 Hi friends, we will discuss about settling of discrete particles before. My name is Mayu Rubale. I am an assistant professor in the Department of Civil Engineering, WIT, Singapore. The learning outcome of this session, the students will be able to determine the settling velocity of discrete particles in sedimentation time. Now let's see what is meant by discrete settling. Discrete settling is a sedimentation of discrete particles in the suspension of very low solid concentrations. It also means a free settling, since the discrete particles have a negligible tendency to flocculate upon each other, as the discrete particles have its own shape, size, and they behave as an individual entity, so there will be no interaction between particles takes place. And in the water treatment plant, we will always focus on the low concentration of solids, that is the discrete particles. As their nature, the particle will not try to settle down with respect to the gravity. Therefore, we use a discrete settling. Now, when particles come into the fluid, that is our water, it will try to accelerate until the frictional force, which is the proper resistance, which is coming through the water to the particle, or what we call it as the drag, particle, which is equal to the gravitational force of particle, that is our Newton's law. So the three types of forces will act upon the discrete particle. The first word we say is the drag force, that is our frictional force. How we can define it? The frictional force, that is Fd, is equal to Cd, that is coefficient drag multiplied by area of particle, that is our Ap multiplied by density of fluid, that is rho multiplied by v square by 2, where v is velocity of fluid. Now second one is gravity force, that is force acting due to gravity. It is notated as Fg, where it is defined as Fg is equal to the density of solids, that is our rho is multiplied by acceleration due to gravity, that is g multiplied by velocity of particle, that is vp. The third word will be the upper force, that is buoyancy force, or we normally say as buoyant force. It is denoted as Fb, where Fb is equal to density of fluid, that is our water, it is noted as rho multiplied by acceleration due to gravity, that is g multiplied by velocity of particle, that is vp. When the particle tries to accelerate in the fluid, the resistance, that is our drag force will try to oppose it. So when the driving force becomes equal to the drag force, the particle will go upon a constant flow. How it happens? The drag force is the upward force, but the force due to gravity is down side and buoyancy force is upside. So when the force of gravity goes on increasing with respect to the buoyancy force, the upper force, that is our Fd will act upon it. As we have designed the sedimentation tank in such a way that the driving force will push the particle down side. So when the design force, that is the implied force, when it gets equal to the drag force, the particle will go and contain a very constant flow. That constant flow we call it as terminal settling flow. It can also be defined as when the drag force is equal to the driving force, the particle reaches a constant value. It is denoted as v small s. Now in the previous two slides, we have seen four equations. By equating them and considering the particle to be a sphere, we will get a formula for a settling velocity which is equal to a root of 4 by 3 multiplied by acceleration due to gravity divided by coefficient drag multiplied by, in bracket, specific gravity of solid minus specific gravity of fluid multiplied by the diameter of particle divided by the density of fluid. When we divide the density of fluid to the upper part, we will get a very simpler equation. Then we say vs is equal to root of 4g in bracket, specific gravity of particles that is s small s minus 1 bracket complete multiplied by the diameter of particle that is small d divided by 3cd that is coefficient drag. Now this all the parameters must be put up in a very standard format that is bigger v that is volume of particle must be calculated in the form of meter cube. The ap that is a cross sectional area of particle must be calculated in the form of meter square. The velocity of particle must be calculated in terms of meter per second. The density of particle must be calculated in the form of kg per meter cube. The density of fluid is also calculated in the form of kg per meter cube. Acceleration due to gravity that is our small g is always put up in the form of meter per second square. The cd that is our coefficient drag is always unitless. Now we will talk about the stocks law. The stocks law is always deals with the particles which are discrete in nature also but we assume that the particle is in the sphere form that is a circular cross section area of the particle. Now again the simplifying the Newton's law equation we can get a simpler and very easier formula. We can say that small v s that is our terminal settling velocity is equal to 1 by 18 multiplied by acceleration due to gravity that is small g multiplied by in bracket the specific gravity of particle that is s small s minus 1 bracket complete multiplied by the diameter of particle square divided by the kinematic viscosity of particle. So we have to understand the difference here we are taking care of viscosity part 2 but in the Newton's law we try to assume the particle is very non viscous but though there the frictional force will attract but the in the stocks law we do not consider the frictional force in that case but here we take the viscosity into the consideration. Here it has to be understandable that in this whole equation all the parameters have different units. Let's see the settling velocity of particle have a unit of millimeter per second the diameter of particle is kept millimeter the kinematic viscosity of water is kept in centi stocks. The specific gravity of particle is unit less and the acceleration due to gravity is taken as 981 centimeter per second square. Now if you see by taking the acceleration due to gravity that is 981 centimeter square and the kinematic viscosity that is 131 integrate the factor that is 1 by 18 g nu will get 418 it means the kinematic viscosity of particle changes with temperature how we can derive the temperature based equation let's see. If you see at a standard 10 degree centigrade the formula is same that is 418 s small s minus 1 d square but what about other temperatures at other temperature we use a very different formula which is inclined to stocks law the formula is small v s is equal to 418 s minus 1 d square multiplied by bracket 3t plus 70 by 100 now let's have some review questions let's see first one when the driving force is lesser than the drag force the particle will achieve terminal velocity in the settling tank true or false second one kinematic viscosity of water is dependent on third one in the stocks law shape of discrete particle is assumed to be so here pause this video for some time being solve these questions and now I will go forward so these are the answers for the first question the first one answer is false the driving force is always greater or equal to drag force when the driving force is equal to drag force only at that time terminal settling velocity will achieve the kinematic viscosity of water is always dependent on temperature and in the stocks law shape of discrete particle is assumed to be spear the references for this discussion is taken from the doctor bc punmia and ashok kumar jane and doctor arun jane from water supply engineering book and the second book I have referred is water supply engineering from santosh kumar gar thank you