How It Works

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General

The cooling system on the 530i is little different from engines of the last 70 years, and very consistent with most cars of the last 20+ years.  A belt-driven impeller-type pump circulates water rapidly through the engine and radiator.  A simple wax-driven thermostatic valve restricts flow through the radiator, effectively regulating engine temperature.
The thermostat is mounted on the outlet of the engine and sees the highest temperature in the coolant circuit. There is a also a "bypass" flow path, shorting out the thermostat/radiator components and allowing reduced coolant circulation through the engine and pump while the thermostat is closed. This allows even temperature distribution throughout the engine block and heads while warming up, in addition to making the passenger compartment heater useful.

Thermostat

The 530i thermostat is a common design, stopping coolant flow completely when below the "cracking" temperature, then opening somewhat linearly over an additional 10 degrees F (approx.)  The thermostat regulates engine temperature (measured at engine coolant outlet) at some point within that range. The purpose of having the valve travel over a temperature "range" instead of snapping open at a precise value is to encourage stable operation, aside from the fact that it allows a practical thermostat design (using a sealed volume of special wax.) Control system engineers will understand that as a fairly low system "gain," the downside being less-precise control.

There may also be a small amount of mechanical and thermal hysteresis which will cause a slight temperature dither during otherwise steady-state operation. Also, temperature distribution over the entire engine will vary due to a number of parameters including engine (pump) speed and heat output, to say nothing of the path taken through the block and heads.

Engine-Driven Fan

To promote effective radiator operation, there is both a mechanical and an electric fan.
The mechanical fan is a tradition on front engine/rear-wheel drive cars, improved the last two decades with a thermally-controlled clutch (to reduce power loss) combined with higher pulley gearing (to allow faster operation at engine idle.)
I'm accustomed to fan clutches that simply slip, applying more-or-less a constant torque to the fan resulting in a fairly constant speed and therefore airflow.  The BMW's fan clutch seems to be designed to additionally lock-up when the air flowing over its fins exceeds a certain temperature.  This is evident only at idle, allowing the fan to run at the fairly-high direct pulley speed.  When you accelerate from a stop it stays locked until about 3,000 rpm has been reached and then continues to run with normal slippage.  The fan makes a lot of noise for the few moments is it locked-up.
As effective as this system might be, the noise and loss of power incurred while accelerating is annoying and unacceptable for a car of this price, in my opinion.

Electric Fan

The thermally-switched electric fan has two speeds, the lower provided by a series resistor to the fan motor. In my experience, the thermal setpoints are high and not typically encountered during normal operation of the 530i even in the hottest U.S. climates.  However, the fan is provided as a last resort to prevent overheating and may be useful as the car gets older and the radiator less-effective.

The A/C system does use the electric fan to ensure air is flowing through the condenser (in front of the radiator.)  The electric fan is set to low speed whenever the A/C is on, and cycles to high speed when the A/C compressor clutch is actually engaged.

Keep in mind that the thermostat is solely responsible for engine temperature regulation. The mechanical and electric fans, although "temperature controlled," are on/off devices and simply shift the effectiveness of the radiator to higher levels to prevent the thermostat from running out of range. These "step functions" are generally momentarily disruptive to the thermostat's ability to maintain a stable temperature. However, in the event the thermostat does run out of range, overheating to some extent is unavoidable.

Temperature Gauge

The BMW temperature gauge has one unusual feature - it reads dead center during normal operation, winter or summer. Normally, you should see a little variation (due to the low system gain with no integral term) mentioned before. My theory is that BMW have tailored the response of the gauge to show exactly normal temperature whenever actual temperature is acceptable, avoiding unnecessary customer inquiries. Some BMW owners might panic if their expensive cars show a small variance in temperature.  This "feature" makes the gauge more-or-less useless as there is already a warning function in the Check Control system.

Practical Tips

Parts Source:  I recommend using exclusively factory OEM parts and antifreeze.
Hoses:  Replace all of them at seven year intervals.   Clean the connections carefully to remove corrosion and sharp edges.  Install "dry," but using soap under the hose clamp to allow the rubber to flow evenly while tightening.  Tighten very gently, especially at plastic fittings.   Check and tighten further if leaks are detected.
Water Pump:  The pump seal is usually the first thing to fail and will initially spray a small amount of coolant on the surrounding parts.
Thermostat:  Replace at 100,000 miles or 10 years at the latest.   My 530i has a 95�C version, not sure if a cooler one could be (or should be) fitted.
Coolant Level Sensor:  This is a well-know failure item, giving a false indication of low coolant.  Check the E34 archives on www.bimmer.org for solutions.
Overheating:  If you are driving and an overheating condition appears, pull over and turn off the engine as quickly as you can.  If you have some water handy, pour it slowly over the radiator fins and do not remove the reservoir cap until the engine has cooled somewhat.  There is no value in leaving the engine running as the cooling system is not likely to magically fix itself.  Turn it off and limit your losses.