MECH 344 Final Exam

Unsteady heat transfer

temperature changes with respect to time at locations within an object

spatially isothermal heat transfer

Resistance to conduction within a solid is small compared to the resistance to convection from the surface of the solid
Resistance is a function of time, not position

characteristic length of a sphere

V = 4/3*pi*r² A_s = 4*pi*r²

L_c = r/3

characteristic length of a cylinder

V = pi*r²*L A_s = 2*pi*r*L

L_c = r/2

characteristic length of a plane wall

convection on both sides:

V = 2*L*A A_s = 2*A

L_c = L
convection on one side:
V = L*A A_s = A

L_c = L

Semi-infinite solid

a solid that is initally at a uniform temperature, T_i, and is assumed to extend to infinity from a surface at which thermal conditions are altered

any object that for a given transient the temperature does not change all the way throung the object (will still be T_i at the other end)

Boundry conditions for 1D diffusion equation

dT/dx_x=0 = 0

dT/dx_x=L = h(T(L,t) - T_infinity)

Biot number

measure of change in T in a solid relative to change in T in a fluid

at what biot number does the object become spatially isothermal

Bi < 0.1

why is temperature imposed for case 1 for semi-infinite solids

imposed temperature on surface because of high h from turbulent flow over surface

B.C.’s:

T(x,0) = T_i, T(0,t) = T_s

velocity boundry layer

a region of flow characteristics by shear stresses and velocity gradients
a consequence of the viscous effects associated with relative motion between a fluid and a surface

what is the critical reynolds number over a flat plate?

Re = 5×10⁵

Thermal boundry layer

region where the temperature gradients are noticible results in heat exchange between fluid and wall

what is the friction coefficient at the leading edge of the plate?

Re_x^-1/2 is in denomenator and approaches 0 so, C_f goes to infinity

Prandlt number

gives a measure of the thickness of the velocity boundry layer to the thermal boundry layer

nusselt number

allows us to have a way to get the coefficient at a particular x on a plate

interpolation equation

Y = [(Y2-Y1)/(X2-X1)](X+X1)+Y1

Wall shear

T_w=0.5pu²C_f

initial temperature equation for case 3 finite difference method

T₀^P+1 = (1 - 2F_o - 2F_oBi)T_o^P + 2F_oT₁^P + 2F_oBiT_infinity

drag equation

D = T_w A

Surface energy balance

method of applying conservation of energy to a control surface (at an instance in time) for which we assume has no volume or mass

(energy storage and generation are not pertinent)

What does fourier’s law give us?

conduction (q) based upon knowledge of temperature distribution in the medium

critical radius

the point at which there is a local minimum for total resistance and a local maximum for the rate of heat transfer

Why is T_i constant at r_i for a thin-walled pipe with insulation?

T_i is an imposed boundry condition and does not change

What happens to heat transfer when the insulation outer radius (r_o​) increases for a thin-walled pipe?

Heat transfer rate (q′) increases until reaching r_crit​, then decreases beyond r r_crit​.

for a thin-walled pipe
Why does heat transfer (q′) remain the same through insulation in steady-state conditions?

Because there is no internal heat generation in the insulation (T is imposed), so all heat entering at r_i must leave at r_o

for a thin-walled pipe
What is the critical radius of insulation (r_crit)?

The radius at which thermal resistance is minimized and heat transfer is maximized.

for a thin-walled pipe
What happens to the outer temperature (T_o) when insulation is increased?

• If r < r_crit, T_o​ decreases because increased surface area improves convection (more efficient heat loss).

• If r > r_crit, T_o​​ decreases due to increased resistance before convecting away.

Where is the maximum heat transfer rate (q′) in a generating solid with insulation?

At r_i​ (the inner surface), since all generated heat must exit there.

How does q′ behave inside a generating solid with insulation?

It increases from the center outward until the generating surface, then remains constant through insulation.

What does r_crit represent for a generating solid with insulation?

The point of minimum total resistance, where heat can escape most efficiently.

What happens to T_i at r_critin a generating solid with insulation?

It is at a minimum because heat is leaving with the least resistance.

What happens to T_i when insulation is added beyond r_crit in a generating solid with insulation?

T_i increases because it becomes harder for heat to escape.

Why does T_o decrease when insulation is added beyond r_crit​ in a generating solid with insulation?

Because the increased surface area improves heat dissipation through convection.

What is the effect on q’ of varying r_o in a generating solid with insulation?

ncreasing ror_oro​ does not result in a maximum heat transfer at r_crit.
(The maximum q′ always occurs at r_i, where all the generated heat must exit.)

Active tip fin heat transfer rate simplification

q_f = Mtanh(mL_c)
L_{c, pin} = L + d/4
L_{c, plate} = L + t/2

heat transfer

thermal energy in transit caused by a difference in temperature

conduction

energy flows through a medium by virtue of a temperature gradient within the medium

happens when more energetic particles transfer energy to less energetic particles due to particle interaction

thermal conductivity, k

material property

function of T

assume constant given a wall with constant properties

steady flow

at any particular point in the flow field no thermodynamic property changes with respect to time

steady heat transfer

at any particular point in the medium temperature does not change with respect to time

convection

thermal energy is transported across (either into or out of a system) a system boundry by a moving medium or a fluid

fluid

a substance that deforms continuously when subjected to a shear force of any magnitude

heat exchanger

a device that causes heat to be exchanged between two fluids that do not mix

convective heat transfer coefficient, h

a function of both the fluid that is flowing and the nature of the flow

determined empirically

thermal radiation

electromagnetic radiation propagated as a result of a temperature difference

surfaces above absolute zero emit this

no medium required

parallel mechanism for heat transfer

usually neglected except for high temperatures

emissive power, E

rate at which energy is released from a surface per unit area

AKA heat flux

irradiation, G

rate at which radiation is incident on a surface

absorptivity + transmissivity + reflectivity = 1

blackbody, Eb

an ideal emitter of thermal radiation

stefan boltzmann law

for prescribed temperature and wavelength, no surface can emit more than a blackbody

emissivity

how efficiently electromagnetic radiation eminates from an object compared to a blackbody

absorptivity

the percentage of radiation absorbed by the recieving object and converted into internal energy

surface energy balance

method of applying conservation of energy to a control surface for which we assume has no volume or mass

energy storage and generatin are not pertinent to the energy balance

1-D

thermodynamic properties change up and downstream but not normal to chosen cross section

free convection

air isnt forced to flow, moves due to p gradient

Overall heat transfer coefficient, U

1/((1/h₁)+(L/k)+(1/h₂))

manufacturer supplied

included conduction and convection

frequently given as U*A

parallel flow

both fluids in heat exchanger flowing in same direction

counter flow

the two fluids in heat exchanger flow in opposite directions

when MUST you counter flow?

if T_C,o>T_H,o

when do you use LMTD?

if 3 of 4 temperatures are known in a heat exchanger

when do you use E-NTU?

if 2 of 4 temperatures are known and the heat exchanger is a concentric tube, cross-flow, or shell-and-tube

what is epsilon (E)

E = (actual heat transfer) / (max possible heat transfer

why do you use (mC_p)min for qmax?

because if a fluid with the larger mCp underwent a change in Tmax the other fluid would have to undergo a change in T greater then change in Tmax to satisfy the first law, which is impossible

natural convection

the result of the motion of a fluid due to density changes arising from a heating process

what does the movement of a fluid result from in natural convection?

the bouyancy forces imposed on a fluid when its density in the proximity of the heat transfer surface is decreased as a result of heating

free boundry flows

no adjoining surface

quiescent fluid

motionless fluid at a location

grashof

bouyancy force over viscous force

when can we ignore natural convection?

\frac{Gr_{L}}{\left(\operatorname{Re}_{L}\right)^2}\ll1

when do we only have free/natural convection?

\frac{Gr_{L}}{\left(\operatorname{Re}_{L}\right)^2}\gg1

when do we have mixed convection?

\frac{Gr_{L}}{\left(\operatorname{Re}_{L}\right)^2}=1

what does the Rayleigh number (Ra) tell us?

where flow transitions from laminar to turbulent flow for natural convection Ra_{L}=10^9