Intro to Chemical Engineering Unit 5 Review: Fluid Mechanics

Key Concepts and Definitions

• Fluids encompass both liquids and gases that continuously deform under applied shear stress
• Density $(\rho)$ represents the mass per unit volume of a fluid, typically expressed in units of $kg/m^3$
• Viscosity $(\mu)$ measures a fluid's resistance to flow or deformation, with units of $Pa \cdot s$ or $N \cdot s/m^2$
• Pressure $(P)$ is the force per unit area exerted by a fluid, measured in $Pa$ or $N/m^2$
  • Absolute pressure considers the total pressure, including atmospheric pressure
  • Gauge pressure measures the pressure relative to atmospheric pressure
• Compressibility describes how a fluid's density changes with pressure, with gases being more compressible than liquids
• Surface tension $(\sigma)$ arises from intermolecular forces at the interface between two fluids or a fluid and a solid, expressed in $N/m$
• Capillary action occurs when surface tension and adhesive forces cause a fluid to rise in a narrow tube or porous material

Fluid Properties and Behavior

• Fluids exhibit unique properties that distinguish them from solids, such as the ability to flow and take the shape of their container
• Shear stress in fluids is proportional to the velocity gradient, with the proportionality constant being the fluid's viscosity
• Newtonian fluids have a constant viscosity independent of shear rate (water, air), while non-Newtonian fluids have viscosity that varies with shear rate (blood, polymers)
  • Shear-thinning fluids decrease in viscosity with increasing shear rate (pseudoplastic)
  • Shear-thickening fluids increase in viscosity with increasing shear rate (dilatant)
• Viscosity is affected by temperature, with liquids becoming less viscous and gases more viscous as temperature increases
• Fluids can be classified as ideal (inviscid and incompressible) or real (viscous and compressible) for simplifying analysis
• Bulk modulus $(K)$ quantifies a fluid's resistance to compression, defined as the ratio of pressure change to volumetric strain

Fluid Statics and Pressure

• Hydrostatic pressure is the pressure exerted by a fluid at rest due to its weight, increasing with depth
  • Hydrostatic pressure formula: $P = \rho g h$, where $\rho$ is density, $g$ is gravitational acceleration, and $h$ is depth
• Pascal's law states that pressure applied to a confined fluid is transmitted undiminished in all directions
• Archimedes' principle explains buoyancy, stating that an object immersed in a fluid experiences an upward force equal to the weight of the displaced fluid
• Manometers measure pressure differences using a U-shaped tube filled with a liquid (mercury, water)
• Barometers measure atmospheric pressure using a column of mercury or other liquid
• Pressure head is the equivalent height of a fluid column that would produce a given pressure: $h = P / (\rho g)$
• Gauge pressure and absolute pressure are related by: $P_{absolute} = P_{gauge} + P_{atmospheric}$

Fluid Dynamics and Flow Types

• Fluid dynamics studies the motion and forces in fluids, considering velocity, pressure, and other properties
• Streamlines are curves tangent to the velocity vector at each point in a flow field, representing the path of fluid particles
• Laminar flow occurs when fluid moves in parallel layers without mixing, characterized by low Reynolds numbers $(Re < 2300)$
  • Velocity profile in laminar flow is parabolic, with maximum velocity at the center and zero at the walls
• Turbulent flow is characterized by chaotic motion and mixing, with high Reynolds numbers $(Re > 4000)$
  • Velocity profile in turbulent flow is flatter, with more uniform velocity across the cross-section
• Transitional flow exists between laminar and turbulent regimes $(2300 < Re < 4000)$, exhibiting characteristics of both
• Compressible flow involves significant changes in fluid density, typically occurring at high velocities (Mach number > 0.3)
• Incompressible flow assumes constant fluid density, valid for most liquids and low-speed gas flows
• Steady flow has fluid properties (velocity, pressure) that do not change with time at a given point, while unsteady flow has time-varying properties

Conservation Laws in Fluid Mechanics

• Conservation of mass (continuity equation) states that mass is neither created nor destroyed in a system
  • For steady, incompressible flow: $\rho_1 A_1 v_1 = \rho_2 A_2 v_2$, where $\rho$ is density, $A$ is cross-sectional area, and $v$ is velocity
• Conservation of momentum (Newton's 2nd law) relates the net force on a fluid element to its rate of change of momentum
  • Momentum equation: $\sum F = \frac{d}{dt} (mv)$, where $F$ is force, $m$ is mass, and $v$ is velocity
• Conservation of energy (1st law of thermodynamics) states that energy is conserved in a system, accounting for work and heat transfer
  • Energy equation: $\frac{v_1^2}{2} + gz_1 + \frac{P_1}{\rho} = \frac{v_2^2}{2} + gz_2 + \frac{P_2}{\rho} + h_L$, where $v$ is velocity, $g$ is gravitational acceleration, $z$ is elevation, $P$ is pressure, $\rho$ is density, and $h_L$ represents head losses
• Bernoulli's equation is a simplified form of the energy equation for steady, incompressible, inviscid flow along a streamline
  • Bernoulli's equation: $\frac{v^2}{2} + gz + \frac{P}{\rho} = constant$
• Head loss $(h_L)$ accounts for energy dissipation due to friction and other irreversibilities in real fluid flows
  • Major losses occur along pipe lengths due to wall friction
  • Minor losses occur at fittings, valves, and other flow disturbances

Fluid Flow in Pipes and Channels

• Pipe flow is characterized by fluid moving through a closed conduit, driven by a pressure difference
• Laminar pipe flow has a parabolic velocity profile, with maximum velocity at the center and zero at the walls
  • Hagen-Poiseuille equation describes laminar flow in a circular pipe: $Q = \frac{\pi R^4 \Delta P}{8 \mu L}$, where $Q$ is flow rate, $R$ is pipe radius, $\Delta P$ is pressure drop, $\mu$ is viscosity, and $L$ is pipe length
• Turbulent pipe flow has a flatter velocity profile, with more uniform velocity across the cross-section
  • Friction factor $(f)$ relates pressure drop to flow rate in turbulent flow: $\Delta P = f \frac{L}{D} \frac{\rho v^2}{2}$, where $D$ is pipe diameter and $v$ is average velocity
• Moody diagram is a graphical tool for determining the friction factor based on Reynolds number and relative pipe roughness
• Equivalent pipe length is used to account for minor losses by expressing them as an equivalent length of straight pipe
• Open-channel flow occurs when a fluid flows with a free surface, such as in rivers or partially filled pipes
  • Manning's equation estimates the average velocity in open-channel flow: $v = \frac{1}{n} R_h^{2/3} S^{1/2}$, where $n$ is Manning's roughness coefficient, $R_h$ is hydraulic radius, and $S$ is channel slope

Dimensional Analysis and Similitude

• Dimensional analysis is a technique for simplifying complex problems by considering the fundamental dimensions of variables (mass, length, time)
• Buckingham Pi theorem states that a physically meaningful equation involving $n$ variables can be reduced to an equation with $n-k$ dimensionless groups, where $k$ is the number of independent dimensions
• Dimensionless numbers are ratios of forces or other quantities that characterize fluid behavior, enabling scaling and comparison of different systems
  • Reynolds number $(Re)$ represents the ratio of inertial to viscous forces: $Re = \frac{\rho v D}{\mu}$, where $\rho$ is density, $v$ is velocity, $D$ is characteristic length, and $\mu$ is viscosity
  • Froude number $(Fr)$ represents the ratio of inertial to gravitational forces: $Fr = \frac{v}{\sqrt{gD}}$, where $g$ is gravitational acceleration
• Similitude is the concept of achieving similar behavior between a model and a prototype by maintaining geometric, kinematic, and dynamic similarity
  • Geometric similarity requires proportional dimensions
  • Kinematic similarity requires similar velocity and acceleration fields
  • Dynamic similarity requires similar force ratios (equal dimensionless numbers)

Applications in Chemical Engineering

• Fluid mechanics principles are essential for designing and analyzing various chemical engineering processes and equipment
• Pumps and compressors are used to transport fluids and increase their pressure, with selection based on flow rate, pressure rise, and fluid properties
  • Centrifugal pumps are common for high flow rates and moderate pressure rises
  • Positive displacement pumps (gear, diaphragm) are used for high pressures and viscous fluids
• Heat exchangers transfer heat between fluids, with design considerations including flow arrangement (parallel, counter), surface area, and pressure drop
  • Shell-and-tube exchangers are widely used, with one fluid flowing through tubes and the other through the surrounding shell
  • Plate heat exchangers offer high surface area and ease of cleaning for viscous or fouling fluids
• Reactors involve fluid flow and mixing to facilitate chemical reactions, with design factors such as residence time, mass transfer, and heat management
  • Continuous stirred-tank reactors (CSTRs) provide good mixing and temperature control for liquid-phase reactions
  • Plug flow reactors (PFRs) are used for gas-phase reactions or when minimal mixing is desired
• Separation processes, such as distillation, absorption, and extraction, rely on fluid mechanics principles for design and optimization
  • Packed columns use a packing material to increase surface area for mass transfer between phases
  • Tray columns employ a series of perforated plates to promote mixing and separation
• Fluidization is used in processes such as catalytic cracking and drying, where a fluid is passed upward through a bed of solid particles, suspending them in a fluid-like state
  • Minimum fluidization velocity is the fluid velocity required to initiate fluidization
  • Fluidized beds offer high heat and mass transfer rates due to the large surface area and mixing of particles