Chapter 5, Problem 5.20P

Interpretation Introduction

Interpretation: The Reynold’s number and the pressure drop per unit length for the given aqueous solution is to be calculated in appropriate units.

Concept introduction: The flow of a particular fluid can be classified by the Reynold’s number which is given as

  $\text{Re}\text{=}\frac{\text{ρ×<mover accent="true"><mo>V</mo><mo>¯</mo></mover><mo>×D</mo>}}{\text{μ}}\mathrm{.......}\text{(1)}$

The notations used in this question are

  $\begin{array}{l}\text{Re}\text{=}\text{Reynold's}\text{Number}\\ \text{ρ}\text{=}\text{Density}\text{of}\text{fluid}\\ \text{<mover accent="true"><mo>V</mo><mo>¯</mo></mover>}\text{=}\text{Average}\text{Velocity}\text{of}\text{fluid}\text{with}\text{which}\text{it}\text{is}\text{flowing}\\ {\text{D}}_{\text{eq}}\text{=}\text{Equivalent}\text{Diameter}\text{of}\text{the}\text{tube/pipe}\text{in}\text{which}\text{flow}\text{is}\text{occuring}\end{array}$

The value or range of Reynold’s number for different types of flow are

  $\begin{array}{l}\text{Re}\text{<}\text{2100}\text{Laminar}\text{Flow}\\ \text{2100}\text{<Re<4000}\text{Transition}\text{Flow}\\ \text{Re>4000}\text{Turbulent}\text{Flow}\end{array}$

The equivalent diameter is the diameter which is used in the case of non-circular channels.

  $\begin{array}{l}{\text{D}}_{\text{eq}}\text{=}{\text{4r}}_{\text{H}}\mathrm{.......}\text{(2)}\\ {\text{r}}_{\text{H}}\text{=}\text{Hydraulic}\text{radius}\end{array}$

  $\begin{array}{l}{\text{r}}_{\text{H}}\text{=}\frac{\text{S}}{{\text{L}}_{\text{p}}}\mathrm{.......}\text{(3)}\\ \text{S}\text{=}\text{Area}\text{of}\text{channel}\\ {\text{L}}_{\text{p}}\text{=}\text{Perimeter}\text{of}\text{channel}\end{array}$

  $\begin{array}{l}\text{S}\text{=W×B}\mathrm{.......}\text{(4)}\\ {\text{L}}_{\text{p}}\text{=}\text{2(W+B)}\mathrm{.......}\text{(5)}\end{array}$

W is the width of the channel and B is the height of the channel.

The pressure drop per unit length in case of Laminar flow is,

  $\frac{\text{ΔP}}{\text{L}}\text{=}\frac{\text{32μ<mover accent="true"><mo>V</mo><mo>¯</mo></mover>}}{{\text{D}}_{\text{eq}}{}^{\text{2}}}\mathrm{.......}\text{(6)}$

  $\begin{array}{l}\text{ΔP}\text{=}\text{Pressure}\text{drop}\\ \text{L}\text{=}\text{Length}\text{of}\text{channel}\\ \text{μ}\text{=}\text{Viscosity}\text{of}\text{fluid}\\ \text{<mover accent="true"><mo>V</mo><mo>¯</mo></mover>}\text{=}\text{Average}\text{velocity}\text{of}\text{fluid}\end{array}$