I'm nearly lost now and just guessing what you are doing.
Every notebook you attach has more invisible ToolTip hidden inside than the last one and this time there were over a dozen. I don't know what you are doing to get that, you almost certainly don't know what you are doing to get that. Usually the first thing I do when I get any notebook from you is to use Notepad to carefully delete all those and the stuff that goes with them and then see if the notebook works or not. Often that fixes at least half your problems.
It looks like you have changed some of your variable names to alpha1 and alpha2 but left some as \[Alpha]1 and \[Alpha]2 which may or may not show up as Greek characters after the select-and-copy-and-pasteToForum-and... Since the warning message in orange that starts with FindRoot::nlnum has some alpha's showing in that, I suspect this is one place to start.
It is usually a good idea to put variable "declarations" that don't depend on other variables before declarations that then depend on those. Mathematica is hiding a vast amount of stuff behind the curtain and one of those things is (mostly) invisibly hiding that it keeps the last value assigned to any variable.
With your newly introduced For loop in this version I'm having a very hard time untangling whether the order you have assigned values to variables might be part of your problem.
Is there any way your problem might be made lots simpler? Or is this as simple as it can get?
EDIT
I have taken one more whack at your code, or at least the first section of it.
I have manually edited out the ToolTip using Notepad.
I have guessed:
you wanted your Greek alpha1 and alpha2 to be english alpha1 and alpha2,
you wanted your r subscript 1 to be r1.
you wanted your k=longexpression==0; to be k=longexpression;
I have reordered your lines so that if x=expression depends on y= expression then y=expression is before x=expression, that includes your h1 For iterator..
I think I've done something else, but I can't spot what it might be now.
Here is the result
In[1]:= rhop = 1.781 10^3;
rhol = 0.781 10^3;
g = 9.80665;
T = 0.4/10^2;
q = Sqrt[((rhop - rhol) g)/T];
bb = 0.0000001;
alpha1 = Pi/6;
R1 = 900/10^9;
R2 = 600/10^9;
l0 = 400/10^9;
alpha2 = 0;
E1 = 1.781072418 ;
For[h1 = 400/10^9, h1 < 1.2 R1 - l0, h1 = h1 + 10/10^9,
r1 = Sqrt[(l0 + h1) (2 R1 - l0 - h1)];
Psi1 = ArcSin[r1/R1] - alpha1;
r2 = Sqrt[(l0 + h) (2 R2 - l0 - h)];
Psi2 = ArcSin[r2/R2] - alpha2;
a = Sqrt[bb*(bb + 2*r1)*(bb + 2*r2)*(bb + 2*r1 + 2*r2)/(4*(bb + r1 + r2)^2)];
T1 = Log[(a/r1) + Sqrt[(a^2/r1^2) + 1]];
T2 = Log[(a/r2) + Sqrt[(a^2/r2^2) + 1]];
k = r1*Sin[Psi1]*(T1+2*Log[1-Exp[-2*T1]])-(r1*Sin[Psi1]+r2*Sin[Psi2])*Log[E1*q*a]+(r1*Sin[Psi1]-r2*Sin[Psi2])*(A+B)-h;
A = \!\(\*SubsuperscriptBox[\(\[Sum]\),\(k=1\),\(2\)]\(\((1/k)\)*\((\((Sinh[k*\((T1-T2)\)])\)/\((Sinh[k*\((T1+T2)\)])\))\)\)\);
B = \!\(\*SubsuperscriptBox[\(\[Sum]\),\(k=1\),\(2\)]\(\((2/k)\)*\((\((Exp[\(-k\)*T2*Sinh[k*T1]])\)/\((Sinh[k*\((T1+T2)\)])\))\)\)\);
Print[{((-(Pi*T) ((2 h1 R1 Cos[alpha1] - r1 h1 Sin[Psi1] + r1^2) + (2 h*R2 Cos[alpha2] - r2 *h *Sin[Psi2] +
r2^2)) - ((rhop - rhol) g) (1/2 *Pi*(-h1^2) (2 l0 (2 R1 - l0) - h1^2) + (4/3 (R1 - l0) (-h1^2) + r1^2 h1^2) +
1/2*Pi* (-h^2) (2 l0 (2 R2 - l0) - h^2) + (4/3 (R2 - l0) (-h^2) + r2^2 h^2)))/(10^(-14)))} /.
FindRoot[r1*Sin[Psi1]*(T1 + 2*Log[1 - Exp[-2*T1]]) - (r1*Sin[Psi1] + r2*Sin[Psi2])*
Log[E1*q*a] + (r1*Sin[Psi1] - r2*Sin[Psi2])*(A + B) - h1 == 0, {h, 600/10^9}]
]
]
FindRoot::lstol: The line search decreased the step size to within tolerance specified by AccuracyGoal and PrecisionGoal but was unable to find a sufficient decrease in the merit function. You may need more than MachinePrecision digits of working precision to meet these tolerances. >>
{-2.54714-3.82639*10^-9 I}
FindRoot::lstol: The line search decreased the step size to within tolerance specified by AccuracyGoal and PrecisionGoal but was unable to find a sufficient decrease in the merit function. You may need more than MachinePrecision digits of working precision to meet these tolerances. >>
{-2.55259-3.87041*10^-9 I}
FindRoot::lstol: The line search decreased the step size to within tolerance specified by AccuracyGoal and PrecisionGoal but was unable to find a sufficient decrease in the merit function. You may need more than MachinePrecision digits of working precision to meet these tolerances. >>
General::stop: Further output of FindRoot::lstol will be suppressed during this calculation. >>
{-2.55755-3.91336*10^-9 I}
{-2.56204-3.95522*10^-9 I}
{-2.56606-3.99598*10^-9 I}
{-2.56962-4.03562*10^-9 I}
{-2.57272-4.07412*10^-9 I}
{-2.57538-4.11148*10^-9 I}
{-2.57761-4.14767*10^-9 I}
{-2.5794-4.18269*10^-9 I}
{-2.58078-4.21651*10^-9 I}
{-2.58471-4.15521*10^-9 I}
{-2.58849-4.09302*10^-9 I}
{-2.59213-4.02997*10^-9 I}
{-2.59562-3.96607*10^-9 I}
{-2.59899-3.90136*10^-9 I}
{-2.60223-3.83587*10^-9 I}
{-2.60535-3.7696*10^-9 I}
{-2.60834-3.7026*10^-9 I}
{-2.61123-7.26977*10^-9 I}
{-2.614-7.13296*10^-9 I}
{-2.61667-6.99483*10^-9 I}
{-2.61923-6.85542*10^-9 I}
{-2.6217-6.7148*10^-9 I}
{-2.62408-6.57301*10^-9 I}
{-2.62637-6.43012*10^-9 I}
{-2.62857-6.28617*10^-9 I}
{-2.63069-6.14123*10^-9 I}
As a check, if I scrape-n-paste everything from the rhop= down to the closing ] then it appears to run.
Please see if you can reproduce exactly this behavior on your end with a fresh start of Mathematica in an empty notebook.
The warnings appear to happen every loop, but it stops complaining after three times. It looks like the closer FindRoot gets to a solution the slower the curve approaches zero so it gives up before it gets an answer accurate enough to make it happy. That kind of error/warning message doesn't make me tend to think this is a terrible programming error. It appears that every solution has a tiny but nonzero complex part. That may really be in your equation or it might just be roundoff errors. If you don't care about the imaginary part you could wrap Re[] around the whole thing you are giving to Print and see if the imaginary part goes away. If you do that then don't forget, just in case the complex part is much bigger sometime.
Look carefully at how I've ordered all your variable assignments and to guess what I'm thinking when I did that. Sometimes you can change the order and it can still work, but sometimes this can bite you when you least expect it.
Is there any way you can check to see whether these answers are correct?
Can you then try doing the same thing to your next part of your original notebook and see if you can get it to work?