.. _section-A5.4:

Appendix 5.4: Air Blast Heat Exchanger Stack Momentum Equation
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

This appendix derives an expression for the air mass flowrate through
the natural convection stack. The stack contains an opening at its base
through which air is drawn in, the air passes over the finned tubes of
the air blast heat exchanger and then rises to be exhausted at the top
of the stack.

The one-dimensional steady-state momentum equation for flow in a channel
of uniform cross section is

.. math::
	:label: A541

	\frac{\text{dP}}{\text{dz}} = \frac{- \text{d}}{\text{dz}} \left( \rho v^{2} \right) - \rho g \sin{\theta} - \tau \frac{P_{\text{w}}}{A}~ ,

where

:math:`p` =pressure

:math:`\rho` =density

:math:`v` =velocity

:math:`\tau` =wall shear stress

:math:`P_{\text{w}}` =wetted perimeter

:math:`A` =flow area

:math:`\theta` =channel inclination relative to horizontal

Integrating :eq:`A541` gives the pressure change along the channel

.. math::
	:label: A542

	\Delta P = - \left( \frac{w}{A} \right)^{2} \left( \frac{1}{\rho_{\text{o}}} - \frac{1}{\rho_{\text{i}}} \right) - \rho_{\text{m}}g\mathcal{l} \sin{\theta} - \frac{K}{2\rho_{\text{m}}} \left( \frac{w}{a} \right)^{2}

where

:math:`w` =channel mass flowrate

:math:`\mathcal{l}` =channel length

:math:`\rho_{\text{o}}` =outlet density

:math:`\rho_{\text{i}}` =inlet density

:math:`\rho_{\text{m}}` =mean density

:math:`K` =flow loss coefficient

Using :eq:`A542`, the pressure change form stack inlet to above the
heat exchanger is

.. math::
	:label: A543

	\Delta P = - \left( \frac{w}{A_{\text{R}}} \right)^{2} \left\lbrack \frac{K_{\text{SI}}}{2\rho_{\text{c}}}\left( \frac{A_{\text{R}}}{A_{\text{SI}}} \right)^{2} + \frac{K_{\text{HX}}}{2\rho_{\text{c}}}\left( \frac{A_{\text{R}}}{A_{\text{HX}}} \right)^{2} \right\rbrack

where

:math:`A_{\text{SI}}` = stack inlet cross-sectional area

:math:`A_{\text{R}}` = riser cross-sectional area

:math:`A_{\text{HX}}` = flow area at heat exchanger

:math:`K_{\text{SI}}` = stack inlet loss coefficient

:math:`K_{\text{HX}}` = heat exchanger loss coefficient

:math:`\rho_{\text{c}}` =inlet air density

The gravity and acceleration terms have been neglected.

Similarly, the pressure change from the start of the riser to the stack
outlet is

.. math::
	:label: A544

	\Delta P = - \left( \frac{w}{A_{\text{R}}} \right)^{2} \frac{\left( K_{\text{SO}} + K_{\text{R}} \right)}{2\rho_{\text{h}}} - \rho_{\text{h}} g\mathcal{l}

where

:math:`K_{\text{SO}}` = stack outlet loss coefficient

:math:`K_{\text{R}}` = riser loss coefficient

:math:`\rho_{\text{h}}` = riser air density

:math:`\mathcal{l}` = riser length

The pressure change from the stack outlet through the outside air back
to the stack inlet is approximately

.. math::
	:label: A545

	\Delta P = \rho_{\text{c}} g \mathcal{l}

The above three pressure changes, :eq:`A543` through :eq:`A545`, must sum
to zero since they are taken around a closed circuit. Solving for the
air flowrate yields

.. math::
	:label: A5.4‑6

	w^{2} = \frac{\left( \rho_{\text{c}} - \rho_{\text{h}} \right) g \mathcal{l} A_{\text{R}}^{2}}{\frac{K_{\text{SI}}}{2\rho_{\text{c}}} \left( \frac{A_{\text{R}}}{A_{\text{SI}}} \right)^{2} + \frac{K_{\text{HX}}}{2\rho_{\text{c}}} \left( \frac{A_{\text{R}}}{A_{\text{HX}}} \right)^{2} + \frac{\left( K_{\text{SO}} + K_{\text{R}} \right)}{2\rho_{\text{h}}}}