p571.html

> restart:#"m17_p57"

An airboat (or fan-boat) is a flat-bottomed vessel propelled by an aircraft-type propeller (driven by a reciprocating engine) to sail in shallow or marshy waters. For a certain propeller of 2 m in diameter spinning at 1200 rpm, thrust and power coefficients can be approximated in terms of advance ratio J as cF=0.050.06J0.1J2, and cP=0.0150.03J2, respectively. If boat drag can be approximated as D=cDAwv02/2, with cD=0.002, A=1.5 m2, w water density, and neglecting air drag (assumed included in boat drag), find:

a) Propeller efficiency versus advance ratio, and advance speed for maximum efficiency.

b) Power demanded by the propeller, and thrust provided, at zero advance speed.

c) Maximum advance speed, and corresponding values of p, F, and .

Considérese una lancha de pantanos con una hélice propulsora aérea de D=2 m de diámetro, que girando a n=1200 rpm produce un empuje F=cFn2D4, con cF=0,050,06·J0,1·J2, siendo J=v0/(nD) la velocidad de avance (v0) adimensional, y  la densidad del aire. La potencia al eje necesaria para mover esa hélice la proporciona un motor de gasolina, y se puede poner como P=cPn3D5, con cP=0,0150,03·J2, y la resistencia al avance de la lancha (incluida la hélice) se estima como D=cDAwv02/2, con cDA=0,003 m2, y w la densidad del agua. Se pide:
a) Calcular el rendimiento propulsivo de la hélice, =Fv0/P, en función de la velocidad de avance relativa, J.
b) Empuje y potencia al eje en el arranque.
c) Velocidad máxima alcanzable, y valores del rendimiento, empuje y potencia correspondientes.
Datos:

> read"../therm_eq.m":read"../therm_proc.m":with(therm_proc):with(RealDomain):

> su1:="Aire":su2:="H2O":dat:=[Di=2*m_,n=20/s_,cDA=0.003*m_^2];

Airboats:

> dat:=op(dat),get_gas_data(su1),Const,SI2,SI1:Wdat:=get_liq_data(su2):eqrho:=rho=p0/(R*T0);eqrho_:=subs(dat,%);eqcF:=c[F]=0.05-0.06*J-0.1*J^2;eqcW:=c[W]=0.015-0.03*J^2;eqJ:=J=v[0]/(n*D);eqF:=F=c[F]*rho*n^2*D^4;eqWdot:=Wdot=c[W]*rho*n^3*D^5;eqD:=D=c[D]*A*rho*v[0]^2/2;

(1)

a) Propeller efficiency versus advance ratio, and advance speed for maximum efficiency.

An airboat (or fan-boat) is a flat-bottomed vessel propelled by an aircraft-type propeller (driven by a reciprocating engine) to sail in shallow or marshy waters.

> eqetap:=eta[p]=F*v[0]/Wdot;eqetap:=eta[p]=c[F]*J/c[W];eqetap_:=subs(eqcF,eqcW,%);Jetamax_:=fsolve(diff(rhs(eqetap_),J)=0,J=0..0.5);etamax_:=subs(J=Jetamax_,rhs(eqetap_));v0max_:=subs(J=Jetamax_,eqrho_,D=Di,dat,J*n*D);plot([rhs(eqcF),rhs(eqcW)],J=0..0.5,c=0..0.05);plot(rhs(eqetap_),J=0..0.5,eta=0..1);

Plot_2d

Plot_2d

i.e. the maximum efficiency is for J=0.29 (eta_max=0.56).

b) Power demanded by the propeller, and thrust provided, at zero advance speed.

We see from cP that it is at J=0:

> J=0;eqF0:=subs(eqcF,J=0,eqF);eqF0_:=subs(eqrho_,D=Di,dat,eqF0);eqWdot0:=subs(eqcW,J=0,eqWdot);eqWdot0_:=subs(eqrho_,D=Di,dat,eqWdot0);

(2)

i.e. the maximum power demandad from the engine is Wmax=4.7 kW, and the corresponding thrust (maximum thrust) is 390 N.

c) Maximum advance speed, and corresponding values of p, F, and .

At start, the propeller yields a thrust of F=390 N without any opposing force (no drag), and the airboat accelerates until drag matches thrust (Fig. E1b); i.e. maximum v0 corresponds to F=D (thrust=drag), what yields:

> eqBM:=F=D;eqF;eqD;subs(v[0]=J*n*D,%);eqD_:=D=subs(c[D]=cDA/A,D=Di,Wdat,dat,rhs(%));eqF_:=subs(eqcF,eqrho_,D=Di,dat,eqF);eqBM_:=rhs(eqF_)=rhs(eqD_);J_:=solve(eqBM_,J)[1];eqvmax:=v[0,max]=subs(J=J_,D=Di,dat,J*n*D);plot(subs(SI0,[rhs(eqF_),rhs(eqD_)]),J=0..0.5);eqWdotmax:=subs(eqcW,J=J_,eqrho_,D=Di,dat,eqWdot);eqFmax:=subs(eqcF,J=J_,eqrho_,D=Di,dat,eqF);eqetamax:=subs(J=J_,eqetap_);

Plot_2d

(3)

i.e. F=D occurs at J=0.28, hence the maximum speed of the airboat is v0=11.4 m/s. At this point, F=190 N, P=3.9 kW, and eta=0.56.