The results of the experiment of dewatering analysis carried out on a municipal sludge were recorded. Similarly, analytical mathematics was used to develop an equation to calculate volume of filtrate. The two results were plotted in graphs and comparisons were made. The results show very close relationships between the two results. In fact in most cases the two results were almost overlapping. For example after 3, 6, 9, 12, 15 hours, the volumes of filtrates are: 0.0037 m ^{3} (model), 0.004 m ^{3} (experiment), 0.005 (model), 0.0055 (experiment), 0.006 (model), 0.0067 (experiment), 0.0066 (model), 0.0065 (experiment) and 0.0065 (model), 0.0064 (experiment). This has shown that the experiment was carried out very well and that, mathematical equations can be used directly in order to calculate the volume of filtrate and the values can be nearly the same with model results.
Wastewater can have a number of definitions. Domestic wastewater is defined as wastewater from household water use, while industrial wastewater is from industrial practices only. In reality, the term “wastewater” clearly encompasses domestic, commercial, industrial, agricultural components and also faecal sludge, these are sometimes covered separately in order to clarify or highlight the importance of the individual components or wastewater streams [
Sludge is a semi-solid material produced by various biological and chemical processes in water and wastewater treatment plants that needs further treatment prior to its disposal into the environment. More importantly, the water in sludge is not only in one form in terms of its binding characteristics to solids. According to [
The form of water in sludge determines the effectiveness of sludge treatment operations to separate the water associated with the solids. Before ultimate disposal, the water content of sludge should be decreased both from environmental and economical point of view [
This was pioneered by Almy and Lewis [
d v d t = k p n / V m (2.0)
This constituted the basic law of filtration, where m and n are indefinite powers of the relationships, which the rate of flow d v d t possesses with pressure P
and volume V respectively. Since Poiseuille’s law and Darcy’s law both deal with flow and considering sand bed as a bundle of capillaries, Carman proposed the following equation:
t = V 2 C μ r 2 P A 2 + μ R V P A (2.1)
Reference [
V = A 2 C r C r A 1 2 p C 1 2 r 1 2 t μ f (2.2)
Since the relationship between V and t has been established to be parabolic, Anazodo substituted f = 1/2 to obtain this equation:
V 2 = P A 5 / 2 t μ C 1 2 r 1 2 (2.3)
Reference [
V 2 = P μ − 1 A 2 b t C r 2 b − 3 (2.4)
So that relationship between V and A should be experimentally determined. Reference [
V 2 = P A t 1.82 t μ C r 1.18 (2.5)
and
V 2 = P A 2 e f f 2.76 t µ ( C r ) 0.24 (2.6)
Halff on his own part developed an equation that established the fact that filtration follows the parabolic relationship until the cake reaches 80% filtration [
C f i = δ γ i μ ( C o ) S (2.7)
Even though the specific resistance parameter is a good measure of sludge filtrability, it does not have constant unit. In the light of the above, a dimensionless number referred to as the sludge dewaterability number was developed by [
Δ H ( C o − C f ) V t C c + H 0 V t (2.8)
where:
Cc = Concentration of cake (kg/m^{3})
C_{f} = Concentration of filtrate (kg/m^{3})
C_{o} = Concentration of sludge (kg/m^{3})
H_{0} = Initial headloss (m)
∆H = Change in headloss (m)
SDN = Sludge Dewaterability Number
t = Time of filtration
V = Volume of filtration (m^{3})
The filtration equation according to [
Q A = Δ P μ ( R f + α c V / A ) (2.9)
considers the overall resistance to flow of filtrate to be made up of contributions from the filter medium R_{f}, and from the cake with specific resistance α. Normally R_{f} is found to depend on the operating pressure; the two intercepts may correspond to different values of R_{f} at the two pressures. Although these porosities and permeabilities are of unused materials, the relative values may be useful for comparing behaviors under filtration conditions. Permeability K_{p} normally is the property reported rather than the resistivity that has been discussed here. It is defined by the equation
Q A = K p Δ P / μ L (2.10)
where L is the thickness. Thus the resistivity of the medium includes its thickness.
The relation of the above equation to resistivity is:
R f = L K p (2.11)
Reference [
Q = d V d t = A Δ P μ R (2.12)
The resistance R is made up of those of the filter cloth R_{f} and that of the cake R, which may be assumed proportional to the weight of the cake. Accordingly:
Q = d V d t = A Δ P μ ( R f + R c ) (2.13)
where:
c = wt of solids/volume of liquid (kg/m^{3})
= viscosity (N sec/m^{2})
P = pressure difference (N/m^{2})
A = filtering surface (m^{2})
V = volume of filtrate (m^{3})
Q = rate of filtrate accumulation (m^{3}/sec).
At constant pressure Equation (2.13) is integrated as:
A Δ P μ t = R f V + α C 2 A V 2 (2.14)
The above equation can be written in linear form as:
t V / A = μ Δ P R f + μ α C 2 Δ P V A (2.15)
The constants R_{f} and a derivable from the intercept and slope of the plot of t/V against V. If the constant pressure period sets in when t = t_{0} and V = V_{0}, the equation becomes:
t − t 0 V − V 0 = μ A Δ P R f + μ α C 2 A 2 Δ P ( V + V 0 ) (2.16)
At a constant rate of filtration Equation (2.16) can be written as:
Q = V t = A Δ P μ ( R f + α c V / A ) (2.17)
The above equation can be re-arranged in linear form as follows:
Δ P Q = Δ P V / t = μ A R f + μ α c A 2 V (2.18)
The constants again are found from the intercept and slope of the linear plot of ∆P/Q against V. After the constants have been determined, Equation (2.18) can be employed to predict filtration performance under a variety of constant rate conditions. For instance, the slurry may be charged with a centrifugal pump with a known characteristic curve of output pressure against flow rate [
In order to carry out this research work, sewage sludge was collected from an open unlined drain along Layin Wanki in Unguwar Sarki area, Kaduna, Kaduna North Local Government of Kaduna State. The sample was collected equally along the length of the open drain in order to get a composite sample. Even though it is a municipal wastewater, but there is need to mention that there are people along the street, who are engaged in washing of clothes, locally. The wastewater from their activities is also included in this sludge. About 15 litres of sludge was poured on to the drying bed which was located on a corridor of the Water Laboratory of the Department of Civil Engineering Kaduna Polytechnic, Kaduna The drying bed was made with a glass so that it is transparent. The length is 1.5 m and width is 0.5 metres. The filter medium consists of coarse sand and gravel. The sand has a depth of 0.3 m and the same depth is the gravel (0.3 m). The depth above the sand (where the sludge is slowly allowed in) is 0.2 m. In this research, the wastewater was allowed to percolate onto the drying bed for 15 days, and the volume of the filtrate was taken (and recorded) after every 24 hours, throughout the duration. From the results obtained, the t/v was calculated and a graph of volume V and t/v, V and t were plotted. A sand layer is placed on top of the gravel. The sand layer enhances drainage and prevents clogging, as it keeps the sludge from lodging in the pore spaces of the gravel. The diameter of the sand is crucial as sand with a larger diameter (1.0 - 1.5 mm) can result in the relatively fast accumulation of organic matter, thereby increasing the risk of clogging, In order to reduce this risk a sand with a smaller diameter (0.1 - 0.5 mm) is use. The municipal sludge was passed through the filter in order to get a dry solid known as cake.
The mathematical equation will be developed using advanced calculus [
V f d C d t = Q 0 C 0 − Q C f (3.0)
where Q_{0} is the rate of sludge flow (m^{3}/sec)
Q is the rate of volumetric increase of the filtrate (m^{3}/sec)
C_{0} is the concentration of the sludge (kg/m^{3})
C_{f} is the concentration of the filtrate (kg/m^{3})
dC is the small change in the cake concentration (kg/m^{3})
V_{f} is the volume of fitrate in m^{3 }
Volume of Filtrate Equation (V_{f})Considering a situation where sludge is applied on the surface of filter and it was dewatered. If Q and C_{f} are considered to be functions of time while C_{0}, Q_{0} are considered to be constant, then integrating, Equation (3.0), we will have:
∫ V f d C = ∫ ( Q 0 C 0 − Q C f ) d t (3.1)
If V_{f} is considered to be constant, after final collection, then Equation (3.1) becomes:
V f ∫ d t = Q 0 C 0 ∫ d t − ∫ Q ( t ) C f ( t ) d t (3.2)
But according to [
V f = Q t
Therefore Equation (3.2), becomes:
V f ∫ d C = Q 0 C 0 ∫ d t − ∫ V f t C f ( t ) d t (3.3)
Integrating Equation (3.3), Equation (3.4) will be:
V f C = Q 0 C 0 t − V f ∫ 1 t C f ( t ) d t (3.4)
Integrating by part: 1 t = d V , U = C f ( t ) , V = log ( t ) , d u = d C f (t)
Therefore: u v − ∫ v d u
V f C = Q 0 C 0 t − V f [ C f ( t ) ⋅ log ( t ) − ∫ log ( t ) ⋅ d C f ( t ) d t ]
V f C = Q 0 C 0 t − V f C f ( t ) ⋅ log ( t ) − V f ∫ log ( t ) ⋅ d C f ( t ) d t (3.5)
Neglecting rate of change in C f , Equation(3.6), becomes:
V f C = Q 0 C 0 t − V f C f ( t ) log ( t ) (3.6)
But [
Q 0 t = V 0
Therefore Equation (3.6) becomes:
V f C = V 0 C 0 − V f C f ( t ) log ( t ) (3.7)
Re-organizing the above equation, Equation (3.7) becomes:
V f ( C + C f ( t ) log ( t ) ) = V 0 C 0 (3.8)
Since according to [
V 0 = V f + V c
Therefore
V f ( C + C f ( t ) log t ) = C 0 ( V f + V c ) (3.9)
Re-organizing the above equation, Equation (3.9), will be :
( C + C f ( t ) log ( t ) ) = C 0 ( V f + V c ) / V f (3.10)
C V f = C 0 ( V f + V c ) − C f ( t ) log ( t ) (3.11)
C V f = C 0 V f + C 0 V c − C f ( t ) log (t)
C V f − C 0 V f = C 0 V c − C f ( t ) log (t)
V f ( C − C 0 ) = C 0 V c − C f ( t ) log (t)
V f = C 0 V c − C f ( t ) log ( t ) / ( C − C 0 )
V f = C 0 V c − C f ( t ) log ( t ) / ( C − C 0 ) (3.12)
Considering the graphs plotted in Figures 1-3, it can be observed that the differences between the model and experimental results are so near to each other, and in fact in
Time | V_{f} (Model) | V_{f} (Experimental) |
---|---|---|
3 | 0.0037 | 0.004 |
6 | 0.005 | 0.0055 |
9 | 0.006 | 0.0067 |
12 | 0.0066 | 0.0065 |
15 | 0.0065 | 0.0064 |
Cake | V_{f} (Model) | V_{f} (Experimental) |
---|---|---|
0.0074 | 0.0037 | 0.0036 |
0.129 | 0.0058 | 0.0059 |
0.134 | 0.006 | 0.0062 |
0.15 | 0.0068 | 0.0069 |
0.173 | 0.0075 | 0.0077 |
Co | V_{f} (Model) | V_{f} (Experimental) |
---|---|---|
0.5 | 0.0037 | 0.0039 |
0.65 | 0.005 | 0.0053 |
0.8 | 0.006 | 0.0064 |
0.88 | 0.0068 | 0.0069 |
0.95 | 0.007 | 0.0075 |
The analysis carried out in this research has shown that mass balance equation can be used in order to develop an equation for the volume of filtrate. Also good and reliable laboratory work can lead to a good result. This can be concluded by comparing the model and experimental results that are almost the same.
From the laboratory analysis, it can be recommended that:
1) The medium of the dry bed filter should be arranged in such a way to avoid clogging within a short time; because this will make the dewatering process to be delayed and cost more, as the filter has to be cleaned within a short period of time.
2) Drops of filtrate should be monitored properly at every time. This will help to monitor the efficiency of the filter.
3) The process should be carried out in-doors to avoid external forces, such as wind that may blow away the drops from entering the collecting vessel.
4) More research should be carried out where period of filtration is increased and the time interval of recording the filtrate is decreased. For example the dewatering process can be increase to a month and the recording time be made after 12 hours. The process should be monitored to see whether it is more efficient, than the one carried out in this research.
Yusuf, S. and Ademiluyi, J. (2018) Comparing Model and Experimental Results of the Volume of Filtrate during Sludge Dewatering. Open Access Library Journal, 5: e4650. https://doi.org/10.4236/oalib.1104650