Example 11.4-4 Transpiration Cooling
In spherical coordinates, vr is the only non-zero velocity component and it depends only on r.
Eq. C in Table 3.4-1 then reduces to 10.5-21
>  restart;
>  A:=r->4*Pi*r^2; eq:=wr=A(r)*vr(r)*rho; wr is the total mass flow rate in the radial direction
and is constant.
>  vr(r):=solve(eq,vr(r));vr:=unapply(vr(r),r); eq. 10.5-22
T depends only on r. Neglecting viscous disipation in eq. C of Table 10.2-3, we have only the two terms
shown in 10.5-23.
>  de:=diff(T(r),r)=k/(rho*Cp*vr(r)*r^2)*diff(r^2*diff(T(r),r),r); eq. 10.5-23 after dividing by rho*Cp*vr
>  s:=dsolve({de,T(R)=T1,T(kappa*R)=Tk},T(r));
>  assign(s);
>  T:=unapply(T(r),r): Making a function of r
>  wr:=R0*4*Pi*k/Cp; Replacing wr with R0 as defined after eq. 10.5-25
>  Thet:=simplify((T(r)-T1)/(Tk-T1),assume=positive);
>  Thetbook:=(exp(-R0/r)-exp(-R0/R))/(exp(-R0/(kappa*R))-exp(-R0/R));
>  simplify(Thetbook-Thet,assume=positive);
>  R0:=phi*kappa*R/(1-kappa); Replacing R0 with phi as defined after eq. 10.5-32
>  simplify(T(r),assume=positive); Not too simple!
>  Tmaybe:=r->(exp(-R0/r)-exp(-R0/R))/(exp(-R0/(kappa*R))-exp(-R0/R)); eq. 10.5-25
>  dif:=simplify((T(r)-T1)/(Tk-T1)-Tmaybe(r),assume=positive);
>  Theta:=r->(T(r)-T1)/(Tk-T1);
>  limit(Theta(r),phi=0,right);
>  %-((1/r)-(1/R))/((1/(kappa*R))-(1/R)); Subtracting off eq. 10.5-26
>  simplify(%); That checks
>  Q:=4*Pi*k*(kappa*R)^2*D(T)(kappa*R); Getting the rate of heat flow to the inner sphere
>  Q-4*Pi*k*R0*(T1-Tk)/(exp(R0*(1-kappa)/(kappa*R))-1); Comparing our result to eq. 10.5-30
>  simplify(%,assume=positive); It checks.
>  Q0:=limit(Q,phi=0,right); Checking eq. 10.5-31
>  epsilon:=simplify((Q0-Q)/Q0,assume=positive); The effectiveness
>  simplify(%-1+phi/(exp(phi)-1),assume=positive); It is rather messy, but
It checks with eq. 10.5-32
>  wr; This allows us to see the relation between phi and wr.
>  eps:=phi->1-phi/(exp(phi)-1); Making a function of eq. 10.5-32
>  plot([eps(phi),phi/2],phi=0....4,color=[red,blue]);

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