Liquids; volume and mass flow rates

In summary, if the mass flow rates are the same, then the density ratios are 2v_{2}/v_{1}. If the volume flow rates are the same, then the height at which liquid is added or drained to equalize the volume flow rates is 1/2.
  • #1
bacon
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Suppose that two tanks, 1 and 2, each with a large opening at the top, contain different liquids. A small hole is made in the side of each tank at the same depth h below the liquid surface, but the hole in tank 1 has half the cross-sectional area of the hole in tank 2. (a) What is the ratio [itex]\rho_{1}/\rho_{2}[/itex] of the densities of the liquids if the mass flow rate is the same for the two holes? (b) What is the ratio of the volume flow rates from the two tanks? (c) To what height above the hole in the second tank should liquid be added or drained to equalize the volume flow rates?

Here is how I first went through parts (a) and (b):

(a) The mass flow rate equation: [itex] Av\rho= constant[/itex]
Since the mass flow rates are the same,
[itex] A_{1}v_{1}\rho_{1} = A_{2}v_{2}\rho_{2}[/itex]
and since [itex]A_{1}=A_{2}/2[/itex] the above equation becomes, with a little algebra,
[itex]\rho_{1}/\rho_{2}=2v_{2}/v_{1}[/itex]
And setting [itex]v_{1}=v_{2}[/itex] I got
[itex]\rho_{1}/\rho_{2}=2[/itex]

(b) The volume flow rate equation: [itex] Av= R [/itex] a constant
[itex]A_{1}v_{1}=R_{1}[/itex]
[itex]A_{2}v_{2}=R_{2}[/itex]
Using the same substitutions as in (a) I end up with,
[itex]R_{1}/R_{2}=1/2[/itex]

Both the answers are correct(from the back of the book), but going back through the problem I cannot justify [itex]v_{1}=v_{2}[/itex]. It made sense at the time I did it but now it does not, so I need conceptual help here.

(c) I can't get very far here.
I know that the volume flow rates are equal so,
[itex]A_{1}v_{1}=A_{2}v_{2}[/itex] and [itex]A_{1}=A_{2}/2[/itex]
This gives [itex]v_{1}=2v_{2}[/itex]
I'm suspecting Bernoulli's Equation may come into play here but I'm fuzzy as to how.
[itex]P_{1} + 1/2\rho_{1}v_{1}^{2}+\rho_{1}gh= a constant[/itex]
[itex]P_{2} + 1/2\rho_{2}v_{2}^{2}+\rho_{2}gy_{2}= a constant[/itex]
But not necessarily the same constant. Solving for [itex]y_{2}[/itex] is what I would want to do but there are too many unknowns.
Thanks for any help.
 
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  • #3
ideasrule, that was the piece I needed. The rest of the problem went well.
Thank you.
 

Related to Liquids; volume and mass flow rates

1. What is the difference between volume and mass flow rates?

Volume flow rate refers to the amount of liquid that passes through a specific point in a given amount of time. It is measured in units of volume per unit of time, such as cubic meters per second. Mass flow rate, on the other hand, refers to the amount of mass that passes through a specific point in a given amount of time. It is measured in units of mass per unit of time, such as kilograms per second. In simpler terms, volume flow rate measures the quantity of liquid while mass flow rate measures the weight of the liquid.

2. How are volume and mass flow rates related?

Volume and mass flow rates are related through density. Density is the measure of how much mass is contained within a given volume. The formula for density is mass divided by volume. Therefore, if the density of a liquid is known, the volume and mass flow rates can be calculated using this formula.

3. How is mass flow rate affected by temperature?

Mass flow rate is not affected by temperature as mass is an intrinsic property of a substance and remains constant regardless of temperature. However, temperature can affect the density of a liquid, which in turn can affect the volume flow rate. As temperature increases, the density of most liquids decreases, resulting in an increase in volume flow rate.

4. What factors can affect the volume and mass flow rates of liquids?

The volume and mass flow rates of liquids can be affected by several factors, such as the properties of the liquid (e.g. viscosity, density), the size and shape of the container, the pressure and temperature of the system, and the presence of any obstructions or restrictions in the flow path.

5. How are volume and mass flow rates measured?

Volume flow rate is typically measured using a flow meter, which can be based on various principles such as mechanical, electromagnetic, or ultrasonic. Mass flow rate can be measured using a mass flow meter, which typically uses a thermal or coriolis principle. Both flow rates can also be calculated using the appropriate formulas and measuring the necessary parameters, such as velocity and cross-sectional area.

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