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Chapter
3 -- Conservation of Mass |
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Updated
--- 11 Sept 06 0930 |
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Hint |
Answer |
Update Date |
3.1 |
a)
Remember that there is no conservation of volume principle. Apply the
conservation of mass principle to your system to find the instantaneous rate
of change. To find the amount of mass at any time you must
integrate the conservation of mass rate equation. To do this, separate
variables to get dm = [function(t)]dt and
then integrate both sides of the equation. Do not assume that t is a
constant when you do the integration. Suggest you use a definite integral,
but you can use an indefinite integral if you evaluate the constants
correctly. b) Think about what mass is actually in this larger system and how you would proceed. |
a) At t = 4 min |
05Sep05 |
3.2 |
Clearly
identify your system, develop a model using explicit assumptions, and show
your work beginning with the rate-form of the appropriate governing equation.
This is how you tackle any problem where you are using the accounting
principle! |
mdot,3=-40 kg/s, mdot,8=100
kg/s, |
05Sep05
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3.3 |
a)
Use the definition of volumetric flow rate to set up the integral you will
need to evaluate. What should you use for dAc? Note that u only
depends on y, so which is better dAc
= d dy or H dx. b) Apply conservation of mass. There is no conservation of volume. c) Consider the definition of mass flow rate when the velocity profile is uniform. |
a) 0.2531 m3/s |
05Sep05 |
3.4 |
a)
How can you perform an integral over this area if you only know the velocity
at discrete points? What if the velocity was assumed to be uniform over each
sub region? |
a) Vdot=8.58 ft3/s
; |
05Sep05
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3.5 |
For
a) you should use the definition of volumetric flow rate ( |
a)
(2/3)Uhb |
05Sep05
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3.6 |
Part
(a) is a math problem; just use the given velocity equation and plug in the values
given to solve for delta. If you define your system as shown and then follow
the CoM method for that system, Parts (b) and (c)
are not too hard as long as you keep track of your units. Remember to
write down all of your assumptions when computing your mass flow rates! |
a)
delta = 4.99 mm b)
Vtop_dot = 0.25 m3/s c)
vave = 1.25 cm/s |
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3.7 |
For
part b) be sure to use the normal velocity to find mdot. |
a)
mdot,1 = 2.25 lbm/s |
05Sep05 |
3.8 |
Remember
that only the component of velocity normal to your system boundary
contributes to mdot. (And
you, if you haven't been drawing system boundaries, you betta' start right
away!) |
a)
471 kg/s |
05Sep05 |
3.9 |
It's
all about choosing your system and making the conservation of mass reflect
what's going on in the picture of that system. Since we've given you hints on
choosing systems, be sure you actaully draw
the system boundaries. Pay particular attention to where your conservation of
mass terms are on those pictures. |
a)
V_dot = 0.00471 m3/s or 74.7 gpm, v = 15 m/s b)
dh/dt = -0.00471 m/s c) d)
h = 0.353 m |
05Sep05 |
3.10 |
For
an open system you will have a mass
flowing across a boundary, so you probably want your system boundary to cut
through the pipe. For a closed system
there can be no matter entering or leaving the system, so you probably want
to choose a system that accounts for all of the gasonline. |
a)
3 m3/min, 6.4 m/s b)
0.06 m/min c)
0.06 m/min |
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3.11 |
Similar
to in class examples. You should whip this baby. |
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05Sep05 |
3.12 |
When
using the ideal gas model, be careful to handle units correctly. |
a)
0.49 lbm |
05Sep05 |
3.13 |
a)
You can base your mass fraction calculations on the actual number of moles or
on a hypothetical amount. Also keep in mind that there is the universal gas
constant (mole basis) and the gas constant specific to each ideal gas (mass
basis). |
mfO2 = 0.169, |
05Sep05 |
3.14 |
It's
probably easiest to assume 1 lbmol of mix for
analysis. |
a)
mfCH4 = 0.484, mfN2 = 0.065 |
05Sep05 |
3.15 |
On
any accounting of species type problem remember that you will always have an
extra equation left over. Use it as a check. |
mdot,2 = 870 kg/h, mfbenzene,3
= 38.7% |
05Sep05 |
3.16 |
Again,
if you want a method that should always work, you can use species accounting
for each species and then any necessary composition equations (∑mfi = 1). (Of course, you also need
any given information as well.) In this problem you should end up with 10
equations and 10 unknowns. |
Stream
6: |
13Sep05 |
3.17 |
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mdot,5 = 29.99 kg/h, mdot,6
= 126.5 kg/h, mfsolid,3 = 93.0% |
05Sep05 |
3.18 |
Is
there anything fundamentally different with accounting for species on a mol basis rather than a mass basis? Practically speaking,
doesn't this just mean writing n and ndot
instead of m and mdot? |
nfCS2 = 0.0626 |
05Sep05 |
3.19 |
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3.20 |
Note
that for stream 2 the weight percents should be oil
10% and solids 90%. You have 14 unknowns and require 14 independent equations
to solve. Source of equations: 3 equations from composition on streams
3, 4, 5; 2 cons. of mass on mixing tee and condenser (only one species); 2
species acctg on evaporator (no solids); 6 species acctg for extractor and filter; 1 constraint eqn about the ratio of hexane mass to bean oil mass in
the filter cake. Note: You can only write as many species equations for any given system as their are species within or crossing the boundary of the system. |
State
3 -- Effluent State 4 -- Filter Cake State 6 -- Oil State 7 -- 2710 kg/h |
14Sept05 |
3.21 |
a)
Start with cons. of mass first. Remember there is no general law
"conservation of volume" |
b)
80 ft |
05Sep05 |
3.22 |
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a)
17.4 kg |
05Sep05 |
3.23 |
Clearly
identify your system, develop a model using explicit assumptions, and show
your work beginning with the rate-form of the appropriate governing equation.
This how you tackle any problem where you are using the accounting principle! |
-5
lbm/s; 3 lbm/s; -17 lbm/s |
05Sep05 |
3.24 |
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At
t = 2 min, m = 146.7 kg and dm/dt = -50.0 kg/min. |
05Sep05 |
3.26 |
a)
Try an open system with one moving boundary and one outlet. |
b)
2.81 cm/s |
05Sep05 |
3.27 |
Standard
open system with a moving boundary.
Nothing tricky here other than keeping your units consistent. |
a)
dh/dt = 0.0143 m/s (up) b)
V2_dot = 0.046 m3/s (out) c)
V2_dot = 0.034 m3/s (out) |
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3.28 |
Clearly
identify your open system and apply conservation of mass. Where
necessary use the definition of mass flow rate to relate density, normal
velocity, and flow area. Remember that volume is not usually
conserved. |
(a)
0.7 (b) 2.05 |
07Sept05 |
3.29 |
Clearly
identify your open system and remember to use the system boundaries defined
in the problem (even if you do not think they are a great choice!). Remember,
this is a Conservation of Mass problem, so your diagram should only contain
mass or mass flow rate terms. Be sure to list all the assumptions you make
about the flows. |
a)
V3_dot = -0.5m3/s b)
v1 = 0.71 m/s |
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3.30 |
Clearly
identify your open system and apply conservation of mass. Where
necessary use the definition of mass flow rate to relate density, normal
velocity, and flow area. Because the outlet velocity profile is non-uniform,
you must integrate the product of local velocity V over the flow area.
[Hint: You can always pick dA = dxdy, but if velocity only depends on y use
dA = Wdy.
Why would dA = Hdx
be a BAD choice? So you want the largest dA
for which the velocity is essentially constant! If V was a function of
both x and y, then you must use dA
= dxdy.]] |
(a)
9.00 m3/s (b) 9 m/s |
07Sept05 |
3.31 |
Clearly
identify your open system and apply conservation of mass. Where
necessary use the definition of mass flow rate to relate density, normal
velocity, and flow area. |
L = 400 m |
07Sept05 |
3.32 |
Select
an open, deforming system that consists of the gasoline in the tank, so the
volume can be written in terms of h and tank diameter D. Apply
conservation of mass to this system. Simplify assuming that the density is
uniform and constant. Be careful with the math. It is suggested that when
required you separate variables and then use a definite integral. It's
possible to use and indefinite integral but you must evaluation the
constants. |
(a)
6.41 min; 5.44 m (b) 8.56 min |
07SEpt05 |
3.33 |
It is very important to understand the chain rule in this problem! For part (b), try assuming that steady-state will exist and then see if you can find a value of h which satisfies the equation you derive. For part (c), you need to use some more of those calculus skills: separate and integrate! |
a)
V_dot = 3h2 b)
h = 5 m c)
h = sqrt(H2 - 100t/3) |
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3.34 |
What
does it mean in the language of CoM if the fluid
stays at the same height? That is the key to Part (a). Remember that your CoM diagram must only show mass or mass flow rates! List
all your assumptions. For Part (b), look at the units to figure out if the
time varying rate is a mass flow rate, a volumetric flow rate, or a velocity.
Part (c) is a calculus problem; due to the equation the child can drink for
an infinite amount of time but the glass will never empty. If you spot that,
the rest is just math. |
a)
1.59 m/s b)
dh/dt = -0.667exp(-t/5) cm/s c)
2.67 cm |
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3.35 |
Part
(a): choose your system so the liquid melt pouring through the top, and the
solid PVC on the right, both cross through your system boundaries. Remember
to start with Conservation of Mass, and keep track of the densities because
they change between the liquid and solid state. Part (b): choose a new system
to make the math a little easier. |
a)
Vdot = (pi/4)(rhos/rhom)v(Do2-Di2) b)
Ve = (rhos/rhom)v(Do2-Di2)/(De2-Di2) |
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3.36 |
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3.37 |
Parts
(a) and (b) are pretty standard. For Part (c), notice that it is only the top
of speed lake that is draining; the bottom of the lake maintains the same
volume throughout. |
a)
h1 = 2m b)
h1 = 2.21 m c)
dh2/dt = -0.99 m/hr d)
t = 1.01 hr |
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3.38 |
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3.39 |
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3.40 |
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3.41 |
Eight
unknowns. Apply species accounting to get 6 equations and then use
composition at 3 and 5 to obtain remaining equations. |
(a)
2 composition & 6 species acctg (b) m3 =2510 lbm/h |
14Sept05 |
3.42 |
Try
four systems: the mixing tee, the humidifier, the room, and the splitter tee.
Other combinations will also work. Remember that for a splitter tee the mass
fractions do not change between the inlet and the outlet. This should help
you generate some additional equations. |
mdot8
= 503.5
kg/min |
11Sept06 |
3.43 |
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3.44 |
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3.45 |
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3.46 |
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3.47 |
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