Heat Calculations (101 Series #2)
The second part of our energy exchange 101 series.
Written by guest author and ally Aaron Sawheb.
At an atomic level, the Internal Energy (U) of a thermodynamic system (system for short) is a measurement of the general movement of the molecules that make up the system, a measurement of how powerful the vibrations of the molecules are, of how fast the molecules are traveling, of how often chemical bonds are formed and broken.
On the other hand, Heat (Q), is a measurement of the transfer of energy between interacting systems. That is, as systems with different state variables (Volume, Pressure, Temperature, etc) interact with one another, there is a net transfer of heat from one system to another, which stops when the two systems reach thermal equilibrium. In other words, If two separate systems with a difference in temperature are brought together, the phenomenon of heat occurs, in which heat is transferred from the hotter system to the colder system until their temperatures are equal.
In the context of an extraction, the quantification of Internal Energy change is not helpful for making process decision—Heat, by contrast, is extremely important when making process decisions. The calculation of heat sets the foundation for sizing equipment, choosing equipment materials of construction, determining if the infrastructure of a facility has the capacity to meet the needs of heat-exchange equipment, the list goes on ad infinitum. So, diverging from the thermodynamics-driven discussion encountered in “EMB 101,” it is important to dive directly into the calculations and their rationale; the sooner one understands heat calculations, the sooner process decisions can be driven by the fundamental science.
Units and Unit Conversion (Note: It is of critical importance that the reader is familiar with all of this before proceeding further.)
Perhaps the most important practical skill to develop in terms of heat calculations is the ability to convert between units of measurement for relevant variables, like: temperature, energy, mass, density, volume, time, and so on. Below is a list of conversion skills that should be considered a pre-requisite to the understanding of the material to follow:
Converting Celsius (°C) to Kelvin (K)
- To convert °C to K, simply add 273 to the value in °C.
- Example 25°C = (25 + 273)K = 298K
Using Density (ρ) to convert between Mass and Volume
- Density is a ratio of mass to volume, and can be written as such, i.e. ρ= m/V
- The standard (S.I.) units of density are kg/m^{3}, which is equivalent to 1 g/L
The Relationship between Energy in Joules (J) and Power in Watts (W)
- Heat is measured in Joules (J), which is equivalent to
- Most importantly
- Example: say one desires to heat up a fluid from one temperature to another within a certain time frame. Calculate the heat required, divide it by the amount of time allowed and convert the denominator from the beginning units (minutes, hours) to seconds.
Determining kWh from a Power measurement in Watts
- After determining the Power required in Watts, convert to KiloWatts (divide the number in Watts by 1,000).
- Multiply by the total number of hours the heat transfer process is occurring and you’ve got a number in kWh.
A Brief Note Concerning Heat Capacity, Latent Heat and Density
Heat Capacity
Heat Capacity, is a measure of how much energy (in the form of heat) is required to raise the temperature of a system. Specific Heat Capacity, on the other hand, is a measurement of how much heat is required to raise a mass (or volume) of a pure component (i.e. a system made of one type of molecule) by a degree of temperature (in K or °C). Heat Capacities are exclusively used for determining heat required to change temperature—for instance, in a heat exchanger.
Latent Heat
Latent Heat has several forms, essentially it is a measure of how much energy is released or absorbed as a volume of molecules changes from one phase to another; i.e. during the process of freezing/melting (Latent Heat of Fusion) or condensing/vaporizing (Latent Heat of Vaporization). In the context of this conversation, only latent heat of vaporization will be utilized. Latent Heats are exclusively used for determining heat required to vaporize or condense fluids—for instance, in an evaporator or condenser.
A More Realistic Idea of Heat Capacity and Density
In the following discussion, Heat Capacity and Density will be regarded as unchanging numbers, but in a more realistic sense, these quantities vary depending on the temperature of a system—more formally stated: the heat capacity and/or density of a component is a function of temperature. For entry-level calculations, however, these considerations are over-kill.
Calculating Heat
There are two primary type of heat calculations, regarding the discussion at hand: the calculation of heat required to change the temperature of a fluid that does not change phases throughout the heating process and the calculation of heat required to change the phase of a fluid. The first type of heat calculation will be referred to as (Q_{Heat}) and the second type will be referred to as (Q_{PC}) (PC = Phase Change). Let’s get right to it.
- Q_{Heat} = Heat Input required to change T_{1 }to T_{2} (J)
- m = The Mass of the Fluid (g)
- C_{P} = The Specific Heat Capacity of the Fluid (J / g*K)
- T_{2 }= The Temperature of the Fluid after the addition of heat (K)
- T_{1 }= The Temperature of the Fluid before the addition of heat (K)
- Q_{PC} = Heat Input required to Boil the Fluid (J)
- L_{V} = The Specific Latent Heat of Vaporization of the Fluid (J/g)
These two equations are all that is needed to calculate the heat energy required to change a system from one temperature to another and/or to change the phase of a fluid. More importantly, since both types of heat are measured in Joules, they can be added together for a combined calculation when a situation involves both temperature change and phase change. Given that the reader has met the “pre-requisites” mentioned earlier in the article, one know has all the information needed to calculate the heat required for some common situations—so, let’s explore a relevant example.
The Example: Evaporating a Cold Solvent
An extraction technician uses Ethanol (EtOH) at -80 °C as a solvent for extraction. The crude EtOH-Extract mixture is fed into a batch evaporator that heats the crude mixture from -80 °C to the boiling point of ethanol (assume 80 °C, for simplification). After heating the crude mixture to 80 °C, heat continues to be added to the evaporator until all the EtOH has been vaporized. The extraction technician wants to know how much heat is required to heat and vaporize 10 L of a crude mixture, and how much it will cost.
Known Variables:
T_{2} = 80 °C = 353K T_{1} = -80 °C = 193K V = 10 L ρ_{Ethanol} = 789 g/L m = 7,890 g C_{P} = 2.44 J/g*K L_{V} = 918.2 J/g |
Unknown Variables:
Q_{Heat} Q_{PC} _{ } Note: Heat is technically in units of Joules, and power is in units of Joules/Time. So, these calculations are for POWER, but if we removed the “per hour” they would be Heat Calculations. This has been done for the sake of saving space. |
So, let’s plug in the variables and see what values we get:
Essentially, we have our answer, but that most-useful practical skill of unit conversion now becomes very important. Let’s assume the extraction tech wants to heat and vaporize this volume of Ethanol every hour:
Let’s further assume, that this process runs non-stop all year—24 hours a day, 365 days a year:
Now, finally, we are able to estimate the cost of the process of evaporating ethanol. If we determine the municipal cost for electricity, we can determine how this cost translate into an annual cost:
Unfortunately, this cost is not an accurate representation of the electric bill that an operation will receive. So, it’s about time we start discussing heat exchangers…
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