 In this lecture we're going to take a look at force convective heat transfer for an internal flow and typically what we'll be looking at is the flow within pipes and so what we'll do we're going to begin by looking at the fluid mechanics or the hydrodynamics for pipe flow. Now if you're looking at pipe flow quite often when we're looking at the equations they're for what we call fully developed pipe flow but it turns out that with pipe flow there is an entrance region where we do not yet have what we would call fully developed pipe flow and so I just want to begin by by going through the entrance region so imagine if we have a long section of pipe as shown here and if we could assume that the velocity coming into that pipe would be what we would call a uniform entrance region or a top-hat profile the velocity profile across the pipe cross-section may look something like this. Now what's going to happen as the flow continues to move along and in this case the flow is moving from the left to the right so it's moving in that direction. Now what will happen is this top-hat profile or a uniform velocity profile the walls recall we have a no-slip boundary condition along the walls and so the effects of viscosity will diffuse towards the middle of the pipe very much like we saw with the hydrodynamic boundary layer the flat plate boundary layer however here what's happening is the boundary layer is growing from both the top and the bottom and consequently our velocity profile will change as we move from left to right so I'm just going to sketch that out now. So what eventually will happen is our boundary layer will grow and grow and grow and eventually we'll get to some point within the flow field where the boundary layers have merged from around the perimeter of the wall so here we're looking at for example a round pipe and when that occurs that denotes what we call the hydrodynamic entry length and then downstream of that point the velocity profile is what we would call fully developed pipe flow and the exact profile is going to depend on whether or not the flow is laminar or turbulent but for right now what we'll do we'll just assume that it's fully developed and the unique nature of fully developed pipe flow is that the velocity profile itself will not change as we go further downstream unless we go through transition and go from laminar to turbulent but we're not talking about that yet so that is the entrance region and so over here we have uniform inlet flow and then once we get through the hydrodynamic entry length we have what we call fully developed pipe flow. Now when we're looking at pipe flow we use a non-dimensional number to characterize what is happening very much like we saw for the flat plate but here we use Reynolds number based on diameter so that is the characteristic length scale the velocity that we use we use a mean velocity which we'll take a look at in a moment and then we have our dynamic viscosity so the mean velocity let's take a look at the definition of the mean velocity the mean velocity can be determined by computing rho ua which is essentially the mass flux that is what is in our numerator divided by rho a in the bottom and that will give us a mean velocity across the pipe so if we were to go and try to expand this integral for the case of pipe flow we would be integrating from zero to r naught so let's say this is our pipe this is the center line the radius is going to go from zero to r equals r outer r naught we have our density the velocity the velocity is going to be a function of r and it can also be a function of x and as we evolve but once we get to fully developed it won't change with x and then we have our da term so the circumference of a circle times dr and we're integrating from zero to r naught and then on the denominator we have our density and the area the cross-sectional area of this remember we're dealing with the pipe so we have something like that it is just going to be pi r outer squared and so immediately what we can see here is a few of these terms are going to cancel out density is gone pi is gone and so this expression will simplify somewhat so that then becomes an expression for the mean velocity in our pipe flow and we'll be using this again later when we look at computing a bulk velocity or bulk temperature for non isothermal pipe flow or pipe flow where we have heat transfer so what we have here we have a couple of different terms um is the mean velocity over the tube cross-section and d that is the pipe or tube diameter and that would be the inner diameter okay so that is the entrance region a way to calculate the mean velocity now for pipe flow just like we saw for the flat plate flow we have a transitional Reynolds number and this is the Reynolds number whereby we can expect the flow to transition from a laminar flow starting to move into a turbulent flow regime where the velocity profile will change and rule of thumb number about 2300 although it can range quite a bit you can even get pipe flow at much higher Reynolds numbers up to about 10,000 if you've really really controlled conditions and you'll still have laminar flow although for industrial applications you'd never be able to get that type of low vibration type facility and consequently 2300 is the number that we often use and with that we also have a way to be able to find the entry length because if you recall from our schematic this is the entry length here and so we have a way to quantify this length depending on whether or not we have a laminar or a turbulent flow so entry length and so this would be H denoting hydrodynamic entry length to get to fully develop flow if we have a laminar flow it is approximately 0.05 times the Reynolds number based on diameter and if we have a turbulent flow so to get to fully develop flow for the hydrodynamic flow field rule of thumb is between 10 and 60 diameters that's how long it will take to get to fully develop flow for turbulent flow fields and and so quite often in flow metering applications we're not talking about that here but quite often they'll say you need to have 10 pipe diameters upstream or downstream of a flow meter and so that would kind of be agreeing with this there you might have something where you have pipe flow coming along and then you have an elbow you want to make sure that you go at least 10 diameters before you would put any kind of flow metering device in order to measure what the flow rate would be in that pipe so anyways that that's just how to get to the fully develop flow regime and here the elbow would be a thing causing an instability that changes the flow field and so it takes a while for the effects of the elbow to wash out so that is the entrance region transitional Reynolds number the mean velocity field and how to compute the entry length in pipe flow looking at the hydrodynamics what we're going to do in the next segment we're going to take a look at the thermal boundary layer the growth of the temperature distribution within the pipe flow