Reynolds Transport Theorem and macroscopic mass balance

CHEE 3363 Spring 2014 Handout 7

�Reading: Fox 4.1, 4.2, 4.3

Learning objectives for lecture

1. State the extensive quantity and the corresponding intensive �

2. State the Reynolds Transport Theorem.�

3.

2

Introduction to control volumes

Classical mechanics: Lagrangian description

- Equations of motion describe the spatial position of a body as a function of time

- Analysis of the motion of a system

Fluid mechanics: Eulerian description

- Difficult to analyze the motion of a complete system -- too many particles!

- Instead, want to analyze the properties of flow through a control volume or at specific points in the flow as fluid particles move through it

t t + ∆t

3

Conservation of mass

Conservation of linear momentum

Conservation of angular momentum

Control volume laws I

dM

dt

∣

∣

∣

∣

sys

= 0

F =dP

dt

∣

∣

∣

∣

sys

T =dL

dt

∣

∣

∣

∣

sys

T = r × Fs +

∫

Msys

r × g dm + Tshaft

linear momentum:

mass:

angular momentum:

torque:

Msys =

∫

Msys

dm =

∫

Vsys

ρdV

Psys =

∫

Msys

vdm =

∫

Vsys

vρdV

Lsys =

∫

Msys

r × v dm =

∫

Vsys

r × vρ dV

4

Control volume laws IIFirst law of thermodynamics

Second law of thermodynamics

Q − W =dE

dt

∣

∣

∣

∣

sys

Esys =

∫

Msys

e dm +

∫

Vsys

eρ dV

e = u +v2

2+ gz

energy:

u: specific internal energy

dS ≥

δQ

T

dS

dt

∣

∣

∣

∣

sys

≥

Q

T

Ssys =

∫

Msys

s dm =

∫

Vsys

sρ dVentropy:

5

Nsys =

∫

Msys

η dm =

∫

Vsys

ηρ dV

Generalized control volume formulationN

(extensive)η

(intensive)

M

P

L

E

S

Select arbitrary piece of fluid (part of system)

Initial shape of fluid system chosen as control volume

- Fixed in space relative to coordinates xyz

After time Δt the system will have moved and changed shape

Conservation laws apply to this piece of fluid

Examine geometry of system/control volume pair at times t0 and t0 + Δt to obtain CV formulations of basic laws

6

Reynolds transport theorem I

Three regions

- I and II make up control volume

- III, which with II is the location of the system at t0 + Δt

Rate of change of Nsys:

x

y

z t0 t0 + Δtx

y

z

systemcontrol volume

I

IIIII

subregion (1) of I

subregion (3) of III

7

dN

dt

∣

∣

∣

∣

s

= lim∆t→0

NCV |t0+∆t

− NCV |t0

∆t+ lim

∆t→0

NIII |t0+∆t

∆t− lim

∆t→0

NI |t0+∆t

∆t

Reynolds transport theorem IIFrom the geometry of the diagram:

a b c

Term a:

8

Ns|t0+∆t= (NII + NIII)t0+∆t = (NCV − NI + NIII)t0+∆t

Ns|t0= NCV |

t0

dN

dt

∣

∣

∣

∣

s

= lim∆t→0

(NCV − NI + NIII)t0+∆t − NCV |t0

∆t

lim∆t→0

NCV |t0+∆t

− NCV |t0

∆t=

∂NCV

∂t=

∂

∂t

∫

CV

ηρdV

dV = ldA cos↵ = l · dA = v · dAt

∆l = v∆t

αdA

v

Reynolds transport theorem III

Term b:

control surface III

system boundary at t0 + Δt

dA is an outward-oriented normal

9

dNIII |t+0+∆t= ηρv·dA∆t

lim∆t→0

NIII |t0+∆t

∆t= lim

∆t→0

∫

CSIII

dNIII |t0+∆t

∆t= lim

∆t→0

∫

CSIII

ηρv·dA∆t

∆t

=

∫

CSIII

ηρv·dA

(from volume of prismatic cylinder)

Reynolds transport theorem IV

10

Term c: lim∆t→0

NI |t0+∆t

∆t= −

∫

CSI

ηρv·dA

Substitute everything back in to obtain:

REYNOLDS TRANSPORT THEOREM:

dN

dt

∣

∣

∣

∣

sys

=∂

∂t

∫

CV

ηρ dV +

∫

CS1

ηρv·dA +

∫

CS1II

ηρv·dA

entire surface

Reynolds transport theorem V

Term 1:

�

Term 2:

�

Term 3:

11

dN

dt

∣

∣

∣

∣

sys

=∂

∂t

∫

CV

ηρ dV +

∫

CS

ηρv·dA

1 2 3

REYNOLDS TRANSPORT THEOREM:

Physical meaning of each term:

Macroscopic mass balanceStart with conservation of mass:

Apply Reynolds Transport Theorem:

Incompressible fluids:

12

Msys =

∫

M(sys)

dm =

∫

V (sys)

ρdV

0 =dM

dt

∣

∣

∣

∣

sys

=∂

∂t

∫

CV

ρdV +

∫

CS

ρv·dA

Example: flow around bend

Problem: Square channel width h, uniform velocity U, 90 degree bend, to obtain output profile as shown with vmax = 2vmin.

Control volume: surround pipe as shown.

Calculate: vmin given U.

Write down conservation of mass:

Assumptions:

13

At (1):

Example: flow around bend 2

Problem: Square channel width h, uniform velocity U, 90 degree bend, to obtain output profile as shown with vmax = 2vmin.

Control volume: surround pipe as shown.

Calculate: vmin given U.

14

At (2):

Example: drain from cylindrical tank 1Problem: Cylindrical tank, radius R, height h; inlet radius ri, outlet radius ro, inlet velocity vi. When tank height is filled to height ht1, outlet pump turned on, exit velocity vo. When tank height is ht2, drain opened such that height remains constant.

Control volume: surround tank as shown.

Calculate:

Write down conservation of mass:

h

(a) Time at which outlet pump is switched on (b) Time at which drain is opened (c) Flow rate into drain

vi

vo

ri

ro

R

CV

Assume:

15

drain

Example: drain from cylindrical tank 2

h

vi

vo

ri

ro

R

CV

(a)

Time at which outlet pump is switched on:

16

drain

Example: drain from cylindrical tank 3

h

vi

vo

ri

ro

R

CV

(b) After outlet pump turned on:

(c) Flow rate into drain = net inflow

17

drain

∫

CS

ρv · dA =∑

CS

ρv · dA

Summary: macroscopic mass balance

18

0 =dM

dt

∣

∣

∣

∣

sys

=∂

∂t

∫

CV

ρdV +

∫

CS

ρv·dA

total change in mass in control

volume with time

flux of mass through control

surface

For uniform velocities that are constant across the control surface: