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1.Apples having an initial homogeneous temperature of 20 0C are water cooled with flowing water of 5 0C until 8 0C. If water flows over the surface of one apple (the average water velocity is constant far from the apple), the convective heat transfer coefficient is h = 10 W/(m2 K). You may describe the apples as homogeneous spheres having a diameter of 8 cm. The thermal conductivity of the apples is 0.4 W/(m K) and the heat capacity of the apples is 3.8 kJ/(kg K). The density of the apples is 820 kg/m3.
a) State the general heat diffusion equation with the appropriate boundary conditions, for the case of one apple in the water stream.
b) Is the heat diffusion equation and boundary conditions the same for cooling a lot of apples in a vessel at once in a water stream?
c) How long does it take before the centre of an apple is cooled down to 8 0C. Is the average cooling time to a centre temperature of 8 0C the same as the average for apples in a vessel were water is flowed through? State clearly your assumptions.
d) Estimate how long it takes before the average temperature of the apples is decreased to 8 0C?
Q2
1Consider a thin electrical heater attached to a plate and backed by insulation. Initially, the heater and plate are at the temperature of the ambient air, T. Suddenly, the power to the heater is activated, yielding a constant heat flux q0'' (W / m2 ) at the inner surface of the
1 From Incropera P5.1 1

The Hague University of Applied Sciences
Academy for Technology, Innovation and Society The Hague, TISH Process and Food Technology, PFT.
Course: Unsteady State Processes PTS42 resit exam assignment 2011-05 Lecturer: Ben C. Bonekamp
plate.
a. Sketch and label, on T-x coordinates, the temperature distributions: Initial, steady state, and two intermediate times.
b. Sketch the heat flux at the outer surface q'' (L,t)as a function of time. x
Plate
Insulation
~
x
T,h x=L
Q3
2A batch of 1,200 kg of tomato serum, initially at 20 0C, is heated in an agitated jacketed vessel. The vessel has a heating area of 4.7 m2 and uses steam at a pressure of 140 kPa. The temperature of the serum, which has a mean heat capacity of 3,300 J/kg/K, increased with time as follows:
a. Which equations describe the above heat transfer problem and why?
b. Determine the overall heat transfer coefficient.
Q4
3A long steel rod 0.305 m in diameter is initially at a temperature of 588 K. It is immersed in an oil bath at 311 K. The surface convective coefficient is 125 W/m2 K. The average physical properties of the steel are k=38 W/m.K and the heat diffusivity is 0.0381 m2/h.
2 After P10.1 of Smith(2003)
3 After P5.3-7 of Geankoplis (2007)
2
Time/min
Temperature/0C
20
55.1
30
67.0
40
76.1
50
83.5

The Hague University of Applied Sciences
Academy for Technology, Innovation and Society The Hague, TISH Process and Food Technology, PFT.
Course: Unsteady State Processes PTS42 resit exam assignment 2011-05 Lecturer: Ben C. Bonekamp
a. State the heat diffusion equation with the appropriate boundary conditions for this case.
b. Calculate the temperature at the center of the rod after 1h.
Q5
4A flat brick wall 1.0 ft thick is the lining on one side of a furnace. The wall is initially at a uniform temperature of 100 0F, and then one side is suddenly exposed to a gas at 1100 0F. The rear side of the wall is insulated. The convection coefficient is 2.6 btu/h/ft2. The heat conductivity of the brick is 0.65 btu/h.ft.0F. The thermal diffusivity is 0.0381 m2/h.
a. State the heat diffusion equation including the appropriate boundary conditions for this case.
b. Sketch accurately the temperature profile in the brick wall as a function of position: initially, after a short time, a longer time and after a very long time.
c. Which solution to the heat diffusion equation is appropriate for short times? When is this solution not valid anymore?
d. Estimate the temperature at a distance of 5 cm from the heated side after 1 min.
e. Estimate the time for the furnace wall at a point 0.5 ft from the heated surface to
reach 500 0F.
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