The type of your boundary conditions - specifically the adiabatic walls - is the only thing that would prevent you to use the conventional Nusselt number correlations for turbulent flow inside pipes (Gnielinski, Dittus-Boelter and Sieder-Tate).
Since heat is only transferred from only one side of the duct and as I suggested before in comments, you can treat that side as "forced convection over a flat plate" problem. I see that you are worried about the effect of boundary layer on the solution, so I made a little experiment to investigate that approach I suggested.
Simplified Problem Setup
I didn't solve your exact problem but I proposed a simple setup only to investigate the flat plat approximation and here are my assumptions:
1 - I took a tiny part of the duct (10 cm), not the whole 50 meters.
2 - I replaced ceramic material with copper, because simply I don't know which ceramic material you have in mind (and I don't have the thermal properties of ceramics), and just for simplification I used copper.
3 - To simplify the problem even further, I assumed that the lower part of the solid copper material is at fixed temperature of 25 C (Since we are only investigating the heat transfer coefficient inside the duct, there is no need to make the problem more complex by introducing the heat transfer coefficient of the ambient air).
Methodology
Total heat transfer from hot air in duct to the lower plate $Q$:
$$ Q = m^oC_p (T_{f_o} - T_{f_i}) = UA_s\triangle T$$
where $T_f$ is air flow temperature, subscripts o and i represents outlet and inlet respectively, $U$ is the overall heat transfer coefficient , $A_s$ is the surface area of the lower side of the duct and $\triangle T$ is the difference between average temperature of flow inside duct and the temperature of the lower side of the copper.
$U$ is calculated as follows:
$$ \frac{1}{U} = \frac{1}{h_i} + \frac{L}{K} $$
where $h_i$ is the heat transfer coefficient of air flow inside the duct and $K$ is the thermal conductivity of the copper.
to calculate $h_i$:
$$ Nu = \frac{h_iL}{K} = 0.037 Re^{0.8} Pr^{\frac{1}{3}} $$
since at first iteration $T_{f_o}$ is unknown, all properties are calculated at $T_{f_i}$ only.
Python 2.7 code:
from __future__ import division
import CoolProp.CoolProp as coolprop
nRe = lambda rho, v, L, mu: (rho*v*L) / mu
nPr = lambda cp, mu, k: (cp*mu) / k
nNu = lambda Re, Pr: 0.037 * (Re**0.8) * (Pr**(1/3))
Rho = lambda T: coolprop.PropsSI('D','P',60*100000,'T',T+273,'Air.mix')
Mu = lambda T: coolprop.PropsSI('V','P',60*100000,'T',T+273,'Air.mix')
Cp = lambda T: coolprop.PropsSI('C','P',60*100000,'T',T+273,'Air.mix')
K = lambda T: coolprop.PropsSI('L','P',60*100000,'T',T+273,'Air.mix')
duct_length = 0.1 # 10 cm
plate_thickness = 0.001 # 1 mm
duct_height = 0.015 # 1.5 cm
mass_flow = 0.03 # 30 g/s
surface_temp = 25 # C
# first iteration
Tavg = 700
velocity = mass_flow/(Rho(Tavg) * (duct_height**2))
Re = nRe(Rho(Tavg), velocity, duct_length, Mu(Tavg))
Pr = nPr(Cp(Tavg), Mu(Tavg), K(Tavg))
Nu = nNu(Re, Pr)
h = (Nu * K(Tavg))/duct_length
U = ( (1/h) + (0.001/398) )**-1
Tfo = Tavg - ( (U * (duct_height *duct_length) * (Tavg - surface_temp)) / (mass_flow*Cp(Tavg)) )
# iteration loop
i = 0
while i < 30:
Tavg = (700 + Tfo) / 2
velocity = mass_flow/(Rho(Tavg) * (duct_height**2))
Re = nRe(Rho(Tavg), velocity, duct_length, Mu(Tavg))
Pr = nPr(Cp(Tavg), Mu(Tavg), K(Tavg))
Nu = nNu(Re, Pr)
h = (Nu * K(Tavg))/duct_length
U = ( (1/h) + (0.001/398) )**-1
Tfo = Tavg - ( (U * (duct_height *duct_length) * (Tavg - surface_temp)) / (mass_flow*Cp(Tavg)) )
i += 1
print 'Outlet temperature = {0} C'.format(Tfo)
Result: Outlet temperature $T_{f_o}$ = 668.42 C
CFD Simulation
The reason for the 10 cm part was just to simplify the CFD geometry and lower computation time. Air was treated as an ideal gas and I used a RANS equations turbulence model (k-epsilon) and here are the results:
Temperature distribution at the center of the duct (left side is the inlet).

Temperature profile of duct outlet.

And finally, the simulation resulted in average outlet temperature $T_{f_o}$ = 684.57 C.
Conclusion
Please keep in mind that I didn't solve your exact problem, I just made a simplified setup to investigate the accuracy of the flat plate assumption.
There's a quite big difference between our assumption and the CFD of the actual case (16.15 C), so if this level of accuracy is not sufficient for you as preliminary calculations (I don't think it is), then you should set up an experimental apparatus to get a Nusselt correlation for the actual case, you won't find much help with the available correlations with those boundary conditions.
Hope that helps.
(I'll double check my calculations and get back if there were any errors.)
we can think to a fluid in phase-transition, so with a negligible thermal resistance
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