diaphragm filter pressThe core element of the filter plate set consists of a set of normal filter plates with concave sides alternating with a set of diaphragm plates with diaphragms on both sides. The grooved portion of the filter plate is combined with the diaphragm plate to form the filter chamber. The filter chamber is equipped with a filter cloth, through which the two-phase flow of sludge passes under the pressure of the feed pump for the purpose of solid-liquid two-phase separation. Due to its high pressure resistance, good sealing, convenient discharging and low water content of filter cake, diaphragm filter press is widely used in metallurgy, chemical industry, coal and sludge treatment. At present, the domestic sludge treatment plant on the diaphragm filter press for each workflow time setting mainIt is a disadvantage of roughness and arbitrariness, which leads to the long working cycle and low efficiency of the filter press. Most of the theoretical researches on the sludge dewatering performance of diaphragm filter press at home and abroad are limited to the establishment of mathematical models for the filtration characteristics of the feed press stage according to Darcy's theorem, but there are fewer researches on the drum press stage, which is complicated in the change of filtration characteristics. Liu Peng et al.[4] studied the measurement method of cake specific resistance during filtration and the influencing factors of constant pressure filtration, but the method was not applicable to the membrane press stage of diaphragm filter press; Zhao Yang et al.[5] showed that different starting points of membrane press had a great influence on filtration, but the method was based on a large number of experiments to summarise the rules and did not establish the mathematical model of filtrate volume change with time in the membrane press stage. However, this method was based on a number of experiments and did not establish a mathematical model of the filtrate volume over time during the membrane press stage, nor did it propose a theoretical calculation method; STICKLAND et al. carried out a filtration study on plate and frame filter presses with fixed chambers, in which the slope of the filtrate volume versus time initially increased, followed by an increase in the corresponding slope pressure and chamber resistance, and then by the formation of a constant cake. This study also did not cover the membrane press stage of the diaphragm filter press. Therefore, the authors of this paper used Fluent software to simulate the filtration stage of the filter press, and based on the results, mathematical modelling of the various working processes of the filter press was established through mathematical fitting methods. The "virtual filtrate volume method" is proposed to describe the time-dependent mathematical model of the filtrate volume in the drum stage, which has complicated filtration changes, so as to optimize the time of each working stage of the diaphragm filter press and improve the working efficiency of the filter press.

1 Diaphragm filter press workflow and principle

The diaphragm filter press duty cycle is divided into the feed stage, the filter press stage, the membrane drum stage, the blowback stage, the discharge stage, and the preparation stage for the next duty cycle, such as cleaning the filter cloth and pressing the filter plate. The diaphragm filter press operating cycle timeline is shown in Figure 1. The blowback, discharge, and preparation phases have a constant time, which is integrated for ease of calculation and is referred to as T0. The feed phase time, T1, starts when the sludge is pumped to the filter press and fills the entire chamber, which is also a constant value. The filter press stage time T2 starts at 0 and stops at t2, when the sludge continues to be fed at the rated pressure P2 and filtered. At T3, the filter press stops feeding, the diaphragm plates on both sides of the diaphragm are filled with water or air at rated pressure p3 (p3 > p2) to expand the diaphragm chamber, compressing the volume of the sludge cake inside the chamber and further filtering the sludge cake until t3.

Since the filtration speed in the filter press and membrane drum stages decreases with time, the efficiency of the filter press in treating sludge begins to decrease in the later stages of operation. Therefore, the efficiency of the diaphragm filter press can be improved by optimising the filter press time T2 and the membrane drum time T3.

2 Based on Fluent simulation of filter press working process

The purpose of this simulation is to simulate the filtration process of the filter press, record the filtrate volume at each time point, and find out the relationship between the filtrate volume and time in the filtration stage, the specific filtration speed[2, 7-8] and time, and the relationship between the filtration pressure and the limiting filtration volume of the sludge, so as to provide a basis for the prediction of the mathematical model for the other working stages of the filter press.

2.1 Establishment of Filter Chamber Models Fluent pre-processing was carried out using Gambit software to establish the geometric model of a single filter chamber of the filter press and mesh the chamber. The calculation method was "Standard k-ε model", the discrete format was "QUICK", the pressure interpolation method was "PRESTO! The pressure interpolation method is "PRESTO!" and the pressure-velocity coupling method is "PISO". The main parameters were as follows: chamber diameter of 300 mm; solid-phase particle diameter of 0.01 mm; chamber thickness of 10 mm; rated filtration pressure of 0.2-3.0 MPa; solid-phase density of 1,051 kg/m3; filtration time of 30 min; porosity of 201 TP3T; two-phase flow rate of 951 TP3T; and coefficient of inertial resistance of 3.5×107 ; The coefficient of inertial resistance is 3.5×107; the coefficient of viscosity resistance is 1.2×1015; and the dynamic viscosity is 0.02 Pa?s.

2.2 Simulation results Under the filtration pressure of 0.2~1.4 MPa, the relationship between the total volume of filtrate V and time t of the four groups is shown in Fig. 2. As can be seen from Fig. 2, the total volume of filtrate gradually increases with time and tends to be close to a certain limit value; the higher the pressure, the faster the filtrate speed, and the greater the ultimate filtration volume.The liquid-phase flow velocity in the outer cross-section of the porous medium is plotted at a filtration pressure of 1.4 MPa. The specific filtration velocity q versus time t is shown in Fig. 4. From Fig. 4, it can be seen that the specific filtration velocity will surge to a certain value at the beginning, continue to rise for a short period of time, reach the maximum value, and then gradually decrease with the increase of time; the main reason for the transient increase of the specific filtration velocity is that the sludge particles are bonded at the beginning of the filtration process, and the diameter of the particles increases, which leads to the decrease of the specific surface area of the sludge layer and the increase of the porosity.

3 Mathematical modelling of two-phase flow filtration

3.1 Conventional filtration calculation methods

The traditional method of measuring specific resistance usually considers the mud cake incompressible and measures the slope of the curve K of dt/dV-V[12] , which is derived as a proportional function of K[6, 13] . The specific resistance r is derived as a proportional function of K [6, 13], which leads to the specific resistance r. However, this method regards the cake as incompressible and the specific resistance does not change with time, which is obviously not consistent with the filtration situation of two-phase flow of sludge in the filter chamber of the diaphragm filter press. In addition, the diaphragm filter press has many workflows, the filtration stage is a constant-pressure feed filtration, and the volume of the filter chamber in the membrane drum stage changes with time, which makes the variation of the specific resistance complicated, and it is difficult to express the relationship between q and t in equation (3).

3.2 Calculation method based on the simulation results By simulating and recording the curve of the total volume of sludge filtrate V(t) versus time t and the curve of the specific filtration rate q versus time t, the expression of the V-t function is obtained by fitting the calculation.

3.2.1 Feed stage During the feed stage, the filtration volume is approximated to be 0 L. The sludge is fed at a certain flow rate to reach the final M1, which is the chamber volume.

From its derivative (Figure 4), it can be seen that: first there will be a short-term increase, and then gradually decrease, and finally converge to 0 L. Therefore, the exponential form is more in line with the change rule of the total amount of filtrate over time, and can be fitted to the V(t)-t curve of the filtration pressure stage by the least squares method. Set 2 / 22 22 2 ( ) e , 0, 0, 0, 0 b t Vt a a b t t = ＞＜＜ ≤ (4) The simulated V2-t curves of 8 groups under the conditions of 0.2~3.0 MPa are fitted by the least squares method, and the results are shown in Table 1. From Table 1, the curves of filtration pressure p2 and parameter a2 are shown in Fig. 5. The ultimate filtration capacity a2 increases with the increase of pressure, but the rate of increase is from fast to slow, and tends to a constant value, after reaching the constant value, the further increase of pressure can not make the mud cake further filtration[14] .

The parameter b2 hardly changes with filtration pressure under certain sludge characteristics and filter press operating parameters. 3.2.3 Membrane Stage Assuming that the filter press continues to be fed at the pressure p3 during the membrane stage, the ultimate filtrate volume a3 = ka2 (where k is the ratio of the membrane filtrate volume to the filtration filtrate volume). However, the membrane stage stops feeding, the diaphragm plate with a certain pressure on the compression of the cake, by reducing the volume of the filter chamber to achieve the purpose of filtration, the limit of the filtrate volume must be less than a3, so a3 is not the limit of the filtrate volume of the membrane stage. Therefore, a3 is not the limiting filtrate volume in the membrane stage. In this paper, a3 is the virtual limiting filtrate volume in the membrane stage, and V3' is the virtual filtrate volume in the membrane stage, whose value is only a mathematical assumption, not the real filtrate volume.

3.2.4 Filter Press Blowback, Unloading and Preparation for Cleaning After the membrane is blown, a blowback process is performed to clean the pipework of any residual slurry and filtrate. This is followed by the unloading process and preparation for the next cycle. The time for this phase is essentially constant. 3.2.5 Optimisation of filter press operating time points As shown in Figure 1, the non-filtration sum time T0+T1=t0 and the filter press filtration time T2+T3=t3, the filter press operating cycle T=t0+t3, with filtration time T2=t2 and membrane drumming time T3=t3-t2. Assuming that the original sludge water content is η0, the final water content of the cake during the drumming process at t=t3 reaches ηf, which is the final water content of the sludge dewatering process. At t=t3, the final water content of the sludge cake in the membrane drum process reaches ηf, which is the standard of sludge dewatering.

In practice, since the expansion of the diaphragm at the membrane expansion stage is not arbitrarily large, the pressure of the diaphragm plate will not be able to act sufficiently on the cake if the cake does not reach a certain thickness, which limits the scope of application of this method to calculate the optimum filter press time. Based on the above problems, the minimum feed quantity Mmin should be set according to the diaphragm expansion performance of the diaphragm plate of the diaphragm filter press, and when the theoretically calculated filtration press time T2 corresponds to the total feed quantity M＞Mmin, T2 calculated by the above method is the optimal filtration press time, and T3 is the optimal membrane drumming time. When the total amount of feed M corresponding to the theoretically calculated filter press time T2 < Mmin, the time T2′ to reach Mmin is the optimum filter press time.

4 Optimisation of filter press working cycle in sludge treatment plant 4.1 Filter press working parameters A sludge treatment plant treats 500 m3 of sludge with a water content of 95% per day, and four XAGZ200/1250-30u diaphragm filter presses are designed to work simultaneously for 24 hours, which cannot actually complete the work task. At present, the working cycle of the filter press in this plant is T=210 min, in which the preparation time is 20 min, the feeding time is 10 min, the filter press time is 120 min, the membrane drum time is 30 min, the blowback time is 10 min, and the unloading time is 20 min. The parameters of the XAGZ200/1250-30u diaphragm filter press are as follows: the area is 200 m2 , the number of chambers is 80, the outer diameter of the filter plate is 10 m2 , and the number of filter plates is 10 m2 . The parameters of XAGZ200/1250-30u diaphragm filter press are as follows: the area is 200 m2 ; the number of filter chambers is 80; the outer diameter of the filter plate is 1 250 mm×1 250 mm; the thickness of the filter chamber is 30 mm; the centre feed, the rated filtration pressure is 0.8 MPa, and the rated pressing pressure is 1.6 MPa. The amount of filtrate during the working process of the filter press was recorded, as shown in Table 2.At the end of the filter press process, the total amount of filtrate was 14.36 m3 . The total amount of filtrate at the end of the drum stage was 15.17 m3 , and the water content of the final cake was 60.4%. The sludge processing rate of the filter press was u=0.083 m3 /min.

4.2 Optimisation of the timing of the various operating phases of the filter press

According to the time setting of the original filter press workflow, each filter press can work for 6 cycles per day, and the daily processing capacity of the 4 filter presses is about 416.64 m3 , which is not able to complete the daily production task. After optimisation, the working cycle of filter press is about 2 h, and the daily operation is 12 cycles, the daily production capacity can be 613.44 m3 . The plant actually operates according to the cycle proposed in this paper, and lets one of the machines rest and standby in turn while completing the task, which not only meets the requirements of the daily production.It also gives the filter presses more downtime and overhaul time, which helps to extend the service life of the filter presses.

5 Conclusions 1) Based on Fluent simulationsfilter pressThe relationship between the total volume of filtrate V and time t under different pressures in the filtration stage was investigated, and a least-squares method was used to fit the functional expression of the curve to obtain the relationship between the total volume of filtrate and the pressure. 2) The concept of virtual filtrate volume V3′ in the membrane stage was proposed, i.e., the total filtrate volume under the assumption of continuing to feed at the membrane pressure p3 without changing the chamber volume. In this way, the mathematical relationship between the actual filtrate volume V3 at the membrane drum stage and the time t is obtained, and the mathematical relationship between the filter press efficiency u and the filter press time t2 is also obtained. 3) Optimising the time of each working stage of diaphragm filter press in a sludge treatment plant, the sludge treatment efficiency was improved by 37.7%.