Google Scholar. Sergey Aleshko , Sergey Aleshko. Nataliia Fialko , Nataliia Fialko. Nikolay Maison , Nikolay Maison.
Nataliia Meranova , Nataliia Meranova. Artem Voitenko , Artem Voitenko. National Aviation University, Kiev, Ukraine. Igor Pioro Igor Pioro. Author Information. Alexander Zvorykin. Sergey Aleshko. Nataliia Fialko. Nikolay Maison. Nataliia Meranova. Artem Voitenko. However, the method can be extended to three-dimensional problems at the expense of E Q U A T IO N S considerable amount of complexity using six dependent variables, namely, 1. NonlineQrity A close look at Eqs S. Ia and b reveals that the convection part the three components of the vorticity vector and three components of the of the momentum equations involve nonlinear terms.
For example, in Eq. Ia , velocity-potential vector. However, this can be treated like the conductivity, k being a function of temperature, T as discussed in Chapter 4.
Starting with a guessed velocity field, one could itera- tively solve the momentum equation to arrive at the converged solution for the velocity components. Therefore, nonlinearity poses no problems as such. It only makes the computa- tions more involved. Pressure gradient The main hurdle to pvercome in the calculation of velocity field is the unknown pressure field.
The pressure gradient behaves like a source term for a momentum eq1. But, there is no equation for obtaining pressure. For a specified pressure field there is no particular difficulty in solving the momentum equations. So, the challenging task is to determine the correct pressure distribution. There- fore, vorticity distribution at the inlet calculated from the inlet velocity distribu- tion can be applied everywhere in the computational domain.
Hence, the equation to solve is Eq. Once the Laplace equation for stream function is solved, the velocity distribution in the domain can be calculated from Eqs 5. For specifying boundary conditions it should be kept in mind that for inviscid flow, since the viscous terms are neglected, the order of the governing equations of motion drops from two to one.
As a result, only one boundary condition, with respect to the velocity field, can be satisfied at the boundaries. Thus; slip is al- lowed parallel to the walls in contrast with the no-slip condition for viscous flows and the normal velocity component is taken as zero. Therefore, the value of stream function is constant along a wall.
These boundary conditions may be ob- tained from x - and y - m o m e n t u m equations and these are generally in terms of gradients. For unsteady flow problems, VI-distributions at given time intervals are used. It is possible to employ a different grid for each dependent variable. In the staggered grid, the velocity components are calculated for the points that are located on the faces of the control volume shaded while the pressure is cal- culated for the regular grid points.
Figure 5. For a typical control volume shown shaded with hatched lines , the discretized continuity equation would contain the differences of the adjacent velocity components and hence a wavy velocity field would not be satisfied. The pressure difference between the successive grid points now becomes the natural driving force for the velocity component located between the grid points.
Hence, a non-uniform or a wavy pressure field will not be treated as a uniform pressure field and cannot arise as possible solutions.
These neighbours, would in turn, bring their nli,,. Ultimately, the velocity correction formula would involvI thlopressure correction at all grid points in the calculation domain and thl resulting pressure-correction equation becomes unmanageable. The omu. Jionof any term would, of course, be unacceptable if it meant that the ultimate solution would not be true solution of the discretized form of the momentum' and continuity equation. In the converged solution we acquire a pressure field such that the corresponding velocity field do satisfy the continuity equation.
This term represents an indirect or implicit influence of the pressure correction on velocity; pressure corrections at nearby locations can alter the neighbouring velocities and thus cause a velocity correction at the point under consideration. We do not include this influence and thus work with a scheme that is only partially, and not totally implicit.
Since the velocity corrections can be assumed to be zero at the previous itera- tion step, Eq. This thin region, in which large veloc- ity gradients exist, is called boundary layer introduced by L. Prandtl in The region outside the boundary layer, where the forces due to friction are small and may be neglected, and where the real or perfect fluid approximation is valid is called potential or in viscid flow region Fig. The method of solution of biharmonic equation in I f!
Finally, the method of computation of boundary layer flow over a flat plate and a wedge by similarity method using shooting technique is discussed. The chapter ends with a treatment of finite-difference methods using x - y coordinates and also X - O J coordinates V on Mises transformation for flat plate boundary layer problems.
Jaluria, Yogesh and Kenneth. Heat and Mass Transfer, Vol. Tadmor, Zehev and Costas G. Welch, J E. LA, From Eq. This is a major advantage of Eq. The fluid enters the channel at a uniform velocity o f 2 m 1 s. The inlet and outlet at the bottom plate are each 4 cm wide, and their centrelines are This chapter deals with the method of computation of flow field of incompressible 12 cm apart.
If the vertical distance between the plates is 20 cm and the fluid by finite-difference techniques. Two types of flow problems have been exit channel is 1 cm high, obtain the streamlines and the velocity distribu- discussed namely. The and inviscid flow, with greater emphasis placed on the former. Both stream velocity at the exit can be taken as uniform assuming it to be a long narrow function-vorticity and primitive variables approaches have been discussed in passage.
Note that creeping flow approximation will no longer be valid here. This is particularly important circumstance in industry, since the flow is induced by means of an external agency in most thermal systems, such as heat exchangers; b when the fluid motion arises simply because of density differ- ences, caused by temperature differences in a body force field, such as gravity, the process is termed natural or free convection. A heated body cooling in still air, the free rise of heated buoyant air due to a fire, circulation of water in a lake, dissipation of heat from the coil of a refrigeration unit to the surrounding air, are some of the examples of natural convection.
There is also another class of convection known as mixed convection in which both natural and forced convection effects are significant and neither mechanisms may be neglected. Manufacturing processes such as wire drawing and extrusion belong to the aforementioned category.
There is an important difference between forced and natural convection. In the former case the flow is externally imposed and is often independent of the temperature field. The flow field can thus be obtained independent of the heat transfer processes and then used in the determination of the temperature field. In natural convection, on the other hand, the flow and heat transfer are coupled, since the flow itself arises due to the temperature difference in the fluid.
As a result, the flow field cannot be obtained independent of the temperature field and the two must be considered simultaneously. This complicates the problem considerabJy, and thus even the simplest problem usually demands numerical.
In the following sections, we shall discuss numerical methods for forced convection in which fluid properties are assumed to be constant, i.
Next, the thermal boundary layer flows will be dealt with. We seek the steady state temperature T x by numerical means. In other words, grid is to be refined. But, refinement of grid is not computationally efficient, as it will require more computer time as a large number of equations will then have to be solved.
The method is attractive in terms of accuracy, but the computation of exponentials is time consuming. The hybrid scheme combines upwind and central differencing of the convective terms to achieve the stability of the upwind method and some of the better formal accu- racy of the central differencing. The scheme of discretization of the convective term is changed from central to upwind when Pee;; A much better accuracy is obtained by the use of a piecewise power-law scheme, discussed by Patankar The power law scheme is a computationally efficient approximation of the exponen- tial method.
However, the power-law scheme is strictly valid for the steady one dimen- sional case and cannot be extended to two or three dimensional or transient prob- lems. In most practical situations, one has to deal with multi-dimensional convec- tion-diffusion problems and therefore, a need for a more general method has arisen.
In Chapter 7, a method known as operator-splitting O-S algorithm employed to alleviate the defects of the upwind scheme at high Peclet numbers will be discussed. The O-S method is essentially an alternative method to upwind differencing.
If the Peclet number is large, a thermal front whose thickness is small will move through the flow region with unit velocity. The density of the fluid in this case air decreases, resulting in a buoyancy effect causing circulation of the fluid.
Here the velocity and temperature distributions are interrelated; the temperature distribution, in effect, produces the velocity distribution. Figure 6. The chapter begins with a detailed study of the possible numerical methods for solving 1-D steady convection-diffusion prob- lems. It is shown that upwind scheme is superior to central differencing when the Peelet number of the flow is high.
The concept of false diffusion arising in upwinding is also introduced. Brief mention is made of exponential, hybrid and power-law schemes. Also, finite-difference formulations using upwinding is described with respect to unsteady, one-dimensional and two-dimensional con- vection-diffusion problems. Next, computation of thermal boundary layer flows both external and internal is taken up. For computa- tion of temperature field of liquid metals 0.
Under the topic of natural convection, the problem of the solution meth- odology of the transient free convection from a heated vertical plate is given in cJetail using both explicit and implicit schemes.
Finally, only the qualitative aspect of the method of solution of the problem of free convection in enclosures is mentioned briefly. First, we shall discuss method- i and then, method- ii. Under certain conditions, the fin is in contact with high temperature or corrosive fluid. A layer of high strength or corrosion resistant J;Ilaterialis coated on both sides of the fin to withstand harsh environment Ju et fll.
The problem of applying ADI method to composite bodies is the interface. The as results have been compared with analytical, central difference and upwind results Muralidhar and Ghoshdastidar, The upwind method fails due to false diffusion see Chapter 6 inherent in its formulation.
The central difference method fails due to matrix ill conditioning and loss of diagonal dominance at high Peclet numbers. The as algorithm is uniformly accurate at low as well as high Peclet numbers. The algorithm has also been successfully applied to solve engineering problems such as Adsorption-Desorption problems in porous region, numerical modelling of enhanced oil recovery using water-injection method and flow past a discrete protruding heater in a vertical channel Ghoshdastidar and Muralidhar, In conclusion, it can be said that OS algorithm is a viable alternative to upwind scheme so far as the convection-diffusion problems in heat transfer are concerned and can be applied effectively to solve practical problems.
SUMMARY This chapter introduces the readers to two new methods of numerical solution developed in recent years of important heat transfer problems of practical significance in some detail. They are: i application of ADI scheme to the solution of transient heat transfer problem in a straight composite fin, and ii alternative to upwind scheme-Operator-Splitting algorithm to solve convection-diffusion problems.
In each case, considerable details have been given to make the readers appreciate the modelling techniques and interpretation of results. For some cases, a separate nomenclature section has been added at the end of the discussion.
Heat transfer by both mixed convection and radiation is considered. Mixed convection effect should be appreciable for low speed air flow over the fin, e. Radiation heat transfer mode is important for large temperature difference between the fin and the surrounding as well as for high emissivity fin material and for low speed air-flow. Consideration of transients is essential as during start-up and shut-down the heat transfer rates vary significantly.
Conjugate heat transfer should be taken into account as it gives rise to non-uniform heat transfer coefficient the consideration of which is more realistic in contrast with the assumption of uniform heat transfer coefficient in conventional fin theory.
Maximum values of X and Yare 1. Uniform grid spacings have been used in X and Y directions. The grid sizes and time-step are chosen after having performed sufficient numerical experimentations. An 11 x 16 grid system has been used.
A lg o rith m Due to non-linearities and mutual coupling, the governing equations are solved simultaneously and iteratively. The iterative procedure involves an overall iteration loop together with sub-iteration loops.
The mixed convection loop see Fig. This is required to solve the fin heat conduction Eq. To solve Eq. This is required to calculate the radiation heat transfer coefficient, h ,.
Equation 8. The algorithm, in the form of a flow chart is shown in Fig. The numerical results have been obtained using the data listed in Table 8. The fin emissivity value at the base temperature T b is used. Average execution time per computer run i. It is clear that with the progress of time, the temperature of the fin increases which is also expected from the physical point of view. This error can be removed by making finer grid in the fin. This is expected as 2 fin temperature as observed from the plots.
The results of the conventional fin theory buoyancy force. With increasing X, the heat flux decreases at first, reaches a mini- Figure 8. It is seen that at greater times. That is, initially heat transfer rate is pared to that due to mixed convection. This is because, the temperature difference less and then gradually it increases.
The solid raw material "raw mixture" is fed into the shell at the upper end and during the process becomes transferred to its lower end, where it is discharged.
I n the majority of high temperature kilns the required reaction temperature of the raw material is achieved by gas or oil combustion in the kiln internally fired kilns above the solid, usually by employing a burner at the end of the kiln and producing a flow of burning gas countercurrent to the solid movement. The present smaller Helmrich and Schtigerl, The rotary kilns are used mainly in the cement and metallurgical industries. Its Figure. The angle, ais called the other applications include production of chemicals and fertilizers such as phos- fill-angle which shows the region of volume containing the solid.
Figure 8. The rotary kilns are also used for burning of toxic or non-toxic waste. Some of the important advantages of rotary kiln reactors as compared to other types, such a fluidized bed reactors are simple design and high flexibility, co or countercurrent gas flow, easy adaptability to changing operating conditions by varying the length, diameter, inclination, speed of rotation, etc. Processes with highly corrosive reaction mixtures can be carried out in rotary kilns.
Very high temperature processes can be performed in rotary kilns and materials of varying consistencies can be handled easily. The main disadvantages are low space-time yield, inefficient energy utilization and difficult control of temperature and com- position of the reaction mixture in the central zones Helmrich and Schiigerl, In spite of the diverse uses of rotary kilns in industry, very few systematic theoretical models of the transport processes in the rotary kilns exist in literature.
There is an urgent need for optimization of rotary kilns. In this section, two applications of rotary kilns are considered, namely i rotary kiln dryer with applications to the non-reacting zone of a cement rotary kiln and ii rotary kiln solid waste incinerator.
The first application involves heat transfer without chemical reaction while the second one involves heat transfer with chemical reaction. Finally, the solid is transferred to the lower end where it heat transfer processes in a rotary kiln reaches the desired temperature and is discharged. In the present study, the kiln can be divided into three sections. In the second section, the liquid evaporates at constant temperature until the feed is Thermal Radiation Among Hot Gas, Refractory Wall and Solid completely dry.
In the third section, the solids are heated to some specified tem- Surface Heat is exchanged among the hot gas, the inner wall of the kiln and the perature and then are discharged from the kiln. The wall is divided For modelling, the non-reacting zone of a cement kiln is considered. The raw into surface elements as shown in Fig. Each axial segment of size equal to materials fed into the kiln contain calcium carbonate CaC03 , silica Si02 , 1 m of the refractory surface is divided into fifteen surface elements of equal shale AI and iron ore Fe The solid surface is divided into five surface elements.
The temperature of mixed according to the type of cement being made. Upon heating by hot gases,. There are three important zones: the preheat zone, the calcining zone, and the burning zone. In the preheat dry zone, the free water is. It is assumed that the surface ele- evaporated and the solid material is heated to,the point around K where.
It is assumed that water is always available on the solid surface. The end of the second section is indicated where the cumulative mv is equal to the total predetermined amount of water to be evaporated per second. Input data are shown in Table 8. Finite difference techniques are used and steady-state thermal conditions are assumed.
A grid independence test has been done. False Transient approach is used to solve the wall conduction equation. The solution is initiated at the inlet of the kiln and proceeds to the exit. In the second section of the kiln, the change in shape factors due to removal of water from the solids is accounted for. The output data consist of refractory wall temperature, solids temperature, gas temperature, the individual lengths of the first, second and third section of the kiln and the total kiln length.
However, the kiln length predicted by the present model turns out to be 22 metres as compared to The deviation of the present predicted length from that of the actual kiln may be attributed to uncertainties in some of the input data such as the composition of the hot gas, contact heat transfer coefficient, the ambient tempera-. This amount of dry solids have to be heated. For higher amount of water in the feed, the wet solid Figure 8.
This means that lower while the reverse is true for the axial solid temperature. This is because the although same amount of dry solid will have to be heated to the same exit tem- higher gas flow rate implies that the gas residence time is less and hence the kiln perature and the fill angle remains same in the third section no matter whether the has to be longer.
This means the solid has to stay for a greater period of time in the solid contains more water or less, the total heat transfer to the solid in the last kiln and therefore, solid temperature is high.
The gas, on the other hand, loses section of the kiln for the solid with high water-content will be greater than that more heat to the solids and hence the gas temperature is low. This can be explained by the fact that for larger solids mass flow rate the fill angle increases which in turn reduces mean beam length as the gas volume decreases and hence the emissivity of the gas decreases.
This gives rise to lower loss of heat by the gas by radiation and hence the gas temperature is high. The solid temperature is less as the heat transfer by radiation to the solid is less and Fig. Using such hot inlet gas would be uneconomic although it would drastically reduce the kiln length. This is expected as the gas flow is not a function of the aforesaid parameters. However, the axial solids temperature is greater for higher angle of inclination of the kiln or for higher rotational speed.
Increase in the kiln inclination angle or the rotational speed increases the predicted kiln length and vice-versa. The temperature increases sharply when the wall is near the flame surface but rises less rapidly as it gets farther from the flame.
The peak temperature is highest just before being covered by waste. The refractory surface temperature decreases while covered by waste. The maximum refractory temperatures are K and K for a equal to and degrees, respec- tively. The maximum temperature is increased when ais greater due to the greater flame surface area. These temperatures were calculated at the inlet and with a equal to degrees. Tem- peratures are shown for equal to 6, and degrees. At equal to 6 de- grees, the wall has just emerged from the waste and consequently the surface temperature is near the minimum.
At equal to degrees, the maximum sur- face temperature of K is reached. At equal to degrees, the surface has been cooled to the minimum by the waste. Except for the thin layer approximately 0. Conse- quently, increasing the refractory wall thickness would not effect the thermal characteristics of the kiln.
The solid waste temperatures are shown in Fig. The temperature increases almost linearly in the entire length of the kiln. However, the rate of temperature increase is greater for a equal to degrees than when a is equal to degrees.
The two curves are terminated at the point where the mass flow rate of the solid waste is equal to zero. The maxi- mum temperature of the solid waste for a equal to degree is K and occurs at 3 m from the kiln inlet. The maximum temperature for a equal to degrees is K and occur at 7. The mass flow rate remains nearly constant near the inlet since the volatalization rate is low.
A rapid decrease occurs at approximately 2. The mass flow rate Fig. Finite difference techniques are used to mode the Summary of Results Heat transfer characteristics of the rotary kiln incinera- heat conduction in the refractory wall and the energy transport by the solid waste tor was modelled so that the interaction of refractory wall conduction, flame material. Numerical results are presented which demonstrate relatively shallow of uniform channel depth where the molten material is further the capabilities of the computer model.
The results of primary importance for heated accompanied by further increase in pressure and the material is subjected design purposes are the refractory wall temperature variation and the length of to high shear, thus enhancing the mixing. However, it may be noted that pressure kiln required to completely burn the wastes. Each of the parameters listed in rise in the metering section will occur when die opening at the exit of the extruder Table 8.
For a widely open valve, the pressure will fall. In the present example, a detailed numerical study of the thermal transport s.
Gupta and P. Chakraborty, B. Ray, G. Biswas, F. Durst, A. Sharma, P. Chakraborty, G. Biswas, P. Koustubh Sinhal, P. Sayan Sadhu and P. Author: P. Year of Publication: Hardcover without floppy diskette ISBN Publisher: Oxford University Press. Reprints: Jan, , June, , October, Ghoshdastidar PI and Dr. Agarwal PI and Dr. Jain PI and Dr. Dipak Mazumdar PI and Dr. Numerical simulation of nanofluids flow and heat transfer in various geometries using i homogeneous model and ii heterogeneous model.
Ghoshdastidar published in in the institution journal. This award was given away by the President of India on December 20, during the inaugural session of the 10th Indian Engineering Congress held in Jaipur.
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