Fundamentals of Bio-transport Phenomena

Fundamentals of Bio-transport Phenomena Youngwoon Choi, Ph.D. School of Biomedical Engineering, Korea University [email protected]

Chapter 1:

Modes of Heat Transfer

Objectives

1. Understand the physical processes and the rate laws describing the following three modes of heat transfer • Conduction • Convection • Radiation 2. Understand the material properties that affect heat conduction in a material.

Modes of Heat Transfer

Conductive heat transfer Through a medium

Convective heat transfer Through a medium

Radiative heat transfer No medium

Conductive Heat Transfer • • • •

Movement of thermal energy through a medium from its more energetic particles to the less energetic. The energy of molecules is transferred by collision process. → Material mediated

Microscopic motions of molecules are involved. Material itself doesn’t move (no macroscopic movement). collision

• •

energy transfer

•

Energy transfer through metallic materials is faster than non-metallic materials. Energy transport through liquid is less effective than a solid. The lowest energy transport through a gas.

Thermal Conductivity Thermal conductivity: Measure of the efficiency of heat conduction often mostly water Free electrons

Molecules are closely packed

Fourier’s Law Why negative? Rate of heat flow

dT qx  kA dx

[W]

qx: rate of heat flow in the x direction A: area perpendicular to the x direction k: thermal conductivity of the medium

Heat flux

dT qx "   k dx

[W/m2 ]

qx” = qx/A heat flux: heat flow per unit tie per unit area Determined experimentally

Unit of thermal conductivity (k) : W/(mK) or Btu/(hr ft °F)

Example 2.1.1 Compare heat fluxes for brick (k = 0.69 W/mK) and wood (k = 0.208 W/mK) in

out

Schematic and given data: 1. Surface temperatures of the wall are 5 °C and 20 °C 2. Thickness of the wall is 50 mm 3. Thermal conductivity of brick is 0.69 W/mK and that of wood is 0.208 W/mK

Assumptions: 1. Variation of thermal conductivity with temperature is negligible 2. Heat transfer is one-dimensional, along the thickness

Thermal Diffusivity U=cpT : thermal energy per unit volume

dT k d   c pT  dU qx "   k    dx cp dx dx Volumetric heat capacity

=k/cp : thermal diffusivity

Flux of energy= x Gradient of energy

Density and Specific Heat Density () : kg/m3 Specific Heat (cp) : J/kg·K Volumetric heat capacity (cp): J/m3·K Density Solid density (mean particle density) : mass per unit volume of just the sold portion in porous media Bulk density : mass of the solid to its total volume (solids and pores together)

※ Porosity : volume of pores occupied by air or water (if present) divided by the total volume of the solid

Convective Heat Transfer • •

The movement of heat through a medium as a result of the net motion of a material in the medium (ex. fluid flow and air flow) → Material mediated Macroscopic movement is involved. Convection over a surface

q12  hA T1  T2 

q1-2: heat flow rate from 1-2 A: area normal to the direction of heat flow T1-T2: temperature difference between surface and fluid h: convective heat transfer coefficient

Convective heat transfer coefficient Experimentally determined

[W/m2°C]

Convection includes conduction

more details in Ch. 6

Radiative Heat Transfer “All matter, when at temperatures above absolute zero, emits radiative energy” Changes in the electron configuration of the atom within the matter causes emission of electromagnetic waves Does not require medium for energy transfer

Stefan-Boltzmann Law q  T 4 for black body A Stefan-Boltzmann constant W   5.676 108 2 4 mK

more details in Ch. 8

Chapter 3

Governing Equation and Boundary Conditions of Heat Transfer

Objectives 1. Identify the terms describing storage, convection, diffusion and generation of energy in the general governing equation for heat transfer. 2. Specify the three common types of heat transfer boundary conditions. 3. Describe heat transfer in a mammalian tissue with blood vessels using the bioheat transfer equation.

Governing Equation for Heat Transfer Energy flow rate by bulk movement of a fluid mass flow (convection) E  mc p T  TR   uA c p T  TR  u: fluid velocity TR: reference temperature

Heat flux due to convection E  u  c p T  TR  A From the 1st law of thermodynamics Energy In -

Energy Out

+

Energy Generated

=

Energy Stored

Governing Equation for Heat Transfer Energy in during time Dt

 q " DyDz  uDyDz c x

p

T  TR  x  Dt

Energy out during time Dt

q

x Dx

" DyDz  uDyDz  c p T  TR 

x Dx

 Dt

Energy generated during time Dt QDxDyDzDt

Energy stored during time Dt DxDyDz  c p DT

Governing Equation for Heat Transfer Using the energy conservation

 q " DyDz  uDyDz c x

p

T  TR  x  Dt

-

q

x Dx

" DyDz  uDyDz  c p T  TR  +

=

x Dx

QDxDyDzDt

DxDyDz  c p DT

Dividing throughout by DxDyDzDt and rearranging:

u (T  TR ) x Dx  u (T  TR ) x   qx Dx  qx DT   cp  Q  cp Dx Dx Dt Using the definition of a derivative:

 Using the heat flux, qx”



qx  T   c p (uT )  Q   c p x x t

  T  k x  x

 T    c ( uT )  Q   c p p  x t 

 Dt

TR is dropped

Governing Equation for Heat Transfer With an assumption that k is a constant

generation

T  (uT ) k  2T Q    t x  c p x 2  c p storage flow or convection

conduction

The general governing equation for energy transfer in one dimensional Cartesian coordinate system with constant thermal properties. The continuity equation (incompressible flow) or mass conservation is often used.

u 

u v w   0 x y z

u 0 x

T T k  2T Q u   t x  c p x 2  c p

in one dimensional problem

Example 3.1.1 1-D slab problem : boundary value problem (BVP) Consider a system such as a slab with temperature variation only in one dimension. The system is at steady-state and having no heat generation. 0 0 0

T1

T T k  2T Q u   t x  c p x 2  c p

T2

L

Simply,

d 2T 0 dx 2

A general solution to this 2nd order ODE is T = C1x+C2