SPECIAL MATHEMATICAL FUNCTIONS

Legendre’s Differential Equation

The differential equation, 

where n is a real constant known as Legendre’s equation of order n. When n is a nonnegative integer, i.e., n = 0, 1, 2, …., one solution of the above equation is called the Legendre polynomial (or Legendre function of the first kind) of degree n and is represented by P_n(x). The functions P_n(x) are expressed as follows:

Legendre polynomials are orthogonal with each other, i.e.,

The second solution of the Legendre equation is denoted by Q_n(x) and is called the Legendre function of the second kind of order n. The functions Q_n(x) are expressed as follows:

where A_n and B_n are suitably chosen constants. In particular:

Therefore, the general solution of Legendre’s differential equation is expressed as follows:

The substitution x = cos θ transforms Legendre’s equation into the following form:

The solution of Eq. (11) is given by

Legendre’s Associated Differential Equation

The differential equation:

where m and n are nonnegative integers, is known as Legendre’s associated equation of order n. Solutions of this equation are called associated Legendre functions. The general solution is:

where P_n^m (x) and Q_n^m (x) are called the associated Legendre function of the first kind and associated Legendre function of the second kind, respectively. Associated Legendre functions of the first kind are defined by:

Associated Legendre functions of the first kind are orthogonal with each other, i.e.,

Associated Legendre functions of the second kind are defined by:

The substitution x = cos θ transforms Legendre’s associated differential equation into the equation:

which is satisfied by P_n^m (cos θ) and Q_n^m (cos θ)

Hermite's Differential Equation

The differential equation:

where n is a real constant, is known as Hermite’s equation of order n. If n is a nonnegative integer, i.e., n = 0, 1, 2, ….. then solutions of Hermite’s equation are Hermite polynomials H_n(x) given by:

Hermite polynomials are orthogonal with each other, i.e.,

Laguerre's Differential Equation

The differential equation:

where n is a real constant, is known as Laguerre’s equation of order n. If n is a nonnegative integer, i.e., n = 0, 1, 2, ….., then solutions of Laguerre’s equation are Laguerre polynomials L_n(x) given by:

Laguerre polynomials are orthogonal with each other, i.e.,

Chebyshev's Differential Equation

The differential equation:

is known as Chebyshev’s equation of order n. The general solution of Chebyshev’s differential equation is:

where T_n(x) and S_n(x) are called the Chebyshev polynomials of the first kind and Chebyshev polynomials of the second kind, respectively. Chebyshev polynomial of the first kind is defined by:

Chebyshev polynomials of the first kind are orthogonal with each other, i.e.,

Chebyshev polynomial of the second kind S_n(x) is defined by:

Chebyshev polynomials of the second kind are orthogonal with each other, i.e.,

Solved Examples For Vapor Compression Refrigeration Cycles

Note: For example 1, values are obtained from the following book:

G. van Wylen, R. Sonntag, C. Borgnakke, Fundamentals of Classical Thermodynamics, 4th Ed., Wiley, 1994

Example 1

A conventional refrigerator uses HFC-134a as its working fluid. Saturated vapor at − 20°C leaves the evaporator and enters the insulated compressor whose compression ratio is 9:1, i.e., the outlet pressure is 9 times the inlet pressure. The compressor is 85% efficient based on an adiabatic reversible compression over the same pressure range. The gas leaving the compressor is cooled and condensed isobaricly to a saturated liquid. The saturated liquid is passed through an insulated throttle valve whose downstream pressure corresponds to a saturation temperature of − 20°C. The mixture from the throttle valve goes to the evaporator, where it absorbs just enough heat from the ice trays and the interior of the refrigerator box to become saturated vapor, which enters the compressor and repeats the cycle. Under summertime conditions (greatest load) it is expected that 120 kJ/min will have to be absorbed in the evaporator.

a) What is the rate of circulation of the refrigerant in kg/min?

b) Determine the power of the compressor.

Solution

The cycle can be schematically represented as follows:

a) Since the rate of heat that must be removed from the low temperature region, Q_c is given, then:

State: 1

At the exit of the evaporator, we have saturated vapor at − 20°C. From HFC-134a table we obtain:

State: 3

At the exit of the condenser, we have saturated liquid. The pressure is

P₃ = P₂ = (9)(0.1337) = 1.2 MPa

Hence,

State: 4

The energy balance around the throttling valve gives:

                                                 

Thus, the circulation rate can be determined from Eq. (1) as follows:

b) The pressure at state 2 is 1.2 MPa. If the compression takes place reversibly and adiabatically:

Example 2

A heat pump is being used to maintain a room at 21°C by removing heat from groundwater and discharging heat to the room. For the cycle shown in the figure below:

a) What is the quality of the fluid leaving the evaporator?

b) What is the maximum quantity of heat that can be delivered to the room per J removed from the groundwater?

c) If the room is to receive 1.7 kW, what is the minimum horsepower rating of the motor driving the compressor?

d) What is the minimum energy that would have to be supplied to provide 1.7 kW into the room using the groundwater as a source?

The fluid properties are as follows:

Solution

a) Since compressor operates reversibly and adiabatically, then:

b) The entropy at state 4 is:

c) The circulation rate of the refrigerant can be calculated from:

d) Power required will be minimum when a Carnot refrigeration cycle is to be operated between evaporator and condenser temperatures. The coefficient of performance of such a heat pump is:

Vapor Compression Refrigeration Cycles

Carnot Refrigeration Cycle (Carnot Heat Pump)

A refrigeration cycle is just a reversed heat engine cycle.

Therefore, heat is transferred from a low temperature level to a high temperature level. However, according to the second law of thermodynamics this cannot be accomplished without the use of external energy. The working mediums that are used in compression refrigeration systems are called refrigerants.

The following figure shows a typical Carnot refrigeration cycle:

1. The refrigerant evaporates at a constant temperature and pressure as heat is absorbed from the low temperature region in the evaporator.
2. The vapor leaving the evaporator is then compressed to a higher pressure and external work is required in this process.
3. Heat is rejected to the high temperature region in the condenser as the refrigerant condenses at constant temperature and pressure.
4. To complete the cycle, the liquid from the condenser is returned to its original state by an expansion process.

The overall process is represented on a T-S diagram in the following figure:

The coefficient of performance, COP is used to measure the performance of a refrigeration cycle. It is the ratio of the refrigeration obtained to the work required, i.e.,

The COP of a Carnot refrigeration cycle is given by:

A ton is a common unit used in practice to describe the refrigeration effect. One ton of refrigeration is the term used to refer to 12,000 Btu/h.

Therefore, a chiller or condensing unit with a cooling capacity of 60,000 Btu/h is said to have a capacity of 5 tons. One ton of refrigeration approximates to 3.5 kW of cooling.

As seen from the T-S graph of a Carnot refrigeration cycle, both the compression and expansion steps in the Carnot refrigeration occur within the two-phase region. The compression of a two-phase mixture, however, is very difficult in practice and is generally avoided. This problem can be eliminated by simply allowing the refrigerant to evaporate completely in the evaporator resulting in the production of a saturated vapor.

On the other hand, the fluid passing through the expander is mostly liquid and its specific volume is relatively low. Thus, the amount of work produced by the expander is not appreciable. For this reason, much less expensive and almost maintenance-free throttling expansion devices are preferred over expanders in practice.

The modified cycle and its representation on a T-S diagram are shown in the following two images:

The vapor compression refrigeration cycles can also be represented on a P-H diagram as shown:

Refrigerants

The design of a vapor compression refrigeration system is greatly influenced by the physical, thermodynamic and chemical properties of the refrigerant used.

The desirable properties for a refrigerant can be summarized as follows:

• Positive evaporating pressures: This prevents leakage of atmospheric air into the system during operation.
• Moderately low condensing pressures: This allows the use of light weight equipment on the high-pressure side of the system.
• Low freezing point.
• High latent heat of vaporization and relatively high critical temperatures: A high latent heat means a high refrigeration effect per kg of refrigerant circulated and low power cost for circulation.
• Low cost.
• Inertness and stability
• Not be toxic, irritating or flammable.

 

However, no single compound meets all these requirements.

Ammonia and sulfur dioxide were the early refrigerants for commercial use due to their high latent heats of vaporization. However, they have the obvious drawbacks of being highly toxic and corrosive, and with NH₃ being flammable.

Research efforts in the 1920’s led to the conclusion that small molecules having C-F bonds (fluorocarbons) were suitable choices, with dichlorodifluoromethane (CCl₂F₂) having the best properties. However, these compounds accumulated in the atmosphere and resulted in destruction of O₃ molecules in the ozone layer.

Hydrofluorocarbons (HFCs) were developed as alternatives to the ozone-depleting refrigerants. HFC-134a (1,1,1,2-tetrafluoroethane - CF₃CH₂F) has become the refrigerant of choice to replace CFC-12 in most refrigeration and auto air conditioning systems.

The Gaussian Integral

The Gaussian or Probability Integral is an essential concept in mathematics particularly in the fields of probability theory, statistics and quantum mechanics.

The Gaussian integral, closely related to the erf function, is the integral of the one-dimensional Gaussian function over (-∞, ∞).

The Gaussian Integral can also be defined as the integral of the exponential of -x² over the entire real line.

It can be calculated using the trick of combining two one-dimensional Gaussians:

In this case, Since the variable in the integral is a dummy variable i.e, it integrates out in the end, we can rename from x to y.

When switching to polar coordinates we get:

Example

Prove the following:

Solution

Convert the integral into the polar coordinates (r, θ) where x² + y² = r², 

and dxdy = rdrdθ:

Evaluating the integrals:

Therefore:

CONTROL VOLUME APPROACH

After studying the various type of flows, we need to address how to determine the velocity in the flow field. Two approaches as used for this:

The Lagrangian Approach

• This is used in solid mechanics and involves describing particle’s motion by position as a function of time.
• Can be used to describe the motion of an object falling due to gravity: s = ½gt². At any time, the distance from the body’s original position is known.
• This approach is difficult to use in fluid mechanics because a fluid is a continuous medium i.e., a single fluid volume changes shape, and different fluid particles within the fluid volume are traveling at different velocities. Thus, due to the nature of fluids, the Lagrangian Approach is generally not a desirable analysis method.

The Eulerian (or control volume) Approach

• This is preferred in fluid mechanics.
• In this method, a region in the flow field is chosen for study. For example, consider flow draining from a sink, as shown below. A control volume is chosen around the region of study and is bounded by the dashed line called the control surface while everything outside is called the surroundings. The control volume or shape is chosen for convenience in solving the problem.

• Generally, the control volume shape is selected so that fluid and flow properties can be evaluated at locations where mass crosses the control surface or, if no mass enters or exits, where energy crosses the control surface. Furthermore, the control surface can move or change shape with time as illustrated above.
• The control volume is to fluids as the free-body diagram is to solids. 

The Eulerian approach is suitable in solving fluid mechanical problems. The aim is to develop equations of fluid dynamics that are conservation equations each developed from a general conservation equation:

• the continuity equation (conservation of mass),
• the momentum equation (conservation of linear momentum),
• the energy equation (conservation of energy).

Let N be defined as a flow quantity (mass, momentum, or energy) associated with a fluid volume or system of particles and n represents the flow quantity per unit mass. Thus:

Using the following image consider a system of particles at two different times: V₁ at t₁ and V₂ at t₂ where V₁ is bounded by the solid line and V₂ is bounded by the double line. V₁ consists of V_A and V_B while V₂ consists of V_B and V_C. The control volume is bounded by the dashed line.

The amount of flow quantity N contained in V₁ is the amount in V_A and the amount in V_B at t₁ which is N_A1 + N_B1. The amount of flow quantity contained in V₂ is the amount in V_B and V_C at t₂, which is N_B2 + N_C2. During the time interval, the change in N is, therefore:

To obtain a specific limiting expression for this last term, let us consider a differential area dA through which fluid particles flow:

The fluid velocity at the differential area, dA is Ṽ, which has components normal and tangential to dA: V_n and V_t. The tangential velocity carries no fluid out of the control volume with it as all fluid leaving dA is in the V_n direction. During the time interval Δt, the mass of fluid crossing dA is:

By substituting this into Equation 3, we get the general conservation equation:

Equation 4 gives us a relationship between the various quantities associated with a system of particles i.e., equation 4 simply means:

KINEMATICS OF FLOW

Types of flow and how to characterize them:

• Closed-conduit flows: Type of flow that is completely enclosed by restraining solid surfaces. Examples include flow through a pipe.
• Open-channel flows: In this flow type there is one surface exposed to atmospheric pressure. Examples include flow in a river and flow in a spillway.
• Unbounded flows: In this flow the fluid is not in contact with any solid surface. Examples are the jet that issues from a tap and the jet from a can of spray paint.

KINEMATICS OF FLOW

In fluid flow situations velocity often needs to be determined. However, velocity generally varies in the flow field. The flow can thus be classified according to how the velocity varies.

One-dimensional flow occurs if the parameters of both the fluid and the flow are constant at any cross section normal to the flow. It can also occur when represented by average values over the cross section. Although the flow velocity can change from point to point it remains constant at each location.

The figure below illustrates velocity distributions for a one-dimensional flow:

Two-dimensional flow occurs when the fluid or flow parameters have a gradient in two directions. For flow in a pipe, the velocity at any cross section is parabolic thus the velocity is a function of the radial coordinate. Additionally, a pressure gradient exists in the axial direction that maintains the flow i.e., a pressure difference between inlet and outlet that causes the fluid to flow. Even though the fluid flows in one direction, due to the pressure and velocity gradient, the flow is classified as two-dimensional. This is illustrated as follows:

Another example of two-dimensional flow occurs when at the constant area section velocity is a function of one variable but at the convergent section, velocity is a function of two space variables. Furthermore, a pressure gradient exists that maintain the flow. This is illustrated as follows:

Another example of a one-directional, two-dimensional flow where gradients exist in two dimensions is illustrated below:

Three-dimensional flow occurs when the fluid velocity or flow parameters vary with respect to all three space variables. Thus, a gradient exists in three directions.

Steady Flow occurs when conditions do not vary with time or when fluctuating variations are small with respect to mean flow values, and the mean flow values do not vary with time.

Unsteady Flow occurs when flow conditions change with time.

Quasi-steady Flow occurs in some unsteady flows, where it is permissible or even necessary to assume that steady flow exists to obtain a solution.