All questions of Heat Transfer for Mechanical Engineering Exam

Temperature variations in cycle of an internal combustion engine.

- An internal combustion engine undergoes a series of processes in a cycle, including intake, compression, combustion, and exhaust.
- These processes result in changes in temperature within the engine.
- During the intake stroke, the air-fuel mixture enters the combustion chamber, causing a decrease in temperature.
- In the compression stroke, the mixture is compressed, leading to an increase in temperature.
- The combustion of the mixture during the power stroke produces high temperatures.
- Finally, during the exhaust stroke, the combustion products are expelled, leading to a decrease in temperature.
- This cycle repeats continuously as the engine operates.

Cooling of bars, blanks and metal billets in steel works.

- When bars, blanks, and metal billets are produced in steel works, they are often heated to high temperatures for shaping and processing.
- After the shaping process, these metal components need to be cooled down to ambient temperature for further handling or storage.
- The cooling process involves transferring heat from the hot metal to the surrounding environment.
- However, this cooling process is typically not periodic but rather a continuous and gradual decrease in temperature.
- The rate of cooling depends on various factors such as the initial temperature of the metal, the ambient temperature, and the surface area-to-volume ratio of the metal.
- The cooling rate can be controlled by adjusting the cooling medium, such as air or water, and their flow rates.
- While the temperature of the metal may change over time during the cooling process, it does not exhibit a periodic variation.

Variation of temperature of a building during a full day period of 24 hours.

- The temperature of a building can vary throughout the day due to various factors such as solar radiation, outdoor temperature, and HVAC system operation.
- During the day, when the sun is up, the building may experience an increase in temperature due to solar heat gain.
- As the sun sets and the outdoor temperature drops, the building may start to lose heat and the temperature may decrease.
- However, this temperature variation is not periodic as it is influenced by external factors such as weather conditions and the time of year.
- The temperature of the building may also be affected by the operation of heating, ventilation, and air conditioning (HVAC) systems.
- The HVAC system may be programmed to maintain a constant temperature or to follow a specific temperature schedule, but this schedule is typically not a perfect periodic variation.

Heat processing of regenerators whose packings are alternately heated by flue gases and cooled by air.

- Regenerators are heat exchangers used in various industrial processes, including combustion systems.
- In some regenerators, the packings or beds are designed to alternate between being heated by hot flue gases and cooled by air.
- This alternating heating and cooling process can result in a periodic variation of temperature within the regenerator.
- When the packings are exposed to flue gases, they absorb heat and their temperature increases.
- After a certain period, the flow is switched, and the packings are exposed to cooling air, causing their temperature to decrease.
- This alternating heating and cooling cycle repeats, resulting in a periodic variation of temperature within the regenerator.
- The purpose of this process is to efficiently transfer heat between the flue gases and the air, maximizing the energy efficiency of the system.

Therefore, among the given options

Explanation:

Gr/Re²
The parameter Gr/Re² is a measure of the relative importance of natural convection in relation to forced convection.

Gr
The Grashof number (Gr) is a dimensionless number in fluid dynamics that is used to determine the relative strength of buoyancy and viscous forces in a fluid flow. It is defined as the ratio of buoyancy forces to viscous forces within a fluid.

Re
The Reynolds number (Re) is a dimensionless number that describes the flow of a fluid. It is the ratio of inertial forces to viscous forces within the fluid.

Interpretation
When the Gr/Re² parameter is high, natural convection is more dominant compared to forced convection. Conversely, when this parameter is low, forced convection is more significant.

Conclusion
Therefore, by calculating the Gr/Re² parameter, one can determine the relative importance of natural convection in relation to forced convection in a given fluid flow situation.

Time Constant of a Thermocouple
The time constant of a thermocouple refers to the time taken for the temperature reading to reach a certain percentage of the final value after a sudden change in temperature.

Explanation of the Correct Answer

Option C: 63.2% of the value of the initial temperature difference
- The correct answer is 63.2% because this percentage corresponds to one time constant in a first-order system.
- In a first-order system, it takes approximately 63.2% of the total time constant to reach 63.2% of the final value.
- After one time constant, the temperature reading of the thermocouple will be about 63.2% of the final value.

Other Options Explained
- Option A: The final value to be measured is not relevant to the time constant. The time constant is about how quickly the temperature reading approaches the final value.
- Option B: 50% of the value of the initial temperature difference is not the correct choice for the time constant of a thermocouple.
- Option D: 98.8% of the value of the initial temperature difference is too close to the final value and does not correspond to a standard time constant in a first-order system.
In conclusion, the time constant of a thermocouple is the time it takes for the temperature reading to reach approximately 63.2% of the final value after a sudden change in temperature.

Nusselt number in case of free convection is the function of Grashoff number and Prandtl number. Let us understand why.

Free Convection:
When heat transfer takes place due to buoyancy forces, it is referred to as free convection. In free convection, the fluid moves due to the density differences caused by temperature differences.

Nusselt Number:
Nusselt number is a dimensionless number that represents the ratio of convective to conductive heat transfer across a boundary. It is defined as:

Nu = hL/k

Where h is the convective heat transfer coefficient, L is the characteristic length, and k is the thermal conductivity of the fluid.

Grashoff Number:
Grashoff number is a dimensionless number that represents the ratio of buoyancy to viscous forces in a fluid. It is defined as:

Gr = gβΔTL^3/ν^2

Where g is the acceleration due to gravity, β is the coefficient of thermal expansion, ΔT is the temperature difference, L is the characteristic length, and ν is the kinematic viscosity.

Prandtl Number:
Prandtl number is a dimensionless number that represents the ratio of momentum diffusivity to thermal diffusivity in a fluid. It is defined as:

Pr = ν/α

Where ν is the kinematic viscosity and α is the thermal diffusivity.

Relationship between Nusselt Number, Grashoff Number, and Prandtl Number:
The Nusselt number in case of free convection is dependent on the Grashoff number and Prandtl number. The relationship between these numbers is given by the following equation:

Nu = C(GrPr)^n

Where C and n are constants that depend on the geometry of the system. The above equation shows that the Nusselt number is directly proportional to the Grashoff number and Prandtl number raised to the power of n. Thus, the Nusselt number in case of free convection is the function of Grashoff number and Prandtl number.

Explanation:

Fins are widely used in various engineering applications for enhancing the heat transfer rate from a surface. The primary function of fins is to increase the effective surface area of a heat transfer surface so that more heat can be transferred from the surface. However, fins can also affect other parameters that affect heat transfer rate. Let's discuss each of the given options one by one:

1. Increasing the Temperature Difference:
Increasing the temperature difference between the surface and the surrounding fluid can enhance the heat transfer rate. However, fins do not directly affect the temperature difference between the surface and the fluid. Therefore, option 1 is incorrect.

2. Increasing the Effective Surface Area:
Fins are designed to increase the effective surface area of a heat transfer surface. By increasing the surface area, the contact area between the surface and the fluid increases, which enhances the heat transfer rate. Therefore, option 2 is correct.

3. Increasing the Convective Heat Transfer Coefficient:
Convective heat transfer coefficient is a measure of the efficiency of heat transfer between a surface and a fluid. Fins can enhance the convective heat transfer coefficient by promoting fluid flow and turbulence around the surface. However, fins do not directly affect the convective heat transfer coefficient. Therefore, option 3 is incorrect.

4. Decreasing the Thermal Conductivity:
Thermal conductivity is a measure of the ability of a material to conduct heat. Fins do not affect the thermal conductivity of the surface material. Therefore, option 4 is incorrect.

Conclusion:
From the above explanation, it is clear that option 2 is correct, and options 1, 3, and 4 are incorrect. Therefore, the correct answer is option C, i.e., 2 only.

Given Data:
- Total emissive power (E) = 32 W/m2
- Insolation (I) = 93 W/m2
- Reflectivity (R) = 0.6
- Absorptivity (A) = 0.1
- Transmissivity (T) = 0.3

Calculating the Radiosity:
The radiosity (J) can be calculated using the following equation:
J = A * I + E - R * E

Substituting the given values:
J = 0.1 * 93 + 32 - 0.6 * 32
J = 9.3 + 32 - 19.2
J = 22.1

Therefore, the radiosity is 22.1 W/m2.

Comparing with the Answer Options:
The correct answer is given as option 'A' (87.8 W/m2), but our calculated radiosity is 22.1 W/m2. This means that the given answer is incorrect.

Conclusion:
The correct answer to the given question cannot be determined based on the provided information.

Understanding the Statements on Fin Design
To analyze the correctness of each statement about fins of minimum weight, let's break down the provided information.
1. Weight Savings of Triangular Fins
- The claim that a triangular fin saves 44% of the weight compared to other shapes is generally supported by empirical studies. Triangular fins are often found to minimize material usage while maintaining efficiency.
2. Weight Difference Between Fin Shapes
- The assertion that the weight difference between a fin shaped like a circular arc and a triangular fin is very large may not hold universally true. While there can be differences, they are not always significant enough to deem one shape vastly superior in all applications.
3. Manufacturing Ease of Triangular Fins
- The statement that triangular fins are easier to manufacture is accurate. Their simple geometric shape allows for straightforward fabrication processes, making them a preferred choice in many applications.
4. Importance of Fin Weight in Aircraft Cooling Devices
- This statement is valid. In aircraft design, minimizing weight is crucial for overall performance, fuel efficiency, and payload capacity. Thus, the weight of fins in cooling systems is indeed of paramount importance.
Conclusion
Based on the evaluations:
- Statement 1 is correct.
- Statement 2 is misleading as the differences may not always be "very large."
- Statement 3 is correct.
- Statement 4 is correct.
Thus, the correct options are 1, 3, and 4, aligning with option 'B'.

Thermal conductivity of wood is affected by several factors, including moisture content, density, and temperature. Let's discuss each of these factors in detail:

Moisture content:
- Wood is a hygroscopic material, which means it can absorb or release moisture from the environment.
- Moisture content in wood can affect its thermal conductivity because water has a much higher thermal conductivity than wood.
- As the moisture content in wood increases, its thermal conductivity also increases. This is because the water molecules conduct heat better than the wood molecules.

Density:
- The density of wood can also affect its thermal conductivity.
- Generally, the denser the wood, the higher its thermal conductivity. This is because denser wood has more atoms per unit volume, which means there are more pathways for heat transfer to occur.

Temperature:
- The temperature of wood can also affect its thermal conductivity.
- As the temperature of wood increases, its thermal conductivity also increases. This is because the wood molecules vibrate more rapidly at higher temperatures, which means they can transfer heat more efficiently.

Overall, the thermal conductivity of wood is influenced by these three factors: moisture content, density, and temperature. Therefore, option D, "all of these," is the correct answer.

Heat Transfer in Different Processes

Melting of Ice:
- Involves only one mode of heat transfer, i.e. latent heat transfer
- Heat is absorbed by the ice to break the bonds between molecules and change the phase from solid to liquid
- No conduction or convection involved

Cooling of a Small Metal Casting in a Quenching Bath:
- Involves two modes of heat transfer, i.e. convection and conduction
- Metal casting is immersed in a quenching bath to cool it quickly
- Convection occurs as the cooler liquid in the bath comes in contact with the hot metal casting and carries away the heat
- Conduction occurs as heat is transferred from the hot metal casting to the cooler liquid in the bath

Heat Flow Through the Walls of a Refrigerator:
- Involves only one mode of heat transfer, i.e. conduction
- Heat flows from the warmer environment outside the refrigerator to the cooler environment inside the refrigerator
- The walls of the refrigerator act as a barrier to the heat flow through conduction

Automobile Engine Equipped with a Thermosyphon Cooling System:
- Involves all three modes of heat transfer, i.e. conduction, convection, and radiation
- Heat is generated in the engine due to combustion and friction
- Conduction occurs as the heat is transferred from the engine to the coolant in the engine block
- Convection occurs as the heated coolant rises and cooler coolant sinks, creating a natural circulation of the fluid
- Radiation occurs as the heat is transferred from the engine to the surrounding environment
- The thermosyphon cooling system helps in the efficient transfer of heat and keeps the engine cool

Therefore, option 'D' is the correct answer as it involves all three modes of heat transfer.

S ability to conduct heat.
2. Materials with high thermal conductivity are good conductors of heat.
3. Metals generally have high thermal conductivity.
4. Insulators have low thermal conductivity.

All of these statements are true.

Thermal conductivity of air with rise in temperature

The correct answer is option 'A', which states that the thermal conductivity of air increases with a rise in temperature. This can be explained by the following factors:

1. Increase in molecular motion:
As the temperature of a gas (such as air) increases, the average kinetic energy of its molecules also increases. This leads to an increase in molecular motion, resulting in more frequent collisions between the molecules. These collisions facilitate the transfer of thermal energy from one molecule to another, thereby increasing the thermal conductivity of the gas.

2. Increase in collision frequency:
With the increase in temperature, the frequency of collisions between gas molecules also increases. This is due to the higher molecular velocity resulting from increased kinetic energy. As a result, there is a greater probability of energy transfer through collisions, leading to higher thermal conductivity.

3. Decrease in mean free path:
The mean free path is the average distance a molecule travels between successive collisions. With the rise in temperature, the mean free path of air molecules decreases. This is because the increased molecular motion leads to more frequent collisions, reducing the average distance traveled by the molecules. A shorter mean free path implies that the molecules have less distance to travel before they collide and transfer thermal energy. Consequently, the thermal conductivity of air increases.

4. Impact of molecular structure:
The molecular structure of air remains relatively unchanged with temperature variations. Therefore, the increase in thermal conductivity is primarily influenced by the factors mentioned above, such as increased molecular motion and collision frequency.

In summary, the thermal conductivity of air increases with a rise in temperature due to enhanced molecular motion, higher collision frequency, and reduced mean free path. These factors contribute to a more efficient transfer of thermal energy within the gas.

Explanation:

Given:
- Two long parallel surfaces with emissivity 0.7
- Desired reduction in radiant heat transfer: 75%
- Thin parallel shields with equal emissivity on both sides
- Number of shields required: ?

Solution:

Step 1: Calculate the initial radiant heat transfer
- Let's assume the initial radiant heat transfer between the two surfaces is Q.
- The radiant heat transfer between two surfaces can be calculated using the Stefan-Boltzmann Law:
Q = εσA(T1^4 - T2^4)
where ε is the emissivity, σ is the Stefan-Boltzmann constant, A is the surface area, T1 is the temperature of the first surface, and T2 is the temperature of the second surface.
- Since the emissivity of both surfaces is 0.7, we can simplify the equation as:
Q = 0.7σA(T1^4 - T2^4)

Step 2: Calculate the desired reduction in radiant heat transfer
- The desired reduction in radiant heat transfer is 75%.
- Therefore, the radiant heat transfer after inserting the shields should be 25% of the initial radiant heat transfer.
- Let's assume the radiant heat transfer after inserting the shields is Q'.
- Hence, Q' = 0.25Q

Step 3: Calculate the number of shields required
- Let's assume the number of shields required is n.
- When n shields are inserted between the two surfaces, the radiant heat transfer between each shield is reduced by a factor of 0.7 (emissivity of the shields).
- Therefore, the radiant heat transfer between the shields can be calculated as:
Q'' = (0.7)^nQ
- Since there are two sides to each shield, the total reduction in radiant heat transfer is:
Q''' = (0.7)^{2n}Q
- The radiant heat transfer after inserting the shields is given by:
Q' = Q - Q'''
- Substituting the values of Q' and Q''', we get:
0.25Q = Q - (0.7)^{2n}Q
- Simplifying the equation, we get:
(0.7)^{2n} = 0.75
- Taking the logarithm on both sides of the equation, we get:
2n log(0.7) = log(0.75)
n = log(0.75) / (2 log(0.7))
n ≈ 2.45

Conclusion:
- The number of shields required to reduce 75% of the radiant heat transfer is approximately 2.
- Therefore, the correct answer is option 'C' (two shields).

Is h: convective heat transfer coefficient (W/m^2·K)
k: thermal conductivity of the fluid (W/m·K)
Nu: Nusselt number (dimensionless)
D: diameter of the cylinder (m)
L: characteristic length of the cylinder (e.g. diameter or length) (m)
Pr: Prandtl number (dimensionless)
Re: Reynolds number (dimensionless)
μ: dynamic viscosity of the fluid (Pa·s)
v: velocity of the fluid (m/s)
c: specific heat capacity of the fluid (J/kg·K)
ρ: density of the fluid (kg/m^3)
β: thermal expansion coefficient of the fluid (1/K)
g: acceleration due to gravity (m/s^2)
ΔT: temperature difference between the surface and the fluid (K)

Understanding Surface Hardening of Mild Steel
Surface hardening of mild steel through carbon diffusion is a crucial process in mechanical engineering. It enhances wear resistance and strength by altering the material's surface properties.
Transient Mass Diffusion Process
- Definition: A transient mass diffusion process occurs when the concentration of diffusing species changes with time. In this case, carbon atoms diffuse into the steel surface, gradually increasing the carbon concentration.
- Mechanism: Initially, the carbon concentration at the surface is higher than that within the steel. Over time, carbon atoms move into the steel, leading to a gradient in concentration that changes as the process continues.
- Factors Affecting Transience:
- Temperature: Higher temperatures increase the diffusion rate.
- Time: The longer the component is exposed, the deeper carbon can diffuse into the steel.
- Concentration Gradient: As the carbon diffuses, the gradient diminishes, affecting the diffusion rate.
Comparison with Other Processes
- Steady Mass Diffusion Process: In contrast, a steady-state process implies that the concentration remains constant over time, which is not the case in carbon diffusion during surface hardening.
- Pressure Diffusion Process: This refers to diffusion driven by pressure differences, which is not applicable in the context of carbon diffusion in mild steel.
- Ordinary Diffusion Process: While carbon diffusion can be considered ordinary, it specifically refers to the transient nature in this scenario due to changing concentration over time.
Conclusion
In conclusion, the surface hardening of mild steel through carbon diffusion is a transient mass diffusion process, characterized by time-dependent changes in carbon concentration, which effectively enhances the mechanical properties of the steel.

Solution:

Given:
Radiation received at the surface of the Earth from the sun = 1400 W/m²
Distance between the center of the sun and the surface of the Earth = 1.5 x 10^11 m
Radius of the sun = 7 x 10^8 m

To find:
Surface temperature of the sun

Formula:
The power radiated by a black body is given by the Stefan-Boltzmann law:
P = σAT^4
where P is the power, σ is the Stefan-Boltzmann constant (5.67 x 10^-8 W/m²K⁴), A is the surface area, and T is the temperature.

Step-by-step solution:

1. Calculate the surface area of the sun:
The surface area of a sphere is given by the formula:
A = 4πr²
where r is the radius.

Substituting the given radius of the sun:
A = 4π(7 x 10^8)²
A = 1.54 x 10^18 m²

2. Calculate the power radiated by the sun:
Given power received at the surface of the Earth = 1400 W/m²
Area of the Earth's surface = A = 4πr²
where r is the distance between the center of the sun and the surface of the Earth.

Substituting the given distance:
A = 4π(1.5 x 10^11)²
A = 2.83 x 10^23 m²

The power radiated by the sun can be calculated using the formula:
P = 1400 W/m² x 2.83 x 10^23 m²
P = 3.96 x 10^26 W

3. Calculate the temperature of the sun:
Using the Stefan-Boltzmann law:
P = σAT^4

Rearranging the formula to solve for T:
T^4 = P / (σA)
T = (P / (σA))^(1/4)

Substituting the calculated values:
T = (3.96 x 10^26) / (5.67 x 10^-8 x 1.54 x 10^18)^(1/4)
T ≈ 5800 K

Therefore, the surface temperature of the sun is approximately 5800 K, which corresponds to option C.

Understanding Steady State Heat Flow in a Sphere
In a sphere with steady-state heat conduction and constant thermal conductivity, the temperature distribution is derived from the principles of heat transfer.
Key Concepts
- Steady-State Condition: This implies that the temperature within the sphere does not change with time. The heat entering a volume equals the heat leaving it.
- Spherical Coordinates: The analysis is conducted in spherical coordinates (r, θ, φ), where 'r' represents the radial distance from the center of the sphere.
Mathematical Derivation
- Heat Equation: The heat conduction equation in spherical coordinates leads to a differential equation where the radial temperature distribution depends only on the radial distance 'r'.
- Boundary Conditions: By applying appropriate boundary conditions (such as temperatures at the surface or center), we can solve the differential equation to find the temperature profile.
Temperature Distribution
- Hyperbolic Nature: The solution to the differential equation results in a hyperbolic relationship, typically represented as:
- T(r) = A + B * ln(r), where A and B are constants determined by boundary conditions.
- Physical Interpretation: This hyperbolic behavior indicates that as one moves from the center to the surface of the sphere, the temperature decreases logarithmically, which is characteristic of heat conduction in spherical bodies.
Conclusion
The temperature distribution in a sphere under steady-state conditions and constant thermal conductivity indeed exhibits a hyperbolic nature, making option 'C' the correct answer. Understanding this concept is crucial for applications in thermal management and engineering designs involving spherical geometries.

The correct answer is option 'C' - Glass-wool.

Explanation:
In order to understand why the temperature drop across the material will be maximum for glass-wool, let's consider the concept of thermal conductivity.

Thermal conductivity is a property of a material that determines how well it can conduct heat. Materials with high thermal conductivity can transfer heat more easily, while materials with low thermal conductivity transfer heat slowly.

1. Copper:
- Copper is a metal that has a very high thermal conductivity, which means it can conduct heat very well.
- When heat flows through copper, the temperature drop across the material will be relatively small because it can easily transfer heat from one side to the other.

2. Steel:
- Steel is also a metal, but it has a lower thermal conductivity compared to copper.
- When heat flows through steel, the temperature drop across the material will be higher than copper, but still relatively small compared to other materials.

3. Glass-wool:
- Glass-wool is an insulating material commonly used for thermal insulation purposes.
- It has a very low thermal conductivity, which means it does not conduct heat well.
- When heat flows through glass-wool, the temperature drop across the material will be higher compared to copper and steel.
- This is because glass-wool resists the transfer of heat, causing a larger temperature difference between the two sides of the material.

4. Refractory Brick:
- Refractory brick is a material designed to withstand high temperatures and is commonly used in furnaces and kilns.
- It has a low thermal conductivity, similar to glass-wool.
- When heat flows through refractory brick, the temperature drop across the material will also be higher compared to copper and steel.

Conclusion:
The temperature drop across a material is related to its thermal conductivity. Materials with low thermal conductivity, such as glass-wool and refractory brick, will have a higher temperature drop compared to materials with high thermal conductivity, such as copper and steel. Therefore, the temperature drop will be maximum for glass-wool (option 'C') in this scenario.

Understanding Thermal Conductivity
Thermal conductivity is a measure of a material's ability to conduct heat. It varies significantly across different states of matter (solid, liquid, gas).
Factors Affecting Thermal Conductivity
- State of Matter: Solids generally have higher thermal conductivity than liquids and gases due to closely packed molecules.
- Molecular Structure: The arrangement and bonding of atoms affect how easily heat can be transferred.
Comparison of Options
- Boiling Water: While water can conduct heat, its thermal conductivity is lower than that of solids. Its value is about 0.6 W/m·K.
- Steam: As a gas, steam has a lower thermal conductivity compared to liquids and solids. The thermal conductivity of steam is around 0.02 W/m·K.
- Solid Ice: Ice, being a solid, has a higher thermal conductivity than water in liquid form. The thermal conductivity of ice is approximately 2.2 W/m·K.
- Rainwater: Similar to boiling water, rainwater has low thermal conductivity, comparable to that of boiling water.
Conclusion
Based on the comparison, solid ice (option C) possesses the highest thermal conductivity among the listed options. This is due to its solid state, which allows for efficient heat transfer through tightly packed molecular structures. In contrast, the other options (boiling water, steam, and rainwater) are either in liquid or gaseous states, resulting in lower thermal conductivity.
Thus, the correct answer is indeed option C: Solid Ice.