ME-701 Heat and Mass Transfer - May 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. 4 Marks
   aGive the answer in brief with neat sketch (within 100 words) a) State Lambert's cosine law and its significance.
   How to answer

   • State Lambert's cosine law: $E_\theta = E_n \cos\theta$
   • Explain diffuse emission intensity independent of direction
   • Significance for diffuse surfaces, irradiation and radiosity calculations
   • Neat sketch of emission from surface element at angle
2. 4 Marks
   bGive the answer in brief with neat sketch (within 100 words) b) Explain the need of NTU method over the LMTD method? State the merits and demerits of NTU method over LMTD method.
   How to answer

   • Need of NTU when outlet temperatures unknown, LMTD needs iterative trial
   • Define NTU, effectiveness, capacity ratio
   • Merits of NTU: direct sizing, no iteration, design vs rating
   • Demerits: charts/correlations, less intuitive for existing exchanger
3. 6 Marks
   cGive the answer in brief with neat sketch (within 100 words) c) Define about diffusion and explain Fick's law of diffusion. Explain the term mass diffusivity.
   How to answer

   • Define mass diffusion: species transport by concentration gradient
   • State Fick's law: $\dot{m}_A = -D_{AB} A \frac{dC_A}{dx}$
   • Explain terms and analogy with Fourier's law
   • Define mass diffusivity $D_{AB}$: units, dependence on T, P, medium
4. 10 Marks
   aConsider a $0.8$ m high and $1.5$ m wide double-pane window consisting of two $4$ mm thick layers of glass ($k = 0.78$ W/m$^o$C) separated by a $10$ mm wide stagnant air space ($k = 0.026$ W/m$^o$C), Determine the steady rate of heat transfer through this double-pane window and the temperature of its inner surface for a day during which the room is maintained at $20^o$C while the temperature of the outdoors is $-10^o$C. Take the convection heat transfer coefficients on the inner and outer surfaces of the window to $h_1 = 10$ W/m$^{2o}$C and $h_2 = 40$ W/m$^{2o}$C, which includes the effects of radiation?
   How to answer

   • Compute area $A=0.8\times 1.5$ and thermal resistances: $R_{conv}+R_{glass}+R_{air}+R_{glass}+R_{conv}$
   • Calculate overall heat transfer coefficient $U=1/\Sigma R''$
   • Steady rate $Q = U A (T_{in}-T_{out})$
   • Inner surface temperature from $Q = h_1 A (T_{room}-T_{s,in})$
5. 4 Marks
   bA copper wire of radius $0.5$ mm is insulated with a sheathing of thickness $1$ mm having a thermal conductivity of $0.5$ W/m-K. The outside surface convective heat transfer coefficient is $10$ W/m$^2$-K. If the thickness of insulation sheathing is raised by $10$ mm, then comment on the electrical current-carrying capacity of the wire.
   How to answer

   • Calculate critical radius $r_c = k/h$
   • Compare wire radius 0.5 mm + 1 mm insulation vs $r_c$
   • Show increasing to 10 mm increases resistance beyond critical, reduces heat loss
   • Conclude current-carrying capacity decreases/increases accordingly
6. 4 Marks
   aA stainless steel tube ($k_s = 19$ W/mK) of $2$ cm ID and $5$ cm OD is insulated with $3$ cm thick asbestos ($k_a = 0.2$ W/mK). If the temperature difference between the innermost and outermost surfaces is $600^o$C, Find the heat transfer rate per unit length?
   How to answer

   • Thermal resistance for composite cylinder per unit length
   • Insert $r_1, r_2, r_3$, $k_s, k_a$
   • Use $Q/L = 2\pi \Delta T / \Sigma [\ln(r_{o}/r_i)/k]$
   • Compute numerical value in W/m
7. 10 Marks
   bDerive expression for temperature distribution, under one dimensional steady state heat conduction for a cylinder.
   How to answer

   • Start from Fourier law in cylindrical coordinates, steady 1-D with no generation
   • Integrate $\frac{d}{dr}(r dT/dr)=0$
   • Apply boundary conditions $T(r_1), T(r_2)$ to get log profile
   • Write heat rate $Q = 2\pi k L \Delta T/\ln(r_2/r_1)$
8. 7 Marks
   aDerive the momentum equation for hydrodynamic boundary layer over a flat plate.
   How to answer

   • Consider control volume in laminar boundary layer, mass and momentum balance
   • Introduce velocity profiles $u(x,y), v(x,y)$, shear $\tau = \mu du/dy$
   • Derive von Karman integral or differential $u\partial u/\partial x + v\partial u/\partial y = -1/\rho dp/dx + \nu \partial^2 u/\partial y^2$
   • State assumptions: steady, incompressible, negligible body force
9. 7 Marks
   bIn a cross-flow heat exchanger, air is heated by water. Air enters the exchanger at $150^o$C and a mass flow rate of $2$ kg/s while water enters at $900^o$C and a mass flow rate of $0.25$ kg/s. The overall heat transfer coefficient is $250$ W/m$^2$.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

   • Determine $C_{min}, C_{max}$, $C_r$, $NTU = UA/C_{min}$
   • Use cross-flow effectiveness relation to get $\varepsilon$
   • Energy balance $Q = \varepsilon C_{min}(T_{h,in}-T_{c,in})$
   • Find exit temperatures $T_{h,out}, T_{c,out}$
10. 7 Marks
    aA long rod $12$ mm square section made of low carbon steel protrudes into air at $35^o$C from a furnace wall at $200^o$C. The convective heat transfer co-efficient is estimated at $22$ W/m$^2$K. The conductivity of the material is $51.9$ W/mK. Determine the location from the wall at which the temperature will be $60^o$C. Also calculate the temperature at $80$ mm from base. Attempt the question with derivative approach.
    How to answer

    • Model as infinite fin: $m = \sqrt{hP/kA_c}$, $\theta/\theta_0 = e^{-mx}$
    • Compute $P, A_c$ for 12 mm square, $\theta_0 = 200-35$
    • Solve $x$ for $T=60^o$C and $T$ at $x=80$ mm
    • Mention derivative approach steps
11. 7 Marks
    bExplain the Significance of Dimensionless numbers. Write down the advantages and limitations of dimensional analysis.
    How to answer

    • Define Reynolds, Nusselt, Prandtl, Grashof numbers
    • Physical significance: ratio of forces/heat transfer mechanisms
    • Advantages: reduces variables, model testing, correlations
    • Limitations: empirical, range restricted, loses detail
12. 7 Marks
    aDerive the momentum equation for hydrodynamic boundary layer over a flat plate.
    How to answer

    • Same as Q4a: control volume momentum balance over flat plate
    • Assumptions: steady, 2-D, incompressible, constant properties
    • Final boundary layer momentum equation
    • Mention Blasius simplification $dp/dx=0$
13. 7 Marks
    bAir at $20^o$C flows through a tube $8$ cm dia with a velocity of $9$ m/s. The tube wall is at $80^o$C. Determine for a tube length of $5$ m, the exit temperature of air.
    How to answer

    • Evaluate bulk mean properties, $Re = \rho V D/\mu$, $Pr$
    • Select Dittus-Boelter/Seider-Tate correlation for constant wall temp
    • Compute $Nu$, $h$, $LMTD$ or $Q = h A \Delta T_{lm}$
    • Energy balance $Q = \dot{m}c_p(T_{out}-T_{in})$ to solve exit air temp
14. 7 Marks
    aExplain the boiling heat transfer phenomenon and regimes of boiling with neat sketch.
    How to answer

    • Define boiling: phase change at heated surface, excess temperature
    • Describe regimes: natural convection, nucleate, transition, film boiling
    • Draw boiling curve $q''$ vs $\Delta T$ with critical heat flux
    • Neat sketch with labels
15. 7 Marks
    bExplain the significance of Shape factor. Determine the shape factor from the base of a cylinder to the curved surface. Also find the shape factor from curved surface to base and the curved surface to itself.
    How to answer

    • Define view/shape factor $F_{ij}$, summation and reciprocity rules
    • For cylinder: $F_{base-base}=0$, find $F_{base-curved}$
    • Use reciprocity $A_1F_{12}=A_2F_{21}$ to get curved to base
    • Find $F_{curved-curved}$ by summation rule
16. 4 Marks
    aDefine Radiosity and irradiation.
    How to answer

    • Define radiosity $J = \varepsilon E_b + (1-\varepsilon)G$
    • Define irradiation $G$: incoming radiation per unit area
    • Units W/m2, difference from emissive power
    • Diagram of surface showing $J, G$
17. 4 Marks
    bWhat is Fouling? What are its effects on heat exchanger performance?
    How to answer

    • Define fouling: deposition of scale, dirt, biological growth
    • Thermal resistance $R_f$ added, reduces $U$
    • Effects: lower effectiveness, higher pressure drop, maintenance
    • Mitigation: cleaning, treatment
18. 6 Marks
    cExplain the condition under which the slope of the hot and cold fluid temperature lines will be equal. Indicate the effect of such condition in the case of counter flow exchangers.
    How to answer

    • Condition $C_h = C_c$ i.e. capacity rates equal gives parallel slopes
    • Derive $dT_h/dx$, $dT_c/dx$ from energy balance
    • Effect in counterflow: constant $\Delta T$, linear profiles, max effectiveness
    • Indicate on T-x diagram