# Analogy with the Theory of Electricity

Equation (4.154) gives conduction for the one-dimensional case with constant thermal conductivity:

<t> = ^0, (4.162)

 3 HEAT AND MASS TRANSFER Where S is the distance corresponding to the temperature difference. For the three heat transfer forms,

• Conduction, Eq. (4.162)

• Convection, Eq. (4.156)

We have, respectively,

0 = ^0 (4.163)

0 = (4.164)

AkA

0 = —^<t>. (4.165)

Following from Ohm’s electrical law (theory of electricity), a heat resis­tance can be defined:

Potential difference = resistance • current temperature difference = heat resistance ■ heat flow

O R ■ ‘i>. (4.166)

The conductance or the coefficient of heat transfer U = I / R, or

<t>=U0. (4.167)

For conduction the heat resistance is the distance divided by the heat con­ductivity, R = 8/XA, and the heat conductance is heat conductivity divided by distance, U = A A/S. For convection and radiation the heat resistance is 1 di­vided by the heat transfer factor, 1/aA, and the heat conductance is the same

As the heat transfer factor, U = a A. A coefficient of heat flow is also used, the

K value, which is the total conductance:

 (4.168) A W ‘ m2K

The following connecting rules are based on the above analogy: heat resistance R in series connection

And in parallel connection

1 = J_ + J_

R R, R,

Heat conductance series connection

1 = J_ +J_

U U1 u2

And in parallel connection

U = U, + Uz An A A23A A34 A 04 sA | FIGURE 4.25 Heat transfer through a wall.

The heat resistance between the fluids on the two sides of the pipe wall in Fig. 4.25 is

R.^ = :d=l=J_ + J_ + ^r (4.169)

U K a1 q:2 z—’ A

Where a — aconv + aracj, and 8 and A are the thickness and heat conductivity of

Consecutive layers.

The resistance between the fluids on the inside and outside of the pipe is obtained by integrating with respect to the radius (Fig. 4.26):

= n = (4’170!

The sum includes concentric cylinder layers, such as the layer between the outer and inner diameters of the pipe or a possible thermal insulation layer. For each layer the corresponding heat conductivity A; is used. The outer heat transfer fac­tor is the sum of the proportions of convection and radiation. (Note: Very thin pipes or wires should not be insulated. Because the outer diameter of the insula­tion is smaller than A/au, the resistance is less than that without the insulation.) The resistance between fluids separated by two coaxial spherical surfaces is

R = — i~ + -?- +———————— 7^——- ‘ (4.171)

Irduau trd~as l J j_________ j_ The heat flow density q of a material depends on the local temperature gradi­ent, according to Fourier’s law:

* = 14.172)

In simple one-dimensional cases, it is easy to determine the temperature gradi­ent and calculate the heat flow from Fourier’s law.

The general case is that of steady-state flow, and the thermal conductivity factor is a function of the temperature. In the unsteady state the temperature of the system changes with time, and energy is stored in the system or released from the system reduced. The storage capacity is

DU ^ 3T — di:=® = mCPTt

= pcpV^. (4.173)