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title: Overall heat transfer coefficient
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# Overall heat transfer coefficient
Unit 1, Lecture 10

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## Overall Heat Transfer Coefficient

- Course: Heat and Mass Transfer (DI05019071)
- Unit 1: Conduction
- Lecture 10: Overall heat transfer coefficient

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Welcome. Today we will simplify complex thermal circuits into a single, highly useful coefficient.
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## Lecture Agenda

- 1. Introduction to Combined Modes
- 2. Defining the Overall Heat Transfer Coefficient (U)
- 3. U for a Plane Wall
- 4. U for a Cylindrical Surface
- 5. Inner vs. Outer Area Basis for U
- 6. Fouling Factors

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We will learn how to define U for both flat walls and pipes, and discuss real-world issues like fouling.
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## Combined Modes of Heat Transfer

- Most engineering problems involve multiple modes of heat transfer.
- Example: Hot fluid inside a pipe, cold air outside.
- Heat transfers by: Convection (inside) -> Conduction (through pipe wall) -> Convection (outside).
- We need a simple way to express this total process.

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Rarely is heat transfer just conduction. It's usually bounded by fluids, requiring us to combine convection and conduction.
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## Defining the Overall Coefficient (U)

- We define the overall heat transfer coefficient, U, such that:
- q = U * A * ΔT_total
- Where ΔT_total is the temperature difference between the two bulk fluids.
- Comparing this to q = ΔT_total / R_total, we see that:
- U * A = 1 / R_total

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U is essentially the inverse of the total thermal resistance times area. It tells us how well heat transfers across the entire assembly.
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## U for a Plane Wall

- Consider a wall of thickness L, conductivity k, with fluids h_in and h_out.
- R_total = (1/h_in*A) + (L/k*A) + (1/h_out*A)
- Since area A is constant: R_total = (1/A) * [1/h_in + L/k + 1/h_out]
- Therefore: 1/U = 1/h_in + L/k + 1/h_out

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For a flat wall, the area is the same everywhere, so it cancels out of the equation beautifully.
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## U for a Cylindrical Surface

- For a pipe, the inner area (A_in) is not equal to the outer area (A_out).
- R_total = 1/(h_in*A_in) + ln(r_out/r_in)/(2πLk) + 1/(h_out*A_out)
- Because area varies, U depends on which area we use as a reference.
- q = U_in * A_in * ΔT = U_out * A_out * ΔT

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Because the inside and outside areas of a pipe are different, we must specify whether our U-value is based on the inside area or the outside area.
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## Inner vs. Outer Area Basis

- Based on inner area (A_in = 2πr_in*L):
- 1/U_in = 1/h_in + [r_in * ln(r_out/r_in)]/k + (r_in/r_out)*(1/h_out)
- Based on outer area (A_out = 2πr_out*L):
- 1/U_out = (r_out/r_in)*(1/h_in) + [r_out * ln(r_out/r_in)]/k + 1/h_out
- Note: U_in * A_in = U_out * A_out

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It doesn't matter which one you use to find the total heat transfer, as long as you multiply it by its corresponding area.
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## Fouling Factor Concept

- Over time, surfaces accumulate deposits (rust, scale, algae).
- These deposits act as additional thermal resistance.
- We account for this using a 'fouling factor' (Rf).
- Rf has units of m²·K/W.

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In the real world, equipment gets dirty. Mineral scale in water pipes creates a significant barrier to heat transfer.
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## Including Fouling in U

- We simply add the fouling resistances to our total resistance.
- 1/U = 1/h_in + Rf_in + L/k + Rf_out + 1/h_out
- Fouling always decreases the overall heat transfer coefficient.
- Designers must overestimate U requirements to account for future fouling.

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When designing a heat exchanger, you must factor in how dirty it will get over years of operation, or it will underperform.
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## Typical Values of U (W/m²K)

- Water to Water: 850 - 1700
- Water to Oil: 110 - 350
- Steam condenser (Water in tubes): 1000 - 6000
- Air to Water: 10 - 50
- Notice that systems involving gases (air) have much lower U values.

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Gases are poor heat transfer fluids compared to liquids or boiling/condensing systems, which is reflected in their low U values.
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## Example Problem

- A steel pipe (k=50) has r_in = 20 mm, r_out = 24 mm.
- Inside water h_in = 1000 W/m²K. Outside air h_out = 15 W/m²K.
- Calculate U based on the outside area (U_out).

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Let's apply the U_out formula to a pipe with water inside and air outside.
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## Summary

- The overall heat transfer coefficient (U) combines conduction and convection.
- For cylinders, U must be referenced to a specific area (usually A_in or A_out).
- Fouling factors account for the thermal resistance of scale/dirt build-up.
- U simplifies the calculation of heat transfer in complex systems.

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The U-value is the standard metric used by industry to rate the performance of heat exchangers and building insulation.
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## Next Lecture Preview

- Topic: Heat transfer from an extended surface (Fins)
- How can we increase heat transfer when convection is poor?
- We will explore the use of fins to increase surface area.

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Next time, we look at what to do when your U-value is too low—we add fins to artificially increase the area.
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