ME-701 Heat and Mass Transfer - May 2024 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
   aExplain Newton's law of cooling.
   How to answer

   • State Newton's law of cooling: q = h*A*(Ts - Tinf).
   • Define convective heat transfer coefficient h.
   • Explain proportionality to temperature difference and conditions of applicability.
2. 3 Marks
   bPhysically, what does the Grashof number represent and how does it differ from Reynolds number?
   How to answer

   • Define Grashof number Gr = g*beta*deltaT*L^3/nu^2.
   • Physical meaning: ratio of buoyancy to viscous forces.
   • Define Reynolds number Re = rho*V*L/mu and its meaning: inertia/viscous.
   • Contrast forced vs natural convection relevance.
3. 3 Marks
   cWrite a short note on black body and gray body.
   How to answer

   • Define black body: perfect emitter/absorber, emissivity 1.
   • Define gray body: emissivity <1 constant with wavelength.
   • Compare absorptivity, emissive power.
   • Give examples/laws applicable.
4. 3 Marks
   dExplain about Nusselt number and discuss about its physical significance.
   How to answer

   • Define Nusselt number Nu = h*L/k.
   • Explain as ratio of convective to conductive heat transfer.
   • Relate to temperature gradient at wall.
   • Mention use in convection correlations.
5. 3 Marks
   eExplain about mass average velocity.
   How to answer

   • Define mass average velocity as sum(mi*vi)/sum(mi).
   • Distinguish from molar average velocity.
   • Explain relevance in mass transfer diffusion.
   • Give mathematical expression.
6. 7 Marks
   aDerive expression for temperature distribution, under one dimensional steady state heat conduction for a plane wall.
   How to answer

   • Start from Fourier's law and steady 1D energy balance with no generation.
   • Show d/dx(k*dT/dx)=0 leading to linear profile.
   • Apply boundary conditions T(0)=T1, T(L)=T2.
   • Obtain T(x)=T1-(T1-T2)*x/L and heat rate expression.
7. 7 Marks
   bHow is the concept of the critical radius of insulation applied in the design of refrigeration systems?
   How to answer

   • Define critical radius r_c = k/h for cylinder, 2k/h for sphere.
   • Explain increase then decrease of heat loss with insulation thickness.
   • Discuss small-diameter refrigeration lines vs large pipes.
   • Conclude design criterion: add insulation only if r > r_c.
8. 10 Marks
   aA cold storage room has walls made of 23 cm of brick on the outside, 8 cm of plastic foam and finally 1.5 cm of wood on the inside. The outside and inside air temperatures are $22^{\circ}C$ and $-2^{\circ}C$ respectively. The inside and outside heat transfer coefficients are respectively 29 and 12 $W/m^{2}K$. The thermal conductivities of brick, foam and wood are 0.98, 0.02 and 0.12 $W/m.K$ respectively. If the total wall area is 90 $m^{2}$, determine the rate of heat removal by refrigeration and the temperature of the inside surface of the brick.
   How to answer

   • Calculate conductive resistances of brick, foam, wood and convective resistances.
   • Find overall heat transfer coefficient and total heat transfer Q = U*A*deltaT.
   • Equate Q to convection/radiation balance to get interface temperature.
   • Report heat removal rate and inside surface temperature of brick.
9. 4 Marks
   bState and prove Kirchhoff's law of radiation.
   How to answer

   • State Kirchhoff's law: emissivity = absorptivity at same T and wavelength.
   • Consider enclosure with small body and surroundings.
   • Apply energy balance at thermal equilibrium.
   • Extend to spectral and directional form.
10. 6 Marks
    aExplain the advantages and limitations of dimensional analysis.
    How to answer

    • Define dimensional analysis and Buckingham Pi theorem.
    • List advantages: reduces variables, model testing, correlation.
    • List limitations: no theory, choice of variables, constants not given.
    • Give heat transfer examples.
11. 8 Marks
    bConsider two long, concentric cylindrical surfaces with emissivities of 1, acting as black bodies, where the outer cylinder has a diameter of 0.5 meters and the inner cylinder has a diameter of 0.1 meters. If the outer cylinder is maintained at a temperature of $600^{\circ}C$ and the inner cylinder at $1100^{\circ}C$, how can you calculate the net radiant heat exchange between the two cylinders?
    How to answer

    • Identify as black-body concentric cylinders, view factor Fij.
    • Use net exchange formula Q = sigma*(T1^4 - T2^4)/((1-e1)/A1e1 + 1/A1F12 + (1-e2)/A2e2).
    • Substitute e=1, diameters 0.1 m and 0.5 m, T=1373 K and 873 K.
    • Compute heat exchange per unit length.
12. 7 Marks
    aA cylindrical brass rod with a length of 15 cm and a diameter of 2 cm is initially at $500^{\circ}C$. The rod is exposed to an environment at $30^{\circ}C$ with a convective heat transfer coefficient of 100 $W/(m^{2}K)$. The properties of brass are: density $\rho = 8500$ $kg/m^{3}$, specific heat $C_{p}=380$ $J/(kg \cdot K)$, and thermal conductivity k = 120 $W/(m \cdot K)$. Determine if the lumped system analysis can be used for this transient heat transfer problem.
    How to answer

    • Compute characteristic length Lc = V/A and Biot number Bi = h*Lc/k.
    • Check criterion Bi < 0.1 for lumped validity.
    • Use brass properties rho, Cp, k given.
    • Conclude whether lumped analysis applicable.
13. 7 Marks
    bIf it can be used, find the time required for the center of the rod to cool to $100^{\circ}C$.
    How to answer

    • Use lumped capacitance relation (T-Tinf)/(T0-Tinf)=exp(-h*A*t/(rho*V*Cp)).
    • Solve for time t to reach 100°C.
    • Substitute area/volume for cylinder including ends.
    • Compute time in seconds/minutes.
14. 7 Marks
    aCalculate the average co-efficient of heat transfer for natural convection for a vertical plate 30.48 cm high at $51.67^{\circ}C$. The surrounding air is at $25^{\circ}C$. Also calculate the boundary layer thickness at the trailing edge of plate.
    How to answer

    • Evaluate film temperature and air properties.
    • Calculate Grashof and Rayleigh numbers for vertical plate.
    • Use natural convection correlation Nu = C*Ra^n to get h_avg.
    • Estimate boundary layer thickness at trailing edge.
15. 7 Marks
    bDerive an expression for logarithmic mean temperature difference in case of parallel flow heat exchanger.
    How to answer

    • Consider differential element in parallel-flow exchanger.
    • Write dQ = -mh*cph*dTh = mc*cpc*dTc = U*dA*deltaT.
    • Integrate from inlet to outlet to get ln(deltaT2/deltaT1).
    • Obtain LMTD = (deltaT1-deltaT2)/ln(deltaT1/deltaT2).
16. 6 Marks
    aExplain various modes of mass transfer and enumerate its applications.
    How to answer

    • Define diffusion, convection mass transfer and phase change.
    • Explain Fick's law and mass transfer coefficient.
    • Give examples: evaporation, absorption, drying.
    • List industrial applications.
17. 8 Marks
    bVessel contains a binary mixture of oxygen and nitrogen with partial pressures in the ratio 0.21 and 0.79 at $15^{\circ}C$. The total pressure of the mixture is 1.1 bar. Determine:

    1. i)Molar concentration
    2. ii)Mass densities
    3. iii)Mass fractions
    4. iv)Molar fractions of each species
    How to answer

    • Use Dalton's law to get partial pressures from ratio and total pressure.
    • Apply ideal gas law to get molar concentrations.
    • Compute densities rho_i = p_i*M_i/(R*T).
    • Derive mass and molar fractions of O2 and N2.
18. 14 Marks
    A new experimental heat sink is designed to cool a high-power electronic component. The heat sink is made of aluminium and has multiple fins. The heat sink operates in a room where the air temperature is $25^{\circ}C$. You have been tasked with developing a dimensionless model using the Buckingham Pi theorem to predict the performance of the heat sink. The variables involved are the thermal conductivity (k), the heat transfer coefficient (h), the fin thickness (t), the fin height (L), the temperature difference between the base of the fin and the air $(\Delta T)$, the density of air $(\rho)$, the specific heat of air at constant pressure (cp) and the dynamic viscosity of air $(\mu)$. Using the Buckingham Pi theorem, determine the dimensionless groups (Pi terms) that govern the heat transfer from the fins.
    How to answer

    • List variables: k, h, t, L, deltaT, rho, cp, mu and their dimensions.
    • Choose repeating variables rho, mu, k, L.
    • Form Pi groups e.g. Nusselt, Prandtl, geometric ratios.
    • Interpret groups governing fin heat transfer.