4. Introduction to Heat & Mass Transfer

4. Introduction to Heat & Mass Transfer This section will cover the following concepts: • A rudimentary introduction to mass transfer. • Mass transfer from a molecular point of view. • Fundamental similarity of heat and mass transfer. • Application of mass transfer concepts: - Evaporation of a liquid layer - Evaporation of a liquid droplet

4. Heat & Mass Transfer

1

AER 1304–ÖLG

Mass Transfer: Mass-Transfer Rate Laws: • Fick’s Law of Diffusion: Describes, in its basic form, the rate at which two gas species diffuse through each other. • For one-dimensional binary diffusion: m ˙A , 1

mass flow of A per unit area


= YA (m ˙ A+m ˙ B) − , 1 mass flow of A associated with bulk flow per unit area

2

dYA ρDAB , dx1

mass flow of A associated with molecular diffusion

(4.1)

AER 1304–ÖLG

• A is transported by two means: (a) bulk motion of the fluid, and (b) molecular diffusion. • Mass flux is defined as the mass flowrate of species A per unit area perpendicular to the flow: m ˙A=m ˙ A /A

(4.2)

m ˙ A has the units kg/(s m2 ). • The binary diffusivity, or the molecular diffusion coefficient, DAB is a property of the mixture and has units of m2 /s. 4. Heat & Mass Transfer

3

AER 1304–ÖLG

• In the absence of diffusion: m ˙ A = YA (m ˙ A+m ˙ B ) = YA m ˙ ≡ Bulk flux of species A

(4.3a)

where m ˙ is the mixture mass flux. • The diffusional flux adds an additional component to the flux of A: dYA −ρDAB ≡ Diffusional flux of A, m ˙ A,diff dx (4.3b) 4. Heat & Mass Transfer

4

AER 1304–ÖLG

• Note that the negative sign causes the flux to be postive in the x-direction when the concentration gradient is negative. • Analogy between the diffusion of heat (conduction) and molecular diffusion. - Fourier’s law of heat conduction: dT ˙ Qx = −k (4.4) dx • Both expressions indicate a flux (m ˙ A,diff or Q˙ x ) being proportional to the gradient of a scalar quantity [(dYA /dx) or (dT /dx)]. 4. Heat & Mass Transfer

5

AER 1304–ÖLG

• A more general form of Eqn 4.1: −→ −→ −→ ˙ A+m ˙ B ) − ρDAB ∇YA m ˙ A = YA (m

(4.5)

symbols with over arrows represent vector quantities. Molar form of Eqn 4.5 −→ −→ −→ N˙ A = χA (N˙ A + N˙ B ) − cDAB ∇χA

(4.6)

where N˙ A , (kmol/(s m2 ), is the molar flux of species A; χA is mole fraction, and c is the molar concentration, kmol/m3 . 4. Heat & Mass Transfer

6

AER 1304–ÖLG

• Meanings of bulk flow and diffusional flux can be better explained if we consider that: m ˙ = m ˙A + m ˙B , 1 , 1 , 1

mixture mass flux

species A mass flux

(4.7)

species B mass flux

• If we substitute for individual species fluxes from Eqn 4.1 into 4.7: dYA dYB m ˙ = YA m ˙ − ρDAB + YB m ˙ − ρDBA dx dx (4.8a) 4. Heat & Mass Transfer

7

AER 1304–ÖLG

• Or: dYA dYB m ˙ = (YA + YB )m ˙ − ρDAB − ρDBA dx dx (4.8b) • For a binary mixture, YA + YB = 1; then: dYA dYB − ρDAB − ρDBA =0 , dx1 , dx1 diffusional flux of species A

• In general


(4.9)

diffusional flux of species B

m ˙i =0 8

AER 1304–ÖLG

• Some cautionary remarks: - We are assuming a binary gas and the diffusion is a result of concentration gradients only (ordinary diffusion). - Gradients of temperature and pressure can produce species diffusion. - Soret effect: species diffusion as a result of temperature gradient. - In most combustion systems, these effects are small and can be neglected.


9

AER 1304–ÖLG

Molecular Basis of Diffusion: • We apply some concepts from the kinetic theory of gases. - Consider a stationary (no bulk flow) plane layer of a binary gas mixture consisting of rigid, non-attracting molecules. - Molecular masses of A and B are identical. - A concentration (mass-fraction) gradient exists in x-direction, and is sufficiently small that over smaller distances the gradient can be assumed to be linear. 4. Heat & Mass Transfer

10

AER 1304–ÖLG


11

AER 1304–ÖLG

• Average molecular properties from kinetic of gases: p 8k T Q1/2 Mean speed B of species A v¯ ≡ molecules = πmA pn Q Wall collision 1 A frequency of A ZA ≡ molecules per unit area = v¯ 4 V 1 λ ≡ Mean free path = √ 2π(ntot /V )σ 2 Average perpendicular distance 2 from plane of last collision to a ≡ plane where next collision occurs = λ 3 4. Heat & Mass Transfer

12

theory

(4.10a) (4.10b) (4.10c) (4.10d)

AER 1304–ÖLG

-

where kB : Boltzmann’s constant. mA : mass of a single A molecule. (nA /V ): number of A molecules per unit volume. (ntot /V ): total number of molecules per unit volume. σ: diameter of both A and B molecules.

• Net flux of A molecules at the x-plane: m ˙A=m ˙ A,(+)x−dir − m ˙ A,(−)x−dir 4. Heat & Mass Transfer

13

(4.11)

AER 1304–ÖLG

• In terms of collision frequency, Eqn 4.11 becomes m ˙A , 1

Net mass flux of species A

=

(ZA )x−a , 1

mA −

Number of A crossing plane x originating from plane at (x−a)

(ZA )x+a , 1

mA

Number of A crossing plane x originating from plane at (x+a)

(4.12) • Since ρ ≡ mtot /Vtot , then we can relate ZA to mass fraction, YA (from Eqn 4.10b) 1 nA mA 1 ZA mA = ρ¯ v = YA ρ¯ v 4 mtot 4


14

(4.13)

AER 1304–ÖLG

• Substituting Eqn 4.13 into 4.12 1 m ˙ A = ρ¯ v (YA,x−a − YA,x+a ) 4

(4.14)

• With linear concentration assumption dY YA,x−a − YA,x+a = dx 2a YA,x−a − YA,x+a = 4λ/3


15

(4.15)

AER 1304–ÖLG

• From the last two equations, we get v¯λ dYA m ˙ A = −ρ 3 dx

(4.16)

• Comparing Eqn. 3.16 with Eqn. 3.3b, DAB is DAB = v¯λ/3

(4.17)

• Substituting for v¯ and λ, along with ideal-gas equation of state, P V = nkB T 3 2 p kB T Q1/2 T DAB = (4.18a) 3 2 3 π mA σ P 4. Heat & Mass Transfer

16

AER 1304–ÖLG

or DAB ∝ T 3/2 P −1

(4.18b)

• Diffusivity strongly depends on temperature and is inversely proportional to pressure. • Mass flux of species A, however, depends on ρDAB , which then gives: ρDAB ∝ T 1/2 P 0 = T 1/2

(4.18c)

• In some practical/simple combustion calculations, the weak temperature dependence is neglected and ρD is treated as a constant. 4. Heat & Mass Transfer

17

AER 1304–ÖLG

Comparison with Heat Conduction: • We apply the same kinetic theory concepts to the transport of energy. • Same assumptions as in the molecular diffusion case. • v¯ and λ have the same definitions. • Molecular collision frequency is now based on the total number density of molecules, ntot /V , p Q 1 ntot Average wall collision Z = frequency per unit area = v¯ (4.19) 4 V 4. Heat & Mass Transfer

18

AER 1304–ÖLG


19

AER 1304–ÖLG

• In the no-interaction-at-a-distance hard-sphere model of the gas, the only energy storage mode is molecular translational (kinetic) energy. • Energy balance at the x-plane; Net energy flow in x−direction

=

kinetic energy flux with molecules from x−a to x

−

kinetic energy flux with molecules from x+a to x

Q˙ x = Z (ke)x−a − Z (ke)x+a • ke is given by


3 1 2 v = kB T ke = m¯ 2 2 20

(4.20)

(4.21)

AER 1304–ÖLG

heat flux can be related to temperature as 3 ˙ Qx = kB Z (Tx−a − Tx+a ) 2

(4.22)

• The temperature gradient dT Tx+a − Tx−a = dx 2a

(4.23)

• Eqn. 4.23 into 3.22, and definitions of Z and a p n Q dT 1 Q˙ x = − kB v¯λ (4.24) 2 V dx 4. Heat & Mass Transfer

21

AER 1304–ÖLG

Comparing to Eqn. 4.4, k is Q p 1 n k = kB v¯λ 2 V

(4.25)

• In terms of T and molecular size, p k3 Q1/2 1/2 B k= T π 3 mσ 4

(4.26)

• Dependence of k on T (similar to ρD) k ∝ T 1/2

(4.27)

- Note: For real gases T dependency is larger. 4. Heat & Mass Transfer

22

AER 1304–ÖLG


23

AER 1304–ÖLG

Species Conservation: • One-dimensional control volume • Species A flows into and out of the control volume as a result of the combined action of bulk flow and diffusion. • Within the control volume, species A may be created or destroyed as a result of chemical reaction. • The net rate of increase in the mass of A within the control volume relates to the mass fluxes and reaction rate as follows:


24

AER 1304–ÖLG

dmA,cv dt , 1

Rate of increase of mass A within CV

= [m ˙ A A]x − [m ˙ A A]x+∆x + m ˙ AV , 1 , 1 , 1 Mass flow of A into CV

Mass flow of A out of the CV

Mass prod. rate of A by reaction

(4.28)

- where - m ˙ A is the mass production rate of species A per unit volume. - m ˙ A is defined by Eqn. 4.1.


25

AER 1304–ÖLG

• Within the control volume mA,cv = YA mcv = YA ρVcv , and the volume Vcv = A · ∆x; Eqn. 4.28 ∂(ρYA ) ∂YA = A∆x = A YA m ˙ − ρDAB ∂t ∂x x = ∂YA A YA m ˙ − ρDAB +m ˙ A A∆x ∂x x+∆x (4.29) • Dividing by A∆x and taking limit as ∆x → 0, ∂(ρYA ) ∂ ∂YA = =− YA m ˙ − ρDAB +m ˙A ∂t ∂x ∂x (4.30) 4. Heat & Mass Transfer

26

AER 1304–ÖLG

• For the steady-flow, d dYA = YA m ˙ − ρDAB =0 m ˙A− dx dx

(4.31)

• Eqn. 4.31 is the steady-flow, one-dimensional form of species conservation for a binary gas mixture. For a multidimensional case, Eqn. 4.31 can be generalized as −→ m ˙A − ∇·m ˙A = 0 (4.32) , 1 , 1 Net rate of species A production by chemical reaction


Net flow of species A out of control volume

27

AER 1304–ÖLG

Some Applications of Mass Transfer: The Stefan Problem:


28

AER 1304–ÖLG

• Assumptions: - Liquid A in the cylinder maintained at a fixed height. - Steady-state - [A] in the flowing gas is less than [A] at the liquid-vapour interface. - B is insoluble in liquid A • Overall conservation of mass: ˙ A+m ˙B m ˙ (x) = constant = m


29

(4.33)

AER 1304–ÖLG

Since m ˙ B = 0, then m ˙A=m ˙ (x) = constant Then, Eqn.4.1

(4.34)

now becomes

dYA m ˙ A = YA m ˙ A − ρDAB dx

(4.35)

Rearranging and separating variables m ˙A dYA − dx = ρDAB 1 − YA 4. Heat & Mass Transfer

30

(4.36)

AER 1304–ÖLG

Assuming ρDAB to be constant, integrate Eqn. 4.36 m ˙A − x = − ln[1 − YA ] + C (4.37) ρDAB With the boundary condition YA (x = 0) = YA,i

(4.38)

We can eliminate C; then m = ˙ Ax YA (x) = 1 − (1 − YA,i ) exp ρDAB 4. Heat & Mass Transfer

31

(4.39)

AER 1304–ÖLG

• The mass flux of A, m ˙ A , can be found by letting YA (x = L) = YA,∞ Then, Eqn. 4.39 reads ρDAB 1 − YA,∞ = m ˙A= ln L 1 − YA,i

(4.40)

• Mass flux is proportional to ρD, and inversely proportional to L.


32

AER 1304–ÖLG

Liquid-Vapour Interface: • Saturation pressure PA,i = Psat (Tliq,i )

(4.41)

• Partial pressure can be related to mole fraction and mass fraction YA,i

Psat (Tliq,i ) M WA = P M Wmix,i

(4.42)

- To find YA,i we need to know the interface temperature. 4. Heat & Mass Transfer

33

AER 1304–ÖLG

• In crossing the liquid-vapour boundary, we maintain continuity of temperature Tliq,i (x = 0− ) = Tvap,i (x = 0+ ) = T (0) (4.43) and energy is conserved at the interface. Heat is transferred from gas to liquid, Q˙ g−i . Some of this heats the liquid, Q˙ i−l , while the remainder causes phase change. Q˙ g−i − Q˙ i−l = m(h ˙ vap − hliq ) = mh ˙ fg (4.44) or

Q˙ net = mh ˙ fg 4. Heat & Mass Transfer

34

(4.45) AER 1304–ÖLG

Droplet Evaporation:


35

AER 1304–ÖLG

• Assumptions: - The evaporation process is quasi-steady. - The droplet temperature is uniform, and the temperature is assumed to be some fixed value below the boiling point of the liquid. - The mass fraction of vapour at the droplet surface is determined by liquid-vapour equilibrium at the droplet temperature. - We assume constant physical properties, e.g., ρD.


36

AER 1304–ÖLG

Evaporation Rate: • Same approach as the Stefan problem; except change in coordinate sysytem. • Overall mass conservation: m(r) ˙ = constant = 4πr2 m ˙

(4.46)

• Species conservation for the droplet vapour: dYA m ˙ A = YA m ˙ A − ρDAB dr


37

(4.47)

AER 1304–ÖLG

• Substitute Eqn. 4.46 into 4.47 and solve for m, ˙ ρDAB dYA m ˙ = −4πr 1 − YA dr 2

(4.48)

• Integrating and applying the boundary condition YA (r = rs ) = YA,s

(4.49)

- yields (1 − YA,s ) exp [−m/(4πρD ˙ AB r) YA (r) = 1 − exp [−m/(4πρD ˙ AB rs )] (4.50) 4. Heat & Mass Transfer

38

AER 1304–ÖLG

• Evaporation rate can be determined from Eqn. 4.50 by letting YA = YA,∞ for r → ∞: (1 − Y

= ) A,∞ m ˙ = 4πrs ρDAB ln (1 − YA,s )

(4.51)

• In Eqn. 4.51, we can define the dimensionless transfer number, BY , 1 − YA,∞ 1 + BY ≡ 1 − YA,s 4. Heat & Mass Transfer

39

(4.52a)

AER 1304–ÖLG

or

YA,s − YA,∞ BY = 1 − YA,s

• Then the evaporation rate is

m ˙ = 4πrs ρDAB ln (1 + BY )

• Droplet Mass Conservation: dmd = −m ˙ dt where md is given by md = ρl V = ρl πD3 /6 4. Heat & Mass Transfer

40

(4.52b)

(4.53)

(4.54)

(4.55) AER 1304–ÖLG

where D = 2rs , and V is the volume of the droplet. • Substituting Eqns 4.55 and 4.53 into 4.54 and differentiating dD 4ρDAB =− ln (1 + BY ) dt ρl D

(4.56)

• Eqn 4.56 is more commonly expressed in term of D2 rather than D, dD2 8ρDAB =− ln (1 + BY ) dt ρl 4. Heat & Mass Transfer

41

(4.57)

AER 1304–ÖLG

• Equation 4.57 tells us that time derivative of the square of the droplet diameter is constant. D2 varies with t with a slope equal to RHS of Eqn.4.57. This slope is defined as evaporation constant K: 8ρDAB K= ln (1 + BY ) (4.58) ρl • Droplet evaporation time can be calculated from: 80

dD2 = −

Kdt

(4.59)

0

Do2 4. Heat & Mass Transfer

8td

42

AER 1304–ÖLG

which yields td = Do2 /K

(4.60)

• We can change the limits to get a more general relationship to provide a general expression for the variation of D with time: D2 (t) = Do2 − Kt

(4.61)

• Eq. 4.61 is referred to as the D2 law for droplet evaporation.


43

AER 1304–ÖLG

D2 law for droplet evaporation:


44

AER 1304–ÖLG