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Thermal Resistance

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Details

For the special case of one-dimensional heat transfer with no internal energy generation and with constant properties, a very important concept is suggest by Eq4 from the Temperature Distribution Through a Plane Wall lesson. In particular, there exists an analogy between the diffusion of heat and electrical charge. Just as an electrical resistance is associated with the conduction of electricity, a thermal resistance may be associated with the conduction of heat. Defining resistance as the ratio of a driving potential to the corresponding transfer rate, it follows from Eq4 from the Temperature Distribution Through a Plane Wall lesson that the thermal resistance for conduction in a plane wall is:

(Eq1)

Rt,cond  ≡   (Ts,1 − Ts,2) qx   =   L kA

Similarly, for electrical conduction in the same system, Ohm's law provides an electrical resistance of the form:

(Eq2)

Re  ≡   (Es,1 − Es,2) I   =   L σA

The analogy between Eq1 and Eq2 can be seen. A thermal resistance may also be associated with heat transfer by convection at a surface. From Newton's law of cooling:

(Eq3)

q = hA(Ts − T∞)

The thermal resistance for convection is then:

(Eq4)

Rt,conv  ≡   Ts − T∞ q   =   1 hA

Circuit representations provide a useful tool for both conceptualizing and quantifying heat transfer problems. The equivalent thermal circuit for the plane wall with convection surface conditions is shown:



The heat transfer rate may be determined from separate consideration of each element in the network. Since q_x is constant throughout the network, it follows that:

(Eq5)

qx  =   T∞,1 − Ts,1 1/h1A   =   Ts,1 − Ts,2 L/kA   =   Ts,2 − T∞,2 1/h2A

In terms of the overall temperature difference, T_∞,1 − T_∞,2, and the total thermal resistance, R_tot, the heat transfer rate may also be expressed as:

(Eq6)

qx  =   T∞,1 − T∞,2 Rtot

Because the conduction and convection resistances are in series and may be summed, it follows that:

(Eq7)

Rtot  =   1 h1A  +  1 kA  +  1 h2A

Radiation exchange between the surface and surroundings may also be important if the convection heat transfer coefficient is small (as it often is for natural convection in a gas). A thermal resistance for radiation may be defined by reference to Eq5 of the Radiation lesson:

(Eq8)

Rt,rad  ≡   Ts − T∞ qrad   =   1 hrA

For radiation between a surface and large surroundings, h_r is determined from Eq6 of the Radiation lesson. Surface radiation and convection resistances act in parallel, and if T_∞ = T_sur, they may be combined to obtain a single, effective surface resistance.