Problem estimating the flow rate in natural draft chimney using Bernoulli equation including heat and friction losses

I am trying to build a simple model for fluid flow and heat transfer in a natural draft chimney. I've found some literature about it (most of them giving the same result as the equations on the Wikipedia article), but most of time they use hypothesis such as constant density or neglecting friction losses.

Considering the following problem :

• The air is initially still at altitude $$z_0$$, pressure $$P_0$$, temperature $$T_0$$, density $$\rho_0$$.
• The air is then heated in an horizontal circular pipe of diameter $$D_A$$, length $$L_A$$, to a temperature $$T_1$$ with pressure $$P_1$$ and density $$\rho_1$$.
• The air enter then in a vertical chimney of length $$L_B$$ and diameter $$D_B$$ through a circular 90° elbow with a negligible gravitational potential change (and negligible point losses, for now). At the entrance of the chimney, the pressure is $$P_2$$ the temperature $$T_2$$ and the density $$\rho_2$$.
• The elbow and the chimney are perfectly isolated.
• The flow leaves the chimney at a pressure $$P_3$$, temperature $$T_3$$, density $$\rho_3$$ and altitude $$z_3=z_0+L_B$$

We also have :

• Ideal gas law so $$P_i=\rho_i r T_i \quad \forall i$$ with $$r$$ the specific gas constant of air
• The outlet pressure of the chimney is equal to ambient pressure at altitude $$z_3$$ in quiet atmosphere given by $$P_3=P_0\exp(-\frac{L_B \, g}{r \, T_0})$$ (obtained using barometric equation with zero temperature lapse rate) where $$g$$ is the gravitational acceleration
• The mass flow rate $$Q_m=\rho v A$$ is constant in the system, where $$v$$ is the local average flow velocity and $$A$$ the cross-sectional area.

I a trying to use a variant of Bernoulli's principle (because of the heat addition) to obtain the mass flow rate in the system. We should have the relation

$$\frac{P_3}{\rho_3}+\frac{1}{2} \, v_3^2+g \, z_3 = \frac{P_0}{\rho_0}+\frac{1}{2} \, v_0^2+g \, z_0 + \underbrace{c_p(T_1 - T_0)}_{\text{heat gained}} - \underbrace{\Delta FL}_{\text{friction losses}}$$

where $$c_p$$ is the specific heat capacity of air and the specific friction losses in a circular pipe of diameter $$D$$ and length $$L$$ are calculated using Darcy–Weisbach equation :

$$\Delta FL = \frac {\Delta P}{\rho}=f_{D}\frac{v^2}{2}\frac{L}{D}$$

considering turbulent regime to compute $$f_D$$. I first tried to use a simpler case with $$D_A=D_B$$. We then have $$P_2=P_1$$, $$T_2=T_1$$ and $$\rho_2=\rho_1$$. We can assume that velocity at point 0 is negligible so $$v_0=0$$. The temperature should remain constant in the chimney while $$P$$ and $$\rho$$ vary to match outlet conditions, with $$\frac{P_3}{\rho_3}=r\,T_3=r\,T_2=r\,T_1$$ and $$\frac{P_0}{\rho_0}=r\,T_0$$. Therefore we have

$$r \, T_1+\frac{1}{2} \, v_3^2+g \, z_3 = r \, T_0+g \, z_0 + c_p(T_1 - T_0) - \Delta FL$$ $$\frac{1}{2} \, v_3^2 = r (T_0-T_1)+g(z_0 - z_3) + c_p(T_1 - T_0) - \Delta FL$$ using $$c_v=c_p-r$$ (see here), $$\frac{1}{2} \, v_3^2 = g(z_0 - z_3) + c_v(T_1 - T_0) - \Delta FL$$ which finally gives us $$v_3 = \sqrt{2 \left( g(z_0 - z_3) + c_v(T_1 - T_0) - \Delta FL \right)}$$ or, considering that the velocity is approximately constant in the pipes, the friction losses can be factorized to obtain $$v_3 = \sqrt{2 \frac{ g(z_0 - z_3) + c_v(T_1 - T_0) }{1+ f_D\frac{L_T}{D}}}$$ (where $$L_T=L_A+L_B$$ and $$D=D_A=D_B$$)

I can either ignore $$f_D$$ or compute from a first guess of $$v_3$$, then iteratively computing $$v_3$$ until convergence.

The problem I have is that this equation makes little sense : for example, the flow rate is non-zero even if the chimney height is zero ($$z_3=z_0$$), or, actually, it makes sense because energy was added to the flow, but it contradicts the fact that the flow the driven by buoyancy forces (which are zero when the chimney has zero height). Moreover, the velocity goes crazy even for very slight temperature differences $$T_1-T_0$$.

What is wrong with my reasoning ?

• We must use the full flow of energy $(w+\frac {v^2}{2})\rho v, dw=Tds+Vdp$. For polytropic gas $w=\frac {\gamma p}{(\gamma -1)\rho}$. From here you can derive the change in the velocity (and temperature) of the gas during heating in the form $\Delta v=\frac {(\gamma -1)q}{\rho (c^2-v_0^2)}$. $v_0$ is gas inlet speed, $c$ is the speed of sound. See L. D. Landau, E. M. Lifshitz (1987). Fluid Mechanics. Vol. 6 (2nd ed.). Butterworth-Heinemann. ISBN 978-0-08-033933-7. – Alex Trounev Jun 28 at 6:11