Equation of Continuity II

Summary of Equations of Change

( 𝑟𝑎𝑡𝑒𝑜𝑓𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑒𝑜𝑓 𝑒𝑛𝑡𝑖𝑡𝑦 )=(𝑛𝑒𝑡 𝑟𝑎𝑡𝑒𝑜𝑓𝑎𝑑𝑑𝑖𝑡𝑖𝑜𝑛

𝑜𝑓 𝑒𝑛𝑡𝑖𝑡𝑦 )+( 𝑛𝑒𝑡 𝑟𝑎𝑡𝑒𝑜𝑓𝑝𝑟𝑜𝑑𝑢𝑐𝑡𝑖𝑜𝑛𝑜𝑓 𝑒𝑛𝑡𝑖𝑡𝑦 )

Summary of Equations of Change

Summary of Equations of Change

𝝅=𝑝𝜹+𝝉The momentum molecular flux,

*

molecular stresses = pressure + viscous stresses

Summary of Equations of Change

𝒒=(h𝑒𝑎𝑡 𝑡𝑟𝑎𝑛𝑠𝑝𝑜𝑟𝑡𝑏𝑦 𝑐𝑜𝑛𝑑𝑢𝑐𝑡𝑖𝑜𝑛)+( h𝑒𝑎𝑡 𝑡𝑟𝑎𝑛𝑠𝑝𝑜𝑟𝑡𝑏𝑦 h𝑒𝑎𝑐 𝑑𝑖𝑓𝑓𝑢𝑠𝑖𝑛𝑔𝑠𝑝𝑒𝑐𝑖𝑒𝑠 )

The energy molecular flux is the partial molar enthalpy of species α

Summary of Equations of Change

Recall: the combined energy flux vector e

Combined Energy Flux Vector

Convective Energy FluxHeat Rate from Molecular Motion

Work Rate from Molecular Motion

Combined Energy Flux Vector:

𝒆=( 12 𝜌𝑣2+𝜌 �̂�)𝒗+ [𝝅 ∙𝒗 ]+𝒒

We introduce something new to replace q:

Combined Energy Flux Vector

Combined Energy Flux Vector:

We introduce something new to replace q:

𝝅=𝑝𝜹+𝝉Recall the molecular stress tensor:When dotted with v: [𝝅 ∙𝒗 ]=𝑝𝒗+[𝝉 ∙𝒗 ]

Substituting into e:

𝒆=( 12 𝜌𝑣2+𝜌 �̂�)𝒗+𝑝𝒗+[𝝉 ∙𝒗 ]+𝒒

Summary of Equations of Change

Recall: Substituting the equation for q into e

Summary of Equations of Change

Recall: Substituting the equation for q into e

partial molar

per unit mass

Summary of Equations of Change

Recall: Substituting the equation for q into e

Summary of Equations of Change

Summary of Equations of Change

Dp

Dt

vg

𝜌𝐶𝑝𝐷𝑇𝐷𝑡

=− (𝛻 ∙𝑞)− (𝜏 ∙𝛻 𝒗 )

𝜌𝐷𝑤 𝐴

𝐷𝑡=(𝛻 ∙ 𝜌 𝐷𝐴𝐵𝛻𝑤𝐴 )+𝑟 𝐴

Simultaneous Heat and Mass Transfer

Example 1.

Hot condensable vapor, A, diffusing through a stagnant film of non-condensable gas, B, to a cold surface at y=0, where A condenses

Find:

Simultaneous Heat and Mass Transfer

Assumptions:

1. Steady-state2. Ideal gas behavior3. Total c is constant4. Uniform pressure5. Physical properties are

constant, evaluated at mean T and x.

6. Neglect radiative heat transfer

Simultaneous Heat and Mass Transfer

𝜕𝑐𝐴𝜕𝑡

+(𝛻 ∙𝑐𝐴𝑣𝑀 )− (𝛻 ∙𝑐𝐷 𝐴𝐵𝛻𝑥𝐴 )=𝑅𝐴

Equations of Change:

Continuity (A)

𝜕𝑐𝐴𝜕𝑡

=− (𝛻 ∙𝑁𝛼 )+𝑅𝐴

𝑑𝑁 𝐴 𝑦

𝑑𝑦=0

Simultaneous Heat and Mass Transfer

Equations of Change:

Energy𝜕𝜕𝑡𝜌(�̂�+1

2𝑣2)=− (𝛻 ∙𝒆 )+(𝜌𝒗 ∙𝒈 )

𝑑𝑒𝑦𝑑𝑦

=0

* Both NAy and ey are constant throughout the film

Simultaneous Heat and Mass Transfer

To determine the mole fraction profile:

Recall: The molar flux for diffusion of A through stagnant B

Concentration Profiles

I. Diffusion Through a Stagnant Gas Film

𝑁 𝐴=−𝑐𝐷𝐴𝐵

𝑑𝑥𝐴𝑑𝑧

+𝑥𝐴(𝑁 𝐴+𝑁𝐵)

Since B is stagnant,

𝑁 𝐴=−𝑐𝐷 𝐴𝐵

(1−𝑥𝐴)𝑑𝑥𝐴𝑑𝑧

Simultaneous Heat and Mass Transfer

To determine the mole fraction profile:

Recall: The molar flux for diffusion of A through stagnant B

𝑁 𝐴 𝑦=−𝑐𝐷 𝐴𝐵

(1− 𝑥𝐴)𝑑𝑥𝐴𝑑 𝑦

Recall: Integration of the above equation

Concentration Profiles

I. Diffusion Through a Stagnant Gas Film− ln (1− 𝑥𝐴 )=𝐶1𝑧+𝐶2

Let C1 = -ln K1 and C2 = -ln K2,

1−𝑥𝐴=𝐾 1𝑧𝐾 2

B.C.

at z = z1, xA = xA1

at z = z2, xA = xA2 ( 1−𝑥𝐴1− 𝑥𝐴1 )=( 1−𝑥𝐴21−𝑥𝐴1 )𝑧− 𝑧 1𝑧 2− 𝑧1

Simultaneous Heat and Mass Transfer

To determine the mole fraction profile:

Recall: The molar flux for diffusion of A through stagnant B

𝑁 𝐴 𝑦=−𝑐𝐷 𝐴𝐵

(1− 𝑥𝐴)𝑑𝑥𝐴𝑑 𝑦

( 1−𝑥𝐴1− 𝑥𝐴 0 )=( 1−𝑥𝐴δ1−𝑥𝐴0 )𝑦𝛿

Using the appropriate B.C.s

at y = 0, xA = xA0

at y = δ, xA = xAδ

Simultaneous Heat and Mass Transfer

To determine the mole fraction profile:

( 1−𝑥𝐴1− 𝑥𝐴 0 )=( 1−𝑥𝐴δ1−𝑥𝐴0 )𝑦𝛿𝑁 𝐴 𝑦=−

𝑐𝐷 𝐴𝐵

(1− 𝑥𝐴)𝑑𝑥𝐴𝑑 𝑦

Evaluating NAy from the equations aboveNote that:

𝑑(1−𝑥¿¿ 𝐴)

𝑑𝑦=−

𝑑𝑥𝐴𝑑 𝑦

¿ 𝑁 𝐴 𝑦=𝑐𝐷 𝐴𝐵

(1−𝑥𝐴)𝑑

(1−𝑥¿¿ 𝐴)𝑑𝑦

¿

Simultaneous Heat and Mass Transfer

𝑑(1−𝑥¿¿ 𝐴)

𝑑𝑦=−

𝑑𝑥𝐴𝑑 𝑦

¿ 𝑁 𝐴 𝑦=𝑐𝐷 𝐴𝐵

(1−𝑥𝐴)𝑑

(1−𝑥¿¿ 𝐴)𝑑𝑦

¿

𝑁 𝐴 𝑦 𝑦=𝑐𝐷𝐴𝐵 ln1−𝑥𝐴1−𝑥𝐴0

( 1−𝑥𝐴1− 𝑥𝐴 0 )=( 1−𝑥𝐴δ1−𝑥𝐴0 )𝑦𝛿

BUT…]

𝑁 𝐴 𝑦∫0

𝑦

𝑑𝑦=𝑐𝐷𝐴𝐵

(1−𝑥𝐴)∫𝑥 𝐴 0

𝑥 𝐴

𝑑(1−𝑥¿¿ 𝐴)¿

Simultaneous Heat and Mass Transfer

𝑁 𝐴 𝑦=𝑐𝐷 𝐴𝐵

𝛿ln (1−𝑥𝐴δ1−𝑥𝐴0 )

]

Simultaneous Heat and Mass Transfer

𝑁 𝐴 𝑦=𝑐𝐷 𝐴𝐵

𝛿ln (1−𝑥𝐴δ1−𝑥𝐴0 )

Rearranging and combining

( 1−𝑥𝐴1− 𝑥𝐴 0 )=( 1−𝑥𝐴δ1−𝑥𝐴0 )𝑦𝛿

𝑁 𝐴 𝑦

𝑐𝐷𝐴𝐵

= 1𝛿ln (1−𝑥𝐴δ1−𝑥𝐴0 ) ln ( 1−𝑥𝐴1−𝑥𝐴0 )= 𝑦𝛿 ln(

1−𝑥𝐴δ1−𝑥𝐴0 )

ln ( 1−𝑥𝐴1−𝑥𝐴0 )=𝑦𝑁𝐴 𝑦

𝑐𝐷 𝐴𝐵

Simultaneous Heat and Mass Transfer

@ y = y, xA = xA

ln ( 1−𝑥𝐴1−𝑥𝐴0 )=𝑦𝑁𝐴 𝑦

𝑐𝐷 𝐴𝐵

( 1−𝑥𝐴1− 𝑥𝐴 0 )=exp[(𝑁 𝐴 𝑦

𝑐𝐷𝐴𝐵)¿𝑦 ]¿

1−( 1−𝑥𝐴1−𝑥𝐴0 )=1−exp [(𝑁 𝐴 𝑦

𝑐𝐷 𝐴𝐵)¿ 𝑦 ]¿

( 𝑥𝐴−𝑥𝐴01−𝑥𝐴0 )=1− exp[( 𝑁𝐴 𝑦

𝑐𝐷 𝐴𝐵)¿ 𝑦 ]¿

Simultaneous Heat and Mass Transfer

@ y = y, xA = xA

( 𝑥𝐴−𝑥𝐴01−𝑥𝐴0 )=1− exp[( 𝑁𝐴 𝑦

𝑐𝐷 𝐴𝐵)¿ 𝑦 ]¿

@ y = δ, xA = xAδ

( 𝑥𝐴𝛿−𝑥𝐴01− 𝑥𝐴 0 )=1− exp[( 𝑁 𝐴 𝑦

𝑐𝐷𝐴𝐵)¿𝛿]¿

Taking the ratios of the two equations

Simultaneous Heat and Mass Transfer

To determine the temperature profile:

Note:

where the enthalpy of mixing is often neglected for gases at low to moderate pressures

Simultaneous Heat and Mass Transfer

To determine the temperature profile:

𝑑𝑒𝑦𝑑𝑦

=0

−𝑘 𝑑2𝑇𝑑𝑦2

+𝑁𝐴𝑦~𝐶𝑝𝐴

𝑑𝑇𝑑𝑦

=0

The general solution is

𝑇=𝑐1+𝑐2 exp(𝑟 2𝑦 ) where

Simultaneous Heat and Mass Transfer

where

At y = 0, T = T0 𝑇 0=𝐶1+𝐶2

At y = δ, T = Tδ

𝑇=𝑐1+𝑐2 exp(𝑟 2𝑦 )

𝑇 𝛿=𝑐1+𝑐2 exp(𝑟2𝛿)

Subtracting the two equations

𝑇 0−𝑇 𝛿=𝑐2 ¿

𝑐2=𝑇 0−𝑇 𝛿

1− exp (𝑟2𝛿¿)¿

Simultaneous Heat and Mass Transfer

𝑐1=𝑇 0(1− exp (𝑟 2𝛿 ))−(𝑇 0−𝑇 𝛿)

1−exp (𝑟 2𝛿)

𝑐2=𝑇 0−𝑇 𝛿

1− exp (𝑟2𝛿¿)¿Since 𝑇 0=𝐶1+𝐶2

𝑇=𝑐1+𝑐2 exp(𝑟 2𝑦 )

𝑇=𝑇 0(1− exp (𝑟 2𝛿 ))−(𝑇 0−𝑇 𝛿)

1−exp (𝑟 2𝛿)+

𝑇0−𝑇 𝛿

1−exp (𝑟2𝛿¿)exp (𝑟2 𝑦 )¿

Simultaneous Heat and Mass Transfer

𝑇=𝑇 0(1− exp (𝑟 2𝛿 ))−(𝑇 0−𝑇 𝛿)

1−exp (𝑟 2𝛿)+

𝑇0−𝑇 𝛿

1−exp (𝑟2𝛿¿)exp (𝑟2 𝑦 )¿

𝑇 (1− exp (𝑟 2𝛿 ))=𝑇 0 (1−exp (𝑟2𝛿 ))− (𝑇 0−𝑇 𝛿 )+(𝑇 0−𝑇 𝛿)(exp (𝑟 2𝑦 ))

(𝑇 −𝑇 0 ) (1−exp (𝑟2𝛿 ) )= (𝑇 𝛿−𝑇 0 )(1−exp (𝑟2 𝑦 ))

where (𝑇 −𝑇 0 )(𝑇 𝛿−𝑇 0 )

=1− exp (𝑟 2 𝑦 )1−exp (𝑟2𝛿 )

Simultaneous Heat and Mass Transfer

If we did not consider mass transfer

𝑒𝑦=−𝑘𝑑𝑇𝑑𝑦𝑑𝑒𝑦𝑑𝑦

=0−𝑘 𝑑2𝑇𝑑𝑦2

=0

−𝑘𝑑𝑇𝑑𝑦

=𝑐1

𝑑𝑇𝑑𝑦

=𝑐1−𝑘

𝑇=𝑐1𝑦−𝑘

+𝑐2

@ 𝑦=0 ,𝑇=𝑇 0

𝑐2=𝑇 0

@ 𝑦=𝛿 ,𝑇=𝑇 𝛿

𝑇 𝛿=𝑐1𝛿−𝑘

+𝑇0

𝑐1=(𝑇0−𝑇 𝛿 ) (𝑘𝛿 )=−𝑘 𝑑𝑇𝑑𝑦

Simultaneous Heat and Mass Transfer

With mass transfer

−k𝑑𝑇𝑑𝑦

∣@ 𝑦=0=𝑇0−𝑇 𝛿

1− exp[(𝑁𝐴𝑦 𝑐𝑃𝐴𝑘 )𝛿 ] (

𝑁 𝐴𝑦 𝑐𝑃𝐴𝑘 )∗𝑘

Simultaneous Heat and Mass Transfer

Comparison of the energy flux with & without the presence of mass transfer:

Rate of heat transfer is directly affected by simultaneous mass transferBUT mass flux is not directly affected by simultaneous heat transfer