Measuring and Improving Cooling System Performance – Part 9: Duct Flow and Pressure Gradients

For the last several months, I've been trying to better understand duct flow as I've worked through this cooling system project. Ducts can be hard to wrap your head around, as I found in a propulsion class my last semester in school, and analysis of them sometimes reveals counterintuitive behavior.
 
For this last post let's get into more of the theory of duct flows and compare to experimental results that illustrate some of their characteristics as well as the implications to car cooling system modification.

We can consider the heat exchanger as a constant-area restriction or station in a longer cooling system duct (as in the Hoerner figure below), or as a collection of tiny ducts all stacked on top of each other. Either way, the physical behavior of the flow is the same; it's just a question of which is more convenient for analysis.

Theory
 
First, consider an ideal (inviscid airflow) duct with equal inlet and outlet area: with no friction, there is no resistance to flow through the duct, no loss of momentum, no change in velocity, and no change in static or total pressure. What goes in comes out with no change to any property. This is impossible, you might say—but it turns out, it is very much possible; we just can't see it. A streamtube somewhere in the air away from the influence of the car (but still referenced to the vehicle inertial frame, which moves with respect to the air) behaves exactly like this ideal "duct" if we define the duct walls as stream surfaces:

You may have read that static pressure gradients (what we call a pressure field) drive fluid from one place to another. Normally, this is true—as in the wind tunnel below, where a large fan at the diffuser outlet creates a static pressure drop that pulls flow through the tunnel.

But in the example of a streamtube somewhere away from the influence of the car with no static pressure gradient, what drives the flow? This is something to be very careful about, since the car's cooling system takes in air that is only moving relative to the car, and thus "flowing" with no pressure field. Technically, the properties of the flow we've been talking about over this whole series (and in car aerodynamics in general) should be labeled more specifically as "relative" e.g. p_0,rel, u_rel since they are relative to the (moving) reference frame of the car and not the (fixed) world frame.
 
What happens now if we add solid walls to the duct subject to friction? Shear stress in the fluid and the no-slip condition at the wall cause an internal force to develop which is equal and opposite to the drag force that now acts on the duct. These same stresses cause a boundary layer to build up along the walls of the duct within which momentum is lost, effectively narrowing the outlet area so it is no longer equal to inlet area. In "fully developed" or steady state flow (where flow properties don't change over time), the friction in a real flow means outlet conditions determine inlet properties and not the other way around. This is due to continuity: mass flow in must equal mass flow out, and if the lost momentum at the outlet necessarily reduces mass flow rate, the duct must take in less mass to compensate. (This is worth reiterating, since it is counterintuitive: outlet flow determines inlet flow, not the other way around. Think of a clogged bathtub drain; any restriction in the drain reduces the flow of water through it until inlet mass flow rate equals outlet mass flow rate. This is one of the reasons why the better approach to adjusting mass flow rate through your car's cooling system is to open or close outlet area rather than block the grill).

Note that I am using here, as I have in this whole series, mass flow rate rather than volume flow rate. As we saw in Part 4, cooling capacity depends on mass flow rate ṁ, and, following aerospace convention, I prefer to work with this measure rather than volume flow rate since mass flow is what really matters in determining drag and cooling capacity (and, in aerospace applications, performance measures like engine thrust). Since density varies with temperature, mass flow through your car's cooling system varies with changes in ambient conditions even if intake volume flow remains constant (see the plot in Part 1). We need to make sure the car has adequate cooling capacity in the worst-case scenario (temperature, altitude, vehicle speed, and engine load), which means designing for the mass flow rate in that condition. At steady state, ṁ is constant anywhere in the cooling system duct, unlike volume flow rate dꓯ/dt. Because density ρ changes as the air passes through the heat exchanger core(s), absorbing energy and raising its temperature, constant mass flow rate means volume flow rate behind the exchanger must be different than in front of it while the car is operating.
 
But there's another reason I prefer to work with mass flow rather than volume flow. This has to do with the terms in the model equation for conservation of momentum in a flow, called the "Navier-Stokes equations" (the relationship is a vector equation, which means it has components in multiple directions and so is comprised of several equations represented in one. We'll restrict ourselves here, as before, to the assumption that flow properties only vary in one dimension, something aerodynamicists call quasi-one-dimensional flow). (And, despite what you may have seen in the news recently, the Navier-Stokes equations have not been "solved." The complete equations are a system with 7 independent variables; there is still no analytical solution and it can only be approximated using numerical methods).
 
Go back and review Part 4 to see how this equation is derived, but writing out a control volume momentum balance for the cooling system gives the force acting on the fluid as it flows through the duct or across the exchanger:

Looking solely at the heat exchanger core with its constant area, this modifies to

And the drag force on the exchanger(s) is the opposite of the friction force acting on the fluid:

Now, let's assume flow velocity and dynamic pressure into the heat exchanger are the same as flow velocity and dynamic pressure out. We can make this assumption because, at steady state and again applying continuity, the averaged velocity over the intake area must be the same as the averaged velocity over the outlet area (since both areas are the same; note that we are approximating any change in density as negligible for now. Recall from our observations in Part 6 that the flow properties into and out of the heat exchanger are not uniform, too—hence the averaging). With no acceleration of the flow, the net force acting on it must be 0. This means the friction force the fluid experiences is balanced by the change in static pressure through the exchanger,

Notice that there is no ṁ term in this equation; that is, it tells us nothing directly about mass flow rate. To correlate the pressure term in this equation to mass (or volume) flow rate through the exchanger, we assume that viscous forces—the term on the lefthand side—on the fluid increase with flow velocity (which they do—recall that they are proportional to u^1.8 as derived in Part 4), and that total pressure loss is equal to static pressure loss because of the constant velocity and density i.e. constant dynamic pressure.

You will find the constant density/constant velocity/constant dynamic pressure assumption (circled in red) in mainstream sources on cooling system analysis, as in this plot from Hoerner's Fluid-Dynamic Drag.

In order to arrive at this equation, we have made lots of simplifying assumptions. These theoretical alterations may or may not be valid in reality; in the case of density, for example, while the cooling system is operating, the change in air density through the exchanger(s) may be significant (since it depends on the difference between ambient temperature and heat exchanger outlet temperature). Additionally, there are other problems for home modifiers when trying to ascertain the properties of the cooling system flow in our cars by experimental measurement.
 
Experiment
 
When I first measured heat exchanger static pressures, the readings on my truck were curious. They showed, on average, no difference in static pressure across the heat exchanger—which I interpreted at the time to mean there was little or no airflow through it. My lived experience should have told me this conclusion was very wrong: the truck has never overheated, even on long road trips in very hot weather, and in fact runs on the cool side, an indication that there is plenty of airflow through the heat exchanger.

I resurrected it after it sat at the back of my parents' driveway for five years and immediately drove it 2500 miles back to Illinois in July weather over multiple mountain ranges, loaded with nearly half a ton of stuff in the back. If there was a problem with cooling system airflow, it would have manifested itself on this trip.

Why didn't those pressure readings give me an accurate picture of the flow through the truck's cooling system? It comes down to probe placement.
 
A subsonic jet outlet into a pressure boundary "adjusts" the flow to that pressure boundary (see Anderson’s Fundamentals of Aerodynamics: "In subsonic flow, a jet that is dumped freely into the surrounding air takes on the same pressure as the surroundings"). This means that probe placement is important; taking measurements from a static probe somewhere in the engine bay does not give us the static pressure of the flow as it exits the heat exchanger but does give us the static pressure boundary value to which the flow out of the exchanger adjusts, which may be quite different. The reference probe in the truck cooling system test was placed in the engine bay well downstream of the exchanger itself.
 
So, the first lesson here is, placing an outlet static pressure probe anywhere behind a heat exchanger that is not directly on its rear face (which is impossible to get to on most cars due to shrouds and fans) will give you a reading of system outlet pressure, not exchanger outlet pressure. It is very easy to get wild (and completely wrong) ideas about the direction and amount of flow from these (as I did), so be careful! Even looking back at my previous posts in this series, I was unable to place probes at the heat exchanger rear face and had to settle for measuring static, total, and dynamic pressure behind the fans at the shroud outlet, so those measurements should be taken as rough approximations only.

The second lesson has to do with the number and location of measurements. As we saw earlier in Part 6 and above in the schematics, flow properties across heat exchangers are not uniform (which means any assumption of quasi-one-dimensional flow is not really valid). Consequently, placing a static pressure probe at various locations on a heat exchanger may give different readings. If we can't get a number of static pressure measurements across both faces of a heat exchanger, "mapping" the whole area, we have no way of knowing the average static pressure gradient across it. Any single measurement on either side will not necessarily tell us what the flow through it looks like as a whole.
 
I think it's a lot more useful to measure parameters across the entire cooling system. Outlet conditions (velocity, static pressure, total pressure) at the cooling system duct exit—wherever we define it—are determined by the losses inside the duct. We can see an example of this in the results of a wind tunnel experiment my last year in school. A small model of an aircraft was placed in a 12" x 12" wind tunnel test section with constant area from inlet to outlet, mounted on an arm that allowed pitch rotation and collected measurement data of normal and axial force components (from which lift and drag—which are referenced to the direction of oncoming flow, not the body axes of the arm—could be calculated).

As losses in the duct (here, the test section of the wind tunnel) increase by raising the angle of attack and consequently the drag force on the model, the loss of momentum in the fluid can be observed in the flow velocity as measured by a ring of static pressure taps at the test section entrance and the ambient static pressure in the laboratory:

The copper tube connects all 4 static pressure taps, one on each side of the tunnel.

The losses through the duct reduce outlet flow velocity, and this reduction in outlet velocity propagates upstream. This is imposed by continuity: the mass flow rate into the test section adjusts to match mass flow rate out, which is determined by the losses in the duct—just as Hoerner says.
 
If we consider the cooling system outlet as the fan shroud, with flow exiting into the engine bay, the pressure boundary there does not tell us the static pressure loss through the duct. In fact, "static pressure loss through the cooling system duct" is all but meaningless here since there are area changes throughout the duct (and other ducts you might otherwise be inclined to measure static pressure change across, such as the engine air inlet duct). When we introduce area changes, it is perfectly possible for air to flow against an adverse pressure gradient (low to high static pressure) and in fact this happens over the external and internal surfaces of a car (or other air vehicle, like aircraft) in many locations. For example, here is the measured ΔC_P from inlet to the AC condenser face on my car before cooling system modification:

Positive ΔC_P means an adverse pressure gradient and negative, a favorable pressure gradient.

Because the duct area increases in the direction of flow, there is a strong adverse pressure gradient that also goes up with vehicle speed. But we can see from previous tuft testing that, despite this (and despite the fact that the pressure coefficient at the inlet opening was greater than ambient—be extremely careful here, as most people erroneously think this indicates air flowing forward, out of an inlet), air still flowed into the cooling system as expected:

Let me reiterate this, since it is a pervasive idea amongst enthusiasts: if you measure higher static pressure at a duct inlet than ambient static pressure, it does not mean air is flowing forward and out of the duct inlet. Revisit Part 2 for an explanation.

If you measure more parameters than just static pressure, you can estimate mass flow rate through the cooling system from system loss coefficient (calculated from total pressure loss and average core dynamic pressure), outlet static pressure, and outlet area by the momentum equation:

Multiply this ratio

…to get ṁ, adjusting units as necessary. Find ρ_∞ from the Ideal Gas Law and u_∞ from measured freestream dynamic pressure. Revisit Part 7 and Part 8 to see how these equations are derived and learn how to calculate cooling system drag from these measurements and adjust it as well as mass flow if desired (which, if you've bothered to read all this, I assume your goal is to modify something in your car's cooling). I plan to keep revisiting all these in the future, and you should too. And as always, if you figure out a better solution for probe placement or a better test methodology for enthusiasts with limited budget and equipment, let me know!
 
Conclusion
 
So, there you have it: the final lesson here is that, perhaps disappointingly but also encouragingly, cooling system airflow and duct flow generally are a lot more complicated than most people think but can be characterized through relatively straightforward measurement data. If you've stuck with this whole series, congratulations! Typed up in a Word document (and without images), it comes to just over 45 pages—essentially a chapter in a book. I'm still going back and revising previous posts, as well as the modifications to my cars, so if you revisit each section you might notice changes from time to time. That's the consequence of being human.

Pictured: not where I thought this project would end up. I went into this thinking that typical "eco" modifications, like a grill block, would be the route to lower drag. Doing the opposite reduced total pressure loss through the cooling system, improved diffuser efficiency (drastically), lowered drag, and increased mass flow and cooling capacity.

Rather than just read it, though, I hope you have gone out and measured cooling system pressures on your own car, modified the system, and remeasured to find the changes in performance. The point here is in the doing, after all. As Marx had engraved on his headstone, "The philosophers have merely interpreted the world; the point, however, is to change it."
 
Go break some eggs.

Previous: Measuring and Improving Cooling System Performance – Part 8: Modification