1. Recap of Heat Transfer Modes
- 2. The Thermal Resistance Network
- 3. Defining the Overall Heat Transfer Coefficient (U)
- 4. U based on Inner and Outer Areas
- 5. The Problem of Fouling
- 6. Fouling Factors (Rf) and their effect on U

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We will combine convection and conduction into a single term, and then look at real-world degradation due to dirt and scale.
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## The Heat Transfer Path

- In a tubular heat exchanger, heat transfers from the hot fluid to the cold fluid via 3 steps:
- 1. Convection from hot fluid to the inner pipe wall.
- 2. Conduction through the pipe wall.
- 3. Convection from the outer pipe wall to the cold fluid.
- q = ΔT_total / R_total

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Heat must navigate three distinct thermal resistances in series. We sum these up to find the total resistance.
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## Thermal Resistance Network

- R_total = R_conv,inner + R_cond,wall + R_conv,outer
- R_conv,inner = 1 / (hi Ai)
- R_cond,wall = ln(ro/ri) / (2π k L)   (For a cylindrical pipe)
- R_conv,outer = 1 / (ho Ao)
- R_total = 1 / (hi Ai) + ln(ro/ri) / (2π k L) + 1 / (ho Ao)

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Just like electrical circuits, thermal resistances in series simply add up.
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## Defining U (Clean Surfaces)

- We define U such that: q = U A ΔT = ΔT / R_total
- Therefore, U A = 1 / R_total
- Since Ai ≠ Ao for a pipe, we can define U based on either area.
- Ui Ai = Uo Ao = 1 / R_total
- Ui = 1 / [ Ai/Ai*hi + Ai*ln(ro/ri)/(2π k L) + Ai/(Ao*ho) ]

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Because the inside area is smaller than the outside area of a pipe, the U value will be different depending on which area you multiply it by. Always specify Ui or Uo.
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## Approximations for Thin Tubes

- For thin-walled tubes (like in most compact heat exchangers), Ai ≈ Ao ≈ A.
- The wall resistance is often very small compared to convection.
- If R_cond,wall ≈ 0, then:
- 1 / U ≈ 1 / hi + 1 / ho
- U ≈ (hi * ho) / (hi + ho)

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In many practical cases, the metal tube is highly conductive and very thin, so its resistance is negligible. The overall U is dominated by the convection coefficients.
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## The Problem of Fouling

- Over time, heat exchanger surfaces accumulate deposits.
- Scaling (minerals), rust, biological growth (algae), or suspended solids.
- This deposit acts as an additional layer of insulation.
- This phenomenon is called 'fouling' and it severely reduces U.

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In the real world, heat exchangers don't stay clean. The buildup of scale adds significant thermal resistance, dropping performance.
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## The Fouling Factor (Rf)

- We account for this degradation using a Fouling Factor, Rf.
- Rf has units of m²·K/W or (hr·ft²·°F)/Btu.
- It represents the thermal resistance of the fouling layer.
- There can be fouling on the inside (Rfi) and the outside (Rfo).

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The fouling factor is determined empirically. TEMA (Tubular Exchanger Manufacturers Association) publishes standard fouling factors for different fluids.
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## U with Fouled Surfaces

- The total thermal resistance now has 5 terms:
- R_total = 1/(hi Ai) + Rfi/Ai + R_wall + Rfo/Ao + 1/(ho Ao)
- 1 / (Uo Ao) = 1/(hi Ai) + Rfi/Ai + ln(ro/ri)/(2πkL) + Rfo/Ao + 1/(ho Ao)
- Fouling always decreases the Overall Heat Transfer Coefficient.

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When designing a heat exchanger, you must design for the fouled condition, otherwise it will fail to meet heat duty after a few months of operation.
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## Typical Values of Fouling Factors

- Distilled water: Rf ≈ 0.0001 m²·K/W
- City water: Rf ≈ 0.0002 m²·K/W
- River water: Rf ≈ 0.0005 m²·K/W
- Fuel oil: Rf ≈ 0.0009 m²·K/W
- Untreated cooling tower water requires frequent cleaning.

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Notice that river water deposits much more scale and dirt than distilled water. The dirtier the fluid, the higher the Rf.
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## Controlling Heat Transfer

- The overall U is dominated by the smallest convection coefficient (hi or ho).
- If hi is water (large) and ho is air (very small), U will be close to ho.
- To improve a heat exchanger, focus on improving the side with the lowest 'h'.
- This is why fins are added to the air side of car radiators!

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Adding fins to the water side of a radiator would do almost nothing. You must add fins to the air side because air has a terrible convection coefficient.
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## Example Application

- Given: hi = 1000 W/m²K, ho = 100 W/m²K (thin tube).
- Clean U ≈ (1000*100) / (1100) = 90.9 W/m²K.
- If a fouling factor of 0.002 m²K/W develops on the inside:
- 1/U_dirty = 1/90.9 + 0.002 = 0.011 + 0.002 = 0.013
- U_dirty = 76.9 W/m²K (a 15% drop in performance).

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This simple example shows how a thin layer of scale can reduce the total heat transfer capacity by 15%.
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## Summary & Next Lecture

- U combines convection and conduction into a single term.
- Uo and Ui differ due to differing surface areas.
- Fouling introduces additional thermal resistance, lowering U.
- Designers must use fouled U-values to ensure long-term operation.
- Next Lecture: The Effectiveness-NTU Method.

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Now we know how to calculate U, and how to use LMTD. Next lecture, we introduce a powerful new method to bypass LMTD's limitations: Effectiveness-NTU.
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