Continuum Mechanics

Continuum Mechanics is the study of the behavior of materials by ignoring its particulate nature. 

A continuum is an area that can keep being divided and divided infinitely with no individual particles. It is a simplification that allows us to investigate the movement of matter on scales larger than the distances between particles.

In continuum, the smallest element of a fluid is NOT a fluid molecule, but rather a fluid particle that contains enough number of molecules to make meaningful statistical averages. Continuum assumes that fluid and flow properties like pressure, temperature, density, velocity, etc. vary continuously throughout the fluid.

This helps to study a wide range of phenomena, from air and water flow to even the evolution of galaxies.

For us to know whether or not the continuum hypothesis can be used, a dimensionless number called Knudsen number is used. The Knudsen number allows for characterizing the boundary conditions of a fluid flow. It is defined as:

If the length scale of the fluidic system is in the same range as the mean free path, i.e., Kn = 1, the fluid cannot be treated as a continuum.

The Knudsen number is very useful when assessing the boundary of fluid flows. Usually, the flow at the boundary of a flow field where the channel walls are fixed in space and the liquid directly in contact is considered to not be moving. This is referred to as the no-slip boundary condition, i.e., there is no relative movement (slip) between the wall and the fluid layer that directly contacts with the wall. 

According to the Knudsen number, flows can be divided into different regimes:

(a) For Kn < 0.01, continuum flow dominates and conventional fluid dynamics equations are applicable, indicating that gas molecules interact with neighboring molecules.

(b) For 0.01 < Kn < 0.1, the slip flow regime occurs when the gas molecules experience slipping at the solid interface.

(c) For 0.1 < Kn < 10, transition flow occurs. This refers to a flow regime in which both slip (continuum) and diffusion flows can occur.

(d) For Kn > 10, Knudsen's (free molecular) flow occurs. This refers to gas molecules that flow with minimal or no interaction with neighboring molecules.

Fluid Properties:

Characteristics of a continuous fluid which are independent of the motion of the fluid are called basic properties of the fluid. Some of the basic properties are as discussed below:

• Density: The density ρ of a fluid is its mass per unit volume (kg/m³).

• Specific Weight: This is the weight of a fluid per unit volume (N/m³).

• Specific Volume: is the volume occupied by unit mass of fluid (m³/kg).

• Specific Gravity: For liquids, it is the ratio of density of a liquid at actual conditions to the density of pure water at 101 kN/m², and at 4°C.

• For gases, the specific gravity is the ratio of its density to that of either hydrogen or air at some specified temperature or pressure.

First and Second Law of Thermodynamics Solved Questions

Note: For the following examples, Appendix B for steam values that I have referred to were obtained from: 

M.D Koretsky, Engineering and Chemical Thermodynamics, Wiley, 2004.

Example 1

2 Kgs of steam at 800 kPa and 600°C is contained in a piston-cylinder assembly. Calculate the work if:

a) The steam expands reversibly and isothermally to 300 kPa.

b) The steam expands reversibly and adiabatically to 300 kPa.

Solution

System: Contents of the piston cylinder assembly

a) The work done by the system is calculated as follows:

It is necessary to plot a P versus V graph and calculate the area under the curve. 

Since expansion occurs at T = 600°C, then the variation of V as a function of P can be determined from the steam table as follows:

The plot of P versus V is shown in the figure below. (Plotted using MathCAD)

The area under the curve can be solved using the trapezoidal rule:

Alternatively, the work done by the system can be solved using the 1st law of thermodynamics: 

From Appendix B

b) The 1st law of thermodynamics simplifies as:

Since the process is reversible and adiabatic (or isentropic), the entropy remains constant at a value of 8.1332 kJ/kg.K. Therefore, the properties at the final state are:

Substituting these values into equation3:

Example 2

Consider the system shown in the figure below:

The piston is made of a non-heat-conducting material and the tank is insulated. Initially, the piston is at the extreme left-hand-side of the cylinder and the cylinder contains 1.5 kg of steam at 200°C and 100 kPa. The cylinder is connected to an infinite source of air at 500 kPa and 250°C, and the valve is opened slightly until the pressure in the cylinder reaches 500 kPa.

a) Calculate the final temperature of the air under these conditions.

b) Calculate the final temperature of the air if a small cooling coil is placed in the cylinder to maintain the steam temperature constant at 200°C.

Solution

Let the subscripts A and S stand for air and steam, respectively.

Since steam occupies the whole volume of the cylinder initially, the volume of the cylinder is calculated from the initial conditions. From Appendix B (Table B.4)

Thus:

a) The steam in the cylinder undergoes a reversible and adiabatic compression,

b) In this case, the steam in the cylinder undergoes a reversible and isothermal compression. Thus, the properties of the steam at the final state are:

Considering the air in the cylinder as a system, the unsteady-state mass and energy balances again lead to: