Problem #1

Problem #1: How Much Does Heat Cost?

Calculating and Comparing the Cost of Heat

by Robert Saunders

We’ve all heard the adage “You have to spend money to make money.” The real trick lies in how you spend your money. Would you rather spend $100.00 for electrical energy or $100.00 for a cord of wood? Well, it depends on what you get for $100.00. If we limit our discussion to buying heat and assume a cord of wood costs $100.00, how much heat will a cord of wood produce and how much heat can we buy from the electric company for $100.00?

The purpose of this article is to learn how to calculate the cost of heat. It does not advocate burning wood for fuel to generate heat over the use of any other fuel. It does attempt to provide the reader with enough information to be able to compare the cost of producing heat using one fuel with that of another fuel. In this case, we use wood for our calculations to compare the cost of wood heat with the cost of heat obtained from electricity. We can determine the cost of heat from electricity directly from our electric bills.

How much heat can we buy for $100.00? The simple answer is: It depends on the cost of a unit of heat. The units of heat we will use are Kilowatt-hours (KWHs) and BTUs. We need to know how much a KWH of heat from wood costs and how much the same KWH of heat costs from the electric company.

Let’s start at the beginning. Heat is nothing more or less than energy. There are different forms of energy, one of which is heat. Energy is measured by the electric company in terms of kilowatt-hours (KWH). That is what you pay for. The number of units of energy you buy each month from the electric company is on your electric bill. It is measured in KWHs and, depending on where you live, you probably pay between 3 cents and 12 cents, or more, for each kilowatt-hour of energy you consume.

ENERGY = POWER x TIME

A Kilowatt is a measure of power. A 100-watt light bulb uses 100 watts of power, as long as it is turned on. One kilowatt = 1000 watts.

A Kilowatt-hour is a unit of energy. If it is on for ten hours, it has used 1 kilowatt-hour of energy. One kilowatt-hour = 1000 watt-hours.

You buy energy, not power. You pay for the amount of energy you use. If your 100-watt light bulb is on for ten hours you will pay for 1 KWH of energy. If you have a hot water heater with three 1500-watt elements that operates for one hour each day, you are using, and will pay for, 4.5 kilowatt-hours each day for hot water. That doesn’t seem like much … as long as you don’t use any hot water. We’ll figure out how much it costs to heat water in another exercise.

A BTU is a unit of heat. A BTU is nothing more and nothing less than a unit of heat. Heat is energy. Therefore, we can convert units of heat in BTUs to units of energy in Kilowatt-hours.

1 KWH = 3412 BTUs

This is very simple stuff but the manufacturers of heating units and air conditioners express values of energy in BTUs per Hour rather than in KWHs per Hour. This is a rate of energy consumption and seems to be designed to confuse most people. A 10,000 BTU room air conditioner removes 10,000 BTUs per hour from the room and a 50,000 BTU heater puts 50,000 BTUs per hour into the room. There should be no confusion … if you have your calculator in your pocket.

When we burn wood we generate heat. Actually, when we burn wood we convert one form of energy “stored” in wood to heat by creating a chemical reaction that gives off heat, along with smoke, steam and ash, etc. However, we are only interested in the amount of heat that we get when we burn wood. Different kinds of wood give off different amounts of heat when it is burned. You can find various estimates of wood heat content for different species of wood from many sources. Some of the sources vary considerably. The following web site has some interesting and useful information on wood heat.

www.woodheat.org

If you click on firewood, there is a list of values for various types of wood.

www.woodheat.org/firewood/firewood.htm

How much Energy is in a cord of wood? You might already have some idea how we are going to figure that out. Let’s look at two kinds of wood, White Oak and White Pine. We will use the following values given in BTUs and convert them to KWHs by dividing the BTUs by the conversion factor

3412 BTU / KWH.

Be aware that these estimates are simply estimates. They vary widely, depending on the source. Nevertheless, it is very useful to be able to perform the calculations.

KWH = BTU ÷ 3412

KWHs per Cord = BTUs per Cord ÷ 3412 BTUs per KWH

Table 1 AMOUNT OF ENERGY IN A CORD OF WOOD
Wood Type	BTUs per Cord	KWHs per Cord
White Oak (Hardwood)	30,600,000	8968
White Pine (Softwood)	17,100,000	5011

How much does wood cost? The short answer is that it depends on where you live and what kind of wood you want to buy. In various parts of the country, split seasoned hardwood is available for less than $50.00 per cord to over $150.00 per cord. For our example, we will use the round number of $100.00 per cord for dry, seasoned, split, mixed wood. We will always assume the same wood conditions, unless otherwise specified.

Another adage we might consider is “A penny saved is a penny earned.” In some areas tree-length firewood can be purchased for $35.00 per cord or less. If you purchase tree-length wood, you will have to cut and split it yourself, and you might be able to earn about $65.00 or more per day in the process, if you can cut and split a cord a day. You will probably want to stack the wood yourself and earn a few more pennies. By the way, a full cord of wood is generally accepted to be equivalent to a 4 ft. by 4 ft. by 8 ft. volume of stacked split wood.

How much does electricity cost? If you read your last electric bill, you can find the price per KWH that you are paying. The retail price varies widely from one region to another. An article on the following web site offers some insight into how much you are spending for electricity each month.

www.stretcher.com/stories/960909b.cfm

Another article from the state of Missouri contains interesting comparisons of cost per KWH in several states.

www.psc.state.mo.us/press/consumer_issues/The_Cost_Of_Electricity_2003-01-09.pdf

According to the latter article, the national average cost of electricity for consumers in 2000 was 8.21 cents per KWH. In the most expensive state, Hawaii, the rate was 16.42 cents per KWH.

The following web site from the U.S. Energy Information Administration contains more information on current and historical retail costs of electricity in the United States.

www.eia.doe.gov/cneaf/electricity/page/fact_sheets/retailprice.html

Calculating the cost of energy from wood. We can find the cost per KWH for wood, as follows:

Cost per KWH (Cents) = [Price per Cord ($) ÷ KWHs per Cord] x 100

Price per Cord ($)	Table 2 Calculated Cost per kilowatt-hour
	KWHs per Cord		Cost per KWH (cents)
	White Oak	White Pine	White Oak	White Pine	Mixed
A	B	C	D	E	F
30.00	8968	5011	0.34	0.60	0.47
50.00			0.55	1.00	0.78
75.00			0.84	1.50	1.17
100.00			1.12	2.00	1.56
125.00			1.39	2.50	1.95
150.00			1.67	3.00	2.34

Example. To find the cost per KWH of wood heat, first find the cost of a full cord of wood in column A. The values given in columns B and C are the heat content in KWHs per cord for White Oak (Hardwood) and White Pine (Softwood). If we divide the price per cord from column A by the value in column B, the cost per KWH for White Oak is found in column D. If we divide the price per cord from column A by the value in column C the cost per KWH for White Pine is found in column E. Column F is the average of costs found in columns D and E.

If a cord of wood costs $100.00:

For White Oak: Cost per KWH = [100 / 8968] x100 = 0.01115 x 100 = 1.12 cents (rounded)

For White Pine: Cost per KWH = [100 / 5011] x100 = 0.01996 x 100 = 2.00 cents (rounded)

Fireplaces. Fireplaces are nice, but modern, open-hearth fireplaces are not very efficient. They may be only about 40% efficient, more or less. Fireplaces have a long history, dating back to ancient times. The art of building very large fireplaces as they were once constructed has probably been lost, although there is some interest in reviving that art. There are many books available describing how fireplaces have been built through the centuries. Some of the early ones were built to provide continuous heat for dwellings, and were fired only once a day. An interesting story is found in the following publication, and is available on the Barnes and Noble web site.

The Forgotten Art of Building a Good Fireplace: The Story of Sir Benjamin Thompson, Count Rumford, an American Genius, and His Principles of Fireplace D
Vrest Orton, Austin Stevens (Illustrator)

Efficiency. Until now we haven’t considered efficiency. Without further explanation we can define efficiency as the energy output of a stove or furnace divided by its energy input.

Efficiency = Output Energy ÷ Input Energy

Wood stoves are less efficient than electric heaters. Let’s assume the efficiency of wood stoves or furnaces varies from 40% to 80%. To find a better estimate of the real cost of energy from wood, we should adjust our cost estimates by the efficiency of the stove or furnace. We find the adjusted cost by dividing the calculated cost by the efficiency. The adjusted cost will always be higher than the calculated cost due to inefficiencies of the heating unit. The following shows the adjustment for a cord of mixed White Oak and White Pine.

Adjusted Cost (Cents) = Calculated Cost per KWH (Cents) ÷ Efficiency

Table 3
Price per Cord ($)	Calculated Cost Per KWH (cents)	Adjusted Cost per kilowatt-hour for Mixed Wood (Cents)
		Estimated Efficiency
		100%	80%	50%	40%
A	B	C	D	E	F
30.00	0.47	0.47	0.59	0.94	1.18
50.00	0.78	0.78	0.98	1.56	1.95
75.00	1.17	1.17	1.46	2.34	2.93
100.00	1.56	1.56	1.95	3.12	3.90
125.00	1.95	1.95	2.44	3.90	4.88

Example. To find the adjusted cost per KWH of wood heat, first find the cost of a full cord of wood in column A. The values given in column B are the same as those found in column F in Table 2. Column C shows that at 100% efficiency, the adjusted cost is the same as the calculated cost. The values shown in column D, E and F are calculated by dividing the values found in Column B by the estimated efficiency (in decimal units) of the stove or furnace.

Adjusted cost per KWH for wood costing $100.00 per cord:

For mixed wood:

At 80% efficiency: Adjusted Cost per KWH = 1.56 cents / 0.80 = 1.95 cents

At 50% efficiency: Adjusted Cost per KWH = 1.56 cents / 0.50 = 3.12 cents

At 40% efficiency: Adjusted Cost per KWH = 1.56 cents / 0.40 = 3.90 cents

Adjustment for moisture in the wood. So far we have only mentioned the efficiency of the stove or furnace. There is more to the subject of efficiency that relates to the wood itself. We assumed that we are burning dry wood. Dry refers to wood that has been cut, split and seasoned for a couple of years. Unfortunately, dry wood contains as much as 20% moisture in the form of water. This water has to be boiled off or vaporized as part of the wood burning process. The heat that is used to vaporize the water is called heat of vaporization. This heat goes up the stack or chimney in the form of steam or vapor and is not available for useful purposes. If wood is burned that is not seasoned or is wet or is otherwise green wood, the amount of wasted heat can be much higher.

To compensate in our calculations and to account for this wasted heat, we need to adjust the number we used for efficiency. Actually, what we need is another number for the efficiency of the burning of the wood. We can refer to that number as the combustion efficiency of the wood. This number is independent of the efficiency of the wood burning apparatus. Here is where we do a little guesswork. Let’s say the moisture content of the wood is 20%. We can assume that is by weight. Therefore the total amount of heat that the wood contains cannot exceed 80% of the total heat content of absolutely dry wood. If the specified heat content is for wood with 20% moisture, and our wood does contain 20% moisture, then we need not make any adjustment. Wood with a 20% moisture content is considered as seasoned, dry wood.

If, however, we are burning wood with an unknown amount of moisture. This might be cut, split, mixed wood that we think was cut last year in the fall, and we need to guess at its moisture content. If we guess that the moisture content is 40%, then we need to adjust our estimates by 20% from that of dry (20%) wood. If we accept that difference of 20% as lost heat from moisture, then we can assume a combustion efficiency of 80%. Inspectors use instruments to accurately measure this factor by measuring the temperature and composition of the flue gases.

To combine the efficiencies of two components in a system, we have to multiply the efficiencies of the two components to obtain the total efficiency of the system

E = E1 x E2

System Efficiency = (Efficiency of the wood burning apparatus) x (Combustion Efficiency)

Example. If our wood burning apparatus has an efficiency of 80% and the combustion efficiency of our wood is 80%, then the system efficiency, using the wood we have, is 64%.

Fortunately, we can go back to the previous example and merely divide the cost results we obtained in that example by the combustion efficiency to account for the moisture content of the wood. In the above example, if we divide the results by 0.80, the Actual costs per KWH are shown below.

Actual Cost (Cents) = Adjusted Cost per KWH (Cents) ÷ Combustion Efficiency

At 80% apparatus efficiency: Actual Cost per KWH = 1.95 cents / 0.80 = 2.44 cents

At 50% apparatus efficiency: Actual Cost per KWH = 3.12 cents / 0.80 = 3.90 cents

At 40% apparatus efficiency: Actual Cost per KWH = 3.90 cents / 0.80 = 4.88 cents

Comparing the costs of energy. Now we can compare the cost of energy from wood and the cost of the same amount of energy from the electric company. Many variables effect the cost and efficiency of burning wood. To determine the cost of burning wood compared to using electricity to generate heat, you have to know the following:

· how much electric heat costs

· how much a cord of wood costs

· the efficiency of your wood burning apparatus, adjusted for the moisture content of the wood

Electric heaters are close to 100% efficient, so we can compare the results shown in the above table, if you know what a KWH costs from your local electric company.

How much heat can you buy for $100.00? Since we now know how much wood heat costs and how much heat costs from the electric company, we can compare the costs and determine how much heat we can buy for $100.00. The calculation to determine how many units of heat we can buy for $100.00 is simple if we divide 100 by the cost of a unit of heat.

Units of heat (KWHs) for $100.00 = ($100.00 x 100 cents per $) ÷ Cost per KWH

If we multiply this result by 3412, we have the answer in BTUs.

Units of heat (BTUs) = Units of heat (KWHs) x 3412

Table 4 Units of Heat Vs Price
At a Price of Cents per Kilowatt-hour	You can buy kilowatt-hours for $100.00	You can buy BTUs for $100.00
A	B	C
1.0	10000	34,120,000
2.0	5000	17,060,000
3.0	3333	11,372,196
4.0	2500	8,530,000
5.0	2000	6,824,000
6.0	1667	5,687,804
7.0	1429	4,875,748
8.0	1250	4,265,000
9.0	1111	3,790,732
10.0	1000	3,412,000
11.0	909	3,101,508
12.0	833	2,842,196

Example. Table 4 is a lookup table if you already know the cost of a unit of heat. Let’s say you already have calculated that your adjusted cost for wood heat is 3 cents per KWH. In column A this value is shaded. The value shown in column B is the amount of heat in KWHs you can get from a cord of wood for $100.00. Column B shows the same amount in BTUs.

To compare the amount of heat you can buy from the electric company for $100.00, find the cost per KWH (shown on your last electric bill) in Column A. The amount you can buy for $100.00in KWHs is shown in column B. The same amount is shown in column C in BTUs.

Cost of heat shown graphically. The above graph shows the amount of heat you can buy for $100.00 as the cost varies from 1.0 cents to 12.0 cents per KWH. The vertical axes are in KWHs and BTUs.

The graph is plotted from the same data found in table 4. It can be used as a quick way to compare the amount of heat you can buy for $100.00, if you know the adjusted cost of a unit of heat. First find the cost per KWH on the x-axis at the bottom of the graph. Follow a line up to the curve. From this intersection, follow a horizontal line to the left-hand side of the chart to find the approximate amount of heat you can buy for $100.00. Likewise follow that intersection to the right-hand side of the curve to find the same value in BTUs.

Example. From the previous example, if your adjusted cost of heat from wood is 3 cents per KWH, find 3 cents on the horizontal axis. Follow a line up to the intersection of the curve. From that point, follow a line to the left until it intersects with the vertical axis. This should be a little more than 3000 KWHs. Do the same on the right-hand side of the chart and you should find a value of approximately 11,000,000 BTUs. Do the same thing for the price per KWH you pay the electric company. If you pay 8 cents per KWH, you should find an estimate of around 1000 KWHs and a little less than 5,000,000 BTUs.

Cost compared to a gasoline generator. If a 5KW gas generator burns 0.5 gallons per hour for 10 hours, the total energy output is 50 KWH at a cost, if the cost of gasoline is $1.50 / Gallon, of around $0.75 per hour; a total of $7.50. The same energy from the electric company might cost from $3.50 to $5.00. However, the gas generator is available when the power goes out, or is otherwise not available.

Cost compared to burning other fuels. We’ll leave this exercise for the reader.

You can find an exercise with propane heat content values on the following web site.

courses.washington.edu/me341/hw3sol.htm

The following web sites contain heat content values for a number of different fuels.

www.cornburner.com/BM620-9.html (look for A Cost Analysis)

home.att.net/~alt.hvac/fuels.htm

www.eia.doe.gov/kids/unitsindex.html

A conversion calculator for converting different units of energy exists in the following web site:

kahuna.sdsu.edu/testcenter/Test/solve/basics/units/unitdaemon/unitdaemon.html

Another Calculator. The following web site has a fuel cost calculator that includes calculations for Hardwood and Softwood. If you enter numbers for Cost per Cord and for Efficiency you should be able to compare the numbers you calculated above with their numbers. If you enter $100.00 per cord and 50% efficiency for both hardwood and softwood, you will notice that their cost estimates per 1,000,000 BTUs are somewhat higher than those from our calculations. Since they don’t specify the BTUs per cord, we have to assume that they used a lower BTU content per cord than the numbers we used for White Oak and White Pine.

www.hearth.com/fuelcalc/findoil.html

Summary. The objective of this exercise was to learn how to calculate certain relationships involved in estimating the cost of heat. If you understood the calculations and the logic, then you probably learned more than you might realize.

Specifically, you learned:

· That heat is energy and that energy can be expressed in different units

· That you can convert from one measurement system to another, i.e., BTUs to KWHs

· That energy estimates vary for different types of wood

· That wood burning stoves and furnaces have different efficiencies

· That the type and condition of wood you burn can have a significant effect on costs

· That you can compare the costs to produce a given amount of heat from any fuels

___________________________________

Any questions or comments are welcome by the author at the following email address:

Nwwpa@newnorth.net