MATLAB: Vpasolve not returning an answer, just Empty sym: 0-by-1

 syms x
 %Define Initial Conditions/Parameters
P = 1;
c_p = 1;
T_m = 1;
T_o = 0;
L = 1;
r_2 = 1;
r_1 = 1;
k = 2;
h_1 = 3;
 %Numerically solve for the mass throughput
vpasolve(P-x*c_p*(T_m-T_o)*(exp(-1*L*(2*pi*r_2*1./(x*c_p))*(1./h_1+(r_2-r_1)./k).^-1)-1) == 0)

Best Answer

Your equation is never satisfied. Hence, "vpasolve" returns empty sym. Here's how to figure that out.
First, declare your equation separately because it's 1) good practice and 2) you can see your equation.
>> eqn = P-x*c_p*(T_m-T_o)*(exp(-1*L*(2*pi*r_2*1./(x*c_p))*(1./h_1+(r_2-r_1)./k).^-1)-1) == 0
eqn =
1 - x*(exp(-(6*pi)/x) - 1) == 0
Try to simplify
>> eqn = simplify(eqn)
eqn =
x*(exp(-(6*pi)/x) - 1) == 1
Now just plot the "lhs" to see if it ever equals "1".
lhs = children(eqn);
lhs = lhs(1);
fplot(lhs)
From the plot ( http://pasteboard.co/hh9ftvyZi.jpg ) you can see your equation is never satisfied. Hence, "vpasolve" returns empty sym.
Related Question