FILTRATION Introduction Separation of solids from liquids is a common operation that requires empirical data to make predictions of performance. These data are usually obtained from experiments performed on a small-scale laboratory equipment. In this lab, you will first perform experiments with a small batch filter and use your experimental data to predict operating conditions for a continuous rotary drum filter commonly used in industrial applications. You will then verify your predictions by experiments on the drum filter. The experiments are performed with a slurry of diatomaceous earth in water. In order to satisfactorily complete this laboratory exercise, it is important to have a thorough understanding of the following concepts: Filtration Filter cake resistance Flow through porous media Batch Filtration (Small Scale) Controlling parameters: Pressure drop, time for filtration, concentration of slurry Performance indicators: Filtrate flow rate, rate of cake formation. Feed (slurry) is poured into a graduated cylinder and filtrate is collected in a conical flask. The filter medium is a canvas cloth fixed at the bottom of the cylinder. The batch filtration can be performed in two regimes: 1. Pressure-driven filtration. A constant vacuum pressure is applied to the conical flask via pump induces liquid flow from the cylinder. 2. Gravity-driven filtration. The pump is disconnected from the conical flask and the fluid flow from the cylinder into the flask is driven by gravity. 1 1
Figure 1. Batch filtration The objective of the batch experiment is to determine the permeability κ and porosity ε of the cake. Once these values have been obtained they can be used to make predictions of the rate at which solids are produced in the continuous filter. These predictions are then tested by running the continuous filter. The pressure (either due to the weight of the slurry or the vacuum pump) is related to the water flow rate by Darcy s law for flows through porous media: Δ (1) Here, κ is the cake permeability, μ is the fluid viscosity, Δp is the pressure drop across the fluid, L is the cake thickness, Q is the volumetric flow rate of the liquid, and A is the area of filter media. In Eq. (1) we neglected the resistance of the filter medium, since in our experiment it is much smaller than that of the cake. Values of the water viscosity μ at various temperatures are available in the literature and values of L, A, and Q can be measured directly. In particular, Q can be measured by the bucket and stopwatch method, (2) Here, V(t) is the volume of the filtrate collected at time t. Therefore, we can solve Eq. (1) for the only remaining unknown, namely the cake permeability κ. Note, however, that several quantities in Eq. (1) are time dependent. Clearly, the cake thickness L grows with time. The pressure drop depends on the height H of the slurry, which 1 2
decreases with time. Finally, solution of equations discussed below shows that Q also depends on time. Therefore, if we solve Eq. (1) for κ using data measured at different moments of time, we will observe some fluctuations in κ due to errors in experimental measurement. A more convenient approach to measurement of κ is to perform a least-squares fit to experimental data. For this, we need to rewrite Eq. (1) as a differential equation by substituting Eq. (2) into Eq. (1): A Δ (3) The height of the cake L can be obtained from the mass balance for the solids. The mass of solids in the cake is 1 (4) Here, c o is the concentration of solids in the slurry (kg of solids per m 3 of liquid), is the density of dry solid, and is the porosity (volume fraction of pores) of the cake: is the volume of pores in the cake of volume LA. In the wet cake, this volume is occupied by water. Therefore, the cake porosity can be determined by comparing weights of the wet and dry cakes. Assuming that is negligible in comparison with V, solving Eq. (4) for L, and substituting the result into Eq. (3), we obtain: 1ϵρ A Δ (5) or Δ, (6) where 1ϵρ. (7) Eq. (6) can be easily solved if Δp is constant. In this case, 2Δ /. (8) Therefore, plotting V vs. t 1/2 and fitting the plot to a straight line, we can obtain the cake permeability κ. Note that in reality Δp is time dependent, Δ Δ t. (9) Here, Δp pump is the pressure drop generated by the vacuum pump, ρ slurry is the slurry density, and H(t) is the height of the slurry column above the cake. Dependence of H(t) on time is directly related to that of V(t). Therefore, Eq. (6) can still be solved albeit the solution becomes more complex. In the pressure-driven experiments, Δp pump = const and, if the contribution of the gravity to Δp is negligible, one can assume that Δp= const. 1 3
Continuous Filtration (Large Scale) Controlling parameters: Pressure drop, concentration of slurry, drum speed. Performance indicators: Filtrate flow rate, rate of cake formation. Rotary drum filter consists of a drum rotating in a tub of liquid to be filtered (see Fig. 2). The liquid to be filtered is sent to the tub below the drum. The drum rotates through the liquid and the vacuum sucks liquid and solids onto the drum. The liquid portion is sucked through the filter media to the internal portion of the drum, and the filtrate is pumped away. The solids adhere to the outside of the drum, which is then blown off the surface by air. The rotating drum continuously takes in feed from slurry tank and blows off the cake form the surface. This establishes a continuous operation. (a) (b) Ω θ Figure 2. Rotary drum filter: (a) the Bird-Young filter installed in the Unit Operations Lab; (b) Schematics of the filter operation. The continuous filter operates at constant p and hence each element of cloth acts like a batch filter. Therefore, the filtrate flow rate can be predicted by the batch filtration model discussed above. Let us compute amount of filtrate V passing through the filter during a short time interval t. Divide the drum surface into N segments of size w l, where w is the width of the drum and l is the distance traveled by a point on the drum surface during time t, Ω. (10) Here, R is the drum radius and Ω is the angular velocity of the drum. Area of each segment is and the total area of the drum surface immersed in the slurry is Ω. (11) Ωτ. (12) 1 4
Here, τ = Nt is the time during which a filter segment remains immersed in the slurry. This time can also be expressed as θ Ω, (13) where θ is the angle corresponding to the portion of the filter immersed in the slurry (see Fig. 2b). The segments on the drum surface are numbered so that the n-th segment spent time Δ1. (14) inside the slurry. Then the amount of filtrate that passed through the n-th segment since its immersion into the slurry is given by Eq. (8) with t = t n-1, i.e. Δ2Δ, (15) where K is given by Eq. (7). Volume of the filtrate passing through this segment during the next t seconds is V n = [V(t n ) V(t n-1 )]. Volume of the filtrate passing through the entire filter immersed in the slurry during time t is Δ Δ Thus the filtrate flow rate is 1 (16) Δ Δt ΔA2Δ Δt A2Δ t 2Δ. (17) The 2 nd equality was obtained by multiplying both the numerator and denominator by N. V(τ) in the right-hand side of Eq. (17) is given by Eq. (8). Once V(t) is known, we can also obtain the cake thickness L on the surface of the filter coming out of the solution. For this, we can assume that L = 0 when the filter segment enters the solution and use Eq. (4) to obtain L(τ). In this experiment, we cannot measure L directly. Instead, we can measure the rate of the cake production. A theoretical prediction for this rate can be easily obtained once L(τ) is predicted and the speed of the drum, RΩ, and the filter width, w, are measured. This prediction can then be verified experimentally. 1 5
Objectives Batch Filtration 1. Determine permeability and porosity of the cake from the batch experiment. 2. Determine effects of pressure drop, slurry concentration, and initial amount of slurry on the filtrate flow rate and the porosity and permeability of the filter cake. Continuous Filtration 1. Use results of the batch filtration experiment to predict dependence of the filtrate flow rate and the rate of the filter cake production on the speed of the drum rotation, pressure drop, and the slurry concentration. 2. Verify your theoretical predictions experimentally. 3. Determine porosity of the cake for various pressure drops and slurry concentrations and compare these results with the results of the batch filtration. 1 6