Every year at the $$proffentap$$ evening in the Recup, traditionally a lot of barrels of beer are

consumed. A typical beer vessel has an internal diameter of $36$ cm and an internal height of

$55$ cm and contains $50$ liters of beer. The vessel is connected to a tap installation as shown in

the figure below. The vessel itself is located in the basement and the tap outlet is $3$ meters

higher. The pipe connecting the tap to the vessel has a diameter of $9$mm to the tap, which

has an outlet diameter of $5$mm$.$ A gas bottle with CO$2$ keeps the absolute pressure at the

surface of the beer constant at $1\mathrm{.}5$ bar. Since beer is mostly water, you may use the

thermodynamic properties of water.

a$)$ What is the net average pressure on the side wall of the vessel when it is just

connected? Also draw the pressure profile as a function of the height on the inside wall

of the vessel.

Calculate the height of the beer first and the constant pressure force is the difference between

the pressure in the vessel and atmospheric pressure. Calculate the hydrostatic force and add

them together. The average pressure is the total force divided by the area. This give a pressure

of $50\mathrm{.}9$ kPa.

b$)$ It is Professor Vanierschot's turn to tap and a student asks for a $1-$liter pint. How long

does it take to fill the glass? The 'minor losses' may be neglected and, as always, you

may consider beer as an ideal fluid!

Calculate the outflow velocity based on Bernouilli between the top of the beer and the outlet

of the tap. This gives a velocity of $6\mathrm{.}94$ m$/$s$.$ Calculate the flow rate and the time to fill the

glass is the volume of the glass divided by the flowrate. This gives a time of $7\mathrm{.}3$ seconds.

c$)$ Is the flow regime in the pipe when tapping laminar or turbulent in the case where the

viscosity of beer is not negligible? For this purpose, calculate the Reynolds number,

where you use the flow rate of question b$.$ The viscosity of water is $1$x$10-6$ m$2$

$/$s$.$ Which

regime do you think is best and why?

The Reynolds number is $34700,$ so the flow is turbulent

d$)$ What is the filling time of the glass if the total loss coefficient of the tap system K $=$

$1\mathrm{.}2?$

Put the losses in the equation of Bernoulli in the right hand side and follow the same procedure

as question b$.$ This gives a time of $10\mathrm{.}9$ seconds.

e$)$ A heat exchanger cools the beer from $20\backslash$deg C to $3\backslash$deg C as it flows through the pipe. What

is the cooling power required? What is the minimum area required for the heat

exchanger if the heat transfer coefficient is equal to $200$ W$/$m$2$K and the average

temperature difference between refrigerant and beer is equal to $15\backslash$deg C$?$ Use the flow

rate found in question d$.$

NAAM:

$2/2$

The heat is given by Q $=$ c$$T $=6\mathrm{.}54$ kW$,$ where $$ is the mass flow rate and the temperature

difference $\backslash$Delta T is $17$ degrees. The area is calculated as A $=$

Q

h$15$

$=2\mathrm{.}19$ m$2$

$.$