# Solving a determinant to get non-zero roots in a heat transfer problem

I am trying to solve a mathematical model of a heat transfer problem. As an intermediate step there is a characteristic equation in the form of a determinant, which needs to be solved for variable `r` to get three real roots. These roots feed the subsequent steps. However, the roots I get are really small and probably leads to `Indeterminate` errors later. I am attaching the entire script below:

``Nuc = 5.33; Nuh = 6.49; mc = 9.305 E - 08; mh = 1.1246 E - 07; k = 16.27; vc = (mc/1.138)/(0.5 2 10^-6); vh = (mh/1.225)/(0.25 2 10^-6); cpc = 1039; cph = 1006.43; Ac = 2 (0.5 + 2) 10^-6; Ah = 2 (0.25 + 2) 10^-6;  C1 = mc cpc; C2 = mh cph;  hc = Nuc 0.0242/(0.8 10^-3); hh = Nuh 0.0242/(0.444 10^-3);  b1 = hc Ac/C1; b2 = hh Ah/C2; bc = k (25 1 10^-6)/(1 10^-3 C1);  λ = k (4.5 10^-6)/(25 10^-3 C1); ν = C1/C2;  bstar = b1 + (b2/ν) + 4 bc; γ1 = b1/bstar; γ2 = b2/(ν bstar); γc1 = (2 b1 + 4 bc)/bstar; γc2 = (2 b2 + 4 ν bc)/(ν bstar);  A = {    {-r + b1 (1 - γ1), b1 γ2, -b1 γc2},    {-b2 γ1, -r - b2 (1 - γ2), b2 γc1},    {r, -r, λ r^2}   }; sol = LinearSolve[Det[A] == 0,r]   r1 = r /. sol[[1]] r2 = r /. sol[[2]] r3 = r /. sol[[3]]  G1 = (4 b1 b2 bc +       2 b1 ((b2/ν) +          2 bc) r1)/(4 b1 b2 bc + (((1/ν) - 1) b1 b2 +          4 bc (b1 - b2)) r1 - bstar r1^2); G2 = (4 b1 b2 bc +       2 b1 ((b2/ν) +          2 bc) r2)/(4 b1 b2 bc + (((1/ν) - 1) b1 b2 +          4 bc (b1 - b2)) r2 - bstar r2^2); G3 = (4 b1 b2 bc +       2 b1 ((b2/ν) +          2 bc) r3)/(4 b1 b2 bc + (((1/ν) - 1) b1 b2 +          4 bc (b1 - b2)) r3 - bstar r3^2); H1 = (4 b1 b2 bc -       2 b2 (b1 + 2 bc) r1)/(4 b1 b2 bc + (((1/ν) - 1) b1 b2 +          4 bc (b1 - b2)) r1 - bstar r1^2); H2 = (4 b1 b2 bc -       2 b2 (b1 + 2 bc) r2)/(4 b1 b2 bc + (((1/ν) - 1) b1 b2 +          4 bc (b1 - b2)) r2 - bstar r2^2); H3 = (4 b1 b2 bc -       2 b2 (b1 + 2 bc) r3)/(4 b1 b2 bc + (((1/ν) - 1) b1 b2 +          4 bc (b1 - b2)) r3 - bstar r3^2);  Num = {    {((H1 - G1)/r1) E^-r1, ((H2 - G2)/r2) E^-r2, ((H3 - G3)/r3) E^-r3},    {E^-r1, E^-r2, E^-r3},    {1, 1, 1}   }; Den = {    {(H1 E^-r1 - G1)/r1, (H2 E^-r2 - G2)/r2, (H3 E^-r3 - G3)/r3},    {E^-r1, E^-r2, E^-r3},    {1, 1, 1}   }; eff = 1 - (Det[Num]/Det[Den])  ``

Any help in pointing out my error will is appreciated. The objective is to evaluate `eff`.