# Correlations for Convective Laminar Heat Transfer of Carreau Fluid in Straight Tube Flow

## Abstract

**:**

## 1. Introduction

## 2. Formulation

#### 2.1. Flow

#### 2.2. Temperature

#### 2.3. Nusselt Number

#### 2.4. Evaluation of the Core Flow Rate

#### 2.5. Evaluation of the Nusselt Number

#### 2.6. Numerical Integration

- (1)
- Further divide a subdomain with the largest predicted error.
- (2)
- Apply a cubature rule to the new subdomain.
- (3)
- Update the integral in the subdomains and obtain the global integral.
- (4)
- Check the error by $\left|\frac{{J}^{p+1}-{J}^{p}}{{J}^{p}}\right|$

#### 2.7. Expressions for Correlations

_{1}at the apparent shear rate into ${\mathrm{Nu}}_{p}$. Since the velocity profiles of the power law and the Carreau model are different from each other, the core flow rates would be different, and thus the Nusselt number would be so also. Thus, the mathematical foundation of the method seems weak and is thought to be a phenomenological methodology. Although the bias from the accurate value is somewhat noticeable, this method efficiently yields a useful number. This will be examined later in this work. To overcome the bias, this work modifies ${n}_{1}\left(\mathsf{\Gamma},\phi ,n\right)$. The modified apparent index takes the form of

## 3. Results

#### 3.1. Evaluation of Nusselt Number

#### 3.2. Correlations

^{−3}to 10

^{8}for all the combinations of 19 n’ and 15 $\phi $’s. An optimization problem seeking $c$, $d,$ and $h$ that minimizes a sum of least squares

_{1}and the modified apparent index n

_{2}. With n

_{1}, the bias is quite noticeable in the rising edge, but it is significantly reduced by n

_{2}. For n = 0.05, the bias is further pronounced with n

_{1}, as shown in Figure 6, since the nonlinearity is augmented with the lower n. Again, the bias has been much reduced by n

_{2}. Thus, the proposed modified method improves on the accuracy of the original method.

_{1}in this study. The Nu in Equation (3) and the numerical result by Cruz et al. also agree very well. The bias between them is less pronounced thanks to increased n compared to those in Figure 5 and Figure 6. As can be seen in Figure 7, the result using n

_{2}can reduce the bias dramatically. The maximum error is 3% by n

_{1}and is reduced to 0.9% by n

_{2}.

## 4. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## Nomenclature

${c}_{p}$ | heat capacity |

c(n) | function for apparent index modification |

d$\left(n\right)$ | function for apparent index modification |

h | heat transfer coefficient |

h$\left(n\right)$ | function for apparent index modification |

${J}^{p}$ | p-th evaluation of the integral ${{\displaystyle \int}}_{0}^{R}\frac{{q}_{c}{\left(r\right)}^{2}}{r}dr$ by the quadratue |

k | thermal conductivity |

K | consistency of the power law model |

n | index of the viscosity model (a constant of Carreau model) |

${n}_{1}$ | apparent power law index |

${n}_{2}$ | modified apparent power law index $={n}_{1}\left(c{\mathsf{\Gamma}}^{d};\frac{\phi}{h},n\right)$ |

Nu | Nusselt number |

${\mathrm{N}\mathrm{u}}_{N}$ | Nusselt number for the Newtonian model |

${\mathrm{N}\mathrm{u}}_{p}$ | Nusselt number for the power law model |

${q}_{c}$ | core flow rate |

${q}_{c,N}$ | core flow rate for Newtonian model |

${q}_{c,p}$ | core flow rate for power law model |

${q}_{c,C}$ | core flow rate for Carreau model |

${q}_{s}^{\u2033}$ | uniform wall heat flux |

Q | flow rate |

R | radius |

r | radial position |

S | sum of least squares |

T | temperature |

${T}_{m}$ | mean temperature |

${T}_{s}$ | surface temperature |

u | velocity in the x-direction |

${u}_{m}$ | mean velocity |

${w}_{i}^{p}$ | quadratue for the p-th evaluation at ${r}_{i}$ |

x | coordinate variable in the flow direction |

Greek | |

$\dot{\gamma}$ | shear (strain) rate |

${\dot{\gamma}}_{w}$ | wall shear rate |

${\dot{\gamma}}_{a}$ | apparent wall shear rate |

$\mathsf{\Gamma}$ | $\mathrm{apparent}\mathrm{shear}\mathrm{rate}\mathrm{in}\mathrm{a}\mathrm{dimensionless}\mathrm{form}\left(\equiv {\dot{\lambda \gamma}}_{a}\right)$ |

${\eta}_{0}$ | zero-shear viscosity (a constant of Carreau model) |

${\eta}_{\infty}$ | infinite shear rate viscosity (a constant of Carreau model) |

φ | ratio of ${\eta}_{\infty}$ to ${\eta}_{0}$ |

$\lambda $ | time constant (a constant of Carreau model) |

$\rho $ | density |

τ | shear stress |

${\tau}_{w}$ | wall shear stress |

subscript | |

a | apparent |

C | Carreau |

i | index for discretized value of n |

k | index for discretized value of φ |

m | mean |

N | Newtonian |

p | power law |

w | wall |

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**Figure 3.**Nu for $0.05\le n\le 0.95$ alongside ${\left(\lambda \dot{\gamma}\right)}^{1-n}$ for various $\phi $.

**Figure 4.**Nu for $0.0\le \phi \le 0.512$ alongside ${\left(\lambda \dot{\gamma}\right)}^{1-n}$ for various $n$.

Apparent Index Method (Cruz et al. 2012) | Modified Apparent Index Method | |
---|---|---|

apparent index | ${n}_{1}=\left(1-\frac{1}{1+\left(\frac{1}{\phi}-1\right){\left(1+{\mathsf{\Gamma}}^{2}\right)}^{\frac{n-1}{2}}}\right)\frac{\left(n-1\right){\mathsf{\Gamma}}^{2}}{{\left(1+{\mathsf{\Gamma}}^{2}\right)}^{\frac{n-1}{2}}}+1$ | ${n}_{2}=\left(1-\frac{1}{1+\left(\frac{h}{\phi}-1\right){\left(1+{c}^{2}{\mathsf{\Gamma}}^{2d}\right)}^{\frac{n-1}{2}}}\right)\frac{\left(n-1\right){c}^{2}{\mathsf{\Gamma}}^{2d}}{{\left(1+{c}^{2}{\mathsf{\Gamma}}^{2d}\right)}^{\frac{n-1}{2}}}+1$ |

$c\left(n\right)=0.63732-0.0057246n$ $d\left(n\right)=1.0047-0.021029n$ $h\left(n\right)=0.95951-0.83184exp(-5.9982n)$ | ||

Nusselt number | $\mathrm{Nu}=\frac{8\left(5{n}_{1}+1\right)\left(3{n}_{1}+1\right)}{31{n}_{1}^{2}+12{n}_{1}+1}$ | $\mathrm{Nu}=\frac{8\left(5{n}_{2}+1\right)\left(3{n}_{2}+1\right)}{31{n}_{2}^{2}+12{n}_{2}+1}$ |

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Kim, S.K.
Correlations for Convective Laminar Heat Transfer of Carreau Fluid in Straight Tube Flow. *Energies* **2022**, *15*, 2368.
https://doi.org/10.3390/en15072368

**AMA Style**

Kim SK.
Correlations for Convective Laminar Heat Transfer of Carreau Fluid in Straight Tube Flow. *Energies*. 2022; 15(7):2368.
https://doi.org/10.3390/en15072368

**Chicago/Turabian Style**

Kim, Sun Kyoung.
2022. "Correlations for Convective Laminar Heat Transfer of Carreau Fluid in Straight Tube Flow" *Energies* 15, no. 7: 2368.
https://doi.org/10.3390/en15072368