QUESTION ANSWERS CONDUCTION HEAT TRANSFER
QUESTION ANSWERS
CONDUCTION HEAT TRANSFER
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Q1. List the basic laws which govern heat transfer.

First law of thermodynamics i.e. Law of conservation of energy

Second law of thermodynamics: Heat flows along negative temperature gradient.

Law of conservation of mass

Newton’s Law of motion

The rate equations
Q2. What is the significance of heat transfer?
Like transfer of a person from one city to another city, heat transfer is heat energy in motion.
Q. Definition of thermal conductivity
From Fourier equation
k=q. with dT=1 K, dx=1 m and A=1 m^{2 }
Thus thermal conductivity is the rate of heat transfer in a solid of unit area, unit thickness with unit temperature difference. It is property of a solid for conducting heat. Its symbol is k. Its units are W/m K.
Q3. What is basic equation of conduction?
Fourier equation is the fundamental equation of conduction.
It is used in following three forms.
a. Difference form of equation
Q=k A (T_{h} T_{l})/(x_{2}—x_{1})
It is used when temperatures at the two ends are fixed and distance between the two ends is also fixed.
b. Differential form of equation
Q=k A dT/dx
It is used when temperature is a function of x.
c. Second order differential equation
d^{2}T/dx^{ 2} + d^{2}T/dy^{ 2}+ d^{2}T/dz^{ 2} + q_{g}/k =(1/α) dT/dt
It is three dimensional non steady state conduction equation.
T is the temperature
t is the time.
K is thermal conductivity
α is thermal diffusivity
q_{g} is heat generated per unit volume within the system
Q4. What is thermal resistance in conduction and convection respectively?
Conductive rate of heat transfer q.= –dT/x/kA
thermal resistance in conduction R _{cond}= x /(kA), measured in K W^{−1}
Convective heat transfer: q = ΔT/(1/hA)
Thermal resistance to convective heat transfer= R _{th conv} = 1/hA
Q5. What is difference between heat and temperature?
Heat is the energy in motion or transition. It flows spontaneously from high temperature to low temperature. It varies with temperature. Rises with the increase of temperature and decreases with fall of temperature. The common symbol for heat is Q. Units of heat are kJ.
Temperature is the measurement of heat energy in a body. It is the degree of hotness or coldness of a body. It decides the direction of heat transfer which is always from high temperature to low temperature. The common symbol for temperature is t or T. Its units are ^{o}C or K.
Q6. Where does internal heat generation found?
When electric current flow in wires/cables.
During chemical reactions
Nuclear reactions
Heat treatment processes.
Q7. What are a temperature gradient, velocity gradient and pressure gradient?
Temperature gradient
dt/dx, dt/dy and dt/dz are temperature gradients i.e. temperature variation w.r.t. x, y, z.
Velocity gradient dv/dx, dv/dy and dv/dz are velocity gradients i.e. velocity variation w.r.t. ‘x’, y, z.
Pressure gradient dp/dx, dp/dy and dp/dz are pressure gradients i.e. pressure variation w.r.t. ‘x’, y, and z.
Q8. Write the temperature distributions for a plane wall, cylinder and a sphere.
Firstly, for a plane wall (T_{X—}T_{h})/ (T_{l}—T_{h}) = x/δ Linear variation
Secondly, for a cylinder (T_{R—}T_{h})/ (T_{l}—T_{h}) = ln(r/r_{i}) / ln (r_{0}/r_{i}) Logarithmic variation
Thirdly, for a sphere (T_{R—}T_{h})/ (T_{l}—T_{h}) = (r_{o}/r) (r–r_{i})/ (r_{o}—r_{i}) Hyperbolic variation
Q9. Discuss factors effecting thermal conductivity of a material.
The main factors are vibration of the lattice and movement of free electrons. Most important out of these two is movement of free electrons. Movement of free electrons is found in pure metals. Therefore, all pure metals are good thermal conductors.
Density, high density means high thermal conductivity
Effect of temperature: On rise of temperature k decreases for metals, increases for non metals, liquids and gases and vice versa.
Pressure effect: unaffected in case of solids and liquids but increases with increase of pressure in case of gases
Chemical composition, for copper, 400, for aluminum 250, for steel, 66, for water 0.66 and for air 0.027 W/mK
Q10. On what factors thermal conductivity of gases depend?
k of gases depends on temperature and pressure.
With the increase of temperature, volume will increase, density will decrease. K is linked with density. Therefore k will decrease will the rise of temperature and vice versa.
When pressure increases, volume will decrease, density will increase and hence k will increase with the increase of pressure and vice versa
Q11. Define critical thickness of insulation.
Fig. Critical Radius of Insulation for conduction Through a Cylinder, R_{cr} = k/h_{0}
Fig. Critical Radius of Insulation for conduction Through a Sphere, R_{cr} = 2 k/h_{0}
e insulation thickness for maximum rate of heat transfer is the critical thickness of insulation.
Adding insulation to a wall type surface always reduces the rate of heat transfer because thermal resistance (R_{th}=x/KA) increases.
While in case of cylindrical and spherical bodies the situation is different. Adding insulation increases thermal resistance in conductance (x/kA) while decreases thermal resistance in convection (R _{th}=1/h A). As a result the overall resistance decreases and the rate of heat transfer increases. The insulation thickness for maximum rate of heat transfer is the critical thickness of insulation. The critical radius (outer radius with insulation) is for a
Cylindrical surface r_{c} = k/h_{o}
Spherical surface r_{c} =2k/h_{o}
Where k is the thermal conductivity of insulation
h_{o} is convective heat transfer coefficient on the outer side of insulation
Q12. Explain the significance of Fourier number.
Fourier number = Fo = αt/L_{c}^{2}.
It signifies the degree of penetration of cooling or heating through a solid.
It is a Non dimensional number.
Q13. Differentiate thermal conductance and thermal resistance.
_{ }When rate of heat transfer is compared with electrical flow of current, the concept of thermal resistance comes into existence. Thermal resistance is R_{th}
Rate of heat transfer q^{.} ΔT R_{th}=ΔT/q^{.}
Charge q^{.}=current= I ΔV R = ΔV/I
Thermal conductance is reciprocal of thermal resistance. Its symbol is C. C=1/R_{th}. Its units are W/K. The unit of thermal resistance will be K/W.
Q14. What is log mean area as applied to a hollow cylinder?
Heat transfer through a cylinder is in the radial direction. Radius is changing from inner radius to outer radius. Therefore area is changing. Thus use mean area. Now mean area can be arithmetic mean or logarithmic mean. Proved by experiments that log mean area gives better results.
For a cylinder, log mean area
ln ( A_{m}) = (A_{o} — A_{i})/ln (A_{o}/A_{i}) = (r_{o}–r_{i)}/ ln (r_{o}/r_{i})
For a sphere, log mean area
ln (A_{m}) = (A_{o} –A_{i})^{0.5}= 4 r_{i} r_{o}
Q15.Discuss thermal diffusivity.
Thermal diffusivity is the ratio of thermal conductivity to product of density and specific heat. In a substance with high thermal diffusivity, heat moves rapidly through the solid. It is because the substance conducts heat quickly relative to its volumetric heat capacity. If it is higher, it takes less time for heat transfer to take place through the solid. Further, units of thermal diffusivity is m²/s. Its symbol is α. The thermal diffusivity indicates the rate of heating and cooling under transient conditions. The physical signiﬁcance is that the inverse of thermal diffusivity is a measure of time. This is the time to establish the thermal equilibrium in the specimen. The rate of change of temperature depends on its numerical value.
Q16. Difference between thermal conductivity and thermal diffusivity?
Thermal conductivity (k) represents its ability to conduct heat. Thermal diffusivity (α) indicates how fast the heat conduction?
Q17. Quote at least four examples of multidimensional heat conduction.

Cooling of I.C. Engines

Heat transfer in air conditioning ducts.

Heat transfer in an industrial chimney.

During various heat treatment processes.
Q18. Name the methods used in the analysis of 2 Dimensional steady state conductive heat transfers.
There are four methods.

Analytical method

Graphical Method

Analogical Method

Numerical Method
Q19. Methods used in the analysis of 3 dimensional steady state conductive heat transfers.
There are three methods.

Analytical method

Analogical Method

Numerical Method
Q20. Discuss conduction shape factor?
Convection q^{.} =h A dT
conduction, q^{.} = k A (dT/dx)
Compare and write conduction equation as q^{.} = k S dT
Here S is the shape factor
S = – k/dx in conduction
S =h in convection
Q21. State the assumptions used in Fourier law of heat conduction.

Conduction under steady state conditions.

Heat flow is 1 dimensional.

Temperature gradient is constant and the temperature profile is linear.

There is no internal heat generation.

The bounding surfaces are at respective constant temperature.

Material is homogeneous and isotropic (The value of ‘k’ is constant in all directions).
Q22. Give a list of the essential features of Fourier Law.

This law is applicable to solids, liquids and gases.

Fourier law is verified experimentally and hence not derivable from first principle.

Heat transfer is vector expression and indicates that the rate of flow of heat in the temperature decreasing direction.

It gives the definition of thermal conductivity, a transport property.Q15. What is thermal contact resistance?.
When two solid bodies come in contact, heat flows from the hotter body to the colder body. There is a temperature drop at the common surfaces in contact. It is due to the thermal contact resistance existing because of imperfect contact between the two contacting surfaces. Further it is due to irregular surfaces. Thermal contact resistance is the ratio of temperature drop and the average heat flow across the contacting surfaces.
Q23. Important applications of heat transfer.
(i) Fins in compressors, in motors, on condensers and evaporators, on transformers
(ii) Heat exchanger like condenser, evaporator and a boiler
(iii)Transfer of Heat from the Sun, from fire, from furnaces, from radiators.
Q. Write the 1dimensional unsteadystate conduction equations.
∂^{2}T/∂x^{2} + q_{g}^{.} / k = (бT/бτ)/ α
Where T is temperature and τ is the time.
Q. Write the 1 dimensional unsteadystate conduction equation in cylindrical coordinates.
∂^{2}T/∂r^{2} + + q_{g}^{.} / k = (бT/бτ)/ α
Q. Write the 1 dimensional unsteadystate conduction equation in spherical coordinates.
(1/r^{2}) ∂/∂r (r^{2}∂T/∂r) + q_{g}^{.} / k = (бT/бτ)/ α
Q. Write the 3 dimensional unsteadystate conduction equations in rectangular coordinates.
∂^{2}T/∂x^{2} + ∂^{2}T/∂y^{2} +∂^{2}T/∂z^{2} + q_{g}^{.} / k = (бT/бτ)/ α
Where T is temperature and τ is time.
Q. Write the 3 dimensional unsteadystate conduction equations in cylindrical coordinates.
∂^{2}T/∂r^{2} + + (1/r^{2}) (^{∂2}T/∂θ^{2}) + ∂^{2}T/∂z^{2} + q_{g}^{.} / k = (бT/бτ)/ α
Where r, θ and z are cylindrical coordinates.
Q. Write the 3 dimensional unsteadystate conduction equations in spherical coordinates.
(1/r^{2}) ∂/∂r (r^{2}∂T/∂r) +(1/r^{2} sin^{2}θ) ∂/∂θ (sinθ ∂T/∂θ)
+(1/r^{2} sin^{2}θ) ∂^{2}T/∂φ^{2} +q_{g}^{.} / k = (бT/бτ)/ α
Where r, θ and φ are spherical coordinates.
Q . Define Biot number and its significance.
Biot number is Bi= conductive resistance/convective resistance = hL_{c}/k_{solid}
Where h is convective heat transfer coefficient W/m^{2}K
L_{c} is the characteristic length= volume/surface area
=δ/2 for a plane wall,
=r/2 for a cylinder
=r/3 for a sphere
= side/6 for a cube
k is thermal conductivity of solid material
If Bi < 0.1, temperature of the body will be constant and heat transferred will only change the internal energy.
PHYSICAL SIGNIFICANCE
When Biot number <0.1 indicates that heat transfer changes the internal energy of the body and finally the heat transfer is by convection to the atmosphere. Therefore body with Bi < 0.1 is considered THERMALLY THIN body and temperature gradients in such a body are considered as zero. Bodies with Bi >0.1 are considered THERMALLY THICK, temperature gradients are there and requires very complicated equations to solve steady as well as unsteady state conditions of heat transfer.
Q. What is rate of change of temperature?
The variation of temperature with respect to time is rate of change of temperature.
Q. What is Newtonian heating or cooling of solids?
Fig. Newtonian Heating and Cooling of Solids
It refers to heating or cooling of solids of infinite conductivity i.e. k is very large. Therefore conductive (internal) resistance of such solids is zero. This type of body is a super conductor and conduction of heat trough such solids is at super fast speed. Thus there will be change in internal energy of the solid and heat transfer will take place only by convection. Since conduction resistance is zero. It uses Lumped parameter analysis for its solution. Under Lumped Parameter analysis the fall or rise of temperature varies exponentially with respect to time. Therefore, under Newtonian heating or cooling, the change of temperature with time is exponential.

What is a lumped system ?
In a lumped system dT/dx=0, dT/dy=0, dT/dz=0.
Temperature is a function of time only T=f(t) i.e. unsteady state heat transfer.
Q. List the assumptions of Lumped capacity analysis.

Solid materials have infinite (very large) value of ‘k’.

The conductance resistance or internal resistance is negligible as compared to convective resistance or external resistance.

Temperature is constant at a given time in such solids.

Rate of change of internal energy equals convective heat exchange at the solid surface.
LUMPED CAPACITY
It the product of Biot number and Fourier number as given below.
hA_{s}t /ρVC = (hV/ (kA_{s})) (A_{s}^{2}kt/ (ρV^{2}C))
= (hL_{c}/k) (αt/L_{c}^{2}))
Q, When temperature becomes uniform in a body?
When the Biot number is small (Bi << 0.1).
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