JNTU
JNTU B.Tech IV HEAT TRANSFER Previous Question Paper (July/August - 2022 - R16)
Answer any five questions. All questions carry equal marks.
Main
Derive the general heat conduction equation for Cartesian co-ordinates from fundamentals.
Answer
Consider a small differential control volume of dimensions dx, dy, and dz in a Cartesian coordinate system. Applying the first law of thermodynamics (energy balance): Energy in + Energy generated = Energy out + Rate of change of internal energy.
For the x-direction, the heat flux entering is qx and leaving is qx+dx = qx + (∂qx/∂x)dx. Applying Fourier's Law, qx = -k(dy dz)(∂T/∂x). The net heat flow in the x-direction is -(∂qx/∂x)dx = (∂/∂x)(k ∂T/∂x)dx dy dz. Repeating this for y and z directions and adding internal heat generation q̇(dx dy dz), the total energy balance becomes:
For constant thermal conductivity k, this simplifies to the general heat conduction equation:
where α = k/(ρcp) is the thermal diffusivity.
A 150 mm steam pipe has inside diameter of 120 mm and outside diameter of 160 mm. It is insulated at the outside with asbestos. The steam temperature is 150°C and the air temperature is 20°C. h(steam side) = 100 W/m2K, h(air side) = 30 W/m2K, k(asbestos) = 0.8 W/m°C and k(steel) = 42 W/mK. How thick should the asbestos be provided in order to limit the heat loss to 2.1 kW/m2.
Answer
Given: r1 = 0.06 m, r2 = 0.08 m, Ti = 150°C, To = 20°C, hi = 100 W/m2K, ho = 30 W/m2K, ksteel = 42 W/mK, kasb = 0.8 W/mK, q/A = 2100 W/m2. Since the heat loss is given per unit area (based on outside surface area Ao), we use the thermal resistance network per unit area.
The total resistance Rtotal = ΔT / (q/A) = (150 - 20) / 2100 = 0.0619 m2K/W. The total resistance is the sum of convective, conductive, and external convective resistances:
Substituting ro = r3 (the outer radius of insulation):
Solving for r3 iteratively: 0.0619 - 0.0333 = 0.1667r3 + 0.0068r3 + 1.25r3 ln(r3/0.08). Simplifying: 0.0286 = 0.1735r3 + 1.25r3 ln(r3/0.08). Assuming r3 ≈ 0.12 m, the calculation yields r3 ≈ 0.115 m. The thickness of asbestos is r3 - r2 = 0.115 - 0.08 = 0.035 m or 35 mm.
Derive expression for temperature distribution, under one dimensional state heat conduction for a composite wall. (Consider three layers)
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A 2000 W heater element of area 0.04 m2 is protected on the backside with insulation 50 mm thick of K=1.4 W/mK and on the front side by a plate 100 mm thick with thermal conductivity of 45 W/mK. The backside is exposed to air at 5°C with convection coefficient of including radiation of 250 W/m2 K. Determine the heat flow into the room under steady conditions.
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Derive the expression for temperature distribution and heat dissipation in an infinite long fin of rectangular profile.
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A long carbon steel rod length 40 cm and diameter 10 mm (K = 40 W/mK) is placed in such that one of its end is at 400°C and the ambient temperature 30°C. The film coefficient is 10 W/m2K. Determine: (i) Temperature at mid length of the fin. (ii) Fin efficiency. (iii) Heat transfer rate from the fin.
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Explain the method of construction of Hiestler charts and explain the application.
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A 10 cm diameter cylindrical bar heated in the furnace to a temperature of 200°C is allowed to cool in an environment with convection coefficient of 150 W/m2K and temperature of 40°C determine the (i) the time required to cool the centre of the bar to 50°C ; (ii) Temperature of the surface at this instant. For the material of the bar, K = 50 W/m K and thermal diffusivity 2 × 10-5 m2/s.
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By using dimensional analysis, obtain the general form of equation for Forced convective heat transfer using Buckingham π-Theorem. Also explain the advantages and limitations of dimensional analysis.
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Derive and explain the Reynolds colburn analogy for laminar flow over a plate.
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Air at 20°C and at atmospheric pressure flows at a velocity of 4.5 m/s past a flat plate with a sharp leading edge. The entire plate surface is maintained at a temperature of 60°C. Assuming that the transition occurs at Re = 5 × 105, find the distance from the leading edge, at which the flow in the boundary layer changes from laminar to turbulent, at this location calculate:
i) Thickness of hydrodynamic and thermal boundary layers,
ii) Heat transfer rate from both sides for unit width of the plate,
iii) The skin friction coefficient.
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Define effectiveness of heat exchanger and discuss its significance.
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In a counter flow double pipe heat exchanger, water is heated from 25°C to 65°C by oil with a specific heat of 1.45 kJ/kg-K and mass flow rate of 0.9 kg/s. The oil is cooled from 230°C to 160°C. If overall heat transfer coefficient is 420 W/m2-K. Calculate the rate of heat transfer, mass flow rate of water and surface area of heat exchanger.
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Derive the relationship for intensity of radiation and emissive power of a black body.
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A thermocouple is used to measure the temperature of a gas stream in a duct, records a value of 250°C, if the duct walls are at temperature of 200°C, calculate the true gas temperature. Assume that the thermocouple has a black surface and the convection coefficient is 25 W/m2K.
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