Dimensions of Units

Now absolute temperature is defined in terms of the laws of ideal gases for which the characteristic equation is pv=K0 where v is the molecular volume and K is a universal constant. By suitably altering the size of a degree the factor K might be made unity and then it would be clear that 0 would represent energy for pv has the dimensions of energy. Mechanical theory identifies it with the kinetic energy of the molecular motion.

Viscosity and Temperature.----Let

the problem be to examine how viscosity will depend upon molecular motion when molecules repel each other with a force varying inversely as the pth power of their distance apart. Let force X L where R is the force at unit distance; let the data for the molecules be m = mass, v = mean velocity of agitation, N = number per unit volume; and let it be supposed that the viscosity pc varies as mxvYNzRn. In the usual way it is found that to the critical temperature of the gas. The index I/2 indicates as before that the molecules are behaving as rigid spheres.

Thermal Conduction.

Further examples are to be found in connection with thermal conduction. Heat itself can be expressed either as energy or, in a secondary way, as the number of degrees change of temperature it can produce in a given mass of water.

dO Conductivity, k, is defined by the equation Heat =lea X time.

To free the equation from dimensional difficulties it would be better to write KO instead of 0 whereK is the gas constant, and define k/K as the thermal con ductivity; because as we have seen KO has the dimensions of energy. On the other hand if Heat is measured as (mass of water X degrees rise of temper ature) the temperature enters on both sides of the equation and, as far as dimensions go, it can be cancelled.

The chief problems involving dimensions concern the transfer of heat through pipe walls be tween the fluids on the inside and outside. The actual transfer in the case of metal pipes is con trolled mainly by the presence of a stagnant layer of fluid on each side. In the body of each fluid it is transferred from point to point (except for the case of very slow motion) mainly by turbulent convection, but in the stagnant layers it is by con duction through media which usually are bad conductors. In de scribing his work on steam condensers Joule adopted a new term at the suggestion of Lord Kelvin (1859)—the coefficient of heat trans mission, C—which he defined as the amount of heat flowing per unit area of the pipe surface in unit time per unit difference of temperature between the fluids, i.e., H = CA — 01)X time where H is positive if If is the temperature of the metal pipe (it is practically constant across the section) Fig. 2 we can write The coefficient of transmission is this quantity mul_tiplied by In usual practice n can be taken as r • 7 c hence The value of is determined in a similar way and thence the over-all coefficient is obtained except for numerical constants.

To find the constants recourse must be made to a few experi ments with various values of the variables. To show the impor tance of a knowledge of C in technical practice the following figures, calculated from the formula are given. They are based partly upon the thermal measurements of Stanton at the National Physical Laboratory but mainly upon those by H. Greenwood made for the Ministry of Munitions in 1918. The values are the over-all coefficients (C) in British Cent. Units per sq.ft. per deg. C per hour; vi and are the velocities in feet/min. inside and outside a pipe of r inch diameter at C. carrying water on both sides.

For a six

inch pipe the values range from 97 to 352. Reference should be made to No. 9 of the Technical Reports published for the Ministry of Munitions (London, H.M.S. Stationery Office) for further information and for charts enabling the coefficient to be calculated easily for various temperatures, diameters, vis cosities and for gases as well as liquids.

To prevent unjustifiable use of the formula and table it must be added that the values are for turbulent flow in clean pipes. A layer of incrustation only .02 inch thick may reduce the values to one half. In many actual interchangers where the velocity is very small and the motion only fortuitously turbulent (if at all) the value falls as low as 2 and is rarely more than 15.

The idea that the passage of heat from solids to liquids moving past them is governed by the same principles as apply, in virtue of viscosity, to the passage of momentum, was originated by Osborne Reynolds though with some misgivings and has been further developed by others. Lord Rayleigh has shown, how ever, that the analogy is not complete owing to the existence of a pressure-gradient term in the momentum problem which has no counterpart in the thermal problem. (Advisory Comm. for Aeronautics, T. 941, 1917.) In the same place, considering the flow of heat between two planes, distance D apart, with fluid flowing in the space between the planes he derives the equation KO [Dvp CA for the heat flow H = 12 Where C is the heat per unit volume. For a given fluid the last term is a con stant and may be omitted and dynamic similarity is attained when vD is constant so that a complete determination of the function can be made by varying either v or D. It is best to vary v not D; variation of the latter would require the roughness of the surfaces to vary in the same proportion.