Quiz 1

What is the basic conservation-law idea?

Accumulation= In - Out + Source - Sink

What are the first questions I should ask when reading a quiz 1 problem?

1. What am I conserving

2. What enters

3. What leaves

4. Is anything generated?

5. Is anything removed/decayed

6. Is it steady or transient

7. Is it perfectly mixed?

8. Is density constant?

9. Is velocity uniform or variable?

10. What geometry/surface does the material cross?

What is the system mass balance for a conservative constituent?

What does QC represent?

A mass flow rate

Never use C alone as the IN or OUT mass rate

What does VC represent?

The mass of constituent stored in the system

d(VC)/dt is the rate of accumulation

What does steady state mean mathematically?

Accumulation=0

so d/dt=0

It does not mean IN and OUT individually each equal zero

What words suggest a transient problem?

• initially

• changes with time

• C(t)

• V(t)

• after __ hours
  begins at t=0

• concentration suddenly changes

• how long until….

• accumulation

• water level rises/falls

Keep the derivative

Do constant flow rates automatically mean steady state?

No

Constant Qin and Qout can still produce changing storage.

Example:

Qin>Qout

means dV/dt>0

even though both flows are constant

What does perfectly mixed mean?

The concentration leaving equals the concentration inside

Cout=Csystem=C

The system is treated as having one uniform concentration

Perfectly mixed + constant volume + conservative contaminant gives what equation?

V*dC/dt= QCin-QC

or

dC/dt= Q/V(Cin-C)

What is mean residence/ detention time?

T=V/Q

It represents a characteristic time for fluid to pass through a system

What useful relationship follows from T=V/Q?

T=V/Q —> Q/V=1/T

therefore:

dC/dt= 1/T(Cin-C)

What is the general solution for a perfectly mixed tank with constant Cin, no reaction, and constant V,Q?

C(t)= Cin + (C0-Cin)e^-t/T

where

T=V/Q

If initially C0=0 and contaminated water begins entering, what does the solution become?

C(t)=Cin(1-e^-t/T)

The concentration rises exponentially toward Cin

If contaminated input stops and clean water enters, what happens?

Exponential decay:

C(t)= C0e^-t/T

where C0 is the concentration when the clean water period begins

What do I do when conditions change at a specific time?

Solve the problem piecewise.

The ending concentration from period 1 becomes the initial concentration for period 2

If the change happens at t=ts:

C=Cse^(-(t-ts)/T)

What does “emits”, produces”, “genrates” or “added at __mass/time” mean?

SOURCE

If the source is given directly as mass/time:

+W

What words indicate a sink?

• decays

• consumed

• metabolized

• removed

• uptake

• degradation

These produce a negative term

What is the sink term for first order decay?

-VkC

because the first order reaction rate is proportional to concentration

k has units of:

1/time

What is the complete mixed-system balance with a source W and first order sink?

d(VC)/dt= QCin-QC +W - VkC

if steady:

0=QCin-QC +W -VkC

Stead mixed system with a source but no decay?

0= QCin -QC +W

therefore:

C=Cin +W/Q

What if k is the unknown?

Write:

0=QCin-QC+W-VkC

Then algebraically isolate k

k=QCin+W-QC/VC

What is the constant-density water-volume balance

dV/dt= Qin-Qout

Include evaporation or other losses as outflow/loss terms

What does a positive dV/dt mean?

dV/dt>0 = storage increase

Negative means storage decreases

0 means constant storage

How do i relate changing volume to changing water depth in a tank/lagoon with constant horizontal area?

V=A*h

so:

dV/dt= Adh/dt

dh/dt= net volumetric inflow/A

What clues suggest i should use the control volume approach?

• piper

• inlet/outlet surface

• velocity

• diamter/radius

• flow through a surface

• well screen

• veolcity profile

• u( r )

• derive velocity from Q

• cylindrica;/circular geometry

What is the general control-volume mass conservation equation?

What does steady do in the control volume equation?

Eliminates the accumulation volume integral

It does NOT eliminate the surface integral

What does incompressible mean?

density=constant

Density can often be canceled

What determines the sign of a control-surface flow term

The outward normal n

An inlet: n*u< 0= IN negative

n * u> )= OUT positive

When can i use Q=uA

When u is uniform over the area or the problem gives an average velocity

Q=uA

When must i integrate the velocity

When velocity varies with position, such as:

u=u ( r)

what is dA for a thin circular ring?

dA= 2 pi r dr

What is the full polar coordinate element

dA= rdr dtheta

If the problem gives u(r ), what should immediately go through my mind?

Variable velocity profile —> Integrate

Do not automatically plug the max velocity into uA

Area of a circular opening?

Area of the side of a cylinder?

Volume of a cylindrical tank?

Can I assume Qin=Qout for a gas if pressure or temperature changes?

No

Gas density can change

p1Q1=p2Q2

Idea gas equation

From PV=nRT

p=PM/RT`

What useful gas-flow relationship follows from mass conservation + ideal gas law?

P1Q1/T1=P2Q2/T2

Recognition clue:

gas + changing P,T —> density may change

What does the continuum assumption mean?

We treat matter as continuously distributed rather than tracking individual molecules/particles.

Whether this is reasonable depends on the scale of the system compared with the particles.

What continuum criterion does Chapter 1 use?

1/n ««« L³

How do I solve a continuum problem using 1/N≪≪≪L

Calculate:

1/N

and

L^3

Then compare them

What does 1/N represent dimensionally?

Approximate volume per particle

If N[particles/m³]

then:

1/N [m³/particles]

What unit check should I perform on every constituent mass balance

Every term should reduce to:

mass/time

Relationship between mg/L and g/m³?

What should I do before combining flow rates with time?

Make the time units agree.

Examples:

1hr=60min=3600s

Don't combine m3/s with hours without converting.

What if a tablet/object loses mass over time and that lost material becomes a contaminant source?

Determin its mass-loss rate first:

dissolution rate=

Then multiply by the active-ingredient fraction if needed.

That becomes the source W in the constituent balance.

This is the central setup in the Fall 2020 chlorinator problem

If a tablet is 30%30\% active ingredient, how do I find active-ingredient source rate?

What if solids/sludge accumulate inside part of a control volume?

Do not set accumulation to zero.

Use:

Accumulation= rate entering region- rate leaving region

Then relate accumulated mass to concentration *accumulated volume

M=CV

What if the accumulating material has constant concentration C

M=CV

so:

dM/dt= CdV/dt

if C is constant

What is the key idea when a boundary layer gets thicker with time

The control volume containing material is growing, so accumulation remains

V= A* Delta

then dV/dt= A* ddelta/dt

What did the membrane-type derivation reduce to in our Chapter 1 work?

If dV/dt is positive and constant, what does V(t) look like?

A straight increasing line.

Constant derivative = constant slope.

If dC/dt becomes smaller as C approaches equilibrium, what shape should C(t) have?

Exponential approach to a limiting concentration.

It rises quickly at first, then flattens.

What does exponential decay look like?

Rapid decrease initially, followed by slower decrease approaching zero:

C=C0​e−t/τ​

Steady state means Qin=Qout. True or false?

Not universally.

Steady constituent concentration means constituent accumulation is zero.

For constant-volume incompressible water:

∑Qin=∑Qout

But always determine what quantity is being balanced.

“Which equation do I use?” master card