heat and mass lecture 1

Heat and Mass Transfer II BE2-HHMT2

Winter Term 2010/11

Dr. Darryl R. Overby Lecturer, Dept. of Bioengineering

Office: RSM 4.33

Dr. Jennifer Siggers (Mass Transport) Dr. Danny O’Hare (Chemical Kinetics)

BE2-HHMT2, Heat&Mass Transfer II D.R. Overby, Revised Jan-17-11

Heat and Mass Transfer II (aka. Transport Phenomena)



5-10 hrs



Schedule of Events:

Lectures: Mondays 11:00, RSM 2.28 Fridays 16:00, RSM 2.28

Tutorials: Mondays 09:00, RSM 3.21C, 3.03, 4.01 (starting Jan 24, 2011)

Office Hours: Wednesdays 12:00 – 14:00, RSM 4.33 or by appointment

Examination: TBA


Lecture 1: Introduction and Basic Definitions Lecture Objectives: 1.  To define transport phenomena and give examples where

transport processes are important for the function of biological systems.

2.  To describe the two fundamental processes of mass transport: convection and diffusion.

3.  To describe the engineering definition of a fluid and its properties, the continuum assumption and the engineering concept of stress.

Reading: Truskey §1.1-1.3, 2.3.3 (1.4-1.8, extra)

BE2-HHMT2, Heat&Mass Transfer II D.R. Overby, Revised Dec-28-10

What is “Transport Phenomena”?


• An engineering science that deals with the motion of mass, momentum and/or energy through a medium.

• Transport phenomena is often defined by sub-fields:

• momentum transport –how fluid flows in response to applied forces. (similar to fluid mechanics)

• mass transport – how chemical species move through a solid, liquid or gaseous medium.

• heat transport – how energy (including heat) moves through a solid, liquid or gaseous medium.

•  There are links between the transport of heat, mass and momentum.

What are Transport Phenomena important for biology?


- Homeostasis and normal physiological function requires substances to be in the right places in the body at the right concentrations.

-  Knowledge of transport phenomena is essential for: -  understanding physiology and mechanisms of disease -  design and operation of medical devices -  development of new therapies (e.g., drug delivery or molecular medicine)

- Let’s look at a few examples. (see Truskey §1.4-1.8).

Example 1: Calcium Transport Within the Cell


Calcium is an important signalling molecule, and it’s concentration is tightly regulated in the cell.

Lodish et al., 2000 (fig 18-29).

Contraction in skeletal muscle cells

Alberts et al., 2002 (fig 16-69).

http://biology.kenyon.edu/courses/biol105/html/calcium2.jpg

Example 2: Low-Density Lipoprotein


LDL is a large molecular complex that transports cholesterol from the liver to tissues throughout the body.

Rose and Afanasyeva, Nat Med, 2003

Hahn and Schwartz, Nature 2009 (Fig1)

Example 3: Gas Transport in the Lungs


The primary job of the lungs is to exchange oxygen and carbon dioxide with the blood.

Weibel, 1984

Sapoval et al., 2002

Weibel, 1973

Fundamental Transport Processes


There are 2 fundamental processes at play in transport phenomena

Diffusion – arises from random motion and collision between molecules.

Convection – transport due to bulk motion of a fluid medium.

Additional Definitions:

Solute – the species of interest that is being transported through the medium

Solvent – the medium itself, typically water for biological applications. Usually (but not always!) solvent molecules outnumber solute.

Solution – mixture of solute + solvent

Diffusion


Hyperlinks: http://galileo.phys.virginia.edu/classes/109N/more_stuff/Applets/brownian/brownian.html

Generated using Lattice Random Walk in 3D in Mathematica by Stephen Wolfram

•  Diffusion time:

€

td ∝L2

DL -- distance molecule diffuses [cm] D -- Diffusivity [cm2/sec]

http://galileo.phys.virginia.edu/classes/152.mf1i.spring02/UVaBrownianDemo.mpg

•  Attributed to Brownian motion (after Robert Brown, 1827)

•  Arises from random collisions or “random walk”

•  Efficient over very small distances (<~ 100 µm)

Typical Values of Diffusivity


Diffusing Quantity Diffusion Coefficient* [cm2/sec] Gases in gases 0.1 – 0.5 Gases in liquids 1x10-7 – 7x10-5

Small molecules in liquids 1x10-5

Proteins in liquids 1x10-7 – 7x10-7

Proteins in tissues 1x10-7 – 7x10-10

Lipids in lipid membranes 1x10-9

Proteins in lipid membranes 1x10-10 – 1x10-12

Truskey, Table 1.1

* binary diffusion coefficient at room temperature

Convection


•  Arises from bulk flow of solvent.

•  Solute is transported with same local velocity as solvent.

•  Efficient over very large distances (>~ 100 µm).

•  Convection time:

L -- distance molecule is convected [cm] V – local solvent velocity [cm/sec]

€

tc ∝LV

Quiz


•  Consider the transport of oxygen (O2) through the body by convection and diffusion.

•  Calculate the time for O2 transport over a distance equal to one cell (10 µm) and over a distance equal to the thickness of tissue (1 mm).

•  Make the following assumptions: •  D = 2x10-5 cm2/sec for O2 •  V = 100 µm/sec, corresponding to the capillary blood velocity

O2 Transport Time L = 10 µm L = 1 mm

Diffusion

Convection

Quiz


•  Consider the transport of oxygen (O2) through the body by convection and diffusion.

•  Calculate the time for O2 transport over a distance equal to one cell (10 µm) and over a distance equal to the thickness of tissue (1 mm).

•  Make the following assumptions: •  D = 2x10-5 cm2/sec for O2 •  V = 100 µm/sec, corresponding to the capillary blood velocity

O2 Transport Time L = 10 µm L = 1 mm

Diffusion 0.05 sec 500 sec

Convection 0.1 sec 10 sec

Diffusion Faster

Convection Faster

Peclet Number (Pe)


•  Used to weigh the relative contribution of convection to diffusion.

•  Defined as a ratio of diffusion time to convection time.

•  If Pe >> 1, convection is faster, and convection dominates transport. •  If Pe << 1, diffusion is faster, and diffusion dominates transport. •  If Pe ~ 1, diffusion and convection are both important.

•  Pe is a dimensionless number, and like other dimensionless numbers (e.g., Reynold’s number) it weighs the relative importance of different physical phenomena (e.g., inertial vs. viscous forces).

€

diffusion timeconvection time

≡td

tc

=L2

DVL

=V LD

≡Pe

Fluid Properties: Stress


€

σ =

σ xx σ xy σ xz

σ yx σ yy σ yz

σ zx σ zy σ zz

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

σyx

x

y

σyy

σxy

σxx

σxx

σyy

σxy

σyx

•  Stress is force per unit area. Units: N/m2, Pa, dynes/cm2 •  Depends on 2 vectors: force and surface orientation •  Stress is a tensor of order 2.

- Often written as σij

i -- direction of orientation vector (x,y,z) j -- direction of force vector

i=j for a normal stress i≠j for a shear stress

Stress tensor is symmetric σij = σji

Fluid Properties: Definition of a Fluid


•  Any substance that deforms continually under shear stress. •  Gasses and liquids are usually fluids. •  Corollary: Shear stress must be zero in a fluid at rest.

• Some substances exhibit both fluid-like and solid-like behaviour (viscoelastic)

Upper plate moves, then stops. Upper plate continually moves.

Fluid Properties: Density, Viscosity


Property Symbol Description Units

Density ρ Mass per unit volume g/cm3

Viscosity (Dynamic viscosity) µ Resistance to flow under shear g/(cm sec)

Kinematic viscosity ν = µ/ρ “Diffusivity” of momentum cm2/sec

Typical values of fluid parameters given in Truskey Table 1.2

The Continuum Concept


€

ρ = limΔV→0

Δ massΔV

•  Analysis requires that we define fluid properties at a point.

•  As ∆V 0, eventually molecular variations dominate.

•  Chose some δV that is much larger than molecules, but much smaller than dimensions of interest (L).

€

δV << L

For continuum assumption to hold,

Extensive vs. Intensive Properties


Extensive property (or extrinsic property) •  Depends on the size or amount of material contained in system •  Cannot be defined at a point •  Examples: mass, volume

Intensive property (or intrinsic property) •  Independent of size or amount of material •  Can be defined at each point •  Examples: density, concentration

Integrating an intensive property over space translates an intensive property into an extensive property.

Note, defining intensive properties implicitly assumes that the continuum assumption is valid (otherwise, properties will fluctuate wildly!)

Lecture 1: Introduction and Basic Definitions Lecture Objectives: 1.  To define transport phenomena and give examples where

transport processes are important for the function of biological systems.

2.  To describe the two fundamental processes of mass transport: convection and diffusion.

3.  To describe the engineering definition of a fluid and its properties, the continuum assumption and the engineering concept of stress.

Reading: Truskey §1.1-1.3, 2.3.3 (1.4-1.8, extra)


heat and mass lecture 1

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