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Aviation History
1927
1927 - 0647.PDF
AUGUST 25, 1927 53 THE AIRCRAFT ENGINEER SUPPLEMENT TO FLIGHT W = F,-, the induction factor (induced pouer per lb. at unit V). Then F,/lliax.K,, = 0-00000872 F,Vlllax.3 F, Vmax. F,;Y F —" «niax.K,,V,,,ax. — — 0-00000872 Vmax. (13) (13a) An equation in this form can be very readily solved for Vmax. when the remainder of the quantities are known. The left side of formula (13a) is a linear in Vmax.. The right side varies as Vmax.4- Any two points determine the curve of the linear portion, while • 00000872 V,lmx.' can be easily plotted out. (See Fig. 3) By assuming any value of V a solution for the left-hand portion is obtained. Plotting any two such points as ordinates on their respective values of V as abscissa> and connecting by a straight line to intersect the curve will give the value of Vmax. desired. Fig. 3 has been included in this report for that purpose. This procedure will be more clear from the examples to follow at the end. If it is desired to calculate F,. from a flight test, where Vmax- is known, the following form is most convenient : F,. = I F,,<-m:lx .K — ' • ) (13£) VWc./' 0-00000872 Vmax.:l MAXIMUM CEILING We are indebted for the following proof to the excellent book on " Aeroplane Performance Calculations," by Mr. Harris Booth. The only change lies in the difference in symbols. It is well first to define " Ceiling." Ceiling is that height at which the rate of climb is zero. There is a ceiling corres- ponding to every point in the curve of power required. It is important to determine the maximum of these ceilings or, as it is sometimes called, the " absolute ceiling." Consider any point B, Fig. 1, which has the co-ordinates Fig. 0 ' fflOO 40006000 8000 3. Curve for the Determination of High Speed. jj in10000 At the ceiling of this particular point (VB)CE = -~ VB (14) Where d,. = relative density of the air at altitude. and [(PK )B ],-E = --L PK1, (15) V dy That is, point B has moved upward and to the right and takes the position substantially as shown by BOE Fig. 1. Since P,, = P,. at ceiling this point BCK may be considered as also lying on the curve of P,, at ceiling. /P~ Now, (V,,,,)CE = V,,,, V J1- (16> Where P,. = relative power of motor at altitude. and (P,,)CI! = P,,P,'Y/|P (17) also (VB)CE = V,,,,A/ ^- (16a) "/• /P~(PKH)OE = P,,P,-\/ v- <17rf) v d,. Since power required and power available are equal at ceiling. That is some point, such as C, Fig. 1, on the curve of power available has moved downward and to the left so as to coincide with BCE. Substituting the co-ordinate of BCE into equations 16a and 17a the co-ordinates of point C are derived. P -pKli i ,v d,. V, = ,PV^V d,. VP, (18) (19) (18«) (19a) The maximum ceiling occurs very near that velocity which makes P,. a minimum, that is at point D, Fig. 1. Differentiating P, (equation 7) against V and placing that differential equal to zero one finds : VMI, = Velocity of Min. Power = 14 ( —!') ... (20) and P, min. = 0095 F,!'4 F,."4W _ 0-095 F,-3'4 F,.1'4 W 0-095 F/ From (19a) M/, P,1/4 (21) (18a) (22) Then Fpemax. K/( K,,, when K() = 0-095 F^F,1'4 1/2 and P/:4 = 0-095 F,-^4 F,.1' (23) Fpemax. K;i If the value of P,. is calculated from equation (23), and if the variation of motor power with altitude for the given engine is known, we may definitely determine the absolute ceiling. The above holds true for every case when the speed of minimum power as given by equation (20) is greater than the 596c
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