Transient Lumped Thermal Capacitance Conduction (Low Biot Number)
Background:
Transient conduction refers to the case where conduction in a medium results in the temperature varying with time. Consider a body of primary dimension L with thermal conductivity k that is exposed to a fluid with a convective heat transfer coefficient h. If the Biot number (h L/k) is smaller than about 0.1, the temperature within the body varies with time but at any instant the body temperature is nearly uniform and the body is considered to have a 'lumped thermal capacitance'. Generally this applies to cases where the thermal conductivity of the material is large and/or the convective heat transfer coefficient between the body and the fluid is small, but as indicated by the Biot number, the size of the body (L) is equally important. In the case of small Biot number and where the heat transfer coefficient 'h' is constant, the body's temperature variation with time is described by:
where T is temperature, t is time, h is convective heat transfer coefficient, r is material density, c is body specific heat, A is body surface area, V is body volume, and subscripts ¥ and i refer to the fluid and initial body temperatures, respectively.
Objectives:
To demonstrate the transient lumped thermal capacitance process and use it to determine the convective heat transfer coefficient for the process.
Materials and Equipment:
Two high thermal conductivity rod (copper and aluminum) of about 1/2 inch (1 cm) diameter and 15 to 20 cm (6 to 8 in.) long with hole in one end for dial thermometer.
Dial thermometer
Supports to hold the rods horizontally
Stove or hot bath to heat rods initially.
Experiment:
First measure the room temperature with the thermometer. Place the copper rod in oven or hot bath and heat to about 100 to 300 oC. Using gloves, remove rod and hang or support horizontally (two strings or support ends on two cup edges).
Insert thermometer in hole and record the temperature and time as the rod cools from about the initial temperature to within about 20 oC of room temperature. If one uses a water bath, wipe off excess water from cylinder before starting measurements.
Repeat this for the aluminum rod.
Results:
Graph your results in terms of the natural log of the left-hand side of the above equation against time (t) on semi-log paper. The result should be approximately a straight line and the slope of the line should equal -hA/rcV. Using the density and specific heat of the rod material and the surface area and volume of the rod, determine the convective heat transfer coefficient 'h'. Repeat this for the aluminum rod.
After obtaining the 'h' values, compute the Biot numbers (h L/k) to determine if each value is below the required value of approximately 0.1. If so, the process was one of lumped thermal capacitance and the computed value of 'h' will be a reasonably accurate determination of the actual natural convection heat transfer coefficient. (If the Biot number is larger than about 1.0, then the process was definitely not one of lumped thermal capacitance and the computed value of 'h' will be in error, and increasingly so the larger the Biot number.)
Compare the values of 'h' with those obtained using an accepted correlation for natural convection from horizontal cylinders. [Note that this will be requested in a subsequent homework assignment].
Comments:
In this experiment, in addition to natural convection, there will be some radiation heat transfer (particularly for the heated rod case); thus the computed value of 'h' will be somewhat high. However, if the rod is fairly shiny the radiation heat transfer should be relatively small. In any case, one can estimate the radiation contribution and determine the approximate error in the 'h' obtained in this experiment.
The natural convection heat transfer coefficient is dependent on the temperature difference (DT) between the surface and fluid which is changing during the experiment. As a result the curve obtained will probably exhibit some curvature, and likely will be somewhat concave upwards. The result is that the convective heat transfer coefficient will be larger for the initial portion of the test (large DT) than near the end where DT is reduced. The 'h' you obtain by using a best fit curve over the entire test will be a weighted average value.