Simple Equations Simplify Convection Cooling

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Power electronic system designers know about natural convection, which starts with the simple statement: Hot air rises. For example, a power semiconductor mounted on a metal plate is exposed to the natural flow of air, which transfers heat from the power device to the surroundings. We also know intuitively that using larger metal plates helps this process, which lowers the temperature.

Unfortunately, when it comes to quantitatively predicting the temperature for a given application, many engineers stumble. It often boils down to gut-feeling, trial and error, or seasoned judgement (the latter usually based on previous trials and error). But thermal management isn't a black art. The rules are pretty simple. A major problem wasn't a lack of available rules, but that the rules were cast in so many diverse forms that engineers didn't know how each compared or which to choose.

Our purpose is not to declare the “best” rule, but to show the different forms in which each rule can be written. Furthermore, if they're all converted into the same format, they become amenable to a simple point comparison. And then we will see how amazingly close these different equations are, with respect to temperature predictions.

The place to start with our temperature measurements is with a metal plate. We call the area of one side of a plate exposed to cooling from both sides as A, and the total exposed area A (obviously, A = 2A).

The simplest case is a square plate made from a very good thermally conducting material, dissipating P watts. After some time, we'll find that the plate stabilizes at a certain temperature rise of ΔT over the ambient.

We expect the temperature rise to be proportional to the dissipation. The proportionality constant is called the thermal resistance, R_th, in °C/W, so:

R_th = ΔT/P (1)

Similarly, we expect that the thermal resistance will vary inversely with the area:

R_th α 1/A (2)

The inverse of the above proportionality constant, h, is in W/°C per unit area, and is called by various names, like convection coefficient or heat transfer coefficient.

R_th=1/(h × A) (3)

Finally:

P = h × A × ΔT = 2 × h × A × ΔT = ΔT/R_th (4)

These are the classical definitions. We kept using the word “expect” above because, historically, these relations were presumed to be true. Later realized, the measured temperature rise wasn't proportional to dissipation, nor inversely proportional to area. However, the above classical equations were still maintained for the sake of consistency. What changed is that no longer were h or R_th considered constants. They were allowed to depend on area and on dissipation — the intention being to indirectly factor in the observed deviations from the expected classical relationships.

Available Equations

As a first approximation, h is often stated at sea level as:

h = 0.006W/in.²/°C (5)

If area is expressed in meters this becomes:

h = 0.006 × 39.37 × 39.37 (since 39.37 in. are in a meter).

i.e., h = 9.3W/m²/°C (6)

As mentioned, h is really a function of almost all the parameters it was imagined to be independent of. Under typical operating conditions, h can therefore vary about 1:4 times from the constant value stated above.

So, in literature we can find the following generalized empirical equation for h, and this becomes our design-by Equation No. 1:

h = 0.00221 × (ΔT/L)^0.25 W/in.²/°C (7)

Where:

L = Length along the direction of natural convection (vertical)

In the case of our simple square plate, L = A^0.5, so we can write this as:

h = 0.00221 × T^0.25 × A^-0.125 W/in.²/°C (8)

Also observe that the above equation uses A which is actually half the area exposed to cooling. So, we can equivalently rewrite it in terms of the actual area involved in the cooling process:

h = 0.00221 × ΔT^0.25 × (A/2)^-0.125 W/in.²/°C (9)

Or,

h = 0.00241 × ΔT^0.25 × A^-0.125 W/in.²/°C (10)

These are all available and published forms of the same equation for h.

Note: The above equation predicts that h has a specified dependency on the exposed area of the plate and also on its temperature differential with respect to ambient. This dependency (i.e., A^-0.125) implies that the cooling efficiency per unit area (i.e., h) of large plates is worse than that of small plates. However, if this sounds surprising, we note that the overall/total cooling efficiency of a plate is h × A, which depends on A^+0.875. So, thermal resistance goes as 1/A^+0.875, and is clearly lower for a large plate than for a small plate as we would expect. Compare this to the ideal 1/A variation expected for thermal resistance.

In literature, we often find the following design-by formula (area in square inches), hereafter referred to as our design-by Equation No. 2:

R_th = 80 × P ^-0.15 × A ^-0.70 (11)

Where A is in square inches

a) We can rewrite our design-by Equation No. 1 in terms of dissipation instead of temperature rise:

h = 0.00221[P/(h × A × 2)]^0.25 × A^-0.125 (12)

So:

h = 0.00654 × P^0.2 × A^-0.3 W/in.²/°C (13)

b) Or in terms of the total exposed area:

h = 0.008 × P^0.2 × A^-0.3 W/in.²/°C (14)

c) We can also now try to see what this will look like in MKS (SI) units. The conversion is not obvious, and so we proceed as follows:

Take an imaginary plate of size 39.37 sq in. × 39.37 sq in. This is a 1 sq m plate. Clearly, the thermal resistance of the plate is in °C/W and is therefore independent of the units used to measure area, and must be unchanged. This means that 1/h × A is independent of units, and so is h × A. Therefore, if in MKS units we assume a similar form:

h = x × ΔT^0.25 × A^-0.125 W/m²/°C (15)

So:

h × A = x × ΔT^0.25 × A_{sq m}^-0.125 × A_{sq m} (16)

Also:

h × A = 0.00221 × ΔT^0.25 × A_{sq in.}^-0.125 × A_{sq in.} (17)

Or:

A _{sq m}^0.875 = 0.00221 × A _{sq in.}^0.875 (18)

x = (39.37 × 39.37/1)^0.875 × 0.00221 (19)

Finally: x = 1.37

So in MKS units:

h = 1.37 × ΔT^0.25 × A^-0.125 W/m²/°C (20)

d) Or, in terms of the total exposed area:

h = 1.49 × ΔT^0.25 × A^-0.125 W/m²/°C (21)

e) We can again express h in terms of P instead of temperature:

h = 1.12 × P ^0.2 × A ^-0.3 W/m²/°C (22)

f) Or, in terms of the total exposed area:

h = 1.38 × P ^0.2 × A ^-0.3 W/m²/°C (23)

g) We can also recast the first design-by in terms of thermal resistance instead of h. We get several different forms:

R_th = 1/(h × 2 × A)=76.5 × P ^-0.2 × A ^0.70 (A in sq. in.) (24)

h) Or in terms of the total exposed area:

R_th = 1/(h × A) = 124.3 × P ^-0.2 × A ^0.70 (A in sq.in.) (25)

i) In MKS units

R_th=1/(h × 2 × A) = 0.45 × P ^-0.2 × A ^-0.70 (A in sq. m) (26)

j) Or, in terms of the total exposed area:

R_th = 1/(h × A) = 0.72 × P ^0.2 × A ^0.70 (A in sq m) (27)

Now we can compare this to our second design-by equation

That is:

h = 80 × P ^-0.15 × A ^0.70 (A in sq in.) (28)

And, the above

R_th=76.5 × P ^0.2 × A ^0.70 (A in sq in.) (29)

And, we see that the two equations, one initially expressed in terms of h and the other in terms of R_th aren't that different at all.

Thermodynamics Theory

We define the following parameters:

Nusselt number (Nu): convection heat transfer/conduction heat transfer

Grashof number (Gr): buoyant flow/viscous flow

Under natural convection (laminar flow) we have the following defining equations (MKS units):

Nu = 3.5 + 0.5 × Gr^¼ (30)

Where:

Where

g = 9.8 (acceleration due to gravity in m/s²)

n = 15.9 × 10^-6 (kinematic viscosity in m²/s)

At T_amb = 40°C, this simplifies to:

Nu = 3.5 + 52.7 × ΔT^0.25 × L^0.75 (32)

The coefficient of cooling is defined as:

h = Nu × k_air/L

Where k_air is the thermal conductivity of air:

k_air = 0.026 in W/meter/°C, so:

h = 0.091 + 1.371 × ΔT^0.25 × A^-0.125 W/m²/°C (34)

This is called our design-by Equation No. 3.

Compare this by the comparable form c) of design-by Equation No. 1 in the earlier section

h = 1.37 ΔT^0.25 × A^-0.125 W/m²/°C (35)

We see they are surprisingly close.

Generalizing Standard Equations

Standard design-by Equation No. 1 is written in a generalized form:

h = b × ΔT^c / A^d (A in sq in.) (36)

In MKS units:

h = B × ΔT^C / A^D (A in sq in.) (37)

Where the conversion between the system of units is:

B = b × (10³ / 25.4)^(2-2d)

C = c

D = d

Considering the complete area involved in the convection process in either system of units, we get:

h = b × ΔT^c / A ^d (A in sq m)

h = B × ΔT^C / A ^D (A in sq m)

Where:

b = b × 2 ^d

c = c

d = d

And:

B = B × 2 ^d = b × (10³ / 25.4)^(2-2d) / 2^d

C = c

D = d

To tabulate these we define four cases:

Case 1: Area used in equation is half of exposed Area/sq in.

Case 2: Area used in equation is full exposed Area/sq in.

Case 3: Area used in equation is half of exposed Area/sq m.

Case 4: Area used in equation is full exposed Area /sq m.

And, the general equation expressing ‘h’ is now written as: h= α × ΔT^β / Area^γ

Example: Start with our first design-by equation and plug in the coefficients under α1, β1 and γ1 (remembering that β1 = β2 = β3 = β, and γ1 = γ2 = γ3 = γ), and then evaluate by means of the conversion equations indicated within the tables, all the corresponding forms. Tables 1 to 4 show 16 possible ways of writing any design-by equation like design-by No. 1.

Our first design-by equation starting point was in terms of h, so we could derive R_th from this. However, we may be interested in starting with design-by No. 2 and computing h from this. What we need here is a generalized way of back-calculating our coefficients. In Tables 5 to 8 we provide the equations for this process. We also compare design-by No. 1 coefficients with those of design-by No. 2 for each format.

Application Note AN-1229, titled “SIMPLE SWITCHER PCB Layout Guidelines,” provides a simple equation for estimating the copper area on a p. c. board:

A = 985 × R_th^-1.43 × P^-0.28 (sq in.)

Since this is not a plate but a copper island on a p. c. board, only one side is exposed to cooling. We have therefore just used Case 3 in Table 8 applied to design-by equation No. 1 and solved for A.

We now have at least 16 different ways of presenting each of the basic two standard design-by equations. There are each eight ways of expressing the coefficient h and eight ways in terms of the thermal resistance. Furthermore, if we express h in terms of L (one side of the square plate in question), we have still eight additional ways of expressing h for each of the two standard design-by equations, bringing the total to 24 ways to express each equation. Each of these ways is exactly equivalent, although at first sight they look very different. Also, relative to each other, the two standard design-by equations are very close.

We can see that the standard Equation No. 2 is always more conservative than standard Equation No. 1, i.e. it calls for slightly larger areas. We'll also see that standard Equation No. 3 is almost coincident with standard Equation No. 1, though it predicts slightly lower temperatures.

But unlike No. 3, design-by Equation No. 1 is easier to manipulate mathematically in closed form to arrive at thermal resistance, etc., due to the absence of a constant term in the expression for h. Thus, our first design-by equation seems to be the best choice for further estimations/manipulations. But whatever the final chosen base equation may be, we know how to manipulate it and see the different forms it takes. We shouldn't be surprised to see seemingly different equations anymore, realizing that they may be very close, if not the same.