Question

A counter-flow heat exchanger is stated to have an overall heat transfer coefficient, based on outside tube area of \(50 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft}^{2} \cdot{ }^{\circ} \mathrm{F}\) when operating at design and clean conditions. After a period of use, scale built-up in the heat exchanger gives a fouling factor of \(0.002 \mathrm{~h} \cdot \mathrm{ft}^{2}{ }^{\circ} \mathrm{F} /\) Btu. Determine \((a)\) the overall heat transfer coefficient of the heat exchanger and \((b)\) the percentage change in the overall heat transfer coefficient due to the scale built-up.

Short answer

Answer: The overall heat transfer coefficient with the scale built-up (Uf) is approximately 45.45 Btu/h·ft²·°F, and the percentage change in the overall heat transfer coefficient due to the scale built-up is approximately 9.09%.

Step-by-step solution

Given Parameters

We are given the following information: - The clean overall heat transfer coefficient (U): \(50 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft}^{2} \cdot{ }^{\circ} \mathrm{F}\) - The fouling factor (Rf): \(0.002 \mathrm{~h} \cdot \mathrm{ft}^{2}{ }^{\circ} \mathrm{F} /\) Btu

Calculate the overall heat transfer coefficient with the scale built-up (Uf)

We can use the following formula to calculate the overall heat transfer coefficient with the fouling factor: \(1/Uf = 1/U + Rf\) Now, we can substitute the given values into this equation to find the overall heat transfer coefficient with scale built up (Uf): \(1/Uf = 1/50 + 0.002\) \(1/Uf = 0.022\) To find Uf, take the reciprocal of the value obtained: \(Uf = 1/(0.022)\) \(Uf = 45.45 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft}^{2} \cdot{ }^{\circ} \mathrm{F}\) So, the overall heat transfer coefficient with the scale built-up is approximately \(45.45 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft}^{2} \cdot{ }^{\circ} \mathrm{F}\).

Calculate the percentage change in the overall heat transfer coefficient due to the scale built-up

To calculate the percentage change, we can use the following formula: Percentage change= \(((U - Uf) / U) \cdot 100\) Now, we can substitute the given and calculated values to find the percentage change: Percentage change = \(((50 - 45.45) / 50) \cdot 100\) Percentage change = \((4.55/50) \cdot 100\) Percentage change ≈ \(9.09\%\) The percentage change in the overall heat transfer coefficient due to the scale built-up is approximately \(9.09\%\).

Key concepts

Counter-Flow Heat Exchanger

A counter-flow heat exchanger is a type of heat exchanger where two fluids flow in parallel but opposite directions. This design is highly efficient for heat transfer as it maximizes the temperature difference between the fluids along the entire length of the exchanger. 

Here is why a counter-flow arrangement is advantageous:

• Maximal Temperature Difference: Since the warmest part of the hot fluid meets the warmest part of the cold fluid, the temperature difference remains high throughout the exchanger.
  
• Efficiency: By maintaining a consistent thermal gradient, it transfers heat more efficiently than other designs like parallel-flow.
  

Counter-flow heat exchangers are widely used in applications where efficiency is critical, such as in refrigeration and air conditioning systems.

Overall Heat Transfer Coefficient

The overall heat transfer coefficient ( U) is a measure of how well a heat exchanger conducts heat. It combines the thermal resistances of the various layers through which heat must pass, including the fluid films and the walls of the heat exchanger.

The overall heat transfer coefficient depends on several factors:

• Material Properties: Different materials have varying abilities to conduct heat.
  
• Fluid Velocity: Increased fluid velocity enhances heat transfer by reducing the boundary layer thickness.
  
• Surface Area: The more area available for heat exchange, the higher the heat transfer rate.
  

In our exercise, the clean overall heat transfer coefficient was initially 50 Btu/h·ft²·°F, showing excellent heat transfer capability when conditions are optimal.

Fouling Factor

The fouling factor ( R_f) quantifies the resistance to heat transfer due to scaling or deposits that form on the surface of heat exchanger tubes. It is an important consideration as it can significantly impair the performance of the heat exchanger.

With time, scale and other deposits can build up on the heat exchanger surface, adding an extra layer of thermal resistance. Here's why fouling is critical:

• Efficiency Reduction: The accumulated scale reduces the effective heat transfer area, causing the overall heat transfer coefficient to drop.
  
• Operational Costs: Increased fouling can lead to higher energy consumption as the system works harder to achieve the same results.
  
• Maintenance Needs: Regular cleaning and maintenance are essential to manage fouling and maintain efficient operation.
  

In our example, the fouling factor was 0.002 h·ft²·°F/Btu, which caused a reduction in the heat transfer coefficient from 50 to 45.45 Btu/h·ft²·°F.

Thermal Efficiency

Thermal efficiency in the context of heat exchangers is considered as the ratio of the actual heat transferred by the exchanger to the maximum possible heat that could be transferred under the given conditions.

Calculating thermal efficiency involves various elements:

• Effectiveness: Measures how close the actual heat transfer rate comes to the theoretical maximum, which is affected by factors such as flow arrangement and fouling.
  
• Temperature Approach: A smaller temperature difference between the fluids at the exit indicates higher efficiency.
  

In our exercise, although not directly calculated, the fouling has reduced efficiency by around 9.09%, indicated by the drop in the overall heat transfer coefficient. Preserving efficiency involves regular maintenance and monitoring of fouling levels.