Biothermodynamics Heat Transfer Exam 1

Lectures 9, 10, 11, 12, and 13

thermodynamics

• “Macro” systems, boundary, surroundings

• Based on conversions between energy types: Kinetic, potential, internal, work, heat, etc.

• Equilibrium states - Properties are spatially uniform - system is in equilibrium with itself

heat transfer

• Infinitesimal control volume, boundary, surroundings

• Based on mechanisms of heat transfer: Conduction, convection, radiation

• Properties are continuously distributed in space and vary with time

NO LONGER using quasi-equilibrium system

cross sectional area

unlike thermodynamic processes, which occur within devices like a piston cylinder, heat transfer occurs across a __________, with a temperature difference driving this exchange

conduction

One mechanism of heat transfer: Occurs by the transfer of the energy of motion between adjacent molecules. Occurs in solids, liquids, and gases. Present in all solid, liquids, and gases in which a temperature gradient exists. Driving gradient - temperature difference driving exchange of heat in the system of interest

conduction in gases and liquids

occurs due to molecular collisions and molecular diffusion

conduction in solids

due to lattice vibrations (energy exchange between bonds) and the flow of free electrons

convection

One mechanism of heat transfer: Occurs by bulk transport and mixing of macroscopic elements. Occurs in liquids and gases

forced convection

fluid/gas is forced to flow past a solid surface by mechanical means

Ex: exercising outside, cool breeze (wind and sweat interact)

natural convection

natural circulation of fluid due to a density difference resulting from a temperature gradient.

Ex: concrete on a hot day

radiation

One mechanism of heat transfer: Transfer of energy through space via electromagnetic waves. All solids, liquids, and gases emit, absorb, or transmit _______ to some degree

thermal radiation

radiation emitted by bodies because of their temperature; Unrelated to x-rays, gamma rays, microwaves, radio waves, television waves

conservation of heat equation

for a rectilinear object

constant flux

First application of conservation of heat: no energy generation (e_gen = 0), 1D heat transfer (dQ/dy=dQ/dz=0), and Steady state (dT/dt=0).

Fourier’s law of heat conduction

second application of conservation of heat: empirical law - gathered data and found an equation to fit.

isotropic material

properties of the material are constant with direction. Ex: constant thermal conductivity

anisotropic

materials change with respect to direction. k=f(x)

diffusivity

how fast heat diffuses (movement without bulk motion) through a material.

α = k/ρc

heat conducted/heat stored per unit volume.

heat flow through a flat plate

conductive heat transfer in hollow cylinder

heat flows in radial direction, inside out. Flow (Q) is indpendent of direction BUT flux (Q/A) is dependent on direction.

conductive heat transfer in hollow sphere

Q(r₂-r₁) /( 4πr₁r₂k) = T₁-T₂

conduction definition

requires a material medium. Solids; Fluids (liquid or gas) in the absence of bulk motion; Quiescent fluid

convection definition

Requires a material medium. Fluids (liquids or gas) in the presence of bulk motion

bulk motion and convection

must now consider both heat transfer and fluid motion. Heat AND momentum transfer

fluid motion

enhances heat transfer. Brings warmer and cooler “chunks” of fluid into contact. Increases conduction. Increased mixing (fluid velocity) leads to increased rate of heat transfer.

heat transfer coefficient (h)

Rate of heat transfer between a solid surface and a fluid per unit surface area per unit temperature difference

Function of:

• type of fluid flow (laminar or turbulent), Re

• dynamic viscosity, μ

• thermal conductivity, k

• density, ρ

• specific heat capacity, c_p

• fluid velocity, v

• geometry of surface, L_c

• roughness of solid surface, C_f

type of fluid flow

Laminar: crystalline, smooth stream

Turbulent: chaotic, splashy

quantified by Reynolds number

dynamic viscocity

a measure of the resistance to flow in a fluid; metric of how sticky a fluid is - how well individual layers stick to each other - internal friction in the fluid

thermal conductivity

how well heat is conducted through the fluid. The amount of heat conducted/transferred within a unit temperature gradient through a unit thickness perpendicular to a unit surface area

specific heat capacity

how much energy the fluid can store per unit mass per temperature

fluid velocity

increases with mixing; the time rate of change of a fluid particle's position in space

geometry of surface

more complex = more complex flow conditions. Quantified by characteristic length

roughness of solid surface

smooth: not as much friction conducted to form boundary layer

rough: more friction through layers of fluid. more energy - more movements/bumps

empirical coorelation

The heat transfer coefficient cannot be predicted theoretically; phenomenological relationship supported by experimentation and not necessarily theory

Nusselt number

used to relate data for the heat transfer coefficient to thermal conductivity of a system

laminar

highly ordered motion and smooth streamlines

turbulent

velocity fluctuations and highly disordered motion. circular pipe: Re < 1200

transition

flow fluctuates between laminar and turbulent before it becomes fully turbulent. circular pipe: Re > 6000

Reynolds number

represents the ratio of inertial to viscous forces within a fluid and is used to indicate the laminar or turbulent nature of a flow

fluid flow

Fluid consists of adjacent layers of molecules. Friction between the moving plate and the adjacent fluid layer initiates fluid movement

movement of fluid layers

First fluid layer attempts to drag adjacent fluid layer along with it due to friction force. Perpetuates throughout fluid

Bottom layer is “stuck” to the stationary plate due to friction → No Slip Condition

no slip condition

velocity of the adjacent plate is stationary. Velocity of plate is same as this type of fluid (?)

assumes that the speed of the fluid layer in direct contact with the boundary is identical to the velocity of this boundary. There is no relative movement between the boundary and this fluid layer, therefore there is no slip.

shear stress

friction force per unit area

Dynamic viscosity (μ)

Measure of a fluid’s resistance to deformation, depends strongly on temperature

• Newtonian Fluids: μ ≠ f(u or y) → Water, oil, saline

• Non-Newtonian μ = f(u or y) → Blood, mayonnaise, oobleck (cornstarch and water)

viscous flow region

Region of flow where frictional effects are significant

inviscid flow region

Region of flow where viscous forces are negligibly small compared to inertial or pressure forces

boundary layers of viscous flow region

• Fluid with uniform (no change with location) velocity approaches a stationary (V=0) flat plate

• Bottom layer of fluid sticks to the plate: No slip condition

• Motionless layer of fluid slows down particles of adjacent layer

• Continues until the diffusion of momentum between the layers becomes insignificant

• X component of velocity varies from u=0 at y=0 to u = V_b at y > δ

  • V =0.99*V_b where V is the velocity of the free stream or inviscid flow region

region of viscous flow

0 ≤ y ≤ δ

boundary layer

region of inviscid flow

y > δ

viscous forces << inertial or pressure forces

thermal boundary layer

• Fluid with a uniform velocity and temperature (T_ꚙ) approaches a stationary plate with a temperature (T_s)

• Bottom layer of fluid reaches thermal equilibrium with plate (no slip condition)

• Bottom layer of fluid exchanges energy with adjacent layers

• Continues until the diffusion of energy between layers becomes insignificant

• Temperature varies from T=Ts at y=0 to T = T_ꚙ at y > δ_t

  • T=T_s+0.99(T_ꚙ-T_s) at y=δ

Prandtl number

relative thickness of the velocity and thermal boundary layers; a dimensionless quantity that correlates the viscosity of a fluid with its thermal conductivity, assessing the relationship between momentum transport and thermal transport capacity.

0.5 ≤ Pr ≤ 1 for gases

2 ≤ Pr ≤ 10⁴ for liquids

dimensionless numbers

Nusselt, Reynolds, and Prandtl numbers

relating dimensionless numbers

Nu=cReⁿPr^m

where c, n, and m are constants.

convective heat transfer assumptions

1. Boundary layer approximations apply

2. Gravity and other body forces are negligible

3. dP/dy=0

4. dP/dx inside and outside of boundary layer is negligable

5. Fully developed flow

6. Upstream velocity is uniform and steady

7. Incompressible flow

8. simple compressible system

boundary layer approximations

1. V_x >> V_y

2. dV_y/dy and dV_y/dx are negligible and dV_x/dy >> dv_x/dx

3. dT/dy >> dT/dx

gravity and other body forces

negligible for convective heat transfer ie only surface forces (shear forces between layers of fluid) are being considered.

pressure

variation of _______ in the y-direction is negligible for convective heat flow. the variation of _______ in the x direction inside and outside the boundary layer is also negligible

fully developed flow

Hydrodynamically: V_avg is a function of r only

Thermally: T(r,x) → T_m(x) and they approach constant values in the x direction

incompressible flow

density does not change with position. dρ/dx = dρ/dy = dρ/dz = 0 so can calculate Re

simple compressible system assumption

the derivation for the Nusselt number comes from an energy balance on open systems with steady flow.

film temperature

T_f = (T_s+T_∞)/2

Fluid properties are assumed constant at this temperature. if we are evaluating heat transfer for the entire surface, we are using an average heat transfer convective coefficient (otherwise use integral)

parallel flow over an isothermal flat plate

critical Reynold’s number: Re_cr = 5×10⁵

Laminar: Re < 5×10⁵ Nu=hL/k = 0.664Re^1/2Pr^1/3

Turbulent Re > 5×10⁵ Nu=hL/k = 0.037Re^0.8Pr^1/3

0.6 ≤ Pr ≤ 60

Zukaukas and Jacob

equations for flow across single cylinder that is smooth, using T_f and Nu_cyl = hD/k = cRe^mPr^1/3

Table 12-3

Re_cr = 2×10⁵

Churchill and Berstein

properities of fluid evaluated at T_f - more exact

Whitaker

for calculating flow across spheres

surface area

Q=hA(T_s-T_∞)

cylinder → 2πrL = πDL (circular cross section)

sphere → 4πr² = πd²