In a heat exchanger there are 200 tubes each \(3.65 \mathrm{~m}\) long and \(30 \mathrm{~mm}\) outside diameter and \(25 \mathrm{~mm}\) bore. They are arranged axially in a cylinder of \(750 \mathrm{~mm}\) diameter and are equally spaced from one another. A liquid of relative density \(0.9\) flows at a mean velocity of \(2.5 \mathrm{~m} \cdot \mathrm{s}^{-1}\) through the tubes and water flows at \(2.5 \mathrm{~m} \cdot \mathrm{s}^{-1}\) between the tubes in the opposite direction. For all surfaces \(f\) may be taken as \(0.01\). Neglecting entry and exit losses, calculate (a) the total power required to overcome fluid friction in the exchanger and \((b)\) the saving in power if the two liquids exchanged places but the system remained otherwise unaltered.

Step by step solution

01

- Calculate the Reynold's Number

First, calculate the Reynold's number for the liquid flowing through the tubes. Since the liquid is flowing at a mean velocity of \(2.5 \, \mathrm{m/s}\), with a tube bore diameter of \(25 \, \mathrm{mm} = 0.025 \, \mathrm{m}\), and using the relative density of \(0.9\), the formula for Reynold's Number \(Re\) is \[ Re = \frac{\rho vd}{\mu} \] where \(\rho\) is the density of the fluid, \(v\) is the flow velocity, \(d\) is the diameter, and \(\mu\) is the dynamic viscosity. Relative density \(0.9\) implies \(\rho = 0.9 \times 1000 \, \mathrm{kg/m^3}} = 900 \, \mathrm{kg/m^3}\). The water's density is \(\rho_w = 1000 \, \text{kg/m}^3\).

02

- Calculate the Friction Factor

The friction factor \(f\) for the flow inside the tubes is given as \(0.01\). This value also applies to the water flowing through the gaps between the tubes.

03

- Calculate Head Loss in Tubes

The head loss \(h_f\) per tube can be calculated using Darcy's equation \[ h_f = f \frac{L}{d} \frac{v^2}{2g} \] where \(L = 3.65 \, \mathrm{m}\) is the length of the tubes. Plug in the given values to find \( h_f \).

04

- Calculate Power Required for Tubes

The total power \(P = \rho Q g h_f\), where \(\rho\) is the fluid density, \(Q = Av\) is the volumetric flow rate for one tube, and \(A = \frac{\pi d^2}{4}\). Multiply by 200 tubes to get the total power required.

05

- Calculate Head Loss Between Tubes

The head loss for water flowing between the tubes is calculated the same way. Assume a cross-sectional area available for flow between tubes and calculate the effective diameter. Then use Darcy's equation to find the head loss for water.

06

- Swap the Liquids and Recalculate

When the liquids are swapped, repeat the calculations above but with the liquid now outside the tubes and water in the tubes.

07

- Calculate the Power Savings

Compare the total power requirements for both scenarios to determine the power savings if the liquids are swapped.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

heat exchanger

A heat exchanger is a device used to transfer heat between two or more fluids without mixing them. In this exercise, we have two fluids flowing in opposite directions within the heat exchanger.
The liquid flows inside the tubes, while water flows in the space between the tubes.
The main challenge is to minimize the power required to overcome fluid friction within this setup.
To achieve this, it’s crucial to understand the flow characteristics of each fluid, calculated using various parameters such as Reynold's Number, friction factor, and head loss.

Reynold's Number

Reynold's Number \(\text{Re}\) is a dimensionless quantity used to predict flow patterns in different fluid flow situations. It helps determine whether the flow is laminar (smooth) or turbulent (chaotic).
For this exercise:

\(\text{Re} = \frac{\rho v d}{\mu}\)

Where:

\(\rho\) = density of the fluid
\(v\) = flow velocity
\(d\) = diameter of the tube
\(\mu\) = dynamic viscosity of the fluid

Given the liquid’s density, the bore diameter of the tubes, and mean velocity, we can calculate \(\text{Re}\).
This determines if the flow inside the tubes is laminar or turbulent.

friction factor

The friction factor (f) quantifies the resistance due to friction in fluid flow through a pipe. It depends on the flow regime and surface roughness of the pipe.
For smooth pipes and turbulent flow, a common approximation of the friction factor is used, such as ===0.01=== in this exercise.
The friction factor directly influences the head loss in the tubes and gaps between the tubes.

head loss

Head loss \((h_f)\) refers to the loss of mechanical energy of the fluid due to friction as it flows through a pipe.
It is calculated using Darcy's equation:

\( h_f = f \frac{L}{d} \frac{v^2}{2g} \)

Where:

\( f \) = friction factor
\( L \) = length of the pipe
\( d \) = diameter of the pipe
\( v \) = velocity of the fluid
\( g \) = acceleration due to gravity

Calculating head loss allows us to determine the energy required to overcome the frictional resistance in the tubes and gaps.

Darcy's equation

Darcy's equation is fundamental for calculating head loss in fluid flow through pipes. The equation is given by:

\( h_f = f \frac{L}{d} \frac{v^2}{2g} \)

Understanding this equation is critical for evaluating the power required to maintain fluid flow within the heat exchanger.
This applies to both situations when either liquid is flowing through the tubes and water in the gaps, or vice versa.
Using Darcy's equation, we can optimize the flow and achieve power savings by swapping the fluids as demonstrated in the problem.