I
ERICH A. FARBER
*
University of Florida, Gainesville, Fla.
HARRY 0. RENNAT University of Wisconsin, Madison, Wis.
Variation of Heat Transfer Coefficient with Length Inclined Tubes in Still A i r A reliable new methog accurately measures surface temperatures and other heat transfer characteristics of cylinder
THE
necessity for the work reported was demonstrated when needed information could not be found in the literature a t the time the cooling system of the atomic submarine was designed. Some time later and independently the Utilities Research Commission, Chicago, Ill., set u p a research project a t the University of Wisconsin to determine temperature distribution along electrical conductors with air approaching from different angles and with different velocities. Only forced convection was treated, and the thermocouples were fastened to the conductors. However, surface temperatures measured by attached thermocouples or any method that disturbs the surface are considerably in error (7, 8). The authors feel that they have developed a reliable method by which it is possible to measure surface temperatures accurately. Statement of Problem
Whenever a surface is surrounded by a cooler fluid-liquid or gas-heat is transferred from the surface to the fluid by convection and radiation. As under ordinary conditions a few molecular layers of stationary fluid cling to the surface, conduction will have to take place through these layers. This conduction is, however, included in the convection relationships. In calculating heat transfer under steady-state conditions from the surface, the following equation is used. p = hA(T,
where q
h
-
T,)
(1)
= heat transferred, B.t.u./hr. = film coefficient, (h, h?),
h, = h, =
T, = T, =
+
B.t.u./hr., sq. ft.; F. convection film coefficient radiation coefficient surface temperature, O F. fiuid bulk temperature, F.
I n determining the film coefficient, h, including both convection (with inherent conduction) and radiation, it is necessary to measure all other quantities in Equation 1 accurately. This requirement determined the course of experimentation taken. If the heat, q, is generated electrically, it can be measured in terms of current
qnd voltage drop with the accuracy and reproducibility desired. The area, A, can be measured with ordinary mechanical gages, if the surface used is held to close tolerances. The fluid bulk temperature, T,, can be measured with thermometers and thermocouples. This method seemed to be accurate and satisfactory as long as no cold or warm air from outside infiltrated the test room. All other surfaces on the room had to be a t ambient temperature. T h e real problem, however, was the accurate determination of the surface T,. Many different temperature, methods had been tried by the authors, until finally the method described below was adopted. Measurement of Surface Temperature
Heat was generated electrically through the resistance of thin-walled metal tubes. Sliding thermocouples (which did not touch the walls) were placed inside the small tubes to record temperature. If the thermocouple touched the surface, it disturbed the surface and the voltage drop across the contact area generated heat within the thermocouple junction, giving a n erroneous reading. Both alternating and direct current were used without any noticeable difference. The thermocouples in the tube read the temperature of the air inside the tube or, in case of a vacuum, the equilibrium temperature with the inside wall surface. Under steady-state conditions, there is no heat exchange between the thermocouple and the inside of the tube, no temperature gradient exists, and the measured temperature is actually the tempeiature of the inside surface of the tube a t this section. In tubes of large diameter it was necessary to fill the tube with some electrically nonconducting material, such as glass wool, to prevent air currents within the tube, between points around the tube, and along the tube. In tubes of small diameter no difference could be observed, whether the tube was filled with air, glass wool, or under vacuum with the thermocouple junction a t
different points in a particular crosssectional plane. This can be explained by the high thermal conductivity of the metal tube and the short distances involved. The results agree well with experimental (optical) surface temperatures obtained for pipes around the circumference (5). If air is in the tube (small diameter), its viscosity overcomes all buoyant forces, thus eliminating circulation. The measurement of the inside surface temperature of the tube has now to be related to the outside surface temperature, the difference being the temperature drop through the tube wall. This temperature drop can be calculated as follows (Figure 1).
where
ql
= hear entering diff‘erential
cylindrical shell, B.t.u.j hr. y2 = heat leaving differential cylindrical shell, B.t.u./ hr . = heat generated within difq, ferential cylindrical shell. B.t.u./hr. k = thermal conductivity, B.t.u./hr., ft., O F. L = length of cylindrical shell, feet r = radius of differential cylindrical shell, feet q”’ = heat generated within unit volume, B.t u./hr., cu. feet ~i = inside tube radius, feet ro = outside tube radius, feet T , = inside surface temperature, ’ F. To = outside surface temperature. ’ F.
Figure 1. VOL. 49, NO. 3
Tube section MARCH 1957
437
THERMOCOUPLE LEA DS5
For the problem a t hand, in steady state no heat is transferred to the space inside the tube; thus d7' ~- = 0 at r =
I
Ti
dr
and
(3)
T
= Ti
at
7
= 7i
I l l
letting
k
=
C, which has been shoicn
by the authors to be a very good assumption, as originally a linear relationship \cas assumed and the solution found by numerical analysis. The deviation from the above assumption appeared in thr fifth place for 1
