Formula Sheet: Heat Exchangers

Heat Exchanger Fundamentals

Heat Transfer Rate

Overall Heat Transfer Rate:

\[q = UA \Delta T_m\]

• q = heat transfer rate (Btu/hr or W)
• U = overall heat transfer coefficient (Btu/hr·ft²·°F or W/m²·K)
• A = heat transfer surface area (ft² or m²)
• ΔT_m = mean temperature difference (°F or K)

Heat Balance - Hot Fluid:

\[q = \dot{m}_h c_{p,h} (T_{h,in} - T_{h,out})\]

• ṁ_h = mass flow rate of hot fluid (lbm/hr or kg/s)
• c_p,h = specific heat of hot fluid (Btu/lbm·°F or J/kg·K)
• T_h,in = hot fluid inlet temperature (°F or K)
• T_h,out = hot fluid outlet temperature (°F or K)

Heat Balance - Cold Fluid:

\[q = \dot{m}_c c_{p,c} (T_{c,out} - T_{c,in})\]

• ṁ_c = mass flow rate of cold fluid (lbm/hr or kg/s)
• c_p,c = specific heat of cold fluid (Btu/lbm·°F or J/kg·K)
• T_c,in = cold fluid inlet temperature (°F or K)
• T_c,out = cold fluid outlet temperature (°F or K)

Heat Capacity Rate

Heat Capacity Rate:

\[C = \dot{m} c_p\]

• C = heat capacity rate (Btu/hr·°F or W/K)
• ṁ = mass flow rate (lbm/hr or kg/s)
• c_p = specific heat (Btu/lbm·°F or J/kg·K)

Minimum and Maximum Heat Capacity Rates:

\[C_{min} = \min(C_h, C_c)\] \[C_{max} = \max(C_h, C_c)\]

Heat Capacity Rate Ratio:

\[C_r = \frac{C_{min}}{C_{max}}\]

• C_r = heat capacity rate ratio (dimensionless)
• Range: 0 ≤ C_r ≤ 1

Log Mean Temperature Difference (LMTD) Method

Parallel Flow Heat Exchanger

Log Mean Temperature Difference:

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

• ΔT_lm = log mean temperature difference (°F or K)
• ΔT₁ = temperature difference at one end (°F or K)
• ΔT₂ = temperature difference at the other end (°F or K)

For Parallel Flow:

\[\Delta T_1 = T_{h,in} - T_{c,in}\] \[\Delta T_2 = T_{h,out} - T_{c,out}\]

Counterflow Heat Exchanger

For Counterflow:

\[\Delta T_1 = T_{h,in} - T_{c,out}\] \[\Delta T_2 = T_{h,out} - T_{c,in}\]

Note: In counterflow, the cold outlet temperature can exceed the hot outlet temperature if C_c < C_h.

Correction Factor Method

Corrected LMTD for Complex Configurations:

\[\Delta T_m = F \Delta T_{lm,cf}\]

• F = LMTD correction factor (dimensionless)
• ΔT_lm,cf = LMTD calculated for counterflow arrangement (°F or K)

Correction Factor Parameters:

\[P = \frac{T_{c,out} - T_{c,in}}{T_{h,in} - T_{c,in}}\] \[R = \frac{T_{h,in} - T_{h,out}}{T_{c,out} - T_{c,in}}\]

• P = temperature effectiveness (dimensionless)
• R = heat capacity rate ratio = C_c/C_h (dimensionless)
• F values are obtained from charts based on P and R for different configurations (1-shell pass/2-tube pass, 2-shell pass/4-tube pass, crossflow, etc.)

Alternative Form:

\[R = \frac{\dot{m}_h c_{p,h}}{\dot{m}_c c_{p,c}}\]

Effectiveness-NTU Method

Heat Exchanger Effectiveness

Effectiveness Definition:

\[\varepsilon = \frac{q}{q_{max}}\]

• ε = heat exchanger effectiveness (dimensionless)
• q = actual heat transfer rate (Btu/hr or W)
• q_max = maximum possible heat transfer rate (Btu/hr or W)

Maximum Heat Transfer Rate:

\[q_{max} = C_{min}(T_{h,in} - T_{c,in})\]

Effectiveness in Terms of Temperatures (when C_h = C_min):

\[\varepsilon = \frac{T_{h,in} - T_{h,out}}{T_{h,in} - T_{c,in}}\]

Effectiveness in Terms of Temperatures (when C_c = C_min):

\[\varepsilon = \frac{T_{c,out} - T_{c,in}}{T_{h,in} - T_{c,in}}\]

Number of Transfer Units (NTU)

Number of Transfer Units:

\[NTU = \frac{UA}{C_{min}}\]

• NTU = number of transfer units (dimensionless)
• U = overall heat transfer coefficient (Btu/hr·ft²·°F or W/m²·K)
• A = heat transfer area (ft² or m²)
• C_min = minimum heat capacity rate (Btu/hr·°F or W/K)

Effectiveness-NTU Relationships

Parallel Flow:

\[\varepsilon = \frac{1 - \exp[-NTU(1 + C_r)]}{1 + C_r}\]

Counterflow:

• For C_r < 1:

\[\varepsilon = \frac{1 - \exp[-NTU(1 - C_r)]}{1 - C_r \exp[-NTU(1 - C_r)]}\]

• For C_r = 1:

\[\varepsilon = \frac{NTU}{1 + NTU}\]

Shell-and-Tube (1-shell pass, 2,4,6... tube passes):

\[\varepsilon = 2\left\{1 + C_r + \sqrt{1 + C_r^2} \frac{1 + \exp[-NTU\sqrt{1 + C_r^2}]}{1 - \exp[-NTU\sqrt{1 + C_r^2}]}\right\}^{-1}\]

Crossflow (both fluids unmixed):

\[\varepsilon = 1 - \exp\left[\frac{NTU^{0.22}}{C_r}\left(\exp[-C_r \cdot NTU^{0.78}] - 1\right)\right]\]

Crossflow (C_max mixed, C_min unmixed):

\[\varepsilon = \frac{1}{C_r}\left[1 - \exp(-C_r(1 - \exp(-NTU)))\right]\]

Crossflow (C_min mixed, C_max unmixed):

\[\varepsilon = 1 - \exp\left[-\frac{1}{C_r}(1 - \exp(-C_r \cdot NTU))\right]\]

Special Case - Condensers and Evaporators (C_r = 0):

\[\varepsilon = 1 - \exp(-NTU)\]

• Valid for all flow configurations when one fluid undergoes phase change at constant temperature

Overall Heat Transfer Coefficient

Thermal Resistance Network

Overall Heat Transfer Coefficient (based on inside area):

\[\frac{1}{U_i A_i} = \frac{1}{h_i A_i} + \frac{\ln(r_o/r_i)}{2\pi k L} + \frac{1}{h_o A_o}\]

• U_i = overall heat transfer coefficient based on inside area (Btu/hr·ft²·°F or W/m²·K)
• A_i = inside surface area (ft² or m²)
• A_o = outside surface area (ft² or m²)
• h_i = inside convection coefficient (Btu/hr·ft²·°F or W/m²·K)
• h_o = outside convection coefficient (Btu/hr·ft²·°F or W/m²·K)
• r_i = inside radius (ft or m)
• r_o = outside radius (ft or m)
• k = thermal conductivity of tube wall (Btu/hr·ft·°F or W/m·K)
• L = tube length (ft or m)

Overall Heat Transfer Coefficient (based on outside area):

\[\frac{1}{U_o A_o} = \frac{1}{h_i A_i} + \frac{\ln(r_o/r_i)}{2\pi k L} + \frac{1}{h_o A_o}\]

Relationship between U_i and U_o:

\[U_i A_i = U_o A_o\]

Flat Plate Heat Exchanger

Overall Heat Transfer Coefficient (flat plate):

\[\frac{1}{U} = \frac{1}{h_1} + \frac{t}{k} + \frac{1}{h_2}\]

• U = overall heat transfer coefficient (Btu/hr·ft²·°F or W/m²·K)
• h₁, h₂ = convection coefficients on each side (Btu/hr·ft²·°F or W/m²·K)
• t = wall thickness (ft or m)
• k = thermal conductivity of wall (Btu/hr·ft·°F or W/m·K)

Fouling Resistance

Overall Heat Transfer Coefficient with Fouling:

\[\frac{1}{U} = \frac{1}{h_i} + R_{f,i} + \frac{t}{k} + R_{f,o} + \frac{1}{h_o}\]

• R_f,i = fouling factor on inside surface (hr·ft²·°F/Btu or m²·K/W)
• R_f,o = fouling factor on outside surface (hr·ft²·°F/Btu or m²·K/W)

For Cylindrical Tubes with Fouling:

\[\frac{1}{U_i A_i} = \frac{1}{h_i A_i} + \frac{R_{f,i}}{A_i} + \frac{\ln(r_o/r_i)}{2\pi k L} + \frac{R_{f,o}}{A_o} + \frac{1}{h_o A_o}\]

Heat Exchanger Performance Parameters

Heat Transfer Area

Total Heat Transfer Area (shell-and-tube):

\[A = N \pi D L\]

• N = number of tubes (dimensionless)
• D = tube diameter (ft or m)
• L = effective tube length (ft or m)

Inside Area of Tubes:

\[A_i = N \pi D_i L\]

Outside Area of Tubes:

\[A_o = N \pi D_o L\]

Actual Heat Transfer Rate

From Effectiveness Method:

\[q = \varepsilon C_{min} (T_{h,in} - T_{c,in})\]

From LMTD Method:

\[q = U A F \Delta T_{lm}\]

Shell-and-Tube Heat Exchangers

Tube-Side Flow

Tube-Side Velocity:

\[V_t = \frac{\dot{m}}{\rho n_p A_t N_t}\]

• V_t = tube-side velocity (ft/s or m/s)
• ṁ = mass flow rate (lbm/s or kg/s)
• ρ = fluid density (lbm/ft³ or kg/m³)
• n_p = number of tube passes (dimensionless)
• A_t = cross-sectional area per tube (ft² or m²)
• N_t = total number of tubes (dimensionless)

Cross-Sectional Area per Tube:

\[A_t = \frac{\pi D_i^2}{4}\]

Number of Tubes per Pass:

\[N_{tubes/pass} = \frac{N_t}{n_p}\]

Shell-Side Flow

Shell-Side Cross-Flow Area:

\[A_s = \frac{D_s B (P_T - D_o)}{P_T}\]

• A_s = shell-side cross-flow area (ft² or m²)
• D_s = shell inside diameter (ft or m)
• B = baffle spacing (ft or m)
• P_T = tube pitch (ft or m)
• D_o = tube outside diameter (ft or m)

Shell-Side Mass Velocity:

\[G_s = \frac{\dot{m}_s}{A_s}\]

• G_s = shell-side mass velocity (lbm/hr·ft² or kg/s·m²)
• ṁ_s = shell-side mass flow rate (lbm/hr or kg/s)

Pressure Drop in Heat Exchangers

Tube-Side Pressure Drop

Tube-Side Pressure Drop:

\[\Delta P_t = n_p \left(\frac{f L}{D_i} + 4n_p\right) \frac{\rho V_t^2}{2g_c}\]

• ΔP_t = tube-side pressure drop (lbf/ft² or Pa)
• n_p = number of tube passes (dimensionless)
• f = Darcy friction factor (dimensionless)
• L = tube length (ft or m)
• D_i = tube inside diameter (ft or m)
• ρ = fluid density (lbm/ft³ or kg/m³)
• V_t = tube velocity (ft/s or m/s)
• g_c = gravitational constant (32.174 lbm·ft/lbf·s² or 1 kg·m/N·s²)
• The factor 4n_p accounts for entrance/exit losses and return losses

Simplified Form (US Customary):

\[\Delta P_t = \frac{f n_p L \rho V_t^2}{2 g_c D_i}\]

Shell-Side Pressure Drop

Shell-Side Pressure Drop:

\[\Delta P_s = \frac{f_s N_b D_s \rho V_s^2}{2 g_c D_e}\]

• ΔP_s = shell-side pressure drop (lbf/ft² or Pa)
• f_s = shell-side friction factor (dimensionless)
• N_b = number of baffles (dimensionless)
• D_s = shell inside diameter (ft or m)
• ρ = shell-side fluid density (lbm/ft³ or kg/m³)
• V_s = shell-side velocity (ft/s or m/s)
• D_e = equivalent diameter (ft or m)

Equivalent Diameter (square pitch):

\[D_e = \frac{4(P_T^2 - \pi D_o^2/4)}{\pi D_o}\]

Equivalent Diameter (triangular pitch):

\[D_e = \frac{4(0.5 P_T^2 \sin(60°) - \pi D_o^2/8)}{\pi D_o/2}\]

Heat Exchanger Selection and Design

Maximum Effectiveness Limits

Maximum Effectiveness for Parallel Flow:

\[\varepsilon_{max} = \frac{1}{1 + C_r}\]

• As NTU → ∞

Maximum Effectiveness for Counterflow:

• When C_r < 1: ε_max = 1
• When C_r = 1: ε_max approaches 1 as NTU → ∞

Temperature Approach

Minimum Temperature Approach (Counterflow):

\[\Delta T_{min} = T_{h,out} - T_{c,in}\]

or

\[\Delta T_{min} = T_{h,in} - T_{c,out}\]

• Whichever is smaller determines the closest approach
• Must be positive for practical heat exchangers

Heat Exchanger Sizing

Required Heat Transfer Area (LMTD Method):

\[A = \frac{q}{U F \Delta T_{lm}}\]

Required Heat Transfer Area (ε-NTU Method):

\[A = \frac{NTU \cdot C_{min}}{U}\]

Special Cases and Condensers/Evaporators

Condensers (Phase Change at Constant Temperature)

Heat Transfer Rate:

\[q = \dot{m} h_{fg}\]

• ṁ = condensate mass flow rate (lbm/hr or kg/s)
• h_fg = latent heat of vaporization (Btu/lbm or J/kg)

Effectiveness:

\[\varepsilon = 1 - \exp(-NTU)\]

• Valid because C_r = 0 for phase change

LMTD for Condenser:

\[\Delta T_{lm} = \frac{(T_{sat} - T_{c,in}) - (T_{sat} - T_{c,out})}{\ln\left[\frac{T_{sat} - T_{c,in}}{T_{sat} - T_{c,out}}\right]}\]

• T_sat = saturation temperature of condensing fluid (°F or K)

Evaporators (Boiling at Constant Temperature)

Heat Transfer Rate:

\[q = \dot{m} h_{fg}\]

LMTD for Evaporator:

\[\Delta T_{lm} = \frac{(T_{h,in} - T_{sat}) - (T_{h,out} - T_{sat})}{\ln\left[\frac{T_{h,in} - T_{sat}}{T_{h,out} - T_{sat}}\right]}\]

Compact Heat Exchangers

Finned Surface Parameters

Total Surface Area:

\[A_{total} = A_{fin} + A_{base}\]

• A_fin = total fin surface area (ft² or m²)
• A_base = exposed base surface area (ft² or m²)

Fin Efficiency:

\[\eta_f = \frac{\tanh(mL)}{mL}\]

• η_f = fin efficiency (dimensionless)
• m = fin parameter = √(hP/kA_c) (1/ft or 1/m)
• L = fin length (ft or m)
• h = convection coefficient (Btu/hr·ft²·°F or W/m²·K)
• P = fin perimeter (ft or m)
• k = thermal conductivity of fin material (Btu/hr·ft·°F or W/m·K)
• A_c = fin cross-sectional area (ft² or m²)

Overall Surface Efficiency:

\[\eta_o = 1 - \frac{A_{fin}}{A_{total}}(1 - \eta_f)\]

• η_o = overall surface efficiency (dimensionless)

Modified Heat Transfer Coefficient:

\[h_{eff} = \eta_o h\]

Regenerative Heat Exchangers

Rotating Regenerator

Effectiveness:

\[\varepsilon = 1 - \exp\left[-\frac{1}{(1 + C_r)}\left(\frac{UA}{C_{min}}\right)\right]\]

• For balanced regenerators with equal heat capacity rates on both sides

Dimensionless Groups for Correlations

Reynolds Number

Reynolds Number:

\[Re = \frac{\rho V D}{\mu} = \frac{V D}{\nu}\]

• Re = Reynolds number (dimensionless)
• ρ = fluid density (lbm/ft³ or kg/m³)
• V = fluid velocity (ft/s or m/s)
• D = characteristic length (tube diameter) (ft or m)
• μ = dynamic viscosity (lbm/ft·s or Pa·s)
• ν = kinematic viscosity (ft²/s or m²/s)

Prandtl Number

Prandtl Number:

\[Pr = \frac{c_p \mu}{k} = \frac{\nu}{\alpha}\]

• Pr = Prandtl number (dimensionless)
• c_p = specific heat (Btu/lbm·°F or J/kg·K)
• μ = dynamic viscosity (lbm/ft·s or Pa·s)
• k = thermal conductivity (Btu/hr·ft·°F or W/m·K)
• ν = kinematic viscosity (ft²/s or m²/s)
• α = thermal diffusivity (ft²/s or m²/s)

Nusselt Number

Nusselt Number:

\[Nu = \frac{hD}{k}\]

• Nu = Nusselt number (dimensionless)
• h = convection heat transfer coefficient (Btu/hr·ft²·°F or W/m²·K)
• D = characteristic length (ft or m)
• k = thermal conductivity of fluid (Btu/hr·ft·°F or W/m·K)

Heat Transfer Coefficient from Nusselt Number:

\[h = \frac{Nu \cdot k}{D}\]

Stanton Number

Stanton Number:

\[St = \frac{Nu}{Re \cdot Pr} = \frac{h}{\rho V c_p}\]

• St = Stanton number (dimensionless)

Common Heat Transfer Correlations

Tube-Side Correlations

Dittus-Boelter Equation (turbulent flow in tubes):

\[Nu = 0.023 Re^{0.8} Pr^n\]

• Valid for: Re > 10,000; 0.7 ≤ Pr ≤ 160; L/D > 10
• n = 0.4 for heating (T_wall > T_bulk)
• n = 0.3 for cooling (T_wall <>_bulk)

Sieder-Tate Equation (turbulent flow with viscosity correction):

\[Nu = 0.027 Re^{0.8} Pr^{1/3} \left(\frac{\mu}{\mu_w}\right)^{0.14}\]

• Valid for: 0.7 ≤ Pr ≤ 16,700; Re > 10,000
• μ = viscosity at bulk temperature
• μ_w = viscosity at wall temperature

Gnielinski Equation (improved accuracy for turbulent flow):

\[Nu = \frac{(f/8)(Re - 1000)Pr}{1 + 12.7(f/8)^{0.5}(Pr^{2/3} - 1)}\]

• Valid for: 3000 ≤ Re ≤ 5×10⁶; 0.5 ≤ Pr ≤ 2000
• f = friction factor from Moody chart or Colebrook equation

Laminar Flow in Tubes (fully developed, constant wall temperature):

\[Nu = 3.66\]

• Valid for: Re <>

Laminar Flow in Tubes (fully developed, constant heat flux):

\[Nu = 4.36\]

• Valid for: Re <>

Shell-Side Correlations

Shell-Side Heat Transfer (simplified):

\[Nu = C Re^m Pr^{1/3}\]

• Constants C and m depend on tube arrangement and baffle configuration
• Typically: C ≈ 0.36, m ≈ 0.55 for crossflow over tube banks

Heat Exchanger Analysis Procedure Summary

LMTD Method (Rating Problem)

Procedure:

1. Calculate heat transfer rate from energy balance: q = ṁ_hc_p,hΔT_h = ṁ_cc_p,cΔT_c
2. Determine outlet temperatures if not given
3. Calculate ΔT_lm for counterflow configuration
4. Determine correction factor F from charts using P and R parameters
5. Calculate corrected mean temperature difference: ΔT_m = F·ΔT_lm
6. Solve for U or verify: q = UA·ΔT_m

ε-NTU Method (Sizing Problem)

Procedure:

1. Calculate heat capacity rates: C_h = ṁ_hc_p,h, C_c = ṁ_cc_p,c
2. Identify C_min and C_max
3. Calculate C_r = C_min/C_max
4. Calculate maximum possible heat transfer: q_max = C_min(T_h,in - T_c,in)
5. Determine required effectiveness: ε = q/q_max
6. Find NTU from ε-NTU relationship for the specific heat exchanger type
7. Calculate required area: A = NTU·C_min/U

Heat Exchanger Types and Configurations

Flow Arrangement Classification

• Parallel Flow: Both fluids enter at the same end and flow in the same direction
• Counterflow: Fluids enter at opposite ends and flow in opposite directions
• Crossflow: Fluids flow perpendicular to each other (mixed or unmixed)
• Shell-and-Tube: One fluid flows through tubes while the other flows through the shell around the tubes
• Plate Heat Exchanger: Thin corrugated plates create channels for fluid flow

Performance Comparison

Effectiveness Ranking (for same NTU and C_r):

• Counterflow > Crossflow (both unmixed) > Parallel flow
• Counterflow is most efficient for given heat transfer area
• Parallel flow has the lowest maximum effectiveness