Thermodynamic Analysis of Unsteady MHD Boundary Layer Flow with Slip Over a Permeable Surface

Thermodynamic Analysis of Unsteady MHD Boundary Layer Flow with Slip Over a Permeable Surface

ABSTRACT

Magnetohydrodynamics (MHD) boundary layer flow problems have become important in industrial manufacturing processes such as plasma studies, chemical engineering, electrochemistry, polymer processing, petroleum industries, MHD power generator cooling of nuclear reactors, and boundary layer control in aerodynamics. Moreover, permeable surfaces are used for boundary layer control in the filtration processes, and also for a heated body to keep its temperature constant. Suction can be utilized on biological chemical processes to remove reactants while blowing is applied to add reactants in the process and to cool the surface body by its ability to decrease the heat transfer rate. In this thesis, both first and second laws of thermodynamics are employed to investigate the inherent irreversibility in an unsteady hydromagnetic, mixed convective boundary layer flow of an electrically conducting, optically dense fluid, over a permeable vertical surface under the combined influence of thermal radiation, velocity slip, temperature jump, buoyancy force, viscous dissipation, Joule heating and magnetic field. The time-dependent governing partial differential equations are reduced to ordinary differential equations by using appropriate similarity variables. A local similarity solution is obtained numerically using the shooting technique coupled with a fourth order Runge-Kutta Fehlberg integration method. The influence of various thermophysical parameters on velocity and temperature profiles, skin friction, Nusselt number, entropy generation rate and Bejan number, are presented graphically and discussed quantitatively. It is found that velocity slip, surface injection, and temperature jump can successfully reduce entropy generation rate in the presence of an applied magnetic field.

Keywords: magnetohydrodynamics (MHD), boundary value problems (BVPs) initial value problems (IVP)

TABLE OF CONTENTS

CERTIFICATION …………………………………………………………………………………………………………………………….. i
DECLARATION AND COPYRIGHT ……………………………………………………………………………………………………. ii
ABSTRACT ………………………………………………………………………………………………………………………………….. iii
ACKNOWLEDGEMENT …………………………………………………………………………………………………………………. iv
DEDICATION ……………………………………………………………………………………………………………………………….. v
CONTENTS ………………………………………………………………………………………………………………………………… vi
LIST OF FIGURES ……………………………………………………………………………………………………………………….. viii
LIST OF TABLES ……………………………………………………………………………………………………………………………. x
CHAPTER ONE …………………………………………………………………………………………………………………………….. 1
1.0 Introduction ………………………………………………………………………………………………………………………. 1
1.1 Definition of terms ……………………………………………………………………………………………………………… 1
1.2 Problem Statement …………………………………………………………………………………………………………… 11
1.3 Study Objectives ………………………………………………………………………………………………………………. 11
1.4 Structure of work ……………………………………………………………………………………………………………… 12
1.5 Significance of Study …………………………………………………………………………………………………………. 12
1.6 Research Methodology ……………………………………………………………………………………………………… 13
CHAPTER TWO ………………………………………………………………………………………………………………………….. 16
Literature Review ………………………………………………………………………………………………………………………. 16
CHAPTER THREE ………………………………………………………………………………………………………………………… 20
Derivation of basic fluid equations ………………………………………………………………………………………………. 20
3.1 The Continuity Equation [34, 42, 51] …………………………………………………………………………………… 20
3.2 Navier – Stoke Equations [34, 42, 51] ………………………………………………………………………………….. 21
3.3 Energy Equation [34, 42, 51] ………………………………………………………………………………………………. 23
3.4 Lorentz Force [9] ………………………………………………………………………………………………………………. 27
3.5 MHD Equations [9] ……………………………………………………………………………………………………………. 28
3.6 Entropy Generation and Second Law [40] ……………………………………………………………………………. 29
CHAPTER FOUR …………………………………………………………………………………………………………………………. 33
4.1: Introduction ……………………………………………………………………………………………………………………. 33
4.2 Model Formulation …………………………………………………………………………………………………………… 33
4.3 Entropy Analysis ……………………………………………………………………………………………………………….. 36
4.4 Numerical procedure ………………………………………………………………………………………………………… 38
vii
CHAPTER FIVE …………………………………………………………………………………………………………………………… 39
5.1 Velocity Profiles ……………………………………………………………………………………………………………….. 40
5.2 Temperatures Profiles ……………………………………………………………………………………………………….. 45
5.3 Skin Friction and Nusselt Number……………………………………………………………………………………….. 50
5.5 Bejan Number ………………………………………………………………………………………………………………….. 58
5.6 Conclusions ……………………………………………………………………………………………………………………… 63
5.7 Study Limitation and Future Work ………………………………………………………………………………………. 64
REFERENCES ………………………………………………………………………………………………………………

CHAPTER ONE

BACKGROUND STUDY

1.0 Introduction

Fluid Mechanics is the study of fluids either at rest (fluids static) or in motion (fluids dynamics and kinematics) and the subsequent effects of the fluids upon the boundaries which maybe either solid surfaces or interfaces with other fluids. It is worth noting that both gases and liquids are classified as fluids according to Batchelor [37]. Fluids, unlike solids, lack ability to offer sustained resistance to a deforming force. Thus, a fluid is a substance which deforms continuously under the action of shearing forces, however small they may be. Deformation is caused by shearing forces – forces that act tangentially to the surfaces to which they are applied [42].

1.1 Definition of terms

Some relevant fluid properties to be considered in this study are highlighted as follows [37, 42, 45, and 51]:
Pressure: Pressure is the stress at a point in a static fluid. The gradient in pressure often drives a fluid flow, especially in ducts. The unit of pressure is N m−2

Temperature: This is a measure of the internal energy of a fluid. If the temperature differences are strong, heat transfer may be important. The unit is in degree Celsius (oC).

Density: The density of a fluid is its mass per unit volume. Density in liquid is nearly constant. This property aids the classification of fluids as either compressible or incompressible. A fluid is termed compressible if its density varies and increases nearly proportionally to the pressure level. Otherwise it is termed incompressible according to White [51]. The unit of density is kgm−3.

Velocity: The fluid velocity is the rate of displacement of the fluid particles with time. The unit is m/s.

Viscosity: A fluid at rest cannot resist shearing forces and if such forces acts in a fluid which is in contact with solid boundary, the fluid will flow over the boundary in such a way that the particles immediately in contact with the boundary have the same velocity as the boundary while successive layers of fluid parallel to the boundary move with increasing velocity. Shear stresses opposing the relative motion of these layers are set up and their magnitude depending on the velocity gradient from layer to layer. For fluids obeying Newton’s law of viscosity, taking the direction of motion as the x-direction and U as the velocity of the fluid in the x-direction at a distance y from the boundary, the shear stress in the x-direction is given by, (1.1)

Where the constant μ is called coefficient of dynamic viscosity and its unit is kgm-1s-1. The ratio of the coefficient of dynamic viscosity to fluid density is called the kinematic viscosity υ; (1.2)

The viscosity property of fluids aids the classification of fluids into Newtonian or non-Newtonian fluids. Viscosity is associated with collective currents that carry momentum from one region of the fluid to another. Consider a fluid where there is, in addition to thermal agitation of the molecules, a collective movement or current of the whole fluid, for example, water running in a canal or pipe under a pressure difference. Traditionally, viscosity is regarded as the most important material property and any practical study that requires the knowledge of fluid response would automatically turn to the basic understanding of viscosity. In general, the Newtonian model describes the rheological behavior of fluids. The Newtonian model is simply a special case with a constant viscosity. However, viscosity is the strong deformation of fluids.

It is the key factor in determining the amount of fluid flowing in channels. It also helps to determine whether the flow regime is laminar, transitional or turbulent. Accurate knowledge of viscosity is very useful for computation of the pressure, velocity, and temperature for a flow system. Viscosity also helps to describe the flow behaviour of shear stress with respect to the rate of deformation of the fluid.

Internal Energy: In thermo-statics, the only energy in a substance is that stored in a system by molecular activity and molecular bonding forces. This is commonly denoted as internal energy.

An identified mass of viscous fluid may be viewed as a thermodynamic system that stores various forms of energy. Whenever any form of this fluid is being deformed, it results in an irreversible transformation of mechanical energy into internal or thermal energy. The internal energy of gas includes the energies of translation, rotation, and vibration of the

dy
dU
x

Molecules as well as, the energy of molecular dissociation and energy of electronic excitation of the molecules. The unit is ????−1.

Specific Heat Capacity: This refers to the measure of the heat energy required to raise the temperature of one gram of a substance by one degree Celsius. There are two distinctly different experimental conditions under which specific heat capacity is measured. It is measured either under constant pressure condition or under constant volume condition. Typical values of the specific heat of gases are not much different from those of liquids. The unit is ????−??−?

Thermal Conductivity: This is the property of fluid that relates the vector rate of heat flow per unit area to the vector gradient of temperature. This proportionality observed experimentally for fluids and solids, is known as Fourier’s law of heat conduction, i.e.:
q =-kT. (1.3)

The minus sign satisfies the convection that heat flux is positive in the direction of decreasing temperature according to Kay and Crawford [45]. The unit is W m−1 K −1.

Table 1.1: Some quantities, symbol and units utilized

Quantity Symbol Units free-stream temperature K free-stream velocity m/s kinematic viscosity  m2/s dynamic viscosity  kg/m-s density  kg/m3 thermal diffusivity  m2/s specific heat cp J/kg-K thermal conductivity K W/m-K

Heat Transfer Mechanisms: Heat tends to move from a high-temperature region to a low temperature region. This heat transfer may occur by the mechanisms of conduction and radiation.

In engineering, the term convective heat transfer is used to describe the combined effects of conduction and fluid flow and is regarded as a third mechanism of heat transfer.

Figure 1.1: Illustration of conduction, convection and radiation heat transfer [45]

Conduction: This is the most significant means of heat transfer in a solid. On a microscopic scale, conduction occurs as hot, rapidly moving or vibrating atoms and molecules interacting with neighbouring atoms and molecules, transferring some of their energy (heat) to these neighbouring atoms. In insulators, the heat flux is carried almost entirely by phonon vibrations.

Electrons also conduct electric current through conductive solids, and the thermal and electrical conductivities of most metals have about the same ratio. A good electrical conductor such as copper, usually also conducts heat well. Thermoelectricity is caused by the relationship between electrons, heat fluxes, and electrical currents.

Convection: This is usually the dominant form of heat transfer in liquids and gases. This is a term used to characterize the combined effects of conduction and fluid flow. In convection, enthalpy transfer occurs by the movement of hot or cold portions of the fluid together with heat transfer by conduction. Commonly, an increase in temperature produces a reduction in density.

Hence, when water is heated on a stove, hot water from the bottom of the pan rise displacing the colder denser liquid which falls. Mixing and conduction result eventually in a nearly homogeneous density and even temperature. Three types of convection are commonly distinguished, they are free convection, forced convection and mixed convection. Regardless of the particular nature of convection heat transfer process, the rate of heat transfer is given by
Newton’s law of cooling

( )  q  h T T w , (1.4)

Where h is the convective heat transfer coefficient, Tw is the geometry surface temperature

Natural or Free Convection: Natural convection is caused by buoyancy forces due to density differences caused by temperature variations in the fluid. At heating, the density change in the boundary layer will cause the fluid to rise and be replaced by cooler fluid that will also heat and rise. This is called free or natural convection. A common example of natural convection is the rise of smoke from a fire. It can be seen in a pot of boiling water in which, the hot and less-dense water on the bottom layer moves upwards in plumes, and the cool and denser water near the top of the pot likewise sinks.

Forced Convection: The fluid movement results from external surface forces such as a fan or a pump i.e. pressure gradient. Forced convection is typically used to increase the rate of heat exchange. Many types of mixing also utilize forced convection to distribute one substance within another. Forced convection also occurs as a by-product of other processes, such as the action of a propeller in a fluid or aerodynamic heating. Fluid radiator systems, and also heating and cooling of parts of the body by blood circulation, are other familiar examples of forced convection.
Mixed Convection: Mixed (combined) convection is a combination of forced and free convections. This is the general case of convection when a flow is determined simultaneously by an outer forcing system (i.e., outer energy supply to the fluid-streamlined body system), and inner volumetric (mass) forces, viz., by the non-uniform density distribution of a fluid medium in a gravity field. The most vivid manifestation of mixed convection is, the motion of the stratified temperature mass of air and water areas of the Earth that are traditionally studied in geophysics. However, mixed convection is found in systems of much smaller scales, i.e., in many engineering devices.

Radiation: This is the only form of heat transfer that can occur in the absence of any form of medium (i.e., through a vacuum). Thermal radiation is a direct result of the movements of atoms and molecules in a material. Since these atoms and molecules are composed of charged particles (protons and electrons), their movements result in the emission of electromagnetic radiation, which carries energy away from the surface. At the same time, the surface is constantly bombarded by radiation from the surroundings, resulting in the transfer of energy to the surface. Since the amount of emitted radiation increases with increasing temperature, a net transfer of energy from higher temperatures to lower temperatures results. The rate of radiation heat exchange between a small surface and large surrounding is given by the expression [6]:
* ( ) 4 4
s sur Q  A T T , (1.5)

Where is the surface emissivity, A is the surface area, * is the Stefan-Boltzmann constant, ??is the absolute temperature of the surface and ????is the absolute temperature of the surroundings.

In differential form, the radiative heat flux within an optically dense medium can be expressed as [6],
4
3 *
4 *
T
k
Q   

, (1.6)
Where k* the mean absorption coefficient.

The First Law of Thermodynamics: The first law of thermodynamics states that energy is always conserved. This is stated as energy can neither be created nor destroyed; it just changes form.

The first law of thermodynamics defines the internal energy as a state function and provides a formal statement of the conservation of energy. However, it provides no information about the direction in which processes can spontaneously occur, that is, the reversibility aspects of thermodynamics processes. For example, it cannot say how cells can perform work while existing in an isothermal environment. It gives no information about the inability of any thermodynamic processes to convert heat into mechanical work with full efficiency, or any insight into why mixtures cannot spontaneously separate, or unmix themselves. An experimentally derived principle to characterize the availability of energy is required to do this.

This is precisely the role of the second law of thermodynamics that will be explained next .

The Second Law of Thermodynamics: The second law of thermodynamics establishes the differences in quality between different forms of energy and explains why some processes can spontaneously occur, whereas others cannot. It indicated a trend of change and is usually expressed as an inequality. The second law of thermodynamics has been confirmed by experimental evidence like other physical laws of nature. The second law of thermodynamics defines the fundamental physical quantity entropy as randomized energy state unavailable for direct conversion to work. It also states that all spontaneous processes, both chemical and physical, proceed to maximize entropy, that is, to become more randomized and convert energy into a less available form. A direct consequence of fundamental importance is the implication that at thermodynamic equilibrium, the entropy of a system is at a relative maximum; that is, no further increase in disorder is possible without changing by some external means (such as adding heat)to the thermodynamic state of the system.

A basic corollary of the second law of thermodynamics is the statement that, the sum of the entropy changes of a system and that of the surroundings must always be positive, that is, the universe (the sum of all systems and surroundings) is constrained to become forever more disordered and to proceed towards thermodynamic equilibrium with some absolute maximum value of entropy. The generality of the second law of thermodynamics gives us a powerful means to understand the thermodynamic aspects of real systems through the usage of ideal systems. What makes this new statement of the second law of thermodynamics valuable as a guide to energy policy is, the relationship between entropy and the usefulness of energy. Energy is most useful to us when we can get it to flow from one substance to another, e.g., to warm a house and to conduct work. Useful energy thus must have low entropy so that the second law of thermodynamics will allow transfer or conversions to occur spontaneously.

1. Energy: This is a scalar quantity which cannot be observed directly but can be recorded and evaluated by indirect measurements. Energy can manifest in various forms. According to Crawford et al. [45], the thermodynamic analysis of energy can be classified into two groups namely:

2. Macroscopic forms of energy: This is the energy which an overall system possesses with respect to a reference frame, e.g. Kinetic and potential energies. The macroscopic energy of a system is related to motion and influence of external effects, such as gravity, magnetism, electricity and surface tension. The energy that the system possesses as a result of its motion relative to some reference form is, kinetic energy. The potential energy of a system is the sum of the gravitational, centrifugal, electrical and magnetic potential energies.

3. Microscopic forms of energy: This is the energy that is related to the molecular structure of a system and the degree of molecular activity, and is independent of outside reference frames. The sum of all the microscopic forms of energy of a system is its internal energy. The internal energy of a system depends on the inherent qualities or properties of the materials in the system, such as composition and physical form as well as the environment variables (temperature, pressure, electric field, magnetic field, etc.

4. Exergy: In thermodynamics, the exergy of a system is the maximum useful work possible during a process that brings the system into equilibrium with a heat reservoir [40]. When the surroundings are the reservoir, exergy is the potential of a system to cause a change as it achieves equilibrium with its environment. Exergy is the energy that is available to be used. After the system and surroundings reach equilibrium, the exergy is zero. Determining exergy was also the first goal of thermodynamics. Energy is never destroyed during a process; it changes from one form to another (see First Law of Thermodynamics). In contrast, exergy accounts for the irreversibility of a process due to an increase in entropy (see Second Law of Thermodynamics). Exergy is always destroyed when a process involves a temperature change. This destruction is proportional to the entropy increase of the system, together with its surroundings. The destroyed Exergy has been called Anergy.

Figure1.2: Illustration of total Energy, Exergy and Anergy of a system

Boundary Layer Flows [31, 34]: A boundary layer is a layer of fluid in the immediate vicinity of a bounding surface where the effects of viscosity are significant. Boundary layer region is the region where the viscous effects and the velocity changes are significant, and the viscid region is the region in which the frictional effects are negligible and the velocity remains essentially constant (i.e. the free stream).In the Earth’s atmosphere, the atmospheric boundary layer is the air layer near the ground affected by diurnal heat, moisture or momentum transfer to or from the surface. On a craft wing, the boundary layer is the part of the flow close to the wing, where viscous forces distort the surrounding non-viscous flow. Laminar boundary layers can be loosely classified according to their structure and the circumstances under which they are created. The thin shear layer which develops on an oscillating body is an example of a Stokes boundary layer, while the Blasius boundary layer refers to the well-known similarity solution near an attached flat plate held in an oncoming unidirectional flow. In the theory of heat transfer, a thermal boundary layer occurs. A surface can have multiple types of boundary layers simultaneously. The viscous nature of airflow reduces the local velocities on a surface and is responsible for skin friction. The layer of air over the wing’s surface that is slowed down or stopped by viscosity is the boundary layer. Boundary layer flows can be laminar or turbulent in nature.

1. Laminar Boundary Layer Flow: The laminar boundary is a very smooth flow, while the turbulent boundary layer contains swirls or eddies. The laminar flow creates less skin friction drag than the turbulent flow but is less stable. Boundary layer flow over a wing surface begins as a smooth laminar flow. The laminar boundary layer increases in thickness as the flow continues back from the leading edge. In laminar flow, any exchange of mass or momentum takes place between adjacent layers in microscopic scale which may not be easily observed and consequently, laminar boundary layers are formed for a very small Reynolds number.

2. Turbulent Boundary Layer Flow: A turbulent boundary layer, on the other hand, is marked by mixing across several layers of it. The mixing is now on a macroscopic scale. Packets of fluid may be seen moving across. Thus there is an exchange of mass, momentum, and energy on a much bigger scale compared to a laminar boundary layer. A turbulent boundary layer forms only at larger Reynolds numbers. The scale of mixing cannot be handled by molecular viscosity alone. Those calculating turbulent flow rely on what is called Turbulence Viscosity or Eddy Viscosity, which has no exact expression. It has to be modelled. Several models have been developed for this purpose.
Figure1.3: Laminar boundary layer flow to turbulence boundary layer flow [34]

3. Magnetohydrodynamics (MHD): This is the study of the magnetic properties of conducting fluids. Examples of such magneto-fluids include plasmas, liquid metals, and salt water or electrolytes. The word magnetohydrodynamics (MHD) is derived from magneto- meaning magnetic field, hydro- meaning water, and -dynamics meaning movement. The field of MHD was initiated by Hannes Alfvén [35] for which he received the Nobel Prize in Physics in 1970. The fundamental concept behind MHD is that magnetic fields can induce currents in a moving conductive fluid, which in turn polarizes the fluid and reciprocally changes the magnetic field itself. The set of equations that describe MHD are a combination of the Navier-Stokes equations of fluid dynamics and Maxwell’s equations of electromagnetism. These differential equations must be solved simultaneously, either analytically or numerically.

Figure (1.4): Illustration of magnetohydrodynamics [9]

1.2 Problem Statement

Thermodynamic irreversibility in fluid flows and thermal systems is associated with the entropy production that destroys the available energy needed for efficient operation of the system. Consequently, many engineering and industrial devices whose operations depend on fluid flow and heat transfer may not perform at optimal efficiency due to the irreversible loss of energy. In order to achieve entropy generation minimization in fluid flows and thermal systems, it is very important to determine the thermo-physical factors that contribute to this inherent irreversibility and control them. This will enhance a better design of the flow and thermal systems for optimal efficiency. Our goal in this thesis is to, examine the thermodynamic inherent irreversibility in an unsteady hydromagnetic boundary layer flow, and heat transfer processes over a slippery permeable surface, in the presence of thermal radiation absorption and buoyancy force. These flow and thermal processes do occur in many engineering and industrial systems devices with hydrophobic open micro-channels such as MHD micro pumps, biological transportation, and drug delivery.

1.3 Study Objectives

The main objectives of the study in this thesis are as follows;

1. Derive a nonlinear mathematical model for hydromagnetic boundary layer flow with slip over a permeable surface.

2. Solve the nonlinear model numerically using shooting method coupled with fourth order Runge-Kutta integration scheme.

3. Determine the effects of various embedded thermo-physical parameters on flow characteristics like velocity and temperature profiles, Skin friction and Nusselt number

4. Determine the effects of various embedded thermo-physical parameters on entropy generation rate, irreversibility ratio and Bejan number.

5. Determine the entropy generation minimization condition for optimal performance of the flow system.

1.4 Structure of work

Chapter one is devoted to background information on boundary layer flows of conducting fluids and its important features with respect to the external magnetic field, buoyancy force, thermal radiation absorption, heat transfer and entropy production. Chapter two deals with a review of relevant literature on MHD boundary layer flow with heat transfer, thermal radiation, and entropy generation. The basic (MHD) equations with respect to conservation of mass, momentum and energy balance including Maxwell equations of electromagnetism and the second law of thermodynamics were outlined in Chapter three. In Chapter four, the model problem for entropy generation rate in an unsteady MHD boundary layer flow past a permeable surface with slip, buoyancy force and thermal is formulated, and analysed. The numerical solutions and graphical results are presented and quantitatively discussed in chapter five. This is also followed by a concluding remark. It is hoped that our findings could be useful in improving the liquid transportation design in micro scale MHD systems by reducing the entropy production and improving the exergy of the system.

1.5 Significance of Study

Magnetohydrodynamics (MHD) flow and heat transfer in the presence of slip is an important topic in many engineering branches, especially in the field of microelectrochemical systems (MEMS), such as micro MHD pumps, rapid mixing of biological fluids in biological processes, biological transportation, and drug delivery. The magnetic field applied by generating a Lorentz force can control the electrically conducting fluid flow in a mixing process. However, as most of the applications of biological transportation via an applied magnetic field are in micro/Nano systems .It is necessary to consider the influence of velocity slip at the boundaries. Permeability is another effect that can act as transpiration of the boundaries in micro-systems, which is an important aspect of micro-mixing of biological samples. In this process, suction is exerted to remove reactants whereas; injection is exerted to add reactants in the process. The entropy based surface micro-profiling (EBSM) technique is used to reduce energy dissipation in convective heat transfer in micro-channels.

1.6 Research Methodology

In our study, the numerical approach in solving model equations with boundary value problems (BVPs) was employed. For this method, we used shooting method together with the fourth order Runge-Kutta-Fehlberg [30]. Shooting method transforms the BVPs into sets of initial value problems (IVP), with certain unknown initial conditions that need to be determined by guessing , after which the fourth order Runge-Kutta-Fehlberg iteration scheme is employed to integrate the set of IVPs until the given boundary conditions are satisfied.

Shooting Method: The shooting method is an iterative algorithm that reformulates the original boundary value problem into a set of initial value problems, with its appropriate initial conditions. The problem requires the solution of the IVP with the initial conditions chosen to approximate the boundary conditions at the end points. If these boundary conditions are not satisfied to the required accuracy, the procedure is repeated again with a new set of initial conditions until the required accuracy is acquired, or a limit to the iteration is reached. The resultant IVP is solved numerically using any appropriate technique for solving the linear ordinary differential equations. In our own case, we used the fourth order Runge-Kutta method, which provides high accuracy results. The solution of the IVP should converge to that of the BVP. The algorithm for the above procedure is implemented in a computer using MATHLAB or MAPLE code. The results are presented usually in graphical form. Consider a two-point boundary value problem:
?′′=?(?,?,?′),?(?)=?,?(?)=? (1.4)

Where, ?Runge-Kutta-Fehlberg Method: One way to guarantee accuracy in the solution of IVPs is to solve the problem twice, using step size and compare answers at mesh points corresponding to large step size. But this requires a significant amount of computation for the smaller step size