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title: Overall heat transfer coefficient - conduction and convection
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# Overall heat transfer coefficient - conduction and convection
Unit 2, Lecture 18

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## Overall Heat Transfer Coefficient

- Course: Heat and Mass Transfer (DI05019071)
- Unit 2: Convection
- Lecture 18: Overall Heat Transfer Coefficient

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In many practical applications, heat transfers through a solid separating two fluids. Today we will combine conduction and convection into a single unified equation using the Overall Heat Transfer Coefficient.
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## Lecture Agenda

- 1. Real-world Heat Transfer Scenarios
- 2. The Thermal Resistance Concept Revisited
- 3. Defining Convection Resistance
- 4. Heat Transfer through a Plane Wall (Convection on both sides)
- 5. The Overall Heat Transfer Coefficient (U)
- 6. U for Cylindrical Geometries (Pipes)
- 7. Fouling and Thermal Contact Resistance

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We will start by reviewing thermal resistance, introduce the resistance formula for convection, and then build complete thermal circuits.
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## Real-world Scenarios

- In most engineering problems, heat transfer involves multiple modes in series.
- Example 1: Heat lost from a room through a wall to the cold outside air. (Inner convection -> Wall conduction -> Outer convection).
- Example 2: Heat exchanger tube. Hot fluid inside, cold fluid outside. (Inner convection -> Tube wall conduction -> Outer convection).

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We rarely deal with isolated conduction or convection. Heat typically moves from a hot fluid, through a solid barrier, and into a cold fluid.
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## Thermal Resistance Revisited

- Recall from Unit 1: Heat transfer can be modeled analogously to electrical current.
- Current (I) -> Heat transfer rate (q)
- Voltage Difference (ΔV) -> Temperature Difference (ΔT)
- Electrical Resistance (R_e) -> Thermal Resistance (R_t)
- q = ΔT / R_total

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This electrical analogy is a powerful tool. It allows us to add up different resistances in series or parallel to find the total heat transfer rate.
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## Convection Resistance

- Newton's Law of Cooling: q = h * A * (T_s - T_\infty)
- Rearranging into the resistance format (q = ΔT / R):
- q = (T_s - T_\infty) / (1 / (h*A))
- Therefore, Convection Thermal Resistance is:
- R_conv = 1 / (h * A)

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By rearranging Newton's law, we can define the thermal resistance due to convection. Notice it depends on the convection coefficient 'h' and the surface area 'A'.
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## Heat Transfer through a Plane Wall

- Consider a wall of thickness 'L', thermal conductivity 'k', area 'A'.
- Hot fluid (T_h, h_1) on one side, cold fluid (T_c, h_2) on the other.
- Thermal circuit consists of three resistances in series:
- 1. R_conv1 = 1 / (h_1 * A)
- 2. R_cond = L / (k * A)
- 3. R_conv2 = 1 / (h_2 * A)

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We simply add these three resistances together. The total resistance is R_total = R_conv1 + R_cond + R_conv2.
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## The Overall Heat Transfer Coefficient (U)

- To simplify calculations, we define the Overall Heat Transfer Coefficient (U).
- Equation: q = U * A * ΔT_overall
- Where ΔT_overall = T_hot_fluid - T_cold_fluid.
- Comparing with q = ΔT / R_total, we get:
- U * A = 1 / R_total   =>   U = 1 / (R_total * A)

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'U' represents the entire system's ability to transfer heat. It wraps all the convection and conduction resistances into a single, easy-to-use value.
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## U for a Plane Wall

- For our plane wall with convection on both sides:
- R_total = (1 / h_1*A) + (L / k*A) + (1 / h_2*A)
- Since U = 1 / (R_total * A), we can factor out 'A':
- U = 1 / [ (1/h_1) + (L/k) + (1/h_2) ]
- Units for U are W/m²·K (same as h).

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For a plane wall, the area is constant, so it conveniently cancels out of the equation for U.
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## U for Cylindrical Geometries

- For pipes, the inner area (A_i = 2πr_i L) is different from the outer area (A_o = 2πr_o L).
- Because A is not constant, U depends on which area it is based on:
- U_i * A_i = U_o * A_o = 1 / R_total
- U_i = 1 / [ (1/h_i) + (A_i*ln(r_o/r_i)/(2πkL)) + (A_i/(A_o*h_o)) ]

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Pipes are tricky because the surface area increases as you move outward. You must specify whether 'U' is based on the inner surface area or the outer surface area.
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## Fouling Factor (R_f)

- Over time, surfaces accumulate deposits (scale, rust, algae).
- These deposits act as additional thermal insulation, reducing heat transfer.
- This added resistance is called the Fouling Factor (R_f).
- R_total = R_conv1 + R_f1 + R_cond + R_f2 + R_conv2

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In real engineering applications like heat exchangers, pipes get dirty. We must account for this by adding a fouling resistance to our thermal circuit.
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## Impact of Fouling on U

- Fouling always increases the total thermal resistance.
- Therefore, fouling always decreases the Overall Heat Transfer Coefficient (U).
- Equipment must be designed with extra capacity (a clean U value vs a dirty U value) to allow for fouling over its operational life.

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Engineers calculate the 'clean' U and the 'dirty' U. Heat exchangers are designed based on the dirty U so they still function properly right before maintenance is required.
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## Summary

- Convection thermal resistance is R_conv = 1 / (hA).
- The Overall Heat Transfer Coefficient (U) combines all series resistances: U*A = 1/R_total.
- For pipes, U must be specified relative to the inner or outer area.
- Fouling adds thermal resistance and degrades performance over time.

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The U-value is arguably the most important parameter in heat exchanger design. It captures the entire thermal performance of a composite wall.
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## Next Lecture Preview

- Topic: Dimensionless numbers and their physical significance
- Key questions:
- - How do we simplify complex convection problems?
- - What are the Reynolds, Prandtl, and Nusselt numbers?

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We've learned what 'h' is and how to use it. Next lecture, we will learn how to actually calculate 'h' using dimensionless numbers.
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