Heat Transfer: Steady-State Conduction

Abstract

Heat transfer involves the flow of thermal energy between objects due to temperature differences. It is an integral part of food processing and preservation techniques. Heating to make foods easier to digest and safe to eat has been practiced since time immemorial. It also helps extend their shelf life and makes them taste better. To achieve this, food processes require that heat be transferred into and out of the food. Some examples include:

Problems

• 7.1 One side of a 1.5 cm thick slice of lean meat (7.8 wt.% fat, 1.5 wt.% ash, 19.0 wt.% protein, 71.7 wt.% water) contacts a hot pan that is maintained at 80^oC while the other side remains at 25^oC. Assuming steady-state heat transfer, draw the equivalent thermal resistance circuit and determine the rate of heat flux through the meat slice.

• 7.2 At steady state, the heat flow through a 0.6 cm thick aluminum sheet is 111408 W. One side of the sheet is maintained at 22 °C while the other is maintained at 18 °C. Determine (a) the area of the sheet, (b) the temperature profile, and (c) the temperature at 0.2 cm, 0.4 cm, and 0.6 cm from the 22 °C surface of the sheet.

• 7.3 To experimentally measure the thermal conductivity of a new food, it was formed into a 4 cm × 4 cm and 1 cm thick block. The edges of the block were insulated, and heat was supplied continuously to one side of the block at a rate of 0.4 W. At steady state, the temperature of both sides was measured with thermocouples and found to be 30 °C and 25 °C, respectively. Calculate the thermal resistance and thermal conductivity of the food. Given the composition of the food as 75 wt.% moisture, 19 wt.% protein, 2.5 wt.% fat, 2.2 wt.% carbohydrates, and 1.3 wt.% ash by weight, estimate the thermal conductivity of the product using the composition data and compare the reliability of the results.

• 7.4 (a) A formulated product has the following composition on dry weight basis: 70.0 wt.% carbohydrates, 15.0 wt.% fat, 1.0 wt.% protein, and 2.0 wt.% ash. Determine the thermal conductivity of the product if 100 g of wet product contains 60 g of water.

  (b) The above food product is packed into an annular cylinder with the temperatures at the outer and inner walls maintained at constant values. The inner and outer walls are of negligible thickness and their radii are 5 cm and 11 cm, respectively. The temperature of the packed food product is monitored at different radial distances inside the cylinder with the help of thermocouples. At steady state, for a flux of 150 W/m², the temperature at various radial positions was recorded. Estimate the product’s thermal conductivity.

  Radial distance, r (cm)	Temperature, T (°C)
  6.0 7.3 8.6 9.8 10.7	6.8 14.8 19.0 25.0 29.2

• 7.5 Single strength juice flows through a stainless-steel pipe of 50-mm inside diameter and 5 m long with a wall thickness is 2 mm. The outside surface temperature is 80 °C and the inside wall temperature is 85 °C. The thermal conductivity of stainless steel is 16 W/m-°C. Calculate the heat losses assuming steady-state flow.

• 7.6 Show that the rate of heat transfer through a spherical shell (thermal conductivity k) with inner radius r₁ and outer radius r₂ is given by the following expression:

  ${\dot{q}}_r=\frac{4\pi k{r}_1{r}_2\left({T}_1-{T}_2\right)}{r_2-{r}_1}$ , where T₁ and T₂ are the inside and outside temperatures.

• 7.7 What length of cylindrical shell will conduct heat at the same rate as a spherical shell, both having the same internal radius and external radius of 1 m and 2 m, respectively. Given that the cylinder and sphere are made of the same material and the inside temperature (T₁) and outside temperature (T₂) are also the same for both the spherical and cylindrical shells.

• 7.8 A refrigerated room is designed with an outer 70 mm thick wall of cement ((k = 0.29 W/m-K) and an inner wall of maple wood (15 mm thick, k = 0.172 W/mK), with the space in between (120 mm) filled with polyurethane foam (k = 0.025 W/mK). If the inner wall temperature is 4 °C and the outer wall is maintained at the ambient air temperature of 34 °C, estimate the rate of heat loss per unit area.

• 7.9 A rigid plastic cooler is built with high density polyethylene (HDPE) (k = 0.51 W/m-K) as a rigid shell and polyurethane foam (k = 0.025 W/m-K) serving as an insulator while it fills the inside of the rigid HDPE walls. You are trying to use this cooler to keep your beverages at a cool 0 °C while the ambient summer temperature is 38 °C. If the HDPE shell thickness is 30 mm and the polyurethane insulation is 140 mm thick, determine the heat flux through the walls of the rigid plastic cooler. Assume heat may only transfer through the walls of this cooler.

• 7.10 Three heat conducting slabs of materials A, B, and C are joined together in parallel. Each slab is 5 cm thick and has an area of 1.5 m². The thermal conductivities of A, B, and C are 0.29, 0.04, and 0.11 W/m-K, respectively. If the temperature at one surface is maintained at 40 °C, calculate the steady-state temperature at the other surface, given that heat flows through the slab at a rate of 500.7 W.