Question

In a shell and tube heat exchanger, steam is condensing on the shell side and a cold fluid is flowing through the tubes in the turbulent flow region, then the Wilson plot is used to:

• A. The linear velocity of cold fluid
• B. Overall temperature difference
• C. Overall heat transfer coefficient
• D. Film heat transfer coefficients

Hint:

Think of the Wilson plot as a way to "dissect" the overall heat transfer resistance into its individual components, particularly the film resistances which are hard to measure directly. Varying one flow rate allows us to see how the overall resistance changes and to back-calculate the individual contributions.

Answer 1

The Correct Option is D

Step 1: Understand the context of a shell and tube heat exchanger with condensing steam and turbulent flow.
In this scenario, heat is being transferred from the condensing steam on the outside of the tubes (shell side) to a colder fluid flowing inside the tubes. The flow inside the tubes is turbulent, which enhances heat transfer. Step 2: Recall the concept of the overall heat transfer coefficient (U).
The overall heat transfer coefficient \( U \) represents the total thermal resistance to heat flow between the two fluids in a heat exchanger. It includes the convective heat transfer resistances on both the shell side and the tube side, as well as the conductive resistance of the tube wall and any fouling resistances. Step 3: Understand the Wilson plot technique.
The Wilson plot is an experimental method used to determine the individual heat transfer coefficients (film coefficients) and fouling factors in a heat exchanger. It relies on varying the velocity of one of the fluids (typically the tube-side fluid in turbulent flow) while keeping other conditions relatively constant. Step 4: Explain the mathematical basis of the Wilson plot.
The overall heat transfer coefficient \( U \) is related to the individual resistances by: \[ \frac{1}{U} = \frac{1}{h_o} + \frac{t}{k_w} + \frac{1}{h_i} + R_{f,o} + R_{f,i} \] where:
\( h_o \) is the convective heat transfer coefficient on the outside (shell side).
\( h_i \) is the convective heat transfer coefficient on the inside (tube side).
\( t \) is the tube wall thickness.
\( k_w \) is the thermal conductivity of the tube wall.
\( R_{f,o} \) is the fouling resistance on the outside.
\( R_{f,i} \) is the fouling resistance on the inside.
In turbulent flow inside the tubes, the heat transfer coefficient \( h_i \) can often be correlated by the Dittus-Boelter equation or similar relations, which have the form \( h_i \propto v^n \), where \( v \) is the fluid velocity and \( n \) is an exponent (typically around 0.8). By varying the tube-side velocity \( v \) and measuring the overall heat transfer coefficient \( U \), we can rearrange the equation to a linear form that allows us to determine \( h_i \) and the combined constant terms (which include \( h_o \), \( t/k_w \), and the fouling resistances) from the slope and intercept of the plot of \( 1/U \) versus \( v^{-n} \). Once these are known, and if one of the film coefficients (e.g., \( h_o \) for condensing steam) is known or can be estimated, the other film coefficient (\( h_i \)) and fouling factors can be determined. Step 5: Evaluate the options in the context of the Wilson plot.
Option 1 (The linear velocity of cold fluid): The Wilson plot uses the variation of velocity as the independent variable but is not directly used to measure it.
Option 2 (Overall temperature difference): The Wilson plot uses measured temperatures to determine \( U \) but does not directly determine the overall temperature difference itself.
Option 3 (Overall heat transfer coefficient): The Wilson plot uses measured parameters to infer the overall heat transfer coefficient at different velocities, but the ultimate goal is to find the individual film coefficients and fouling factors.
Option 4 (Film heat transfer coefficients): The Wilson plot is a primary method for experimentally determining the individual film heat transfer coefficients (like \( h_i \) and potentially \( h_o \) if conditions allow) and fouling factors.
Step 6: Select the correct answer.
The Wilson plot is used to determine the film heat transfer coefficients.