Formula sheet

Thermal Energy Systems Formula Sheet

Thermal energy formulas for energy balance, heat duty, heat exchangers, LMTD, effectiveness, efficiency, exergy, flow, pressure drop, pumping power, and heat recovery.

Branch: Energy Engineering
Content: Formula sheet
Updated: May 29, 2026
Revision: v1.0.8 · reviewed

This formula sheet collects common first-pass calculations for thermal energy systems and heat exchangers. Use it for screening, design review, commissioning checks, and troubleshooting. Detailed design still requires verified fluid properties, equipment standards, fouling assumptions, safety review, controls review, and operating data.

State the boundary before calculating. Heat duty, efficiency, savings, and exergy destruction are only comparable when the same components, auxiliary loads, and operating conditions are included.

Steady-flow energy balance

General steady-flow energy balance:

\displaystyle \dot{Q}-\dot{W}+\sum \dot{m}_{in}h_{in}-\sum \dot{m}_{out}h_{out}=0

For one inlet and one outlet with no shaft work and negligible kinetic and potential energy changes:

\dot{Q}=\dot{m}(h_{out}-h_{in})

For an adiabatic turbine or expander:

\dot{W}_{out}\approx \dot{m}(h_{in}-h_{out})

For an adiabatic compressor or pump:

\dot{W}_{in}\approx \dot{m}(h_{out}-h_{in})

Use a consistent sign convention and state whether heat into the system is positive.

Heat duty

Sensible heat duty:

\dot{Q}=\dot{m}C_p(T_{out}-T_{in})

Heat duty from volumetric flow:

\dot{Q}=\rho Q C_p(T_{out}-T_{in})

Phase-change heat duty:

\dot{Q}=\dot{m}\Delta h

Heat flux:

\displaystyle q''=\frac{\dot{Q}}{A}

Use property values at appropriate temperature, pressure, phase, and composition.

Heat exchanger balance

For a two-stream exchanger with negligible external heat loss:

\dot{Q}_{hot}+\dot{Q}_{cold}=0

Magnitude of heat transfer:

\dot{Q}=\dot{m}_h C_{p,h}(T_{h,in}-T_{h,out})

\dot{Q}=\dot{m}_c C_{p,c}(T_{c,out}-T_{c,in})

The two heat rates should match within measurement uncertainty and heat-loss assumptions. A mismatch can indicate bad data, heat loss, phase change, fouling, or incorrect properties.

LMTD method

Overall heat-transfer equation:

\dot{Q}=UA\Delta T_{lm}

Log-mean temperature difference:

\displaystyle \Delta T_{lm}=\frac{\Delta T_1-\Delta T_2}{\ln(\Delta T_1/\Delta T_2)}

For multipass or crossflow exchangers:

\dot{Q}=UA F \Delta T_{lm}

where $F$ is a correction factor.

Thermal resistance form:

\displaystyle \frac{1}{U}=R_i+R_{wall}+R_f+R_o

The detailed resistance expression depends on geometry. Cylindrical walls, fins, contact resistance, and fouling require appropriate forms.

Effectiveness-NTU method

Heat-capacity rate:

C=\dot{m}C_p

Minimum heat-capacity rate:

C_{min}=\min(C_h,C_c)

Maximum possible heat transfer:

\dot{Q}_{max}=C_{min}(T_{h,in}-T_{c,in})

Effectiveness:

\displaystyle \epsilon=\frac{\dot{Q}}{\dot{Q}_{max}}

Number of transfer units:

\displaystyle NTU=\frac{UA}{C_{min}}

The relationship between $\epsilon$ , $NTU$ , and heat-capacity ratio depends on exchanger flow arrangement.

Thermal efficiency

Heat-engine thermal efficiency:

\displaystyle \eta_{th}=\frac{W_{net}}{Q_{in}}

Equivalent form:

\displaystyle \eta_{th}=1-\frac{Q_{out}}{Q_{in}}

Carnot efficiency:

\displaystyle \eta_{Carnot}=1-\frac{T_C}{T_H}

Temperatures must be absolute. Real cycles operate below the Carnot limit because of irreversibilities and practical constraints.

Coefficient of performance

Refrigerator coefficient of performance:

\displaystyle COP_R=\frac{Q_L}{W_{in}}

Heat-pump coefficient of performance:

\displaystyle COP_{HP}=\frac{Q_H}{W_{in}}

For the same device:

COP_{HP}=COP_R+1

Reported COP depends on boundary, source temperature, sink temperature, part-load operation, defrost, pumps, fans, controls, and auxiliary loads.

Exergy and irreversibility

Exergy destruction:

I=T_0S_{gen}

where $T_0$ is reference environment temperature.

Maximum work from heat transfer from a reservoir at temperature $T$ to an environment at $T_0$ :

\displaystyle W_{max}=Q\left(1-\frac{T_0}{T}\right)

This expression applies to ideal reversible conversion from a reservoir at constant temperature. Real heat recovery and power conversion are lower.

Second-law efficiency can be expressed conceptually as:

\displaystyle \eta_{II}=\frac{\text{useful exergy output}}{\text{exergy input}}

Always state the reference environment and boundary when reporting exergy results.

Flow and Reynolds number

Volumetric flow:

Q=vA

Mass flow:

\dot{m}=\rho Q

Reynolds number:

\displaystyle Re=\frac{\rho vD}{\mu}

Common pipe-flow screening:

Re<2300 \quad \text{laminar}

Re>4000 \quad \text{turbulent}

Thermal equipment often depends on flow regime, but transition can be affected by geometry, roughness, fittings, vibration, multiphase flow, and non-Newtonian behaviour.

Convective heat transfer

Nusselt number:

\displaystyle Nu=\frac{hL}{k}

So:

\displaystyle h=\frac{Nu\,k}{L}

Grashof number for buoyancy-driven convection:

\displaystyle Gr=\frac{g\beta(T_s-T_\infty)L^3}{\nu^2}

These dimensionless numbers are used with correlations appropriate to geometry, flow regime, boundary condition, and property range. Do not mix correlations outside their valid domain.

Pressure drop and pumping power

Darcy-Weisbach head loss:

\displaystyle h_f=f\frac{L}{D}\frac{v^2}{2g}

Pressure drop:

\Delta p=\rho gh_f

Minor loss:

\displaystyle h_m=K\frac{v^2}{2g}

Hydraulic power:

P_h=\rho gQH

Input power with efficiency:

\displaystyle P_{in}=\frac{\rho gQH}{\eta}

Increasing velocity can increase heat transfer but also increases pressure drop and pumping or fan power. Optimize thermal and hydraulic performance together.

Heat recovery

Recovered heat rate:

\dot{Q}_{rec}=\dot{m}C_p(T_{in}-T_{out})

Fuel or purchased heat reduction from recovered heat:

\displaystyle \dot{Q}_{fuel,saved}=\frac{\dot{Q}_{rec}}{\eta_{heater}}

Simple annual energy saving:

E_{saved}=\dot{Q}_{rec}t_{operating}

This assumes the recovered heat is usable whenever it is available. In real systems, timing, temperature level, storage, controls, fouling, and part-load operation can reduce savings.

Thermal storage

Sensible thermal storage:

Q=mC_p(T_{max}-T_{min})

Phase-change storage:

Q=m\Delta h_{phase}

Heat loss through an envelope:

\dot{Q}_{loss}=UA(T_{storage}-T_{ambient})

Storage performance depends on charge rate, discharge rate, stratification, heat loss, cycling, material compatibility, safety, and control logic.

Performance measurement

Measured heat rate from fluid data:

\dot{Q}=\dot{m}C_p\Delta T

Measured thermal efficiency:

\displaystyle \eta=\frac{\text{useful output}}{\text{input}}

Percent performance degradation:

\displaystyle D=\frac{P_{clean}-P_{actual}}{P_{clean}}\times 100\%

For heat exchangers, degradation may be tracked by reduced $U$ , higher pressure drop, lower duty, or larger temperature approach. Use consistent flow and inlet temperature conditions before comparing.

Practical checklist

Use these formulas with a short engineering checklist:

Define boundary, operating mode, and load case.
Build mass and energy balances before sizing equipment.
Check heat duty, temperature approach, and heat-transfer area.
Check flow regime, pressure drop, pump or fan power, and control range.
Include fouling, cleaning access, corrosion, leakage, and degradation.
Compare first-law efficiency with exergy or second-law interpretation.
Validate performance using measured flow, temperature, pressure, power, and fuel data.

Thermal calculations should describe both energy quantity and energy quality. A design that transfers the required heat can still be inefficient, unreliable, or difficult to control if pressure drop, fouling, exergy loss, and operation are ignored.

REF

Disciplines