^{1}

^{*}

^{1}

This study was completed by an extensive mathematical analysis. New equation to sludge filtration processes has been proposed for use in routine laboratory. The equation has been suggested to replace Ademiluyi’s cake filtration equation in view of the limitations of the latter. The new equation can be used for sludges whose compressibility factor is more than one but Ademiluyi’s cake filtration equation can only be used for sludges whose compressibility coefficient is less than one. The new sludge filtration equation was derived using tan
^{n}
*θ* reduction method. The generalized equation thus obtained resembles Ademiluyi’s equation in the mode of parameter combination except the presence of summation notation in the new equation.

Sludge is produced from the treatment of waste water in on-site (e.g. septic tank) and off-site (e.g. activated sludge) systems. This is inherently so because a primary aim of waste water treatment is removing solids from the waste water. Sludge is also produced from the treatment of storm water, although it is likely to be less organic in nature compared to waste water sludge [

Bucket latrine and vault latrines store faecal sludge, which needs to be collected and treated. These two types of latrine are not discussed in this research work because no treatment is involved at the latrines.

The characteristics of sludge vary widely from relatively fresh faecal materials generated in bucket latrines to sludge which has undergone bacterial decomposition for over a year in a double pit latrine. The treatment required is therefore dependent on the characteristics of the sludge. Sludge maybe contaminated with heavy metals and other pollutants, especially when industrial waste are disposed into the sewer. Prevention of contamination of the sludge by industrial wastes is preferable. A conversion process to produce oil from sludge has been developed, which can be suitable for heavily contaminated sludge. The costs of treatment of sludge are generally of the same order as the costs of removing the sludge from the waste water. Also sludge volume is usually less than 1% of the total plant influent, sludge handling costs are 21% - 50% of total plant operating and maintenance costs. Sludge contains solids, oil, fat, protein, phosphates, carbohydrate, nitrogen, water, etc with a specific gravity of 1.02, 1.06 (for organic fraction) and 2.5 for inorganic fraction [

Sewage sludge treatment describes the processes used to manage and dispose of sewage sludge produced during sewage treatment. Sludge is mostly water with lesser amount of solid materials removed from liquid sewage.

The various types of sludge treatment are; stabilization, thickening, dewatering, drying and incineration. The latter is most costly, because fuel is needed and air pollution control requires extensive treatment of the combustion gases. It can be used when the sludge is heavily contaminated with heavy metals or other undesirable pollutants. The organic carbon in the sludge, once stabilized is desirable as a soil conditioner, because it provides improved soil structure for plant. Also, faecal sludge contains essential nutrients (nitrogen and phosphorus) and is potentially beneficial as fertilizers for plants. Biosolids produced from dried or treated sludge act as a fertilizer for crop harvesting. The treated sludge can also be used as top dressing on golf course fairways and a soil substitute in final landfill cover. Also recovers struvite in the form of crystalline pellets from sludge dewatering stream, the resulting crystalline product is sold to the agriculture turf and ornamental plants sectors as fertilizer under the registered trade name “Crystal Green” [

In the course of waste water treatment process, some amount of sludge is usually generated. The bulky and compressible nature of this waste water solid coupled with its high water content of about 97.5% [

Sludge dewatering is an important process finding application in many manufacturing industries and in plants designed for water and waste treatments. Three main concepts have been suggested to evaluate the filterability of sludge. These include specific resistance [

Equation (1) has been modified several times [

One of the ways of achieving sludge dewatering process is through vacuum filtration which is a mechanical process. Many researchers, [

The parabolic relationship between filtrate volume and time of filtration does not hold throughout the filtration cycle [

Equation (2) has some limitations. Ademiluyi’s theory is not applicable during the early period in cake filtration before the vacuum has assumed a constant value and before enough cake has formed to become the dominant filter resistance. Also his theory is only applicable for sludge whose Terzaghi’s compressibility coefficient has been found to be less than 1 (one) (Ademiluyi, 1985). That is, the equation can only been used for sludges whose Terzaghi’s compressibility coefficient is very much less than 1. The research in sludge filtration should continue until an acceptable equation which governs sludge filtration phenomenon is derived [

The basic equation Carman and Tiller’s equations describing sludge dewatering is given as

where p is the vacuum pressure (kN/m^{2}), v is the volume of filtration (m^{3}), t is the time taken to obtain filtrate (s), A is the area of filtration (m^{2}), R_{m} is the medium or septum resistance (m^{−2}), w is the mass of dry solid deposited per unit area (kg/m^{2}), R is the average specific resistance (m/kg) and µ is the viscosity of filtrate (poise).

Equation (3) is frequently the starting point for development of filtration equations. Accounting for the hydrostatic pressure and compressibility coefficient and by appropriate substitution Ademiluyi transformed Equation (3) to

And gave the solution of Equation (4) as

where, H is the driving head at any time (t) in (m), β is the vacuum pressure (kN/m^{2}), H_{0} is the initial head (m), γ is the specific weight of filtrate (N/m^{3}), exponent, s is the compressibility characteristic called the coefficient of compressibility (cm^{2}/g).

^{n}θ Reduction Method

Equation (5) can be integrated by reducing the expression to a simple integratable form by the application of tan^{n}θ reduction method. In order to derive the new sludge filtration equation, the tan^{n}θ method of reduction was employed.

Thus;

Therefore,

where ^{+}

Application of Equation (10), in solving Equation (5)

Thus,

Let

Where s is a positive integer

Let n in Equation (10) =

Substituting Equation (20), (19) in Equation (10) gives

Since

Also

Similarly,

Substituting Equations ((24)-(27)) in Equation (23) gives

Let

Let

Also

Let

Substituting Equations ((40) and (52)) in Equation (28) gives

Substituting for

Substituting for

Substituting for

and

in Equation (61)

Equation (64) can be written in the form

where

Equation (61) is the new sludge filtration equation obtained. It can also be written in the form in Equations ((64) and (65)).

In applying the new Equation (61) in sludge filtration theories, it is expected that measured time, t(s) are measured after the formation of the cake so that the septum resistance become negligible as it is required by the theory. The implication is that time t, was measured after the formation of the cake. Many parameters are involved in the new Equation (61) therefore data were stored in the computer using soft ware allowing later use of the soft ware to evaluate the data using the new Equation (61). The change in driving head H, can be measured directly or by measuring the filtrate volume to the drop in head. All parameters in Equation (61) were gotten from Ademiluyi, J.O, 1985 experimental results except alpha

Equation (61) derived in this work shows that the plot of t versus _{1} and intercept

The result of the experiments, [

shows that t increases with decreasing in numerical value of

The correlation coefficient ‘r’ ranging from 0.998 to 1.000 shows a linear relationship between t and

Generally, the new theory is valid only for cake filtration which is indeed the basis of this investigation. The high correlation coefficients R^{2} ranging from 0.996 to 1.000 with the graphs is a true test of valid linear relationship between t and^{2} of 0.989 and 0.983 shows that both predicted time and measured time are in agreement. In ^{2} of 0.998 and 0.998 using the new equation and 0.987 and 0.982 using Ademiluyi’s equation also confirmed that both predicted time from the new equation and measured time from Ademiluyi’s equation are in agreement.

It must be noted that the compressibility attribute of sludge cake previously suggested by Ademiluyi and that of Carman which has been shown to be in error [

One of the problems in Carman’s equation is the concentration, c which is the mass of dry cake per unit volume of filtrate. This parameter is very difficult to evaluate. To this end, the parameter is taken to be the initial solid content contrary to the theoretical prediction. In the new theory however, the concentration term (So) is the actual initial solid content which can easily be measured in the laboratory.

The problem of which areas to be used during sludge dewatering is not found in the new theory unlike Carman’s theory in which some previous workers advocate effective area of funnel while another school of thought proposed total area. Since the area needed is the area of the filter column used in calculating the driving head H, the problem of area is solved. To calculate driving head at any filtration time, the area of the column should be used not either the total area or effective area of the filter bed.

Effect of dilution on the specific resistance (R^{*})

The results of the investigation of the effect of dilution on the specific resistance

Figures 14-18 shows the graphs of specific resistance with time, at different fil-

tration tested, specific resistance at any height of the sludge cake increases with increasing filtration time. The increase in specific resistance with time can be explained. During cake filtration, more sludge solids which were once on suspension settle at any piezometric point with time. This increasing sedimentation

of initially suspended solids decreases the porosity of cake at the time of sedimentation by blocking the settleable pores. This process will basically increases the specific resistance.

The results of the experimental data collected to investigate the effect of vacuum pressure on the specific resistance is displaced in

The results of the experimental data collected to investigate the effect of piezo-

metric positions on the specific resistance is displaced in

The following conclusions are drawn from the study;

1) A new equation to sludge filtration processes has been recommended for use in routine laboratory investigation. This equation is given as:

2)

The new theory has been recommended for sludge dewatering studies since the experimental analysis is not rigorous as the traditional theory. Besides, the theory accounts for the compressibility attribute of sludges and the hydrostatic pressure, which are believed to influence sludge dewatering. Also in the derivation of the new basic equation, the compressibility attribute of sludge called compressibility factor is considered rather than compressibility coefficient in Ademiluyi’s theory.

Contribution to KnowledgeThe new theory is applicable to sludge whose compressibility factor is greater than 1 (one) unlike Ademiluyi’s theory which is applicable for sludge whose compressibility coefficient is much less than 1 (one) only.

The controversy among previous writers as regards the area of filter bed to be used (whether effective area or total area of filter bed has been resolved since the new theory makes use of the area of filter column and not that of the filter bed in the evaluation of the driving head (H). The concentration term (So) in the new equation is the actual initial solid content of the sludge as distinct from the Carman’s equation in which the concentration term is the mass of dry cakes deposited per unit volume of filtrate. Also in the derivation of the new equation, specific resistance parameter has been treated as a local variable parameter rather than the traditional average value used in sludge filtration studies.

3) Carman’s theory requires the linear plot of t/v versus v. Linear regression analysis cannot be used in drawing this straight line since the variables involved are not independent [

4) It has been shown both analytically and experimentally (Data from Ademiluyi’s experiment) that

a) Specific resistance increases with time of filtration. This is due to the increasing sedimentation of initially suspended solids which decreases the porosity of cake by blocking the available pores, this finding is in agreement with Ademiluyi’s theory.

b) Specific resistance increases toward the filter septum. This may be due to the corresponding decrease of porosity towards the filter septum, in agreement with the finding of Ademiluyi but not in agreement with experimental results obtained from other researchers; Anazodo and Carman, where specific resistance was found to be constant throughout the filtration cycle and along the cake height.

5) It has been found also that specific resistance increase with increasing in vacuum pressure. This finding is in agreement with the finding of both Ademiluyi, and Coackley. The reason for this is that as filtration pressure is increased, the porosity of the cake decreases thereby increasing the specific resistance.

In view of the variable nature of the specific resistance

Limitations of the new equation are,

a) The equation can only be used for sludge whose compressibility factor is more than 1 (one).

b) It is difficult to evaluate its dimensional homogeneity since some of the variables in the new equation have an exponent ‘s’ which is not a dimensionless pure number

I would like to thank a great number of people who have helped me in bringing this thesis into existence. I express my gratitude to my project supervisor, Engr. Prof. J.O. Ademiluyi who in his unique way of inspiring his students drove me to height I did not believe is attainable. My friend Dr. Dennis Agbekaba, my brothers, Udegbunam Ikechukwu, Hon. James I.K Ademu, My Boss, Engr. Jang C. Tanko, Late Dr. Isani Edwin for his guidance during my undergraduate studies. My children, Chidera, Adaeze, Arinze and most importantly, my dear wife, Angel Cynthia Udegbunam for always being there for me.

Udegbunam, O.E. and Ademiluyi, J.O. (2017) Cake Filtration Equation Using tan^{n}θ Reduction Method. Journal of Water Resource and Protection, 9, 851-872. https://doi.org/10.4236/jwarp.2017.97057

A = Area of filtration (m^{2})

C = Mass of dry cakes deposited per unit volume of filtrate (kg/m^{2})

g = Acceleration due to gravity (m/s^{2})

H_{o} = Initial driving head (m)

H = Driving head at any time t, (m)

h_{m} = mercury rise in manometer (m)

h_{f} = Head loss (m)

L = Thickness of the cake (m)

L_{m} = Thickness of the medium (m)

P = applied vacuum pressure (kN/m^{2})

P_{L} = Hydraulic pressure (N/m^{2})

P_{s} = Compressive drag pressure of solids (N/m^{2})

q or dv/dt = Flow rate (m^{3}/s)

^{2}/g)

R = Average specific resistance (m/kg)

R^{1} = Cake resistance (m^{−2})

R_{1} = Local flow resistance (m/kg)

R_{m} = Medium or septum resistance (m^{−2})

s = Compressibility factor (cm^{2}/g)

S_{o} = Initial solid content of sludge (kg/m^{3})

t = Time taken to obtain filtrate (s)

v = Volume of filtrate (m^{3})

W = Mass of dry solids deposited per unit area (kg/m^{2})

β = Vacuum pressure (kN/m^{2})

p_{f} = Density of filtrate (kg/m^{3})

p_{m} = Density of mercury (kg/m^{3})

γ = Specific weight of filtrate (N/m^{3})

µ = Viscosity of filtrate (poise)

[n/2] = greatest integer value of n/2_{ }

tan^{n}θ = Modified Reduction formula