Here are some sizing formulas to ensure baseboards near the beginning of a series circuit are not oversized and those near the end are not undersized.

Issue: 6/02

The "classic" method of sizing finned-tube baseboard convectors piped in a series circuit is as follows:

A. Assume there will be a 20 degrees F temperature drop across the circuit.

B. Subtract half of this 20 degrees F drop, (10 degrees F) from the boiler outlet temperature to get the average temperature of the water in the circuit.

C. Look up the Btu/hr output per foot of baseboard element in the manufacturer's literature.

D. Divide this heat output per foot into the design heating load of each room to get the necessary length of baseboard.

This method selects the size of each baseboard assuming it operates at a hypothetical average circuit temperature that is always 10 degrees F below the supply temperature. In many systems, this simply isn't the case. Baseboards near the beginning of the circuit operate at inlet water temperatures higher than the average circuit temperature. Those near the end operate at lower inlet temperatures. For circuits operating with small temperature drops (perhaps less than 5 degrees F), the error incurred in sizing to an average circuit temperature is arguably small. However, the greater the circuit's temperature drop, the more pronounced the error becomes. The net effect is that baseboards near the beginning of the series circuit are often oversized, while those near the end are undersized. Most complaints stem from the latter.

Overheating can be partially regulated by closing the damper on the baseboard enclosure. Unfortunately, little can be done to correct for underheating at the end of the circuit that won't create another problem farther upstream.

Proper sizing of series-connected baseboards requires a method that accounts for the fluid temperature at each baseboard. Such a method would allow a designer to consider longer circuits that operate at higher temperature drops, without fear that baseboards near the end of the circuit will have insufficient heat output.

This article presents the analytical model for such calculations, as well as a spreadsheet that can be used to for routine design of series baseboard circuits. ## The Analytical Model

The 2000 ASHRAE Systems and Equipment Handbook lists the following theoretical model for heat output of a finned-tube baseboard convector (Equation 1).

Where:

q = heat output (Btu/hr)

a = a constant determined by testing

Tw(bar) = average water temperature in the baseboard (degrees F)

Tair = room air temperature (degrees F) Figure 1.
As the baseboard element being considered gets shorter, the temperature drop along it becomes less. The inlet and outlet temperatures get progressively closer to the average temperature. Eventually, the average temperature approaches the water temperature at a specific position along the baseboard element.

A differential equation can be written to represent the heat output of an infinitesimal length of baseboard (dL) shown in Figure 1. This equation contains specific constants for the model given in Equation 1 based on curve fitting of published baseboard performance data (Equation 2).

Where:

Tw = fluid temperature at a specific location along the baseboard element (degrees F)

B = published heat output rating of baseboard at 200 degrees F fluid temperature and 1 gpm flow rate (Btu/hr/ft)

L = length of baseboard element (feet)

f = fluid flow rate through baseboard element (gpm)

c = specific heat of fluid at the average system temperature (Btu/lb/degree F)

D = density of fluid at average system temperature (lb/cubic foot) When integrated, Equation 2 yields the following relationship for the outlet temperature of a baseboard having length L, and operating with user-defined inputs (Equation 3).

Where:

Tout = outlet temperature of the baseboard element (degrees F)

Tair = air temperature entering the bottom of the baseboard enclosure (degrees F)

Tin = fluid temperature entering the baseboard element (degrees F)

B = published heat output rating of baseboard at 200 degrees F fluid temperature and 1 gpm flow rate (Btu/hr/ft)

L = length of baseboard element (feet)

f = fluid flow rate through baseboard element (gpm)

c= specific heat of fluid (Btu/lb/degree F)

D = density of fluid at average system temperature (lb/ft cubed)

For example: Assume a baseboard with a 15-foot element has a rated heat output of 650 Btu/hr/ft using 200 degrees F water at a flow rate of 1 gpm. The baseboard is installed in a room where the air temperature at floor level is 65 degrees F. Water at 160 degrees F enters the element at a flow rate of 3.5 gpm. The average water temperature in the system is 150 degrees F. What is the outlet temperature of the baseboard element?

Before using Equation 3, it's necessary to reference the density and specific heat of water at the assumed average circuit temperature of 150 degrees F. The density of water at 150 degrees F is 61.2 lb/ft cubed. The specific heat of water at 150 degrees F is 1.00 Btu/lb/degree F.

Substituting these values into Equation 3 yields the answer 156.4 degrees F. Once the outlet temperature is determined, the heat released from the baseboard can be found using Equation 4.

Where:

Q = heat release from the baseboard (Btu/hr)

Tout = outlet temperature of the baseboard element (from Equation 1) (degrees F)

Tin = fluid temperature entering the baseboard element (degrees F)

f = fluid flow rate through baseboard element (gpm)

c = specific heat of fluid (Btu/lb/degree F)

D = density of fluid at average system temperature (lb/ft cubed)

After the outlet temperature of one baseboard is determined, there are a couple of options:

1. The outlet temperature can become the inlet temperature to the next (downstream) baseboard. This is a simplifying assumption that neglects the effect of any heat loss from the piping connecting the baseboards.

2. If the heat loss of the interconnecting piping is expected to be significant, it should be calculated using ASHRAE data, or other established methods. This heat loss can then be used to determine the temperature drop along the piping, and hence the inlet temperature to the next baseboard.

In either case, the calculation procedure is to repeat the use of Equations 3 and 4 while moving from one baseboard to the next in the downstream direction. ## Finding the Right Length

The previous procedure will accurately analyze the thermal performance of a completely specified series baseboard circuit, one in which all baseboard lengths are known. However, a more typical design task is selecting the required length of baseboards for specified room loads. Procedures for doing so should also account for the temperature drop from one baseboard to the next. Such a method has been developed, and is presented as a simple spreadsheet in Figure 2.(Note: Fig. 2 could not be accurately reproduced online. Please see print edition.)

The spreadsheet shown can accurately size up to 10 baseboards connected into a series circuit. The formulas used assume water is used as the system fluid and do not account for the heat loss of any piping between baseboards. The heat output of each baseboard is calculated based on the average water temperature in that baseboard using Equation 5.

q' = heat output of baseboard per foot of element lengths(Btu/hr/ft)

B = published heat output rating of baseboard at 200 degrees F water temperature and 1 gpm flow rate (Btu/hr/ft)

f = fluid flow rate through baseboard element (gpm)

Tw(bar) = average water temperature in the baseboard (degrees F)

Tair = room air temperature (degrees F)

The length of each baseboard element is then established by dividing this output (per foot of element length) into the specified room. All calculated lengths are rounded up to the next whole foot.

The spreadsheet syntax is based on Appleworks 6.0, but can be readily be adapted to Excel or other formats. All cells shown in green contain input information that is typically changed by the user. Cells shown in red contain either calculated results, data needed by the formulas, or headings. These cells should be locked to prevent accidental changes.

The user enters the rating information for the baseboard and the circuit flow rate in the "SYSTEM INPUTS" frame, as well as room names, entering air temperatures, and room heating loads in column B, C, and D, respectively. Because the calculations are sequential, it's extremely important that the room information is entered in the same order as the baseboards will be connected into the circuit. The top row is for the first baseboard on the circuit, the second row for the second baseboard, and so forth.

The example shown in Figure 3 is for a series circuit containing five baseboards. (Note: Fig. 3 could not be accurately reproduced online. Please see print edition.) The assigned names, in order along the circuit, are Living room, Dining room, Kitchen, Bedroom 1, and Bedroom 2. The design heating load of each of these rooms is set at 8,000 Btu/hr. Notice how the baseboard lengths (in column 1) get longer in the direction of flow, even though the room loads are the same. The calculations are compensating for the drop in fluid temperature along the circuit.

## Summary

The analytical models presented can be used to simulate the thermal performance of finned-tube baseboard under a variety of operating conditions. When used in sequential calculations, these models properly distribute the finned-tube element based on room load as well as the local fluid temperature in the circuit. The spreadsheet approach allows the designer to investigate the possibility of using longer series baseboard loops with confidence that the baseboards near the end of the circuit will deliver the proper output. Longer baseboard circuits capable of operating with high temperature drops have the potential to reduce flow rates and thus decrease the installation and operating costs of piping and circulators. Evaluation of such situations with less accurate means runs the risk of improper sizing near both ends of the series circuit.