This study investigated the factors influencing municipal solid waste generation (MSWG) in Bauchi town, the administrative headquarters of Bauchi state, Nigeria. The study used quantitative method, while experiments and questionnaire surveys, were used as the major instruments for data collection. Data on solid waste generation and the socioeconomic attributes of residents from 400 households were collected from residential zones within Bauchi town, the study area. Descriptive statistics, Correlations and Standard Multiple Regressions (SMR) were computed for data analysis using SPSS 2.2 software. Twelve (12) socioeconomic factors were computed in SMR to determine the significant factors of MSWG in the study. The results showed that five factors, namely: household size, income, education, house head age and occupation have exerted significant influence on MSWG in the study area. The results also showed that F (5:362) = 84.058 at p ≤ 0.01 and adjusted R² = 0.531; which indicated that the factors in the SMR model have sufficiently explained the variance in MSWG in Bauchi town. The study concluded that the five significant predictor factors have adequately explained the variance of MSWG in the study area. Therefore, the factors have implications for planning of effective waste management system in Bauchi town, Nigeria

Gana, Bukar Abba

Harir, Adamu Isa

Kasim, Rozilah

Salihu, B. H

English

UTHM Institutional Repository

© 2007 Computational & Experimental Mechanics  Editors: A. K. Ariffin, N. A. Nik Mohamed and S. Abdullah 32   INCOMPRESSIBLE FLUID FLOW AND ENERGY EQUATIONS SIMULATION ON DISTRIBUTED PARALLEL COMPUTER SYSTEM   Manshoor. B1, Wan Hassan, M.N2, Alias. N3 1Faculty of Mechanical & Manufacturing Engineering Universiti Tun Hussein Onn Malaysia, Batu Pahat, Johor 2Faculty of Mechanical Engineering Universiti Technologi Malaysia, Skudai, Johor 3Faculty Science Universiti Technologi Malaysia, Skudai, Johor    ABSTRACT  This paper is concerned with the development of an efficient scheme for solving the finite difference Navier-Stokes and energy equations using distributed parallel computer system. The numerical procedure is based on SIMPLE (Semi Implicit Method for Pressure Link Equations) developed by Spalding. The hybrid scheme which is combination of the central difference and up wind scheme is used in obtaining a profile assumption for parameter variations between the grids points. Parallelization method used on this distributed parallel computer system is Domain Decomposition Method (DDM). The accuracy of the parallelization method is done by comparing with a benchmark solution of a standardized problem related to the two dimensional buoyancy flow in a square enclosure.   Keywords: simple algorithm; parallel algorithm; domain decomposition method; navier-stokes equations   INTRODUCTION   Parallel computing or also known as parallel processing refers to the concept of speeding up the excitation of a program using multiple processors by dividing the program into multiple fragments that can execute simultaneously, each on its own processor. A program being executed across n processors might execute n times faster than it would use a single processor. Here we can say that the parallel processing is differs from multitasking in which a single processor execute several programs at once. This project deals with a development of parallel algorithms in order to solve an incompressible flow simulation using SIMPLE method that originally put forward by Patankar and Spalding (Davis G. de Vahl, 1983). The analysis of an Incompressible Fluid Flow and Energy Equations Simulation on Distributed Parallel Computer System  33 incompressible flow become more complicated and need a high performance computer to solve the problem.  To overcome this problem, parallel computer was used and to determine the performance of this parallel computations, the corresponding parallel algorithms was developed and it based on method of parallelization known as Domain Decompositions Method. Control volume approach is selected and the matrix formed will solved using matrix tri-diagonal solver.    GOVERNING EQUATIONS   Two-dimensional incompressible laminar constant-density flow and energy equation is governed by set of partial differential equations (M.C.Melaaen, 1993). The continuity, momentum and energy equations in their primitive form are shown in equations (1-4) where the equation for conservation of mass is given by:                                                      0=∂∂+∂∂yvxu                                                  (1) The conservation of momentum in x and y directions are governed by the u-momentum equation expressed as:                                uSyuxuxPyuvxuu +⎟⎟⎠⎞⎜⎜⎝⎛∂∂+∂∂+∂∂−=∂∂+∂∂22221ρµρ                (2)  as well as the v-momentum equation:                              vSyvxvyPyvvxvu +⎟⎟⎠⎞⎜⎜⎝⎛∂∂+∂∂+∂∂−=∂∂+∂∂22221 νρ                   (3)  The conservation of energy express as:                                           ⎟⎟⎠⎞⎜⎜⎝⎛∂∂+∂∂=∂∂+∂∂2222yTxTckyTvxTuPρ                            (4)  The terms on the left-hand side of equations (2) and (3) is convective terms and the terms on the right-hand side include the pressure gradient and viscous terms. In all the above equations, u and v is the x and y components of the velocity, P are the pressure and ρ is the density.        A. K. Ariffin, N. A. Nik Mohamed and S. Abdullah 34  DISCRETISATION   In order to numerically solve the velocity and pressure fields that obey the discretized momentum and continuity equations, the finite difference method was applied. This method involves integrating the continuity and momentum equations over a two-dimensional control volume on a staggered differential grid shown in Figure 1 (S.V.Patankar and D.B.Spalding, 1972). This yields the governing equations in their discretized form as shown in equations (5-6). The staggered grid evaluates the scalar variables, in this case only the pressure, which are stored at the scalar nodes and located at the intersection of two unbroken grid lines. The u-velocity components are stored at the east and west cell faces of the scalar control volume and are indicated by the lower case letters e and w. The v-velocity components are stored at the north and south cell faces of the scalar control volume which are indicated by the lower case letters n and s. These velocity components are located at the intersection of a dashed and unbroken line that construct the scalar cell faces and are indicated by arrows.   FIGURE 1 Staggered Grid showing locations of flow variables  The horizontal arrows shown indicate the locations of the ue and uw velocity components and the vertical arrows indicate the locations of the vn and vs velocity components. After the process of discretization, the discretized of u-momentum equation becomes:           ( ) JiJiJIJInbnbJiJi bAppuaua ,,,,1,, **** +−+= ∑ −              (5)  and discretized of v-momentum equation becomes:                         ( ) jIjIJIJInbnbjIjI bAppvava ,,,1,,, **** +−+= ∑ −          (6)  Incompressible Fluid Flow and Energy Equations Simulation on Distributed Parallel Computer System  35 SOLUTION PROCEDURE OF THE SIMPLE ALGORITHM  The SIMPLE method proceeds by a cyclic series of guess and correct operations. The flow chart of the algorithm showed in Figure 2. Treat the corrected pressure p as new guessed p*, return to step 2 and repeat the whole procedure until a converged solution is obtained.    FIGURE 2 Flow chart of the parallelized SIMPLE with DDM  A. K. Ariffin, N. A. Nik Mohamed and S. Abdullah 36  PARALLEL IMPLEMENTATION  A parallel implementation can provide a further reduction in computing time. Parallel implementation also makes solution possible to problems that would require too much memory to solve on a single processor. During to solve this problem, the parallel implementation is based on message passing (distributed memory systems) using the Parallel Virtual Machine (PVM) software. Portability is ensured because PVM is available on many types of parallel computers.  The implementation uses a layer of subroutines on top of PVM, symbolically denoted by start: start entire parallel application, stop: stop parallel application, send: send a message and receive: receive a message. Communication process is the most important process in parallel implementation. For the send and receive subroutines, it consists of communication process between a data or function that will be send or receive. According to the pseudo code solution in Figure 3, the communication process occurs between the master and slave during to their sending and receiving the data or function.   find out if I am MASTER or SLAVES  if I am MASTER      initialize array      send each SLAVES starting info and subarray       do until all SLAVES converge        gather from all SLAVES convergence data        broadcast to all SLAVES convergence signal   end do    receive results from each SLAVE  else if I am SLAVE      receive from MASTER starting info and subarray    do until solution converged       update time       send neighbors my border info       receive from neighbors their border info       update my portion of solution array           determine if my solution has converged         send MASTER convergence data         receive from MASTER convergence signal   end do     send MASTER results endif   FIGURE 3 Pseudo code solutions     Incompressible Fluid Flow and Energy Equations Simulation on Distributed Parallel Computer System  37 COMMUNICATION   Basically this finite difference problem is same with the solution of the problem in this project. From top to bottom of the Figure 4; the one-dimensional vector X, where N=4; the task structure, showing the 4 tasks, each encapsulating a single data value and connected to left and right neighbors via channels; and the structure of a single task, showing its two inports and outports.    FIGURE 4 A parallel algorithms for the finite difference problem   We first consider a one-dimensional finite difference problem, in which we have a vector ( )0X of size N and must compute ( )TX , where;   ( ) ( ) ( ) ( )42:0,10 11titititiXXXXTtNi ++ ++=<≤−<<  That is, we must repeatedly update each element of X, with no element being updated in step t+1 until its neighbors have been updated in step t. A parallel algorithm for this problem creates N tasks, one for each point in X. The ith task is given the value ( )0X and is responsible for computing, in T steps, the values ( ) ( ) ( )Tiii XXX ...,,,21 .  Hence, at step t, it must obtain the values ( )tiX 1−  and ( )tiX 1+ from tasks i-1 and i+1. We specify this data transfer by defining channels that link each task with “left” and “right” neighbors, as shown in Figure 4, and requiring that at step t, each task i other than task 0 and task N-1. i. sends its data ( )TiX on its left and right outports,  ii. receives ( )tiX 1− and ( )tiX 1+  from its left and right inports, and iii. use these values to compute ( )1+tiX .  Notice that the N tasks can execute independently, with the only   constraint on execution order being the synchronization enforced by the receive operations. This synchronization ensures that no data value is updated at step t+1 until the X X X X0 1 2 3Xi“right”“left” A. K. Ariffin, N. A. Nik Mohamed and S. Abdullah 38  data values in neighboring tasks have been updated at step t. Hence, execution is deterministic. Figure 5(a) and 5(b) below showed the algorithms for the sending and receiving data from master and slaves.   C     broadcast data to slaves        call pvmfinitsend  (PVMDEFAULT, info)       call pvmfpack (INTEGER4, nproc, 1, 1, info)       call pvmfpack (INTEGER4, tids, nproc, 1, info)       call pvmfpack (INTEGER4, n, 1, 1, info)       call pvmfpack (REAL8, data, n, 1, info)       msgtype = 1       call pvmfmcast (nproc, tids, msgtype, info)        C     wait for results from slaves        msgtype = 2       do 30 i = 1,nproc       call pvmfrecv (-1, msgtype, info)       call pvmfunpack (INTEGER4, who, 1, 1, info)       call pvmfunpack (REAL8, result(who+1), 1, 1, info)       if (who.eq.0)       then       write (*,1000) result(who+1), who, (nroc-1)       else       write (*,1000) result(who+1), who, 2*(who-1) 30    continue    C     receive data from master        msgtype = 1       call pvmfrecv  (mtid, msgtype, info)       call pvmfunpack (INTEGER4, nproc, 1, 1, info)       call pvmfunpack (INTEGER4, tids, nproc, 1, info)       call pvmfunpack (INTEGER4, n, 1, 1, info)       call pvmfunpack (REAL8, data, n, 1, info)  C     determine which slave I'm (0...nproc-1)        do 5 i = 0,nproc       if (tids(i).eq.mytid) me = i 5     continue  C     do calculation with the data        result = work (me, n, data, tids, nproc)        C     send the result to the master        call pvmfinitsend (PVMDEFAULT, info)       call pvmfpack (INTEGER4, me, 1, 1, info)       call pvmfpack (REAL8, result, 1, 1, info)       msgtype = 2       call pvmfsend (mtid, msgtype, info) C     broadcast data to slaves                              (a)                                                                     (b)  FIGURE 5 Algorithm master to send and receive data to and from slaves and algorithm slaves to receive and send data from and to master  VALIDATION OF THE RESULTS   Tables 1 to 3 show the comparison between the results from the present simulation and the literature results obtained by de Vahl Davis. The results of de Vahl Davis are the standards against which all other codes are evaluated. Maximum horizontal velocity on the vertical midplane of the cavity, Umax, maximum vertical velocity on the horizontal midplane of the cavity, Vmax, and an average of Nusselt number were compared at Rayleigh numbers of 103, 104, 105 and 106. The comparison had been done between the benchmarking results obtained by de Vahl Davis which in serial processor and the present study that are simulation using serial processor and parallel processor or parallel computer.  From the tables, it showed that all these results are in excellent agreement with the benchmark results of de Vahl Davis. Percentage error for the three Incompressible Fluid Flow and Energy Equations Simulation on Distributed Parallel Computer System  39 methods of solution was below than 3% compare with benchmark result. Besides that, the result that was showed in the forms of contour maps of non-dimensional temperature and velocities was also compared with the results that obtained by de Vahl Davis.   TABLE 1 Comparison of the numerical result of present study for Umax   Ra 103 104 105 106      G. de Vahl Davis  3.649 16.193 34.620 64.593 Present study:           i) Serial processor  3.652 16.163 34.871 65.812         % error  0.082 % 0.185 % 0.725 % 1.880 %      ii) Parallel processor  3.592 16.376 34.852 65.847         % error  1.560 % 1.131 % 0.670 % 1.941 %   TABLE 2 Comparison of the numerical result of present study for Vmax   Ra 103 104 105 106      G. de Vahl Davis  3.697 19.167 68.590 216.360 Present study:           i) Serial processing  3.704 19.675 69.482 220.641         % error  0.189 % 2.650 % 1.300 % 1.978 % ii) Parallel processing  3.715 19.642 69.680 221.282         % error  0.487 % 2.478 % 1.589 % 2.275 %  TABLE 3 Comparison of the numerical result of present study for ____Nu    Ra 103 104 105 106      G. de Vahl Davis  1.118 2.243 4.519 8.800 Present study:           i) Serial processing  1.120 2.282 4.583 8.983         % error  0.23 % 1.74 % 1.42 % 2.08 %      ii) Parallel processing  1.123 2.272 4.594 9.008         % error  0.47 % 1.31 % 1.67 % 2.36 %   PARALLEL COMPUTING RESULTS   In order to achieve the objective of this project, parallel execution time was studied to determine the performance of the parallel computations. DDM were used during to obtain the results of the parallel simulation. Since using the parallel computer, there are involved with send and receive a data from master and slave. Therefore the parallel execution time consists of computational time and A. K. Ariffin, N. A. Nik Mohamed and S. Abdullah 40  communication time. Table 4 showed the result of execution time between sequential and parallel solution and the tabulated results of computational time and communication time.  TABLE 4 Execution time for computational solutions  Sequential time Parallel time tcomp tcomm tp Ra. no (tseq) (tp)    103 32.8 s 9.43 s 8.41 s 1.02 s 9.43 s 104 135.75 s 41.39 s 34.62 s 6.78 s 41.39 s 105 2040.26 s 612.06 s 522.82 s 89.24 s 612.06 s 106 163602.04 s 49080.61 s 41923.02 s 7157.60 s 49080.61 s  Execution time is not always the most convenient metric by which to evaluate parallel algorithm performance. As execution time tends to vary with problem size, execution times must be normalized when comparing algorithm performance at different problem sizes.  Other parameter that was used to measure a performance of parallel computations is speed-up and efficiency. From the speed-up, we know that how fast the parallel computer solves the problem under consideration. It is sometimes useful to know how long processors are being used on the computation, which can be found from the efficiency.    TABLE 6 Results for speed-up and efficiency  Ra. no 103 104 105 106 Speed-Up 3.478 3.279 3.333 3.333 Efficiency 86.95 % 81.97 % 83.32 % 83.32 %   DISCUSSION AND CONCLUSION   From the results that were obtained, we can see that execution time for parallel computation was decrease compare with sequential computation. By using sequential computation, total execution time that we need to complete our simulation at Rayleigh number 106 is 163602.04 seconds or 2726.7 minutes or 45.45 hours. For parallel computation, we were reduced an execution time for the simulation at Rayleigh number 106 to 49080.61 seconds or 818.01 minutes or 13.63 hours. In theory, by using 4 processors to solve this problem, we can reduce an execution time until 75%. However, it cannot be achieved since the communication process was occurring.  As a conclusion, a parallel algorithm has been developed to simulate an incompressible flow for the problem of natural convection that occurred in a square cavity with specified boundary conditions. The simulations of the incompressible flow using SIMPLE method on parallel computer are agreement Incompressible Fluid Flow and Energy Equations Simulation on Distributed Parallel Computer System  41 with the benchmark result. Thus, the simulation is successful. Percentage errors for the computational solution which are simulation by serial and parallel computer are below than 3% compare with benchmark result by de Vahl Davis. Therefore it has proved that clustering personal computers together can provide adequate computing power for large engineering problems.   REFERENCES   Date A. W (1985). “Numerical Prediction of Natural Convection Heat Transfer in Horizontal Annulus.” Int. J. Heat Mass Trasfer.  Davis G. de Vahl (1983). “Natural convection of air in a square cavity: a benchmark numerical solution”. Int. Journal Numerical Mech. Fluid 3: 249-264.  Dongarra, J. & Eijkhout, U. (2000). “Numerical linear algebra algorithms and software.” Journal of Computational and Applied mathematics. 123(2): 489-514.  Geist, A. et al. (1994). “PVM: Parallel Virtual Machine. A Users’ Guide and Tutorial for Networked Parallel Computing”. Massachusetts: The MIT Press.  Patankar S. V (1980). “Numerical Heat Transfer and Fluid Flow.” McGraw-Hill Inc, New York.  Patankar S.V and Spalding D. B (1972). “A Calculation Procedure for Heat, Mass and Momentum Transfer in 3-Dimensional Parabolic Flows.” Int. J. Heat Mass Trasfer.  Spalding D. B (1972). “A Novel Finite Difference Formulation for Differential Expressions Involving Both First and Second Derivatives.” Int. J. Num. Methods Eng. 3: 551-559.  Versteeg H. K and Malalasekera M. (1995). “An Introduction to Computational Fluid Dynamics.” Pearson, Prentice Hall. 

The factors influencing municipal solid waste generation in bauchi town, Nigeria

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