TP2 - Heat Transfer Flashcards
What’s natural convection?
When fluid motion is caused by buoyancy forces, which are induced by differences in density due to variation of temperature of the fluid
What’s buoyancy force?
The upwards force exerted by a fluid on a body, completely or partially immersed on it, and it is equal to the weight of the fluid displaced by this body
FB = fluid density * g * body volume
Thus net force = weight - buoyancy force
What’s the Grashof number?
A dimensionless number measuring the relative magnitudes of the buoyancy force and the opposing viscous force acting on a fluid.
Gr = buoyancy forces / viscous forces
Gr = [𝑔𝛽 (Τ𝑠 − 𝑇∞ )𝐿𝑐^3] / 𝑣2
Where:
𝑣: kinematic viscosity of the fluid, [𝑚2/𝑠
𝐿𝑐: characteristic length of the geometry, [𝑚]
𝛵𝑠 : surface temperature (K)
𝑇∞: temperature away from the surface (K)
𝛽𝑖𝑑𝑒𝑎𝑙, 𝑔𝑎𝑠 : Volume expansion coef. for ideal gases [1/K]
What’s the Rayleigh number?
The product of Grashof number and Prandtl number
Ra = Gr*Pr
Gr = [𝑔𝛽 (Τ𝑠 − 𝑇∞ )𝐿𝑐^3] / 𝑣a
Where
a - thermal diffusivity
𝑣: kinematic viscosity of the fluid, [𝑚2/𝑠
𝐿𝑐: characteristic length of the geometry, [𝑚]
𝛵𝑠 : surface temperature (K)
𝑇∞: temperature away from the surface (K)
𝛽𝑖𝑑𝑒𝑎𝑙, 𝑔𝑎𝑠 : Volume expansion coef. for ideal gases [1/K]
What’s the equation for the Prandtl number?
Pr = v / a = mu* Cp/k
v: kinematic viscosity of the fluid, [𝑚2/𝑠]
𝐿𝑐: characteristic length of the geometry, [𝑚]
𝛵𝑠 : surface temperature (K)
𝑇∞: temperature away from the surface (K)
𝜇: dynamic viscosity [kg/m·s] or [N·s/𝑚2]
α: thermal diffusivity [𝑚2/s]
How’s Nusselt number calculated?
Nu = hLc/k
Nu = C(GrPr)^n = CRa^n
What’s forced convection?
Fluid motion is generated by an external source (like a pump, fan, suction device, etc.) => More efficient; Higher heat transfer rates than natural convection
Heat transfer is complicated because:
- involves fluid motion
- depends on the thermophysical properties of the fluid (μ, k, ρ, cp)
- the geometry and roughness of the solid surface
- type of flow (laminar, turbulent)
The higher the fluid velocity, the higher the convection heat transfer
What are the velocity and temperature boundary layers / profiles like?
(Basic forced convection)
Velocity - The fluid layer adjacent to the surface sticks onto the surface
⇒ 𝒗𝒔 = 𝟎 (no-slip condition)
⇒ retards the fluid layers above
⇒ responsible for the velocity profile
Temperature
The fluid layer adjacent to the surface obtains the temperature of the solid
⇒ T𝑠 = 𝑇𝑓 (no-temperature jump condition)
⇒ pure conduction ( motionless fluid)
What’s drag force?
A body which is forced to move within a fluid experiences some resistance. The force that the fluid exerts on the body is called drag force.
Types of forces:
- wall shear/friction forces (viscous effects while fluid flows)
- pressure forces (depend on the shape of the body)
Plate aligned parallel to the flow -> shear only Plate perpendicular -> pressure only
What are the types of forces exerted on fluids in forced convection?
Types of forces:
- wall shear/friction forces (viscous effects while fluid flows)
- pressure forces (depend on the shape of the body)
Plate aligned parallel to the flow -> shear only Plate perpendicular -> pressure only
How is drag force calculated?
D = 0.5CrhoV^2A
How are fluid velocity, plate length and kinematic viscosity used to calculate Re?
R.E. = V*L/v
How does forced convection flow for parallel flow over isothermally heated plates?
There are three regions of flow: laminar, transition and turbulent.
How does forced convection flow for flow across cylinders and spheres?
Flow exhibits a complex flow pattern; both pressure and frictional drag forces are significant
- Low velocities Re <1: fluid wraps up the cylinder
- Higher velocities Re = 10: boundary layer detaches from the surface and moves to the back
- Higher velocities 103
What happens during forced convection in internal flow?
Internal flow: Liquid is within a confined space = > the boundary layer growth has a limit
Velocity:
• The fluid layer adjacent to the walls has 𝑣𝑠 = 0 (no-slip condition on top/bottom walls)
- We are interested in the V.avg
- There is shear stress and friction on the pipe walls
Which type/shape of pipe can withstand the largest pressure differences, inside and out, without undoing significant distortion?
Circular pipes
How do velocity and temperature profiles vary over a heated, vertical plate?
In general, the fluid velocity is very low and hence the heat transfer rate is slow.
The velocity on the surface edge and far away from it, is zero.
The temperature is maximum on the edge but decreases as the fluid flows away from the surface on the surface edge and far away from it, is zero.
Moving away from the plate: (Hot) fluid travels upwards. Velocity increases then decreases and temperature decreases.
The opposite takes place for a cold plate, and the fluid moves down.
How is the volume expansion coefficient (beta) calculated for an ideal gas?
β = 1 / T
What does the Grashof number show?
Gr represents the natural convection effects in a momentum equation.
Determines the type of the flow (vertical plate: turbulent >109, laminar < 109)
Gr = buoyancy forces / viscous forces
What is the characteristic length, Lc, for a vertical plane?
It is equal to the vertical length, L.
The width is irrelevant.
What is the characteristic length, Lc, for a inclined plane?
It is equal to the tilted length, L.
The width is irrelevant.
What is the characteristic length, Lc, for a horizontal cylinder or sphere?
D, diameter
What is the characteristic length, Lc, for a vertical cylinder?
L, length/height of the cylinder
What plate orientation allows the most effective heat transfer?
When the plate surface is facing the oncoming fluid flow, so that the plate does not act as a barrier.
How is film average temperature calculated?
Tf = (𝑇𝑠 + 𝑇∞)/2
What’s the equation for heat transfer by convection?
Q = ℎ𝐴(𝑇𝑠 − 𝑇∞)
What’s the equation for heat transfer by radiation?
Q = ε𝜎A(𝑇𝑠⁴ − 𝑇∞⁴)
How is heat transferred in the absence of buoyancy forces?
Heat transfer between a hot surface and the fluid will occur due to conduction, and not natural convection.
Its rate would be much lower.
What is Fourier’s law of conduction equation?
Q = -kAdT/dx
What are finned surfaces also known as?
Heat sinks
Energy is first transferred from the hot plate to the heat sinks by conduction, and then to the surrounding air via convection.
How does fin size/length effect convection between fins?
Long surface fins: channel flow fully develops and, if the fins are close enough, the channels merge boundary layers from both sides.
Short surface fins or fins with large spacings between experience natural convection from 2 independent plates. There is no merging of channel flows.
How does fin spacing effect heat sink design?
Closely packed fins will allow greater surface area for heat transfer, however heat transfer coefficient will decrease (due to extra resistance from additional fins)
Widely spaced fins will allow higher heat transfer coefficients but have a smaller surface area.
Thus an optimum fin spacing is used to maximise natural convection.
What is the critical length, Lc, for parallel vertical fins?
It is usually fin spacing, S, however plate length L can be used also.
Ra.s gives the Rayleigh number for the fin whilst Ra gives the Rayleigh number for the plate
How is optimum fin spacing calculated (when fin thickness t<
S opt = 2.714*(S³L/Ra.s)¹/⁴
= 2.714*L/Ra¹/⁴
What is the Nusselt number, using optimum fin spacing.
Nu = h*S opt/k = 1.307 = constant
How is total heat transfer rate calculated for a finned surface?
𝑄 = ℎ2𝑛𝐿𝐻(𝑇𝑠-𝑇∞) = ℎ2𝑛𝐴(𝑇𝑠-𝑇∞)
where A is the surface area of 1 fin, n is the total number of fins, and this must be multiplied by 2 to consider both sides of the fin.
Total heat transfer area = 2nA
How does fluid move within a vertical enclosure?
Fluid by the hot surface rises and that by the cold surface moves down, so air can circulate.
How does fluid move within a horizontal enclosure?
If the hot surface is at the top, and cold at the bottom, then no convection currents form as the light/hot fluid is above the heavy/cold fluid.
If the Nusselt number is 1, heat transfer will only be by pure conduction.
If the cold surface is on top, and hot is at the bottom, heat is initially transferred by pure conduction.
Once buoyant forces overcome fluid resistance (Ra>1708), natural convection and air circulation begin
How does fluid move within a horizontal enclosure when the hot plate is on top?
If the hot surface is at the top, and cold at the bottom, then no convection currents form as the light/hot fluid is above the heavy/cold fluid.
If the Nusselt number is 1, heat transfer will only be by pure conduction.
How does fluid move within a horizontal enclosure when the hot plate is on the bottom?
If the cold surface is on top, and hot is at the bottom, heat is initially transferred by pure conduction.
Once buoyant forces overcome fluid resistance (Ra>1708), natural convection and air circulation begin
How is the Rayleigh number calculated for enclosures?
Ra = Prg𝛽(T1 - T2)Lc³/v²
Lc is the distance between hot and cold plates.
Fluid properties are chosen considering the average fluid temperature.
How is heat transferred in an enclosure calculated?
Convection:
Q = hA(T1 - T2) = (kNu/Lc)A(T1 - T2)
Conduction:
Q = (k/Lc)*A(T1 - T2)
If Nu = 1, heat is only transferred by conduction
How is the Nusselt number calculated, considering the Rayleigh number?
When 10⁴ < Ra < 4*10⁵:
Nu = 0.195Ra¹/⁴
When 10⁵< Ra < 4*10⁷:
Nu = 0.068Ra¹/³
How are critical length and heat transfer calculated for concentric horizontal cylinders?
Lc = (Do - Di)/2
Q = (2pik/ln(Do/Di))/(Ti - To)
How is total heat transfer rate between plates calculated (considering radiation)?
Qt = Q conv + Q rad
The heat transfer coefficients for natural convections are typically low (compared to forced convection), therefore radiation should be taken into account for the overall heat transferred.
How is heat transfer by radiation found?
Q = 𝜎𝜀A(T₁⁴ - T₂⁴)
What is forced convection and why is it more complicated?
Fluid motion is generated by an external source (like a pump, fan, suction device, etc.)
More efficient; Higher heat transfer rates than natural convection
Heat transfer is complicated because:
- involves fluid motion - depends on the thermophysical properties of the fluid (μ, k, ρ, cp)
- the geometry and roughness of the solid surface
- type of flow (laminar, turbulent)
The higher the fluid velocity, the higher the convection heat transfer
𝑄′ = ℎ𝐴𝑠 𝑇1 − 𝑇2
How do velocity and temperature profiles vary for forced convection and external flow?
Velocity:
The fluid layer adjacent to the surface sticks onto the surface
- 𝒗 = 𝟎 (no-slip condition)
- retards the fluid layers above
- responsible for the velocity profile
Temperature:
The fluid layer adjacent to the surface obtains the temperature of the solid
- T𝑠 = 𝑇𝑓 (no-temperature jump condition)
- pure conduction (motionless fluid)
What is drag force, and what resistive forces are exerted on the fluid?
A body which is forced to move within a fluid experiences some resistance. The force that the fluid exerts on the body is called drag force.
Types of forces:
- wall shear/friction forces (viscous effects while fluid flows)
- pressure forces (depend on the shape of the body)
Plate aligned parallel to the flow -> shear only Plate perpendicular -> pressure only
How is drag force calculated?
𝐷 = 1/2𝐶𝜌𝑽2𝐴𝑓
Where:
C - drag coefficient
- combines the wall shear and (geometric) pressure drag
- depends on velocity, viscosity, shape and size
- determined experimentally
Af is frontal area
How is Re calculated for forced convection parallel flow over isothermally heated plates?
Re = V*L/v
Where:
V is fluid velocity
L is plate length
v is kinematic viscosity
What are the 3 regions of flow for parallel fluid flow over isothermally heated plates (forced convection)?
Laminar
Transition
Turbulent
How does fluid flow around cylinders and spheres during forced convection?
Flow exhibits a complex flow pattern; both pressure and frictional drag forces are significant.
Low velocities Re <1: fluid wraps up the cylinder
Higher velocities Re = 10: boundary layer detaches from the surface and moves to the back
Use in heat exchangers design which involves external and internal flow of liquids
Higher velocities 103
How is Re calculated for flow across cylinders and spheres (forced convection - external flow)?
Re = V*D/v
Where:
V is fluid velocity
D is external diameter
v is kinematic viscosity
What is internal flow?
How does the velocity vary?
Liquid is within a confined space - the boundary layer growth has a limit
Velocity
• The fluid layer adjacent to the walls has 𝑣𝑠 = 0 (no-slip condition on top/bottom walls)
• We are interested in the Vavg
• There is shear stress and friction on the pipe wall
What shape of pipe can withstand high pressures?
Circular pipes withstand large pressure differences between inside and outside without going significant distortion, whereas non-circular pipes cannot.
What is the critical length, Lc, for non-circular pipes?
Lc = Dh = 4*Ac/P wet
