Calculations for 5AT Advanced Steam Locomotive

Introduction: Funding was obtained at the start of the project for the undertaking of the Fundamental Design Calculations (FDCs) for the 5AT locomotive. This task has now been completed by David Wardale (as of November 2004) after some two and a half years of work.  It is instructive to read Dave Wardale's summary of his work which are presented on a separate page.

The Fundamental Design Calculations cover the basic engineering design for the locomotive, and form the technical foundation from which the detailed design and manufacturing drawings etc will be undertaken. The FDCs are extremely detailed and in the case of the most critical items such as the reciprocating components, they extend almost to the level of "detail design".

The introductory "General Calculations" which define the performance criteria upon which the remaining FDCs are based, are reproduced on this website (see below). Several notes are included on this page, some describing criteria or suppositions upon which individual FDCs have been based, others summarizing conclusions that have been derived from the calculations.

Summary of FDC Calculations: A summary of completed calculations is tabulated below:

 Item No


No of Pages*

Lines of Calcs


General Calculations

FDC 1.1
Determination of Target Power & Tractive Effort – Speed Characteristics


Latest update 26 Aug 2002; Posted on website.

FDC 1.2
Determination of Target Load – Speed - Gradient Curves


Latest update 26 Aug 2002; Posted on website.

FDC 1.3b
Preliminary Basic Calculations amendment (b)


Latest update 22 Jan 2007; Posted on website.

FDC 1.3.F.Rev2
Preliminary Basic Calculations (final version) Revision 2


Latest update 21 Jan 2007; Posted on website.

FDC 1.4
Tractive Effort Diagrams


Latest update 3 Jul 2004; Posted on website

Reciprocating Components of the Engine and Drive Gear

FDC 2.1
Pistons, Rings Rods and Tail Rods


Latest update 14 Apr 2003

FDC 2.2
Crosshead and Slidebars


Latest update 3 Jul 2003

FDC 2.3
Connecting Rods


Latest update 16 Mar 2003

Crankpins, Coupling Rods, Driving & Coupled Axles, and Crankpin & Axle Roller Bearings


Latest update 3 Jul 2004

Piston Valves, Valve Rings, Valve Spindles and Packings (see note below)


Latest update 5 Jul 2003

Valve Gear


Completed 23 June 2003

Cylinders and Cylinder Liners


Latest update 3 Jul 2004

Valve Liner Cooling Steam Calculations


Completed 13 Mar 2004

 FDC 8
Wheel Balancing (see note below)


Latest update 30 Apr 2004

Feedwater Heating (see note below)


Last update 22 Aug 2003

FDC 10
Combustion Air Heating


Completed 19 Aug 2003

 FDC 11

FDC 11.1
Boiler Strength (see note below)


Completed 28 Feb 2004

FDC 11.2
Boiler Combustion System


Completed 28 Nov 2003

FDC 11.3
Combustion Gas + Steam Flow and Heat Transfer


Completed 28 Feb 2004

FDC 12
Exhaust System (see note below)


Completed 12 Apr 2004

 FDC 13
Spring Rigging (see note below)


Last updated 2 Aug 2004

FDC 14


Completed 4 July 2004

FDC 15

Completed 11 Aug 2004

FDC 16
Leading Bogie + Engine Stability


Completed 6 Nov 2004

FDC 17
Specification of Proprietary Equipment 


Completed 20 Oct 2004

 FDC 18
Performance Predictions (see notes below)


Completed 22 Feb 2005



[* page totals exclude illustrations but include listings of references.]

Notes on FDC4 - Piston Valves, Valve Rings, Valve Spindles and Packings:

The following notes by David Wardale precede the calculations for the valves, and are reproduced here for the benefit of those who are interested:

For a high degree of internal streamlining the largest practical piston valves are necessary. The largest valve diameter in common use was 14 inches (355.6 mm) on American locomotives with large cylinders, and this figure is used for the 5AT, rounded down to 350 mm on account of the 5AT's much smaller cylinders. To reduce cylinder clearance volume, which would otherwise be excessive with such small cylinders, a single 350 mm dia. valve is substituted by two 175 mm dia. valves per cylinder, giving the same port openings as the larger single valve. This arrangement has the general advantage that whilst port openings are proportional to the valve diameter, the steam pressure forces on the valve heads and stem are proportional to the square of the diameter, therefore such forces per valve are reduced by a factor of 4 when the valve diameter is halved, allowing twin valves to be correspondingly lighter.

The design is based on that for the Chinese Railways modified QJ Class, which was a development of that successfully used on the SAR 26 Class No. 3450, which gave good thermal performance and very low wear, and which was in turn based on principles laid down by Porta in his (unpublished) 1975 paper "The Mechanical Design of Piston Valves". The principal features of the piston valves and spindles are as follows:

  1. Extreme lightness for minimum bearing pressure and hence low wear, and minimum inertia load on the valve gear.
  2. Guarantee of near-to-perfect steam tightness by elimination of the usual valve tail rod, thereby allowing the valve ring retainers to bear on the valve liners and cover the ring gaps (the nominal ring retainer - liner bearing pressure is considerably lower than that between the tail rod and its bush, e.g. ~ 50% lower in the case of the modified QJ). Even bearing of the valve on the liner is made possible by having the valve spindles pin-jointed at both ends.
  3. Multiple narrow rings, minimising the sealing duty of any individual ring, reducing ring-liner friction and wear, and enabling steam tightness to be maintained for extended periods of service without inspections or maintenance.
  4. Ring-controlled events, with thin tapered inlet edge lands for optimum inlet flow coefficient.
  5. Use of (some) bronze valve rings which deposit bronze on the liner surfaces to reduce wear[6].
  6. Ring retainers are bolted on, for ease of renewal.
  7. Small diameter stem joining valve heads, which minimises loss of steam chest volume (connecting tube on SAR loco. no. 3450 reduces steam chest volume by ~ 15%).
  8. Long steam lap and provision of exhaust lap. The resultant length of the ring retainers favours the fitting of multiple rings whilst maintaining lands of adequate width for low stress and low valve - liner bearing pressure.
  9. Exhaust diffusers are fitted to allow exhaust steam to sweep and cool the ring retainers. This function is aided by all features which allow short cut-off to be used, giving high expansion ratio in the cylinders and consequently moderate exhaust steam temperature.
  10. Insulated valve heads, to inhibit heat transfer between live and exhaust steam across the heads.

In contrast, the BR 5MT piston valves are heavier, do not seal 100%, have too few rings, have a poor inlet flow coefficient at small port openings resulting in large indicator diagram triangular losses and poor breathing at low cut-offs. They do not have the whole of the ring retainer cooled by exhaust steam; they have excessive heat transfer across the valve heads, and they have insufficient lap.

Notes on FDC 8 - Wheel Balancing

The following important note by David Wardale concludes the calculations for the balancing of the reciprocating parts of the locomotive, and are reproduced here for the benefit of those who are interested:

Summary: The 5AT locomotive can be satisfactorily balanced for 200 km/h operation by all criteria given in these calculations (items [39] / [44], [46] / [47], {[50] & [61] - [65]} and [143] / [144]. The key to this is the utmost lightness of the reciprocating parts.

For 200 km/h the maximum permissible dynamic augment ([61] - [65]) limits the amount of the reciprocating mass which can be balanced to some 19.7% ([101]), this fraction being quite small even for the lightweight components concerned. Conversely, according to the criteria of [39] / [44] and [46] / [47] the reciprocating parts need no balancing at all. However the present recommendation is that the reciprocating balance be set at [100] - [101], which will give the least unbalanced forces on the locomotive's structure at a dynamic augment no greater than that of the present BR 5MT at speed [51]. If a lower maximum speed, V, is mandated than 200 km/h, the reciprocating balance can be increased accordingly, in the ratio of (200 / V)2.

It should be noted that these calculations have mostly been based on 'worst case' conditions, e.g. design speed (200 km/h) which is greater than the maximum continuous operating speed (180 km/h), an engine-tender mass [32] which is less than the minimum which will occur in practice, due to the necessity of always running with a reserve of fuel and water in the tender, and so on. Therefore, even under the most extreme conditions likely to be found in service, the unbalanced forces acting on the locomotive, the dynamic augments on the track, and the amplitudes of vibrations will all be less than given by these calculations, i.e. the riding of the locomotive will be smoother and its impact on the track less severe.

Notes on FDC 9 - Feedwater Heating:

The following notes by David Wardale precede the calculations for feedwater heating, and are reproduced here for the benefit of those who are interested:

(9.1) The overall feedwater heating system on the 5AT is as follows:

  1. A 'hot well' and auxiliary heater in the tender. The hot well is a compartment in the tender water tank, at the front end of the well between the bogies if this is fitted, otherwise at the front of the body of the tank, in free communication with the rest of the tank and from which the feed pipe to the boiler feedwater pump is run, see Fig. 9.1. Exhaust steam condensate from the feedwater heater is piped back to the tender and mixes with tender water in the auxiliary heater at the entrance to the hot well[1]. By this means condensate is effectively mixed with the boiler feed when the feed pump is working and the hot well is continually replenished by tender water passing through the auxiliary heater, but if the engine steams for short periods without the pump operating and uncondensed exhaust steam reaches the tender it avoids undue rise in the water temperature in the hot well. Any exhaust steam from auxiliaries which is similarly recovered (see [1.3.(134]) may be piped back directly to the hot well. By mixing tender water and condensate / exhaust steam immediately upstream of the feed pipe, recovery of heat energy in the condensate / exhaust steam is accomplished together with the recovery of water.
  2. A 'closed' or 'surface' type (shell & tube) heat exchanger ('heater') situated between the feedwater pump and the boiler: feedwater under pressure is pumped through the tubes and is heated by exhaust steam condensing on the outside of the tubes in the heater shell.
  3. A Chapelon-type economizer at the front of the boiler barrel.[2] This is a partition at the front of the barrel formed by an intermediate tubeplate and in free communication (by an overflow hole(s)) with the rest of the boiler. This free communication means zero pressure difference between the two and there may therefore be no need for pressure-tight joints where the boiler tubes pass through the intermediate tubeplate a good fit of the tubes in the holes to stop gross flow of water between the two sections may suffice, e.g., for parallel tubes, to be achieved by drilling all tubeplates to the same template and aligning them up with some dummy tubes in place when welding to the barrel. The boiler clack valves are situated in the economizer section, all incoming feedwater entering the boiler at the front where the combustion gasses are at their lowest temperature, and therefore acting to give the highest temperature difference, and consequently highest heat transfer, in this part of the boiler, improving its absorption efficiency.

(9.2) The importance of feedwater heating in general, and the advantages of the surface type heater over mixing types (which include exhaust steam injectors) are given in Ref. [3]. To summarize, feedwater heating becomes ever more essential as the power : weight ratio of steam locomotives is pushed towards its limit, as is the case with the 5AT, and obtaining the highest feedwater temperature entirely from exhaust steam heat mandates the surface type heater.

Notes on FDC 11 - Boiler Strength Calculations:

The following note appears towards the end of the boiler strength calculations:

The 5AT boiler will therefore be approximately 1,4 metric tons lighter than that of the BR 5MT using either type A or type E superheater elements. This reduction in mass, despite an increase in pressure from 1551 kPa to 2130 kPa, is due primarily to the welded construction eliminating butt straps and all lap joints, a smaller volume of water in the barrel due to a larger tube bundle, thinner inner firebox plates due to the use of steel rather than copper, and to the use of a U-section foundation ring instead of the solid type.

Notes on FDC 12 - Exhaust System

The following note by David Wardale introduces the calculations for the exhaust system:

The exhaust system, dynamically connecting the boiler and cylinders, is thermodynamically the heart of the locomotive[1] and must therefore be as good as possible, within practical limitations. That the exhaust entrains sufficient combustion air to sustain the combustion rate necessary to match the steam demand throughout the boiler's evaporative range is a cardinal point for good performance from any steam locomotive, and that it does this with the minimum of exhaust steam energy is the key to optimum performance[2]. This point is especially important on the 5AT as the locomotive is to operate mostly at high speed with full throttle and low cut-off, giving high heat conversion to mechanical work in the cylinders and therefore limiting the amount of energy available for draughting work in the exhaust steam (it is common for locomotives to steam adequately at long to medium cut-offs but not at short, for this reason), and this is compounded on the 5AT by the use of piston valves with exhaust lap, delaying release.

The following note on the subject of triple exhaust is also instructive:

Triple exhaust. The number of chimneys is a compromise between maximising the length : diameter ratio of each individual chimney in order to increase ejector efficiency, which multiple chimneys achieve by reducing individual chimney diameter within the available height restriction, and minimising complication within the smokebox. If therefore a triple exhaust gave sufficient increase in blast nozzle tip area, its complication could be justified. The available length [40] would allow triple chimneys 'in line' to be fitted for oil firing only (i.e. no self-cleaning plates or spark arrestor, and allowing for the rearmost part of the back chimney bell mouth and diffuser being set behind the front of the superheater header). The foregoing calculations have therefore been exactly repeated for a triple exhaust, the total primary, secondary and mixture mass flow rates being divided by 3 to get the figures per chimney, and the preliminary data being based on the optimum values calculated for the double exhaust. Note that as the diffuser outlet diameter is smaller for each chimney in a triple exhaust, and also because the chimneys can be set lower down (by item [24]), their length is > [95], being ~1 800 mm. However the calculated total blast nozzle tip area ~164 cm2, which is only some 2% (than) for a double Lempor. This small increase is insufficient to justify the extra complexity of a triple exhaust, which is therefore not considered further, a double exhaust being sufficient for the prevailing conditions.

Notes on FDC 13 - Spring Rigging

The following notes by David Wardale introduce the calculations for the locomotive's suspension:

[Item 3]: Since the steam era there have been changes made to railway vehicle suspension. However the steam locomotive is different in a number of ways from other railway vehicles, e.g. (i) in its chassis layout, with coupled axles running in the mainframe rather than in bogies, (ii) being subject to periodically applied vertical loads due to connecting rod angularity and the rotating masses for balancing the reciprocating parts, and (iii) in its comparatively high centre of gravity and resultant rolling tendency, which requires relatively hard springing. Given this, it is considered that the 5AT project is not the place to experiment with alternative spring arrangements, rather the type of spring and spring rigging that have proved themselves suitable for steam locomotives during millions of km of safe running and at speeds > 100 mph (160 km/h) should be retained and optimised.

{Item 2]: The 5AT will have a 3-point suspension with compensated springing for the 3 coupled axles. The 3 suspension points are the bogie centre and the two centres of support of the coupled wheel springing, one at each side of the locomotive. The mass transfer to the bogie is to be at the bogie pivot, i.e. nominally at the locomotive's lateral centre, rather than by mass transfer pads at each side of the bogie as on the BR 5MT[1], i.e. the bogie gives zero lateral support to the locomotive. The coupled wheel springs on each side of the locomotive are not cross-compensated with each other, therefore they provide (all) the lateral support. This 3-point compensated suspension system is that almost universally used with steam traction: compared to the uncompensated springing in general use in the UK, to which it is inherently superior, it gives more constant axle loads with track irregularities and consequently better riding and traction qualities and reduced spring peak dynamic loads and stresses. It should be pointed out that the spring system for a 4-6-0 is a very simple example of the 3-point principle, which all compensated spring rigging, even on large multi-wheeled engines, seeks to achieve[2].

Notes on FDC 18 - Performance Predictions

In this retrospective set of calculations, Wardale (with the assistance of Dr David Pawson) uses two software packages, "Perform" and "Perwal" written by Professor W. Hall to produce performance predictions for the 5AT and compares the results with his own hand calculations. Wardale introduces the calculations with a cautious qualification:

"As there is no transparency in the computer programs, their validity cannot be directly verified. They require the use of data, such as discharge coefficients, that has to be estimated, and however good the theory behind the programs, the results are clearly only as accurate as the estimation of this data. Good correlation between performance figures obtained from these programs and test results for B.R. locomotives is said to exist, depending on the values used for the various coefficients (i.e. the coefficients have been chosen such that there is agreement between calculation and test). This would point to an indirect verification of the programs' validity when applied to First Generation Steam (FGS). However when using them to predict performance where there are no test results available for comparison, as in the present case, there is no way of confirming that any estimated input data is correct.

Although with the reservation given in item 3 above the programs appear applicable to FGS, this does not necessarily guarantee applicability to SGS such as the 5AT, where engine design shows a number of advances over that of FGS designed to improve steam flow and reduce heat transfer, i.e. to make the power generation process approach closer to the isentropic ideal. Such advances are, however, at least partly accounted for in Perwal by the input data (in respect of such items as discharge coefficients, expansion and compression indices, valve motion and cylinder cover temperature). Steam flow and heat transfer processes in an engine are complex, and there must be doubt that any analytical approach is better than an approximation. However as the calculations concerned are not design calculations, which of necessity must be accurate, a degree of uncertainty can be accepted, i.e. the present work can be taken as giving at least a reasonable guide to the expected 5AT performance. It also follows that any error will tend towards under-estimating performance of SGS, i.e. the 5AT should perform as well as or better than Perwal predicts."

The concluding lines of calculation include the following non-consecutive paragraphs:

Comparing the 5AT target Maximum Indicated Power-Speed curve from the data of FDC 1.1, which was estimated from the performance recorded by SAR 26 Class No. 3450 shows that the two curves agree well above about 100 km/h, where the maximum difference is only some 3%.  Below 100 km/h the 5AT performance predicted by Perwal is significantly better than the target curve, being some 15% higher at 70 km/h and 33% at 40 km/h. ..... To summarise: Perwal supports the previous estimation of the 5AT maximum power above about 100 km/h and predicts significantly higher power below this speed.

The difference in maximum drawbar power predicted by the two methods is within the accuracy of the calculations.

The highest cylinder thermal efficiency according to Perwal is at the highest speed (200 km/h) and lowest cut-off (10%) of the ranges analysed. This is attributable to the good cylinder internal streamlining and high superheat.

The indicator diagrams given by Perwal for each condition analysed are uniformly good, 'fat' diagrams being produced even for the extreme condition of 200 km/h and 10% cut-off. The diagrams are extremely regular, with little difference between front and back of the cylinders. Also as speed is increased at constant cut-off, successive indicator diagrams are drawn one over the other and show relatively little diminution in area with increasing speed. All this is testament to good internal streamlining, which not only gives good 'breathing' at high speed but also determines the lowest speed at which maximum evaporation (i.e. the full capacity built into the boiler) can be used. For the 5AT Perwal predicts this to be ˜ 36 km/h in full forward gear, at which the indicated tractive effort (155,3 kN) is 99% of the nominal starting tractive effort (157,0 kN). Because maximum indicated tractive effort is effectively constant below 36 km/h, the indicated power is approximately proportional to speed, making the power - speed characteristic a straight line below 36 km/h.

The indicator diagrams and 'more detail' diagrams show no evidence of compression loops at any speed and cut-off (which, if present, might have been attributable to the long exhaust lap). The 'more detail' diagrams show a little too much lead at short cut-offs and low speeds, showing as the cylinder pressure rising to steam chest pressure just before dead centre. At all likely running Cut-off-Speed combinations the lead achieves steam chest pressure in the cylinders at exactly dead centre, i.e. is perfect. No evidence of lack of lead, showing as a triangular loss on the indicator diagram near dead centre, is evident at any combination of speed and cut-off within the locomotive's rated maximum cylinder steam flow...... All diagrams do show the admission line starting at steam chest pressure at dead centre. From the above it is considered that the lead chosen (which is fixed for all engine working conditions) is correct.

It is suggested that little, if any, improvement in the diagrams would be achieved by using poppet valve gear.

As Webmaster, I hope I will be excused for taking the liberty of copying a line from the letter from Wardale which accompanied these calculations.  I think it makes a fitting epilogue to the FDCs by providing an glimpse of Wardale's motivation in undertaking the huge task that these calculations represent:

"Perhaps no-one can imagine as well as I the experience that the 5AT would give as it accelerates at full power (according to Fig. 2. above) from low to high speed. In my own mind I can see it, and hear it. The 'stack talk' would be out of this world, and that's what it is all about. Forget about economics and efficiency.  How would you apply these to the Mona Lisa or Shakespeare or Salisbury Cathedral?"

"General Calculations" published on this website:

The General Calculations for the 5AT are made available on this site in PDF (Adobe Acrobat) format and can be downloaded from this page. If you don't have Acrobat Reader software, please click the Acrobat icon (at right) to download it free directly from Adobe's website. At present, it intended only to present the General Calculations for the 5AT locomotive on this website.

These "General Calculations" form the basis of the Fundamental Design Calculations for the locomotive which are currently being worked on by David Wardale. All calculations for the FDCs will be presented in tabular format similar to that used for the General Calculations, and will follow a logical stepped sequence which has been formulated by Dante Porta over the period of his career - indeed funding for these calculations has been provided specifically for the purpose of recording Porta's design methodology for posterity even if the 5AT locomotive never eventuates.

The General Calculations will be seen to demonstrate the elegance of Porta's design methods and the relative ease with which they may be followed by anyone with a grasp of engineering principles. The calculation methodology aims to provide not only for the logical and comprehensive design of the 5AT, but for the design any steam locomotive of any size or arrangement.

The General Calculations are broken down into four sections as follows:

  4. Section 1.3.F. Rev 2: PRELIMINARY BASIC CALCULATIONS (final version 2nd revision).

Please click on each of the above links to access these calculations.

Get Acrobat ReaderNote: These calculations are presented in Adobe Acrobat PDF format. If you don't have Acrobat Reader software, please click the Acrobat icon (at right) to download it free directly from Adobe's website.

Page updated: 11th Feb 2007