Which correlation for heat exchange in a partially adiabatic channel?

I'm modelling a heat exchanger inside a channel. Talking about convective one:
Geometry is simple, square of 1.5 cm but only one side exchanges heat.

The "problem" is that only one side exchanges heat with environment, the others are adiabatic.

At the moment I think best correlation to use is a simple internal correlation (e.g. Gnielinski), rather than an external correlation because this doesn't take into account boundary layer which can't grow up since there is the "roof" of the channel (characteristic length is no more a constant) making Nusselt number constant! but I'm not sure about this... I've never handled a problem like this!

an example of my "fear":

• $Re_D = 61000$, certainly turbulent flow
using Gnielinski (internal flow correlation) I get $Nu_D = 125$
and since $L/D=3000>>60$ the most of the pipe is in Fully Developed region >that ensure me to average Nu == local Nu:
$Nu_{average}=Nu_D=125$
• while if I use an external correlation, at the end of the pipe, I get:
$Re_L = 2e+8$, certainly turbulent flow
$Nu_L=0.0296*Re_L^{4/5}*Pr^{1/3} = 1.1e+5$
$Nu_{average}=(0.037*Re_L^{4/5}-871)*Pr^{1/3} = 134384$

this is the numerical proof about my doubt

• flowing fluid is air at $T=700 [°C]$ , $p=10 [bar]$ , $m=36.56 [g/s]$ ;
these result in $\rho=3366.18 [g/m^3]$ , $u_{mean}=48 [m/s]$ , $\mu=4e-5 [Ns/m^2]$ , $Pr=0.734$ , $k=6.8e-2 [W/mK]$
• walls are ceramic-made: adiabatic ones have infinite thickness while the only one which exchanges heat is 1 [mm] thick
• channel is 50 [m] long and hydraulic diameter is 1.5 [cm]
• ambient is at constant temperature of 675 [°C], we can think to a fluid in phase-transition, so with a negligible thermal resistance, so don't worry about ambient...

EDIT: I want to share an online PDF which try to talk about it (here) inter alia at pagg. 46 and 264

• I don't think Gnielinski's correlation for turbulent flow in pipes would apply here even if we treated the problem with its hydraulic diameter as a pipe due to the given boundary conditions, however, you should give more details about the problem; data about fluid inside the duct, ambient conditions, wall material type, wall thickness and length of the duct. – Algo Jul 25 '15 at 18:20
• How about considering the uninsulated part of the duct as a flat plate subjected to turbulent flow $$Nu = 0.037\, Re_{L}^{0.8} Pr^{1/3}$$ and after obtaining the heat transfer coefficient of the flow over the plate $h_{flow}$you might get overall heat transfer coefficient $U_i$ combining $h_{ambient}\ , h_{flow}$ and $k$ of the material, and finally sub. in: $$Q = U_iA_i\theta_{LMTD}$$ However, what do you mean by we can think to a fluid in phase-transition, so with a negligible thermal resistance? – Algo Jul 26 '15 at 9:49
• I don't "like" flat plate correlation because it doesn't take into account boundary layer behaviour (as I'll better explain in the OP). Just think about at the end of the channel: Nusselt number in flat case will be higher than higher because characteristic length is 50[m] – mattia.b89 Jul 26 '15 at 10:13
• Are air mass flow rate and ambient temperature given? – Algo Jul 26 '15 at 10:18
• How can you not worry about ambient? What is the "heat exchange" that's taking place here? – Algo Jul 26 '15 at 10:27

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.)

• first, I'm truly thrilled from your precision, completeness and comprehensiveness in your answer! second, I edit opening post to include/enhance my thoughts about doubt using a flat plat correlation – mattia.b89 Aug 10 '15 at 8:17
• @mattia.b89 thank you. May I ask why are you bounded by this setup? what is your application? – Algo Aug 10 '15 at 10:25
• it's a channel of a SOFC; what do you think about numerical explanation of my doubt in the opening post ? – mattia.b89 Aug 10 '15 at 10:47
• @mattia.b89 what's $e$ in your explanation? – Algo Aug 11 '15 at 7:28
• if you mean e.g. $Re_L=2e+8$; $e$ stands for exponent, it's the scientific notation, so $Re_L=2e+8=2*10^8$ – mattia.b89 Aug 11 '15 at 7:37