Newton's law of cooling: The rate of heat loss from a warm body cooled by a fluid stream is proportional to the difference in temperature between the surface of the body and the fluid far away from the body.

The object of this experiment is to determine the average heat transfer coefficient at the surface of a solid sphere from measurements of the temperature history in a transient thermal test.

The rate of heat loss from a surface to a fluid is controlled by diffusion processes in the fluid, modified by fluid motion. The rate should scale directly with overall temperature difference if the fluid motion and thermal properties are temperature independent and provided that the thermal boundary conditions remain similar as the temperature difference changes.

In this experiment, a warm solid body is placed in a constant-velocity air stream and allowed to cool. The body is made of a metal having a large thermal conductivity so that internal temperature gradients are small and a single temperature T can be used to characterize the solid. The upstream fluid temperature T_t is constant. A heat balance on the solid, neglecting radiation losses, gives the relation

where q is the rate of heat loss from the body which has surface area A and volume V, ρ is the density and c_p the specific heat of the body, t is time, and h is the convective heat transfer coefficient. If the properties of the solid body are known, and if the fluid temperature and solid temperature histories are measured, then the instantaneous h can be determined. According to Newton's law of cooling, h should not change with time.

The assumption of negligible internal temperature gradients is valid provided that the Biot number hD / k_s « 1, where D is a characteristic length of the solid and k_s is the solid conductivity. This can be converted into a rough estimate for the upper limit of Reynolds number Re permissible for this experiment as

where k_f is the fluid conductivity and where an approximate relation between the Nusselt number Nu = hD / k_f and the Reynolds number for a sphere was used. For copper and air this inequality becomes Re« 8.7 · 10⁸.

Natural convection and radiation will become important at low Reynolds number. Then the effective heat transfer coefficient determined from Eq (10.1) will no longer be independent of temperature. The lower limit on Reynolds number for Newtonian cooling depends on the size of the body; the smaller the body the lower the limit.

Air jet

Solid metal spheres as provided (may be copper, brass, aluminum)

Thermocouple wire

Thermocouple indicator

Pitot tube

Manometer

Clock with second hand

Propane torch

Drill a shallow hole in each solid. Insert and solder a thermocouple in each hole.

The air jet is provided by a squirrel-cage fan blowing through a short section of duct as shown in Fig. 10.1. If the duct is short, a packet of soda straws placed in the end of the air duct will help to form a uniform-velocity profile at the exit. Use a 4-in. or larger jet with a 1-in. sphere. Air velocity can be measured using the Pitot tube attached to the apparatus. Taking p_s to be the measured static pressure, p_t to be the measured total (dynamic plus static) pressure from the flow facing Pitot tube and ρ the air density, the velocity v is computed:

Set a constant jet velocity. Measure and record the velocity.

Measure the air temperature.

Heat one of the solids to approximately 150.C with a torch or in a lab oven.

Suspend the solid, by the thermocouple wire, in the air jet.

Log the temperature history between 150.C and 50 .C.

Plot the measured temperature and the calculated heat transfer coefficient versus time. Data reduction is less tedious and the accurary of the determination of an instantaneous h can be improved if an A/D data acquisition system is used. Then the derivative in Eq (10.1) can be approximated with finite differences of closely spaced data as in fig. 10.2.

Repeat the procedure at different velocities and with different solid bodies

Considerations

1. Does the measured heat transfer coefficient vary with body temperature at a constant air velocity? Compare natural convection (zero jet velocity) and forced convection.

2. How do the results for the sphere compare with correlations available in the literature?

3. Does the presence of the thermocouple influence the rate of heat loss? Compare the results for the sphere obtained with the thermocouple normal to the air stream (horizontal jet) with those obtained with the thermocouple downstream from the sphere (vertical jet). See Fig. lO.1

4. How does h vary with orientation for the asymmetric solids?

5. Can the results for all of the solids be described by a single correlation of Nusselt number with Reynolds number?

6. What is the effect on the heat transfer coefficient if the uniform velocity is disturbed by placing another, unheated body just upstream from from the solid in question? What if the body upstream is also heated?

Fig. 10.1

Fig. 10.2. Data from a one-inch sphere in 20.C air.