ME-701 Heat and Mass Transfer - Nov 2023 Question Paper

Under each question, How to answer lists the points an answer should cover, outlined by CampusPrep from the syllabus. Check the details against your textbook.

1. 2 Marks
   aWhat is Fourier's law of conduction?
   How to answer

   • State Fourier's law: $q = -kA(dT/dx)$
   • Define thermal conductivity k and negative sign for heat flow down temperature gradient
   • Give one-dimensional vector form
2. 3 Marks
   bWhat is critical thickness of insulation? Explain.
   How to answer

   • Define critical thickness / radius of insulation
   • Explain increase then decrease of heat loss due to conduction vs convection resistance
   • Give expressions for cylinder and sphere and condition for minimum loss
3. 3 Marks
   cExplain about radiation shield.
   How to answer

   • Define radiation shield purpose to reduce radiative exchange
   • Explain insertion of low emissivity shield between surfaces
   • Give heat transfer reduction formula with n shields
4. 3 Marks
   dWrite a short note on boundary layer thickness and displacement thickness.
   How to answer

   • Define velocity boundary layer thickness $\delta$ as $u=0.99U_\infty$
   • Define displacement thickness $\delta^*$ integral definition
   • Give physical significance and formula
5. 3 Marks
   eDefine absorptivity, reflectivity and transmissivity.
   How to answer

   • Define absorptivity $\alpha = G_{abs}/G$
   • Define reflectivity $\rho = G_{ref}/G$
   • Define transmissivity $\tau = G_{tr}/G$ and $\alpha+\rho+\tau=1$
6. 10 Marks
   aDerive expression for temperature distribution, under one dimensional steady state heat conduction for a cylinder.
   How to answer

   • Start from general heat conduction equation in cylindrical coordinates
   • Apply steady state, 1-D radial, no heat generation assumptions
   • Integrate to get $T(r)=C_1 \ln r + C_2$ and apply boundary conditions for heat rate
7. 4 Marks
   bA hollow cylindrical insulation has an internal diameter of $16\,cm$ thickness of $8\,cm$. The inner surface is at $-10^{\circ}C$ while the outer surface is at $35^{\circ}C$. Determine the radius at which the temperature is $0^{\circ}C$.
   How to answer

   • Use logarithmic temperature profile $\frac{T-T_i}{T_o-T_i}=\frac{\ln(r/r_i)}{\ln(r_o/r_i)}$
   • Substitute $T=0^{\circ}C$, $T_i=-10^{\circ}C$, $T_o=35^{\circ}C$, $r_i=8\,cm$, $r_o=16\,cm$
   • Solve for $r$
8. 10 Marks
   aA reactor wall $320\,mm$ thick is made up of an inner layer of fire brick ($k = 0.84\,W/m^{\circ}C$) covered with a layer of insulation ($k = 0.16\,W/m^{\circ}C$). The reactor operates at a temperature of $1325^{\circ}C$ and the ambient temperature is $25^{\circ}C$.

   1. i)Determine the thickness of fire brick and insulation which gives minimum heat loss.
   2. ii)Calculate the heat loss presuming that the insulating material has a maximum temperature of 1200°C. If the calculated heat loss is not acceptable, then state whether addition of another layer of insulation would provide a satisfactory solution.
   How to answer

   • Write heat loss per unit area through composite wall in terms of $x$ and $0.32-x$
   • Differentiate for minimum loss / discuss interface temperature limit $1200^{\circ}C$
   • Comment on adding extra insulation if limit exceeded
9. 4 Marks
   bExplain its importance and derive the expression to calculate critical thickness of insulation for a cylinder.
   How to answer

   • Explain trade-off between conduction and convection resistance
   • Derive $r_c = k/h$ for cylinder from $dQ/dr=0$
   • State importance for small diameter wires vs large pipes
10. 7 Marks
    aTwo large parallel plates of $1m \times 1m$ spaced $0.5m$ apart in a very large room whose walls are at $27^{\circ}C$. The plates are at $900^{\circ}C$ and $400^{\circ}C$ with emissivities $0.2$ and $0.5$ respectively. Find the net heat transfer to each plate and to the room.
    How to answer

    • Write radiosity equations for two plates and large room enclosure
    • Solve for net heat exchange using emissivities 0.2, 0.5 and temperatures
    • Compute heat to each plate and to room walls
11. 7 Marks
    bDerive a general expression for shape factor in case of radiation between two surfaces of same area.
    How to answer

    • Define shape factor $F_{ij}$ and summation/reciprocity rules
    • Consider two surfaces of same area with enclosure
    • Derive general expression for $F$
12. 7 Marks
    aA solid spherical steel ball with a diameter of $10\,cm$ and a thermal conductivity of $45\,W/(m \cdot K)$ is initially at a uniform temperature of $300^{\circ}C$. The ball is suddenly quenched in a large oil bath that is maintained at $25^{\circ}C$. The convective heat transfer coefficient between the ball surface and the oil is $250\,W/(m^2 \cdot K)$. Assume the density of steel is $7800\,kg/m^3$ and the specific heat is $500\,J/(kg \cdot K)$. Determine if the lumped capacitance method is applicable for the transient analysis.
    How to answer

    • Calculate Biot number $Bi = hL_c/k$ with $L_c=R/3$
    • Check $Bi < 0.1$ criterion for lumped validity
    • State applicability conclusion
13. 7 Marks
    bIf applicable, calculate the time required for the ball to reach $50^{\circ}C$.
    How to answer

    • Use lumped temperature-time relation $(T-T_\infty)/(T_i-T_\infty)=\exp(-hAt/\rho Vc)$
    • Substitute time constant $\tau = \rho Vc/hA$
    • Solve for $t$ to reach $50^{\circ}C$
14. 8 Marks
    aA horizontal pipe of $6\,cm$ diameter is located in a room. The temperature of air is $20^{\circ}C$. The surface temperature of the pipe is $150^{\circ}C$. Calculate the free convection heat loss per meter length of the pipe.
    How to answer

    • Evaluate film temperature and properties of air
    • Calculate Grashof, Prandtl and Nusselt numbers for horizontal cylinder
    • Compute $Q/L = h\pi D (T_s-T_\infty)$
15. 6 Marks
    bDerive the momentum equation for hydrodynamic boundary layer over a flat plate.
    How to answer

    • Consider control volume in laminar boundary layer over flat plate
    • Apply mass and x-momentum balance with shear stress
    • Obtain von Karman momentum integral equation
16. 8 Marks
    aIn a cross flow heat exchanger, air is heated by water. Air enters the exchanger at $15^{\circ}C$ and a mass flow rate of $2\,kg/s$ while water enters at $90^{\circ}C$ and a mass flow rate of $0.25\,kg/s$. The overall heat transfer coefficient is $250\,W/m^2 \cdot K$. If the heat exchanger has a heat transfer area of $8.4\,m^2$, find the exit temperatures of both the fluids.
    How to answer

    • Compute heat capacity rates $C_h$, $C_c$ and $C_{min}/C_{max}$
    • Calculate NTU $=UA/C_{min}$ and effectiveness for cross-flow
    • Obtain $Q$ and exit temperatures $T_{h,o}$, $T_{c,o}$
17. 6 Marks
    bExplain about convective mass transfer and mass transfer coefficient.
    How to answer

    • Define convective mass transfer and concentration boundary layer
    • State Fick's law analogy and mass transfer coefficient $h_m$
    • Give Sherwood number correlations
18. 6 Marks
    aThe Logarithmic Mean Temperature Difference (LMTD) of a counter flow heat exchanger is $20^{\circ}C$. The cold fluid enters at $20^{\circ}C$ and The hot fluid enters at $100^{\circ}C$. Mass flow rate of the cold fluid is twice that of the hot fluid. Specific heat at constant pressure of the hot fluid is twice that of the cold fluid. Find the exit temperature of the cold fluid?
    How to answer

    • Write energy balance $C_h(T_{h,i}-T_{h,o})=C_c(T_{c,o}-T_{c,i})$ with given ratios
    • Write LMTD expression for counter flow $\Delta T_{lm}=20^{\circ}C$
    • Solve for cold fluid exit temperature
19. 8 Marks
    bDescribe the distinct regimes observed in film boiling and nucleate boiling, emphasizing the differences in heat transfer mechanisms and surface conditions that define each regime.
    How to answer

    • Describe nucleate boiling: bubble formation, high heat transfer, surface roughness/wettability role
    • Describe film boiling: vapor film, Leidenfrost point, low heat transfer
    • Contrast mechanisms, heat flux vs superheat and surface conditions