SOLUTIONS MANUAL FOR
CHEMICAL
ENGINEERING FLUID
MECHANICS
by
Ron Darby
Raj P. Chhabra
, SOLUTIONS MANUAL FOR
CHEMICAL
ENGINEERING FLUID
MECHANICS
by
Ron Darby
Raj P. Chhabra
, Chapter 1
1-1 Write equations that define each of the following laws: Fick’s, Fourier’s, Newton’s, and
Ohm’s. What is the conserved quantity in each of these laws? Can you represent all of these
laws by one general expression? If so, does this mean that all of the processes represented by
these laws are always analogous? If they aren’t, why not?
Solution:
dCA
(a) Fick: 1 − D: n Ay = -DAB , 3 − D : n A = -DAB∇CA
dy
d ( ρCv T ) dT
Fourier: 1 − D : q y = -α = -k , 3- D : q = -k∇T
dy dy
k
α= = Thermal Diffusivity
ρCv
d ( ρv x )
Newton: 1 − D: (τ )
yx m = -ν
dy
dv
dy
(
= -µ x , 3- D: τ m = -µ ∇v + ( ∇v )
+
)
ν = Kinematic viscosity
(3–D form is not analogous to other laws)
de
Ohm: 1 − D : i y = -k e , 3- D : i = -k e∇e
dy
(b) What is the conserved quantity?
Fick: Mass of species A
Fourier: Heat
Newton: x-momentum
Ohm: Charge
dCQ
(c) General Expression: Q y = -k T applies to 1- D forms of all laws
dy
3 − D : Q = -k T ∇CQ applies to all except momentum
d) All but momentum equations are analogous. (in 3–D)
1
, 1-2 The general conservation law for any conserved quantity Q can be written in the form of
Eq. (1-12). We have said that this law can also be applied to ‘‘dollars’’ as the conserved
quantity Q. If the ‘‘system’’ is your bank account,
(a) Identify specific ‘‘rate in,’’ ‘‘rate out,’’ and ‘‘rate of accumulation’’ terms in this
equation relative to the system (i.e., each term corresponds to the rate at which dollars
are moving into or out of your account).
(b) Identify one or more ‘‘driving force’’ effects that are responsible for the magnitude of
each of these rate terms, i.e., things that influence how fast the dollars go in or out. Use
this to define corresponding ‘‘transport constants’’ for each ‘‘in’’ and ‘‘out’’ term
relative to the appropriate ‘‘driving force’’ for each term.
Solution:
Bank
Σ($)in Σ($)out
Balance:
Rate of Rate of Rate of Rate of
∑ − + =
$in $ out Generation of $ Accumulation of $
Rate of
∑ = Paycheck + Gifts + Investment Income + ....
$in
Rate of
∑ = School + Rent + Food + Transport + Clothing + Entertainment + ...
$ out
Rate of
∑ Genetation of $ = Interest ( This can also be considered a "Rate in " term )
Rate of X = k x ⋅ ( DF ) x , DF = driving force
( DF )Paycheck ≅ k1 ( Education + Ingenuity + Experence + ...)
( DF )Gifts ≅ k 2 ( Generosity of Parents )
( DF)Investment ≅ k 3 ( Amount of Savings / Portfolio )
( DF )School ≅ k 4 ( Where + no. hours )
( DF)Rent ≅ k 5 (Size, Loation of Apt.)
( DF)Food ≅ k 6 ("Taste",e.g. Steak vs.Hamburger )
2
CHEMICAL
ENGINEERING FLUID
MECHANICS
by
Ron Darby
Raj P. Chhabra
, SOLUTIONS MANUAL FOR
CHEMICAL
ENGINEERING FLUID
MECHANICS
by
Ron Darby
Raj P. Chhabra
, Chapter 1
1-1 Write equations that define each of the following laws: Fick’s, Fourier’s, Newton’s, and
Ohm’s. What is the conserved quantity in each of these laws? Can you represent all of these
laws by one general expression? If so, does this mean that all of the processes represented by
these laws are always analogous? If they aren’t, why not?
Solution:
dCA
(a) Fick: 1 − D: n Ay = -DAB , 3 − D : n A = -DAB∇CA
dy
d ( ρCv T ) dT
Fourier: 1 − D : q y = -α = -k , 3- D : q = -k∇T
dy dy
k
α= = Thermal Diffusivity
ρCv
d ( ρv x )
Newton: 1 − D: (τ )
yx m = -ν
dy
dv
dy
(
= -µ x , 3- D: τ m = -µ ∇v + ( ∇v )
+
)
ν = Kinematic viscosity
(3–D form is not analogous to other laws)
de
Ohm: 1 − D : i y = -k e , 3- D : i = -k e∇e
dy
(b) What is the conserved quantity?
Fick: Mass of species A
Fourier: Heat
Newton: x-momentum
Ohm: Charge
dCQ
(c) General Expression: Q y = -k T applies to 1- D forms of all laws
dy
3 − D : Q = -k T ∇CQ applies to all except momentum
d) All but momentum equations are analogous. (in 3–D)
1
, 1-2 The general conservation law for any conserved quantity Q can be written in the form of
Eq. (1-12). We have said that this law can also be applied to ‘‘dollars’’ as the conserved
quantity Q. If the ‘‘system’’ is your bank account,
(a) Identify specific ‘‘rate in,’’ ‘‘rate out,’’ and ‘‘rate of accumulation’’ terms in this
equation relative to the system (i.e., each term corresponds to the rate at which dollars
are moving into or out of your account).
(b) Identify one or more ‘‘driving force’’ effects that are responsible for the magnitude of
each of these rate terms, i.e., things that influence how fast the dollars go in or out. Use
this to define corresponding ‘‘transport constants’’ for each ‘‘in’’ and ‘‘out’’ term
relative to the appropriate ‘‘driving force’’ for each term.
Solution:
Bank
Σ($)in Σ($)out
Balance:
Rate of Rate of Rate of Rate of
∑ − + =
$in $ out Generation of $ Accumulation of $
Rate of
∑ = Paycheck + Gifts + Investment Income + ....
$in
Rate of
∑ = School + Rent + Food + Transport + Clothing + Entertainment + ...
$ out
Rate of
∑ Genetation of $ = Interest ( This can also be considered a "Rate in " term )
Rate of X = k x ⋅ ( DF ) x , DF = driving force
( DF )Paycheck ≅ k1 ( Education + Ingenuity + Experence + ...)
( DF )Gifts ≅ k 2 ( Generosity of Parents )
( DF)Investment ≅ k 3 ( Amount of Savings / Portfolio )
( DF )School ≅ k 4 ( Where + no. hours )
( DF)Rent ≅ k 5 (Size, Loation of Apt.)
( DF)Food ≅ k 6 ("Taste",e.g. Steak vs.Hamburger )
2
