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title: Overall heat transfer coefficient
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# Overall heat transfer coefficient
Unit 4, Lecture 30

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## Overall Heat Transfer Coefficient

- Course: Heat and Mass Transfer (DI05019071)
- Unit 4: Heat Exchanger
- Lecture 30: Overall Heat Transfer Coefficient & Fouling

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Welcome. Today we focus on 'U', the overall heat transfer coefficient, which dictates how easily heat moves from one fluid to another.
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## Lecture Agenda

- 1. Heat Transfer Process in a Tube
- 2. Thermal Resistance Network
- 3. Overall Heat Transfer Coefficient (U)
- 4. U based on Inside vs. Outside Area
- 5. Introduction to the Fouling Factor
- 6. Design Considerations for Fouling

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We'll build the thermal resistance network, derive U, and then discuss the real-world problem of fouling or scaling in pipes.
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## Heat Transfer Process in a Tube

- Heat transfers from hot fluid to cold fluid through three mechanisms:
- 1. Convection from hot fluid to the inner wall.
- 2. Conduction through the solid tube wall.
- 3. Convection from the outer wall to the cold fluid.

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Heat encounters three hurdles: the inner fluid boundary layer, the metal wall itself, and the outer fluid boundary layer.
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## Thermal Resistance Network

- Total thermal resistance R_total = R_conv,i + R_wall + R_conv,o
- R_conv,i = 1 / (h_i * A_i) (Inner convection resistance)
- R_wall = ln(r_o / r_i) / (2 * pi * k * L) (Cylindrical wall conduction resistance)
- R_conv,o = 1 / (h_o * A_o) (Outer convection resistance)

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We can model this using an electrical resistance analogy. The total resistance is the sum of the inner convection, wall conduction, and outer convection.
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## Overall Heat Transfer Coefficient (U)

- Q = (T_h - T_c) / R_total = U * A * (T_h - T_c)
- Therefore, U * A = 1 / R_total
- U is the overall heat transfer coefficient.
- It represents the total ability of the system to transfer heat.

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By equating our resistance network to Newton's law of cooling, we define U. It is simply the inverse of the total thermal resistance times area.
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## U based on Inside vs. Outside Area

- Since the inner area (A_i) and outer area (A_o) of a tube are different, U depends on which area is used.
- U_i * A_i = U_o * A_o = 1 / R_total
- 1 / (U_i * A_i) = 1/(h_i*A_i) + R_wall + 1/(h_o*A_o)
- When specifying U, one must specify if it is U_i or U_o.

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For thick-walled tubes or finned tubes, the inner and outer areas differ significantly, so U_inner will not equal U_outer.
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## Simplifying U for Thin-Walled Tubes

- If the tube wall is very thin and highly conductive, R_wall approaches 0.
- Also, A_i is approximately equal to A_o.
- The equation simplifies to: 1/U = 1/h_i + 1/h_o
- U is dominated by the smaller convection coefficient (the larger resistance).

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In many practical cases, the metal wall resistance is negligible compared to the fluid resistances, simplifying our calculations greatly.
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## The Fouling Factor (Rf)

- Over time, surfaces accumulate deposits (scale, rust, algae).
- These deposits act as additional insulation, reducing heat transfer.
- The effect of these deposits is represented by a fouling factor, R_f.
- R_f has units of m^2 K / W.

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In the real world, pipes get dirty. Scale, algae, and rust build up and form a layer of insulation that ruins heat exchanger performance.
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## Impact of Fouling on Heat Transfer

- Fouling adds additional thermal resistances to our network.
- New equation for U based on outer area:
- 1 / U_o = (A_o/A_i)(1/h_i) + (A_o/A_i)R_f,i + A_o*R_wall + R_f,o + 1/h_o
- As R_f increases, U decreases, and Q drops.

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We account for dirt mathematically by adding fouling resistances to our thermal circuit. This directly lowers the overall U value.
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## Typical Values of Fouling Factors

- Distilled water: R_f = 0.0001 m^2 K/W
- City water: R_f = 0.0002 m^2 K/W
- River water: R_f = 0.001 m^2 K/W
- Fuel oil: R_f = 0.0009 m^2 K/W
- Notice that dirtier fluids have much higher fouling factors.

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Engineers use tables to estimate fouling based on the fluid. River water will foul a heat exchanger ten times faster than distilled water.
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## Design Considerations for Fouling

- Engineers must oversize heat exchangers to account for future fouling.
- A heat exchanger designed with a clean U value will fail to meet requirements after a few months.
- Periodic cleaning schedules must be planned to restore U to design levels.

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Because fouling is inevitable, we deliberately build heat exchangers larger than strictly necessary, providing a buffer so they still work when dirty.
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## Summary

- U incorporates all convective and conductive resistances.
- U must be referenced to a specific area (inside or outside).
- Fouling adds thermal resistance and degrades performance over time.
- Heat exchangers must be oversized to compensate for fouling.

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To summarize, the overall heat transfer coefficient U dictates performance, but it degrades over time due to fouling.
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## Next Lecture Preview

- Topic: Effectiveness of Heat Exchanger (NTU Method)
- - What to do when outlet temperatures are unknown
- - Definition of heat exchanger effectiveness
- - The Number of Transfer Units (NTU)

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Next lecture, we will introduce the NTU method, a powerful alternative to LMTD used when fluid outlet temperatures are unknown.
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