I have doubts on the use of Bernoulli equation when multiple volumetric flow rates are involved.

Consider for istance the device in picture. Water flows in the main tube, while in the tank below there is insecticide that is sucked into the main tube. enter image description here The volumetric flow rate of water is $Q_{W}$, the desired volumetric flow rate of insecticide is $Q_I$. So at point $3$ the flow rate would be $Q_W+Q_I$.

On textbook the speeds in point $2$ and $3$ are determined as

$$v_{2,\mathrm{water}}=\frac{Q_W}{\pi (\frac{d_1}{2})^2}$$ $$v_{2,\mathrm{insecticide}}=\frac{Q_I}{\pi (\frac{d'}{2})^2}$$ $$v_{3,\mathrm{mixture}}=\frac{Q_W+Q_I}{\pi (\frac{d_2}{2})^2}$$

And I'm ok with that. The problems are with the use of Bernoulli equation in this context (which I found on textbook).

Firstly B.E. is used for insecticide between point $1$ and $2$, and I don't have problems with that.

But the equation is used also between point $2$ and $3$ for water and then mixture, that is

$$p_2+\frac{1}{2} \rho_{\mathrm{water}} v^2_{2,\mathrm{water}}=p_3 +\frac{1}{2} \rho_{\mathrm{water}} v^2_{3,\mathrm{mixture}}$$

The point is that in $3$ there is no just water but there is a mixture, so the fluid is different: is it allowed to use Bernoulli equation between the two points anyway?

My guess would be no because I could also choose to use Bernoulli equation between $2$ and $3$ but considering the insecide instead of water

$$p_2+\frac{1}{2} \rho_{\mathrm{insecticide}} v^2_{2,\mathrm{insecticide}}=p_3 +\frac{1}{2} \rho_{\mathrm{insecticide}} v^2_{3,\mathrm{mixture}}$$

And $v_{2,\mathrm{water}}\neq v_{2,\mathrm{insecticide}}$ so this is in contrast with the previous equaton (at least the different values of $\rho$ solve the problem, which I doubt).

Also for the use of Bernoulli in $3$ I guess that it is not correct to use one particular value of $\rho$, since again the fluid is a mixture.

So my question is: is there any reason for wich it is correct to use Bernoulli equation between $2$ and $3$ in this case? And how should it be used?


2 Answers 2


In principle you can't apply Bernoulli to what is in effect a (simple) network of pipes but in some cases approximations will do.

Bernoulli pipe.

Let's apply Bernoulli's equation to the left and middle sections of the pipe:

$$P_1+\frac12 \rho v_1^2=P_2+\frac12 \rho v_2^2$$

As liquids are incompressible ($A$ is the cross-section of the pipe):


So with a simple substitution:

$$P_2=P_1-\frac12 \rho \Bigg(\Big(\frac{A_1}{A_2}\Big)^2-1\Bigg)v_1^2$$

Since as $A_1 > A_2 \implies P_2 < P_1$

Now apply Bernoulli to the insecticide flow. At point $1$ we'll assume flow velocity to be negligible and pressure to be atmospheric ($P_0$):

$$P_0=P_2+\frac12 \rho v_i^2+\rho gh$$

$$\implies \frac12 \rho v_i^2=P_0-P_2-\rho gh$$

With the above equation:

$$\frac12 \rho v_i^2=P_0-P_1+\frac12 \rho \Bigg(\Big(\frac{A_1}{A_2}\Big)^2-1\Bigg)v_1^2-\rho gh$$

In order to have any upward insecticide flow at all, i.e. $v_i > 0$, then:

$$\frac12 \rho \Bigg(\Big(\frac{A_1}{A_2}\Big)^2-1\Bigg)v_1^2 > P_1-P_0+\rho gh$$

If that condition is not met, then the middle section of the pipe will actually be leaking water into the insecticide dispenser!

If it is met and assuming $v_i\ll v_1$ then your approach should work approximately. But for larger $v_i$ you would need to apply the Cross method for networks of pipes.

I have doubts on the use of Bernoulli equation when multiple volumetric flow rates are involved.

So your doubts are justified but for small dispensed volumes of insecticides your approximate approach should work.


(Subscripts $w,i,m$ correspond to water, insecticide, and mixture respectively.)

First, $Q_m=Q_w+Q_i$ only if $\rho_w=\rho_i=\rho_m$. Otherwise you must equate mass flow rates, $\dot{M}_m=\dot{M}_w+\dot{M}_i$, to find $v_m$, assuming that mixture density is uniform over the cross-section at point 3.

Second, the form of Bernoulli equation you have written down is applicable along a streamline, only if density remains constant along that streamline. So you must find a streamline over which density change is negligible compared to some reference density, say mean density over that streamline. I don't know which of the Bernoulli equations you have written down are valid.


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