# Pipe Flow

Pipe flow calculator accounting for heat exchange and pressure drop

Pipe parameters

# 1.   Parameters

Symbol Parameter name Unit
$$d_i$$ internal pipe diameter mm
$$d_e$$ external pipe diameter mm
$$L$$ pipe length m
$$\varepsilon$$ pipe surface roughness mm
$$p_i$$ inlet pressure bar
$$T_i$$ inlet temperature K
$$\dot{m}$$ inlet mass flow rate kg/h
$$T_w$$ pipe temperature K

# 2.   Geometry properties

The flow area is:

\begin{equation*} A_f = \frac{\pi d_i^2}{4} \end{equation*}

The fluid volume is:

\begin{equation*} V = A_f L \end{equation*}

The internal and external surface areas are calculated as:

\begin{equation*} A_i = \pi d_i L \end{equation*}
\begin{equation*} A_e = \pi d_e L \end{equation*}

The mass of the pipe is:

\begin{equation*} m_p = \rho_p \frac{\pi \left( d_e^2 - d_i^2 \right)}{4}L \end{equation*}

where $$\rho_p$$ is the density of the pipe material (steel, aluminum etc.)

# 3.   Pressure losses

In general the pressure loss in pipes and components depends on the upstream fluid density $$\rho$$ and the flow velocity

\begin{equation*} v = \frac{\dot{m}}{\rho A_{f}} \end{equation*}

This pressure loss can be calculated from the formula:

\begin{equation*} \Delta p=c_{d}\frac{\rho v^{2}}{2}, \end{equation*}

where $$c_{d}$$ is the flow drag coefficient. For pipes the drag coefficient depends on the pipe geometry and the Darcy friction factor $$\zeta$$ [Wikip-DFF]:

\begin{equation*} c_{d}=\zeta\frac{L}{d_i} \end{equation*}

where $$L$$ is the length of the pipe and $$d_i$$ is the internal diameter of the pipe.

The friction factor depends on the Reynolds number $$Re={\rho v d}/{\mu}$$ and the relative surface roughness $$\varepsilon/d$$. It can be determined using the Moody chart

Moody chart

In the laminar region the friction factor depends only on the Reynolds number:

\begin{equation*} \zeta = \frac{64}{Re} \end{equation*}

In the turbulent region the relation is more complex and is given by the Colebrook equation [Colebrook39]:

\begin{equation*} \frac{1}{\sqrt{\zeta}} = -2.0 \log_{10} \left(\frac{\epsilon/d_i}{3.7} + {\frac{2.51}{Re \sqrt{\zeta} } } \right) \end{equation*}

In the limit of high Reynolds numbers the friction factor depends solely on the relative surface roughness.

As the Colebrook correlation is implicit in $$\zeta$$, it is not suitable for direct calculations. Different approximations have been developed amongst which special attention deserves the Churchill correlation [Church77], which covers all flow regimes: laminar, transitional and turbulent:

\begin{equation*} \zeta = 8\left[\left(\frac{8}{Re}\right)^{12}+\frac{1}{\left(\Theta_{1}+\Theta_{2}\right)^{1.5}}\right]^{\frac{1}{12}} \end{equation*}
\begin{equation*} \Theta_{1} = \left[2.457\cdot\ln\left(\left(\frac{7}{Re}\right)^{0.9}+0.27\frac{\varepsilon}{d_i}\right)\right]^{16} \end{equation*}
\begin{equation*} \Theta_{2} = \left(\frac{37530}{Re}\right)^{16} \end{equation*}

# 4.   Heat Exchange

Heat exchange can be computed with or without iteration. The steps of the algorithm are:

• if we have $$Re$$, we can find $$Nu=f\left(Re,Pr\right)$$ as follows:

• if $$Re<=2300$$, the flow is laminar and $$Nu=3.66$$
• for $$1e4<=Re<=1e6$$, the flow is turbulent and the Nusselt number is calculated as [VDI]
\begin{equation*} Nu=\frac{\left(\xi/8\right)Re\cdot Pr}{1+12.7\sqrt{\xi/8}\left(Pr^{\frac{2}{3}}-1\right)} \end{equation*}
\begin{equation*} \xi=\left(1.8\cdot\log_{10}Re-1.5\right)^{-2} \end{equation*}
• the transition range is for $$2300<Re<1e4$$, in which the value of $$Nu$$ is obtained through linear interpolation
• $$Re>1e6$$ are values outside the range of validity
• from $$Nu$$ the convection coefficient $$\alpha$$ is found as

\begin{equation*} \alpha=\frac{\lambda\cdot Nu}{d_{i}} \end{equation*}

where $$\lambda$$ is the thermal conductivity of the fluid

• the heat flow rate is thus

\begin{equation*} \dot{Q}_{w}=\alpha A \Delta T \end{equation*}

where $$\Delta T$$ is the temperature difference

• from

\begin{equation*} \dot{Q}_{w}=\dot{m}\cdot h_i-\dot{m}\cdot h_o \end{equation*}

we find the outlet enthalpy $$h_o$$ and obtain the value of the outlet temperature $$T_o$$ from the fluid state determined by the outlet pressure $$p_o$$ and $$h_o$$

In the case of computation with iteration, we guess $$T_o$$, compute the fluid properties at mean temperature

\begin{equation*} T_{m}=\frac{T_{i}+T_{o}}{2} \end{equation*}

and use logarithmic mean temperature difference (LMTD) to caculate the heat exchange:

\begin{equation*} \Delta T=\frac{T_{o}-T_{i}}{ln\frac{\left(T_{w}-T_{i}\right)}{\left(T_{w}-T_{o}\right)}} \end{equation*}

For computation without iteration, $$\Delta T=T_i-T_w$$

# 5.   References

 [Church77] Churchill, S.W. (November 7, 1977). "Friction-factor equation spans all fluid-flow regimes". Chemical Engineering: 91–92.
 [Colebrook39] Colebrook, C.F. (February 1939). "Turbulent flow in pipes, with particular reference to the transition region between smooth and rough pipe laws". Journal of the Institution of Civil Engineers (London).
 [VDI] VDI (Verein Deutscher Ingenieure), Heat Atlas, Springer-Verlag, 2010, Part G1: Heat Transfer in Pipe Flow, p.697, eq(26) and (27)