Our Approach to Heat Transfer

    q is heatflux

    hg is the gas side heat transfer coefficient

    Taw is adiabatic wall temperature

    Twg is the temperature on the hot side of the wall

    k is the thermal conductivity of the wall material

    t is the thickness of the wall

    Twc is the temperature on the cool side of the wall

    hc is the cooling side heat transfer coefficient

    H is the overall heat transfer coefficient

    At this point in design, we’ve already discovered what we need our engine to be capable. This feeds us, from fuel properties and engine parameters, conditions inside the engine at steady state, and the allowable temperature of our selected engine material. From this, we can rely on Dr. Bartz’s derivation of Colburns Nusselt Number formula…

where:

Dt is the hydraulic diameter

u is the viscosity

Cp is the specific heat of the gas

Pr is the Prandtl number

(Pc)ns is the combustion stagnation pressure

g is the acceleration caused by gravity

c* is the characteristic velocity

R is the radius of curvature of the nozzle

At is the area at the throat of the engine

A is the throat of the point being examined.

𝞂 is a correction factor that looks like…

where:

    M is the local mach number

    Tc ns is the combustion stagnation temperature

    gamma is the ratio of specific heats

Formulas (3) and (4) describe what is happening on the hot side of the wall. For the coolant side of the wall, we’ve used the Sieder-Tate equation to make our model. This equation is…

Where:

    Nu is Nussalt’s Number

    Re is Reynolds Number

    uw is the viscosity at the cool side of the wall.

    Do note, in our design, we are staying below the critical temperature of the fluid. This allows us to use the Seider-tate equation and avoid some multi-phase flow calculations. There is a more efficient way of ripping heat off the wall with nucleate boiling, but there is added complexity in that which we thought best to avoid.

    These equations above can tell us all we need to understand the general struggle for design. Using a few approximating formulas from Design of Liquid-Propellant Rocket Engines, and fuel data from providers, you can now generate a substantial heat-transfer model for design. Now that the wall isn’t melting, we must consider how the creature is being affected by stress. The most severe point occurs at the throat on the hot side of the wall, where the heat flux is highest. The stress can be found by adding the compressive stress between the cooling passage and combusting gas and the thermal stress. For a coaxial shell cooling jacket approach, Design of Liquid-Propellant Rocket Engines provides this equation.

    Before I get into our design. It’s important to note that the set of equations above is for pure regenerative cooling. Our engine, and others like it, see the most success with a combination of both regenerative cooling and film cooling. Film cooling is where you use some of your fuel to create a film boundary between the combustion chamber wall and the combusting gases. There are no reliable heat transfer models for this. For gas film cooling approaches, one could use the Hatch-Papell correlation. Unfortunately it comes up short under a variety of conditions. There is a decent formula to consider experimentally that I will cover when we get a bit closer to testing and it is more relevant. For us, we approached film cooling as a mechanism to provide the appropriate temperature ratio in Bartz’s correction factor formula. In addition, it should lower the heat flux, giving us an unknown but greater than one factor of safety. When we start to get to testing, we can flesh out the model to more accurately represent what is happening with the film cooling.

    Since we are laying some groundwork for future generations of OTRA members, we decided to go a bit overkill in how we made our model. We started by using some pretty textbook compressible flow approaches to find the gas properties across the combustion chamber. These numbers however will play an important role for finding the stresses across the jacket, as well as developing the appropriate boundary layer for solving the heat transfer problem. Our values for the current model we are analyzing are…

To generate Taw — the adiabatic wall temperature — we need to include an additional recovery factor. To base it entirely off of the combustion chamber, you find that

Where r is a local recovery factor. To quote Huzel and Huang, “[This] factor is a ratio between the frictional temperature increase to the increase caused by adiabatic compression.” Depending on whether you are looking for Taw in a laminar or turbulent region, you can find r based on Prandtl’s Number to the .5 or .33 power respectively. The Mlocal — the local Mach number — is dependent on the area ratio. Due to some ITAR concerns, we won’t be publishing the numbers or schematics for the engine, but do know that our approach had us start with an internal engine geometry to optimize performance.

Now that we know the rules, we can play the design game for the combustion chamber wall c:

Step 1: How to not explode

The first thing we must consider is the stresses. The only term we can’t really play with here is the Radius, as that has already been optimized. E,a, v, & k are all be effected based on material selection. Based on how the combusting gas is behaving, we can discover q via the bartz correlation and from q the thickness of the wall.

The initial design concerns we can pull from this is that

a.) We want the smallest heat flux we can to reduce thermal stress and reduce cooling

requirements

b.) The material we choose must have a good combination of E, a, v, & k to reduce

thermal stress.

c.) The pressure difference between the cooling pressure and the combusting gas must

be minimized safely. This one is dependent on the behavior of the injector. It’s a

good rule of thumb to have a change in pressure across the injector of about 60psi.

Step 2: Figure out the heat flux

The Bartz correlation defines what the gas side heat transfer coefficient will be based on how the combusting gas behaves as well as a temperature ratio. This temperature ratio comes from the required change in temperature for the material of the combustion wall. (The ratio could be considered as (Tglasious point of material * S.F.):(Tcombustion) for all practical purposes. With this, you can generate what your heat flux should be.

The goal here is to keep the heat flux as low as possible while maintaining the appropriate temperature ratio. From formula (1), the smaller the gas side heat transfer coefficient is, the lower the heat flux must be, as the temperature change requirement is static. This provides us with a new knob to play with for design.

Step 3: Determine Thickness of the wall

From formula (1), the thickness is dependent on the thermal conductivity of the material, and the required change in temperature. For us, that change in temperature only required that we bring the temperature down to the critical temperature of the fuel multiplied by a safety factor. If you follow that guide line, you’ll get the ‘perfect’ wall thickness, however it will be thinner than you might be able to manufacture. The other step to consider in this section of the game is manufacturability. If you thicken the wall, you will bring the required cool temp of the wall on the far side down.

The balance to be struck here is between a manufacturable thickness and how hot you predict your coolant temperature to be on the far side of the cooling jacket. For reference, in Modern Engineering for Design of Liquid-Propellant Engines, Huzel and Huang say that the typical total temperature increase for a thrust-chamber design is about 100-400℉.

We approached this by selecting a thickness at the throat of the wall, and then altering the thickness across the nozzle and combustion chamber as a function of heat flux to optimize our wall thickness values.

Step 4: Keepin it cool

The design of the cooling jacket is dependent on a lot of factors. Stress, the prevention of uncontrollable hot zones, and manufacturability are some of the key concerns have here at OTRA. We decided, since we aren’t dealing with a very high pressure rocket engine, that the best approach for us would be running a coaxial shell cooling jacket. Manufacturability here comes to welding flanges, and the gap distance. Based strictly off of formula (5), you can design a jacket that will do the job. We discovered that often it called for shell gaps that required a large change in pressure to power. Some toying with is required.

Much like with the thickness, we based our cooling jacket gap as a variable from the heat flux across the engine. The solutions to the heat transfer model came out as something like this.

Step 5: Safety Check!

Solving for the stresses we found…

This yielded us a safety factor of 1.55! Not the greatest, but certainly exciting to see something workable. We wound up going through and scrapping a variety of different designs and material options based on the results we got this way. We are planning on toying with the variables for a bit longer, but the numbers shown here won’t be too far away from the final numbers.

This is the primary way we are currently tackling design currently of our combustion chamber and jacket here at OTRA. In addition, we are doing some cross comparisons with the Rocket Propulsion Analysis software, as well as digging our teeth into an alternative method for heat transfer proposed by V.M. Ievlev. We are also digging into multiphase flow to see if we can come up with some rough estimates for a film cooling model.

I hope you enjoyed this insight into life here at OTRA!

-David Minar

Engine Development Lead