 We're now going to work an example problem involving calculating pressure drop in pipe flow and so in the next couple of segments we'll be taking a look at this type of problem where we have to estimate pressure drop. So let me write out the problem statement. Okay so in this problem what we have is a syringe filled with a fluid and the fluid is being forced out of the syringe through a very small diameter needle section that has a diameter of 0.25 millimeters and we are given the volumetric flow rate coming out of the syringe and we are asked to determine what is the force required in order to maintain a steady flow rate and the viscosity of the fluid is given so what we are going to do in order to solve this is we are going to determine the pressure drop within the needle section of the syringe so within this section here and we will then apply the energy equation and we will use the pressure drop from the energy equation so what we will do we will refer to this as being section one and this here where the fluid is coming out is section two and we will assume that this here is p atmospheric pressure that the fluid is going into so in coming up with the solution to this problem the first thing we need to do if we're going to look for the pressure drop in this section here we need to look at the Reynolds number and we need to determine if the flow is laminar or turbulent so that will be the first step in our solution procedure and so when we write out the Reynolds number based on diameter we have rho vd over our dynamic viscosity now the velocity in the Reynolds number here this is the average velocity in our tube or our pipe and so we need to determine the average velocity and that is the volumetric flow rate divided by the cross sectional area of the section that is generating the pressure drop so you'll notice what i'm doing here is this is the diameter and this is in centimeters and so the answer that we're going to obtain is going to be an answer in centimeters per second so the velocity of the fluid in the needle is about eight meters per second that's a fairly high velocity now let's check the Reynolds number to determine if we're dealing with a laminar flow or a turbulent flow and when we plug in the velocity the density the diameter and the kinematic viscosity the dynamic viscosity we obtain a Reynolds number of 916.73 which is less than the 2300 that we said would indicate the transition from laminar to turbulent therefore we're dealing with a laminar flow so with that we can then go to determine what the pressure drop will be in the section of pipe and in order to get the pressure drop we need to obtain the friction factor so the friction factor for laminar flow it was a very clean relationship it's just 64 divided by the Reynolds number based on diameter which we then obtain a number of 0.0698 now alternatively you can also get that from the moody diagram and i won't do it here because it's a relatively straightforward calculation fairly simple but you could also obtain that number from the moody diagram and then once we have the friction factor here we take that and we will feed it into the Darcy Weisbach equation which will then tell us what the head loss is in this section of the syringe so we're going to determine the head loss due to laminar flow in that section so in the Darcy Weisbach equation again using our average velocity across the cross section of our tube or pipe so we get a number of 14.18 meters now if I come back to the schematic what I'm going to do is I'm going to neglect head loss in the larger diameter part of the syringe or of the needle and so we'll neglect head loss here and the rationalization for that is the diameter is quite a bit larger than it is in the needle and consequently the velocity is going to be very low and consequently the head loss in that section will be quite small so with that we then go to the steady flow energy equation and we're dealing with the laminar flow and consequently our kinetic energy coefficient alpha 1 and alpha 2 is going to be 2.0 and we're assuming v1 is approximately equal to 0 so writing out the steady flow energy equation between states 1 and 2 so when we look at this equation we can make one cancellation to begin with we're assuming that the velocity in the larger cylinder section is approximately equal to 0 and the other thing we can say is p2 is equal to p atmosphere because that's where the flow out of the needle is coming into is the atmosphere and so rearranging now what I'm going to do I'm going to pull p2 over to this side so we have p1 minus p2 and if p2 is at atmosphere then that would be a measurement of p1 at gauge pressure divided by the density and then on the right hand side we have the following terms so we get that now we say nothing in the problem statement about the orientation of our needle or our syringe and so we're going to assume that it is horizontal and consequently the elevation z1 will equal z2 and with that this term disappears as well and so quite quickly it's starting to simplify when you plug in the values we've solved for v2 we know alpha 2 we've solved for the head loss and so the only unknown in this equation is going to be p1 gauge and we then obtain a number of 184 about 185 kPa is required in order to force the fluid out through the small needle now the question asked what is the force and so what we need to do is we need to take that pressure and multiply it by the area of the syringe section or of the piston section and so with that we get force is p1g times the cross sectional area at section one and we can plug in the values and we obtain 14.53 newtons so that would be the force required in order to force fluid out of the syringe this is an example of estimating pressure drop using a flow field where you'd have very low Reynolds number and consequently we had laminar flow and then you apply the energy steady flow energy equation to be able to solve it so that is an example of friction factor for laminar flow