Transport Phenomena Numerical (Mass, Energy, Momentum)

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Transport Phenomena
Numerical (Mass, Energy, Momentum)

2. 2
HELLO!
I’m Mujeeb UR
Rahman
Chemical Engineering student @Mehran
University of Engineering & Technology
Jamshoro, Pakistan.
You can find me at SlideShare
@MujeebURRahman38
ResearchGate @Mujeeb UR Rahman, Academia @Mujeeb UR

3. Transport Phenomena
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1.2 The mass balance: exercise 1. (example)
Q.1: We take a classic brass kettle with a capacity of 2.2 liters of water
and start pouring in tap water (density: ρ = 1kg/L) until we filled it with
0.9 liters of water. Then, we turn on the heat on maximum power and
after some time the water starts boiling. As soon as the water starts
boiling (denoted as t = 0), we measure a mass flow over time out of the
kettle of 0.3 g/s. This mass flow is stable until the brass kettle is
completely dried out. For simplicity, we ignore evaporation during the
warming phase, i.e before the water boils.

4. Transport Phenomena
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Given Data:
𝜌 = 1kg/L
V(o) = 0.9L
𝜙𝑚 = 0.3g/s  0.0003kg/s
𝑉 𝜌 𝑀
𝜙𝑚
Now solve:
𝑑𝑀
𝑑𝑡
= −𝜙𝑚 [kg/s] = [kg/s]
∫ 𝑑𝑀= ∫ −𝜙𝑚 . 𝑑𝑡
M = −𝜙𝑚. 𝑡 + 𝐶1
Initial condition : M(o) = 0.9kg
Since = 𝜌 1kg/L and V(o)= 0.9L
M(o)= −𝜙𝑚. 0 + 𝐶1 , 𝐶1 = M(o)
M = −𝜙𝑚. 𝑡 + 𝑀(𝑜)
𝑑𝑀
𝑑𝑡
= In = out + Production
𝑑𝑀
𝑑𝑡
= −𝑜𝑢𝑡 = −𝜙𝑚
So, when is the brass kettle empty?
Empty M = 0
M = −𝜙𝑚 . 𝑡 + 𝑀(𝑜)
0 = −𝜙𝑚 . 𝑡 + 𝑀(𝑜)
t =
𝑀(𝑜)
𝜙𝑚
=
0.9
0.0003
= 3000 𝑠𝑒𝑐𝑜𝑛𝑑𝑠

5. Transport Phenomena
1.2 The mass balance: exercise 2 (example)
Q.2: The swimming pool we are looking at is 25 meters in length, 10
meters in width and 2 meters in depth. The water level is 20 cm below
the terrace of the backyard, so the water level is 1.8 meters. On Sunday
morning, the professor throws 1000 gram of the cleaning product
(chlorine in tablet form) into the pool. Then, after the concentration of
the product is the same everywhere in the water, the professor turns on
the pumps to refresh the water.
The water level is 20 cm below the terrace
at all times and fresh water (with no chlorine
in it) is flowing in with Φv=0.006 m3/s. The
water can flow out at the other side of the
pool through a sink.

6. Transport Phenomena

7. Transport
Phenomena

8. EXAMPLE 2.1A: ENERGY FORMS (ELEMENTARY)
▪ Energy comes in many forms: kinetic energy, potential from gravity,
internal energy, bonds in molecules, etc. Here we are going to develop
a feel for energies, by converting one into the other and by seeing what
we get.
Consider a mass m of 10 kg. It is drop from a height of 100 m (without
initial velocity). We will neglect friction from the air.
1. Compute the velocity with which the mass impacts on the earth.
2. If all this energy is finally converted into internal energy of the mass, i.e,
heat, what will the temperature rise of the mass be? The specific heat
is cp=2500 J/kgK.
▪ Chemical energies, stored in bonds between molecules, is very large.
Consider 1 gram of petrol. When combusted, it will release 45 kJ of
energy.
1. What would the velocity of this 1 gram be if all combustion energy
would be used as kinetic energy for the mass?
2. What height could it reach (assume gravity’s acceleration stays 9.81
m/s2 ‘for ever’)?
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9. Transport
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10. Transport
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11. EXAMPLE 2.1B: ENERGY OF WATERFALLS (MEDIUM)
▪ In the theory lecture we used the Victoria waterfall as an example
to show various forms of energy. Here we will calculate how energy
per second is associated with the falls.
The fall has a height of 108m. The average water flow rate is 1088
m3/s.
▪ Assuming all potential energy can be converted to electricity, what
would be the power of the “Victoria Power Station”? For reference:
a standard modern fossile fuel plant has an electric power output of
1000-2000 MWatt.
▪ Similarly, calculate the power of a hypothetical “Niagara Power
Station”. The Niagara falls have a height of 51m and an average
flow rate of 2400 m3
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12. Transport
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13. EXAMPLE 2.1C: MAKING A CUP OF TEA (ADVANCED)
▪ Time for a cup of tea. So, you need to get boiling water. Easy
enough, you take a water cooker with an (electrical) power of
2200 Watt. You poor in 0.5 liter of water from the tab. This water
has a temperature of 15oC. We would like to know, how long it
takes for the water to boil.
1. Set up an energy balance for the water in the cooker. Determine
which kind of energy you would like to deal with. Is this a steady
or unsteady balance?
2. Turn this energy balance into an equation for the temperature of
the water. To make life simple, we will approximate the problem
by stating that the heat capacity of water is a constant with a
value of 4200J/kgK. Moreover, we will compute when the
temperature of the water has increased to 100oC, but does just
not boil.
3. Solve the temperature equation and find how long it takes for the
water to just/just not boil.
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15. BONUS: EXAMPLE 2.1D: COMPARING ENERGY CONCENTRATIONS
(ELEMENTARY)
▪ In this exercise we are going to compare energy concentrations of various
different cases. We will try to express the energy concentration in terms of
energy per kg.
1. What is your kinetic energy per kg if you are running at 5m/s?
2. What is your potential energy per kg when on top of the Eiffel Tower in Paris
(top is 324m high)
3. What is your kinetic energy per kg when flying in a Boeing cross Atlantic
(cruise speed 900km/h); what is your potential energy per kg (altitude
10km)?
4. Figure out what the energy in regular CocaCola is per kg (try to look it up on
a bottle or tin can)
5. Try to find the chemical energy stored in a kg of gasoline.
6. What is the nuclear energy released when a kg of hydrogen is fused
completely to Helium (as happens in our sun)? Four hydrogen atoms are
needed to form one atom of Heliem.
▪ These numbers give you hopefully some feeling for energy densities: kinetic
and potential are really small, chemical is hugh and nuclear ‘goes through
the roof’.
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17. 1. What is your kinetic energy per kg if you are running at 5m/s?
𝐾𝑖𝑛𝑒𝑡𝑖𝑐 𝐸𝑛𝑒𝑟𝑔𝑦 𝐸𝑘 =
1
2
𝑚𝑣2
𝐾𝑖𝑛𝑒𝑡𝑖𝑐 𝐸𝑛𝑒𝑟𝑔𝑦 𝑝𝑒𝑟 𝑘𝑔( 𝐸𝑘/𝑘𝑔) =
1
2
𝑣2
Required:
𝐸𝑘
𝑘𝑔
= ? ?
Given: v = 5m/s putting in eq.
𝐸𝑘
𝑘𝑔
=
1
2
5 2
=
25
2
= 12.5𝐽/𝑘𝑔
2. What is your potential energy per kg when on top of the Eiffel Tower in
Paris (top is 324m high)?
𝑃𝑜𝑡𝑒𝑛𝑡𝑖𝑎𝑙 𝐸𝑛𝑒𝑟𝑔𝑦 𝑃𝑘 = 𝑚𝑔ℎ
𝑃𝑜𝑡𝑒𝑛𝑖𝑎𝑙 𝐸𝑛𝑒𝑟𝑔𝑦 𝑝𝑒𝑟 𝑘𝑔( 𝑃𝑘/𝑘𝑔) = 𝑔ℎ
Required:
𝑃𝑘
𝑘𝑔
= ? ?
Given: h = 324m putting in eq.
𝑃𝑘/𝑘𝑔 = (9.81)(324)
𝑃𝑘/𝑘𝑔 = 3,178.44 J/kg
Ek/kg = ? ?
V = 5 m/s
h
=
324m
Pk/kg = ? ?
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Phenomena

18. 3. What is your kinetic energy per kg when flying in a Boeing cross
Atlantic (cruise speed 900km/h); what is your potential energy per kg
(altitude 10km)?
𝐾𝑖𝑛𝑒𝑡𝑖𝑐 𝐸𝑛𝑒𝑟𝑔𝑦 𝑝𝑒𝑟 𝑘𝑔( 𝐸𝑘/𝑘𝑔) =
1
2
𝑣2
𝑃𝑜𝑡𝑒𝑛𝑖𝑎𝑙 𝐸𝑛𝑒𝑟𝑔𝑦 𝑝𝑒𝑟 𝑘𝑔( 𝑃𝑘/𝑘𝑔) = 𝑔ℎ
Required: 𝐸𝑘/𝑘𝑔 and 𝑃𝑘/𝑘𝑔
Given: v = 900km/h and height (h) = 10km
Converting km/h to m/s = 250 m/s
Km to m = 10000 m
𝐸𝑘/𝑘𝑔 =
1
2
𝑣2
1).
𝐸𝑘
𝑘𝑔
=
1
2
250 2 31,250𝐽/𝑘𝑔
2).
𝑃𝑘
𝑘𝑔
= 9.81 10000 98 100 𝐽/𝑘𝑔
v = 900km/h
h
=
10km
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19. 4. Figure out what the energy in regular CocaCola is per kg (try to look
it up on a bottle or tin can)?
 Sugar contain there’s around 10.6 g of sugar per 100ml.
Converting g to kg = 10.6/1000 = 0.0106kg.
This means in regular CocaCola Tin contains 100ml per 0.0106kg of
sugar.
5. Try to find the chemical energy stored in a kg of gasoline?
 The mass energy equivalence relation is,
E = mc2
Here, m is the mass of gasoline and c is the speed of light.
The chemical energy of gasoline is 46MJ/kg. The chemical energy of
1kg of gasoline is,
E = 46 MJ/kg
= 46 𝑀𝐽
106𝐽
1 𝑀𝐽
46 × 106𝐽
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Phenomena

20. 6. What is the nuclear energy released when a kg of hydrogen is
fused completely to Helium (as happens in our sun)? Four hydrogen
atoms are needed to form one atom of Heliem.
Four hydrogen nuclei have a total mass of 6.693 x 10-27 kg. They fuse
into a helium nucleus of mass 6.645 x 10-27 kg. The mass lost is
0.048 x 10-27 kg. Thus, only the fraction, 0.048/6.693 = 0.00717 of
the original hydrogen mass was converted to energy.
In other words, when you start with 1 kg of hydrogen, a mass of
0.00717 kg will be converted to energy during the fusion. This gives
an energy output of:
E = mc2 = 0.00717 x (3.0 x 108)2 = 6.45 x 1014 Joules
It is still a huge amount of energy, even though most of the kilogram is
still there afterwards (now in the form of helium).
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21. 2.1 Forms of energy: exercise (example)
 Imagine you are dropping a basketball from a balcony on the
second floor. From the moment the ball leaves your hands, to the
moment it lies still on the floor below, different types of energy are
converted into each other. In this question you will identify the
different types of energy in this process.
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Thanks!