Chapter 7 Formulae
7.0.0.1 Energy Efficiency \(\mu\) (as %):
\[\mu = \frac{energy \, output}{energy \, input} \times 100 \]
7.0.0.2 Conservation of Energy Formula
(Closed System)
\[\Delta U = Q - W \]
Where:
\(\Delta U\) : as a change in internal energy
\(Q\) : the net quantity of heat supplied to the system by its surroundings
\(W\) : denotes the net work done by the system.
(Open System)
\[ \dot{Q} -\dot{W} = \sum \dot{m_{in}} - \dot{h_{out}}\]
Where:
\(\dot{m}\) : is the change in mass with respect to time (“flow”)
7.0.0.3 Heat transferred by:
7.0.0.4 1. Conduction
\[ \dot{Q}= KA \frac{T_{1}-T_{2}}{1} \]
Where:
\(K\) : is the thermal conductivity constant (obtained by experimentation in W/m.K.)
\(A\) : is the area of the surface
\(T\) : is for the temperature of the system
7.0.0.5 2. Convection
\[\dot{Q}= hA ({T_{1}-T_{2}}) \]
Where:
\(h\) : convective heat transfer coefficient
\(A\) : is the area implied in the heat transfer process
\(T\): is for the temperature of the system
7.0.0.6 3. Radiation
\[\dot{Q}= \varepsilon \sigma A ({T^{4}-T_0^{4}})\]
Where:
\(\varepsilon\) : is the emissivity of the system
\(\sigma\) : is the constant of Stephan-Boltzmann \(5.670367(13)\times10^{-8} W \cdot m^{-2} \cdot K^{-8} )\)
\(A\): is the area involved in the heat transfer by radiation
\(({T^{4}-T_0^{4}})\) : is the difference of temperature between two systems
7.0.0.7 PMV
The PMV index is expressed by P.O. Fanger as
\[ PMV = (0.303e ^{0.036M} + 0.028) L \]
where:
\(PMV\) : Predicted Mean Vote Index
\(L\) : thermal load - defined as the difference between the internal heat production and the heat loss to the actual environment - for a person at comfort skin temperature and evaporative heat loss by sweating at the actual activity level.
7.0.0.8 HDD (Heating Degree Days)
\[HDD (T_{ref}) =\frac{1}{24}\sum_{8760}^{i=1} max (T_{ref}- T_{ext, i} , 0)\]
Where:
\(T_{ref}\) : reference temperature
\(T_{ext}\) : exterior temperature
\(i\) : inlet temperatures of hot/cold fluid
7.0.0.9 CDD (Cooling Degree Days)
\[CDD (T_{ref}) =\frac{1}{24}\sum_{8760}^{i=1} max (T_{ref, i}- T_{ext} , 0)\]
####Thermal Balance
\[Q= Q_{in} - Q_{out} = Q_{walls} + Q_{windows}+ Q_{roof}+ Q_{ceiling}\]
7.0.0.10 Ventilation and Air Leakages
\[\dot{Q} = \dot{m}cp\Delta T\]
Where:
\(cp\) : surface pressure coefficient
\(\Delta T\) : temperature difference
####Overall Heat Transfer Coefficient (U)
\[Q = U\Delta T\]
\[U = \frac{1}{1/h1 + La/Ka + 1/h2}\]
Where:
\(q\) : heat transfer (W, J/s, Btu/h)
\(A\) : heat transfer area (\(m^2, ft^2\))
\(k\) : thermal conductivity of material (W/m K or W/m oC, Btu/(hr or ft2/ft))
\(dT\) : temperature gradient - difference - in the material (K or oC, oF)
\(s\) : material thickness (m, ft)
7.0.0.11 Heat Balance
\[Q = Q_{heating/cooling} + Q_{envelope} + Q_{internal}+ Q_{air}\]
7.0.0.12 Heat through envelope
\[Q_{envelope} = HDD\times Q_{x}\]
\[Q_{x} = A_{ceiling} \times U_{ceiling} \times A_{floor} \times U_{floor} A_{window} \times U_{window} + ( A_{wall} - A_{window }) \times U_{wall}\]
7.0.0.13 Heat through air exchange
\[Q = \dot{m} _{leakage} \times Cp \times HDD\]
Where:
\(V\) : Volume of the room
7.0.0.14 Internal Gains
\[Q_{internal} = Q_{occupants} + Q_{appliances}\]
7.0.0.15 Solar Gains
\[Q_{solar} = AI [T+U(\sigma^\alpha)]\]
Where:
\(A\) : Area
\(I\):irradiation
\(TU(\sigma^\alpha)\) : Coefficient that depends on the transmissivity and the absorbed radiation by the surface, through each radiation enters the room
7.0.0.16 Hot water modelling
Changing Product Temperature - Heating up the Product with Steam
The amount of heat required to raise the temperature of a substance can be expressed as:
\[\Delta U = m c_{p} \Delta T\]
Where:
\(\Delta U\) : quantity (difference) of energy or heat (kJ)
\(m\) : mass of substance (kg)
\(c_{p}\) : specific heat of substance (kJ/kg K)
\(\Delta T\) : temperature (difference) rise of substance
\(c_{p}\) 4.18 kJ/kg.K
7.0.0.17 Pipe Losses
\[q_{p}= \pi (T_{2}-T_{1})/ ln (D_{out}/D)\]
7.0.0.18 Enthalpy
The enthalpy change associated with the change in temperature and specific humidity of the present state and a reference state, by:
\[\Delta h = h-h_{ref} = [Cp_{dry\, air}T+ w(Cp_{water\, vapour}T+h_{fg})]-[Cp_{dry\, air}T+ w(Cp_{water\, vapour}T+h_{fg})]\]
Since the reference conditions are typically considered to be ° C and dry, enthalpy air from the reference state is calculated by:
\[h = Cp_{dry\, air}T+ w(Cp_{water\, vapour}T+h_{fg})\]
This expression is usually analyzed according to a sensitive component \(\Delta h_{sen} = (Cp_{dry\, air}+ w \times Cp_{water\, vapour})T\) and is a latent component \(h_{lat} =(h_{fg})w\). Since the specific humidity takes very low values in climatization situations, the psychrometric chart is defined by the axis of dry temperature and specific humidity. Generally, the following values are considered for specific heat and enthalpy of phase:
\[h = 1.01\times T + w(1.9T + 2480)\]
7.0.0.19 Relative humidity
The relative humidity is calculated by the ratio between the effective vapor pressure and the maximum vapor pressure. The maximum vapor pressure corresponds to the saturated vapor pressure temperature (pvsat (T)).
\[RH = \frac{p_v}{p_{vsat}(T)}\times 100 \%\]
7.0.0.20 Saturated Vapor Pressure
The saturated vapor pressure is obtained directly from the water vapor tables, being a function of temperature, and can be obtained by the following correlation, where T is in ° C and pvsat in kPa.
\[p_{vsat}(T)= 10^{(28.59051-8.2log(T+273.16)+0.0024804(T+273.16) - \frac{3142.31}{(T+273.16)})}\]
7.0.0.21 Specific humidity
While the relative humidity establishes a relationship between the volume and the vapor the maximum possible vapor volume, the specific humidity establishes a mass ratio between water vapor and dry air present in the mixture, being defined by:
\[w= 0.622\frac{p_{v}}{p_{atm}-p_{v}}\]
where patm is the atmospheric pressure, which assumes the normal value of 101,325 kPa.
7.0.0.22 Dry temperature and wet temperature
The dry and humid temperature are related by the following expression:
\[p_{v} = p_{vsat}(T_h) - 0.000666(T_s-T_h)\]
7.0.0.23 Lighting Concepts
Luminous Flux (\(\Phi\ : lm\setminus m^2\))
7.0.0.24 Illuminance from a Light Source
\[E = \frac{lcos\Phi }{d^2}\]
Where: \(E\) : illuminance from a certain place (lux)
\(d\) : distance to the light source
\(\Phi\) : Angle from the light source
\(I\): light source luminous intensity (lm)
7.0.0.25 Lighting Service (L)
Amount of time that the activity takes place
\[L = E\times A\times \Delta T(lm. s)\]
Where:
\(E\) : Required level of illuminance in a certain place (lux)
\(A\) : Area which requires a certain level of illuminance (\(m^2\))
\(\Delta\) : time period
7.0.0.26 Inverse Square Law
(The intensity of illumination produced by a point source varies inversely as square of the distance from the source.)
\[E = \frac{I}{d^2}\]
Where:
\(I\): intensity of illumination
\(d\): distance from the source
7.0.0.27 Cosine Law (Lambert’s Law)
\[E_H = \frac{I}{d^2}\cos\Theta\]
\(I\): intensity of illumination
\(d\): distance from the source
\(\Theta\): angle from the light source
7.0.0.28 Cosine Cubed Law
\[E_H = \frac{I}{d^2}\cos^3\Theta\]
7.0.0.29 Useful Lumen Output (ULO)
\[ULO = (n\times N\times F)\times(UF)\] Where:
\(n\): lamp number per fixture
\(N\): total fixture number
\(F\): Individual Lamp Lumem output
\(UF\):utilisation factor
7.0.0.30 Illumination Average Area
(rate of the portion of Lumen Output with influence in the lighting of an area)
\[E = (n\times N\times F \times UF \times LLF)/A\]
Where:
\(E\): Illuminance Average (in lux)
\(n\): lamp number per fixture
\(N\): total fixture number
\(F\): Individual Lamp Lumem output
\(UF\):utilisation factor
\(A\):Area
\(LLF\): Light Loss Factor
7.0.0.31 Efficacy Index
\[P(W)/100 (lux)/m^2\]
(it shoud be \(<5\))
7.0.0.32 Electrical Potential
\[P = U\times I\]
Where:
\(P\) : Power (Watt)
\(U\): voltage (volts)
\(I\): current (Amperes)
7.0.0.33 Shape Factor
(indicator of the compacness of a building)
\[FF= \frac{A}{V}(m^2/ m^3)\]
Where:
\(A\): Area \((m^2)\)
\(V\): Volume \((m^3)\)
7.0.0.34 Opportunity Cost
\[Opportunity \ Cost = Return \ on \ most \ Profitable \ Investment \ Choice - Return \ on \ Investment \ Chosen \ to \ Pursue\]
7.0.0.35 Future Value
\[Future\ Value = Present \ Value \cdot (1+i)^n\]
7.0.0.36 Present Value
\[Present\ Value = \frac{C}{(1+i)^n}\]
\(C\) - Net amount of money (cash-flows) that goes in or out of a project
\(n\) is the number of compounding periods between the present date and the date where the sum is worth C
\(i\) is the interest rate for one compounding period (the end of a compounding period is when interest is applied, for example, annually, semiannually, quarterly, monthly, daily).
7.0.0.37 Annuity
Let \(a_{n}\) denote the present value of the annuity. As the present value of the \(j^{th}\) payment is \(v^{j}\), where v = 1/(1+i) is the discount factor, the present value of the annuity is:
\[a_{n}= v + v^{2} + v^{3}+...+ v^{n}\]
\[= v \times\left [ \frac{1-v^n}{1-v} \right ]\]
\[= \frac{1-v^n}{i}\]
\[= \frac{1-(1+i)^{-n}}{i}\]
The accumulated value of the annuity at time \(n\) is denoted by \(s_{n}\)
This is the future value of ane at time \(n\). Thus, we have:
\[s_{n} = s_{n} \times (i+1)^n\]
\[s_{n} =\frac{(1+i)^{n}-1}{i}\]
If the annuity is of level payments of P, the present and future values of the annuity are \(Pa_n\) and \(Ps_n\), respectively.
Given that A = annual repayment amount, the present value of one annual repayment amount paid in t years time is
\[P= \frac{A}{(1+i)^t}\],where \(i\) is the annual rate of interest expressed as a decimal or fraction
Given that A = annual repayment amount, the present value of one annual repayment amount paid in t years time is
Loan amount = sum of the present value of all the repayments (assuming payment at the end of each payment period)
Where:
\(P\) = Loan amount
\(A\) = periodic repayment amount,
\(t\) = the number of payment periods,
\(i\) = the interest rate for the payment period expressed as a decimal or fraction
\[P =\frac{A}{(1+i)} + \frac{A}{(1+i)^2} + \frac{A}{(1+i)^3} + .... \frac{A}{(1+i)^t}\]
\[A =\frac{1}{(1+i)} + \frac{1}{(1+i)^2} + \frac{1}{(1+i)^3} + .... \frac{1}{(1+i)^t}\]
\[\Rightarrow A = P\frac{i(1+i)^t}{(1+i)^t-1}\]
This type of calculation is so common that it is convenient to derive a formula to shortcut the calculation for the regular repayment \(A\). By considering the general case of an amortised loan with interest rate \(i\), taken out over \(t\) years, for a loan amount of \(P\), a geometric series can be used to derive the general formula:
\[A= P\frac{i(1+i)^t}{(1+i)^t-1}\]
7.0.0.38 Net Present Value (NPV)
\[NPV_{i, N} \sum_{n=0}^{N}\frac{C_n}{(1+i)^n} - Investment\]
When cash inflows are even:
\[NPV = C × 1 − (1 + i)^{-n} − Initial \ Investment\]
In the above formula, \(C\) is the net cash inflow expected to be received in each period; \(i\) is the required rate of return per period; \(n\) are the number of periods during which the project is expected to operate and generate cash inflows.
When cash inflows are uneven:
\[NPV = C1 + ...+C_3 − Initial \ Investment\]
\(C_1:(1 + i)^1\)
\(C_2:(1 + i)^2\)
\(C_3:(1 + i)^3\)
Where, \(i\) is the target rate of return per period;
\(C1\) is the net cash inflow during the first period;
\(C_2\) is the net cash inflow during the second period;
\(C_3\) is the net cash inflow during the third period, and so on …
7.0.0.39 Internal Rate of Return (IRR)
The Internal Rate of Return (IRR), corresponds to finding out what is the rate of return on the project that makes the NPV equal to 0.
\[IRR= \sum_{n=0}^{N}\frac{C_n}{(1+i)^n} = 0\]
7.0.0.40 Payback
\[Payback \, Period = \frac{Amount\,Invested}{Estimated\,Net\,Cash\,Flow}\]
7.0.0.41 Accounting Equation
\[Assets = Liabilities + Owner' s\ Equity\]
\(Assets = Liabilities + Equity\)
\(PV \ System = Accounts \ payable + None\)
\(1000 = 1000 + 0\)
\[Assets = Liabilities + Capital + Revenue - Expenses - Drawings\]
7.0.0.42 Levelized Cost of Energy (LCOE)
\[LCOE = \frac{sum \ of \ costs\ \ over \ lifetime}{sum \ of \ eletrical\ energy\ produced \ over \ lifetime} = \frac{\sum_{t=1}^{n}\frac{I_t+M_t+F_t}{(1+r)^n}} {\sum_{t=1}^{n}\frac{E_t}{(1+r)^n}}\]
\(I_t\) = Investment expenditures in year t (including financing)
\(M_t\) = Operations and maintenance expenditures in year t
\(F_t\) = Fuel expenditures in year t
\(E_t\) = Electricity generation in year t
\(r\) = Discount rate
\(n\) = Life of the system
7.0.0.43 Cost-Optimaly Methodology
The global cost must be calculated according to EN15459 as indicated in the formula:
\[C_{g}(\tau) = C_I \sum{_j}[ \sum_{i=1}^{\tau}(C_{a,i(j)} \cdot R_d(i)) - C_{f\tau}(j)]\]
Where:
\(Cg(t)\) are the Global costs referring to the starting year τ=0,
\(Cl\) are the Initial investment costs,
\(Ca,i(j)\) are the annual costs year i for energy-related component j (energy costs, operational costs, periodic or replacement costs, maintenance costs),
\(Rd(i)\) is the discount rate for year i (depending on interest rate),
\(Vf,τ(j)\) is the final value of component j at the end of the calculation period (referred to the starting year τ=0 ),
7.0.0.44 Capital Asset Pricing Model (CAPM)
\(CAPM = R_f + \beta \times R_m\) and \(\beta=R_f+(r_m−r_f)\)