1842.1 



THE CIVIL ENGINEER AND ARCHITECT'S JOURNAL. 



155 



of expan6ion=^/i ; pressure of steam at termination of expansion=rs. 

 Then, efficiency of whole stroke of piston=« (0+6) log.--(-l)—6f 



or put a^l p—l 



1 



and then efficiency of the whole stroke =s(l+6) (log. — f-l) — 6 



To find from this what would be the efficiency were the expansion 

 valve close to the cylinder port, put 6=0 ; and hence — 



Efficiency when valve is close to cylinder port=s (log. — 1-1) 



Do.when valve is at a distance (6) from do.=:s (1+6) (log. — 1-1)— 6 



which formulae assign the amount of injury entailed upon an engine 

 by any given position of expansion valve, and, therefore, the amount 

 of preference due from this cause to one valve over another. But by 

 substituting for (b) its value, in practical cases we find that the amount 

 of the injury is by no means of sufficient importance to warrant the 

 exclusion of all expansion valves that do not cut oft' close to the port ; 

 and by making a comparison of the three, namely, the slide inserted 

 betwixt the D valve and the face of the cylinder, the slide valve in 

 the inside of the cylinder steam jacket, and the double-beat conical 

 valve (the three which seem to be at the present time of the greatest 

 importance). Taking into consideration simplicity of arrangement, 

 simplicity of execution, facility to keep in repair, and efficiency of 

 action, including facility of altering the degree of expansion, we are 

 led to believe that the double-heat expansion valve is the one which, 

 in the aggregate, is the best. This valve (6) in well constructed 

 engines does not exceed one-thirtieth ; but as an average case, take 

 one-fifteenth of the volume of the cylinder, and by substituting this 

 value of (6) we find the loss of effect of the steam from having one- 

 fifteenth of the whole volume of the cylinder betwixt the cylinder 



0-06- — sxO-06' (log. i+1) 



1-06-s (log. +l)_0-OG- 



or -when s:=l to nearly 4i per cent, of the whole power developed 

 by the engine, which points out the propriety of putting the valve as 

 close to the D valve casing as possible. 



Adopting then the double beat conical valve, the economy of fuel 

 as produced by expansion may be thus estimated. Let the volume of 

 steam cylinder when there is no expansion^a ; volume of steam 

 betwixt cylinder port and expansion valve = 6; pressure of steam at 

 commencement of expansion=/) ; pressure of steam at termination of 

 expansions*. Then s ; p : : a ' volume of cylinder when expan- 

 sion is carried to extent (s) .' . increased volume of cylinders— but (6) 



must be taken proportional to the volume of the cylinder .'.a : — 

 : : 6 ; increased volume betwixt cylinder ports and expansion valve 



port and the expansion valve to be- 



ditto 



= — and by substituting these values of 



(a and 6) in the formula s (a + b) log. P + 1) — bj>, we find that with 



the expansion (8)the efficiency ofa given quantity of steam is (ap+bp) 



p bp^ 



(log. - + 1)—-^ . Put in this a =1 p = 1 i) = ^x volume of 

 s s 



steam cylinder, which, in well constructed engines, it will not 



exceed, and efficiency = 1-04 X (log. - + i)_0'04X- 



s s 



Let s=l which is the case of no expansion, and efficiency = 1 



K s= 5 which is the case of the steam 



being cut off, when stroke J finished & efficiency= 1'285 



" s=| " " I " =1'4 



» 8=4 " « i " =1-68 



" g=i " « i " =2-06 



« «=i « " i " =2-32 



" s=i " « J « =2-88 



And hence the following conclusions : — 



If the work done by a ton of coal when the steam is used without 

 expansion be represented by unity, the work done by the same fuel 

 when the steam is used expansively to the extent of reducing its 

 pressure to I of its original pressure, is represented by 1.285. 

 And when to the extent of reducing 



its pressure to . . . . § the work is represented by 1-4 

 " " " * « " 1-G8 



" " " ^ " « 2-06 



" " " i " " 2-32 



" " " I " « 2-88 



In a former part of this paper it was mentioned that it is advan- 

 tageous to construct the D valve with a small extent of cover, 

 notwithstanding its efiect of diminishing the available volume of the 

 cylinder, and for the following reasons : — when the piston of the steam 

 cylinder is near either end of its stroke, the effective leverage of the 

 crank to turn the paddle shaft is small ; and although the amount of 

 power developed by a given quantity of steam is at every part of the 

 stroke precisely the same, yet, in consequence of the varying leverage 

 of the crank, that portion of the power applied to overcome resistance 

 at the extremity of the paddle wheel arm is by no means the same, 

 being diminished at no two parts of the stroke by the same quantity, 

 to overcome friction ; for as the leverage of the crank decreases, that 

 of the friction on some of the journals, in reference to the power, 

 increases. And beyond a certain point, it is very questionable if (the 

 engine being deprived of all momentum,) any amount of power that 

 could be applied to the crank pin would bring the crank to the per- 

 pendicular position, and therefore for a certain portion of the stroke 

 at each end, the steam not only does not increase the efficiency of the 

 engine, but detracts from it, by the amount of power necessary to 

 overcome the friction produced by its application ; and hence the 

 propriety of the plan adopted by some makers, of constructing the D 

 valve with so much cover as causes the eduction to commence before 

 the termination of the stroke, and the average vacuum effected before 

 the return stroke has advanced to that point when the steam begins 

 to be effective. The extent of cover necessary to effect this object is 

 quite beyond the power of analysis to assign, and perhaps no two 

 engineers have from their observation arrived at exactly the same 

 conclusion ; but I should say that constructing the D valve with only 

 sufficient cover on the eduction side to keep it tight, and with from 

 0-25 to 0'3 of the breadth of the port on the steam side, would 

 probably with the average of cases be not far from the truth, with 

 which construction the cylinder would be in communication with the 

 condenser from -jir to -n; of the whole time, for eduction before the 

 termination of the stroke. 



Greenock Foundry, 15/4 ^pril, 1842. 



ON THE SKEW ARCH. 



An analytical investigation of the principal properties o/ihe Skem 

 Arch. By a ]Vorkman, 



Let A B C D, a 6 c rf, be the plan of the semicylindrical arch, the ele- 

 vation on the skew of which is represented by A G B 6 F a ; draw a K 

 perpendicular to the axis H E R, and G E perpendicular to A B, and 

 N n n' parallel o a K ; let the spirals, which commence from n n', 

 intersect the skew elevation in the points S s, join S and s, and produce 

 this line to meet the axis G E produced if necessary in O, the line 

 E O is stated by Mr. Buck to be a constant quantity, the investigation 

 of which will form the subject of this paper. 



The subject of inquiry seems to demand the complete solution of 

 the two propositions. 



First. To determine the intersections s and S of the spirals com- 

 mencing at II n', with the skew elevation a F 6 and A G B. 



Second. To determine the point where the line joining S and 

 produced meets the axis G E. 



Z 2 



