Heat Transfer and -Capacity

[Tom4K]

Heat Transfer and -Capacity

Heat transfer and heat capacity modeling should be addressed here.
I would like to show some helpful information here, esp. some interesting numbers for nice materials, like the best and
the worst heat transfering materials, typical airospace materials, and so on.

When studying the material, you will see, that *diamonds* are by far the most valuable material in this context...
So the ultimate heatshield would consist of pure diamonds.....

After that a few thoughts about how we could model all this.

I link two pictures of the validation model I made in Matlab/Simulink :

Model Top-View

Simulation Result



Material constants


Specific heat capacity (c_p) [J/(kg K)]
--------------------------------------------------
Hydrogen : 14300
Helium : 5190
*Diamond* : 5000
Water : 4180
Ethanol, Glycerin : 2430
Methane : 2160
Water vapor (@100°C) : 2080
Water ice (@0°C) : 2060
Wood ˜ : 1700
Butane : 1660
Air (dry) : 1010
Aluminium : 896
Concrete : 880
Carbondioxid : 846
Glas : 670 – 840
Argon : 523
Iron : 450 – 550
Steel : 500
Copper : 382
Silver : 235
Mercury : 139
Lead : 129


Melting temperatures (@1.013 bar) [K]
-----------------------------------------------
*Diamond* : > 4500
Tantalum hafnium carbide : 4488
Tungsten : 3695 (Wolfram)
Titanium : 1941 (boiling: 3533)
Iron : 1809
Silicon : 1683
Steel : 1670
Copper : 1357
Gold : 1337.33 (boiling: 3243)
Silver : 1234.93 (boiling: 2483)
Aluminium : 933.48
Magnesium : 923
Lead : 600.61 (boiling: 2022)
PUR(Polyurethane) : 533.65
Sodium : 370.87 (boiling: 1156)
Gallium : 302.9146
Water : 273.15
Mercury : 234.43 (boiling: 629.88)
Nitrogen : 63.15 (boiling: 77.36)
Oxygen : 54.36 (boiling: 90.20)
Hydrogen : 14.01 (boiling: 20.28)
Helium : 0.0001785 (boiling: 4.22)



Typical connection heat transfer coefficients (alpha) [W/(m^2 K)]
-----------------------------------------------------------------------------------
Still Air -> Wall : 8
Stormy Air -> Wall : 290
Still Water -> Tank : 580 - 2300
Slow cool Water -> Pipe : 2300
Fast boiling Water -> Pipe : 12000
Condensing water vapor -> Wall : 6000 - 12000

=> you see, take what you want.


Mass densities [kg/m^3]
-----------------------
Hydrogen : 0.0898
Helium : 0.179
Aerogel : 1.0
Air : 1.2 // At sea level
Tungsten hexafluoride (g) : 12.4 // One of the heaviest known gases
Liquid hydrogen : 70 // At approximately -255 °C
Cork : 240
Lithium : 535 // Least dense metal
Wood : 700
Ice : 916.7 // At temperature < 0 °C
Sodium : 970
Water (fresh) : 1000 // Maximum, occurs at approximately 4 °C
Liquid oxygen : 1141 // At approximately -219 °C
Nylon : 1150
Plastics : 1175 // Approximate;for polypropylene, PETE/PVC
Glycerol : 1261
Sand : 1600 // Between 1600 and 2000
Magnesium : 1740
Silicon : 2330
Concrete : 2400
Glass : 2500
Quartzite : 2600
Granite : 2700
Aluminium : 2700
*Diamond* : 3500
Titanium : 4540
Vanadium : 6100
Zinc : 7000
Chromium : 7200
Tin : 7310
Steel : 7850
Iron : 7870
Copper : 8940
Silver : 10500
Lead : 11340
Mercury : 13546
Uranium : 19100
Tungsten : 19300
Gold : 19320
Platinum : 21450
Iridium : 22420
Osmium : 22570 // Densest natural element on Earth



Modeling heat bodies

- Each heat body consists of two parts: surface zone and inside zone
- Surface and Inside have the same heat capacity C (we can discuss this)
- Surface and Inside have different temperatures Ts and Ti when not in equilibrium
- Surface and Inside are insulated to each other but always connected together
- Parts of the Surface can be connected to different other Surfaces or vacuum
- Each of these connections lead to a different heatflow to the Surface
- Heatflow to the Inside can only occur due to temperature difference to Surface
- Surface connection to vacuum yields to black body radiation as a heatsink
- Surface connection to atmosphere yields to frictional heat flow into the Surface
- Surface connection to a different Surface of another heat body leads to heat exchange between the Surfaces
- Heat transfer is linear and calculated by the means of heat transfer coefficients

<to be continued>

 

[Tom4K]

Discussion

A few words of how to model this kind of stuff:

So this is for us real time, quasi stationary, junkies.

This professional thing with finite elements, CAD-calculation, local heat flow with conductivity and so on...
we do not need this, and, we don't want this !
We do not want to have sleepless nights, because of brain headake and turn-around-times in the minutes region.
You don't want to calculate how much heat flows through that tiny connection points between hot and cold zones and so on.
We don't want to argue about hot spots, melting areas because of local heat... and so on.

No, we want to keep it simple, but realistic. The big picture

So we know, that low temperature tanks, e.g. for liquid hydogen, always are insulated, and it's outer suface temperature is much higher then
the inner wall temperature, and crossing an insulation leads to a dramatic steep temperature drop and so on.

So let us just model it as two zones with constant temperature, and a heat transfer through an infinite small insulation area.

This works well as I can see. And this works well if I think about KSP .

So we have the freedom to choose our heat transfer coefficients like we want, and most of all: that they work.

What we try here is simplifying that heat thing to a body consisting of 2 zones with constant temperature.
This is only realistic if we have two seperate zones, which are softly isolated to each other, in that way, that heat flow inside each zone
is much fast than heat flow between the two zones.

Let's say, we have two of those parts connected together, an "front" and an "rear" part.
The "front" part is bombarded with a heatflow, the rear part is in the shadow of the front part and is connected to vacuum in addition.
You will surely recognize: This could be a capsule protected by a heatshield.

Front part (the heatshield) is denoted with index 1 here, back part with index 0 (the capsule).

The bombardment (from airspeeds of several km/s) shall introduce a heatflow of :

Q1_dot_surface [W] on the heatshields front surface.

It depends on velocity, surface area of shield, air density, viscosity and so on, which is not part of this chapter.

I am adding the units in formulas here, because the quantities can be seen in that way, so it is easier for my eyes to understand what happens.
As I have written before, you can just ignore units if working in the SI system, because everything transforms correctly
until the very last equation in a beautiful manner.


The temperature of the heatshield changes per second this way now :

T1_surface_dot [K/s] = ( ( Q1_dot_surface [W] - Q1_dot_ablative(T1_surface,v_vehicle) ) + (T1_inside - T1_surface)[K] * alpha_1[W/K] ) / C1_surface[J/K]

T1_inside_dot [K/s] = ( (T1_surface - T1_inside)[K] * alpha_1[W/K] ) / C1_inside[J/K]

I would model Q1_dot_ablative(T1_surface,v_vehicle) as depending of surface temperature and vehicle speed... perhaps also ambient pressure...
Place it in a nice table then. Q1_dot_ablative is zero for non-ablative shields... or when it's empty
=> game over if you forget to fire away this hot Bratpfanne (frying pan) as fast as possible.


The temperature of the capsule changes this way now :

T0_surface_dot [K/s] = ( (T1_inside - T0_surface)[K] * alpha_12[W/K] + Q_dot_blackbody(T0_surface) + Q_dot_aircooling(v,p) ) / C0_surface[J/K]

T0_inside_dot [K/s] = ( (T0_surface - T0_inside)[K] * alpha_2[W/K] ) / C1_inside[J/K]

Q_dot_aircooling(v,p) could be again a characteristic map, describing the (negative) heatflow of the capsule cooled by the lower atmosphere by the means of air convectional cooling.

It is important that the capsules surface is connected to the heatshields body, and not to its bombarded surface.
I think, you know what I mean.
Black body radiation we already discussed. It may be negligible here, I haven't checked yet.

In this example, C1_inside is the heat capacity of the inside zone.
The heat capacity describes, how much energy (in Joule) is needed to change the bodies temperature (by one Kelvin).

So we can write :
C1_surface[J/K] = c1_surface[J/K/kg] * m1[kg] / 2
C1_inside[J/K] = c1_inside[J/K/kg] * m1[kg] / 2

With the full heatshield mass m1 = m1_surface + m1_inside .

We can choose 50% of mass to be inside, or heat capacities to be equal, leading to a symetrical solution.

If inside is made of aluminum the specific heat capacity is (see table above) : c1_inside = 896 J / (kg K) .

The "alpha" values are the so called heat transfer coefficients. They depend on material and properties of connection
(you know that from your last CPU you personally inserted, burning in your PC... at least in my case).


A simple estimation for the heat flow between two heat bodies (alpha12, example: aluminum cylinder)

Generally speaking, the alpha12 should be choosen from experimental data.
So here we have a lot of free room the choose parameters we like for our simulation. It remains realistic.
But it is benefit and also a big disadvantage at the same time, because a variable may me hard to find for a nice and realistic behavior.


Rigid connection

If you want to connect two cylinders with a rigid connection (that means they become one body), you can use this formula :

Two cylinders with length L1 and L2 and arbitrary Diameter D with heatflow from front to rear surface only :

alpha = 2*lambda / (L1+L2)

(lambda can be found in the appendix)



Different models for different heat body types

Heat-Body type Heatshield : Has 1 surface and an inside. Surface connects to Air in front and inside connects to protected body
Heat-Body type Capsule : Has 2 surfaces (side, rear) and an inside
Heat-Body type Fuel Tank : Has 3 surfaces (side, front, rear) and an inside connected to fuel

So for a standard fuel tank without heat protection, each surface volume and the inside volume could have the same specific heat capacity.
E.g. you can calculate the heat capacity for the hole cylinder, then divide by two, giving half the the surfaces, half to the inside.
For a cylinder, you can calculate the surface areas, and distribute the heat capacity according to their relative dimensions...




Final words

So the model approach described here is clearly a simplification.
And it is only allowed, if the heatflow is influenced much more by the part connections than by the heatflow inside of the material.

So it is very important, to choose the heat transfer coefficients reasonably.
So there must clearly be an isolation between the heatshield and the capsule, because otherwise,
the hot shield will burn your... paws.



Sources:
Diamonds heat capacity :

diamond

webbook.nist.gov

6 J/(mol K) => 1 mol <=> 12.0107g => 0.4996 J / (g K) = 4996 J/ (kg K)

Thermal conductivity and resistivity - Wikipedia

en.wikipedia.org

https://en.wikipedia.org/wiki/Heat_transfer_coefficient )


<to be continued>

 

[Tom4K]

Appendix


Heat transfer coefficient vs. thermal conductivity

*Caution*: do not mix up heat transfer coefficient (alpha) with thermal conductivity (lambda) !

Conductivity is the ability of a material to transport heat inside
("Fouriers law" : Q_dot [W] = lambda[W/K/m] * Delta_T [K] * A_surface [m^2] / waylength_of_heattransport [m] )

Heat transfer is the ability of transporting heat from one material to another and describes the separation area between two materials.

As an example:
Silver has a higher heat conductivity than gold, but gold on gold contact has a higher heat transfer coefficient than silver on silver.


Heat conductivity table

Heat conductivity (lambda) [W/(K m)]

*Diamond* : 1000 - 2500
Silver : 429
Copper : 401
Gold : 314
Aluminium : 236
Aluminium alloys : 75-235
Magnesium : 170
Silicon : 163
Magnesium : 156
Sodium : 133
Zinc : 110
Brass : 120
Iron : 80.2
Platinum : 71
Tin : 67
Lead : 35
Titanium : 22
Steel : 15 - 58
Stainless steel : 15 - 25
Mercury : 8.3
Glas : 0.76
Sand : 0.58
Concrete : 0.8 - 2.1
Rubber : 0.16
Wood : 0.09 - 0.19
Cork : 0.035 - 0.046
Wool : 0.035 - 0.045
Plastic (PUR) : 0.021 - 0.035
Aerogel : 0.017 - 0.021
Vacuum insulation: 0.004 - 0.006
Vacuum 0

 

[Kiwi Shark]

But really, thanks for keeping all of these refs and numbers coming. I wonder just how in depth the thermodynamics will get. I'd be content with a fairly simple flux in flux out model but accounting for the metallurgical composition of my atmospheric landers would be bananas in its own right. Aerogel is cool though, we can have some aerogel as a treat.

 

[averageksp]

But really, thanks for keeping all of these refs and numbers coming. I wonder just how in depth the thermodynamics will get. I'd be content with a fairly simple flux in flux out model but accounting for the metallurgical composition of my atmospheric landers would be bananas in its own right. Aerogel is cool though, we can have some aerogel as a treat.

Aerogel would be cool, I agree