Multiphase Flow in Fluid Dynamics - Energy Coupling

[Tom Hardy]

Hello, I have a general question regarding energy coupling in multi phase flow. My question comes is based on the text from this book:

https://books.google.co.uk/books?id=CioXotlGMiYC&pg=PA33&lpg=PA33&dq=thermal+coupling+parameter+continuous+dispersed&source=bl&ots=s9uWpatRmM&sig=-aRSlKoQXHc7gTlZTsl30ROncfo&hl=en&sa=X&ved=0ahUKEwjh6-DfhdzSAhVELMAKHfKOBHAQ6AEIGjAA#v=onepage&q=thermal coupling parameter continuous dispersed&f=false

(page 33)

This is kind of obscure so I'll provide a quick background:

Basically, the premise of the question is that there is a flow of fluid, the fluid has a continuous phase, and within the continuous there exists a dispersed phase, so something like water flowing with little droplets of oil inside the water. The idea is that, suppose the water is flowing through a pipe and into a control volume. The coupling parameter will compare the effect of the dispersed phase on the control volume to the continuous phases effect. So, a flow of fluid with a continuous phase containing n drops per volume flows into a cubic control volume of length L (a volume of $L^3$).

Energy coupling is related to temperature. So, in energy coupling the two things that are to be compared are:

• The energy (heat) released by the droplets (the convective heat transfer from droplets to water)
• The energy provided by the continuous phase (the heat energy of the water)

Thus, the coupling parameter will be:
$$
\Pi_{\text{Energy}} = \frac{\text{Heat Released by Droplets}}{\text{Enthalpy Flux from Water}}
$$
The heat released by the droplets comes from Newtons law of cooling:
$$
mC_d\frac{dT_d}{dt} = Nu k_c \pi D (T_c - T_d)
$$
D is the diameter of the droplet, Tc and Td are the temperatures of the continuous/dispersed phases. Nu is the Nusselt number. Again, there are n droplets per unit volume therefore the total heat released from the droplets is:
$$
nL^3Nu k_c \pi D (T_c - T_d)
$$
The heat flux due to the continuous will just be:
$$
\rho_c u C_c T_c L^2
$$
Which is the density of the continuous phase multiplied by the volume flowing per unit time multiplied by specific heat capacity and it's temperature. The coupling parameter is therefore:
$$
\Pi_{\text{Energy}} = \frac{nL^3Nu k_c \pi D (T_c - T_d)}{\rho_c u C_c T_c L^2}
$$

Ok, so that's all fine and dandy. The issue is when they simplify the above formula. So there exists a thermal response time constant (I do not believe it matters where it comes from, this is basically an algebraic exercise at this point):
$$
\tau_T = \frac{\rho_dC_dD^2}{12k_c}
$$
This can be included in the above equation such that :
$$
\Pi_{\text{Energy}} = \frac{nL^3Nu k_c \pi D (T_c - T_d) \rho_d C_d D^2 \tau_T}{\rho_c u C_c T_c L^2 \tau_T }
$$
This can be cleaned up, Nu/2 can be approximated to 1 at low Reynolds numbers. piD^2/6 multiplied by density of a droplet (spherical) gives the mass of that droplet, multiply that by n and you end up with the total droplet density per unit volume. The above equation simplifies to:
$$
\Pi_{\text{Energy}}= \frac{T_dCLC_d}{uC_c T_C \tau_T } \bigg(1-\frac{T_c}{T_d}\bigg)
$$
Where C is the ratio of total dispersed phase density and continuous phase density.

Here is the problem, the book quotes the final result as:
$$
\Pi_{\text{Energy}}= \frac{CL}{u\tau_T } \bigg(1-\frac{T_c}{T_d}\bigg)
$$
Which implies that:
$$
\frac{C_dT_d}{C_c T_c} = 1
$$

I don't see any justification for that though...does anyone know where the extra terms went?

Thanks.

 

[Chestermiller]

I worked on a Physics Forums thread very similar to this about a month ago. It was a mass transfer problem rather than a heat transfer problem, but the basic setup was the same. Here is the reference: https://www.physicsforums.com/threads/cocurrent-diffusion-cant-be-this-hard.904956/
Here is the corresponding analysis for your problem:
If the total volumetric flow rate of continuous and discontinuous phases is F and the volume fraction of disperse phase is f, then Eqns. 1 and 2 can be rewritten as:
$$f\rho_D C_D\frac{dT_D}{dt}=\alpha \phi\tag{1}$$
$$(1-f)\rho_C C_C\frac{dT_C}{dt}=-\alpha \phi\tag{2}$$where t is the cumulative residence time (t = V/F), $\alpha$ is the heat transfer surface per unit volume and $\phi$ is the heat flux through the interface. The heat flux is given by:
$$\phi=\frac{k_CNu}{2R}(T_C-T_D)\tag{3}$$where R is the discontinuous phase drop radius.
The discontinuous phase drop radius R in Eqn. 3 is related to the volume fraction of drops f and the surface area per unit volume $\alpha$ by the equations:
$$4\pi R^2 n=\alpha\tag{4}$$
$$\frac{4}{3}\pi R^3 n=f\tag{5}$$where n is the number of drops per unit volume. If we divide Eqn. 5 by Eqn. 4, we obtain:
$$R=\frac{3f}{\alpha}\tag{6}$$
If we add equation 1 and 2, we find that $$f\rho_D C_DT_D+(1-f)\rho_C C_CT_C=f\rho_D C_DT_{D0}+(1-f)\rho_C C_CT_{C0}\tag{7}$$
Combining Eqns. 1-3 and 6 yields:
$$\frac{dT_D}{dt}=\frac{k_C\ Nu\ \alpha^2}{6f^2\rho_D C_D}(T_C-T_D)\tag{8}$$
$$\frac{dT_C}{dt}=-\frac{k_C\ Nu\ \alpha^2}{6f(1-f)\rho_C C_C}(T_C-T_D)\tag{9}$$
If we subtract Eqns. 8 and 9, we obtain: $$\frac{d\ln(T_D-T_C)}{dt}=-\left[\frac{1}{f\rho_D C_D}+\frac{1}{(1-f)\rho_C C_C}\right]\frac{k_C\ Nu\ \alpha^2}{6f}\tag{10}$$
Hope this helps.