For the shrink that fits - Alice's adventures in tolerance land

More thoughts about the group body casting.

Having stared at the casting after looking directly at sun for a few minutes I am now pretty sure that it is an assembly of two parts: the bulk plus a small internal sleeve that holds the dispersion screen. Putting these two parts together could be done in a number of ways, but to avoid the possible consequences of heating the casting to soldering or brazing temperatures, I think that a shrink fit is probably the best bet.

From the engineeringtoolbox:

Shrink-fits are assembled by heating them to temperatures where the expansion exceeds the interference. Required temperature heating can be calculated as

dt = δ / α d_i         (1)

where

dt = temperature heating (^oC, ^oF)

δ = diametric interference (mm, in)

α = coefficient of linear expansion (m/m^oK, in/in^oF)

d_i = initial diameter of hole before expansion (mm, in)

Diametric interference can be calculated as

δ = dt α d_i         (2)

So assuming the sleeve is at room temperature and the body is heated in boiling water:

Nominal hole diameter d_{i =} 58mm
Temperature heating dt = 80^oC
Coefficient of linear expansion α for C23000 brass = 18.7x10^-6 (°C)^-1
Coefficient of linear expansion α for C87850 brass = 18.5x10-6 PER °C (20-100 C)

Therefore:

for C23000
δ (80^oC) = 80^oC x 0.0000187 (°C)^-1 x 58mm = 0.086768mm

for C87850
δ = 0.08584mm

...or more or less the same.


So over-sizing the sleeve diameter by 0.08mm (which is pretty close to 0.003") will allow it to be shrink fit into place with minimal force.

And after drinking the potion mark "drink me" Alice can now go down the rabbit hole to tolerance land. 

Interference H7/s6 S7/h6  - Medium drive fit for ordinary steel parts or shrink fits on light sections, the tightest fit usable with cast iron. 

The tolerance for the sleeve and the hole diameters is important as the two add together. So a symmetrical tolerance of +/- t will possibly result in a total tolerance of 2t.

So allowing for a minimum of 0.05 interference which should still provide a 'medium drive' shrink fit, the diameter of the hole should be 58 +0.015/-0 and the sleeve diameter should be 58.08 +0/-0.015

So in the worst case scenario, the hole is 58.015 and the sleeve is 58.065 which still results in 0.05 interference.

Brazing

From copper.org

http://www.copper.org/applications/plumbing/techcorner/soldering_brazing_explained.html

A major difference between brazed and soldered joints is in the amount of joint overlap or fill necessary to develop full strength of the joint. In a brazed joint, full insertion of the tube to the back of the fitting cup is still highly recommended; however, complete fill of this joint space throughout this entire length is not necessary to achieve full joint strength. According to the American Welding Society (AWS), it is suggested that the brazing filler metal penetrate the capillary space at least three times the thickness of the thinnest component being joined, which is usually the tube. This is known in the industry as the AWS 3-T Rule.

Because of the increased strength of brazing alloys, even this rather small amount of fill penetration will result in a properly fabricated brazed joint stronger than the tube and or fitting themselves. However, unlike a solder joint, where the cap or fillet provides minimal additional strength, a brazed joint should be fabricated so that a well-developed fillet or "cap" of filler metal is provided between the tube and fitting on the face of the fitting. This fillet, or cap as it is often referred to in the trade, permits the stresses developed within the joint (by thermal expansion, pressure or other cyclic reactions such as vibration or thermal fatigue) to be distributed along the face of the fillet. In a brazed joint fabricated without the well-developed concave fillet, all stress would be concentrated at the sharp point of contact between the tube, braze alloy (filler metal), and the fitting, possibly leading to development of a stress fracture in the tube at that point. Creation of the fillet when fabricating the brazed joint greatly minimizes this possibility.

______________

Besides the strength of the filler metal in the joint, the overall strength of the joint or assembly (tube, fitting and joint) following the joining operation must also be considered when choosing whether to use soldered or brazed joints. As discussed, by definition the temperature that defines the difference between soldering and brazing of copper is approximately 840°F/449°C. This temperature is much more important than just an arbitrary definitional threshold. It is important because 700°F/371°C is the temperature at which copper begins to anneal, or be changed from hard temper (rigid) to annealed temper (soft). With this change in temper comes an inherent loss in strength - hard temper copper is stronger than annealed temper copper. The overall amount of annealing that occurs, and thus strength that is lost, is determined by the temperature and the time the material spends at that temperature. The higher the temperature, the less time it takes to change from hard temper to soft temper.

Since brazing temperatures must exceed the melting point of the brazing alloys, between 1,150°F/621°C and 1,550°F/843°C, the process of making a brazed joint causes the base metals to anneal or soften, resulting in a reduction in the overall strength of the assembly. While a brazed joint is demonstrably stronger than a solder joint, the Rated Internal Working Pressure, that is the 24/7 allowable working pressure of the system, is lower for annealed tube (see Copper Tube Handbook, Tables 3a through 3e).

Boiler research

The following is based on two avenues of research, one from the model steam boiler community, the other from the copper industry - the latter is perhaps more pertinent finally. 


The massively skilled Don R. Giandomenico, aka rcdon, built a model steam boiler based on instructions from "Model Boilers & Boilermaking” by K.N. Harris (1967). 



To make the boiler shell (outer cylinder of the pressure vessel) I am using a piece of solid drawn (seamless) type “L” copper tubing (seen below). This tubing is 6.125” in OD and has a wall thickness of .140”. According to the Harris book (page 31) it is satisfactory to have a seamless boiler shell at 5.845” ID x 0.094” wall operating at a working pressure of 100 PSI. This is calculated by multiplying the the working pressure (P) by the internal diameter of the shell in inches (D). You then divide this value by two times the derated tensile strength of the material being used (t). T is equal to the thickness of the boiler shell in inches:

                                      (P X D) ÷ (2t)= T

       The normal tensile strength of copper is around 25,000 pounds per square inch which in this case is derated by the factor of 8 times for a safety margin (3,125 X 8 = 25,000 PSI). This means that the boiler can handle 8 times the stress that would be applied to it under normal operating conditions. Knowing these values I can then plug them into the equation:

     (100 x 5.845) ÷ (2 X 3125) = T ................... 584.5 ÷ 6250 = 0.094” thick

       The boiler shell I am going to use would effectively handle a working pressure of 150 PSI. In fact, the manufacturers listed burst pressure of this type of tubing is at around 2,690 PSI !!!! I will have no trouble trusting this tubing at 80 PSI. Of course copper starts to lose it’s strength at elevated temperatures so it is important to keep it within it’s operating temperature.

The Aurora is rated to 1.5 atmospheres (or 1.5 bar) with an operating pressure of between 0.8 and 1 atmosphere. 

From Harris p.28:

In all boilers it is usual to allow a comparatively high factor of safety, that is to say that if a boiler is required to work at 100 lb. per sq. in., its plates, stays, etc., are calculated on a basis of its bursting at anything from six to ten times this pressure. A good all round factor for model work is eight and that will be the one adopted in what follows.

p.31

This brings us to the strength of boiler shells. In calculations relating to the strength of a boiler shell; so far as the plate is concerned it makes no difference whether it is a rolled plate with a longitudinal joint or a solid drawn tube, that is to say so far as the stressing of the shell is concerned. What does have to be taken into account, however, is the strength of the longitudinal joint, a solid drawn tube having no joint is the strongest form.

The next best arrangement (model practice) is probably one with an inside butt strap riveted and hard soldered, the rivets being only to hold the whole issue together during the brazing or silver soldering operation. If all the contacting surfaces of butt -strap and boiler shell are clean and well fluxed and a proper job is made of the soldering, which entails on the one Land plenty of heat and on the other the avoidance of over-heating, the value of the joint should be about 95 per cent. of that of a solid drawn tube. A joint made with a double butt -strap, see sketches of of joints and double or treble riveted, which means either two or three rows of rivets each side of the joint, should have a strength equal to about 80 per cent. of that of a solid tube.

A double riveted lap joint will have around 75 per cent. and a single riveted lap joint 55 per cent of the strength of a solid drawn tube, so that it is very obvious that it pays handsomely to use the butt-strap plus brazing technique in making longitudinal joints. Best of all, of course, is
to use a solid drawn tube.

As I am not interested in riveting, the approximately 7" diameter boiler must be butt-strap and (hard) brazed construction as 7" nominal tube is not to be had. t, the derated tensile strength for copper is 25,000psi / 8 = 3125psi. The construction technique further derates the thickness by a factor of 100/95%.

T = (P x D)/2t x 100/95

T - thickness of the shell
P - working pressure in psi
D - internal diameter of the shell in inches 
t - derated tensile strength of the material

The working pressure is 1 bar or 14.7 psi so:

T = (14.7psi  x 7")/(2 x 3125psi) x 100/95

T = 0.017in which is around 26 gauge or not very thick at all...

Too thin in fact to make it easy to braze the fittings in place.


_______

The Copper tube handbook contains all the tables, formulas and recommendations discussed here.

Further reading on the copper.org site suggests that brazing or welding will anneal the copper considerably lowering its tensile strength. The brazing temperature threshold is 840F (450C) with most brazing alloys considerably higher than that (~1200F (650C)). Copper begins to anneal  at 700F (370C). The hotter temperature and the longer the heat is applied, the quicker the annealing takes place.

Since brazing temperatures must exceed the melting point of the brazing alloys, between 1,150°F/621°C and 1,550°F/843°C, the process of making a brazed joint causes the base metals to anneal or soften, resulting in a reduction in the overall strength of the assembly. 

Consequently, copper.org offers tables of the maximum safe working temperatures and pressures for various diameters of annealed tube.

Working pressure for the machine is 1 bar gauge (which is 2 bar absolute pressure i.e. 1 atmosphere above atmospheric pressure). At 2 bar, water boils at 120C (250F). So from tables 14.3 A, B and C:

Calculated Rated Internal Working Pressures for Annealed Copper Tube

Nominal Tube type and diameter inches	S = 4800psi 250F	Wall thickness inches	Inside diameter inches
6 K	277	0.192	5.741
8 K	295	0.271	7.583
6 L	201	0.14	5.845
8 L	216	0.2	7.725
6 M	175	0.122	5.881
8 M	183	0.17	7.785

Based on maximum allowable stress in tension (psi) for the indicated temperatures (°F).

Raising the temperature to 300F results in a lowering the safe working pressures by around 6psi.

An explanation of the formula used to calculate these values is here. 

Furthermore:

In designing a system, joint ratings must also be considered, because the lower of the two ratings (tube or joint) will govern the installation. Most tubing systems are joined by soldering or brazing. Rated internal working pressures for such joints are shown in Table 14.4a. These ratings are for all types of tube with standard solder joint pressure fittings and DWV fittings. In soldered tubing systems, the rated strength of the joint often governs design.

From table 14.4a - the working pressure rating for brazed joints on all diameters of tube for use with saturated steam is 120psi gauge. Soldered joints, with all alloys and for all diameters and for use with saturated steam are limited to 15psi gauge.

The very conservative working pressure ratings give added assurance that pressurized systems will operate successfully for long periods of time. The much higher burst pressures measured in tests indicate that tubes are well able to withstand unpredictable pressure surges that may occur during the long service life of the system. Similar conservative principles were applied in arriving at the working pressures for brazed and soldered joints. The allowable stresses for the soldered joints assure joint integrity under full rated load for extended periods of time. Short-term strength and burst pressures for soldered joints are many times higher. In addition, safety margins were factored into calculating the joint strengths.

So what are the conclusions to be drawn from this? The numbers from copper.org are far more conservative than those from Harris. However, the desired working pressure of 1 bar (14.7 psi) is exactly on the money for soldered systems! My guess, given that espresso machines are low-pressure steam systems, is that the boiler is soft soldered!

Once again copper.org gets the last word:

Temperature and pressure are directly proportional for steam. As the pressure in the system is increased, the temperature increases accordingly. Saturated steam, a condition where steam contains as much water as it can and still be a vapor, at 15 psig has an absolute pressure at sea level of 29.7 psia (pounds per square inch absolute). At this pressure it would have a corresponding temperature of approximately 250°F which is the maximum recommended temperature for soldered joints as shown in Table 4 of the Copper Tube Handbook. Therefore, rather than the allowable pressure of the soldered joints controlling the rating, the allowable temperature is the controlling factor, leading to the rating of 15 psig regardless of the solder alloy used.

...

As with any piping system, the pressure rating of the system is controlled by the lowest allowable pressure of the tube, fitting, joint or joining material. For steam systems constructed using copper tube of Types K or L, the maximum allowable pressure at which the system could be designed would be 120 psig. As shown in Tables 3a and 3b of the Copper Tube Handbook, the lowest maximum operating pressure for Type L copper tube is 127 psig (corresponds to 12-inch nominal Type L tube in annealed form). Since this is more than the allowable pressure for the brazed joint, the 120 psig allowable for the joint is the controlling factor, regardless of the fact that smaller diameter tubes have higher allowable pressures. However, to use copper tube and fittings in a steam system at this pressure the joints must be brazed.


As long as these temperature and pressure limits are met, copper tube and fittings can be used in both high- and low-pressure steam systems. The system must still be designed and installed to meet the requirements of all applicable local, state and federal construction and safety codes for steam applications.